Understanding Pore Space through Log Measurements (Volume 76) (Developments in Petroleum Science, Volume 76) [1 ed.] 9780444641694, 0444641696

Table of contents :
Front Cover
Understanding Pore Space through Log Measurements
Understanding Pore Space through Log Measurements
Copyright
Dedication
Contents
Preface
Acknowledgments
1 - Pores and pore space
1.1 The pore space of rocks
1.1.1 The pore space of granular rocks
1.1.1.1 Pore space of some granular rocks not falling within the above categories
1.1.1.1.1 Intercrystalline pore space
1.1.2 The pore space of carbonate rocks—an example of the porosity classification
1.1.3 The pore space of coals
1.1.4 Pore space of shale reservoir rocks
1.1.4.1 Organic pores in shale reservoir rocks
1.1.4.2 Bitumen pores in shale gas reservoirs
1.1.4.3 Kerogen pores in shale gas reservoirs
1.1.4.4 Organic pores in shale oil reservoirs
1.1.5 The pore space of tight sedimentary rocks
1.1.6 The pore space of nonsedimentary rocks
1.2 Classification of pores by size
1.3 Pores and pore throats
1.4 Logs and pore space
1.4.1 Pore attributes used in petrophysical models of pore networks
1.4.2 Pore throat models
1.4.3 A popular and successful model of pore space – the capillary tube bundle for modeling flow of an incompressible fluid throu ...
1.4.4 A generic treatment of transport of an incompressible fluid through a medium such as a porous rock
1.4.5 Modeling the orientation of pores in space
1.5 The fractal model of pore space
1.6 Use of log data
1.6.1 Understanding the degree of connectivity between two pore size classes using NMR log data
1.6.2 Insights from high-resolution resistivity imaging tool data into the pore size heterogeneity
References
Further reading
2 - Inversion of log data to the gross attributes of pore space
2.1 Estimation of the bulk porosity of laminated formations using deterministic approach
2.1.1 Computation of effective porosity using shallow resistivity, neutron capture gamma ray spectrometry, formation bulk density ...
2.1.2 Computation of effective porosity using shallow resistivity, gamma ray, formation bulk density, and neutron porosity data
2.1.2.1 The wet clay content of shale
2.1.3 Total porosity
2.1.3.1 Pore volume not shared with shale, per unit rock volume
2.1.4 The different ways clay/shale is manifest within clastic rocks
2.1.5 The Thomas – Stieber approach
2.1.5.1 Evaluation of ϕmax
2.1.5.2 Simple analysis ignoring structural shale: computation of laminated and dispersed shale volume fractions in the formation
Gamma activity of laminated formation
2.1.5.3 Analysis considering structural shale also
Computation of the structural laminated and dispersed shale volume fractions in the formation
2.1.5.4 Using NMR data
2.1.5.5 Computation of representative resistivity of sand and laminated shale volume using tensor resistivity data
2.2 Stochastic inversion of log data for laminated formation
2.2.1 Basic response equations that lead to the forward model
2.2.1.1 Total porosity of the rock
2.2.1.2 Total porosity of rock within the flushed zone
2.2.1.3 The equation for sand conductivity in the uninvaded zone
2.2.2 The forward model
2.2.3 Essential constraints
2.2.4 Additional outputs computed
2.2.5 Horizontal permeability and vertical permeability
2.2.6 Usage of high-resolution data
2.2.6.1 Solution when constraints are present
2.2.6.2 Solution when constraints are present
2.3 Evaluation of microporosity
2.4 Evaluation of blocky (nonlaminated) reservoirs
Conclusions
Appendix 1
Bulk density and hydrogen index of wet clay and silt and the value of the silt index of shale
Appendix 2
Hydrogen index of dry clay and silt
Appendix 3
Resistivity along and normal to formation bedding
The inversion process
Concluding remarks
Appendix 4
Computation of total porosity from bulk density and magnetic resonance logs
Case of water base mud
Case of oil base mud or SOBM
Appendix 5
The dual water equation and input parameters
Input parameters which are usually given and how they are used
Computation of the value of Cbw
Source of the input parameters
Appendix 6
High-resolution data
High-resolution bulk density and photoelectric factor
High-resolution model
Output of the high-resolution model
Improved-resolution acoustic slowness – multishot processing of sonic data
Signal processing for the different subarrays
S,T plane and S,T plots
Concluding remarks
Improved-resolution density and neutron porosity logs from conventional 2-detector tool data
Deconvolution technique
Alpha processing
Alpha processing for improving the vertical resolution of conventional neutron porosity logs
Alpha processing for improving the vertical resolution of conventional density logs
Enhancement of vertical resolution of gamma ray logs
High-resolution NMR data
References
Further reading
3 - Pore attributes of conventional reservoirs
3.1 The pore space of intergranular rocks
3.2 Attributes of pore space
3.2.1 Pore shape, pore size, and pore throat size
3.2.2 Pores as the building blocks of pore space
3.2.3 Geometry of the pore body
3.2.4 Size of a pore
3.2.5 The concept of pore class
3.2.6 Surface area to volume ratio
3.2.7 The characteristic length scale of pore space
3.2.8 Hydraulic radius measure of the pore space
3.2.9 The pore shape factor
3.2.10 T2 log mean
3.3 Distribution of incremental porosity over pore radius
3.3.1 Computation of CP(r) using NMR data
3.3.2 Distribution of incremental porosity over pore throat radius
3.3.3 Ratio of pore size to pore throat size
3.3.4 Hard data on the distribution of pore throat size over incremental porosity (CPT(R))
3.3.5 Obtaining CPT(R) from mercury injection data
3.3.6 Obtaining CPT(R) from log data
3.4 Computation of CPT(R) from the NMR data
3.4.1 The linear conversion work flow
3.4.2 Nonlinear conversion work flow
3.5 Pore shape factor through integrating NMR and MICP (mercury intrusion data)
3.5.1 Frequency distribution of pore radius
3.6 A simple visualization of constriction and its effect on the gross permeability of pore space
3.6.1 Model prediction of permeability
3.6.2 Timur–Coates permeability predictor from the perspective of constriction
3.7 Fractal attributes of pore space
3.7.1 The fractal model of the pore space, based on a pore–pore throat assemblage visualization of the physical pore space
3.7.2 A fractal model of the pore space
3.7.3 Permeability from the perspective of the fractal model of the pore space
3.7.4 Cumulative pore volume
3.7.5 Representative hydraulic tortuosity and cumulated surface area to cumulated volume of the capillaries
3.7.5.1 Brookes–Corey permeability and the fractal model of pore space
3.8 Electrical formation factor from the perspective of the fractal model of the pore space
Appendix 1
Relation between tortuosity (τ), porosity (ϕ), and formation factor (F)
References
Further reading
4 - Pore space attributes of nonconventional reservoirs
4.1 CBM reservoirs
4.1.1 The components of the space occupied by fluids in coals
4.1.1.1 Hodot's pore classification scheme
4.1.2 Characterization of the pore space of coals – cleats and fractures that are not cleats
4.1.2.1 Extraction of fractures and cleats from borehole images
4.1.2.1.1 The process for low-angle events
4.1.2.1.2 The process for high-angle events
4.1.2.2 Computation of fracture/cleat aperture
4.1.2.3 Computation of cleat density from images
4.1.2.4 Computation of cleat density from acoustic logs
4.1.2.4.1 Stoneley reflection coefficient, transmission coefficient, and energy loss
4.1.2.5 Cleat volume per unit rock volume
4.1.2.6 Noncleat fracture volume per unit rock volume
4.1.2.7 Cleat orientation and the direction of maximum principal horizontal stress
4.1.3 Characterization of the pore space of coals using NMR data
4.1.3.1 Partitioning of the pore space of coals using NMR data by pore size
4.1.3.2 Interpore-class connectivity
4.1.3.2.1 Degree of connectivity between mesopores and macropore assemblages
4.1.3.2.2 Degree of connectivity between micropore and mesopore assemblages
4.1.3.3 Partitioning of the pore space of coals as bound fluid and free fluid volumes using NMR data
4.1.4 Permeability of coal—measurement
4.1.4.1 Measurement of the permeability of coal using log data
4.1.4.1.1 Permeability from stoneley full waveform inversion
4.1.4.2 Modeling the permeability of coal using log data
4.2 Shale reservoirs
4.2.1 Pore size encountered within shale reservoirs
4.2.2 Differentiation of pore classes for shale reservoirs
4.2.2.1 Bound fluid and free fluid porosity
4.2.3 Multidimensional inversion of NMR echo data using maximum entropy principle
4.2.3.1 Application of MEP
4.2.4 Presentation of the results of inversion
4.2.5 Porosity partition
4.2.5.1 Shale reservoirs bearing oil
4.2.6 The method of diffusion editing
4.2.7 The method of Laplace Inversion with regularization
4.2.8 D-T2 plots (more familiarly known as D-T2 maps) – forward models
4.2.8.1 Gas diffusion and the role played by adsorption of gas in the modeling
T1, T2 are relaxivity for gas shales.
4.2.8.2 Relaxation of bulk gas
4.2.8.3 Relaxation of gas within organic pores
4.2.8.4 Relaxation of gas within inorganic nanopores
4.2.8.5 Adsorption of methane gas into kerogen and the role played by this adsorption in relaxing spins
4.2.9 D-T2 maps and other plots related to the results of echo data inversion—field examples
4.2.9.1 Burst mode activation
4.2.10 Partitioning of total gas into free and adsorbed gas components using only NMR data
4.2.10.1 Partitioning of methane into free and adsorbed methane using relaxation data from echo amplitude inversion
4.3 Characterization of fractured reservoirs
Appendix 1
Kherroubi's work flow for trace extraction for low-angle events
Extraction of pixels which form part of a fracture
Obtaining fracture traces from the extracted pixel sets
Finding the set of vectors (segments) that best fits a trace when placed end to end
Computing the main fracture/cleat orientation mentioned above
Obtaining the final cleat/fracture traces of high confidence
Appendix 2
A derivation of Eq. (4.21)
Appendix 3
Computation of kerogen volume, gas volume, and total porosity in shale gas reservoir
The density of the adsorbed methane (ρgad)
Computation of adsorbed methane volume per unit rock volume
Total gas volume within the kerogen pores and its partition
Total porosity
References
Further reading
5 - Log measurements commonly used for finding the Bulk porosity of conventional reservoirs
5.1 Pore space attributes of conventional reservoirs
5.2 Measurement of bulk porosity
5.2.1 Measurement of formation density for bulk porosity
5.2.1.1 Single scattering of gamma photons
5.2.1.2 Multiple-scattered photons play a big role in the formation density measurement
5.2.1.2.1 Dual detector data processing using single window (per detector) count rates
5.2.1.2.1.1 Graphical method of solving the detector response equations
5.2.1.2.1.2 Computing density and density correction without using graph
5.2.1.2.1.3 Final correction to computed bulk density to account for electron density of water not being half of the bulk density of water
5.2.1.2.2 Computation of ρb using multiwindow count rate inversion
5.2.1.2.2.1 Photon energy spectrum versus number of scatterings
5.2.1.2.2.2 Photoelectric index
5.2.1.2.2.3 Equations of “spine”
5.2.1.2.2.4 Ratio of count rate in lithology window to that in compton window
5.2.1.2.2.5 The quantity (SCHC) of Eq. (5.49)
5.2.1.2.2.6 The quantity (HCH) of Eq. (5.49):
5.2.1.2.2.7 The quantity (SSC) of Eq. (5.49)
5.2.1.2.2.8 Effect of mud cake
5.2.1.2.2.9 The borehole effect
5.2.1.2.3 General method of simultaneous inversion of multienergy window count rates
Appendix 1
Simple approach to the ratio (SSC)
Measurement of slowing down length and diffusion length of neutrons for bulk porosity
Kinematics of neutron transport
Apparent hydrogen index
Slowing down length, diffusion length, and migration length
The two-group model of neutron transport – group 1 transport and slowing down length
Energy partitioning defining the groups – rationale
The thermal neutron energy spectrum
Neutron flux distribution for point source in a homogeneous isotropic medium for group 1
In the two-group model of neutron transport, inelastic collisions of neutrons are ignored
Removal cross section for group 1
L1 is known as slowing down length of neutrons
Physical meaning of the term “neutron slowing down length”
Neutron flux distribution for point source in a homogeneous isotropic medium for group 2
Diffusion length
Physical meaning of diffusion length
Neutron migration length
Neutron detectors
Neutron detector count rates
Group 1 neutron flux and epithermal neutron detector counts
Group 2 neutron flux and thermal neutron detector counts
Effect of the borehole and factoring-in of the borehole effect
Standard conditions
How the count rate data are inverted to porosity
Apparent water-filled limestone porosity for standard conditions
Appendix 2
Mean squared displacement of fast neutrons
Appendix 3
Solution of Eqs. (5.79) and (5.103)
Solution of Eq. (5.79)
Solution of Eq. (5.103)
Measurement of porosity using acoustic wave slowness data
Slowness porosity relations
Elastic moduli and wave propagation speeds
Porosity dependence of shear wave propagation speed for rocks where the pores are largely interconnected
Empirical and semiempirical relations
Porosity dependence of compressional wave propagation speed for rocks whose pores are largely connected to one another
Model-based inversion of compressional slowness to porosity for sandstones
Modeling of p and q: some insights
The case of spherical pores
Model-based inversion of acoustic slowness to porosity using Xu – White scheme
Evaluation of the exponents p and q
Fluid properties and grain properties input to the model-based inversion described above
Empirical relations connecting porosity and compressional slowness in rocks
The Raymer–Hunt–Gardner relation
Extending Raymer–Hunt–Gardner equation to rocks containing clay
A relation like Raymer–Hunt–Gardner relation for shear wave speed
Wyllie's relation
A generalization of Wyllie's relation
The acoustic body wave slowness measurement
Monopole excitation
Dipole excitation
Recorded waveforms time evolution of wave field illustrated for the slow formation case
Recorded dipole waveforms for the case of a fast formation
The importance of frequency of bandwidth of excitation
Appendix 4
A brief discussion on STC and DSTC
Nondispersive STC or simply STC
Dispersive STC
The effect of noise
Quadrupole excitation
Appendix 5
An illustration of array receiver signal processing using semblance
Tool modes interference or otherwise with the borehole modes
The dispersion model used in quadrupole waveform data inversion
The inversion method
References
Further reading
6 - Log measurements essential for characterizing the pore space of unconventional reservoirs
6.1 Measurement of total porosity using nuclear magnetic resonance
6.1.1 NMR theory
6.1.1.1 Perturbation
6.1.1.1.1 The purpose of applying the perturbation
6.1.1.2 Perturbation in the classical picture of perturbation
6.1.1.2.1 Dephasing of spins
6.1.1.2.2 Rephasing of spins
6.1.1.2.3 Acquisition of spin echoes
6.1.1.2.4 Temporal decay of echo amplitude
6.1.1.2.5 Effect of spatial variation of the B0 field
6.1.1.2.6 Bloch's equations with diffusion for the case of the static magnetic field varying spatially
6.1.1.2.7 Effect of the spin flips (180 degrees tipping of spins) on phase
6.1.1.2.7.1 Evaluation of A((2n−1)τ)
6.1.1.2.8 The amplitude of the nth echo
6.1.2 Pore space attributes and the relaxation of transverse magnetization
6.1.2.1 Surface relaxation (spin–lattice relaxation) and bulk relaxation (spin–spin relaxation)
6.1.2.2 Relaxation of the transverse magnetization of a pore saturated with a grain-wetting fluid
6.1.2.3 Evaluation of porosity
6.1.2.3.1 Input data
6.1.2.4 Obtaining CPBj from mj(T2j)
6.1.2.5 Quality control of the inversion
6.1.3 Total porosity and the bin porosities
6.1.4 Estimation of total porosity directly from the echo data
6.1.5 Obtaining total porosity using the formation density and NMR data
6.2 NMR and the porosity of CBM reservoirs
6.2.1 Coal pores
6.2.2 The T2 relaxation spectra of coals
6.2.3 Total porosity and gas volume
6.2.4 Porosity available within the rock for holding free gas
6.2.5 Cleat volume per unit rock volume
6.3 Porosity of shale gas/shale oil reservoirs
6.3.1 Estimation of effective porosity
6.3.1.1 Estimation of vcbw
6.4 Estimation of elemental concentration in rocks
6.4.1 Neutron capture gamma spectrometry
6.4.1.1 The capture gamma yield
6.4.1.2 Relative yield of capture gamma rays of an element
6.4.2 Inelastic gamma spectrometry
6.4.2.1 Inelastic gamma yield
6.4.2.2 Relative yield of inelastic gamma rays of an element
6.4.3 Computation of the average density of solid part of the formation
6.4.4 The acquisition of inelastic and capture gamma ray spectra
6.4.4.1 Transforming a pulse height spectrum to an energy spectrum fit for spectral decomposition
6.5 Characterizing the pore space of CBM reservoirs using image data
6.5.1 Cleat/fracture aperture and volume, and matrix porosity of CBM reservoirs
6.5.1.1 Estimation of cleat/fracture aperture and volume
6.5.1.1.1 Extraction of fracture segments from image data
6.5.1.1.2 Estimation of cleat/fracture aperture
6.5.1.1.3 Cleat volume per unit rock volume
6.5.1.1.4 Fracture volume per unit rock volume
6.5.1.2 Evaluation of matrix pore volume
6.5.1.2.1 Removal of the isolated conductive anomalies
6.5.1.2.2 Removal of the isolated resistive anomalies
6.5.1.2.3 Matrix conductivity and matrix pore volume
6.6 Characterizing the pore space of shale reservoirs using image data
6.6.1 Shale pores
6.6.2 Delineation of fractures within shale reservoirs using image data
6.6.3 Borehole electric images and distribution of organic matter
6.7 Generation of high-resolution electrical images of the borehole wall
6.7.1 Sensors
6.7.2 Position of each sensor in space
6.7.3 Data acquisition
6.7.4 Process flow for creating borehole images from the button current maps
6.7.4.1 EMEX correction
6.7.4.2 Data equalization
6.7.4.3 Speed and depth corrections
6.7.4.4 Magnetic declination correction
6.7.4.5 The button conductivity matrix
6.7.4.6 Normalization
6.7.4.7 Scaled button conductivity data
Appendix 1
Porosity calibration for NMR
Appendix 2
Certain aspects of relaxation of magnetization within fluid-filled porous media (after Brownstein and Tarr (1979))
Cylindrical geometry
Spherical geometry
Appendix 3
Computation of porosity using density and NMR log data (after Freedman et al., 1997)
Appendix 4
Kerogen property model and hydrocarbon property model used (Mosse et al., 2016)
Kerogen property model
Hydrocarbon property model
Appendix 5
Decomposition of acquired gamma ray spectra using the standard spectra
Spectral decomposition
Appendix 6
Permeability prediction for CBM reservoirs using image data
Analysis
Appendix 7
Histogram equalization
Proof of Eq. (A7.5):
References
Further reading
7 - Characterizing pores and grains using logs
7.1 Pore facies
7.1.1 Pore size distribution
7.1.1.1 Pore size distribution from NMR
7.1.2 Pore shapes from logs
7.1.2.1 Pore aspect ratio and acoustic logs
7.1.2.1.1 Inversion of acoustic slowness data to pore aspect ratio in sandstones
7.1.2.1.1.1 Differential effective medium model
7.1.2.1.1.2 Spherical pores
7.1.2.1.1.3 Evaluation of representative grain-shear modulus
7.1.2.2 Inversion of shear wave slowness log data to pore aspect ratio
7.1.2.3 Inversion of compressional wave slowness log data to pore aspect ratio
Appendix 1
Case of dry frame
The case of dry frame with no compliant pores and stiff pores spherical in shape
Appendix 2
Forward model of the bulk modulus of water saturated rock
Grain modulus
Inversion of acoustic slowness data to pore aspect ratio in carbonates
Differential effective medium model
Evaluating αP from shear wave velocity
Evaluating αP from shear wave velocity and compressional wave velocity
Aspect ratio of intergranular pores
Pore aspect ratio, Grain aspect ratio, and dielectric logs
Pore aspect ratio and grain aspect ratio
Appendix 3
Klein and Swift model for the dispersion of the dielectric permittivity of brines
Appendix 4
Wu's tensor
References
Further reading
8 - Archie's cementation exponent
8.1 Introduction
8.2 Prediction of the value of Archie cementation exponent “m” using effective medium theories
8.3 Approaches used in modelling Archie's m parameter—General Remarks
8.4 The approach for computing “m” using single-frequency dielectric data and using Archie's equation
8.5 The approach for computing “m” using multifrequency dielectric data and using Archie's equation
8.6 The approach for computing “m” from grain attributes obtained through multifrequency dielectric data inversion
8.6.1 Inversion of Archie “m” from single-frequency dielectric data
8.6.2 Inversion of Archie “m” from multifrequency dielectric data
8.6.3 Work flow for the generation of the dispersion model of dielectric permittivity ε
8.6.4 The dispersion model for rock conductivity
8.6.5 Applicability of the model to clayey rocks
8.6.6 The role of contribution to complex permittivity, coming from unconnected pores
8.6.7 Approaches to estimation of Archie “m” based on differential effective medium theory
8.6.7.1 A symmetric formulation of the effective medium of a composite
8.6.7.2 An illustration of modeling “m” using the differential effective medium theory
8.6.7.3 Salient features of Sheng's model for sedimentary rocks
8.6.7.4 Sheng's model and Archie's “a”.
8.6.7.5 Sheng's model and Archie's “m”.
Appendix 1
Derivation of Eq. (8.22) which is in fact modified Maxwell Garnett mixing law
Case of aligned inclusions (identical spatial orientation of principle axes of inclusions)
Appendix 2—differential effective medium theory for aligned inclusions case
Appendix 3
Depolarization factors
Introduction
Depolarization factors of an ellipsoid in general and a spheroid in particular
Approaches to model “m”: in case of Shaly rocks using the Bergman spectral density representation of the effective permitti ...
Sum rules to be obeyed by (s')
Constraint on g(s')
Further necessary condition that the spectral density function is expected to satisfy
Appendix 4
Application of Maxwell's Equations applied to binary mixtures: results in the Quasi-Static limit—Bergman's theorem
The poles of f (s) are real numbers
The fact of positive residues of the poles of f(s) arises naturally in this analysis
The fact of every pole of f(s) being a simple pole arises naturally in this analysis
The essence of the analysis made, and its importance
An approach to Archie “m” through NMR data analysis
Percolation theories and Archie “m” factor
Appendix 5: logarithmic mixing law for effective permittivity of a mixture
Charge distribution within the medium
Behavior of the mixing law in the zero-frequency limit
Approaches to estimate “m” through fractal model of pore space
Concluding remarks
References
Further reading
9. Permeability of unimodal pore system
9.1 Introduction
9.2 Response of local pressure field, local fluid velocity field, and average fluid velocity field to changes in driving pressure
9.3 A simple model of pore space presented
9.4 Flow through a capillary
9.4.1 How pore space attributes influence permeability
9.4.1.1 Preamble
9.5 Surface area or representative pore dimension or characteristic length scale driven approaches to permeability
9.5.1 The “bundle of capillary tubes” model of pore space
9.5.2 Forward model of permeability in terms of pore space attributes
9.5.3 The stream tube model of flow through a porous medium whose grain- and pore-arrangement of a macro level volume segment, is ...
9.6 Depiction of pore space in the model
9.6.1 Stream tubes
9.7 Elemental stream tube
9.8 Integral representation of macroscopic permeability (or simply “permeability”), and the concept of microscopic permeability ...
9.9 Integral representation of permeability
9.10 Average permeability field ks, also referred to as the “effective permeability factor”
9.11 Integral representation of average permeability field (effective permeability factor) ks and relation between permeability ...
9.12 The elemental stream tube permeability factor field
9.12.1 Explicit representation of permeability factor of a streamline (elemental stream tube) in terms of some of its attributes
9.12.2 Decomposition of permeability into macroscopic pore space attributes, namely, hydraulic tortuosity, hydraulic constriction ...
9.13 Insights from the simplest possible pore space model and the role played by “hydraulic radius” in permeability modeling—der ...
9.14 Concept of hydraulic radius
9.14.1 How microscopic streamline attributes and macroscopic pore space attributes are related in a porous medium
9.14.1.1 Precise definition of the term “representative tortuosity (τ) of pore space”
9.14.2 Hydraulic Constriction factor (Cs) of connected pore space, which is the macroscopic counterpart of streamline attribute C(S)
9.15 Generalized Kozeny-Carman equation
9.15.1 Conventional form of Kozeny-Carman equation predicting permeability
9.15.2 Well known equations for permeability prediction from log measurements. The equations discussed below are for water wet roc ...
9.15.2.1 The SDR equation for permeability
9.15.2.2 Timur equation for permeability
9.15.2.3 Coates equation for permeability
9.15.2.4 Berg's equation
9.15.2.5 Computation of average grain size and grain size distribution from NMR data
9.15.2.6 Relation between representative or effective pore dimension and the representative or effective grain dimension—Van Baaren' ...
9.15.2.6.1 Van Baaren's equation
9.16 From Kozeny-Carman equation, to Van Baaren's equation
9.17 The RGPZ equation
Appendix 1
Hydraulic constriction factor
Appendix 2
Estimation of bound fluid volume used in Coates equation from NMR data
Appendix 3
A brief derivation of RGPZ equation
Rock attributes which control permeability, and their inversion from log measurements –some challenges
Pore space attributes, and log measurements (in the context of nonfractal modeling of pore space)
Calculation of Cs
Prediction of effective permeability factor ks, and permeability k using log measurements (in the context of nonfractal mod ...
Eq. (9.21) and the Kozeny-Carman equation—interpreting the fitting-constant C in Kozeny-Carman equation
Why the “bundle of capillary tubes” models of pore space succeed in predicting the fluid transport properties of porous media
The pores and pore throats model of pore space
Fractal dimension of connected pore space and its influence on permeability
Some important relations
Pore volume Vp and porosity ϕ
Representative hydraulic tortuosity τ
Ratio of cumulative surface area of connected pore space, to cumulative volume of connected pore space Spv
Relationship between Df, Dt and ϕ, and between Df, F, and ϕ
Relation of the results from fractal model of pore space, as arrived at above, with Brooks-Corey equation, but with pore sp ...
Relation of the results from fractal model of a pore space comprised of pores and pore throats, with the Brooks-Corey equation
Permeability from Brooks-Corey equation—the impact of the fractal model
Estimation of rmax, rmin, and γ1 (pore size to pore throat size ratio) from NMR log data
Estimation of rmax from NMR log data
Estimation of rmin from NMR log data
Estimation of γ1 (pore size to pore throat size ratio)
Generation of a valid permeability predictor as a continuous log
Estimation of pore size heterogeneity index λ from logs
Computation of electric formation factor log
Estimation of the (fractal) dimension of pore radius fractal (Df) from logs
Estimation of Df from λ
Estimation of Df from rmax, rmin
Estimation of Df from rmax, and NMR (1T2)mean
Estimation of Df from rmax, and k
Estimation of Df from F
Significance of the parameter Df
The fractal dimension of the pore radius fractal and Archie's a,m parameters
Estimation of the fractal dimension for lengths of capillaries (Dt) using log measurements
Estimation of τ the hydraulic tortuosity associated with connected pore space, assuming pore space is not a fractal object
Estimation of τ the tortuosity associated with connected pore space, assuming pore space is fractal
Appendix 4
Fluid transport through a perfectly rigid framework of grains, with no isolated pore space
Formal averaging procedure applied on local fluid velocity field and local fluid pressure field, that are related through l ...
The tensor field α҃ (r) is entirely defined by the medium properties
Appendix 5
Stoneley wave slowness and Stoneley mobility
Appendix 6
Estimation of γ1 (pore size to pore throat size ratio)
References
Further reading
10. Permeability and electrical conductivity of rocks hosting multimodal pore systems and fractures
Preamble
10.1 Pore size nomenclature
10.1.1 Micropores, mesopores, and macropores
10.2 Pore classification using NMR T2 distribution
10.2.1 Relation between pore sizes and NMR T2: the fast diffusion limit
10.2.2 Limits of validity of a T2 threshold-based porosity partition
10.2.3 Model-based porosity partition for the case of rocks having intragranular porosity
10.2.4 Case where vugs are not sparsely distributed in the rock
10.3 Pore classification using well bore images
10.3.1 Preconditioning of data
10.3.2 Dip picking
10.3.3 Closing the data gap
10.3.3.1 Training pattern classification
10.3.3.2 Score class prototype
10.3.3.3 Simulation to populate the pixels which have no data
10.3.3.4 Image rescaling
10.3.4 Extraction of fracture segments
10.3.4.1 Extraction of low apparent dip fracture segments
10.3.4.2 Spatial orientation of traces
10.3.4.3 Differentiating bedding planes and facture planes
10.3.4.4 Main orientation of fractures
10.3.4.5 Obtaining high confidence fracture traces
10.3.4.5.1 Extraction of high apparent dip fracture segments
10.3.5 Matrix extraction
10.3.6 The problem of computing the pore volume contribution of heterogeneities
10.3.6.1 The challenge of extraction of heterogeneities
10.3.7 An efficient methodology for extracting heterogeneities
10.3.7.1 Classification of the mosaic pieces
10.3.7.2 Connectedness attribute of a conductive heterogeneity
10.3.7.3 Final porosity partition
10.3.7.4 The mosaic image and the heterogeneity image
10.3.7.5 Classification of heterogeneities and quantification of the porosity associated with them
10.3.7.6 Porosity association of the different types of spots (heterogeneities)
Appendix 1
10.4 Porosity partition using acoustic logs
10.4.1 Challenges of porosity partition using acoustic logs
10.4.2 A work flow of porosity partition using acoustic logs
What is a “vug” in the model?
Sensitivity of model results to the shape of the vug
Appendix 2
The self-consistent theory of Berryman in the long wavelength limit
Introduction to Kuster–Toksoz model
The Kuster–Toksoz estimates
Invoking the assumption of long wavelength limit
Self-consistent estimates in the long wavelength limit
Invoking the long wavelength limit
10.5 Electrical conductivity of an unfractured composite hosting dual porosity
Modeling σ2 for fully water saturated component 2 case
Evaluation of σ the electrical conductivity of a composite hosting dual porosity
Case of partial saturation
10.6 Permeability of an unfractured composite hosting dual porosity
Base rock permeability
Permeability of the composite rock
10.7 Electrical conductivity of fractured rocks
Partial saturation case
10.8 The variable cementation exponent method of computing water saturation
Case of connected pore space having tortuosity unity
Pore volumes
Rationale behind Equation (10.89)
Level by level evaluation of the Archie cementation exponent of the formation
Discussion
Computation of water saturation
Appendix 3
Rationale behind Equation (10.89)
10.9 Permeability of rocks hosting connected vugs/fractures
Alternate simpler way of estimating Qcon
Permeability kb of the base rock
Permeability of gross rock
10.10 Bray – Smith method of computing permeability
10.11 Another approach to permeability modeling which also relies on NMR log data
References
Further reading
Index
A
B
C
D
E
F
G
H
I
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Z
Back Cover

Citation preview

DEVELOPMENTS IN PETROLEUM SCIENCE 76

Understanding Pore Space through Log Measurements

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DEVELOPMENTS IN PETROLEUM SCIENCE 76

Understanding Pore Space through Log Measurements K. Meenakshi Sundaram Mentor Petrophysicist and Consultant, Tata Petrodyne Ltd

Soumyajit Mukherjee Professor of Geology, Indian Institute of Technology Bombay, India

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2022 Elsevier B.V. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the Publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher, nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-444-64169-4 ISSN: 0376-7361 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Candice Janco Acquisitions Editor: Jennette McClain Editorial Project Manager: Howi M De Ramos Production Project Manager: Bharatwaj Varatharajan Cover Designer: Miles Hitchen Typeset by TNQ Technologies

Dedication This book is dedicated to all those who taught, mentored, and inspired me. Also, to those who worked alongside, and learned from me, and also challenged me. But above all, this book is dedicated to those who made me believe that Reach is always farther than Grasp. e KMS I dedicate this book to Prof. Dr. Devang Khakhar (Ex-Director, Department of Chemical Engineering, IIT Bombay). e SM

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Contents Preface Acknowledgments

1.

Pores and pore space 1.1

The pore space of rocks 1.1.1 The pore space of granular rocks 1.1.2 The pore space of carbonate rocksdan example of the porosity classification 1.1.3 The pore space of coals 1.1.4 Pore space of shale reservoir rocks 1.1.5 The pore space of tight sedimentary rocks 1.1.6 The pore space of nonsedimentary rocks 1.2 Classification of pores by size 1.3 Pores and pore throats 1.4 Logs and pore space 1.4.1 Pore attributes used in petrophysical models of pore networks 1.4.2 Pore throat models 1.4.3 A popular and successful model of pore space e the capillary tube bundle for modeling flow of an incompressible fluid through porous rocks 1.4.4 A generic treatment of transport of an incompressible fluid through a medium such as a porous rock 1.4.5 Modeling the orientation of pores in space 1.5 The fractal model of pore space 1.6 Use of log data 1.6.1 Understanding the degree of connectivity between two pore size classes using NMR log data 1.6.2 Insights from high-resolution resistivity imaging tool data into the pore size heterogeneity References Further reading

2.

xvii xix

1 1 4 4 6 12 13 13 14 16 17 18

20 20 20 21 22 23 24 26 28

Inversion of log data to the gross attributes of pore space 2.1

Estimation of the bulk porosity of laminated formations using deterministic approach

31 vii

viii Contents 2.1.1

Computation of effective porosity using shallow resistivity, neutron capture gamma ray spectrometry, formation bulk density, and neutron porosity data 32 2.1.2 Computation of effective porosity using shallow resistivity, gamma ray, formation bulk density, and neutron porosity data 38 2.1.3 Total porosity 40 2.1.4 The different ways clay/shale is manifest within clastic rocks 44 2.1.5 The Thomas e Stieber approach 45 2.2 Stochastic inversion of log data for laminated formation 55 2.2.1 Basic response equations that lead to the forward model 55 2.2.2 The forward model 65 2.2.3 Essential constraints 66 2.2.4 Additional outputs computed 68 2.2.5 Horizontal permeability and vertical permeability 68 2.2.6 Usage of high-resolution data 69 2.3 Evaluation of microporosity 75 2.4 Evaluation of blocky (nonlaminated) reservoirs 76 Conclusions 76 Appendix 1 76 Appendix 2 78 Appendix 3 79 Appendix 4 86 Appendix 5 90 Appendix 6 93 References 113 Further reading 115

3.

Pore attributes of conventional reservoirs 3.1 3.2

3.3

The pore space of intergranular rocks Attributes of pore space 3.2.1 Pore shape, pore size, and pore throat size 3.2.2 Pores as the building blocks of pore space 3.2.3 Geometry of the pore body 3.2.4 Size of a pore 3.2.5 The concept of pore class 3.2.6 Surface area to volume ratio 3.2.7 The characteristic length scale of pore space 3.2.8 Hydraulic radius measure of the pore space 3.2.9 The pore shape factor 3.2.10 T2 log mean Distribution of incremental porosity over pore radius 3.3.1 Computation of CP(r) using NMR data 3.3.2 Distribution of incremental porosity over pore throat radius

117 117 118 118 119 119 120 120 121 121 122 122 124 124 125

Contents

3.3.3 3.3.4

Ratio of pore size to pore throat size Hard data on the distribution of pore throat size over incremental porosity ðCPT ðRÞÞ 3.3.5 Obtaining CPT(R) from mercury injection data 3.3.6 Obtaining CPT(R) from log data 3.4 Computation of CPT(R) from the NMR data 3.4.1 The linear conversion work flow 3.4.2 Nonlinear conversion work flow 3.5 Pore shape factor through integrating NMR and MICP (mercury intrusion data) 3.5.1 Frequency distribution of pore radius 3.6 A simple visualization of constriction and its effect on the gross permeability of pore space 3.6.1 Model prediction of permeability 3.6.2 TimureCoates permeability predictor from the perspective of constriction 3.7 Fractal attributes of pore space 3.7.1 The fractal model of the pore space, based on a poreepore throat assemblage visualization of the physical pore space 3.7.2 A fractal model of the pore space 3.7.3 Permeability from the perspective of the fractal model of the pore space 3.7.4 Cumulative pore volume 3.7.5 Representative hydraulic tortuosity and cumulated surface area to cumulated volume of the capillaries 3.8 Electrical formation factor from the perspective of the fractal model of the pore space Appendix 1 References Further reading

4.

ix 125 127 128 129 131 131 132 134 134 135 141 143 144

145 147 148 149 150 155 155 157 158

Pore space attributes of nonconventional reservoirs 4.1

4.2

CBM reservoirs 4.1.1 The components of the space occupied by fluids in coals 4.1.2 Characterization of the pore space of coals e cleats and fractures that are not cleats 4.1.3 Characterization of the pore space of coals using NMR data 4.1.4 Permeability of coaldmeasurement Shale reservoirs 4.2.1 Pore size encountered within shale reservoirs 4.2.2 Differentiation of pore classes for shale reservoirs 4.2.3 Multidimensional inversion of NMR echo data using maximum entropy principle

161 161 162 170 173 177 178 179 183

x Contents 4.2.4 4.2.5 4.2.6 4.2.7

Presentation of the results of inversion Porosity partition The method of diffusion editing The method of Laplace Inversion with regularization 4.2.8 D-T2 plots (more familiarly known as D-T2 maps) e forward models 4.2.9 D-T2 maps and other plots related to the results of echo data inversiondfield examples 4.2.10 Partitioning of total gas into free and adsorbed gas components using only NMR data 4.3 Characterization of fractured reservoirs Appendix 1 Appendix 2 Appendix 3 References Further reading

5.

188 188 191 195 197 203 207 209 209 214 216 223 225

Log measurements commonly used for finding the bulk porosity of conventional reservoirs Pore space attributes of conventional reservoirs 227 Measurement of bulk porosity 227 5.2.1 Measurement of formation density for bulk porosity 227 Appendix 1 257 Appendix 2 285 Appendix 3 288 Appendix 4 322 Appendix 5 332 References 341 Further reading 342 5.1 5.2

6.

Log measurements essential for characterizing the pore space of unconventional reservoirs 6.1

Measurement of total porosity using nuclear magnetic resonance 6.1.1 NMR theory 6.1.2 Pore space attributes and the relaxation of transverse magnetization 6.1.3 Total porosity and the bin porosities 6.1.4 Estimation of total porosity directly from the echo data 6.1.5 Obtaining total porosity using the formation density and NMR data

345 345 367 377 379 380

Contents

6.2

NMR and the porosity of CBM reservoirs 6.2.1 Coal pores 6.2.2 The T2 relaxation spectra of coals 6.2.3 Total porosity and gas volume 6.2.4 Porosity available within the rock for holding free gas 6.2.5 Cleat volume per unit rock volume 6.3 Porosity of shale gas/shale oil reservoirs 6.3.1 Estimation of effective porosity 6.4 Estimation of elemental concentration in rocks 6.4.1 Neutron capture gamma spectrometry 6.4.2 Inelastic gamma spectrometry 6.4.3 Computation of the average density of solid part of the formation 6.4.4 The acquisition of inelastic and capture gamma ray spectra 6.5 Characterizing the pore space of CBM reservoirs using image data 6.5.1 Cleat/fracture aperture and volume, and matrix porosity of CBM reservoirs 6.6 Characterizing the pore space of shale reservoirs using image data 6.6.1 Shale pores 6.6.2 Delineation of fractures within shale reservoirs using image data 6.6.3 Borehole electric images and distribution of organic matter 6.7 Generation of high-resolution electrical images of the borehole wall 6.7.1 Sensors 6.7.2 Position of each sensor in space 6.7.3 Data acquisition 6.7.4 Process flow for creating borehole images from the button current maps Appendix 1 Appendix 2 Appendix 3 Appendix 4 Appendix 5 Appendix 6 Appendix 7 References Further reading

xi 381 381 382 385 385 385 385 389 390 390 395 398 399 403 403 413 413 414 414 415 415 416 417 418 422 423 427 429 431 434 439 442 445

xii Contents

7.

Characterizing pores and grains using logs 7.1

Pore facies 7.1.1 Pore size distribution 7.1.2 Pore shapes from logs Appendix 1 Appendix 2 Appendix 3 Appendix 4 References Further reading

8.

447 447 455 465 469 486 487 492 492

Archie’s cementation exponent 8.1 8.2

Introduction Prediction of the value of Archie cementation exponent “m” using effective medium theories 8.3 Approaches used in modelling Archie’s m parameterd General Remarks 8.4 The approach for computing “m” using single-frequency dielectric data and using Archie’s equation 8.5 The approach for computing “m” using multifrequency dielectric data and using Archie’s equation 8.6 The approach for computing “m” from grain attributes obtained through multifrequency dielectric data inversion 8.6.1 Inversion of Archie “m” from single-frequency dielectric data 8.6.2 Inversion of Archie “m” from multifrequency dielectric data 8.6.3 Work flow for the generation of the dispersion model of dielectric permittivity ε 8.6.4 The dispersion model for rock conductivity 8.6.5 Applicability of the model to clayey rocks 8.6.6 The role of contribution to complex permittivity, coming from unconnected pores 8.6.7 Approaches to estimation of Archie “m” based on differential effective medium theory Appendix 1 Appendix 2ddifferential effective medium theory for aligned inclusions case Appendix 3 Approaches to model “m”: in case of Shaly rocks using the Bergman spectral density representation of the effective permittivity of a binary mixture Appendix 4 An approach to Archie “m” through NMR data analysis Percolation theories and Archie “m” factor

495 500 501 501 501 503 503 511 522 524 527 528 530 551 556 559

564 581 589 592

Contents

xiii

Appendix 5: logarithmic mixing law for effective permittivity of a mixture 594 Approaches to estimate “m” through fractal model of pore space 602 Concluding remarks 607 References 609 Further reading 611

9.

Permeability of unimodal pore system 9.1 9.2

9.3 9.4 9.5

9.6 9.7 9.8

9.9 9.10 9.11

9.12

Introduction Response of local pressure field, local fluid velocity field, and average fluid velocity field to changes in driving pressure A simple model of pore space presented Flow through a capillary 9.4.1 How pore space attributes influence permeability Surface area, or, representative pore dimension or characteristic length scale driven approaches to permeability 9.5.1 The “bundle of capillary tubes” model of pore space 9.5.2 Forward model of permeability in terms of pore space attributes 9.5.3 The stream tube model of flow through a porous medium whose grain- and pore-arrangement of a macro level volume segment, is isotropic, and thus, macroscopic permeability field is a scalar Depiction of pore space in the model 9.6.1 Stream tubes Elemental stream tube Integral representation of macroscopic permeability (or simply “permeability”), and the concept of microscopic permeability density or local permeability density Integral representation of permeability Average permeability field ks , also referred to, as the “effective permeability factor” Integral representation of average permeability field (effective permeability factor) ks and relation between permeability k and effective permeability factor ks The elemental stream tube permeability factor field 9.12.1 Explicit representation of permeability factor of a streamline (elemental stream tube) in terms of some of its attributes

613

614 616 617 620 621 621 626

628 628 628 629

630 631 631

632 633

634

xiv Contents 9.12.2

Decomposition of permeability into macroscopic pore space attributes, namely, hydraulic tortuosity, hydraulic constriction factor, and hydraulic pore radius 9.13 Insights from the simplest possible pore space model and the role played by “hydraulic radius” in permeability modelingdderivation of a generalized Kozeny-Carman equation 9.14 Concept of hydraulic radius 9.14.1 How microscopic streamline attributes and macroscopic pore space attributes are related in a porous medium 9.14.2 Hydraulic Constriction factor (C s ) of connected pore space, which is the macroscopic counterpart of streamline attribute CðSÞ 9.15 Generalized Kozeny-Carman equation 9.15.1 Conventional form of Kozeny-Carman equation predicting permeability 9.15.2 Well known equations for permeability prediction from log measurements. The equations discussed below are for water wet rock only 9.16 From Kozeny-Carman equation, to Van Baaren’s equation 9.17 The RGPZ equation Appendix 1 Appendix 2 Appendix 3 Estimation of rmin from NMR log data Estimation of g1 (pore size to pore throat size ratio) Appendix 4 Appendix 5 Appendix 6 References Further reading

10.

636

638 638

640

640 641 643

646 663 665 667 668 670 703 703 711 720 728 730 731

Permeability and electrical conductivity of rocks hosting multimodal pore systems and fractures Preamble 10.1 Pore size nomenclature 10.1.1 Micropores, mesopores, and macropores 10.2 Pore classification using NMR T2 distribution 10.2.1 Relation between pore sizes and NMR T2: the fast diffusion limit 10.2.2 Limits of validity of a T2 threshold-based porosity partition

735 735 737 738 738 742

Contents xv

10.2.3

Model-based porosity partition for the case of rocks having intragranular porosity 10.2.4 Case where vugs are not sparsely distributed in the rock 10.3 Pore classification using well bore images 10.3.1 Preconditioning of data 10.3.2 Dip picking 10.3.3 Closing the data gap 10.3.4 Extraction of fracture segments 10.3.5 Matrix extraction 10.3.6 The problem of computing the pore volume contribution of heterogeneities 10.3.7 An efficient methodology for extracting heterogeneities Appendix 1 10.4 Porosity partition using acoustic logs 10.4.1 Challenges of porosity partition using acoustic logs 10.4.2 A work flow of porosity partition using acoustic logs Appendix 2 10.5 Electrical conductivity of an unfractured composite hosting dual porosity Modeling s2 for fully water saturated component 2 case Evaluation of s the electrical conductivity of a composite hosting dual porosity Case of partial saturation 10.6 Permeability of an unfractured composite hosting dual porosity Base rock permeability Permeability of the composite rock 10.7 Electrical conductivity of fractured rocks Partial saturation case 10.8 The variable cementation exponent method of computing water saturation Case of connected pore space having tortuosity unity Pore volumes Rationale behind Equation (10.89) Level by level evaluation of the Archie cementation exponent of the formation Discussion Computation of water saturation

742 746 747 747 747 747 753 758 761 762 770 772 772 773 778 788 789 792 793 794 794 796 798 799 801 801 801 802 804 804 807

xvi Contents Appendix 3 10.9 Permeability of rocks hosting connected vugs/fractures Alternate simpler way of estimating Qcon Permeability kb of the base rock Permeability of gross rock 10.10 Bray e Smith method of computing permeability 10.11 Another approach to permeability modeling which also relies on NMR log data References Further reading

Index

808 811 816 817 818 818 819 822 824

825

Preface This book explains how log measurements explain pore space. The idea is to familiarize industry persons, students of geophysics, petrophysics, and petroleum engineering, who study well logging at the graduate level, and those embarking on research in topics related to petrophysics. We start from the basic concepts that lead to models for the important microscopic and macroscopic pore space attributes. This book, while having this stated goal, is also aimed at researchers, with two objectives: (i) to make it less timeconsuming for a researcher to imbibe published materials; (ii) to familiarize the researcher with the principles behind some of the instrumentation involved in data acquisition and processing. The focus throughout this book has been to restrict to the important techniques, and keep the discussion generic. It is assumed that a reader has a background of classical physics, quantum mechanics, vector analysis including vector calculus, and a basic knowledge of tensors. It is presumed that the reader has some grounding in the nuclear magnetic resonanceebased techniques for eliciting pore space attributes. Basic knowledge of nuclear physics is also assumed. Familiarity with the techniques of morphological image processing, effective medium theories, and log data inversion for formation attributes can be helpful in following some of the chapters with ease. The list of books given below is recommended for appreciating some of the topics covered in this book. Coates, G.R., Xiao, L.Z., Prammer, M.G., 2000. NMR Logging Principles and Applications, Gulf Publishing Company, Houston. Dunn, K.-J., Bergman, D.J., LaTorraca, G.A. (ed.), 2002. Nuclear Magnetic Resonance: Petrophysical and Logging Applications, Pergamon Press, New York. Ellis, D.V., Singer, J.M., 2008, Well Logging for Earth Scientists, second ed, Springer. Griffiths, D.J., 2013, Introduction to Electrodynamics, fourth ed, Pearson. Tuck, C.C, 1999, Effective medium Theory: Principles and Applications, Clarendon Press.

xvii

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Acknowledgments KMS and SM thank the Elsevier team (Amy Shapiro, Howell Angelo M. De Ramos, proofreaders) for outstanding support. CPDA grant (IIT Bombay) provided infrastructure to SM.

xix

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Chapter 1

Pores and pore space 1.1 The pore space of rocks 1.1.1 The pore space of granular rocks Inorganic granular rocks are formed due to the compaction and lithification of sediment whose grains are clastic (e.g., Mukherjee and Kumar 2018) and also include oolitic rocks (e.g., Dasgupta and Mukherjee 2020). Organic granular rocks are formed due to the compaction and lithification of grains composed of the skeletal remains, fecal fragments, radiolarii, algal fragments and coral fragments, and other remnants connected with marine flora and fauna. The sedimentary grains giving rise to inorganic granular rocks do not have internal void spaces. On the other hand, the grains of the sediment which gives rise to organic granular rocks can have internal void space. The reason for this is the fact that a grain, in the context of organic granular rocks, can itself be an aggregate of smaller units, e.g., fecal fragments. It can also have dissolution pore space, as, for example, in the case where a grain is a skeletal fragment or aggregates of skeletal fragments. Rocks such as bound-stones can have pore space, which is intra- as well as interfragmentary. The term ‘fragment’ here would refer to a unit of the bound-stone, which is the result of the binding of these fragments. Pore space is defined by pores that are the elementary units, whose aggregate is the pore space. Usually, sediments which compact and lithify into granular rocks have a size distribution of their grains. As a result, we have a grain size distribution and correspondingly, a pore size distribution. Often the grain size can be divided into two classes. The size class of the lower range, say < 40 microns, overlaps with what is generally referred to as ‘matrix.’ When the larger grains float within the matrix, they do not contribute to the pore space. Such rocks where the matrix grains are the framework grains are called as mud-supported rocks. The opposite is the case where the larger grains comprise the framework and the matrix is distributed as partings, or lenses, or as a pore-filling coating of the framework grains. Such rocks are called as grain-supported rocks.

Understanding Pore Space through Log Measurements. https://doi.org/10.1016/B978-0-444-64169-4.00007-9 Copyright © 2022 Elsevier B.V. All rights reserved.

1

2 Understanding Pore Space through Log Measurements

It is not unusual to find rocks, which are partly grain supported and partly mud supported. These are texturally immature rocks found in certain depositional settings such as in fans. In general, the nature of the distribution of the grain size of a rock defines its textural maturity. Besides the intergranular and intragranular pore space, micropores arise inside overgrowths or cement (Fig. 1.1). Fig. 1.1 shows the typical pore types that arise within sandstones. One further category of pore space existsdthe dissolution pore space. It arises due to the dissolution of grains during neogenesis or diagenesis. Examples of this are moldic porosity, vugular porosity, solution channels, and leached zones. The phenomenon of grain-dissolution dilates the original pores, and pores occluded by neogenetic calcite precipitation, by subsequent dissolution that spreads as patches. These can be described as leached-pore porosity patches. Dissolution pore space also arises due to the pressure solution phenomenon giving rise to stylo-lamination. The stylo-laminations are anastomosed and also are associated with tension gashes, which arise from the necessity of stress-release. The pore fabric encountered is often the result of multiple cycles of porosity creation and destruction in the course of cementation, neogenesis, and diagenesis. By the same token, the volume of the intergranular pore space postlithification is often reduced owing to the occlusion of pores due to overgrowths, secondary cementation, grain alteration, and authigenesis, during diagenesis. Dolomitization, which is a special case of grain alteration, results in the increase in the size of the calcite grains without any significant alteration of the morphology of the grain-to-grain contacts. This usually results in the

FIGURE 1.1 Idealized representation of different pore types such as intergranular, dissolution, fracture, and micropores, that can arise within sandstones. Adapted from Pittman (1979).

Pores and pore space Chapter j 1

3

enhancement of the pore volume. In the cases mentioned above, secondary cementation, spar deposition in limestones, authigenesis of clay minerals on the grain surfaces, and structures that straddle the pore body reduce the pore volume. Example for the former is the authigenesis and/or generation of montomorillonite as grain coats/alteration of fine mica grains which were originally part of the matrix, and the authigenesis of chlorite or its genesis as the alteration product of mafic minerals. An example of the latter is the authigenesis of illite as threadlike structures which straddle the pore-body space. The pore space that is intimately connected with grain contacts is the gateway that connects two contiguous pores. Secondary cementation and authigenesis, which occur during diagenesis, often constrict this important connecting pore volume (the connecting pore volume is called as pore throat). This in turn hampers the degree of connectivity of the pores for fluid transport and thereby hinders the pore space to transport fluid. The quantitative measure of the ability of fluid flow is called as permeability. Thus, diagenetic processes as mentioned above reduce permeability. Mud-supported rocks fall into a class by themselves. Mud-supported rocks have clay-size and silt-size particles. The pores within the mud are largely flat pores. The pore structure complicates when a mud-supported rock hosts carbonaceous matter including bitumen and kerogen. Bitumen hosts pore system in some cases but post its formation. Kerogen, however, hosts multi-nanometer-sized pores. The pore space within kerogen is called as ‘kerogen pore space.’ Kerogen floats inside the clay matrix, in a mudsupported rock containing carbonaceous organic matter. Kerogen pores are also commonly referred to as organic pores. In a mud-supported rock hosting organic matter, we have inorganic pores, which are pores present within the matrix not occupied by organic matter and kerogen pores within the organic matter, in addition to pores within bitumen, if present. Additionally, submicron-sized inter-kerogen pores arising within the interstices between microvolumes of kerogen material can also exist. Kerogen pores host adsorbed methane.

1.1.1.1 Pore space of some granular rocks not falling within the above categories 1.1.1.1.1 Intercrystalline pore space Cementation during neogenesis and diagenesis of carbonate rocks can result in significant pore volume occupied by calcite crystals. These crystal assemblages sometimes host submicron to nanometer-sized pores, which are known as intercrystalline pores. Such pores can also develop during the postdepositional dolomitization process in the case of some carbonates. Intercrystalline pores are flat and have a very low degree of connectedness.

4 Understanding Pore Space through Log Measurements

1.1.2 The pore space of carbonate rocksdan example of the porosity classification The different varieties of pore space discussed above are exemplified in carbonate rocks. Choquette and Pray classification (Choquette and Pray 1970) is useful (Fig. 1.2). Appearance of different pore types in carbonates is given at Fig. 1.3 as an example.

1.1.3 The pore space of coals Coals are made up of macerals and finely distributed silt and clay, made up of inorganic minerals including calcite in some cases, in addition to clay and iron

FIGURE 1.2 Choquette and Pray classification of limestone porosity. Reproduced from Choquette and Pray (1970).

Pores and pore space Chapter j 1

5

FIGURE 1.3 Panel (A) is an example of an intergrain porosity. Panel (B) is an example of moldic porosity. Panel (C) represents intragrain-dissolution porosity. Panel (D) is an example of intrafossil porosity. Panel (E) presents micropores-hosted porosity. Panel (F) shows intergrain, graindissolution porosity. Panel (G) is an example of vuggy porosity. Panel (H) is an example of a stylolite hosting some porosity. Panel (I) presents porosity arising from a microfracture. All the panels are SEM photographs. Reproduced from Lu et al. (2021).

6 Understanding Pore Space through Log Measurements

minerals. Some coals also contain bitumen. Barring low-rank coals, coals contain cleats as well as natural fractures. Cleats are separation type fractures. Of these face cleats are throughgoing fractures which form as parallel fractures. These are the dominant cleats and are first formed during coalification. Butt cleats are fractures that terminate orthogonally on the face cleats. These are formed subsequent to the formation of face cleats. Cleats are the pore space in coals that are responsible primarily for the diffusive and hydraulic transport of methane present within coals. They are also pore space that acts as storage space for methane gas within coal. Cleat aperture ranges from a few microns to hundreds of microns. Cleats have finite length, as well as tortuosity and rugosity of the cleat-walls. Fractures which are not connected with cleat formation also occur within many coal seams. These are of tectonic origin and are formed after a coal seam is formed. The aperture of these fractures can range from micrometer to millimeter. The cleat and noncleat pore spaces account only partially for the total pore space of coals. A large part of the remaining pore space not accounted by cleats and noncleat fractures is named as the adsorption pores. These are nanometer-sized pores that occur mainly within the bitumen present in some coals, in some cases, and within the maceral assemblage. These are also called as adsorption pores because they host the adsorbed methane within the coals. Apart from these pore systems, nanopores and micropores of size less than a micrometer are present within the finely divided silt and clay present within coals. The fractional porosity contributed by these pore systems is nonnegligible, in the case of medium and low-rank coals. Such coals contain nonnegligible fine silt and clay content. This fine silt and clay host intergranular pore assemblages, which account for a part of the total pore volume available within such coals. The intergranular pore space mentioned above is also called as the pore space hosted by inorganic pores. Pores larger than these, which also hold gas, come under the categories of gas pores and sink-holes. These pores play the role of passages through which gas can diffuse and also hydraulically transport, in a coal, in addition to playing the role of storage volume for the gas. Fig. 1.4 shows the gas pores, natural fractures which are not cleats, and some examples of inorganic pores. Fig. 1.5 shows an example of the pore structures within different coal macerals. These constitute the major part of organic pores within coals. When coals have pyrobitumen within them the pyrobitumen sometimes hosts nanopores.

1.1.4 Pore space of shale reservoir rocks Shale reservoir rocks host intergranular pores, dissolution pores, and organic pores. Of these classes of pores, organic pores dominate. The pores mentioned

Pores and pore space Chapter j 1

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FIGURE 1.4 SEM photographs that show examples of some of the pore types seen in coals, excluding nanopores and cleats. The pore type is indicated on the respective panels. Reproduced from Jienan et al. (2019).

above constitute together the storage space for gas. In terms of size, micropores dominate the total pore assemblage of shale reservoir rocks. Mesopores and a small amount of macropores are also present. Fig. 1.6 shows organic pores developed with a honeycomb pattern in an organic-rich shale. The pore diameters can be compared against the scale bar. This is not a unique pattern of development of organic pores within organicrich shales (shale reservoirs). Because of the small size and abundance, the cumulated surface area of these pores is often large and these pores host adsorbed methane on their surfaces. Fig. 1.7 shows examples of inorganic pores found in organic-rich shale (shale reservoir rock). The inorganic pore assemblage is similar for shale oil and shale gas reservoirs.

8 Understanding Pore Space through Log Measurements

FIGURE 1.5 SEM photograph showing the pore structures of different coal macerals: (A) Collodetrinite, (B) Collotelinite, ((C) Telenite, (D) Semifusinite, (E) Fusinite, and (F) Inertodetrinite. The scale bar is 10 nm. The average pore diameter is around 3 nm. Reproduced from Zhang et al. (2020b).

1.1.4.1 Organic pores in shale reservoir rocks Organic matter within shale reservoir rocks includes kerogen, bitumen, and pyrobitumen. Bitumen-hosted pores and pyrobitumen-hosted pores and kerogen pores are prominent within gas shales. For the case of oil-shales, the intraparticle pores in kerogen are not prominent, while pore space for oil is mainly within the bitumen, which in turn fills the porosity available in the shale during maturation.

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FIGURE 1.6 SEM photograph showing organic pores developed within an organic-rich shale with a honeycomb pattern. The pore size distribution can be appreciated against the scale bar shown in the figure. Reproduced from Er et al. (2016).

1.1.4.2 Bitumen pores in shale gas reservoirs Solid bitumen and pyrobitumen, which occur within shale gas reservoirs, are reported to host pores, in some cases. This type of organic pore-hosted porosity contains gas. An example from Nie et al. (2018) is shown at Fig. 1.8. Another example is shown at Fig. 1.9, which also pertains to gas shale. Fig. 1.10 gives a detailed view of pyrobitumen pores as another example. The pores are well connected and act as storage for gas. The cause of the porosity development within bitumen is release of gas when liquid hydrocarbons crack and when the shale (and thereby the organic matter within it) comes within the gas window during burial (Loucks et al., 2009). 1.1.4.3 Kerogen pores in shale gas reservoirs Intraparticle porosity within kerogen begins to develop when the organic-rich shale enters the dry gas window. This does not happen when the shale falls within the oil window. The porosity evolution for the Bakken shale is shown in Fig. 1.11. 1.1.4.4 Organic pores in shale oil reservoirs Solid bitumen, which can transform to liquid hydrocarbon, can arise, when the kerogen, partially converted to bitumen through catagenesis, is Type II typical of marine paleoenvironment. At the oil window, the shale pores are filled with kerogen and bitumen. When oil is expelled into this porosity due to the conversion of bitumen to oil, the oil may replace bitumen volume as small voids. Kerogen in oil window contains little intraparticle porosity. Hence oil storage is conceivable only within the bitumen, which in turn was pore-filling organic matter. Thus, pore structures present within shale reservoir rocks are complex. The pores differ widely in respect of their genesis, shape, and size. Many pore

10 Understanding Pore Space through Log Measurements

FIGURE 1.7 SEM photographs showing examples of inorganic pores found within organic-rich shales. (A) Intergranular pores; (B) fracture porosity and intergranular porosity. The pore at the lower left, that has a near-square cross section, is an intergranular pore. Clastic grains appear light gray and organic matter dark gray. The clastic grains have concaveeconvex contacts relationship with organic matter. Intergranular contacts between clastic grains are seen to be point-line contacts. Organic pores of diameter 10 microns). Normal sandstones can have all or some of the pore types mentioned above, and in addition contain intergranular pores of size >10 microns in abundance. The classification of Lonoy (2006) of pores in carbonate rocks (which is of relevance here), categorizes pores as pore types as per the mode of occurrence of the pores. Pore size classes are then defined for specific pore types. This is illustrated at Table 1.1.

14 Understanding Pore Space through Log Measurements

TABLE 1.1 Classification of carbonate rock porosity (Lonoy, 2006). Pore type Interparticle

Pore size (in mm)

Pore distribution

Pore fabric

Micropores (10e50)

Uniform

Interparticle, uniform micropores

Patchy Uniform

Interparticle, patchy micropores Interparticle, uniform mesopores

Patchy Uniform

Interparticle, patchy mesopores Interparticle, uniform macropores

Patchy

Interparticle, patchy macropores

Uniform

Intercrystalline, uniform micropores

Patchy Uniform

Intercrystalline, patchy micropores Intercrystalline, uniform mesopores

Patchy Uniform

Intercrystalline, patchy mesopores Intercrystalline, uniform macropores

Patchy

Intercrystalline, patchy macropores

Mesopores (50e100) Macropores (>100) Intercrystalline

Micropores (10e20) Mesopores (20e60) Macropores (>60)

Intraparticle moldic

Micropores (20e30)

Moldic micropores Moldic macropores

Vuggy Mudstone microporosity

Vuggy Micropores (13%. The average pore radius continues to maintain 50-micron value for porosity 7.5% while the average pore throat radius reduces to 18.4 microns. This shows that when the porosity reduces to a value below a certain threshold, the pore size to poreethroat size ratio increases, even when the

20 Understanding Pore Space through Log Measurements

mean pore size value does not reduce with reduction of porosity This points to a change in the nature of the 3D structure of the pore network, reducing the effective radius of pore throats, when porosity reduces below a certain threshold value. In all the panels the pore throat distribution is highly skewed. This makes the average pore throat value vulnerable to changes in the nature of connection in 3D space, of the pores. This change is apparently happening noticeably when the porosity value falls below a certain threshold value.

1.4.3 A popular and successful model of pore space e the capillary tube bundle for modeling flow of an incompressible fluid through porous rocks Starting from cylindrical pores joined by cylindrical pore throats, the model is simplified by assuming that the pore space can be visualized successfully as a bundle of capillaries. Here the radius of cross section varies where a cylindrical pore and cylindrical throat join. The topology of this bundle is such that no two capillaries connect. Thus, the bundle of capillaries is a union of disjoint subspaces, each of which is a capillary. Consider this bundle, such that any point within the connected pore space in a rock, belongs to one of the disjoint subspaces, viz., a capillary. This model of the pore space further supposes that the capillaries have a topology which mimics the collection of flow tubes when a flow occurs within the pore space. Thus, this is intimately connected to the fluid flow geometry and reflects it. This way of visualization implicitly assumes that the pore space is isotropic, with regard to macroscopic attributes such as mean pore radius, mean pore throat radius, and permeability to fluid flow. In many models of fluid transport through the connected pore space of rocks, a further simplification is assumed by making the value of the ratio of the poreethroat radius to the pore radius equal to 1.0, leading to the unconstructed bundle of capillaries model of the connected pore space of rocks.

1.4.4 A generic treatment of transport of an incompressible fluid through a medium such as a porous rock A general treatment of fluid transport through porous rocks (Berg, 2014) leads to the concepts of mean radius of a flow tube, and a flow constriction coefficient (or a flow constriction factor). The visualizations of the pore space in the manner indicated in the foregoing have been proven to be successful since that they anticipate permeability predictors such as the KozenyeCarman relation (Carman, 1937).

1.4.5 Modeling the orientation of pores in space Pores can have geometric shapes that intrinsically possess one or more axes of symmetry (principal axis/axes). The orientation of these axes, relative to the

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plane of bedding, controls the anisotropy of macroscopic rock properties. The rock properties referred to above include the hydraulic transport, electric, acoustic, and mechanical properties of the rock at the macroscopic level. Boreholes drilled for oil and gas are generally vertical while rock strata have low dips. Log measurements carried out within boreholes generally sense the properties of the rock perpendicular to the borehole axis. Hence, log measurements generally sense rock properties as measured along the bedding planes. Pore systems are often heterogeneous from the perspective of pore size. Larger pores are often considered as inclusions within a mass of pores lacking intrinsic axes of symmetry. In this case, the pores have a single axis of symmetry, the pore system is considered to be: i) transversely isotropic in a macroscopic sense, with the axis of symmetry of each pore parallel to that of any other pore, and perpendicular to the bedding planes, or ii) the symmetry axes of the different pores are randomly oriented in space. For case i) the macroscopic electrical hydraulic transport acoustic and mechanical properties of a composite of the inclusions and the groundmass of the pores hosting the inclusions exhibit anisotropy. For case ii), the macroscopic electrical hydraulic transport acoustic and mechanical properties of a composite of the inclusions and the groundmass of the pores exhibit isotropic behavior. The abovementioned properties are expressed quantitatively as tensors. The values of the elements of such tensors considered in a coordinate frame depend upon the orientation of the principal axes of the inclusions relative to the reference coordinate system. Generally, the two coordinate axes are considered to lie on the bedding plane, and the direction normal to this plane defines the third axis. This makes the orientation of the principal axes of the inclusion relative to the bedding planes as a crucial model input. In many cases, the distribution of the orientation of the principal axes of the inclusions or pores in space can be realistically considered to be random (case (ii) above). When this is factored into the modeling, the rock is heterogeneous but isotropic in the macroscopic sense. In that case the tensors mentioned above reduce to scalars (rank zero tensors).

1.5 The fractal model of pore space Even for the case of granular rocks, modeling pore space as a network of pores as nodes and pore throats as connections is not the only way of visualization of pore space. One approach which is found to be highly successful in modelling pore space, is the one that models the connected pore space of a rock, as a fractal object. One of the significant successes of this manner of visualization of pore space of rocks has been the Brookes-Corey relation (Brookes and Corey 1964) for absolute permeability and the effective permeability for one phase in the presence of the other phase, for rocks whose pore fluids comprise two phases. Modeling fractures as fractal objects have also proven useful. In

22 Understanding Pore Space through Log Measurements

such a visualization of fractures, the tortuosity and the rugosity of the walls of fractures arise as manifestation of the fractal nature of the system, at differing scales of visualization. Scaling relations of fracture length to fracture aperture based suggested by a fractal perspective are used in the prediction of fluid flow efficiency of fracture systems (Bandyopadhyaya et al., 2019).

1.6 Use of log data The recovered model can include the following microscale attributes of the pore space, such as a) pore size (given prior knowledge of the pore geometry, from lab data, pertaining to the relevant rock stratum being investigated), b) pore aspect ratio, assuming a pore geometry (e.g., spheroidal geometry). The recovered model can also include the following macroscale attributes of the pore space, such as A) representative tortuosity of connected pore space, B) the distribution of the surface-area-to-volume ratio of different pore size classes available within the rock, C) representative value of the ratio of the surface area to volume, of the pore space, D) total pore volume, E) Archie cementation exponent representative of the pore volume, and F) the hydraulic permeability of the pore volume. Step (i) can also be a forward model of the gross attributes of the pore space and the grain network, or models of flow paths for fluids and electric current through pores and so on. Step (ii) would recover the model of hydraulic permeability and the representative tortuosity of pore space for fluid transport and electric current, respectively, and so on. When step (i) is a model built on a) pore shape and size, b) grain shape and size, and c) the complex electrical permittivity of the grain material, the recovered model retrieved would be X) the complex permittivity of the pore fluids, and Y) the tortuosity of the pore network. Similarly, when the forward models involve the acoustic properties of the pore fluids and those of the solids regime, the recovered model would be the bulk modulus of elasticity of the pore fluid, the relative concentrations of the components of pore fluids, and so on. The modeling of the pore network and pore level attributes, e.g., shape, size, and orientation of principal axes of grains and pores in space, is crucial for the whole process of the use of log data to yield useful outputs as discussed above. Without resorting to inversion, it is possible to also find from log data, total porosity, and predict the value of absolute hydraulic permeability with a good degree of reliability, when the full complement of the required log data acquisition suite is deployed. It is possible to gain a qualitative understanding of the degree of connectivity between pore classes differentiated by their mean sizes using NMR data. Such data allow one to differentiate groups of pores, which share a common value of the ratio of cumulated surface area of the group of pores to their cumulated volume, from other pore groups that are characterized by a different value of this ratio. When the

Pores and pore space Chapter j 1

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representative pore shape applicable to a given such pore group is known from lab investigation of cores cut out the rocks analyzed and studied, it is possible to generate the distribution of the cumulated pore volume for pores sharing a pore size range, over the different pore size ranges. Knowledge of the resistivity of the pore fluid allows one to obtain the value of the electrical formation factor of the pore space using resistivity and porosity data of clean to moderately clayey water bearing reservoir rocks. This in turn leads to the value of the Archie cementation exponent valid for the pore space of the rocks in question. Knowledge of the porosity and the electrical formation factor, obtained from logs, leads to the estimate of electrical tortuosity. This is an important data because the hydraulic tortuosity, geometric tortuosity, and electrical tortuosity are close in magnitudes for intergranular-porous rocks. High-resolution electric image logs enable one to quantity the orientation of cleats and fractures. The data acquired by these tools, when integrated with other log measurements, e.g., shallow resistivity logs, reliably estimate fracture aperture, cleat aperture, and so on. Inversion of data acquired by advanced acoustic logging tools results in reliable estimate of cleat/fracture density. Cleat/fracture density can also be obtained from the analysis of high-resolution electrical images.

1.6.1 Understanding the degree of connectivity between two pore size classes using NMR log data NMR logs offer the best way to understand the degree of connectivity between two pore size classes. For example, consider the micropore class and macropore class. In the absence of connectivity between these two pore groups, there will be a gap between the distributions of the pore size of these two groups as found on NMR T2 distributions. However, in the presence of connectivity between these two pore groups, diffusive coupling (diffusion of fluid molecules between the pores of these two pore groups) of these two pore groups will result in the following. The macropores and the micropores coupled through diffusive coupling are not represented at the proper T2 ranges that they individually belong to. Instead, a pore size intermediate between that of microand macropore is now represented on the NMR T2 distribution (Carneiro et al., 2014). This effect has been successfully modeled (Ramakrishnan et al., 1999). The more the degree of interpore group connectivity, more is the effect of the diffusive coupling of the pore groups on the NMR T2 distributions. Thus, an understanding of the degree of connectivity between pore size classes is obtained by analyzing the NMR T2 distributions. Fig. 1.15 is an example from a limestone. The pore size distribution obtained from NMR T2 distribution is compared to the MICP data. The smearing effect, due to the connectivity between the two pore groups differentiated by pore size, is clear. Note the way the apparent pore size distribution

24 Understanding Pore Space through Log Measurements

FIGURE 1.15 An example from Indiana Limestone. The existence of two distinct pore size classes is evident from the pore radius distribution derived from MICP data (red curve; gray in print version). This distribution has been derived, assuming a value of unity for the pore-radius-topore-throat-radius ratio. The pore radius distribution derived from NMR T2 is shown by a blue (black in print version) curve. This distribution has been derived by assuming circular crosssections for the pores. The difference between the distribution in blue (black in print version) and that in red (gray in print version) is due to the diffusive coupling effect between pores as the spins relax. The diffusion coupling results from the diffusion of fluid molecules from pores of onepore size group to those of the other pore size group. The diffusion, in turn, is due to the connectivity in space existing between the two pore groups indicated. Reproduced from Carneiro et al. (2014).

from NMR T2 distribution differs from the true pore size distribution is reflected in the MICP data.

1.6.2 Insights from high-resolution resistivity imaging tool data into the pore size heterogeneity Besides NMR log data, the button current data of high-resolution resistivity imaging tools offer insights into pore size heterogeneity. Under conditions of conductive drilling mud used, low to zero clay content, and low to zero hydrocarbon saturation within the flushed zone, the following steps are implemented. The button current data are converted to resistivity data for the formation investigated by a button electrode. These resistivity data are inverted to porosity data by using the equation:  1=m Rmf f¼ (1.1) Rb Here f: Button porosity of the rock volume investigated by a button electrode (button porosity), Rmf: mud filtrate resistivity, Rb: resistivity of the rock volume

Pores and pore space Chapter j 1

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investigated by a button electrode, and m: Archie cementation exponent. Eq. (1.1) is a different form of Archie Equation (Archie, 1952). It is implicitly assumed that the magnitudes of m and f are independent. This assumption implies that the degree of pore connectivity for a volume of the rock is independent of the total pore volume associated with that volume of the rock. This condition is satisfied for the case of porous sedimentary rocks. The value of f, computed using Eq. (1.1), has a distribution over the different button electrodes, and thus, over the volume elements of the rock investigated by the different button electrodes. This distribution, when expressed as a histogram over the button locations in 3D, represents the heterogeneity of the porosity within the rock, for a volume of rock cumulatively investigated by all the button electrodes. Physically, this region comprises the formation rock surface exposed as the borehole wall and extending radially, a short distance into the formation rock. It can be reasonably assumed that the rock volume investigated by a button electrode is the same, for any button electrode. In that case, if we consider any peak within the distribution, the value of the area under that peak would be proportional to the cumulated pore volume of those volume elements, whose porosity values lie within the porosity interval on the x-axis, spanned by the peak. Porosity histograms mirror the pore size histograms. To elaborate, consider two volume elements of the rock, of equal volume. Let the porosity of one volume of rock, which we denote as A, be fA . Let the porosity of the other volume, denoted as B, be fB . Let fB > fA . Since A and B are of equal volume and fB > fA , the pore volume of B > pore volume of A. Therefore, it can be expected that the mean pore size for B is greater in value, than the mean pore size for A. The reason is the fact that the mean pore size for a volume of rock and the cumulated pore volume for that volume of rock often go hand in hand for porous rocks which have interconnected pore assemblage. Thus, it is to be expected that for porous rocks, the porosity histograms and the pore size histograms are similar. Fig. 1.16 is an example of the button porosity histograms for the case of a carbonate rock free of fractures and stylo-laminations, hosting primary pores and dissolution pores (completely accounting for the secondary porosity developed within the rock). It is easily seen that this button porosity histogram represents local porosity distributed within the rock. In light of the analysis presented, note that the mean pore sizes of the respective pore assemblages, which lead to the middle peak and the right-most peak (Fig. 1.16), are, respectively, higher than the mean pore size of the pore assemblage that was responsible for peak at the left-most position. As stated, secondary porosity within the carbonate rock is wholly accounted by the grain-dissolution pores. By the very mechanism of the genesis of grain-dissolution pores, the dimensions of the grain-dissolution pores would have to be much greater than that of the intergranular pores, which constitute the primary porosity developed within the rock. Using the

26 Understanding Pore Space through Log Measurements FIGURE 1.16 Example of a porosity histogram of a carbonate rock having primary porosity and secondary porosity. Reproduced from Newberry et al. (1996).

above reasoning the left-most peak represents primary porosity within the rock and the other two peaks the secondary porosity. In a button porosity histogram (Fig. 1.16), let AP stand for the value of the area under a peak P of the button porosity histogram. Let RP stand for the volume of the rock investigated together by those button electrodes whose button porosity histogram is the peak P. Let VP stand for the value of the total pore volume present within RP. Then, AP fVP . Because of this proportional relation of AP to VP, button porosity histograms such as the one depicted at Fig. 1.16 lead to a quantitative understanding of the partition of total pore volume of a rock into primary-pore volume and secondary-pore volume. Further, since the area under a peak is proportional to the fractional pore volume, of the total pore volume of the rock cumulatively investigated by all the button electrodes taken together, the relative magnitudes, of the areas under the different peaks in a button porosity histogram such as depicted at Fig. 1.16, give an understanding of the partition of total pore volume of a rock into primary-pore volume and secondary-pore volume. From the foregoing, it is clear that the ratio between the magnitude of the area under the left-most peak and the sum of the magnitudes of the areas, respectively, under the middle and the right-most peaks should be equal to the ratio between the magnitudes of the primary-pore volume and the secondarypore volume, respectively. Fig. 1.16 presents how data acquired using highresolution electrical imaging tools help in understanding the pore space heterogeneity of rocks.

References Archie, G.E., 1952. Classification of carbonate reservoir rocks and petrophysical considerations. AAPG Bulletin 36, 278e298. Bandyopadhyaya, K., Mallik, J., Ghosh, T., 2019. Dependence of fluid flow on cleat aperture distribution and aperture-length scaling: a case study from Gondwana coal seams of Raniganj Formation Eastern India. International Journal of Coal Science and Technology 17, 133e146.

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Berg, C.F., 2014. Permeability description by characteristic length tortuosity constriction and porosity. Transport in Porous Media 103. Brooks, R.H., Corey, A.T., 1964. Hydraulic Properties of Porous Media, Hydrology Paper No. 3. Colorado State University. Carman, P.C., 1937. Fluid flow through a granular bed. Transactions of the Institution of Chemical Engineers 15, 150e167. Carneiro, G., et al., 2014. Evaluating pore space connectivity by NMR diffusive coupling. In: SPWLA 55th Annual Symposium Held at Abu Dhabi. UAE. Choquette, P.W., Pray, L.C., 1970. Geologic nomenclature and classification of porosity in sedimentary carbonates. AAPG Bulletin 54 (2), 207e250. Dasgupta, T., Mukherjee, S., 2020. Sediment Compaction and Applications in Petroleum Geoscience. In: Advances in Oil and Gas Exploration & Production. Springer. ISSN: 2509-372X. Er, C., et al., 2016. Pore formation and occurrence in the organic-rich shales of the Triassic Chang7 Member, Yanchang Formation, Ordos Basin, China. Journal of Natural Gas Geoscience xx, 1e10. Jiang, T., et al., 2021. Three-dimensional morphology and connectivity of organic pores in shales from the Wufeng and Longmaxi formation at the Southeast Sichuan Basin in China. Geofluids 2021. Jienan, P., et al., 2019. Characteristics of multiscale pore structure of coal and its influence on permeability. Natural Gas Industry B6, 357e365. Lei, Q., et al., 2020. Describing the full pore size distribution of tight sandstone and analysing the impact of clay type on pore size distribution. Geofluid 2020. Article ID 5208129. Lindquist, W.B., Venkatarangan, A., Dunsmuir, J., Wong, T., 2000. Pore and throat size distributions measured from synchrotron X-ray tomographic images of Fontainebleau sandstones. Journal of Geophysical Research 105 (B9), 21509e21527. Lønøy, A., 2006. Making sense of carbonate pore system. AAPG Bulletin 90 (9), 1381e1405. Loucks, R.G., et al., 2009. Morphology, genesis and distribution of nanometer-scale pores in silicious mud stones of the Missisipian Barnett Shale. Journal of Sedimentary Research 79 (12), 848e861. Lu, H., et al., 2021. Pore structure characteristics and permeability prediction model in a cretaceous carbonate reservoir. North Persian Gulf Basin, Geofluids 2021. Article ID 8876679. Mukherjee, S., Kumar, N., 2018. A first-order model for temperature rise for uniform and differential compression of sediments in basins. International Journal of Earth Sciences 107, 2999e3004. Nie, H., Jin, Z., Zhang, J., 2018. Characteristics of three organic pore types in the Wufeng Longmaxi Shale Of The Sichuan Basin, Southwest China. Scientific Reports 8. Article no. 7014. Newberry, B.M., Grace, L.M., Stief, D.D., 1996. Analysis of carbonate dual porosity systems from borehole electrical images. In: SPE 35158, Presented at the Permian Basin Oil and Gas Recovery Conference, Society of Petroleum Engineers. Pittman, E.D., 1979. Porosity, diagenesis and production capability of sandstone reservoirs. In: Scholle, P.A., Schluger, P.R. (Eds.), Aspects of Diagenesis: Society of Economic Paleontologists and Mineralogists, vol. 26. Special Publication, pp. 159e173. Ramakrishnan, T.S., et al., 1999. Forward models for nuclear magnetic resonance in carbonate rocks. The Log Analyst 40, 260e270. Zargari, S., et al., 2015. Porosity evolution in oil prone source rocks. Fuel 153, 110e117. Zhang, Z., Shi, Y., Li, H., Jin, W., 2016. Experimental study on the pore structure characteristics of tight sandstone reservoir in Upper Triassic Ordos Basin, China. Energy Exploration and Exploitation 34 (3), 418e439.

28 Understanding Pore Space through Log Measurements Zhang, K., et al., 2020. Effect of organic maturity on shale gas genesis and pores development: a case study in marine shales in the upper Yangtze region South China. Open Geosciences 12 (1). Zhang, Z., Qin, Y., You, Z., Yang, Z., 2020. Pore structure characteristics of coal and their geological controlling factors in Eastern Yunnan And Western Guizhou China. ACS Omega 5 (31), 19565e19578.

Further reading Archie, G.H., 1942. The electrical resistivity log as an aid in determining some reservoir characteristics. Transactions of the AIME 146, 54e62. Avseth, P., Mukerji, T., Mavko, G., 2005. Quantitative Seismic Interpretation. Cambridge University Press. Brie, A., Pampuri, F., Marsala, A.F., Meazza, O., 1995. Shear sonic interpretation in gas-bearing sands. In: SPE Paper-30595, Presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas, USA, (1995). ¨ ber die elastizita¨t poro¨ser medien. Vierteljahrsschrift der Naturforschenden Gassmann, F., 1951. U Geselschaft in Zu¨rich 96, 1e23. Translated into English as: Elasticity of porous media, available at: http://sepwww.stanford.edu/sep/berryman. Hodot, B.B., 1996. Coal and Gas Outburst. Coal Industry Press, Beijing. Janjuhah, H.T., Sanjuan, J., Salah, M.K., 2019. An overview of the porosity classification in carbonate reservoirs and their challenges: an example of macro-micro porosity classification from offshore Miocene carbonate in Central Luconia, Malaysia, World Academy of Science, Engineering and Technology. International Journal of Geological and Environmental Engineering 13 (5). Juliao, T., Suarez-Ruiz, I., Marquez, R., Ruiz, B., 2015. International Journal of Coal Geology 147e148, 126e144. Keys, R.G., Xu, S., 2002. An approximation for the Xu-White velocity model. Geophysics 67, 1406e1414. Kuster, G.T., Toksoz, M.N., 1974. Velocity and attenuation of seismic waves in two phase media: part 1: theoretical formulation. Geophysics 39, 587e606. Li, K., 2004a. Theoretical development of the brooks-corey capillary pressure model from fractal modelling of porous media. In: SPE 89429, pp. 1e6. Li, K., 2004b. Characterization of rock heterogeneity using fractal geometry. In: SPE 86975, pp. 1e7. Lucia, F.J., 1983. Petrophysical parameters estimated from visual description of carbonate rocks: a field classification of carbonate pore space. Journal of Petroleum Technology 35 (03), 629e637. Paper Number: SPE-10073-PA. Lucia, F.J., 1995. Rock-fabric/petrophysical classification of carbonate pore space for reservoir characterization. AAPG Bulletin 79, 1275e1300. Lucia, F.J., 1999. Carbonate Reservoir Characterization. Springer-Verlag, Berlin. Mavko, G., Mukerji, T., Dvorkin, J., 2009. The Rock Physics Handbook e Tool for Seismic Analysis of Porous Media. Cambridge University Press. Shi, X., et al., 2018. Micrometer-scale fractures in coal related to coal rank based micro-CT scanning and fractal theory. Fuel 212, 162e172. Silva, F., Beneduzi, C., 2017. Using sonic log for fluid identification in siliciclastic reservoirs. In: 15th International Congress of the Brazilian Geophysical Society & EXPOGEF, Rio de Janeiro Brazil.

Pores and pore space Chapter j 1

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Sundaram, K.M., et al., 2011. Optimization of dry frame modelling using realistic approach and its impact on fluid substitution: a deep water case study. In: SPWLA-INDIA 3rd Logging Symposium, Mumbai, India. Xu, S., White, R.E., 1995. A new velocity model for clay-sand mixtures. Geophysical Prospecting 43, 91e118.

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Chapter 2

Inversion of log data to the gross attributes of pore space 2.1 Estimation of the bulk porosity of laminated formations using deterministic approach The porosity analysis of laminated formations entails the determination of the effective porosity and total porosity of the component lithofacies, which constitutes the laminated formation. The process involves two steps. First, the gross porosity and shale volume are determined. Next, these data are used to characterize the shale distribution within the formation. Also characterized is the effective porosity of the laminae hosting reservoir development, and the microporosity, which is assumed to be confined to the shale present within the formation. Two methods are described for implementing the first step. One method is applicable when neutron capture gamma spectrometry data are available. The other one applies when it is not. The former method is expected to yield more accurate results than the latter. Consider a clastic-laminated rock, with the lithology of the laminae being (i) sandstone with dispersed shale, and (ii) only shale. The thickness of the laminae is less than 8 inches which is the minimum thickness resolvable by the log measurements, except high-resolution images acquired against the formations. The effective porosity of the rock volume probed by the measurements is denoted by fe. Effective porosity is the intergranular pore volume not occupied by clay bound water, per unit rock volume. The intergranular pore volume per unit rock volume is also known as the total porosity. Clay bound water includes the interlattice water present within clay minerals, and the water present within the electrical double layer. The double layer, in turn, includes the Stern layer and the Guoy layer present in the proximity of the surface of the clay minerals within the rock. Here the term clay exclusively means the clay-sized detritus (0.25 g cm3 is: HIhc ¼ rhc þ 0:3 (Schlumberger, 1989).

(2.19)

36 Understanding Pore Space through Log Measurements

Another relation for HIhc for hydrocarbons with density >0.25 g cm3 is:  4  2:5rhc HIhc ¼ 9 (2.19a) 16  2:5rhc (Schlumberger, 1989). A popular relation for hydrocarbons with density >0.25 g cm3 is: HIhc ¼ 2:2  rhc

(2.20a)

(Schlumberger, 1989). Substituting for HImf and HIhc, respectively, from Eq. (2.18) and Eq. (2.19) into Eq. (2.17):   rmf ð1  PÞ  rhc  0:3  4N ¼ f4e þ Vcl 4ncl g  4e ð1  Sxo Þ   (2.20) rmf 1  PÞ  HIma The quantity within the curly brackets in Eq. (2.16) is the value of 4N for the case of the entire effective porosity being saturated with mud filtrate. It is called the neutron porosity corrected for hydrocarbon ð4Nhcc Þ. Rearranging Eq. (2.20): 4Nhcc ¼ 4e þ Vcl 4ncl ¼ 4N þ D4N 



rmf ð1  PÞ  rhc  0:3  Here D4N ¼ 4e ð1  Sxo Þ  rmf ð1  PÞ  HIma

(2.21)  (2.22)

D4N is called the hydrocarbon correction to neutron porosity for hydrocarbons denser than 0.25 g cm3. Similar formulae exist for light hydrocarbons. Eq. (2.22) conforms with the hydrocarbon correction formula found in the technical document “Well logging and interpretation techniques” (Dresser, 1981). The porosity of a limestone with the same hydrogen index HI as the original rock, and the same shale content as the original rock, but which has its effective porosity fully saturated with mud filtrate, is denoted by 4NLst. Note that, 4nshLst denotes the porosity of a limestone fully saturated with mud filtrate, which has the same hydrogen index as wet shale.   rmf ð1  PÞ  rhc  0:3   4NLst ¼ 4e þ Vsh 4nshLst  4e ð1  Sxo Þ (2.23) rmf ð1  PÞ The parameter rma is an important input parameter for the calculation shown above spanning Eqs. (2.6)e(2.10). The source for this can be core studies. In the absence of core data, neutron capture gamma spectrometry becomes important. Such spectrometry data are also inverted for dry weight fraction of clay (Wdcl) here, and grain density ðrgrain Þ here. This is the density of the solid part of the rock. The symbol rdcl stands for the density of dry clay. rdcl ¼

rcl  wclp 1  wclp

(2.24)

Inversion of log data to the gross attributes of pore space Chapter j 2

37

The value of wclp comes from the core data or from area knowledge.   rgrain  Wdcl rdcl rgrain ¼ Wdcl rdcl þ ð1  Wdcl Þrma from which one gets; rma ¼ ð1  Wdcl Þ (2.25) To continue the analysis, consider a unit volume of rock. The value of the wet clay volume fraction in the rock (Vcl) can be computed from the dry weight fraction of clay in the rock (Wdcl) obtained using neutron capture gamma spectrometry data, as follows. The volume of dry clay per unit rock volume is denoted as Vdcl. Vdcl ¼ Wdcl ð1  4t Þ=rdcl

(2.26)

Vdcl ¼ Vcl ð1  wclpÞ

(2.27)

From Eqs. (2.26) and (2.27): Vcl ¼

Wdcl ð1  4t Þ rdcl ð1  wclpÞ

(2.28)

Here, 4t stands for total porosity and is given as, 4t ¼ 4e þ Vcl wclp

(2.29)

Substituting for 4t from Eq. (2.29) into Eq. (2.28) and rearranging:  x (2.30) Vcl ¼ ð1  4e Þ: 1þ x Wdcl Here; x stands for the quantity r ð1wclpÞ . dcl The value of x is known. The computation of effective porosity is illustrated in the following, for the case of hydrocarbon density exceeding 0.25 g cm3. The computation starts with assuming Sxo ¼ 1. Then, 4D ¼ 4e þ Vcl 4Dcl

(2.31)

4N ¼ 4e þ Vcl 4ncl

(2.32)

The values of 4e , respectively, obtained by solving Eqs. (2.30e2.32), are averaged out to obtain an initial value of 4e ð4einitial Þ. The following algorithm is implemented to evaluate Vcl ; 4e ; Sxo (which are, respectively, the wet clay volume fraction, effective porosity, and flushed zone mud filtrate saturation). Two input data are required for running the algorithm. These are the resistivity of the flushed zone ðRxo Þ and the resistivity of clay ðRcl Þ. The resistivity of the flushed zone is available as a log, e.g., the microspherically focused resistivity log. The resistivity of clay is not directly available from the log data. Its value is assumed as the resistivity of shale with maximum clay content. Such a shale level is identified by a high gamma

38 Understanding Pore Space through Log Measurements

reading, high apparent neutron porosity, and low resistivity. The algorithm can be represented as: 1 4e ¼ 4einitial 2 Sxo ¼ Sxo ðRxo; Vcl ; Rcl ; 4e Þ 3 D4D ¼

½1:07044e ð1Sxo Þðrmf ð1:110:15PÞrhc 0:3Þ ðrma rmf Þ # "

4 D4N ¼ 4e ð1 Sxo Þ

ðrmf ð1PÞrhc 0:3Þ ðrmf ð1PÞHIma Þ

(Ref Eq. (2.12))

(Ref Eq. (2.21))

5 4Dhcc ¼ 4D  D4D (Ref Eq. (2.12)) 6 4Nhcc ¼ 4N þ D4N (Ref Eq. (2.21)) Solve Eqs. (2.13) and (2.30) for 4e, denote the value of 4e obtained, as 4e1 ; Solve Eqs. (2.21) and (2.30) for 4e, denote the value of 4e obtained, as 4e2 ð4 þ4 Þ 7 4e ¼ e1 2 e2 8 Is the modulus of ð4e 4eprev Þ > ε1 or ðSxo Sxoprev Þ > ε2 ? (The subscript “prev” refers to the value obtained in the previous iteration. For the first iteration it refers to the initial value assigned. ε1 and ε2 refer to small preset positive numbers used to test the convergence of the values 4e and Sxo , respectively.) 9 If yes, then go to step 2. Else, consider the output values of Vcl ; 4e ; Sxo , respectively, as wet clay volume fraction, effective porosity, and flushed zone mud filtrate saturation. In case of gas, 4e obtained above is further corrected for excavation effect through the formula:    
4ecor ¼ 4e þ K 242e HIfl  0:044e 1  HIfl ; with HIfl ¼ Sxo HImf þ ð1  Sxo ÞHIhc (2.33) (Schlumberger, 1989). 4ecor is used as 4e in all subsequent calculations.

2.1.2 Computation of effective porosity using shallow resistivity, gamma ray, formation bulk density, and neutron porosity data The value of gamma ray log against clean formation is identified by the minimum gamma ray reading on log(GRclean). The maximum gamma ray reading on logs, which also corresponds to maximum value of apparent neutron porosity on the neutron porosity logs, is picked up. This presumably represents the gamma ray reading corresponding to 100% shale and is denoted by GRshale. The gamma ray reading against the formation at the depth level being analyzed is denoted by GR. The volume of shale present within unit

Inversion of log data to the gross attributes of pore space Chapter j 2

39

volume of rock is denoted by Vsh. An approximate value of VGR is computed as: Vsh ¼

ðGR  GRclean Þ ðGRshale  GRclean Þ

(2.34)

2.1.2.1 The wet clay content of shale The volume of silt per unit volume of shale is called the Silt Index (Isilt) and is given by: ðVsh  Vcl Þ Vsh

(2.35)

Vcl ¼ Vsh ð1  Isilt Þ

(2.36)

Isilt ¼

Substituting for Vsh from Eq. (2.34) into (2.36): Vcl ¼

ðGR  GRclean Þ ð1  Isilt Þ ðGRshale  GRclean Þ

(2.37)

The relation between the hydrogen index and density, for the composite of solids excluding dry clay, is assumed to be known. For any value of rma therefore, the value of HIma is known. The following algorithm is implemented to evaluate rma ; 4e ; Sxo (which are, respectively, the matrix density, effective porosity, and flushed zone mud filtrate saturation). Two input data are required for running the algorithm. They are the resistivity of the flushed zone (Rxo) and the resistivity of wet clay (Rcl). Initialization: A value denoted as rmai less than the anticipated value of matrix density is assigned to rma . Initial value of Sxo is assigned as 1.0. rma ¼ rmai Sxo ¼ 1 ðr r Þ

4Dcl ¼ ðr mar cl Þ (Ref Eq. (2.13a)) ma

mf

ðHIcl HIma Þ (Ref Eq. (2.16a)) 4Ncl ¼ ðHI mf HIma Þ

4D ¼ 4e þ Vcl 4Dcl (Ref Eq. (2.31)) 4N ¼ 4e þ Vcl 4ncl (Ref Eq. (2.32)) The values of 4e obtained by solving Eqs. (2.31) and (2.32) are averaged out to obtain an initial value 4einitial . 1 4e ¼ 4einitial 2 Sxo ¼ Sxo ðRxo; Vcl ; Rcl ; 4e Þ

40 Understanding Pore Space through Log Measurements

3 D4D ¼

½1:07044e ð1Sxo Þðrmf ð1:110:15PÞrhc 0:3Þ ðrma rmf Þ " #

4 D4N ¼ 4e ð1 Sxo Þ

ðrmf ð1PÞrhc 0:3Þ ðrmf ð1PÞHIma Þ

(Ref Eq. (2.12))

(Ref Eq. (2.21))

5 4Dhcc ¼ 4D  D4D (Ref Eq. (2.12)) 6 4Nhcc ¼ 4N þ D4N (Ref Eq. (2.21)) 7 Substitute the values of 4Dhcc and 4Nhcc computed at above, respectively, into Eqs. (2.13) and (2.21). Solve Eq. (2.13) for 4e and denote the solution as 4e1 . 8 Solve Eq. (2.21) for 4e, denote the value of 4e obtained, as 4e2 . e2 Þ 9 4e ¼ ð4e1 þ4 2 10 rma ¼ ½rbhcc ðVcl rcl þ4e rmf Þ=ð1 4e Vcl Þ (Ref Eq. (2.9b)) 11 Substitute the values of 4e obtained as above, Sxo and Vcl into Eq. (2.9), and invert the resulting equation for rma and thus compute the value of rma . 12 Is the modulus of ðrma rmaprev Þ > ε3 or ðSxo Sxoprev Þ > ε4 ? or ð4e 4eprev Þ > ε5 ? (Here, in the subscripts, prev refers to the value obtained in the previous iteration. For the first iteration it refers to the initial value assigned. The symbols ε3 ; ε4 ; ε5 refer to small preset positive numbers used to test the convergence of the values of rma ; Sxo , respectively.) 13 If yes, then go to step 2. Else output the values of rma ;4e ;Sxo , respectively, as wet clay volume fraction, effective porosity, and flushed zone mud filtrate saturation. In the case of gas, 4e obtained through a similar algorithm is further corrected excavation effect through the formula: 4ecor ¼ 4e þ Kð242e HIfl 0:044e Þð1 HIfl Þ; with; HIfl ¼ ½Sxo HImf þð1 Sxo ÞHIhc  (2.33) (Schlumberger, 1989). 4ecor is used as 4e in all subsequent calculations. The flowchart in Fig. 2.1 illustrates the computation discussed above.

2.1.3 Total porosity Total porosity ð4t Þ is the difference between rock volume and the solids volume with the difference considered for unit volume of rock.

2.1.3.1 Pore volume not shared with shale, per unit rock volume The analysis so far computed the effective porosity ð4e Þ. It is defined as the difference between the total porosity and the volume of clay bound water, with the difference considered for unit volume of rock. In the analysis of laminated formations, it is important to compute a porosity ð4E Þ defined by the pore volume not shared with shale, per unit rock volume. The distinction between 4e and 4E lies in the fact that 4e includes the

Inversion of log data to the gross attributes of pore space Chapter j 2

41

FIGURE 2.1 Flowchart of the algorithm implemented to evaluate rma ; 4e ; Sxo .

micropore volume present within shale while the 4E does not. Fig. 2.2 presents the distinction. In what follows, the case of hydrocarbon denser than 0.25 g cm3 is considered. The following analysis illustrates this case only. For the lighter hydrocarbons appropriate modifications can be made. Referring to the bottom panel of Fig. 2.2, and using the mixing laws:   rb ¼ ð1  4E  Vsh Þrma1 þ Vsh rsh þ 4E rmf   (2.38)  1:07044E ð1  Sxo1 Þ rmf ð1:11  0:15PÞ  rhc  0:3 Here rma1 represents the density of the composite of solids excluding solids associated with shale within the rock. Sxo1 is the saturation of mud filtrate in 4E . Eq. (2.38) is rearranged:   ð1  4E  Vsh Þrma1 þ Vsh rsh þ 4E rmf ¼ rb þ 1:07044E   (2.39) ð1  Sxo1 Þ rmf ð1:11  0:15PÞ  rhc  0:3 We define the density porosity of shale as the porosity of a rock composed of grains of density rma1, which, when saturated fully with mud filtrate, would have a bulk density ¼ rsh, and is denoted as 4Dsh . ðr  rsh Þ  4Dsh ¼  ma1 rma1  rmf

(2.40)

42 Understanding Pore Space through Log Measurements

Define the quantity 4D1 as: ðr  rb Þ  4D1 ¼  ma1 rma1  rmf

(2.41)

Substituting for rb from Eq. (2.38) into Eq. (2.41) and using Eq. (2.40):
  1:07044E ð1  Sxo1 Þ rmf ð1:11  0:15PÞ  rhc  0:3   4D1 ¼ 4E þ Vsh 4Dsh þ rma  rmf (2.42) We define the quantity rbhcc1 as the bulk density of a rock with a porosity 4E composed of grains with density rma1, volume of shale per unit rock volume equal to Vsh, and with the pore volume excluding shale, saturated with mud filtrate. rbhcc1 ¼ ð1  4E  Vsh Þrma1 þ Vsh rsh þ 4E rmf 
 rma1 ¼ rbhcc1  Vsh rsh þ 4E rmf =ð1  4E  Vsh Þ

(2.43) (2.43a)

The apparent density porosity for bulk density rbhcc1 and matrix density rma1 is denoted as 4Dhcc1 . It is the porosity of a rock having a bulk density of rbhcc1 , composed of grains of density rma1, and fully saturated with mud filtrate. Thus, rbhcc1 ¼ ð1  4Dhcc1  Vsh Þrma1 þ 4Dhcc1 rmf 4Dhcc1 ¼ 4D1  D4D1 where; D4D1 ¼

(2.44)


  1:07044E ð1  Sxo1 Þ rmf ð1:11  0:15PÞ  rhc  0:3   rma1  rmf (2.45)

4Dhcc1 ¼ 4E þ Vsh 4Dsh

(2.45a)

For the case of neutron response, we define an apparent neutron porosity 4N1 as the porosity of a rock composed of grains having hydrogen index for hydrogen index HIma1. This corresponds to the grains of density rma1 and which has the same hydrogen index of the candidate formation fully saturated with mud filtrate. 4N1 ¼ 

4N1 ¼ 

ðHI  HIma1 Þ  HImf  HIma1

(2.46)

  HImf  HIhc ðHI  HIma1 Þ ðHI  HIma1 Þ  ¼ 4E þ Vsh  sh   4E ð1  Sxo1 Þ   HImf  HIma1 HImf  HIma1 HImf  HIma1 (2.47)

Inversion of log data to the gross attributes of pore space Chapter j 2

43

ðHIsh HIma1 Þ Here, ðHI is the apparent neutron porosity of shale for the matrix of mf HIma1 Þ hydrogen index HIma1 and is denoted as 4Nsh .

ðHIsh  HIma1 Þ  4Nsh ¼  HImf  HIma1

(2.48)

HIsh is the water-filled limestone porosity of shale at formation temperature and is the x-coordinate of the shale point (Appendix 1). The x-axis represents neutron porosity for limestone matrix, available as log data, corrected for the formation conditions. Eq. (2.47) is written as:   HImf  HIhc  (2.49) 4N1 ¼ 4E þ Vsh 4Nsh  4E ð1  Sxo1 Þ  HImf  HIma1 And, for the case of hydrocarbon denser than 0.25 g cm3,   rmf ð1  PÞ  rhc  0:3  4N1 ¼ 4E þ Vsh 4Nsh  4E ð1  Sxo1 Þ  rmf ð1  PÞ  HIma ðHIsh  HIma1 Þ   D4N1 4N1 ¼ 4E þ Vsh  HImf  HIma1



rmf ð1  PÞ  rhc  0:3  D4N1 ¼ 4E ð1  Sxo1 Þ  rmf ð1  PÞ  HIma1

(2.50)

(2.51)  (2.52)

4Nhcc1 ¼ 4N1 þ D4N1

(2.53)

4Nhcc1 ¼ 4E þ Vsh 4Nsh

(2.54)

4Nhcc1 is the apparent neutron porosity of a rock having a porosity 4E composed of grains of hydrogen index HIma1, volume of shale per unit rock volume ¼ Vsh, and with the pore volume excluding shale that is saturated with mud filtrate. The relation between the hydrogen index and density, for the composite of solids excluding those in shale, is assumed to be known. For any value of rma1 therefore, the value of HIma1 is known. The following algorithm is now implemented to compute the values of rma1 ; 4E; and Sxo1 . Initialization: A value denoted as rma1i less than the anticipated value of rma is assigned to rma1 . The initial value of Sxo1 is assigned as 1.0. rma1 ¼ rma1i Sxo1 ¼ 1 ðrma1 rsh Þ (Ref Eq. (2.40)) 4Dsh ¼ ðr r Þ ma1

mf

44 Understanding Pore Space through Log Measurements ðHI HI

Þ

4Nsh ¼ ðHImfsh HIma1 (Ref Eq. (2.48)) ma1 Þ

4D1 ¼ 4E þ Vsh 4Dsh (Ref Eq. (2.42)) 4N1 ¼ 4E þ Vsh 4Nsh (Ref Eq. (2.50))

The values of 4e obtained by solving Eqs. (2.42) and (2.50) are averaged out to obtain an initial value of 4E . This is denoted by 4Einitial : 1 4E ¼ 4Einitial 2 Sxo1 ¼ Sxo1 ðRxo; Vsh ; Rsh ; 4E Þ 3 D4D1 ¼

½1:07044E ð1Sxo1 Þðrmf ð1:110:15PÞrhc 0:3Þ ðrma1 rmf Þ " #

4 D4N1 ¼ 4E ð1 Sxo1 Þ

ðrmf ð1PÞrhc 0:3Þ ðrmf ð1PÞHIma1 Þ

(Ref Eq. (2.45))

(Ref Eq. (2.52))

5 4Dhcc1 ¼ 4D  D4D (Ref Eq. (2.45)) 6 4Nhcc1 ¼ 4N1 þ D4N1 (Ref Eq. (2.53)) 7 Substitute the values of 4Dhcc1 and 4Nhcc1 computed at above, respectively, into Eqs. (2.45a) and (2.54). 8 Solve Eq. (2.45a) for 4E , denote the value of 4E obtained, as 4E1 . 9 Solve Eq. (2.54) for 4E , denote the value of 4E obtained, as 4E2 . E2 Þ 10 4E ¼ ð4E1 þ4 2 11 rma1 ¼ ½rbhcc1 ðVsh rsh þ4E rmf Þ=ð1 4E Vsh Þ (Ref Eq. (2.43a)) 12 Is the modulus of ðrma1 rma1prev Þ > ε6 or ðSxo1 Sxo1prev Þ > ε7 or the modulus of (ð4E 4Eprev Þ > ε8 ? (Here, in the subscripts, prev refers to the value obtained in the previous iteration. For the first iteration it refers to the initial value assigned. The symbols ε6 ; ε7 ; and ε8 refer to small preset positive numbers used to test the convergence of the values of rma1 ; Sxo1 ; and 4E , respectively.) 13 If yes, then go to Step 2. Else, take the output values of rma1 ;4E; and Sxo1 . In the case of gas, 4E obtained through a similar algorithm is further corrected excavation effect through the formula:    (2.55) 4Ecor ¼ 4E þ K 242E HIfl1  0:044E 1  HIfl1 
(2.56) Here; HIfl1 ¼ Sxo1 HImf þ ð1  Sxo1 ÞHIhc The following flowchart (Fig. 2.2) illustrates the algorithm.

2.1.4 The different ways clay/shale is manifest within clastic rocks Clay occurs within rocks in three forms. Dispersed clay forms within sands during deposition as the clay mineral-rich grains. The size difference between the dispersed clay grains and the framework grains allows the latter grains to form to line or fill pore throats. Postdepositional processes such as burrowing or diagenesis result in dispersed clay, which lies inside the pore bodies owing

Inversion of log data to the gross attributes of pore space Chapter j 2

45

FIGURE 2.2 Flowchart of the algorithm to calculate rma1 ; Sxo1 ; 4E .

to the microscopic size of the authigenic clay aggregates or extremely fine grains which alter to clays. In simple petrophysical models, the dispersed clay is considered as comprised of clay minerals and fine subsilt size grains, which are not made of clay minerals. The term dispersed shale is used for the composite of the clay minerals and the subsilt size detritus as mentioned above. Fig. 2.3 illustrates these concepts. Sometimes, during deposition fragments of shale or claystone which are the result of ablation during sediment transportation, with a grain size equal to or greater than that of the framework grains, are deposited simultaneously. Alternatively, during diagenesis, selective replacement can alter framework grains, for example, feldspar to kaolinite, a type of clay mineral. The claystone or shale grains mentioned above as well as the products of diagenesis are grouped under the term structural shale. Laminar shales form during deposition, interspersed with sands.

2.1.5 The Thomas e Stieber approach The model uses a single term namely shale to encompass the occurrence of clay and fine silt as described above. Thus, we have dispersed shale, structural shale, and laminar shale. In simple language, dispersed shale occurs as a porefill and thus consumes the intergranular pore space, which would otherwise be available in the absence of this shale. Laminar shale is like shale replacing a volume of rock originally free from the laminar shale. The model used in the

46 Understanding Pore Space through Log Measurements

FIGURE 2.3 How clay occurs within clastic rocks and the respective petrophysical models are shown. The picture at right top reprinted with permission from Pittman (1979). The picture at right middle reprinted with permission from Clavaud et al. (2005). The graphics at bottom right reprinted with permission from Formation Evaluation Manual, Institute of Petroleum Engineering, Heriot Watt University, Edinburgh, UK (2012).

deterministic inversion of the log data presumes that the three forms of shale mentioned above are the same electrochemically and physical property wise including gamma activity and hydrogen index. In the Thomas e Stieber Approach, a laminated clastic formation is modeled as interbedded shale and sand. It is assumed to contain mostly porefill clay and practically no grain-replacing clay/shale. This assumption is reasonable from a geological perspective, for the case of laminated formations. The properties of laminar shale and pore-fill clay-rich material are considered to be the same. This assumption (Thomas and Stieber, 1975) is central to the analysis of laminated formations. Though it is a simplification, it is widely used in thin bed analysis.

Inversion of log data to the gross attributes of pore space Chapter j 2

47

In the analysis the same term “shale” is used for laminar shale as well as for the clay-rich pore-fill. The reason for the common notation is the fact that their properties are assumed to be the same in the model, as already mentioned above. Another geologically relevant assumption is that the intergranular porosity of every clean sandstone layer within a laminated depositional interval is constant. A laminated depositional interval means a pack bounded above and below by thick shale. The intergranular porosity ð4max Þ of clean sandstone layers is the maximum effective porosity any sandstone layer within the pack can have. The symbol 4tsd represents the total porosity and is interchangeably used with 4max in deterministic processing. The effective porosity of a sand layer is denoted by 4Esd. The pore volume present within shale per unit volume of shale is denoted by 4sh.

2.1.5.1 Evaluation of 4max The total porosity of the rock ð4t Þ will be evaluated. Fig. 2.4 is a cross plot between 4t and Vsh. The figure is a plot of total porosity against shale volume from the gamma ray log. This plot is typical of rocks containing negligible fraction of structural shale. The term “structural shale” is defined below: During transportation of sediment, fragments of shale or claystone, with a grain size equal to or greater than that of the framework grains, are ablated. During deposition, these fragments of shale or claystone, with a grain size equal to or greater than that of the framework grains, are deposited simultaneously along with the other detrital fragments which ultimately take the structural role of framework grains. Alternatively, during diagenesis, selective replacement can alter framework grains, for example, feldspar to kaolinite, a type of clay mineral. The claystone or shale grains mentioned above as well as the products of diagenesis are grouped under the term structural shale. Thus structural shale grains replace original framework grains of a clean rock, in the morphological sense. The color bar is the total porosity of the sand layers. Vlam in Fig. 2.4 indicates laminated shale volume in the rock in percentage. The term “p” indicates sand volume in the rock as a percentage. The intersection of the northeasterly bounding line of the points scattered with the ordinate gives the value of 4max . The y- coordinate of the point marked as “pure shale” gives the value of 4sh . If the presence of structural shale is ignored, then the y- coordinate of the point marked as on the plot gives the value of the product of 4sh and 4max while the x-coordinate of the point marked as “Disp” on the plot gives the value of 4max . Even if structural shale is present, the trend of points having low volume of shale would intersect the ordinate at the value of 4max .

48 Understanding Pore Space through Log Measurements

FIGURE 2.4 Total Porosity (PHIT) e Shale Volume (VSH) cross plot. Shale volume computed using gamma ray log is plotted on the x-axis. Total porosity is plotted on the y axis. On the plot, p followed by a number (say 20) means, percentage of sand in the rock is 20%. Similarly, on the plot Vlam followed by a number (say 40) means, percentage of laminar shale in the rock is 20%. Reprinted, with permission from Van Der Val and Stromberg (2012).

The value of 4max is found out by cross-plotting the hydrocarbon-corrected bulk density rbhcc1 against the hydrocarbon-corrected hydrogen index. The values of rbhcc1 is readily obtained for any depth level, since, at this point of the analysis, the values of 4E , Vsh, and Sxo1 are known for any depth level. The equation used for computing the values of rbhcc1 is Eq. (2.43). The hydrogen index of the formation corrected for hydrocarbon is denoted here as HIhcc1. HIhcc1 ¼ HIma1 ð1  4E  Vsh Þ þ Vsh HIsh þ 4E rmf

(2.57)

HIsh is the water-filled limestone porosity of shale at formation temperature. It is the x-coordinate of the shale point (Fig. 2.5). The x-ordinate of the shale point represents the neutron log porosity of shale for limestone matrix corrected for formation temperature. In laminated clastic formations, the clean sandstone lamina shares a common value of intergranular porosity ð4max Þ. Hence, 4max is the porosity corresponding to the clean points, on a cross plot of rbhcc1 against HIhcc1. The volume of laminated shale per unit volume of rock is denoted by VLam. The volume of dispersed shale per unit volume of rock is denoted by VDis. The volume of structural shale per unit volume of the rock is denoted by VStr. The volume of shale present within a unit volume of rock, exclusively as a sandSd. stone layer, is denoted by Vsh

Inversion of log data to the gross attributes of pore space Chapter j 2

49

FIGURE 2.5 An example of a density neutron cross plot with both density and neutron corrected for hydrocarbon. Representation of the cross plot of hydrocarbon-corrected formation bulk density against the hydrocarbon-corrected formation hydrogen index. The point scatter is taken from Hussain and Ahmed (2012).

2.1.5.2 Simple analysis ignoring structural shale: computation of laminated and dispersed shale volume fractions in the formation In the simplified analysis presented below, consider the volume of structural shale to be insignificant and hence ignored. This is the usual case with many types of sand-shale interbedded formations. A more rigorous analysis that takes into account the structural shale involving stochastic inversion of log data against thin beds is discussed further in this chapter. When structural shale is ignored, the following equations apply: Vsh ¼ VLam þ VDis 4e ¼ 4E þ Vsh 4sh  Vcl 4cl ¼ 4E þ ðVLam þ VDis Þ4sh  Vcl 4cl

(2.58) (2.59)

4t ¼ 4E þ Vsh 4sh ¼ 4E þ ðVLam þ VDis Þ4sh

(2.60)

4t ¼ 4e þ Vcl 4cl

(2.61)

4E ¼ 4Esd ð1  VLam Þ

(2.62)

Sd ¼ VDis =ð1  VLam Þ Vsh

(2.63)

Sd 4Esd ¼ 4max  Vsh ¼ 4max  VDis =ð1  VLam Þ

(2.64)

Substituting for 4Esd from Eq. (2.63) into Eq. (2.62): 4E ¼ 4max ð1  VLam Þ  VDis

(2.65)

50 Understanding Pore Space through Log Measurements

Substituting for 4E from Eq. (2.65) in Eq. (2.60): 4t ¼ 4max  VLam ð4max  4sh Þ  VDis ð1  4sh Þ

(2.65a)

Gamma activity of laminated formation We denote the gamma ray count against rock, which is 100% sand, 100% shale, and 100% clean sandstone, respectively, as GRsd, GRsh, and GRclean. The gamma ray API against the formation is denoted as GR. Then, Sd GRsd ¼ GRclean þ Vsh GRsh

Substituting for

Sd Vsh

(2.66)

from Eq. (2.63) into Eq. (2.66), we have,

GRsd ¼ GRclean þ VDis =ð1  VLam ÞGRsh

(2.67)

GR ¼ ð1  VLam ÞGRsd þ VLam GRsh

(2.68)

Substituting for GRsd from Eq. (2.67) into Eq. (2.68): GR ¼ ð1  VLam ÞGRclean þ ðVDis þ VLam ÞGRsh ¼ ð1  VLam ÞGRclean þ Vsh GRsh (2.69) Eq. (2.65) and Eq. (2.69) are solved for VLam and VDis since Vsh is known. 2.1.5.3 Analysis considering structural shale also Computation of the structural laminated and dispersed shale volume fractions in the formation The computation assumes that the value of 4t is already determined, either from petrophysical analysis discussed earlier in this chapter, or using the combination of density log and NMR-derived Hydrogen Index data. Appendix 4 presents the details. The value of 4t derived this way is robust and is preferable to the value of 4t obtained by any other means. The relevant equations are: 



 33000ms  ðrma1 rb Þ ðrma1 rsh Þ B  Vsh r r A þ HINMR ðrma1 rmf Þ ð ma1 mf Þ (2.70) 4t ¼ ðA þ BÞ   rmf  rhc  A¼ (2.71) rma1  rmf   HImf  HIhc B¼ (2.72) HImf

Inversion of log data to the gross attributes of pore space Chapter j 2

51

Sd ; and V Sd , respectively, denote the total porosity of sand, Let 4tsd ; VDis Str volume of dispersed shale present within unit volume of sand, and volume of structural shale present within the unit volume of sand.   Sd Sd Sd þ VDis 4tsd ¼ 4max  VDis 4sh þ VStr 4sh (2.73)

A unit volume of sand contains ð1 4max Þ volume of grains. Out of this, the volume of structural shale grains that replaced the framework grains of the clean sand is VStr. The GR API of unit volume of sand (GRsd) is given as,   Sd Sd Sd GRsh þ VDis GRsh (2.74) GRsd ¼ ð1  4max Þ  VStr GRma1 þ VStr Here GRma1 denotes the GR API of unit volume of the framework grains of clean sand. GRma1 is related to GRclean by: GRma1 ¼

GRclean ð1  4max Þ

4t ¼ ð1  VLam Þ4tsd þ VLam 4sh

(2.75) (2.76)

Substituting for 4tsd from Eq. (2.73) into Eq. (2.76):  Sd  Sd Sd 4t ¼ ð1  VLam Þ4max  ð1  VLam ÞVDis þ ð1  VLam Þ VDis þ VStr 4sh þ VLam 4sh (2.77) Thus, Sd VDis ¼ ð1  VLam ÞVDis

(2.78)

Sd VStr ¼ ð1  VLam ÞVStr

(2.79)

Sd and ð1 V Sd Substituting for ð1 VLam ÞVDis Lam ÞVStr , respectively, from Eq. (2.78) and Eq. (2.79) into Eq. (2.77):

4t ¼ ð1  VLam Þ4max  VDis þ ðVDis þ VStr Þ4sh þ VLam 4sh ¼ 4max  VDis  4max VLam þ ðVDis þ VStr þ VLam Þ4sh

(2.80)

Vsh ¼ ðVDis þ VStr þ VLam Þ

(2.81)

4t ¼ 4max  VDis  4max VLam þ Vsh 4sh

(2.82)

Eq. (2.82) is a linear equation in two variables VDis and VLam. Substituting GRsd from Eq. (2.74) into Eq. (2.68):

    Sd Sd Sd GRma1 þ ð1  VLam Þ VStr GRsh þ VLam GRsh GR ¼ ð1  VLam Þ ð1  4max Þ  VStr þ VDis (2.83)

52 Understanding Pore Space through Log Measurements

Using Eqs. (2.78) and (2.79) in Eq. (2.83) and simplifying: GR ¼ ð1  VLam Þð1  4max ÞGRma1  VStr GRma1 þ ðVDis þ VStr þ VLam ÞGRsh (2.83a) GR ¼ GRma1 ð1  4max Þ  VLam GRma1 ð1  4max Þ  ðVsh  VDis  VLam ÞGRma1 þ Vsh GRsh (2.84)

In deriving Eq. (2.84) from Eq. (2.83), Eq. (2.81) was used. Eq. (2.84) simplifies to: GR ¼ ½GRma1 ð1  4max  Vsh Þ þ Vsh GRsh  þ ðGRma1 4max ÞVLam þ GRma1 VDis (2.85) Eq. (2.85) is linear in two variables VDis and VLam. Eqs. (2.82) and (2.85) are solved to obtain the values of VDis and VLam, respectively. The value of VStr is obtained from Eq. (2.81). The pore volume in unit volume of sand not shared with shale is denoted by 4Esd . Its value is given by: Sd ¼ 4max  4Esd ¼ 4max  VDis

VDis ð1  VLam Þ

4E ¼ ð1  VLam Þ4Esd ¼ ð1  VLam Þ4max  VDis

(2.86) (2.87)

2.1.5.4 Using NMR data The total porosity can be obtained from the NMR T2 distribution and the bulk density logs as per Appendix 4. The equation for total porosity obtained by this method is:
    rgrain  rb  4CBW þ 4Cap rw  rmf   4t ¼ A rgrain  rmf (2.88)     HINMR  4CBW þ 4Cap HIw  HImf þB HImf l m ;B ¼ lþm lþm   rmf  rhc  l¼ rgrain  rmf   HImf  HIhc m¼ HImf

Here; A ¼

(2.89)

(2.90)

(2.91)

Inversion of log data to the gross attributes of pore space Chapter j 2

53

The NMR T2 distribution also gives the bound fluid volume per unit rock volume, which is the sum of the microporosity within the clay-sized grain aggregates including the clay bound water and the porosity hosted by the capillary pores. This porosity is the cumulative porosity hosted by all pores with 0.3 msec < T2  T2cutoff . A sample of the graphical representation of the pore sizes and grain sizes and the depiction of porosity hosted by capillary pores is available at Fig. 2.6. In Fig. 2.6 the pore size and grain size distributions are compared for two thin bedded sand units exposed within the well section of a well drilled in Offshore Sabah Malaysian Deep Water. This comparison is for representational purpose. Interested reader can refer to the original paper for the context of the comparison shown at Fig. 2.6. In addition to this the pore volume hosted by shale due to its internal porosity is also obtained (Cao-Minh et al., 2008) within laminated formations as follows. In laminated formations the T2 distribution is bimodal. A cutoff value w 13T2cutoff partitions the T2 distribution into that of the shale pore volume and that representing the pore volume present  in sand. Thus, the area under the T2 distribution for the T2 ranges 0.3- 13T2cutoff  1 as4NMR 0:3 ms; 3T2cutoff gives the value of Vsh 4sh . Thus,

 1 Vsh 4sh ¼ 4NMR 0:3 ms; T2cutoff 3

denoted here

(2.92)

FIGURE 2.6 Pore size and grain size distributions of two sands exposed in well drilled through a thin bedded reservoir in Offshore Sabah Malaysia Deep Water. Reprinted with permission from Kantaatmadja et al. (2015).

54 Understanding Pore Space through Log Measurements

For very porous formations, the internal porosity of sand and that of shale are nearly the same. In that case, Vsd represents the volume of sand per unit rock volume.  4NMR 0:3 ms; 13T2cutoff Vsh Vsh (2.93) ¼ ¼ 4NMR ð0:3 ms; 3000 msÞ Vsd 1  Vsh

2.1.5.5 Computation of representative resistivity of sand and laminated shale volume using tensor resistivity data It is assumed in the model that (i) the sandstone layers within a pack are electrically isotropic and that (ii) the shale interbedded with the sand layers is transversely isotropic with the symmetry axis normal to bedding. It is assumed that the dispersed shale and the structural shale are isotropic electrically, and that they affect the sand resistivity identically. A shale-sand lamination pack is, therefore, transversely isotropic, with the axis of symmetry normal to the bedding planes. The following discussion assumes that the values of true formation resistivity measured normal (RV) and parallel (RH) to the bedding planes are available. Appendix 3 presents how these data are obtained. The resistivity of the shale beds along and normal to the bedding are, respectively, denoted by Rshh and Rshv. Their values can be picked up from logs and hence are known. The resistivity of the individual sand beds is not known. Consider a hypothetical lamination pack identical to the formation investigated by the resistivity tools, except for the fact that each sand within this pack has the same resistivity, equal to Rsd. The value of Rsd is unknown at this point of the analysis. The following equations apply. 1 ð1  VLam Þ VLam ¼ þ RH Rsd Rshh

(2.94)

RV ¼ ð1  VLam ÞRsd þ VLam Rshv

(2.95)

Eqs. (2.94) and (2.95) can be solved for VLam and Rsd. For RH sRsh ; the usual case in Hagiwara (1997), Schoen et al. (1999) and Mollison et al. (1999) is: Rsd ¼

R0sd 9 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ffi =  0 0 0 Rsd Rsd Rsd Rshh 1 þ 12 1  þ 4 Rshh Rshh  1 Rshv  1 ; : Rshh 8 15%. Y¼

sd CECrdcl Vdcl CECrdcl Vclsd ð1  WCLPu Þ ¼ ð1  4tsd Þ ð1  4tsd Þ

(2.152)

(Note that Y is counterion charge per unit grain volume of the sand for the uninvaded zone). From Eq. (2.119),

Inversion of log data to the gross attributes of pore space Chapter j 2

 sd  ðVsh  VLam Þ sd Vclsd z VDis þ VStr ¼ ð1  VLam Þ ðVsh  VLam Þ WCLPu ð1  VLam Þ ðVuwa þ Vuhc Þ þ ðVsh  VLam ÞWCLPu ¼ ð1  VLam Þ

63

(2.153)

4tsd ¼ ðVuwasd þ Vuhcsd Þ þ

aV H CECr

WCLPu ¼ ð1þaVQ H CECrdcl Q

dcl Þ

(2.154)

(see Eq. (2.134).

We thus note that since CEC and rdcl are inputs given to the model, WCLPu is a known parameter in the model. Appendix 5 discusses how these inputs are given. Substituting Vclsd from Eq. (2.153) into Eq. (2.152): Y¼

CECrdcl ð1  WCLPu Þ ðVsh  VLam Þ ð1  4tsd Þ ð1  VLam Þ

(2.155)

Substituting 4tsd from Eq. (2.154) into Eq. (2.155) and simplifying: Y¼

CECrdcl ð1  WCLPu ÞðVsh  VLam Þ 1  VLam  ½ðVuwa þ Vuhc Þ þ ðVsh  VLam ÞWCLPu 

(2.156)

We can compute Y using uninvaded zone parameters because Vsh is understood to be in the context of uninvaded formation. Also note that Y has the same value for both the flushed zone and the uninvaded zone. This is because the dry clay volume is the same irrespective of whether it is considered for the flushed zone or the uninvaded zone. The value of the Archie cementation exponent (m) is that computed from Eq. (2.151) using the value of Y from Eq. (2.156). To put things in perspective, we state the following. The Dual Water Model views the effect of clay on the sand conductivity, as manifest in two ways. i) The clay water conductivity alters the equivalent conductivity of the water within the pore space from the conductivity of the far water. The equivalent conductivity of the water occupying the pores is equal to that of a mixture of far water and the clay bound water. ii) The way electric current results within the far water is as per the ionic mobility and charge in an ionic solution. However, the way electric current results within the clay bound water is more complicated. The mobility of the counterions is constrained by the electrical interaction with the clay mineral surfaces through the electric field of the clay mineral surfaces that acts across the exclusion layer. Therefore, the condition current flow through an electrolyte saturating a pore space, with no surface field effects present (which condition is essential for Archie Law to hold) is not

64 Understanding Pore Space through Log Measurements

satisfied here. Hence, the apparent Archie cementation exponent for the case of sand containing clay cannot equal the case of a clean sand, albeit the same total porosity of the sand containing clay. Point (i) is addressed through the specification of the contents of the square braces in Eqs. (2.126) and (2.148). Point (ii) is addressed through specifying the value of m, the apparent Archie cementation exponent, through Eq. (2.151). Eq. (2.150) can be concisely stated as: Csd ¼ Csd ðVuwa ; Vuhc ; Vsh ; VLam ; CEC; rdcl ; WCLPu ; Cw ; a; m; nÞ

(2.157)

In Eq. (2.157), the only unknowns are Vuwa, Vuhc, Vsh, and VLam. We state first the NMR response equation:     HINMR ¼ 4CBW þ 4Cap HIw þ 4FF sxoff HImf þ 4FF 1  sxoff HIhc (2.157a) (See Eq. (A4.6) of Appendix 4).Here, 4CBW is the pore volume per unit rock volume occupied by the clay bound water and the intergranular microporosity of the clay-sized particles. The symbol 4Cap represents the pore volume per unit rock volume occupied by the capillary bound water. Neither of these two pore volumes are drained by hydrocarbons or occupied by mud filtrate. The sum of these two pore volumes ð4CBW þ4Cap Þ is called the bound fluid volume per unit rock volume and is a standard output data of the NMR logs. This sum ð4CBW þ4Cap Þ is denoted by 4BF. The total porosity excluding 4BF is called as the free fluid porosity and is designated as 4FF at Eq. (A4.6) of Appendix 4, above. We have,   (2.158) 4BF ¼ 4CBW þ 4Cap   4FF ¼ 4t  4CBW þ 4Cap ¼ ð4t  4BF Þ ¼ ½ðVxwa þ Vxhc Þ  4BF  (2.159) Substituting ð4CBW þ4Cap Þ from Eq. (2.158) and for 4FF from Eq. (2.159) in the cited Eq. (A4.6) of Appendix 4:   (2.160) HINMR ¼ 4BF HIw þ 4FF sxoff HImf þ 4FF 1  sxoff HIhc We also have, since hydrocarbon and mud filtrate are not present within the pore space accounted for by 4CBW and 4Cap, respectively: 4FF sxoff ¼ Vxwa   4FF 1  sxoff ¼ Vxhc

(2.161) (2.162)

Substituting for 4FF sxoff and 4FF ð1 sxoff Þ; respectively, from Eqs. (2.161) and (2.162) into Eq. (2.160): HINMR ¼ 4BF HIw þ Vxwa HImf þ Vxhc HIhc

(2.163)

4t ¼ ðVuwa þVuhc Þ þ ðVsh VLam ÞWCLPu þ VLam 4sh (see Eq. (2.124)).

Inversion of log data to the gross attributes of pore space Chapter j 2

65

4t ¼ 4max  VDis  4max VLam þ Vsh 4sh (see Eq. (2.82)). Substituting 4t from Eq. (2.124) into Eq. (2.82) and rearranging: ðVuwa þ Vuhc Þ þ ðVsh  VLam ÞWCLPu þ VLam 4sh ¼ 4max  VDis  4max VLam þ Vsh 4sh Substituting 4t from Eq. (2.96) into Eq. (2.82) and rearranging: 4max ð1  VLam Þ  VDis ¼ ðVxwa þ Vxhc Þ

(2.164)

Since ðVxwa þ Vxhc Þ ¼ ðVuwa þ Vuhc Þ, 4max ð1  VLam Þ  VDis ¼ ðVuwa þ Vuhc Þ

(2.165)

2.2.2 The forward model The forward model is now stated as: rb ¼ ð1 VLam Þrma1  ðVxwa þ Vxhc Þrma1  ðrma1 rsh ÞðVsh VLam Þ þ ðVxwa rmf þ Vxhc rhc Þ (See Eq. (2.83)) HI ¼ ð1 VLam ÞHIma1  ðVxwa þ Vxhc ÞHIma1  ðHIma1 HIsh Þ ðVsh VLam Þ þ ðVxwa HImf þ Vxhc HIhc Þ (See Eq. (2.84)). HI comes from neutron tool and is equal to the equivalent water-filled limestone porosity. GR ¼ GRma1  GRma1 ðVxwa þVxhc Þ þ Vsh ðGRsh GRma1 Þ (see Eq. (2.89)) DT ¼ ð1 VLam ÞDTma1  ðVxwa þVxhc ÞDTma1  ðDTma1 DTsh Þ ðVsh VLam Þ þ ðVxwa DTmf þVxhc DThc Þ (see Eq. (2.90)) HINMR ¼ 4BF HIw þ Vxwa HImf þ Vxhc HIhc (see Eq. (2.163)) Cxo ¼ Cxo ðVxwa ; Vxhc ; Vsh ; VLam ; CEC; rdcl ; WCLPx ; Cmf ; a; m; nÞ (see Eq. (2.136))   VLam 1 ¼ ð1  VLam ÞCsd Vuwa ; Vuhc ; Vsh ; VLam; Isilt ; 4ucl ; Cw ; a; m; n þ RH Rshh (2.166) RV ¼

ð1  VLam Þ  þ VLam Rshv  Csd Vuwa ; Vuhc ; Vsh ; VLam; Isilt ; 4ucl ; Cw ; a; m; n

(2.167)

Csd ¼ Csd ðVuwa ; Vuhc ; Vsh ; VLam ; CEC; rdcl ; WCLPu ; Cw ; a; m; nÞ (see Eq. (2.157)) m ¼ mdw þ cdw ð0:258Y þ0:2ð1 e16:4Y ÞÞ (see Eq.(151)) dcl ð1WCLPu ÞðVsh VLam Þ (see Eq. (2.156)) Y ¼ 1VLamCECr ½ðVuwa þVuhc ÞþðVsh VLam ÞWCLPu 

Here, CEC stands for the cation exchange capacity of dry clay. The subscript “dcl” denotes dry clay.

66 Understanding Pore Space through Log Measurements

Since rdcl Vdcl is the mass of dry clay per unit sand volume Eq. (2.106) can be stated as, CECrcl Vsh ð1Isilt Þ Y ¼ 1VLam ðV xwa þVxhc þVsh ð1Isilt Þ4

xcl Þ

(see Eq. (2.106))

4max ð1 VLam Þ  VDis ¼ ðVuwa þVuhc Þ (see Eq. (2.126)) 4max ð1 VLam Þ  VDis ¼ ðVxwa þVxhc Þ (see Eq. (2.127)) Vsh ¼ VLam þ VDis þ VStr (see Eq. (2.81)) Note that the procedure for computing various input parameters to the model other than those where the values are directly assigned by the interpreter has been dealt with at the appropriate places in the foregoing analysis. Further; Vma1 þ VLam þ VDis þ VStr þ Vuwa þ Vuhc ¼ 1

(2.168)

2.2.3 Essential constraints Eqs. (2.164) and (2.165) are the outcomes of the fact that the pore space not shared with shale within the rock can only be occupied by dispersed shale or fluids. This implies: 0  VDis  4max ð1  VLam Þ

(2.169)

The solids volume within the rock not shared with dispersed or laminated shale can be occupied by only the grains of the principal mineral (present as the sole solids component in clean rock) and structural shale grains. Thus, ð1  4max Þð1  VLam Þ ¼ VStr þ Vma1

(2.170)

Here Vma1 stands for the cumulative volume within the rock, of the grains of the principal mineral (present as the sole solids component in the clean rock). This implies, 0  VStr  ð1  4max Þð1  VLam Þ

(2.171)

Additionally, the volume of hydrocarbon must satisfy the following constraints: 0  Vuhc  ð4max Þð1  VLam Þ  ðVuwa þ VDis Þ

(2.172)

0  Vxhc  ð4max Þð1  4Lam Þ  ðVxwa þ VDis Þ

(2.173)

In the case of water base mud used for drilling: Vxwa  Vuwa

(2.174)

The system of equations mentioned above under subsection 2.2.2 as the forward model, subject to the constraints defined at equation and in Eq. (2.130) to Eq. (2.136), is usually solved, using a Nonlinear Weighted Regularized Least-Squares Method. This involves the following. The solution to the problem is defined as the set of parameters ðVuhc ; Vuwa ; Vxhc ; Vxwa ; Vsh ; VLam ; VDis Þ that minimizes a weighted error

Inversion of log data to the gross attributes of pore space Chapter j 2

67

function involving the match between the forward model and actual data, constraints, and a “reference model.” The solution set mentioned above is the set fVuhc ; Vuwa ; Vxhc ; Vxwa ; Vsh ; VLam ; VDis g. A solution vector ! p is defined, whose components are the ele! ments of the solution set mentioned above. A solution vector d is defined whose components are the input measurements against a depth. The operator F is an operator matrix expressing the response equations plus Eq. (2.168). The result of F acting upon the solution vector is the theoretical forward model of the log measurements, for the case of no measurement error. This forward model is called here, as the zero-error forward model, for convenience. Note that, all the relevant equations (including the response equations involving the log measurements plus those not directly involving the log measurements, but still are a part of the forward model) are absorbed into the operator. For this reason, the zero-error forward model is generated, as F! p. The Nonlinear Weighted Regularized Least-Squares Method entails the following: Evaluate p that minimizes Eð! p Þ, where E is defined as below.   2 2    2 ! ! ! ! ! E p ¼ Wd F p  d þ Wc G p  h þ Wp ! (2.175) p ! p ref The solution is sought under the general condition that no element of the solution vector can be less than zero. Here, in general G the operator ensconces the equality constraint expressions, which connect the solution vector with the ! vector h of the quantities, which are on the other side of the equality p pertaining to constraint expressions. The vector ! p ref is the solution vector ! the depth level preceding the depth level currently processed. The vector ! p ref is called here as the “reference model.” This is the prior model or a prior estimate of ! p which estimate is improved by using the measurements data against the current level, through the process of the minimization of the objective function, ! p . Multiple ! p can exist which locally minimize ! !2 2 ! ! p ! p ref Þ2 in Eð! pÞ ½Wd ðF p  d Þ þ Wc ðG p  h Þ . The term Wp ð! ! further ensures that the convergence of the minimization of Eð p Þ occurs at the appropriate local minimum of Eð! p Þ so that physically plausible models which do not drastically differ between contiguous depth levels are inverted from the p ! p ref Þ2 represents the distance in solution space, input data. The term Wp ð! of the extracted model from inversion, from the prior model. Wd is a diagonal matrix whose diagonal elements are the weights given to individual measurements, and is called as the weight matrix for data. Wc is a diagonal matrix whose diagonal elements are the weights given to individual equality constraints in general, and is called as weight-matrix for constraints. Wp is a diagonal matrix whose diagonal elements are the weights given to individual deviations of the respective elements of ! p from those of ! p ref .

68 Understanding Pore Space through Log Measurements

The solution is obtained using the Newton Iterative Process. In case the nonlinear equations present in the set of input equations are linearized, then the solution is obtained using the MarquardteLevenberg technique combined with the Singular Value Decomposition Method (Mezzatesta et al., 1995 and Mezzatesta, 1996).

2.2.4 Additional outputs computed The following important outputs are generated as follows, from the solution set fVuhc ; Vuwa ; Vxhc ; Vxwa ; Vsh ; VLam ; VDis g. sd ¼ Vsh

Vsh  VLam 1  VLam

4t ¼ ðVxwa þ Vxhc Þ þ Vsh 4sh ¼ ðVuwa þ Vuhc Þ þ Vsh 4sh

(2.176) (2.177)

4tsd ¼ 4t ð1  VLam Þ

(2.178)

Vuwasd ¼ Vuwa ð1  VLam Þ

(2.179)

Vxwasd ¼ Vuwa ð1  VLam Þ

(2.180)

Swtsd ¼

Vuwasd 4tsd

(2.181)

Swbsd ¼

sd Vsh 4sh 4tsd

(2.182)

Computation of Cbw is elaborated at Appendix 5 

  4m Swtsd  Swbsd Swbsd tsd n S Csd ¼ Cw þ Cbw a wtsd Swtsd Swtsd Rsd ¼

1 Csd

(2.183) (2.184)

The value of Rsd as above will be close to the value of Rsd computed from RH and RV because RH and RV are the input channels for the stochastic inversion and their equations are part of the forward model.

2.2.5 Horizontal permeability and vertical permeability The abbreviations kH and kV are the values of permeability, respectively, in the horizontal (along bedding planes) and vertical (normal to the bedding planes) directions. Note:

Inversion of log data to the gross attributes of pore space Chapter j 2

ksd ¼ C44tsd

 1  Swirrsd 2 Swirrsd

kH ¼ ð1  VLam Þksd þ VLam ksh H  ð1  VLam Þ VLam 1 kV ¼ þ ksd ksh V

69

(2.185) (2.186) (2.187)

2.2.6 Usage of high-resolution data Vertical resolution of 1 ft for conventional logs and vertical resolution of 0.4 inches for borehole images is delivered in the industry. Data having 1 ft vertical resolution are considered as normal resolution data. Data with better vertical resolution fall under the category of high-resolution data. We examine how the high-resolution log data is leveraged in the characterization of laminated formations. Presently technology is available, which generates bulk density and photoelectric factor log data at 6.5 inches vertical resolution and resistivity data at 0.5 inches vertical resolution. Neutron porosity data at 6-inch vertical resolution can also be generated. It is also possible to generate acoustic slowness log data at 6 inches vertical resolution. Combining these data with spectral gamma ray log data and implementing stochastic inversion, it is possible to generate petrophysical data of the individual beds, which make up a laminated formation. The technology involved in the generation of the high-resolution data mentioned above is presented briefly in Appendix 6. By the term high-resolution data, we mean the data at a vertical resolution of 6 inches. When high-resolution NMR data are also included in the inversion, the vertical resolution is slightly reduced as it becomes 7.5 inches. Note that while carrying out inversion, resistivity at 0.5-inch vertical resolution data is resampled to obtain resistivity data at 6-inch vertical resolution. The inversion of the high-resolution data is carried out using a forward model that involves a model of the rock as a mixture of minerals and fluids. The mineral model includes the different types of clay minerals and nonclay minerals inclusive of the principal minerals present within the rock. Let m denote the number of types of measurements, and let n denote the number of minerals plus fluids that comprise together, the rock model. Let mi denote the value of the ith type of measurement, Vj denote the volume of the jth mineral (the mineral itself is denoted as simply j for convenience) per unit rock volume. Let εi denote the random noise (assumed to be zero mean Gaussian noise) associated with the ith type of measurement. The measurements are called as tool responses. Thus, mi is called as tool response for tool “i”. The tool response for tool i when the measurement is

70 Understanding Pore Space through Log Measurements

made against a hypothetical rock of zero porosity and comprising entirely mineral j is denoted by rij. A tool response equation is defined as the relation connecting a measurement value with the volumes of the different minerals and fluids which together constitute the rock. The set of measurements generally inverted include resistivity, bulk density, volumetric photoelectric factor, acoustic slowness, uranium corrected gamma ray, thorium concentration, potassium concentration, and total gamma ray (when spectral gamma ray data are unavailable). NMR and elemental capture spectral gamma ray data are valuable additions to the input set of measurements where available. The ith response equation is: mi ¼

n X

rij Vj þ εi

(2.188)

j¼1

Note resistivity response equation is linearized in order to ensure that every response equation is linear. There are m equations of the type of Eq. (2.150) corresponding to the m types of measurements. Further, there is an additional equation, which reflects the fact that the sum of the volumes (retrieved model) is unity: n X

Vj ¼ 1

(2.189)

j¼1

Eq. (2.151) is considered as the ðm þ 1Þth equation. These equations can be expressed as: M ¼ RV þ ε

(2.190)

Here, M is the column vector of the measurements and unity, V is the column vector of the volumes, and ε is the noise contribution to the measurements organized as a column vector. R is the matrix ½rij . The value of rij for i ¼ mþ1 is necessarily unity for j ¼ 1 to n. We now discuss one method of inversion (Quirein et al., 1986) as an example. The discussion assumes no constraints. Here: The objective function to be minimized is, " #2 m n X X C¼ Di mi  rij Vj (2.191) i¼1

j¼1

The solution space is restricted to positive numbers and zero. Di are prechosen coefficients. In matrix notation: C ¼ ðM  RVÞT ðM  RVÞ ¼ ðRV  MÞT DðRV  MÞ (Quirein et al., 1986).

(2.192)

Inversion of log data to the gross attributes of pore space Chapter j 2

71

Here, D is the mm diagonal matrix, with diagonal elements Dii ; i ¼ 1 to ðm þ 1Þ. Normally, Dii should be equal to s1i . However, even for this case there is a problem. Different log measurements have different ranges of values. For example, the neutron porosity log measurement in units of v/v ranges 0e1. The gamma ray log measurement in units of GAPI counts ranges 0e400. The expected raw standard deviation for the neutron porosity measurement in units of v/v can be 0.01 in a reasonably good hole, while that for the gamma ray measurement can be 5 GAPI. It is clear that these numbers are dependent on the value ranges and not necessarily convey data spread or uncertainty in the mathematical sense. So, in order to bring the values of the elements Dii (which are the weights for generating the weighted data misfit function’s value e or the residual norm), s1i should be normalized in some way. This normalization is carried out in the following way. The maximum valued element among the set 1 1 1 1 s1 ; s2 ; s3 ; .; sm

is denoted as Wmax and is used to normalize

1 si

values.

Hence, since this step factors-in, the variation in the measured value ranges that differ from measurement type, it can be expected

 type  to measurement  that Dii should be equal to

1 si

1 Wmax

.

Now, every type of log measurement does not carry the same intrinsic confidence level of its adequacy for inversion, for a given measurementenvironment, downhole. For example, when a cave is present, the environment is beyond the design envelop of the density sensor. In order to overcome the different log measurements i is this shortcoming, the value  of D ii for   modified from the expected Dii ¼

1 si

1 Wmax

to:

   1 1 ðWMÞi si Wmax

(2.193)

In Eq. (2.154), (WM)i for measurement types 1 to m are numbers lying between 0 and 1, and are called as the weight multipliers for the different measurement types. These are input to the inversion by the analyst carrying out the inversion. The value of (WM)i for i ¼ mþ1 is always 1 because the volumes comprising the solution should always sum to 1.0. The weighted least square solution is:  1 V ¼ RT DR RT DM (2.194) Notwithstanding the foregoing, the approach discussed above is still an approach, where the search for the most likely solution nearest to the true solution vector relies solely on the minimization of the weighted sum of squares of the data misfit, measurement type-wise. The following fact is recognized in stochastic inversion. Any inversion method, whose sole aim is to minimize the weighted sum of squares of the data misfit with respect to the

72 Understanding Pore Space through Log Measurements

forward model, does not ensure a reasonable and useful solution. When the cost function (objective function) is built solely on the residual norm (weighted sum of least squares of the data misfit of data with respect to the forward model), the model retrieved at successive depth levels can show considerable variation. While the variation itself is unrealistic and physically implausible, some of the retrieved models could be models that are themselves physically implausible. For the above reason, inversion methods usually further build-in certain strategies, in addition to the strategy of minimizing the residual norm. Here, the term “residual norm” stands for the weighted sum of squares of the data misfit with respect to the forward model. The additionally built-in regularization strategies mentioned above provide additional bounds for the solution by leveraging prior models. Prior models are given estimates of the solution vector elements. The leverage is realized through a balance between minimization of the residual norm and the minimization of the distance of the solution from a prior model, in the solution space. The objective function stated at Eq. (2.192) is modified popularly as per the Tikhanov Regularization Scheme. In its simplest form the Tikhanov regularization scheme seeks to minimize: C ¼ ðRV  MÞT DðRV  MÞ þ M T aIM

(2.195)

Here, a is a damping factor and I is the identity matrix. The generalized Tikhanov objective function is, with Ma standing for a prior model: C ¼ ðRV  MÞT DðRV  MÞ þ aðMa  MÞT Wm ðMa  MÞ

(2.196)

Here, Wm is a diagonal matrix whose diagonal elements are the weights for computing the total model deviation from the prior model Ma.

2.2.6.1 Solution when constraints are present The equality constraints (those that are expressed as equations and which are also called as holonomic constraints) are treated as additional response equations. The weights allotted to the equality constraints are kept high. For the case of inequality constraints (those that are expressed as in equations and which are also called as nonholonomic constraints) the search for the solution is bounded by these inequality constraints. A method of inversion which uses the above concepts and which involves an explicit specification of a prior model is the LevenbergeMarquardt method of damped least squares. This method is well suited for multiparameter inversion where the model can be ill-conditioned since it is often the case that enough types of measurements are not available, to ensure uniqueness of the solution. The method strikes a balance between minimization of the residual

Inversion of log data to the gross attributes of pore space Chapter j 2

73

norm and the minimization of the distance of the solution from a prior model, in the solution space through the implementation of damped least square optimization, employing a damping factor. The LevenbergeMarquardt method of damped least squares is discussed in the following for the unconstrained inversion. In this (the damped least square optimization) method, the objective function to be minimized is defined as: 4a ¼ ðRV  MÞT ðWd ÞT ðWd ÞðRV  MÞ þ aðIðV  Va ÞÞT ðIðV  Va ÞÞ (2.197) Eq. (2.156) is equivalent to: 4a ¼ ðWd ðRV  MÞÞT ðWd ðRV  MÞÞ þ aðIðV  Va ÞÞT ðIðV  Va ÞÞ (2.198) Here Wd is a (mþ1)  (mþ1) diagonal matrix whose elements are the inverses of the standard deviation in the respective measurements. The value of the diagonal element W(mþ1)(mþ1) of Wd is set to a sufficiently high value. Va is the prior model, which is set as the final solution vector for the previous depth level. a is the damping parameter. Its value is preset and does not vary during the inversion. a is a dimension-less number. I is the Identity matrix. The prior model plays a key role in the inversion. Multivariate inversion with limited input data is complex. For such an inversion the error function (cost function) can have a number of local minima in the parameter space (solution space). A prior model is therefore a requirement because, as mentioned above, such a model is an initial estimate of the model sought to be recovered through inversion. Note that even with zero measurement noise, inversion cannot lead to the true physical attributes of the formation. The recovered model is still an estimation. The prior model is an initial estimate, which is improved upon by the data available at the current depth level being processed. By using the prior model, we are ensuring that the convergence of the process of inversion does not arbitrarily happen but happens toward the appropriate local minimum. Convergence to the appropriate local minimum means that the culmination of this convergence process is a likely solution vector, that is not unphysical, or physically implausible. Since the model sought to be retrieved is of the subsurface, uncontrolled variation in different depth levels of the retrieved model would be unphysical. Therefore, the best option would be to consider the retrieved model for a depth level preceding the current level under process, as the prior model Va. For a given Va the solution V that minimizes 4a is:
1
T  R M V ¼ Va þ RT R þ aI (2.199)

74 Understanding Pore Space through Log Measurements

2.2.6.2 Solution when constraints are present The equality (holonomic) constraints are treated as part of the forward model. The objective function is modified to:
 4a ¼ ðWd ðRV  MÞÞT ðWd ðRV  MÞÞ þ ðWc ðGV  HÞÞT ðWc ðGV  HÞÞ 
þ aðIðV  Va ÞÞT ðIðV  Va ÞÞ (2.200) Here, G is the matrix of the coefficients of the equality constraint expressions, which connect the solution V with the vector H of the quantities, which are on the other side of the equality constraint expressions. The inequality (nonholonomic) constraints are taken into account by accordingly limiting the search volume within the solution space. We consider now, an example for a laminated sand-shale formation, to illustrate the input measurements normally used and the model retrieved. Input measurements: High-Resolution Resistivity, High-Resolution Bulk Density, High-Resolution Photoelectric Factor, High-Resolution Neutron Porosity, High-Resolution Compressional Slowness, Uranium Corrected Gamma Ray Log, Thorium Concentration Log, Potassium Concentration Log. Solution Set (Model retrieved): Volume of quartz ðVquar Þ, volume of kaolinite ðVkaol Þ, volume of illite ðVilli Þ, volume of chlorite ðVchl Þ, volume of water in the uninvaded zone ðVvuwa Þ, and volume of oil in the uninvaded zoneðVuoil Þ, all per unit rock volume within the zone of investigation reflecting the measurements. The retrieved model is at the vertical resolution of the input measurements. Since the input measurements are at high vertical resolution, the output for the sand and the shale layers are independently retrieved with no mutual interference. Using proper discriminators, it is possible to retrieve the properties of the sands, from the output solution set. As a result, we can compute the cumulated volumes per unit volume of sand in respect of ðVquar Þ; ðVkaol Þ; ðVilli Þ; ðVchl Þ; ðVvuwa Þ; and ðVuoil Þ, which are, respectively, denoted as: sd sd sd sd sd sd Vquar ; Vkaol ; Villi ; Vchl ; Vvuwa ; Vuoil :

The following outputs for the sand are now computed as: sd sd sd Vclsd ¼ Vkaol þ Villi þ Vchl

(2.201)

sd sd 4sd e ¼ Vuwa þ Vuoil

(2.202)

sd sd 4e ¼ ð1  VLam ÞVuwa þ ð1  VLam ÞVuoil þ VLam 4sh e

(2.203)

Inversion of log data to the gross attributes of pore space Chapter j 2

75

sd Vuwa ¼ ð1  VLam ÞVuwa

(2.204)

sd Vuoil ¼ ð1  VLam ÞVuoil

(2.205)

The best source of 4sh e is the cumulative porosity hosted by pores whose T2 ranges 3e3000 ms, which is obtained as a routine output from NMR, when the same is considered against shale.

2.3 Evaluation of microporosity High-resolution NMR data are the best source for the computation of microporosity within rocks. High-resolution NMR data have a vertical resolution of 7.5 inches. Sand layers and shale layers are easily demarcated from a highresolution NMR T2 distribution log. Sand layers show presence of area under the T2 distribution beyond 3 ms T2 value, as a significant fraction of the area under the total T2 distribution, while the opposite is the case for the shale layers. Each lithology has a typical T2cutoff value, defined by the property that only pores whose T2 exceeds the cutoff value of T2cutoff can be drained by hydrocarbon fluids. It is understood that the above statement is in the context of drainage of reservoir rocks, normally encountered. The cumulative pore volume given  by the area under the T2 distribution curve for the range of T2 value 0.3-

1T 3 2cutoff

is a fair estimate of the total microporosity within the

rock (Cao-Minh et al., 2008). Applying this partitioning on the T2 distribution, the microporosity present within the sand lamina and the shale lamina can be estimated, at a vertical resolution of 7.5 inches. In case NMR data are not available, the microporosity can be computed in the following manner. Total clay volume per unit rock volume denoted as Vcl is: Vcl ¼ Vkaol þ Villi þ Vchl

(2.206)

Total shale volume per unit rock volume denoted as Vsh is: Vsh ¼

Vcl ðVkaol þ Villi þ Vchl Þ ¼ ð1  Isilt Þ ð1  Isilt Þ

(2.207)

Denoting the microporosity of rock by 4micro : 4micro ¼ Vsh 4sh

(2.208)

Here 4sh is the total porosity of shale. The method of obtaining 4esh has already been discussed earlier under the discussion on the ThomaseStieber model. Eq. (2.166) offers a method of computing the microporosity within a rock, to a fair degree of approximation.

76 Understanding Pore Space through Log Measurements

2.4 Evaluation of blocky (nonlaminated) reservoirs The methodology of evaluation of blocky reservoirs is identical to the method of stochastic inversion discussed above.

Conclusions The different ways in which diverse log data can be used to evaluate porous rocks from the perspective of the gross porosity mainly and also fluid saturation for the different fluids have been discussed. Among the log measurements density log is the most straightforward to interpret and has the least number of assumptions when viewed from the above perspective. The utility of formation density data goes beyond petrophysical analysis. The source of the data need not always be logs. Some of the applications which bring out the importance of these data beyond the scope of reservoir evaluation can be found by the interested reader from Mukherjee (2017, 2018a, 2018b, 2018c).

Appendix 1 Bulk density and hydrogen index of wet clay and silt and the value of the silt index of shale Fig. A1.1 is a cross plot of bulk density versus neutron porosity, sourced from Mahmoud et al. (2017). The cross plot has been used here purely for representational purpose. The trends and points marked on the cross plot are not a part of the original cross plot cited. The lithology of the formation whose data are plotted is hydrocarbon bearing sandstone, water bearing sandstone, and shale. In Fig. A1.1, the green (dark gray in print version) points represent highly shaly rock and shale. The most northerly points among the green (dark gray in print version) points represent shale and highly shaly rock composed of sandstone replaced by shale lamina to variable degrees. The sandstone part of the rock itself is such that, its entire pore space of the sandstone parts of the rock is entirely occupied by shale. The 100% shale point is identified as the eastern extremity of the line bounding this cluster of points. From core studies, the density of dry clay solids is determined, as well as the quantitative clay minerology. The hydrogen index of dry clay mineral for clay minerals is available in published literature. From these data the hydrogen index of clay can be computed (see Appendix 2). This enables the dry clay point to be plotted. The line joining the water point and the dry clay point defines the compaction trend of clay. This has been indicated in Fig. A1.1 as “Dry clay to water trend.” Again, from core studies, the representative density of the nonclay solids within shale can be determined. Further, the minerology of the nonclay solids

Inversion of log data to the gross attributes of pore space Chapter j 2

77

FIGURE A1.1 Cross plot of bulk density versus neutron porosity. Adapted from Mahmoud et al. (2017).

78 Understanding Pore Space through Log Measurements

composite is also determined (Appendix 2). From this information, it is possible to compute the hydrogen index of the silt content within shale. This allows us to plot the silt point. This is particularly apt because in our model (Refer Section 2.1.5 on the ThomaseStieber Approach) silt-sized grains occur only within shale in the rock. In case only density but not the mineralogy of the silt is available, the hydrogen index of silt is to be assumed from prior knowledge of the area. The silt point has been denoted as Q in Fig. A1.1. The intersection of this line joining the silt point and the 100% shale point, with the dry clay to water trend line, defines the wet clay point. The wet clay point is marked as Cl in Fig. A1.1. The bulk density and neutron porosity values on the cross plot, corresponding to the points Cl, give the wet clay bulk density and neutron porosity attributes. The silt index Isilt of shale is computed as: Isilt ¼

Distance between the points Sh and Cl Distance between the points Q and Cl

(A1.1)

Appendix 2 Hydrogen index of dry clay and silt From core studies, the densities of the composite of dry clay solids and that of the silt are known. Let rdcl and rslt , respectively, denote the density of the composite of dry clay solids and the density of the composite of the silt-sized grains. From core studies, the clay minerals and their dry weight fractions of the dry clay-composite are known. Also known from the core studies are the minerals present as silt-sized grains within the rock. Let Wcli be the dry weight fraction of clay mineral i in the dry state within the dry clay solids composite, and let ridcl the density of the clay mineral in the i be the dry volume fraction of clay mineral i within the dry dry state. Let Vdcl j clay solids composite. Let Wslt be the dry weight fraction of mineral j occurring as grains of silt size within the silt-sized grains composite, and let j rjslt be its density of the mineral. Let Vslt be the dry volume fraction of silt j i mineral i within the silt. Let HIdcl and HIslt denote the hydrogen index of clay mineral i in dry condition and the hydrogen index of silt mineral j, respectively. Let HIdcl and HIslt, respectively, denote the hydrogen index of the dry clay and the hydrogen index of silt. We have, i Vdcl ¼

Wcli =ridcl Wcli rdcl ¼ i 1=rdcl rdcl

(A2.1)

Inversion of log data to the gross attributes of pore space Chapter j 2 j j Wslt rslt =rjslt Wslt ¼ j 1=rslt rslt X i i HIdcl ¼ Vdcl HIdcl

j Vslt ¼

79

(A2.2) (A2.3)

i

HIslt ¼

X j j Vslt HIslt

(A2.4)

j

Appendix 3 Resistivity along and normal to formation bedding Consider the formation to be transversely isotropic with the symmetry axis normal to the bedding. The data acquisition uses a set of arrays of coils mounted on a metallic mandrel with due insulation between the mandrel and the coils. An array is defined as a set of three sets of coils, which are, respectively, the set of transmitter coils carrying alternating current, the set of balancing receiver coils, and the set of main receiver coil for that array. Each set of coils is three coils oriented in mutually perpendicular directions. Each array has a transmitter to receiver spacing different from that of the other arrays. The geometry of an array is depicted at Fig. A3.1. The amplitude of the signal sensitive to conductivity is denoted as a set of nine measured voltage amplitudes Vij ; i ¼ x; y; z; j ¼ x; y; z. Here, the first subscript of Vij conveys the transmitter orientation while the second subscript conveys the receiver orientation. By the term measured voltage amplitude is

FIGURE A3.1 Illustration of the coil arrangement of an array for a given transmitter to receiver spacing. Reprinted with permission from Wu et al. (2007).

80 Understanding Pore Space through Log Measurements

meant, the amplitudes of the R-Component and X-Components of the receiver 3 2 Vxx Vxy Vxz 7 6 7 signal of the main receiver. Like the matrix 6 4 Vyx Vyy Vyz 5 we have a Vzx Vzy Vzz matrix of the x-Components of the receiver signals. From these components, the abovementioned elements of the signal amplitude matrices the apparent conductivity values corresponding to the respective R, X signal pairs are computed. These conductivity values are organized into the matrix 2 3 sxx sxy sxz 6 7 6 syx syy syz 7. The elements of this matrix constitute components of a 4 5 szx szy szz tensor called the raw conductivity tensor, or the measurement tensor. The physical model of the data acquisition is presented in Fig. A3.2. The formation model for any array i is a medium bounded by parallel planar surfaces. The thickness of this slab is much greater than the transmitter to main receiver spacing of the array. The angles q and 4 are the dip and the azimuth of any of the bounding plane relative to a Cartesian coordinate system, whose axes are along the directions of the X-Transmitter coils, Y-Transmitter coils, and the Z-Transmitter coils, respectively. The formation is assumed to be a transversely isotropic medium whose axis of symmetry is normal to the bounding planes. For convenience, these planes are also interchangeably referred to as bedding. The conductivity of the formation measured in the direction normal to the bounding planes is called as the vertical conductivity (sv ). The conductivity of the formation measured in the direction parallel to the bounding, and along any azimuthal direction, is called as the horizontal conductivity ðsh Þ. For any given array, the values of the elements of its conductivity tensor are affected by not only sv and sh, but also by the eccentricity of the tool within the borehole ðdecc Þ in Fig. A3.2, the radius of the borehole, conductivity of the fluid within the borehole, the relative dip and azimuth of the bedding with respect to the borehole axis, the azimuth of the tool eccentricity, the conductivity of the filtrate of the borehole fluid, invading the formation, the invasion profile, and the metallic mandrel. The factors mentioned above, with the exception of the formation conductivity as measured along and normal to bedding, are the environmental factors. Due to this reason, the model which can be used in inverting the raw conductivity tensors to sv and sh and hence to their inverses denoted as RV and RH is bound to be complex. Different approaches to the inversion of the raw conductivity tensor are discussed here. One approach (Wu et al. 2007) is hereby illustrated. In this approach, the raw conductivity tensor of each array is inverted to the following model ðhd; Rha ; Rva ; q; F; decc ; jÞ. Here, the first two quantities are the apparent horizontal and

Inversion of log data to the gross attributes of pore space Chapter j 2

81

FIGURE A3.2 Illustration of a tool in the borehole, which is inclined relative to the normal to the bedding plane. The borehole is assumed to have a circular profile. Reprinted with permission from Wu et al. (2007).

vertical resistivity of the formation as inverted from the array raw conductivity tensor. Here hd stands for the borehole diameter, which is common for all the arrays, decc stands for the tool eccentricity, defined as the distance between the axis of the borehole and the center of the main receiver coil of the candidate array. j denotes the azimuth of the tool eccentricity. Rha and Rva stand for the resistivity measured normal to, and along the symmetry axis, in a transversely isotropic formation penetrated by a borehole identical to the borehole in which the data being inverted have been acquired. The orientation in space of this borehole relative to the formation is also identical to that of the original borehole in which the data being inverted have been acquired. Further, the values of tool eccentricity and the azimuth of the tool eccentricity are identical to the values of the corresponding parameters for the case of acquisition of the original data. However, the formation is uninvaded by the borehole fluid filtrate in the present case, whereas formation is invaded by borehole fluid filtrate for the case of acquisition of the original data sought to be inverted. Thus, the first stage of the inversion does not consider invasion of the formation by the borehole fluid filtrate. Owing to this reason, Rha and Rva vary from array to array, since the effect of invasion on the values of the components of the raw conductivity tensor of an array depends on the transmitter to main receiver coil spacing of the array. The inversion process is briefly discussed below.

82 Understanding Pore Space through Log Measurements

The inversion process The modeled conductivity tensor is called here, as si;j;k . Its matrix representation is ½sijk . Here, the indices i, j, and k of a tensor element are the array spacing, transmitter orientation (x,y,z), and receiver orientation. The raw conductivity tensor components for the different arrays are organized as the conductivity tensor smi;j;k. Its matrix representation is ½smijk . smi;j;k is also called as measured conductivity tensor. Again, the indices i, j, and k of a tensor element are the array spacing, transmitter orientation (x,y,z), and receiver orientation. Either tensor is a tensor whose matrix representation has the number of rows equal to the number of arrays and the number of columns equal to 9. Each row would represent an array and the 9 elements of the tensor 2 3 sxx sxy sxz 6 7 6 syx syy syz 7 for an array would be the row elements of ½sijk  for that 4 5 szx szy szz array. Similar remarks would apply for the matrix ½sijk  of the forward modeled conductivities. Consider the seven-dimensional space spanned by the parameter set ðhd; Rha ;Rva ;q;F;decc ;jÞ. Let x denote a vector in this seven-dimensional space. As a part of the quick computation of the forward model, where the modeled conductivity tensor is a function of x for the vector x for each array, the sevendimensional space is discretized as a grid. The values of sjk for each i (i denotes an array) are precomputed. The values of sjk for each i for any given x are obtained through the implementation of a quick interpolation routine. In this way, the elements of ½sijk  are determined for any given x. The receiver coil responses are computed through the solution of Maxwell’s equations using the Born Approximation. Note that the response function of a coil is not a purely geometric function totally independent of the medium’s conductivity distribution. Hence, Born Approximation is used. The strategy is to look at coil response as a sum of the response due to an average medium property plus the result of the effect of variations of medium conductivity from this average background. The Born response function acts upon the deviation of the medium’s conductivity distribution from the background, to generate the contribution of this effect of variations of medium conductivity from this average background, to the receiver coil response. Fig. A3.3 shows, as an example, the Born response functions corresponding to R-Signal of the different transmitter orientations relative to one another. They are, thus, the response functions that link the different component voltages, of the forward model of R-Signal Voltage Matrix 2 3 Vxx Vxy Vxz 6 7 6 Vyx Vyy Vyz 7 to the transmitterereceiver coil orientations for an array 4 5 Vzx Vzy Vzz

Inversion of log data to the gross attributes of pore space Chapter j 2

83

FIGURE A3.3 Born approximationebased response functions for receiver coil responses for an array. The first letter of the pairs naming each response function indicates the transmitter orientation. The second letter indicates the receiver coil orientation. Each transmitterereceiver pair has positive (red; gray in print version) and negative (blue; dark gray in print version) responses. Reprinted with permission from Anderson et al. (2008).

and hence to the components of the modeled conductivity tensor 3 2 sxx sxy sxz 7 6 6 syx syy syz 7. 5 4 szx szy szz The precomputation uses the Born response functions for the response of each receiver coil. The solution vector consisting of the elements hd (common to all arrays), Rha, and Rva for the different arrays, q; F; decc ; and j (common to all arrays), which constitutes the recovered model, is denoted as xs. The inversion process entails determining the vector xs that minimizes the value of the cost function, E, with E defined as:  X 2   2 E¼ wi;j;k smi;j;k  si;j;k þ f x  xp  (A3.1) i;j;k

Here, as already mentioned above, smi;j;k and si;j;k are the measured and modeled conductivity tensors respectively. wi,j,k is the weighting function. xp is  2 the prior model that is assumed. f ðx  xp  Þ is a penalty function of the distance of x from xp in parameter space (solution space). The problem of multivariate inversion sought to be solved as detailed above is complex. The cost function E is bound to have many local minima, if P it is limited to wi;j;k ðsmi;j;k  si;j;k Þ2 . xp is an initial estimate of xs. The term i;j;k

84 Understanding Pore Space through Log Measurements

 2 f ðx  xp  Þ is added to ensure that the local minimum of E corresponds to x that minimizes E being the most likely and physically meaningful solution vector xs. The robustness of the inversion almost entirely hinges on how efficient is the algorithm that results in the prior model xp which is an initial estimate of xs. The inversion improves upon this initial estimate, using the information available, which is the measured tensor of conductivity. xp is obtained using an algorithm, which employs a coarse-grid search strategy. In the next step, the recovered model from the inversion discussed above is used to generate the forward models of the conductivity tensor, for the case of borehole present and for the case of no borehole present, for the different arrays. As already mentioned, the recovered model includes the apparent conductivity and hence resistivity of the formation for different arrays, and the parameters relevant to the tool position within the borehole, borehole size, and the relative dip and azimuth of formation bedding relative to the tool. Let skh and skv denote, respectively, the apparent conductivity of the formation measured along and normal to the bedding for the array k. Note that since invasion is ignored, the values would be different from array to array. The forward model for the different components sklm ; l ¼ x; y; z; m ¼ x; y; z of the conductivity tensor notated here as skbh of array k is computed for the case of the presence of borehole as:   sklm ¼ sklm hd; skh ; skv ; q; F; decc ; j (A3.2) Thus, the forward modeled conductivity tensor skbh in the presence of borehole for the array k is also computed. The forward modeled conductivity tensor for the case of no borehole is notated here, as sknbh for the array k. The components s0k pq ; p ¼ x; y; z; q ¼ x; y; z are computed as:  k k  0k s0k (A3.3) pq ¼ spq sh ; sv ; q; F Thus, the forward modeled conductivity tensor sknbh for the case of no borehole for the array k is also computed. Consider the tensor Dskbh whose components are the borehole effects for different transmitter and receiver orientations for array k. The components of Dskbh are computed as:   (A3.4) Dskuv ¼ skuv  s0k uv When the components of the tensor Dskuv are subtracted from the components of the raw conductivity tensor, also called as the measured conductivity tensor for the array k, we get the borehole-corrected components of the measured conductivity tensor for the array k. The measured conductivity tensor for the array k is denoted as smk . The tensor whose components are the borehole-corrected components of the measured conductivity tensor for the array k is notated here as skbhc .

Inversion of log data to the gross attributes of pore space Chapter j 2

85

The components of skbhc for all k (all arrays) are grouped together forming the components of a tensor denoted sbhc i;j;k whose matrix representation has a structure identical to the matrix representation of the si;j;k . sbhc i;j;k is inverted using the same technique as has been discussed in the foregoing analysis but with the computed conductivity values at the grid points, as per the case of no borehole present, to recover the model ðskh ; skv for k ¼ 1 to N; q; FÞ, where N is the total number of arrays. Denoting the reciprocal of the conductivity by R, the model recovered is: 

  xnoinvasion ¼ Vector Rkh k¼1to N ; Vector Rkv k¼1to N ; q; F (A3.5) Here the subscript for the LHS of Eq. (A3.5) recalls that the results are for the case of an uninvaded formation which would give rise to the measured conductivity tensor, when no borehole is present. The inverted Rkh ; Rkv for any array k would be the same as the Rkh0 ; Rkv0 where k Rh0 ; Rkv0 mean the following. Consider the hypothetical case of the conductivity measurement tensor smk0 for the array k pertaining to the case of data acquired in a borehole normal to the bounding planes (bedding) of the formation. Let ðsmk0 Þbhc be the tensor smk0 after it is borehole-corrected. Let Rkh0 ; Rkv0 be the apparent formation resistivity along and normal to the bedding, respectively, inverted from smk0 : Rkh and Rkv are whatRkh0 and Rkv0 , respectively, represent. 
 At this point; the total recovered model is : hd; decc ; j; q; F; Vector Rkh k¼1to N ; Vector


k  Rv k¼1to N ; q; F

(A3.6)

Note that, as indicated above, Vector½Rkh k¼1to N ; Vector½Rkv k¼1to N are vectors of the resistivity for the case of borehole being normal to formation. Thus, at this stage the effect of formation dip and the effect of borehole have been neutralized through the inversion, and the recovered model now has only the effect of invasion of the borehole fluid filtrate into the formation. This effect varies with the array spacing. The array spacing of the different arrays used spans a variety of values. The design of the tool described at Wu et al. (2007) makes it possible to consider six arrays. The array spacing for these arrays considered are, respectively, 15, 21, 27, 39, and 54 inches. The array spacing of array k determines the vertical resolution of Rkh . It also determines the radial response of Rkh . Rkh data are brought to the same resolution of around 48 inches (4 ft). The resolution matched Rkh data are inverted using the radial Born response functions of the different arrays, to obtain i) the uninvaded zone resistivity along the bedding, at 4 ft resolution, denoted as RH , ii) the flushed zone resistivity along the bedding, at 4 ft resolution, denoted as Rxo , and iii) the diameter of invasion, denoted as Di. Rkv data of all arrays are brought to a vertical resolution of 4 ft. The Rkv data for the longer spacing at

86 Understanding Pore Space through Log Measurements

4 ft vertical resolution are corrected for the invaded zone, using the Rkv data at 4 ft vertical resolution, to obtain the RV data, which are the resistivity of the uninvaded zone of the formation measured in a direction normal to the bedding. RV data are also at 4 ft vertical resolution. RV and RH data as mentioned above are the data RV and RH used in Section 2.1.4.5 in the main text.

Concluding remarks The technology for obtaining robust RH and RV for the general case of borehole not normal to bedding works best when the conductivity of the borehole fluid is not too high compared to the conductivity of the formation, measured along the bedding or normal to the bedding. Special mandrel design is resorted to when this situation is apprehended. The ranges of conductivity contrast between formation and the borehole fluid that dictate the limits of usability of the technology are dependent on the tool design and the inversion methods. The technology is generally used when deep invasion is not apprehended, and when oil base mud or SOBM is used for drilling, and for mud salinity of water base mud used for drilling not exceeding 40,000 ppm.

Appendix 4 Computation of total porosity from bulk density and magnetic resonance logs The analysis is loosely on the lines of Freedman (1997). It is assumed that the density and NMR tools investigate the flushed zone. Rocks contain micropores that include those which host clay bound water exclusively. Rocks also contain micropores, whose pore entry pressure is beyond displacement pressures encountered in the course of normal drainage and imbibition. This category of micropores is called as capillary pores. What has been referred to above, as imbibition includes the process of imbibition of mud filtrate invasion when a well is drilled. The remaining pore space is called as the free fluid porosity following NMR nomenclature when a unit volume of rock is considered. The water within the capillary pores is called the capillary bound water. Mud filtrate invasion is therefore confined to the free fluid porosity ð4FF Þ. It is assumed that the bulk density of water (rw ) within the capillary pores equals to that of the clay bound water. Further, the mud filtrate saturation in total porosity is denoted as sxot, and the mud filtrate saturation in free fluid porosity as sxoff here. The pore volume occupied per unit rock volume by the clay bound water is denoted by 4CBW . The pore volume occupied by the capillary bound water per unit rock volume is denoted by 4Cap.   4t ¼ 4CBW þ 4Cap þ 4FF (A4.1)

Inversion of log data to the gross attributes of pore space Chapter j 2

87

The equation for bulk density can be stated as,     rb ¼ ð1  4t Þrgrain þ 4CBW þ 4Cap rw þ 4FF sxoff rmf þ 4FF 1  sxoff rhc (A4.2) Here rgrain is the density of the solids composite within the rock.      rb ¼ ð1  4t Þrgrain þ 4CBW þ 4Cap rw  rmf þ 4CBW þ 4Cap rmf      þ4FF sxoff rmf þ 4FF 1  sxoff rhc ¼ ð1  4t Þrgrain þ 4CBW þ 4Cap rw  rmf      þ 4CBW þ 4Cap rmf þ 4FF rmf  4FF 1  sxoff rmf  rhc ¼ ð1  4t Þrgrain þ      4CBW þ 4Cap rw  rmf þ 4CBW þ 4Cap þ 4FF rmf  4FF    1  sxoff rmf  rhc (A4.3) 4FF sxoff ¼ 4t sxot

(A4.4)

Substituting for 4FF sxoff from Eq. (A4.3), and substituting for ð4CBW þ4Cap þ4FF Þ from Eq. (A4.1) into Eq. (A4.2):      rb ¼ ð1  4t Þrgrain þ 4CBW þ 4Cap rw  rmf þ 4t rmf  4t ð1  sxot Þ rmf  rhc (A4.5) The hydrogen index of the formation, as computed from NMR, after correcting for the effect of wait time, is denoted by HINMR. Consider that the hydrogen index of clay bound water and capillary bound water are equal and denote either of them by HIw. We have,     (A4.6) HINMR ¼ 4CBW þ 4Cap HIw þ 4FF sxoff HImf þ 4FF 1  sxoff HIhc      HINMR ¼ 4CBW þ 4Cap HIw  HImf þ HImf 4CBW þ 4Cap   þ HImf 4FF  HImf 4FF þ 4FF sxoff HImf þ 4FF 1  sxoff      (A4.7) HIhc ¼ 4CBW þ 4Cap HIw  HImf þ HImf 4CBW þ 4Cap þ 4FF     4FF 1  sxoff HImf  HIhc Substituting 4FF sxoff from Eq. (A4.3), and substituting for ð4CBW þ4Cap þ4FF Þ from Eq. (A4.1) into Eq. (A4.9):      HINMR ¼ 4CBW þ 4Cap HIw  HImf þ HImf 4t  4t ð1  sxot Þ HImf  HIhc (A4.8)

88 Understanding Pore Space through Log Measurements

Case of water base mud When water base mud is used for drilling the hole, consider the values of rw and rmf are close enough. This is because the second term of the RHS of Eq. (A4.5) can be neglected in comparison with the other terms. Eq. (A4.5) simplifies:   rb ¼ ð1  4t Þrgrain þ 4t rmf  4t ð1  sxot Þ rmf  rhc (A4.9) Consider, a formation having a bulk density of rb logged with density and NMR tools. Now consider, a rock, having the same mineral makeup as that of the original formation rock, and such that its bulk density would be equal to rb when its pore space is fully saturated with mud filtrate. Let FD denote the porosity of this rock.   rgrain  rb   FD ¼ (A4.10) rgrain  rmf Substituting rb from Eq. (A4.9) into Eq. (A4.10) and simplifying: FD ¼ 4t þ 4t ð1  sxot Þl   rmf  rhc  Here; l ¼  rgrain  rmf

(A4.11) (A4.12)

Similarly, when the water base mud is used for drilling the hole, it can be assumed that the values of HIw and HImf are close enough. This is because the first term on the RHS of Eq. (A4.8) can be neglected in comparison with the other terms. Eq. (A4.8) then simplifies:   (A4.13) HINMR ¼ HImf 4t  4t ð1  sxot Þ HImf  HIhc   HImf  HIhc Let m ¼ (A4.14) HImf Eq. (A4.13) can be stated as,  HINMR ¼ 4t  4t ð1  sxot Þm HImf Eqs. (A4.11) and (A4.15) are solved for 4t :  lFD þ m HIHINMR mf 4t ¼ lþm

(A4.15)

(A4.16)

Inversion of log data to the gross attributes of pore space Chapter j 2

89

Case of oil base mud or SOBM In this case, the assumption stated at the beginning of the analysis in this Appendix for the case of water base mud may not be valid always.    (A4.17) Let r0b ¼ rb  4CBW þ 4Cap rw  rmf Eq. (A4.8) can be stated as,

  r0b ¼ ð1  4t Þrgrain þ 4t rmf  4t ð1  sxot Þ rmf  rhc

(A4.18)

We have, F0D ¼ 4t þ 4t ð1  sxot Þl Here, F0D

  rgrain  r0b  ¼ rgrain  rmf

(A4.19)

(A4.20)

The quantity ð4CBW þ4Cap Þ is not affected by hydrocarbons or filtrate because the original fluids occupying this porosity are neither drained by hydrocarbon nor replaced by filtrate. It is a routine output of NMR processing and is called the bound fluid volume, and hence is known. Hence the value of r0b is known.    0 (A4.21) ¼ HINMR  4CBW þ 4Cap HIw  HImf Let HINMR Eq. (A4.8) can be stated as,

  0 HINMR ¼ HImf 4t  4t ð1  sxot Þ HImf  HIhc

(A4.22)

Referring to remarks made above regarding why r0b is a known quantity, 0 note that HINMR is also a known quantity. From Eqs. (A4.19) and (A4.22):  HINMR0 0 lFD þ m HImf (A4.23) 4t ¼ lþm  
  rgrain  rb  4CBW þ 4Cap rw  rmf   4t ¼ A rgrain  rmf (A4.24)     HINMR  4CBW þ 4Cap HIw  HImf þB HImf Here; A ¼

l m ;B ¼ lþm lþm

(A4.25)

90 Understanding Pore Space through Log Measurements

Appendix 5 The dual water equation and input parameters The Dual Water Equation for the conductivity of the sand within the rock in the uninvaded zone is stated as (see Eq. 2.106): 

  4m Swtsd  Swbsd Swbsd tsd n S Csd ¼ Cw þ Cbw (A5.1) a wtsd Swtsd Swtsd The following analysis parameters apply exclusively for the uninvaded zone. Here, Swtsd and Swbsd are the saturations of the far water and the bound water in the total volume of water given by 4t Swtsd. The abbreviations Cw and Cbw stand for the conductivity of the pore water and the conductivity of bound water, respectively. The quantity within the square braces stands for the representative conductivity of the pore water. Cw is a model input given by the analyst. Cbw is computed as follows. The parameter b denotes the conductivity of bound water for unit counterion charge per unit volume of bound water. It is dependent only on temperature (T) and is computed as: b¼

T þ 8:5 ; where T is in units of degrees centigrade 22 þ 8:5

(A5.2)

The cation exchange capacity of the dry clay present within the sand is denoted as CEC here. Let Vbwsd denote the volume of bound water per unit volume of sand. Let Qbwsd and xbwsd , respectively, denote the cumulative counterion charge within unit volume of sand and the counterion charge within unit volume of bound water within the sand, respectively. xbwsd is thus the counterion charge concentration within the bound water. The parameter VQH denotes the volume of bound water associated with unit counterion charge, when the far water salinity value exceeds 10,000 kppm. VQH is called as the clay water volume factor. The volume of bound water associated with unit counterion charge regardless of far water salinity is given by: Vbwsd ¼ aVQH Qbwsd

(A5.3)

Here, a denotes the expansion factor, which is a known function of far water salinity and temperature. The value of a is essentially 1.0 when the far water salinity exceeds 10,000 kppm. The value of VQH the clay water volume factor is a known function of temperature. The value of VQH at 25 C is 0.28 cubic meters per 1 milli-ion equivalent of counterion charge. An ion equivalent is the amount of charge in a mole of protons (6.023  10 23 protons times 1.602  10 -19 coulombs/proton). The value of VQH is independent of the salinity of the far water and depends only on the temperature. Ï

Ï

Qbwsd ¼ CECrdcl Vdcl

(A5.4)

Inversion of log data to the gross attributes of pore space Chapter j 2

91

Substituting for Qbwsd from Eq. (A5.4) into Eq. (A5.3): Vbwsd ¼ aVQH CECrdcl Vdcl

(A5.5)

Wet clay porosity for the uninvaded zone, denoted by WCLPu, is the clay bound water volume per unit wet clay volume: Vbwsd ¼ Vclsd WCLPu

(A5.6)

sd Vdcl ¼ Vclsd ð1  WCLPu Þ

(A5.7)

Equating the RHS of Eqs. (A5.5) and (A5.6): sd Vclsd WCLPu ¼ aVQH CECrdcl Vdcl

(A5.8)

sd from Eq. (A5.7) in Eq. (A5.8), canceling like terms, Substituting for Vdcl and rearranging:

aVQH CECrdcl  WCLPu ¼  1 þ aVQH CECrdcl

(A5.9)

rclsd ¼ rdcl ð1  WCLPu Þ þ WCLPu

(A5.10)

(Assuming that the density of clay bound water is 1.0.)

Input parameters which are usually given and how they are used Usually the values of far water salinity, temperature, rdcl , and CEC are given as model inputs. They are used in the following manner. The value of VQH ; the clay volume factor, is computed from temperature, and the value of a the expansion factor is computed from the temperature and salinity of far water. Using the values of these two parameters mentioned above, the value of WCLPu is computed using Eq. (A5.9). Computation of the value of Cbw The counterion charge concentration within the bound water is given by: +

xbwsd ¼

Qbwsd 1 1 ¼ ¼ Vbwsd Vbwsd =Qbwsd aVQH

(A5.11)

Since the parameter b denotes the conductivity of bound water for unit counterion charge per unit volume of bound water, we have: Cbw ¼ bxbwsd

(A5.12)

Substituting for xbwsd from Eq. (A5.11) in Eq. (A5.12): Cbw ¼

b aVQH

(A5.13)

92 Understanding Pore Space through Log Measurements

Source of the input parameters Static bottom-hole temperature is found out from the maximum temperature recorded in different logging runs through a Horner plot. The bottom-hole temperature data are used to generate a geothermal gradient, to enable computation of formation temperature against any depth desired. Salinity of formation water gives the salinity of far water. The salinity is obtained from produced water salinity in offset wells or from pump out samples collected through wireline sampling. SP against clean formation or Pickett Plots are other sources of formation water resistivity. Hence formation water conductivity and salinity which are, respectively, the same as the far water conductivity and salinity. Special core analysis gives the clay mineral species and the respective dry weight fractions within dry clay present within sand. Let Wcli and ridcl stand for the dry weight fraction within the dry clay of the wet clay present within the sand and the dry clay density of the clay mineral species i, respectively. The former quantity comes from the special core analysis and the latter is available from published literature. The value of rdcl the density of dry clay of the wet clay present within sand is computed as, rdcl ¼ P i

1 Wcli =ridcl

(A5.14)

i

Let CEC stand for the cation exchange capacity of the dry clay mineral species i. CEC, the cation exchange capacity of the dry clay, is computed as, X CEC ¼ Wcli CEC i (A5.15) i

When core data are not available for offset wells or the candidate well, the following procedure can be adopted. From the Thorium versus Potassium cross plot and/or Potassium versus PEF cross plots from spectral gamma logs, the clay minerals are identified. A thick sand interval, which is not finely laminated, is selected. Using these clay minerals as part of the rock model and inverting the log data using a multimineral forward model for stochastic inversion, the volumes of wet clay of each clay mineral found within the sand i can be found out. Let Vwet and WCLPi denote the wet volume of clay mineral i per unit sand volume and the wet clay porosity of clay mineral, i, respectively. The former quantity is computed by the stochastic inversion while the latter quantity is available from published literature. The value of Vdcl, the volume of dry clay per unit rock volume, is given by: X   i 1  WCLPi (A5.16) Vdcl ¼ Vwet i

The weight of dry clay per unit rock volume is given by: X   i Wdcl ¼ 1  WCLPi ridcl Vwet i

(A5.17)

Inversion of log data to the gross attributes of pore space Chapter j 2

93

The value of rdcl is obtained by dividing Eq. (A5.17) by Eq. (A5.16): P i Vwet ð1  WCLPi Þridcl Wdcl i rdcl ¼ ¼ P i (A5.18) Vdcl Vwet ð1  WCLPi Þ i

Let

i Vdry

denote the dry clay volume for the clay mineral i. We have,   i i 1  WCLPi (A5.19) Vdry ¼ Vwet i Vdry ridcl Wcli ¼ P i i i Vdry rdcl

(A5.20)

The value of CEC is now obtained by substituting for the value of Wcli obtained from Eq. (A5.20) into Eq. (A5.15).

Appendix 6 High-resolution data We first discuss briefly, a methodology for generating deep resistivity data at a vertical resolution of 0.5 inches, out of deep resistivity data at a vertical resolution of 2 ft, through deconvolution. This technique requires the button resistivity data of a microimager tool, which offers the average resistivity (after removing bad button data and after proper normalization), of the buttons against any given sampled depth. These data are labeled as SRES, which is a resistivity data at 0.5 inches vertical resolution. This resolves all beds with > ¼ 0.5 inch thickness. The deep resistivity data at 2 ft resolution are called as RDeep for convenience. Consider a laminated formation with true resistivity R(z). Here z stands for depth. Beds thinner than 0.5 inch go undetected SRES. Hence R(z) is modeled as a box curve (square log) with 0.5 inch minimum thickness occurring for any bed to be 0.5 inches. The data R(z) at this point are unknown. However, the boundaries of the “beds” of the box curve is known. This bed boundary information is known because the boundaries can be extracted from SRES as follows. The boundaries can be either manually picked, or automatically obtained by thresholding the second derivative of SRES. Hence, we know the points of excursions of R(z) in depth z. The vertical response function of conventional deep resistivity is known, or can be modeled for a single bed having infinitely thick side-beds. Using this information, and the points of excursion of R(z), the function F(z) can be generated, which has the property:

94 Understanding Pore Space through Log Measurements

RDeep ¼ FðzÞ  RðzÞ

(A6.1)

Here, “” stands for convolution. Note that the individual thin bed boundary information is embedded in F(z). RðzÞ ¼ F 1 ðzÞ  RDeep

(A6.2)

High-resolution bulk density and photoelectric factor Bulk density and photoelectric factor logs at 6.5 inch resolution are possible to be generated using the three-detector density tool technology. The measurement is illustrated with the three-detector density tool principles for the tool developed by M/s Schlumberger and in wide use. Conventional two detector technology uses two gamma detectors, the one nearer to the source, being a collimated detector being located around 6 inches from the source, which is also collimated, and the one farther to the source being an uncollimated detector located around 16 inches from the source. For the range of electron density normally encountered within geological formations, the source to detector spacing for either detector far exceeds the mean free path of a gamma photon within the formation. A large fraction of the detected gammas for the case of either detector, therefore, comprises multiple scattered gammas. The angle of scattering for each collision of the gamma ray photon with an electron is also low. In the case of three-detector technology, the short-spaced detector is uncollimated, as well as the long-space detector. A third detector is placed very near the source, in-between the source, and the short-spaced detector, at a distance comparable with the mean free path of a gamma photon within the formation. As a consequence, the counted gamma photons for the case of this third detector comprise mostly single-scattered gamma photons. The response of a detector which mainly detects multiple scattered gamma photons is in the simplest terms as: 4ðxÞ ¼ 40 esrx

(A6.3)

Here 40 is proportional to source strength, x is the representative total path length for a gamma photon from source till it reaches the detector, and s is the scattering cross section for a gamma photon for the representative angle of scattering in a collision. r is the electron density of the formation. The single-scattered gamma photons that reach the third detector are mostly high-angle scattered gamma photons. Therefore, single-scattered gamma photons are also called as backscattered gamma photons. Also, the depth of penetration is around 2 cm only. The general relation for singlescattered gamma ray count and the formation electron density is,

Inversion of log data to the gross attributes of pore space Chapter j 2

4ðxÞ ¼

 40 sð4Þrx r02  e rv P  P2 sin2 4 þ P3 2 2 4pr1

95

(A6.4)

A sin b 1 for a narrowly collimated detector and where, b is P ¼ ð1þEð1þcos 4ÞÞ r22 the acute angle between the axis of the collimation of the detector and the line joining the source and the detector, r1 is distance between the source and the active scattering volume, r2 is distance between the active scattering volume and the detector, 4 is angle of scattering. E stands for gamma photon energy and r0 stands for the classical electron radius. v is the value of the active volume and is fixed for a given source detector spacing. For the case of a single-scattering event into the third detector: The value of 4 is close to 180 degrees. PyA sin rb2 cos q and is fixed for a given source and detector geometry. A is a 2 constant. Hence,

4ðxSC Þ ¼ Krersðr1 þr2 Þ ¼ K40 rersSC xSC

(A6.5)

Here, xSC is the single-scattering path length from source to detector, xSC ¼ (r1 þ r2). s is the differential cross section of scattering relevant to the single scattering, integrated over the appropriate collimation solid angles for the source and detector collimations, respectively. K is a constant whose value depends on the source and detector collimation design and the source to detector spacing. 40 is proportional to the source strength. From Eq. (A6.3):  d 4ðxÞ ln ¼  sx (A6.6) dr 40 Eq. (A6.6) indicates that the counts sensitivity to electron density is independent of the electron density are negative, and thus detector counts increase as electron density decreases, and vice versa. From Eq. (A6.5):  d 4ðxSC Þ K (A6.7) ln ¼ eKsx dr 40 r When the value of xSC is small Eq. (A6.7) becomes:  d 4ðxSC Þ K ln ¼ dr 40 r

(A6.8)

Integrating both sides of Eq. (A6.8) it is easy to see that detector count rate has a linear relationship with electron density. The analysis from this point onward follows that of Eyl et al. (1994). A source detector configuration is considered (see Fig. A6.1). The source is a Cesium-137 gamma source and there are two detectors which use NaI(Tl) scintillator crystal, located in line with the source, at

96 Understanding Pore Space through Log Measurements

FIGURE A6.1 Pad layout of the three-detector tool. Reprinted with permission from Eyl et al. (1994).

distances of 6 and 16 inches, respectively. The source is collimated, but none of the two detectors is collimated. These detectors are called as short-spaced detector and long-spaced detector, respectively, here. There is a third detector which is collimated and located very near to the source, at a distance of the order of the mean free path of a gamma photon within the formations normally encountered. The third detector mentioned above utilizes a cerium-doped gadolinium orthosilicate crystal. This satisfies the requirement of the faster response time for the third detector, in view of the high range of count rates expected at the location of the third detector. The detector counts are implemented in three energy windows for each detector. Thus, out of the three detectors, only the third detector, which is nearest to the source, is collimated. Note that the short-spaced detector is deliberately not collimated. A shortspaced detector which is uncollimated results in much higher count rate than a collimated detector. It also results in a deeper response into the formation the case of a collimated short-spaced detector. Count rates for each detector are acquired in three energy windows. Thus, there is a maximum of nine window counts available. The third detector referred to above is more familiarly known as the backscatter detector. The gamma photons detected by the backscatter detector, as already mentioned above, are single-scattered gamma photons with scattering angle is close to 180 degrees. These photons originate from 1 to 2 cm into the formation. The backscattering probability is proportional to the electron density at the locale of scattering. The detector responses (the energy window counts) are empirically parameterized as nonlinear functions of formation and mud cake properties according to the laws of photon transport through matter. Mud cake and tool standoff are equivalent in their effect on the tool response except for their individual geometries. The following expression gives the energy window count rate for any given the jth energy window against depth level: 
j a2j rn a3j ðrn Pen Þa4j hmcn ðDrÞn a5j hmcn DðrPe Þn Wjn ¼ a1j rm ð1 þ f Þ þ g (A6.9) n e Here ðDrÞn ¼ rmcn  rn ; DðrPe Þn ¼ rmcn Pemcn  rn Pen . aij, i ¼ 1 to 5 are energy-window-specific constants. mj is the energy-window-specific average

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number of collisions a gamma photon makes before it is detected and counted, with its energy falling within the energy window. f and g are functions of the physical properties of the mud cake that are not window-specific. Eq. (A6.9) is the theoretical forward model for no measurement noise. The energy window counts used in the processing to be briefly described shortly are the result of preprocessing applied on the raw energy window counts data. The following describes the preprocessing involved. During preprocessing, the calibrated count rates for each energy window are filtered and (vertical) resolution matched. The resolution-matching factors in the vertical resolution of each detector as well as the difference in the vertical resolution of the detectors whose energy window count rates are subjected to the preprocessing. If the energy window counts of the backscatter detector and the short-spaced detector alone are considered, the preprocessing results in data with 6 inches vertical resolution. The inverted model has the vertical resolution of 6.5 inch vertical resolution. This is the high-resolution output. This does not mean, however, that the data from the long-spaced detector are not used in the inversion. It does get used in the sense that low-resolution model data are used in the process of inversion. By low-resolution data we mean the model retrieved through the inversion of the energy window count rate data of all the three detectors. Obviously, preprocessing would result in 16 inch vertical resolution data and the model retrieved would have a vertical resolution of 16 inch plus. This model is called as the low-resolution model inverted out of the count rate data relevant to the energy windows of the three detectors. The actual count rate post preprocessing, for the jth energy window with the tool against the nth depth level, is designated as mjn. Consider the column vector mn of the energy window counts of a group of N energy windows. Let pn be the column vector of the 5 physical properties inverted viz., frn ; Pen ; rmcn ; Pemcn ;hmcn g. Then pn, even in the case of no-measurement noise, is still not the vector of the true values of the physical parameters mentioned above. It is a realization of a random vector which is the sum of the vector of the true values of the physical properties mentioned above, and the random vector of a zeromean multivariate normal distribution. Any inverted model (pn) is but an estimate of the true values of the physical properties sought. The covariance matrix of the realization of the normal distribution referred to above in pn is designated as Cpn here.

 (A6.10) mn ¼ fj ðpn Þ þ εjn Here, fj is a function which returns Wjn when it acts upon pn, and thus embodies Eq. (A6.9). The two terms appearing at the RHS of Eq. (A6.10) are the column vectors, respectively, of the quantities shown within the square braces, for j ¼ 1to N.

98 Understanding Pore Space through Log Measurements

εjn is the measurement noise for the count rate for depth level n and energy window j. εjn, j ¼ 1to N is a realization of the random variable viz., the measurement noise. Measurement noise is assumed to be Gaussian and of zero mean. It is assumed that the standard deviation of εjn, j ¼ 1 to N is a fair estimate of the standard deviation of the measurement noise. The covariance matrix of εjn, j ¼ 1 to N is notated as Cmn. We use the notation f ðpn Þ for the column vector ½fj ðpn Þ and the notation εn for the column vector ½εjn . Eq. (A6.10) is now stated as, mn ¼ f ðpn Þ þ εn

(A6.11)

Consider that Cmn is known. It is also assumed that the covariance matrix Cpn of the difference between pn and the vector of the true values of the physical properties frn ; Pen ; rmcn ; Pemcn ; hmcn g is known. It is assumed that a realization of pn is known. This of course would not be the vector of the true values of the physical properties frn ; Pen ; rmcn ; Pemcn ; hmcn g but would be an estimate of it. This estimate is taken to be the model inverted at a depth level just preceding the current depth level. The prior model called as pbn here is the estimate referred to above. pbn is assumed to be the model frn ; Pen ; rmcn ; Pemcn ; hmcn g retrieved against the depth level preceding the current depth level whose energy window counts are under inversion.b p n is, thus, not the vector of the true values of the physical properties inverted. pbn is the sum of the vector of the values of the true values of the physical properties and the vector of random numbers which are a realization of the multivariate normal distribution referred to, above. Finally, εn and pbn are assumed to be mutually independent. The inversion problem is cast as an update problem. The goal is to find a better estimate of the vector of the true values of frn ; Pen ; rmcn ; Pemcn ; hmcn g, than pbn , using the information provided by the newly available measurements, namely, mn . One of the ways in which the goal can be achieved is, theoretically, by obtaining the “maximum-likelihood” estimate. The “maximum-likelihood” estimate of the vector of the true values of frn ; Pen ; rmcn ; Pemcn ; hmcn g is the vector of frn ;Pen ; rmcn ; Pemcn ; hmcn g, which minimizes qðpn Þ where:  T     1 1 T 1 qðpn Þ ¼ ½mn  f ðpn Þ ðCmn Þ ½mn  f ðpn Þ þ pbn  pn pbn  pn Cpn 2 (A6.12) Thus, theoretically, the problem is to find a pbn ¼ minqðpn Þ, with qðpn Þ defined as above. qðpn Þ is a quadratic cost function which is a sum of two terms. The first term measures the difference between the actual measurement vector mn and the theoretical measurement vector f ðpn Þ. The second term measures the distance of the current estimate pn from the prior model pbn which

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is the solution vector (retrieved model) against the depth level preceding the current depth level. Minimizing qðpn Þ of the form defined at Eq. (A6.12) would be inadequate. This minimization against a level depends upon the model retrieved against a previous depth level and therefore is recursive in nature. In reality, caves and hole washouts can result in differences in the environmental effects on the window counts, not factored-in, in the prescription of qðpn Þ as at Eq. (A6.12). The objective function to be minimized is therefore prescribed as a modified form of qðpn Þ which is qðpn Þ plus an additional term involving the distance of pn from an “average” vector p. The objective function referred to above is designated as 4ðpn Þ. 1 4ðpn Þ ¼ ½mn  f ðpn ÞT ðCmn Þ1 ½mn  f ðpn Þ 2 (A6.13)  T     1 þ pbn  pn pbn  pn þ ðp  pn ÞT bIðp  pn Þ Cpn Here, I is the identity matrix, and bI is the diagonal weight matrix. p is an “average” value of the vector of the physical parameters sought to be inverted. This vector is computed based on the knowledge of the possible range of values of the estimated parameters (model) (Eyl et al., 1994). This addition is thus in the nature of a constraint, limiting the solution space to physically meaningful model, which is at the same time not far removed from the prior model, pbn even when caves and hole washouts occur. The minimization of the objective function is carried out using the GausseNewton Method. The model estimate pnþ1 obtained against depth level nþ1 post inversion using the model estimate pn obtained from the retrieved model against depth level n is related to pn as (Eyl et al., 1994): pnþ1 ¼ pn þ un

(A6.14)

Here, un is white noise.

High-resolution model High-Resolution Model is retrieved when the input energy counts inverted in the manner of the preceding analysis originate from the backscatter detector and the short-spaced detector. However, this does not mean that the inversion uses only the data from two detectors. In fact, the high-resolution inversion uses the results of the low-resolution inversion, which is an inversion of the energy window count rates from all the three detectors, resulting in a lowresolution retrieved model at 16 inches plus, vertical resolution. The results of the low-resolution inversion are used in the high-resolution inversion in the following manner (Eyl et al. (1994)). The low-resolution inversion is used as a

100 Understanding Pore Space through Log Measurements

first step of the high-resolution inversion where it provides an initial lowresolution estimate of pn (the model). The low-resolution inversion provides the low-resolution data misfit (reconstruction errors or the difference between the theoretical forward model of the energy window count rates and the actual energy window count rates). These data are used by the high-resolution inversion to compute possible offsets to be applied on the component values of the retrieved model prior to outputting the retrieved model.

Output of the high-resolution model The final output of the high-resolution model is the retrieved model fr; Pe ; rmc ; Pemc ; hmc g against depth, at a vertical resolution of 6.5 inches.

Improved-resolution acoustic slowness e multishot processing of sonic data Modern sonic tools have arrays of receivers recording waveforms generated due to the propagation of different body waves, surface waves along the borehole wall, and wave modes, which propagate axially along the borehole wall. Multishot processing is a method of processing of the receiver waveforms, which results in high-resolution slowness data. Here, the generation of compressional slowness data of the formation using multishot processing method for improved vertical resolution is briefly discussed. The vertical resolution of the data generated by processing the waveforms recorded by an array of receivers is equal to the length of the array. Stacking of waveforms from the receivers of the full array improves the signal-to-noise ratio, and thereby the noise in the slowness data extracted by processing the N

ðN 2

1Þ

waveforms. The signal-to-noise ratio is directly proportional to stack 12stack d 2 . Here, Nstack is the number of receivers stacked and d is spacing between neighboring receivers. When there is no oversampling, the signal-to-noise ratio can be increased only by increasing the number of receivers stacked (Nstack). However, the penalty is degradation of vertical resolution because as Nstack increases, it means that the array length has increased. It is, however, possible to improve signal-to-noise ratio through oversampling (bringing in redundancy of receiver data) while at the same time reducing the number of receivers stacked, thereby improving the vertical resolution, since the effective receiver array length is now reduced. This is the essence of the multishot approach to improvement of vertical resolution. Consider a transmitter and an array of N receivers with a constant interreceiver spacing (d). Consider a subarray of n receivers, of the abovementioned array. Let each subarray comprise successive receivers only so that the interreceiver spacing remains as Dz. The maximum number of subarrays of the type above possible is (m ¼ Nnþ1).

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In every subarray, let the receivers be numbered such that the receiver nearest to the transmitter for a subarray j would carry the number 1j. Let a reference receiver be identified for each subarray and let that necessarily be the receiver nearest to the transmitter. Thus, the reference receiver for subarray j would be receiver 1j. The waveform recorded by the kth receiver of subarray j will be designated as Wkj ðtÞ. Here t stands for time since the firing of the transmitter. It is assumed that the length of the subarray is small enough so that the formational acoustic slowness does not change for the entire length of the subarray between the first and the last receiver of the subarray. Finally, and most importantly, the receivers of a subarray are physically different, for different vertical positions of the sonic tool, but the vertical intervals spanned by the subarray of receivers are identical. Fig. A6.2 illustrates the concept. Fig. A6.2 shows seven sample positions of the array. Of interest are the first four sample positions. Numbering the receivers of the array as 1, 2, 3, .12 with 1 standing for the receiver nearest to the transmitter, we note that four

FIGURE A6.2 Multishot for the case of a receiver array having 12 receivers and the chosen subarrays spanning 3 receivers (red rectangle; gray in print version). Different positions of the total array as the logging tool is moved up and waveforms are acquired against depth sample points are shown. Adapted from Thompson and Burns (1989).

102 Understanding Pore Space through Log Measurements

subarrays of receivers, namely, (7,8,9), (5,6,7), (3,4,5), and (1,2,3), share a common vertical span. The reference receivers for these subarrays are, respectively, 7, 5, 3, and 1. The subarray number j for these subarrays would be 1, 2, 3, and 4, respectively. It is easy to see, from Fig. A6.2, that when the number of receivers in a subarray increases, the number of such subarrays possible, and hence redundancy (oversampling), also increases, and hence signal-to-noise ratio improves, but the vertical resolution degrades.

Signal processing for the different subarrays Define:  Wkjbp ðtÞ ¼ Wkj ðt þ ðk  1ÞSc d

(A6.15)

Here, Sc is the compressional slowness of the formation opposite to the common interval spanned by the selected subarrays of receivers. d is the interreceiver spacing. (k1)Scd gives the time move-out of the kth receiver with respect to the reference receiver. Wkjbp ðtÞ is the receiver waveform of the kth receiver, corrected for move-out. If the true value of Sc is substituted in Eq. (A6.15) and Wkjbp ðtÞ evaluated, the wave packet for compressional within the waveform Wkjbp ðtÞ would overlap in time, the compressional wave packet within the waveform of the reference receiver. In fact, Wkjbp ðtÞ is the waveform, which would be recorded by the receiver k, if it were to be positioned at the location of the reference receiver. The process of generating Wkjbp ðtÞ as per Eq. (A6.15) is known as backpropagation of receiver k of subarray j to the reference receiver of subarray j. Wkjbp ðtÞ is called as the backpropagated waveform of receiver k of subarray j. Semblance between a set of waveforms is a quantity defined as the coherent energy in a time window of length Tw at arrival time T, normalized by the total energy of all the waveforms within the same time window in which the coherent energy has been computed. The notation pj ðSc ; TÞ is used here to denote the semblance relevant to subarray j. The semblance pj(Sc,T), between the different backpropagated waveforms for all the receivers save the reference receiver, for the subarray j in the time window (T,T þ Tw) is computed as (Kimball and Marzetta, 1984): i2 Pt¼TþTW hPk¼n bp i2 Pt¼TþTW hPk¼n  j  k¼1 Wkj ðtÞ k¼1 Wkj ðt þ ðk  1ÞSc d t¼T t¼T pj Sc ; T ¼ P 2 i ¼ P 2 i hP hP t¼TþTW k¼n bp t¼TþTW k¼n W ðtÞ W ðt þ ðk  1ÞS dÞ kj c k¼1 kj k¼1 t¼T t¼T (A6.16)

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Wkj ðt þðk 1ÞSc dÞ is obtained from Wkj ðtÞ using the following relation.
 Wkj ðt þ ðk  1ÞSc dÞ ¼ F 1 Akj ðf Þej2pfSc ðk1ÞDz (A6.17) For dispersive backpropagation required when the receiver waveforms concern propagation of dispersive waves, the generalized form of Eq. (A6.17) given below can be used:
 Wkj ðt þ ðk  1ÞSc dÞ ¼ F 1 Akj ðf Þej2pfSðf Þðk1Þd (A6.18) Here, the operator F stands for the Fourier Transform of the operand function. The operator F1 stands for the inverse Fourier Transform of the operand function. f stands for frequency. Note that the parameter Sc of Eq. (A6.17) is now replaced with S(f) in Eq. (A6.18). S(f) is the dispersion curve, usually applied as a parameterized function of frequency, in terms of borehole parameters. Here S means phase slowness. S(f) is the function which tells how S phase slowness of a wave of frequency f varies with f. The value of Sc which is most likely the true value of the compressional slowness of the formation is that value of Sc which maximizes pj(Sc,T), when pj(Sc,T) is evaluated for different values of T.

S,T plane and S,T plots Consider a 3D plot with the different possible values of Sc represented by the x-axis, those of T by the y-axis and pj(Sc,T) for different combinations of these values, plotted on the z-axis. The xy plane will henceforth be referred to as the S,T plane. Let TW the time window width be large enough to cover the compressional wave packet. The graph of pj(Sc,T) would be a surface s having the following properties. s would be nearly a plane located close to and above the S,T plane, except for the region R, within which region, s would be a 3D positive peak. The lower and upper bounds of T, namely, Tl and Tu, respectively, for the region R are as follows. Tl is the time of onset of the compressional wave packet at the reference receiver. Tu is the time marking the end of the compressional wave packet. The lower and the upper bounds of Sc for the region R are denoted as Sl and Su, respectively. Sl would be that value of compressional slowness, for which the beginning parts of the compressional wave packet found on at least some of the backpropagated receiver waveforms would have some time-overlap with the ending part of the compressional wave packet found in the waveform of the reference receiver. Su would be that value of compressional slowness, for which the ending parts of the compressional wave packet found on at least some of the backpropagated receiver waveforms would have some time-overlap with the beginning part of the compressional wave packet found in the waveform of the reference receiver.

104 Understanding Pore Space through Log Measurements

The three-dimensional positive peak of s referred to above is called as the “semblance peak.” We shall now focus on the projection of the semblance peak onto the S,T plane. The projection of the semblance peak onto the S,T plane is best represented by a set of contours having the following properties. Each contour is a closed contour. Every point on a closed contour has a one to one correspondence with a point on the semblance peak in such a way, that each such points on the semblance peak are at the same height above the S,T plane and thus have the same value of semblance associated with them. Thus, the closed contours on the S,T plane are equal-semblance contours (iso-semblance contours). The set of these contours is called as the ST plane of compressional wave. While carrying out multishot processing, generally all the wave packets within a waveform are backpropagated, employing dispersive backpropagation where necessitated. Please note that in general the ST plane of even dispersive modes can be generated using Eq. (A6.18) for the backpropagation, provided the dispersion curve is available. Choosing the method of backpropagation is not difficult because the physics of acoustic wave propagation is well understood. For instance, it is known that monopole compressional is nondispersive, and that monopole shear is dispersive due to the overlap of pseudo Rayleigh mode. Flexural mode is known to be highly dispersive. Stoneley mode is weakly dispersive but is not usually relevant to multishot processing. The set of all contours for all the wave/mode types present within the receiver waveforms are collectively called as the ST plane, in the industry. The semblance values associated with any contours enclosed by a contour say l always exceeds the semblance value attached with the contour l. Let T0j, S0j denote the coordinates of the center of the contour associated with the maximum value of semblance, respectively. Then, T0j is the travel time of the fluid compressional wave packet from the transmitter till the reference receiver of the subarray j. S0j is the formation compressional slowness. The coordinates T0j, S0j are called as the time and slowness of the “coherence peak,” respectively, for compressional wave, for the array j. Both the coordinate values are estimates because noise would always be present. To mitigate the effect of noise, data redundancy available due to the presence of data from the other chosen subarrays is leveraged in the following manner. i) Process each subarray for the wave packet of interest, or all the wave packets present in the waveforms, using dispersive backpropagation, where required. For the present case nondispersive backpropagation would suffice. The result would be an ST plane for each subarray. ii) Shift the ST planes for the subarrays, in time, to account for the differences in transmitter receiver spacing. iii) Poststep ii) above, find out the average value of the semblance values of the different subarrays at every value of time and slowness. This results in

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a set of contours in the ST plane, each of which is marked by a given value of semblance. This set of contours is the set of contours that represent the set of subarrays. This set of contours is called as the average ST plane of the subarrays. iv) The result of step (iii) is now used to extract the coherence peak coordinates. The above workflow for each sampling depth results in the slowness logs at the vertical resolution equal to the length of a subarray. The illustration case of three receivers forming a subarray would lead to slowness logs at vertical resolution 2D, which is one foot. This is because usually the interreceiver spacing (receiver offset) is 6 inches.

Concluding remarks It is important to note that the multishot technique is a tradeoff between the standard deviation of the slowness obtained and the vertical resolution achieved. Reducing the subarray length improves the vertical resolution. But this comes at the cost of increased random error in the output slowness which expresses itself as the standard deviation of the output slowness. The total number of receivers in modern acoustic tools ranges 8e13 and interreceiver spacing of 6 inches. For the case of waveform data of average quality, the number of receivers in a subarray used is 5 receivers. The vertical resolution of the acoustic slowness output achieved is 24 inches. The illustrated example of three receivers in a subarray yields acceptable quality of acoustic slowness data at a vertical resolution of 12 inches, when the input waveform data quality is very good. This is usually the case. The reasons for this are i) modern drilling practices widely adopted, result mostly in smooth boreholes drilled, and ii) the deployment of state of the art acoustic tools that ensure maximum signal to noise ratio is common. Multishot processing involving subarrays of two receivers is feasible for small intervals where data quality is very good and borehole totally free of rugosity and is circular. Improved-resolution density and neutron porosity logs from conventional 2-detector tool data The conventional formation density log has a vertical resolution of 16 inches. The vertical resolution of the neutron porosity log ranges 18e24 inches depending upon formation porosity. Here, we discuss two methods of generating logs with improved vertical resolution, from the detector counts data of the conventional density or neutron porosity tool.

Deconvolution technique The term deconvolution means convolving the measured data with the inverse of the vertical response function of the measurement to the formation property

106 Understanding Pore Space through Log Measurements

being measured. The vertical response function will henceforth be referred to as VRF. One of the conditions for deconvolution using the inverse of the true vertical response function VRF is the following. The Fourier spectrum of VRF in the spatial frequency domain should not have zero crossings. This is not satisfied by the density and neutron porosity measurements. However, it is satisfied by the detector count rate data for both the types of logs. Theoretically, therefore, density and neutron data at improved vertical resolution can be generated by deconvolving detector counts data and using the deconvolved detector counts to generate density and neutron data. However, it has been found that suitably designed VRFs still deliver on the technique, both in respect of recovering the true formation properties, and improvement of vertical resolution in respect of, both density and neutron porosity. In this context, the technique (Smith, 1990) is discussed in brief taking neutron porosity measurement as an example. In this technique, Gaussian VRFs are assumed for the long-spaced detector and short-spaced detector counts. An example for the case of neutron tool is given at Fig. A6.3. M* referred to at Fig. A6.3 is defined as, ðM  Þ2 ¼ L2s þ L2e þ ð0:66LÞ2

(A6.19)

Here, Ls is the fast neutron slowing down length corresponding to the neutron energy interval 4.6 MeV to 1.46 eV, Le is the slow neutron thermalizing length corresponding to the neutron energy interval 1.46e0.025 eV, and L is the thermal neutron diffusion length corresponding to mean thermal energy 0.025

FIGURE A6.3 Variation of the vertical response of the far detector (long-spaced detector) with distance from measure point (midpoint of the source to detector spacing) for different formation porosities. In the figure S stands for source and D stands for detector. M* denotes effective migration length. Reprinted with permission from Smith (1990).

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eV. The mean squared straight-line distance a neutron travels from source to its point of capture is related to M  by, ðxÞ2 ¼ 6ðM  Þ2

(A6.20)

Here, x denotes distance traveled from source till capture, by a neutron. While applying the deconvolution on individual detector counts data, the width of the VRF Gaussian peak and its location with respect to the midpoint of the source to detector spacing are adaptively adjusted, based on the neutron porosity value of the formation from the standard resolution log. The resulting deconvolved detector counts are used as input to the process which uses the ratio of the deconvolved counts channels and converts this ratio to effective migration length and ultimately to water-filled limestone porosity. Implementation of the above method results in a neutron porosity log having a vertical resolution of around 9e12 inches for the porosity range 40%e12%. Higher the formation porosity, the better the vertical resolution. Similar technique can be used for resolution improvement of conventional 2-detector density data as well as the photoelectric factor data derived from the conventional 2-detector density tool. Fig. A6.4 shows the overlay of

FIGURE A6.4 Overlay of density and neutron porosity logs with improved vertical resolution generated through the technique discussed above (red; gray in print version), over the standard resolution logs (black). Reprinted with permission from Schlumberger Oil Field Review, July Issue (1991).

108 Understanding Pore Space through Log Measurements

the standard resolution and the improved resolution logs generated through the technique discussed above. The deconvolution benefit of improved vertical resolution comes at a cost, which is the introduction of statistical fluctuations in the deconvolved logs. The deconvolution technique can also be used for improving the vertical resolution of PEF (photoelectric factor) data by using a suitable VRF (or VRFs) as the case may be.

Alpha processing This method is the most popular technique used in the industry for the resolution enhancement of conventional 2-detector neutron porosity and density logs. This technique leads to logs having the vertical resolution equal to the source to detector spacing of the short-spaced detector (near detector). In essence, the method mathematically superimposes the rapid changes of the short-spaced detector (near detector), on the slowly changing, but accurate long-spaced detector (the far detector), to produce a log with a vertical resolution equal to the spacing of the short-spaced detector from the source, while ensuring a depth of investigation comparable to that of the long-spaced detector.

Alpha processing for improving the vertical resolution of conventional neutron porosity logs The following is the brief outline of Alpha Processing of Neutron tool data for resolution enhancement (Galford et al., SPE 15541, 1989). The long-spaced detector counts log and the short-spaced detector counts log are depth matched. The vertical resolution of the short-spaced detector counts is matched to that of the long-spaced detector count log. The depth and resolution matched detector counts are next used as inputs for computing the count rates for standard laboratory conditions through transforming functions which account for the tool environment being different from the standard laboratory conditions. This would include accounting for the borehole fluid not being the fresh water, the hole diameter not being the standard diameter, the temperature not being the standard temperature, effect of the mud cake, and so on. The corrected counts are now used to compute a porosity, which is called as the environmentally corrected porosity index. A continuous calibration factor for the short-spaced detector forward model connecting porosity and short-spaced detector counts is computed. This enables a porosity near to the true water-filled limestone response of the formation to be computed for any given short-spaced detector count rate. The entire process is equivalent to the calibration factor correcting the raw short-spaced detector count rate to produce an accurate water-filled limestone neutron porosity of the formation. This in turn is equivalent to an accurate

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hydrogen index of the formation. The notable feature of this output is the fact that it retains the high statistical precision (because of good short-spaced detector count rate) and improved vertical resolution (equal to the source e detector spacing of the short-spaced detector).

Alpha processing for improving the vertical resolution of conventional density logs Alpha Processing of Neutron tool data for resolution enhancement of density logs is briefly outlined below. Let nss denote the short-spaced detector counts log. Let rss denote the apparent density of the formation, which, when input to the forward model based on the photon transport theory as applied to the source-short-spaced detector system, predicts a short-spaced detector count rate equal to nss. rss is called the short-spaced density. Let rb denote the formation density log data at standard vertical resolution (16 inches vertical resolution, equal to the distance of long-spaced detector from the source). rss is at a vertical resolution equal to the distance of the short-spaced detector from the source (6 inches vertical resolution). The following steps are now executed. i) rss data are depth matched with rb data. The depth matched data are denoted here as rdm ss . (Also see panel 1 of Fig. A6.5). ii) rdm ss data now on-depth with rb data are smoothed to match the vertical res . resolution of rb data. The resulting rss is denoted here as rstd ss (Also see panel 2 of Fig. A6.5). res and rdm is computed. This difference is iii) The difference between rstd ss ss denoted here, as rresidue . (Also see panel 3 of Fig. A6.5)   std res (A6.21) rresidue ¼ rdm ss  rss iv) Formation density data at the vertical resolution of rss (6 inches vertical resolution - equal to the distance of the short-spaced detector from the source) denoted here by rhires are computed as:   std res (A6.22) rhires ¼ rb þ rresidue ¼ rb þ rdm ss  rss (Also see panel 4 of Fig. A6.5).

Enhancement of vertical resolution of gamma ray logs Gamma ray data are improved in their vertical resolution through deconvolution. As stated above, deconvolution is the process where the inverse of the true vertical response function of the sensor making the measurement, or an assumed vertical response function, is applied over the measurement. This

110 Understanding Pore Space through Log Measurements

FIGURE A6.5 Basic steps of vertical resolution enhancement in respect of formation density data through alpha processing. “Near detector density” signifies short-spaced density and the term enhanced density signifies high-resolution density. Reprinted, with permission, from, Schlumberger Oil Field Review, July Issue (1991).

recovers the property measured, as true values against the different layers while resolving their boundaries in the depth domain. Best results are obtained if the deconvolution is based on the true vertical response function. One of the conditions for deconvolution using the inverse of the true vertical response function VRFactual is the following. The Fourier spectrum of VRFactual in the spatial frequency domain should not have zero crossings. This condition is met with in the case of gamma ray counts data enabling the design of an effective deconvolving filter which is robust. Fig. A6.6 illustrates the technique of enhancement of vertical resolution. Looking at the vertical resolution-enhanced gamma ray log shown as solid curve at bottom right of Fig. A6.6, it is noted that even beds which are 12 inches thick are fully resolved.

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FIGURE A6.6 Top Left: Convolution of a cusp with a box function gives rise to a Gaussian-like response function that simulates the gamma sensor’s VRF (Vertical Response Function). Bottom Left: Synthetic gamma ray log obtained by convolving the simulated formation gamma reading (dotted) with the (simulated) VRF. Bottom right: Deconvolved gamma ray log (solid curve) obtained by (i) filtering the synthetic gamma ray log with a smoothing filter that preserves the vertical resolution of the input data while suppressing noise, and (ii) convolving the filtered synthetic gamma ray log with a simple spatial three point deconvolution filter (response function at top right) for the cusp shown at top left. Synthetic gamma ray log is also shown by dashed curve. Reprinted with permission from Jacobson et al. (1990).

From Fig. A6.6, it is seen that the gamma ray readings as per the deconvolved log are near the true gamma ray readings even for the case of beds of thickness 12 inches, which demonstrates the efficacy of the deconvolution in sharpening the gamma ray log to a vertical resolution of 12 inches (1 ft).

High-resolution NMR data From the perspective of the hardware, the vertical resolution of an NMR log is determined by the antenna length, the signal-to-noise ratio of the echoacquisition system the logging speed, and the number of depth levels stacked. The sampling interval in depth is usually 6 inches. The number of depth levels stacked while generating NMR of normal vertical resolution is 5. In the state of the art, antenna lengths, per se, allow beds of 6 inch vertical thickness to be resolved by the NMR measurements. Signal-to-noise ratio is improved through adoption of techniques such as combining overlapping phase alternating pairs (PAPs) of the CPMG pulse-echo sequences, ensuring improved signal-to-noise ratio. However, in the conventional way of

112 Understanding Pore Space through Log Measurements

processing pulse-echo data, the echo data against five contiguous depth levels are stacked. The vertical resolution achieved as a result of conventional processing of echo data is around 24 inches. In the case of high-resolution processing of echo data, the vertical depth stacking of the echo data is done away with. The penalty of such an approach would be a high degree of noise, forcing a high degree of regularization, if the number of T2 components used is the same as that used in the case of conventional processing of echo data. The noise in the output T2 distribution can be reduced, if the number of T2 components used in the inversion of the echo data is kept low. However, such a step would lead to loss of information content in the T2 distribution, if the T2 components or bins as they are called are not properly chosen for their T2 values. In high-resolution processing, the number of T2 components used is around 5. However, the T2 values of the T2 components are chosen appropriately. Normal inversion usually employs 30e50 T2 components, in order to encompass the entire range of possible relaxation times of formation pores through surface relaxation as well as bulk relaxation of fluids. The T2 distribution from such an inversion is first scanned for the range of T2 values, present within the T2 distribution against a given depth. Now, the T2 components used in the inversion of the echo data, for generating a T2 distribution log at high-resolution. To give an example, suppose the T2 distribution against a depth level from the normal-resolution processing has meaningful T2 between T2 values 25 and 350 msec. Then, for the depth levels stacked for generating the normal-resolution T2 distribution for the depth level mentioned above, the T2 components used in the inversion of the echo data, for generating a T2 distribution log at high-resolution, will be derived from the range 25e350 msec only. Thus unwanted T2 components are not used in the inversion. This way, optimal placement of the bins in time is ensured. As a result, the benefit of low noise in the inverted distribution which comes from the small number of T2 components involved is obtained with no loss of information. In other words, vertical resolution up to 6 inches is generated by doing away with stacking of echoes against multiple depth levels. The attendant noise which would have been present had the full number of T2 components been deployed in the inversion is avoided, owing to the low number of T2 components used for the inversion of echo data to T2 distribution. This in turn has been made possible with no attendant loss of information because of the leverage of the normal resolution data as explained above. Thus, this approach can be conceived as an adaptive inversion of echo data to T2 distribution. The validity of the technique is brought out by the good match of the total porosity and logarithmic mean T2 from the high-resolution processing upscaled to normal resolution, through averaging of the results with their counterparts from the normal-resolution processing seen in practice.Fig. A6.7 summarizes the technique discussed above.

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FIGURE A6.7 Vertical stacking of echo trains from multiple depth levels and inversion are used to get a normal-resolution T2 distribution (middle). An equivalent T2 distribution is formed with a reduced number of T2 components (T2 bins) (as explained in the discussion above), used to invert individual echo trains, thus achieving vertical resolution of the order of sampling interval (usually 6 inches). When upscaled to normal resolution, the total porosity and the T2 Log Mean match with those from normal-resolution processing. Reprinted with permission from Allen (2000).

References Allen, D., et al., 2000. Oil field review Autumn 2000 issue. Oil Field Review 10 (3). Anderson, B., et al., 2008. Triaxial induction e a new angle to an old measurement, oil field review Summer 2008 issue. Oil Field Review 20 (3). Bassiouni, Z., 1994. Theory, Measurement, and Interpretation of Well Logs. SPE Textbook Series 4. Cao-Minh, C., Clavaud, J., Sundararaman, P., Froment, S., Caroli, E., Billon, O., Davis, G., Fairbairn, R., 2008. Graphical analysis of laminated sand-shale formations in the presence of anisotropic shales. Petrophysics 49 (5), 395e405. Clavaud, J.B., Nelson, R., Guru, U.K., Wang, H., 2005. Field example of enhanced hydrocarbon estimation in thinly laminated formation with a triaxial array induction tool: A laminated sand-shale analysis with anisotropic shale. In: SPWLA 46th Annual Logging Symposium. June 26-29 (2005). Dresser, A., 1981. Well Logging and Interpretation Techniques. Dresser Industries Inc., Addison. Eyl, K.A., et al., 1994. High resolution density logging using a three detector device. In: 69th SPE Annual Technical Conference and Exhibition, SPE 28407. Freedman, R., 1997. Gas-corrected porosity from density-porosity and CMR measurements, in How to use borehole nuclear magnetic resonance. Oilfield Review 9 (2), 54. Galford, J.E., Flaum, C., Gilchrist Jr., W.A., Duckett, S.W., 1989. Enhanced resolution processing of compensated neutron logs, SPE-15541-PA, Society of Petroleum Engineers (SPE). SPE Formation Evaluation 4 (2), 131e137. Hagiwara, T., 1997. Macroscopic anisotropy approach to thinly laminated sand/shale sequences; sensitivity analysis of sand resistivity estimate and environmental corrections, SPE 38669. In: Presented at SPE Annual Technical Conference and Exhibition. Hussain, R.A.M., Ahmed, M.E.B., 2012. Petrophysical evaluation of shaly sand reservoirs in palouge-fal oil field, Melut Basin, South East of Sudan. Journal of Science and Technology 13 (2). Jacobson, L.A., et al., 1990. Resolution enhancement of nuclear measurements through deconvolution. In: Presented at SPWLA 31st Annual Logging Symposium.

114 Understanding Pore Space through Log Measurements Kantaatmadja, B.P., et al., 2015. Thinly bedded reservoir study, application of sand-silt-clay (SSC) e SHARP-Thomas Stieber Juhasz models, in a deepwater field, Offshore Sabah, Malaysia, SPE paper 176302-MS. In: Presented at the SPE/IATMI Asia Pacific Oil and Gas Conference and Exhibition 2015. Kimball, C.V., Marzetta, T.L., 1984. Semblance processing of borehole acoustic array data. Geophysics 49, 274e281. Mahmoud, M., et al., 2017. Reservoir characterization utilizing the well logging analysis of Abu Madi formation, Nile Delta, Egypt. Egyptian Journal of Petroleum 26, 649e659. Mezzatesta, A.G., Eckard, M., Strack, K.M., 1995. Integrated 2-D interpretation of resistivity logging measurements by inversion methods. In: SPWLA 36th Annual Logging Symposium Transactions. Mezzatesta, A.G., 1996. Reservoir Evaluation Through Optimally Designed Resistivity Logging. PhD Dissertation, University of, Houston, Houston Texas USA. Mezzatesta, A.G., Rodriguez, E.F., Mollison, R.A., Frost, E., 2002. Laminated shaly sand reservoirs e an interpretation model incorporating new measurements. In: Presented at SPWLA 43rd Annual Logging Symposium. Mollison, R.A., et al., 1999. A model for hydrocarbon saturation determination from an orthogonal tensor relationship in thinly laminated anisotropic reservoirs. In: Presented at SPWLA 40th Annual Logging Symposium. Mukherjee, S., 2017. Airy’s isostatic model: a proposal for a realistic case. Arabian Journal of Geosciences 10, 268. Mukherjee, S., 2018a. Moment of inertia for rock blocks subject to bookshelf faulting with geologically plausible density distributions. Journal of Earth System Science 127, 80. Mukherjee, S., 2018b. Locating center of pressure in 2D geological situations. The Journal of Indian Geophysical Union 22, 49e51. Mukherjee, S., 2018c. Locating center of gravity in geological contexts. International Journal of Earth Sciences 107, 1935e1939. Quirein, J., Kimminau, S., La Vigne, J., Singer, J., Wendel, F., 1986. A Coherent Framework for Developing and Applying Multiple Formation Evaluation Models. Paper DD, SPWLA 27th Annual Logging Symposium. June 9-13. Schlumberger, 1989. Log Interpretation Principles/Applications. Schlumberger Limited. Schoen, J.H., Mollison, R.A., Georgi, D.T., 1999. Macroscopic electrical anisotropy of laminated reservoirs: A tensor resistivity saturation model. In: SPE Paper 56509, Presented at SPE Annual Technical Conference and Exhibition. Pittman, E.D., 1979. Porosity, diagenesis and production capability of sandstone reservoirs. In: Scholle, P.A., Schluger, P.R. (Eds.), Aspects of Diagenesis: Society of Economic Paleontologists and Mineralogists, 26. Special Publication, pp. 159e173. Schlumberger, 1967. Well Evaluation Conference Middle East, Schlumberger. Text Book. Smith, M.P., 1990. Enhanced vertical resolution processing of dual-spaced neutron and density tools using standard shop calibration and borehole compensation procedures. In: Presented at SPWLA 31st Annual Logging Symposium. Thomas, E.C., Stieber, S.J., 1975. The distribution of shale in sandstone and its effectupon porosity. In: SPWLA 16th Annual Logging Symposium Transactions, Paper T. Thompson, D., Burns, D., 1989. Multi-Shot Processing for Better Velocity Determination, Earth Resources Laboratory Industry Consortia Annual Report; 1989-90, ERL Industry consortia Technical Reports. Publisher e Massachusetts Institute of Technology Earth Resources Laboratory. http://hdl.handle.net/1721.1/75145.

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Van der Wal, J., Stromberg, S., 2012. Improved workflow for evaluation of thinly bedded sandstones - revisiting the normalised Qv equation of Juhasz. In: Devex Conference. Wu, P., et al., 2007. Borehole effects and correction in OBM for dip and anisotropy for triaxial induction tools. In: SPE 110623, Presented at the SPE Annual Technical Conference and Exhibition 2007.

Further reading Gartner, M.L., 1989. A new resolution enhancement method for neutron porosity tool. IEEE Transactions on Nuclear Science 36 (1). Herrick, D.C., Kennedy, W.D., 1996. Electrical properties of rocks: effects of secondary porosity, laminations, and thin beds. In: SPWLA 37th Annual Logging Symposium, Paper C. Introduction to petrophysics of reservoir rocks. Bulletin of the American Association of Petroleum Geologists 34 (5), 1950, 943e961. Kamel, M.H., andMobarouk, W.M., 2002. An equation for estimating water saturation in clean formation utilizing resistivity and sonic logs: theory and application. The Journal of Petroleum Science and Engineering 36, 159e168. Kennedy, W.D., Herrick, D.C., 2004. Conductivity anisotropy in shale-free sandstone. Petrophysics 45 (01), 38e58. Klein, J.D., Martin, P.R., 1997. The petrophysics of electrically anisotropic reservoirs. The Log Analyst 38 (03), 25e36. Kunz, K.S., Gianzero, S., 1958. Some effects of formation anisotropy on resistivity measurements in boreholes. Geophysics 23 (4), 770e794. Moran, J.H., Gianzero, S., 1979. Effects of formation anisotropy on resistivity-logging measurements. Geophysics 44 (7), 1266e1286. Poupon, A., Clavier, C., Dumanoir, J., Gaymard, R., Misk, A., 1970. Log analysis of sandshale sequences a systematic approach. The Journal of Petroleum Technology 22 (07), 867e881. Saxena, K., Klimentos, T., 2004. Field study of integrated formation evaluation in thinly laminated reservoirs. In: SPWLA 45th Annual Logging Symposium. Shray, F., Borbas, T., 2001. Evaluation of laminated formations using nuclear magnetic resonance and resistivity anisotropy measurements. In: SPE Paper 72370, Presented at SPE Eastern Regional Meeting 2001. Stromberg, S., et al., 2007. Reservoir quality, net-to-gross and fluid identification in laminated reservoirs from a new generation of NMR logging tools - examples from the gharif formation. In: Southern Oman, Transactions of the 1st Annual SPWLA Middle East Regional Symposium. Wang, H., et al., 2006a. Determining anisotropic formation resistivity at any relative dip using a multiarray triaxial induction tool. In: SPE Annual Technical Conference and Exhibition 2006, Paper SPE 103113. Wang, H., et al., 2006b. Triaxial induction logging: theory, modeling, inversion and interpretation. In: SPE International Oil and Gas Conference and Exhibition 2006, Paper SPE 103897. Waxman, M.H., Thomas, E.C., 1974. Electrical conductivities in Shaly sands-I. The relation between hydrocarbon saturation and resistivity index; II. The temperature coefficient of electrical conductivity. Journal of Petroleum Technology 213e223. Transactions, AIME, 257, (1974).

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Chapter 3

Pore attributes of conventional reservoirs 3.1 The pore space of intergranular rocks A rock is a bicontinuum of spatially intertwined grains and pore spaces. Here cement is considered as a part of grain-assemblage. The pore space is best visualized as a 3D matrix composed of two elementsdpores and pore throats. The space occupied by the solid phase of the rock is a subspace of the space occupied by the rock. This will be designated as the grain-subspace or simply grain space. The pore space emerges as that subspace, whose union with the grain-subspace is the whole space occupied by the rock. If we consider any two points located within the pore space, it is possible to conceive any number of lines, which lie entirely within the pore space. For any one of such lines linking the pair of points mentioned above, a surface of minimum area having the following properties can be defined. i) The surface encloses the line joining the pair of points mentioned above, or, otherwise, the surface contains the line joining the pair of points mentioned above. ii) Further, the surface is necessarily the interface between pore space and grain space. This subspace of the pore space is called as a pore channel. The wider portions of pore channels are known as the pore bodies and the narrower portions of pore channels are called as pore openings or pore throats or ports. Pore space can also be imagined as a 3D matrix of larger-sized entities called pores interconnected to each other through narrow channels or openings called pore throats or ports. The essential pore attributes are the pore size, the pore throat size, preshape, and the number of the ports open for a pore to be connected with neighboring pores. There is an entirely different way in which pore space can be imagined. This way is imagining pore space as a fractal object. A key attribute then becomes the fractal dimension of the pore space.

3.2 Attributes of pore space Irrespective of the model, used for depicting pore space, there are certain important gross attributes of the pore space. These are (i) the total volume of

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118 Understanding Pore Space through Log Measurements

the pore space, which is also called as the pore volume, (ii) the cumulative area of the surface of the solid which encloses the pore space that would be wetted if the entire pore space were to be saturated with a liquid that wets the surface of the solid. In simple terms, for a clastic rock, this would be the wetted area of the grains when the pores are fully saturated with a fluid that would wet the grain medium. (iii) The permeability of the pore space, and (iv) the electrical formation factor. Out of the models for pore space, the model of pore network with connecting ports between pores, called as the pore throats, and the model of pore space as a bundle of nonjointed capillaries are the most productive models for predicting the fluid and ion transport properties of pore space. The fractal model of pore space is also very useful in this respect. Depending on the model used, certain key attributes of the model are defined. The way log data can be used to quantify these attributes forms the bulk of the discussion presented in this chapter. The focus is on introducing the concepts. Detailed analyses based on these concepts are presented in Chapters 7, 9, and 10. In the model of pore space as a network of pores connected through pore throats, the key attributes are (i) representative pore size, (ii) dominant pore shape, (iii) pore size distribution, and (iv) pore throat size distribution. In the model of pore space as a bundle of capillaries, the key attributes are (i) representative tortuosity, (ii) representative pore channel constriction factor, and (iii) hydraulic radius. Fractal models of pore space are of two types: (i) where the pore space as a whole is a fractal, and characterized by a single fractal dimension and (ii) where the pore space is a collection of capillaries, where the dimension of the cross section and the length of the capillary are individually considered to be fractal objects. The different attributes indicated above will be briefly discussed.

3.2.1 Pore shape, pore size, and pore throat size Fig. 3.1 gives a fair idea of the sense conveyed by the terms pore body, pore throat, and pore shape.

3.2.2 Pores as the building blocks of pore space The term pore stands for the expression of the pore space at the smallest scale and thus is a building block of the pore space. A pore in turn comprises of pore body and pore throat. Pore throats are connecting channels that are shared by contiguous pores.

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FIGURE 3.1 Representation of primary pores and secondary pores under a polarizing microscope, a laser scanning microscope, and SEM and BSEM photographs, respectively. Panel a depicts the primary intergranular pores, with pore throat and quartz overgrowth also seen; panel b depicts the primary pores, the dissolution pores, and the fractures pores. It is seen that the pore size is smaller than that for the case of panel a; panel c depicts the primary pores, fractures, and pore throat. It is seen that the size of pores is smaller than that for the case of panel a and panel b; a0 ec0 are laser scanning confocal microscopic images of a, b, and c; panel d is a BSEM image depicting the larger pores; panel e is a BSEM image that shows the primary and dissolution pores in feldspar; panel f is an SEM and BSEM combined image depicting the smaller minerals/intercrystalline minor pores. Pr.p stands for primary pores; Q.o stands for quartz overgrowth; P.throat stands for pore throat; Dp stands for dissolution pores; Fr.p stands for fracture pores. Figure reproduced with permission from Kashif et al. (2019).

3.2.3 Geometry of the pore body For intergranular pores the best approximation for the pore body geometry is a spheroid. Accordingly, pore body geometry is generally taken as spheroidal. The ratio of the minor axis-length to the major axis-length of the generating ellipse is defined as the aspect ratio in the pore models.

3.2.4 Size of a pore This is defined relative to the geometry of the pore body. For cylindrical pores, it is the radius of the cylinder if the geometry is of a right circular cylinder. For

120 Understanding Pore Space through Log Measurements

a cylindrical pore of elliptical cross section, it is the length of the minor axis of the ellipse. For slit-shaped pores it is the distance between the opposite walls of the pore. For a spherical pore it is the radius, and for a spheroidal pore it is the length of the minor axis of the generating ellipse. The term absolute pore body radius refers to the radius of the largest sphere which can be fitted within the space occupied by a pore (Hartmann et al., 1999). The term absolute pore throat radius refers to the radius of the largest circular disk oriented perpendicular to the fluid flow direction that would fit within the narrowest space connecting two contiguous pores (Coalson et al., 1994) (Fig. 3.2). Absolute pore body radius and absolute pore throat radius are commonly referred to as simply pore radius and pore throat radius, respectively.

3.2.5 The concept of pore class A pore class is defined as a set of pores, which have pore shape and pore size within narrow limits which define the pore class.

3.2.6 Surface area to volume ratio The term surface area to volume ratio (Spv) refers to the area of wetted surface of the solid phase enclosing a pore or cumulated area of wetted solid phase enclosing a set of pores. It is defined in this way when the pore volume is saturated with a fluid which wets the solid phase, to the volume of a pore or cumulated volume of a set of pores, according to the context.

FIGURE 3.2 The concepts absolute pore body radius and absolute pore throat radius. Reprinted with permission from Coalson et al., (1994).

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Spv of pores belonging to a pore class would lie within narrow limits. The ratio of the cumulated surface area (solid phase wetted area) for a pore class when the pore volume is saturated with a fluid which wets the solid phase, to the cumulated pore volume of the pore class, is known as the Spv defining the pore class. Spv for the pore space in a rock is defined as the ratio between the cumulated wetted solid phase surface area when the entire pore volume is saturated with a fluid which wets the solid phase, to the pore volume.

3.2.7 The characteristic length scale of pore space It is closely associated with Spv. It takes into account the pore space participating in fluid transport. It resembles how the characteristic length scale of pore space has been defined for the ionic current transport through the pore space (as in Johnson et al., 1986) as: R jVPðrÞj2 dS 2 ¼R (3.1a) L jVPðrÞj2 dVp Note that the weighting factor jVPðrÞj2 eliminates contribution to L2 from isolated regions of the pore space that do not contribute to the fluid transport. The integration for the numerator of RHS of Eq. (3.1a) is over the interface between the solid phase and the fluid phase. The integration for the denominator of the RHS of the same equation is over the pore volume. Drawing analogy from the similarity of the equations governing the fluid transport and electric charge transport at a macroscale: 2 S ¼ mð4Þ L Vp

(3.1b)

Here, the function m(4) is the hydraulic analogue of the function m(4) defined at Eq. (3.3) of Johnson et al. (1986). 4 denotes the effective porosity that contributes to transport. L stands for the characteristic length scale of the pore space. S and Vp, respectively, stand for the total surface area of the interface between the fluid and solid phases and the pore volume. Intergranular pore space has generally low-fraction of noneffective porosity. Further, m(4) is close to 2.0 since the bundle of capillaries model of intergranular pore space is a useful model. L would then physically represent the representative pore radius (representative radius of cross section of a capillary in the capillary bundle model of the pore space).

3.2.8 Hydraulic radius measure of the pore space V

The ratio Sp is called as hydraulic radius associated with the pore space (Berg, 2014). In the bundle of capillaries model of the pore space, the above equals twice the value of the representative capillary radius.

122 Understanding Pore Space through Log Measurements

It can be seen that the hydraulic radius is a static measure. On the other hand, the measure “characteristic length scale” discussed above is a better measure theoretically since it focusses on the pore space participating in fluid transport through a porous rock. Further, when fluid pressure is time-variant, it is the only measure which captures the dynamic behavior of the porous rock and the dynamic fluid transport property exhibited by porous rocks/sediments.

3.2.9 The pore shape factor Spv is inversely related to the characteristic dimension, which is a measure of the pore size. For example, if a spherical pore is considered, its Spv is given by 3 where r is the pore radius. In general, pore shape factor is defined through the r equation Spv ¼

a r

(3.1)

Here a denotes the pore shape factor. The factor equals 3 for spherical pores. Its value for the cylindrical pores is 2. The characteristic dimension that is a measure of the pore size for this case is the radius of cross section of the cylindrical pore. Similarly, for the case of slot pores the pore shape factor equals 1.0, with the characteristic pore dimension which is a measure of its size, being the distance between the opposite walls of the pore, or more clearly, the slot-width or the aperture. Consider now a unit volume of a rock, whose total pore space is a union of m number of pore classes. Consider a pore class i among the m pore classes, having a cumulated pore surface area of Si and a cumulated pore volume of 4i. Let Spvi denote the surface area to volume ratio of the pore class. Let T2i denote the NMR T2 bin value characterizing this pore class. Let r denote the surface relaxivity of the solid phase. We have, 1 ¼ rSpvi T2i

(3.2)

Let the fractional contribution to the pore volume, by pore class i, be denoted as ci. ci ¼

4i 4

(3.3)

where the denominator of the RHS denotes the porosity of the rock.

3.2.10 T2 log mean The weighted geometric mean of T2i where the weighting factor for T2i is 4i is called as the T2 log mean and denoted by the symbol T2LM. Here the mean is

Pore attributes of conventional reservoirs Chapter j 3

123

calculated over the m pore classes whose union is the total pore space in a unit volume of rock, as introduced above. By definition, !41 m m m  m  Y Y Y Y c 1ðc Þ 1 4i T2LM ¼ T2i ¼ T2ici ¼ ¼ (3.4) T2i i rSpvi i i¼1

Substituting for T2LM ¼

m  Y i¼1

rSpvi

i¼1

i¼1

i¼1

1 T2i

ci

from Eq. (3.2) in Eq. (3.4): ! ! ! m m  m  Y Y Y ci ci ci ½r ¼r Spvi Spvi ¼ i¼1

i¼1

i¼1

(3.5) The last equality in Eq. (3.5) comes because ci over all i sum to 1.0. It is assumed usually that the pore shape factor remains within narrow limits across pore classes. For conventional reservoirs, this is largely true. For reservoirs with intergranular porosity, even with some diagenetic alteration of pore shape allowed for, this assumption holds true. For reservoirs which have pore shape heterogeneity, such as carbonate reservoirs, this assumption is still valid when dissolution pores are not well connected, and the ground mass of the pore network which is connected dominates the total pore fabric. Hence, it is assumed that a single value of pore shape factor is representative of the parameter across pore classes. We retain the symbol a for this. Let ri denote the representative pore dimension for the pores of the pore class i. We have, in line with Eq. (3.1), a Spvi ¼ (3.6) ri Substituting for Spvi from Eq. (3.6) in Eq. (3.5) we get, ! m  ci m Y Y a ½ri ci ¼ arrLM T2LM ¼ r ¼ ar r i i¼1 i¼1

(3.7)

Here, rLM stands for the weighted geometric mean of the representative radii for the different respective pore classes, where the weighting factor is the fractional cumulated pore volume of a pore class in the total pore volume. rLM is called as the logarithmic mean pore radius. When the NMR spin relaxation mechanism is dominantly surface relaxation, T2LM can be considered as the representative NMR transverse relaxation time for the entire pore volume. When we consider the characterization of pore attributes from logs, NMR is possibly the only reliable input, which helps invert log data to the pore attributes introduced above.

124 Understanding Pore Space through Log Measurements

3.3 Distribution of incremental porosity over pore radius A graph or histogram of ci over ri is an example of the distribution of incremental porosity over pore radius. A distribution of ci over T2i is called as the T2 distribution. Here, we are implicitly assuming that ci equals the ratio of the fractional contribution of NMR bin porosity of a bin to the total porosity measured by NMR. The latter equals the area under the bin amplitude versus bin T2 of the distribution inverted by the NMR software, from the echo-sequence recorded. A T2 distribution is an example of the distribution of incremental porosity over T2i. Consider the pore volume per unit rock volume, hosted by pores, for which, the associated value of T2 lies within the interval ðT2 ;T2 þ DT2 Þ. Let this pore volume be denoted by D4ðT2 ; T2 þ DT2 Þ. Substituting for Spvi from Eq. (3.6), into Eq. (3.2) and rearranging: ri ¼ arT2i

(3.8)

From Eq. (3.8) the pore radius of the pores for which, the associated value of T2 lies within the interval ðT2 ; T2 þ DT2 Þ, lies within the interval ðarT2 ; arðT2 þ DT2 ÞÞ. This interval can be expressed as ðr; r þDrÞ where the value of Dr is given by: Dr ¼ arDT2

(3.9)

Let the function CP(r) be defined as: D4ðr; r þ DrÞ ¼ CP ðrÞDr 4

(3.10)

Thus CP(r) is the density function of the distribution of incremental porosity over pore radius.

3.3.1 Computation of CP(r) using NMR data From Eqs. (3.9) and (3.10): D4ðT2 ; T2 þ DT2 Þ ¼ CP ðrÞarDT2 4

(3.11)

D4ðT2 ; T2 þ DT2 Þ ¼ CT2 ðT2 ÞDT2 4

(3.12)

But,

Equating the RHS of Eqs. (3.11) and (3.12): CP ðrÞ ¼

1 CT ðT2 Þ ar 2

(3.13)

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125

Eq. (3.13) conveys that, if we start with a T2 distribution C T2 ðT 2 Þ and treat it as a distribution CT2(r) and multiply this distribution by

1 ar

, the distri-

bution CP(r) is generated.

3.3.2 Distribution of incremental porosity over pore throat radius Consider pores whose pore throat radius lies within the interval ðR; R þ DRÞ. Let the pore volume per unit rock volume hosted by such pores be D4. Let the function CPT (R) be defined as: D4ðR; R þ DRÞ ¼ CPT ðRÞDR 4

(3.14)

3.3.3 Ratio of pore size to pore throat size NMR data, or any other log data, do not lead to pore throat size distribution, directly. NMR data can be inverted for the pore size distribution. The ratio of pore size to pore throat size, for each pore class, allows one to convert a pore size distribution to a pore throat size distribution. The ratio of pore size to pore throat size for a pore class i is denoted by the symbol li here. Here, ri Ri ¼ (3.15) li Substituting for ri from Eq. (3.8) into (3.15): Ri ¼

arT2i li

(3.16)

In general, the pore size to pore throat size ratio would be distributed over the pore size, or equivalently, over the pore throat size. Denoting this distribution by l(Ri), Eq. (3.16) is stated as, Ri ¼

arT2i lðRi Þ

(3.17)

From Eq. (3.17) it is noted that the pore throat radius of the pores for which, the associated value of T2 lies within the interval ðT2 ; T2 þ DT2 Þ, ranges within the interval (R1, R2) where, R1 is the solution of the equation R1 ¼

arT2 lðR1 Þ

(3.18)

126 Understanding Pore Space through Log Measurements

R2 is the solution of the equation R2 ¼

arðT2 þ DT2 Þ lðR2 Þ

(3.19)

The interval (R1, R2) can be expressed as ðR1 ; R1 þDR1 Þ where the value of DR1 is given by, Generalizing R1 to R, arDT2  DR ¼  dlðRÞ R dR þ lðRÞ

(3.20)

Substituting for DR from Eq. (3.20) in Eq. (3.14) we get, D4ðR; R þ DRÞ ¼ CPT ðRÞDR ¼ CPT ðRÞ  4

arDT2

 þ lðRÞ R dlðRÞ dR

(3.21)

Also, D4ðT2 ; T2 þ DT2 Þ D4ðR; R þ DRÞ ¼ ¼ CPT ðRÞ  4 4

arDT2



(3.22)

R dlðRÞ dR þ lðRÞ

D4ðT ;T þDT Þ

2 2 2 While, ¼ CT2 ðT2 ÞDT2 (see Eq. 3.12). 4 Equating the RHS of Eqs. (3.12) and (3.22):

arDT2  ¼ CT2 ðT2 Þ CPT ðRÞ  dlðRÞ R dR þ lðRÞ 

 CPT ðRÞ ¼

(3.23)

R dlðRÞ dR

þ lðRÞ

arDT2

CT2 ðT2 Þ

(3.24)

Eq. (3.24) conveys that, if we start with a T2 distribution CT2 ðT2 Þ and treat it as a distribution CT2 ðRÞ, we generate the distribution CPT ðRÞ through:   dlðRÞ R dR þ lðRÞ CT2 ðRÞ (3.25) CPT ðRÞ ¼ arDT2

Pore attributes of conventional reservoirs Chapter j 3

127

3.3.4 Hard data on the distribution of pore throat size over incremental porosity ðCPT ðRÞÞ A mercury injection process involves injecting mercury into a fully dried core plug, at a progressively increasing injection pressure. The percentage of the pore volume drained by mercury is recorded. A plot of the injection pressure versus the percentage of the pore volume drained by the mercury is made, which is effectively, the plot of capillary pressure of the lowest sized pore throat (among the pore throats of the pores drained by mercury, for a given value of mercury saturation) against the mercury saturation. Fig. 3.3 is such a plot. The y-axis on the left is the injection pressure in units of psi. The x-axis is the percentage mercury saturation. The minimum value of pore throat radius of a pore drained by mercury at a given injection pressure PC is denoted by R(PC). R(PC) is related to PC through Washburn’s (1921) equation: PC ¼

2s cos q RðPC Þ

(3.26)

FIGURE 3.3 An example of capillary pressure versus mercury saturation plot. Reproduced from Hartmann et al. (1999).

128 Understanding Pore Space through Log Measurements

Here, s denotes the interfacial tension for the mercury-air system, and q is the angle of contact for mercury in the mercury-rock-solid phase-air system. Eq. (3.26) shows that the plot of capillary pressure versus mercury saturation is equally a plot of pore throat radius for the narrowest pore throat among those belonging to pores drained by mercury at a given injection pressure versus mercury saturation at that value of injection pressure. Accordingly, the RHS scale of the y-axis is of the value of R(PC) in microns. Further, the x-axis is the cumulated pore volume of all pores that have the value of the pore throat radius R(PC). Thus, Fig. 3.1 is equally the cumulated plot of the distribution of the fractional contribution of a pore class to the overall pore volume, over the pore throat radius characterizing a pore class. Let D4ðR; R þDRÞ be the fractional contribution of a pore class to the overall pore volume for unit rock volume. D4ðR; R þDRÞ be the incremental porosity due to the set of pores with pore throat radii within the interval ðR; R þ DRÞ. Therefore, the plot of Fig. 3.1 is also the cumulated plot of D4ðR; R þDRÞ over the pore throat radius. This plot is easily inverted to a distribution of incremental porosity due to a pore class over the representative pore throat radius characterizing that pore class. The concepts introduced above are put forth in clear mathematical terms by defining a distribution (CPT), which can be called as incremental porosity density distribution over the pore throat radius.

3.3.5 Obtaining CPT(R) from mercury injection data As per the definition of the distribution CPT over R (Eq. 3.14): D4ðR; R þ DRÞ ¼ CPT ðRÞDR 4

(3.27)

The distribution CPT over R is obtained from the plot of mercury injection pressure versus the mercury saturation using the following analysis. Differentiating both sides of Eq. (3.26): dPC 2s cos q ¼ dR ½RðPC Þ2

(3.28)

The change DPC in the value of PC, corresponding to the change in pore throat radius from R to R þ DR, is: DPC ¼

dPC DR dR

(3.29)

Pore attributes of conventional reservoirs Chapter j 3

Substituting for

dPC dR

129

from Eq. (3.28) in Eq. (3.29): DPC ¼ 

2s cos q ½RðPC Þ2

DR

(3.30)

D4ðR; R þDRÞ is therefore given by the incremental mercury volume in the rock sample, per unit sample volume, when the injection pressure increases by DPC where DPC is given by Eq. (3.29). Let V(PC) stand for the volume of injected mercury within the sample per unit volume of sample, at a stage of injection when the injection pressure is PC. D4ðR; R þ DRÞ ¼

dV DPC dPC

Here, DPC is given by Eq. (3.30). Hence,   dV 2s cos q DR D4ðR; R þ DRÞ ¼  dPC ½RðPC Þ2

(3.31)

Now, by definition, D4ðR; R þ DRÞ ¼ CPT ðRÞDR and D4ðR; R þ DRÞ ¼ 4CPT ðRÞDR 4

(3.32)

Further, dV dSHg ðPC Þ ¼4 dPC dPC

(3.33)

Here, SHg ðPC Þ stands for the mercury saturation for injection pressure PC. dV from Eq. (3.33), Substituting for D4ðR; R þDRÞ from Eq. (3.32) and for dP C in Eq. (3.31), simplifying and rearranging: CPT ðRÞ ¼

 2s cos q d  SHg ðPC Þ 2 R dPC

(3.34)

The function within the square braces in the RHS of Eq. (3.34) is the MICP plot.

3.3.6 Obtaining CPT(R) from log data Consider a formation bearing hydrocarbons. From petrophysical inversion of log data, one obtains the function SW where z stands for depth. In case the depth of occurrence of the free water level (zFWL) is known, the value of the water saturation SW is obtained as a function of h, the height above the free water level. This function is denoted by SW(h). Here h stands for the quantity (zFWLz), which is the height above the free water level. Assuming that the formation rock is water wet, which is normally the case, the saturation

130 Understanding Pore Space through Log Measurements

distribution of the nonwetting phase namely hydrocarbon fluid, denoted as SHC(h), over h is obtained: SHC ðhÞ ¼ 1  SW ðhÞ

(3.35)

Assuming that the accumulation of hydrocarbons within the formation was the outcome of primary drainage of the pores by hydrocarbon fluid, the excess pressure that forced the hydrocarbons into the formation at any height h above the free water level is the difference between the pressure inside the hydrocarbon phase and the water phase within the formation, at height h above the free water level. This pressure is of course the capillary pressure. The term capillary pressure in the present context stands for the pressure difference between the hydrocarbon phase and water phase across the meniscus separating the two phases when the meniscus has the radius equal to the pore throat radius. Note that the radius of the pore throat of a pore is the minimum dimension connected with a pore as a whole (a pore comprises pore body and pore throats which connect the pore to the surrounding pores). It is implicitly assumed that the character of the pore size distribution and, therefore, the pore throat size distribution is independent of height above the free water level. Let R(h) stand for the minimum value of the pore throat radius, for the pores drained by hydrocarbon, at a height h above the free water level. Let PC(h) denote the capillary pressure at height h free water level. We have, from the reasoning given above, PC ðhÞ ¼ pHC ðhÞ  pW ðhÞ

(3.36)

Here, p denotes phase pressure. SHC(h) is equivalent of SHg. Thus, we have in the function of PC(h) versus SHC(h) the equivalent of PC versus SHg. Using the interfacial tension and contact angle relevant to the hydrocarbon, water pair, the reasoning and the computation based on that reasoning, given above for obtaining CPT(R) from MICP data, can be applied for obtaining CPT(R) from SW(h) data, using further the steps mentioned above. Using Eq. (3.11), the NMR transverse relaxation time distribution and the pore throat size distribution and assuming a form of the function l(ri) and ar therefore of l(T2i), the quantity lðT can be evaluated. There would be wells 2i Þ which have penetrated the same formation and which might have been cored. The MICP data from cores would be available. If NMR data are also available, ar the quantity lðT can be evaluated from the MICP data and the NMR data. The 2i Þ ar value of the quantity lðT can then be applied on the transverse relaxation time 2i Þ

distribution data in uncored wells, for the same formation to i) convert transverse relaxation distribution into pore throat size distributions and ii) generate MICP curves of the type given at Fig. 3.3.

Pore attributes of conventional reservoirs Chapter j 3

131

3.4 Computation of CPT(R) from the NMR data In porous rocks with intergranular porosity the major component of the porosity, and when destruction of porosity due to diagenetic processes is of a low degree at the most, the ratio of pore size to pore throat size is stable across different ranges of pore size. In such cases, lðri Þ can be assumed to be a constant distribution and it can be replaced with a single parameter l for any pore class. In that case, the conversion of NMR transverse relaxation time distribution to a pore throat size distribution is called as a linear conversion. Nonlinear conversion as the general case mentioned is also practiced. A linear conversion work flow can first be attempted, and the results inspected. If the results are unsatisfactory, a nonlinear conversion work flow can be implemented. When satisfactory results are obtained with a linear conversion work flow, we infer that the candidate formation is a strong case for such work flow being valid.

3.4.1 The linear conversion work flow This work flow is valid when the assumption of a constant distribution l(ri) which is equivalent to considering a constant distribution l(Ri) is valid. In this case the distribution can be replaced by a single parameter l for any pore class as already mentioned. Then, arT2i ¼ CT2i l

(3.37)

arT2 ½rðRÞ ¼ CT2 ½rðRÞ l

(3.38)

Ri ¼ Eq. (3.37) is generalized to, R¼

Here, C ¼ ar l , and T2[r(R)] is the transverse relaxation time for a pore of radius r and pore throat radius R. The linear conversion workflow utilizes Eq. (3.38). A value of C is selected. The distribution of incremental porosity over T2 which is also called simply as the T2 distribution is considered, with the value of T2 on the logarithmic scale, and the value of incremental porosity on the linear scale. This is converted to the distribution ½CPT ðRÞmodel by shifting T2 distribution with the value of T2 on the logarithmic scale, and the value of incremental porosity on the linear scale, to the right, by the value logC. This distribution [CPT(R)]model is cumulated over R to obtain the distribution “cumulated ½CPT ðRÞmodel”. The MICP plot is considered, with mercury saturation on the y-axis and injection pressure on the x-axis. This is further converted to the plot of mercury saturation on the y-axis and pore throat radius on the x-axis.

132 Understanding Pore Space through Log Measurements

This conversion is implemented as follows. For every data value SHg of the mercury saturation, in the MICP data, the corresponding value of injection pressure PC(SHg) is noted. The pore throat value of the narrowest pore throat among the pore throats corresponding to the pore volume drained by mercury [Rmin(SHg)] is calculated:  2s cos q Rmin SHg ¼  PC SHg

(3.39)

Thus, the MICP data set of (PC(SHg),SHg) is converted to the data set (Rmin(SHg),SHg). This data set is the “cumulated ½CPT ðRÞMICP generated from mercury intrusion process (mercury injection).” A comparison of real data from a well from China, in respect of ½CPT ðRÞMICP, and “cumulated ½CPT ðRÞmodel” generated from the NMR T2 distribution, for different values of the parameter C is given in Fig. 3.4 (Xu et al., 2019). Note that the right-hand side panel of the figure is exclusive to one single core plug. The NMR T2 distribution for this core plug is in black, in the left-hand side panel of Fig. 3.4. “cumulated ½CPT ðRÞmodel” generated from the NMR T2 distribution for different values of the parameter C is presented as pink (gray in print version) and red (dark gray in print version) curves at the right-hand side panel.

3.4.2 Nonlinear conversion work flow When linear conversion work flow yields unsatisfactory results, a nonlinear conversion work flow is attempted. Among the forms of the relationship

FIGURE 3.4 Left: The NMR T2 distribution for 20 core plugs pertaining to a well in China. Right: “cumulated ½CPT ðRÞmodel” instances, generated from the NMR T2 distribution of a particular core plug, using different values of the parameter C, presented as pink (gray in print version) and red (dark gray in print version) curves. The black curve is the “cumulated ½CPT ðRÞMICP” for the same core plug (shown as “PSTD from HPMI”). The NMR T2 distribution of this particular core plug is the curve in black, at the left panel. The quantity Rmin(SHg) is denoted by r in the right-hand panel. From Fig. 3.4, as Xu et al. (2019) point out, we note Coptimal ¼ 0.0126. Reproduced from Xu et al. (2019).

Pore attributes of conventional reservoirs Chapter j 3

133

between T2 and R a power law type of relationship has a connection with imagining the pore space as the fractal. T2 has a relation with the pore size. Also, a power law type of relation between T2 and pore size is suggested by a fractal model of the pore space. In this scheme, a pore throat size is a minimum measure connected units of pore space that together comprise the pore space. Hence a power law type of relation between pore throat size and T2 is appealing. Case studies involving this approach are mentioned at Xu et al. (2019). The nonlinear conversion work flow given here is an example of the application of a power law type of model of T2 in terms of R. The work flow is after Xu et al. (2019). The relation between T2 and R is assumed to be: arT2 ¼ Rn

(3.40)

Taking logarithms on both sides of Eq. (3.40) and rearranging: 1 log R ¼ log C 0 þ log T2 n

(3.41)

Here; C 0 ¼ ðarÞn

(3.42)

1

The values of CPT(R) for the different mercury injection data points to the injection process are obtained in terms of the SHg values of the mercury intrusion data. The value of R is obtained from the magnitudes of injection pressure corresponding to the respective values of SHg and [PC(SHg)] as follows: R¼

2s cos q  PC SHg

(3.43)

Thus, we have multiple pairs of (SHg, R). For every value of SHg, the value of R is known. Suppose we have m such pairs. Let Ri denote the value of R for the ith pair (ith data). Let T2i correspond to the T2 value of the bin whose incremental porosity contribution is (4SHg), where 4 stands for the total porosity. The values of the parameters C 0 and n are obtained now, using the least square error minimization process defined as: 2 i¼m  X 1 0 Minimize log C þ log T2i  log Ri n i¼1

(3.44)

Once the model parameters C 0 and n are recovered through the inversion process given by Eq. (3.44), the Ri forward model data are obtained from the T2i data by substituting the respective values T2i, i ¼ 1 to m in Eq. (3.41), while the CPT(R) data are obtained as the incremental porosity corresponding to the T2i bin divided by 4. In this way, the distribution CPT(R) is computed from the NMR T2 distribution.

134 Understanding Pore Space through Log Measurements

3.5 Pore shape factor through integrating NMR and MICP (mercury intrusion data) From the nonlinear conversion work flow and nonlinear conversion work flow discussed above, it can be seen that generally, the work flows yield the value of (ar) as: ar ¼ ðC 0 Þ

n

(3.45)

Eq. (3.45) is obtained by inverting Eq. (3.42) for (ar). The RHS of Eq. (3.45) is determined since the parameters C 0 and n are evaluated already through the inversion process (3.44). Hence, the value of (ar) is known. The value of C for the case of the linear conversion work flow is directly the value of (ar). For the case where the linear conversion work flow gives credible results, it can be deduced that the intergranular porosity component of the total porosity is high, and also that diagenetic alteration of the pore fabric is low. Under these circumstances, a credible value of l can be arrived at from the analysis of images of thin sections of the rock, obtained through optical microscopy. When there is a strong control on the value of the NMR relaxivity of the solid phase of the rock, a value of the pore shape factor a of high confidence can be computed by dividing the value of the quantity (ar) by r.

3.5.1 Frequency distribution of pore radius The distribution of incremental porosity as the fraction of total porosity over the pore radius has been discussed. Multiplying the incremental porosity fraction of this distribution with the total porosity yields the distribution of incremental porosity over the pore radius. For any interval of pore radius ðr; r þ DrÞ, this distribution will give the pore volume per unit rock volume, hosted by the pores belonging to this class of pore radii defined by the above interval. Let this pore volume be denoted by D4ðr; r þ DrÞ. Let np ðr; r þDrÞ denote the number of spherical pores that can populate this pore radius class. We have, np ðr; r þ DrÞ ¼

D4ðr; r þ DrÞ 4 3 3 pr

(3.46)

From Eq. (3.10): D4ðr; r þ DrÞ ¼ 4CP ðrÞDr

(3.47)

Substituting for D4ðr; r þDrÞ from Eq. (3.47) in Eq. (3.46): np ðr; r þ DrÞ ¼

4CP ðrÞDr 4 3 3 pr

(3.48)

135

Pore attributes of conventional reservoirs Chapter j 3

Note that the value of the RHS of Eq. (3.48) is determined for any value of r, since the distribution CP(r) is known (see discussion above) at this point. Fixing a value of Dr as a class interval, let the ith class  be characterized by  the pore radius class ðri ; ri þDrÞ and population

4CP ðri ÞDr =43 pri3 .

Consider the fractional (normalized) frequency distribution: 4CP ðri ÞDr=43 pri3  Fðri ; ri þ DrÞ ¼ i¼M  P 4 3 4CP ðri ÞDr=3 pri

(3.49)

i¼1

Here M stands for the number of classes given by the ratio of the range of pore radius to the value of Dr chosen. The probability density of the pore radius denoted as PPR ðrÞ is related to the fractional frequency or normalized frequency distribution as: Fðr; r þ DrÞ ¼ PPR ðrÞDr

(3.50)

Pores are interconnected through pore throats. The number of connections a pore has with contiguous pores (coordination number) has a bearing on the number of pore throats available within the pore space. From the distribution of the pore population over different pore radius classes, the corresponding pore throat population is computed using the model of number of pore throats corresponding to the pores within a pore class. Also, the pore throat radius range for a pore class defined by the pore radius interval ðr; r þDrÞ is computed as ðrlðrÞ; ðr þ DrÞlðr þ DrÞÞ. Here, l(r) is the model for the pore throat radius for a given pore radius. It is reasonably assumed that the coordination number of a pore is independent of the pore radius. Hence, the number of pore throats npt ðr; r þDrÞ present, corresponding to the pores within the pore class defined by the interval of pore radius ðr; r þ DrÞ, is given as, npt ðr; r þ DrÞ ¼ xnp ðr; r þ DrÞ

(3.51)

Here, x is a constant, which is a model input. The distribution  of npt over the  pore throat radius is, as Eq. (3.51) indicates, the distribution xnp

r rþDr lðrÞ; lðrþDrÞ

.

The probability density of the pore throat radius is then computed on the same lines as for the probability density of the pore radius.

3.6 A simple visualization of constriction and its effect on the gross permeability of pore space The simplest visualization of pore space is as a bundle of tortuous capillary tubes, each of uniform radius and of circular cross section, and without any

136 Understanding Pore Space through Log Measurements

junction existing, between any two capillaries, anywhere within the pore volume. In this model, the radii of pore throat and of the pore body of the pore network model are of the same value. The radii of the capillary tubes are in general different. The incremental porosity due to capillaries whose radius of cross section lies within the interval (r,r þ dr) is distributed over r. The parameter r is conventionally considered to simulate the pore radius. A bunch of flow lines within the flow of fluid in a porous rock is simulated by the flow through a capillary in this model. Consider a porous rock with isotropic pore structure, when considered at a scale much larger than the size of a pore. It is clear that if a sample volume of such a rock is saturated with fluid, a pressure gradient setup fluid moves as if fluid is moving through a bundle of capillaries, which stretch between the fluid inlet face (surface) and the fluid exit face (surface). It further becomes evident that more realistically, any one of the capillaries would have an area of cross section which would vary along the length of the capillary. Consider a small length of a capillary, whose axis lies on a plane. Fig. 3.5 is a depiction of how a section of the capillary by a plane containing the axis of this small length of capillary would look like. The nomenclature “pore channel” would be interchangeably used in this chapter for a capillary. After Muller-Huber et al. (2016), the radius of a capillary (pore channel) is considered as variable along the length of the pore channel. The total capillary is considered as an assemblage of units in series. A unit is composed of two subunits which can be considered as the left half and the right half of a unit. The section of a half (the left half) of a unit in the plane of

FIGURE 3.5 Section of a subunit of a capillary with the axis of the pore in the plane of the page. The origin of the rectangular coordinate system coincides with the point O shown in the figure. The x-axis is along the axis of the pore. The y-axis is orthogonal to the x-axis and lies in the plane of the page. Our figure, not taken from anywhere.

Pore attributes of conventional reservoirs Chapter j 3

137

the page, which contains the axis of the unit, is presented in Fig. 3.5. The other half (the right half) of the unit will be the reflection of the left half on a plane normal to the axis of the unit whose cross section has the radius equal to the maximum radius of cross section of the unit. The two halves of a unit are called as subunits, here. Conceptually, a subunit represents the combination of half of the pore body and a pore throat. The minimum radius of cross section for a subunit would be R the pore throat radius. The maximum radius of cross section for the subunit is the pore body radius r. The function a(x) specifies the radius of cross section, which is (after Miller-Huber, 2016): aðxÞ ¼ Rebx

(3.52)

Here, x denotes the distance along the axis, of any cross section of a subunit, from the end which is the narrowest. b is the pore shape factor. Eq. (3.52) conveys that a pore is of the geometric shape of the solid of revolution of a generating segment of an exponential, rotated on the axis of the pore, which is same as the axis of the pore channel of which the pore is a part. Fig. 3.5 presents the subunit of a pore channel. Fig. 3.6A presents a section of a unit of pore channel on a plane containing the axis of the unit. The pressure-drop when a flow occurs, across either subunit of a unit of a capillary is the same, so also is the volume and surface area of either subunit. Therefore, the pressure-drop across a capillary, the surface area of a capillary, as well as the volume of a capillary are obtained by multiplying the respective quantity associated with a subunit with the number of subunits in a capillary. This number is denoted by Ns. In Fig. 3.5, AB and CD represent the intersection of two cross sections of the pore with the coordinate plane. They are separated by a distance of Dx. The radius of cross section at positions AB and CD are, respectively, denoted as aðxÞ; aðx þ DxÞ. Let the pressure at AB and CD are, respectively, denoted as pðxÞ;pðx þ DxÞ. Let q denote the flowing fluid volume per unit volume across any cross section of

FIGURE 3.6 The section of a unit of a capillary in the plane containing the axis of the unit. This is the horizontal axis of symmetry of the figure. Our figure, not taken from anywhere.

138 Understanding Pore Space through Log Measurements

the pore. Let the flow direction be from the left to toward the right in Fig. 3.5. We have, by the HagenePoiseuille law (1839): q¼

p ½pðxÞ  pðx þ DxÞ ½aðxÞ4 8h Dx

(3.53)

Here, h denotes the dynamic viscosity of the fluid, which is an incompressible fluid. Eq. (3.53) can be restated: dp 8h ¼ q½aðxÞ4 dx p

(3.54)

Substituting for a(x) from Eq. (3.52) in Eq. (3.54): dp 8h 4 4bx ¼ qR e dx p

(3.55)

Referring to Fig. 3.5 it can be seen that the pressure-drop across the pore is: Z l  8h 4 4bx 8h qR e qR4 1  e4bl dx ¼ Dp ¼ (3.56) 4pb 0 p l¼

r ¼ ebl R

(3.57)

Here, l is the pore size to pore throat size ratio. Eq. (3.57) arises from the fact that when x ¼ l, the radius of the cross section is R for a subunit of the type shown in Fig. 3.5. We assume for simplicity that a total channel spanning the end faces of a sample of rock is made up of identical pairs of subunits that form a set of units in series. Further assume that the entire pore space of a volume of rock is the union of N capillary volumes. Here each capillary volume is that enclosed by a capillary. We assume that the internal structure of any capillary is identical to that of any other capillary. Consider a sample of the rock (Fig. 3.7). The gray-shaded volume is a capillary. This is only a representation and the cross section of the capillary at the end faces carries no quantitative geometric meaning.

FIGURE 3.7 Representation of a capillary (flow channel) in a rock sample. Adapted from MullerHuber et al. (2016).

Pore attributes of conventional reservoirs Chapter j 3

139

Referring to Figs. 3.5 and 3.7, Ns ¼

Le l

(3.58)

Let Q denote the total discharge (volumetric flow rate which is volume of fluid flowing past a cross section of the capillary per unit time), and DP the pressure-drop across the opposite faces of the sample. q¼

Q N

DP ¼ Ns Dp

(3.59) (3.60)

s : tortuotisity of the capillary: Le ¼ Ls

(3.61)

Substituting for Dp from Eq. (3.56) and for Ns from Eq. (3.57) into Eq. (3.60): DP ¼

 Le 8h qR4 1  e4bl l 4pb

(3.62)

Substituting q from Eq. (3.59) into Eq. (3.62): DP ¼

Q Le 8h 4  4bl R e þ1 N l 4pb

(3.63)

Let Ss, vs denote the surface area and the volume of a capillary segment respectively. Let Sc, vc, respectively denote the surface area and the volume of the capillary made up of capillary segments such as the capillary segment mentioned above. Let S, Vp, respectively, denote the cumulated pore surface area and the total pore volume of the sample. Referring to Fig. 3.5: Z l Z l pR2  2bl vs ¼ p½aðxÞ2 dx ¼ pR2 e2bx dx ¼ (3.64) e 1 2b 0 0 Le pR2  2bl pR2  2bl e 1 ¼ e 1 2b l 2b Z l Z l 2pR  bl e 1 Ss ¼ 2paðxÞdx ¼ 2pR ebx dx ¼ b 0 0 vc ¼ N s

Sc ¼ Ns

2pR  bl e 1 b

Vp ¼ Nvc ¼ N

Le pR2  2bl e 1 l 2b

(3.65)

(3.66) (3.67) (3.68)

140 Understanding Pore Space through Log Measurements

The ratio of the rock’s solid phase surface area wetted to the total pore volume when the rock is fully saturated with a fluid that wets the solid phase is denoted as Spv. Spv ¼

NSc Vp

(3.69)

Substituting for Sc, Vp, respectively, from Eqs. (3.67) and (3.68) into (3.69) and simplifying: Spv ¼

2 2 R ðebl þ 1Þ

(3.70)

Porosity 4 is given by: 4¼

Vp Vp ¼ L3 AL

A; face area ¼ L2

(3.71)

Substituting for Vp from Eq. (3.68) in Eq. (3.70): 4¼

NLe pR2  2bl e 1 ALl 2b

(3.72)

Taking logarithms of both sides of Eq. (3.57) and rearranging: bl ¼ ln l

(3.73)

Using Eqs. (3.72) and (3.73) is restated as: 4¼

NLe pR2  2 l 1 AL 2 ln l

(3.74)

Using Eqs. (3.22) and (3.73) is restated as: DP ¼

 QLe 8h R4 1  l4 N 4p ln l

(3.75)

Substituting bl from Eq. (3.73) into Eq. (3.70) and rearranging: Spv ¼

4 1 R ðl þ 1Þ

(3.76)

Eq. (3.76) is same as (A.18) of Muller-Huber et al. (2016), but with different symbols.   4 1 R¼ (3.77) l þ 1 Spv   4l 1 r¼ (3.78) l þ 1 Spv

Pore attributes of conventional reservoirs Chapter j 3

141

3.6.1 Model prediction of permeability k being the permeability, Darcy Law states: Q¼

kA DP hL

(3.79)

Substituting DP from Eq. (3.75) into Eq. (3.79) and simplifying: k¼

4pLN ln l 1  R2 R2 8ALe 1  l4

Dividing Eq. (3.80) by Eq. (3.74):  2 k L 1 1   2 R2 ¼ ðln lÞ2 4 4 Le 1l l 1 Eq. (3.81) can be written as,   k 1 L 2 ðln lÞ2 8  2 R2  ¼ 4 8 Le l  1 1  l4     k 1 L 2 ðln lÞ2 8l2 1 L 2 ðln lÞ2 8r 2 2 R ¼     2  2 ¼ 4 8 Le 8 Le l 1 l 1 l2  l12 l2  l12

(3.80)

(3.81)

(3.82)

(3.83)

Let s denote the tortuosity of the capillaries. s¼

  Substituting

L Le

Le L 1 and ¼ Le s L

(3.84)

from Eq. (3.74) into Eq. (3.83) and rearranging, 2 3 k¼

1 4 26 r 48  8 s2

ðln lÞ2 7  5 l2  1 l2  l12

(3.85)

Eq. (3.85) is same as (A.26) of Muller-Huber et al. (2016), but with different symbols. Substituting for r from Eq. (3.77) into Eq. (3.85): 2 3 2   2 2 1 4 4l ðln lÞ 1 6 7  k¼ 2 (3.86) 48  5 8 s lþ1 S pv 2 2 1 l  1 l  l2

142 Understanding Pore Space through Log Measurements

Eq. (3.86) is rearranged as: 2  2 4 4 1 6 k¼ 2 48  8 s Spv 

k¼

1 4 1 2 s2 Spv

2

2 6 48 

3

2   ðln lÞ 2  7 5 l lþ1 l2  1 l2  l12 2

(3.87)

3

2   ðln lÞ 2  7 l 5 lþ1 l2  1 l2  l12 2

(3.88)

Eq. (3.88) is same as (A.27) of Muller-Huber et al. (2016), but with different symbols. Eq. (3.88) has the form of the KozenyeCarman equation. Note that it reduces to KozenyeCarman equation for the case of l (no constriction). Also note that the assumption of all capillaries being identical in their internal structure and having the same tortuosity is a homogenization of the set of different capillaries of different tortuosity values and internal diameter. Thus, the pore radius, pore throat radius, and tortuosity of the analysis above are amenable to interpretation as the representative pore radius and representative pore throat radius of the pore system. The pore size to pore throat size l is also amenable to interpretation as the representative pore size to pore throat size ratio of the pore system. Finally, the analysis of Muller-Huber et al. (2016) is in a way a bridge between the pore network model and the capillary bundle models of pore space. The next link of the bridge would be introduction of junctions (junctions of two capillary segments or common subspaces of two capillaries) into the model. Eq. (3.88) can be written as:  2 4 1 k ¼ 0:5 (3.89) Cc s2 Spv The effect of constriction is ensconced in the factor Cc. 2 31 2   2 ðln lÞ 2 6 7   Cc ¼ 48 5 l  2 lþ1 l  1 l2  l12

(3.90)

Generalizing 0:5 Cc as C the general calibration coefficient for fitting the KozenyeCarman permeability predictor to actual data assumes the physical meaning, in this scheme of analysis as the role played by the pore size to pore throat size ratio l on permeability.

Pore attributes of conventional reservoirs Chapter j 3

143

3.6.2 TimureCoates permeability predictor from the perspective of constriction The predictor (also known as the Coates equation) (Coates and Dumanoir, 1974) is widely popular. It can be rationalized starting from KozenyeCarman equation by relating Spv to the volume of immovable water BV that sticks to the solid phase of a water wet rock, to the volume of capillary bound water which is not drained by nonwetting fluid in a drainage experiment, even for high values of injection pressure, when nonwetting fluid is progressively injected into a rock fully saturated with wetting fluid. After Muller-Huber et al. (2016), denoting the thickness of the film by t the cumulated pore surface volume which is same as the total area S of the solid phase wetted by the pore fluid at 100% saturation of wetting fluid is: S¼

BV t

FFV ¼ Vp  BV z Vp

(3.91) (3.92)

Here FFV denotes the maximum volume drained in an intrusion experiment referred to above. Spv ¼

S BV z Vp tFFV

(3.93)

Note that a capillary pore or a capillary holding wetting fluid not displaced by nonwetting fluid even at high intrusion pressure is a case of, the value of the capillary dimension (radius of cross section for a capillary of uniform (radius or the maximum radius of cross section for the case of constricted capillary bundle model)) being equal to or less than t. This dimension, mentioned above, in turn, equals the pore radius in the pore body and pore throats assemblage model of pore space. Dividing the numerator and the denominator of Eq. (3.93) by the porosity, we obtain the RHS of Eq. (3.92) as the ratio BVI FFI of the bound fluid index to the free fluid index. From this:     1 FFI (3.94) ¼t Spv BFI   Substituting for S1pv from Eq. (3.43) into Eq. (3.36): 3 20 1   2 2  2 2 ðln lÞ 2 7 4t FFI 6B  C k ¼ 4 @8 (3.95) l 5 A  2 lþ1 s2 BFI l  1 l2  l12

144 Understanding Pore Space through Log Measurements

It is assumed that the hydraulic tortuosity and electrical tortuosity of the pore space are approximately of the same value. We then have (see Appendix 1), 4 1 ¼ ¼ 4m s2 F

(3.96)

Here, F, m, respectively, stand for electric formation factor and the Archie cementation exponent. Substituting for s42 from Eq. (3.96) into Eq. (3.95) and rearranging, Eq. (3.95) is stated:   FFI 2 k ¼ f ðlÞt2 4m (3.97) BFI 1 0   2 2 ðln lÞ 2 C B A l  : (3.98) Here; f ðlÞ ¼ @8  2 lþ1 l  1 l2  l12 Eq. (3.97) provides the insight that the premultiplier of the TimureCoates permeability predictor ensconces the effect of the thickness of the capillary bound water layer and the pore size to pore throat size ratio. Note that Eq. (3.97) resembles the predictor in its form, except that the exponent of porosity is usually considered to have a value of 4 in that predictor. Further discussion of the topic of constriction from a more formal perspective can be found at Chapter 9.

3.7 Fractal attributes of pore space The objective here is to introduce some of the results that follow from a fractal model of pore space. The detailed discussion of the topic of fractal model of pore space and the various results which flow out of the different models is in Chapter 9. Pore space can be visualized as the union of two distinct subspaces: the pore assemblage and the pore throat assemblage. Physically pores and pore throats occur contiguous to one another with a pore throat separating two pores and vice versa. A simple way of visualizing pore throats, from the perspective of fractal models of pore space, is to consider that the geometry of a pore and a pore throat are similar and that a pore throat separating a pore differs from the pores involved, only in scale. Further, every pore has a fixed number of pore throats surrounding it, in a physical rock. These considerations suggest the following. Suppose a fractal which when visualized at a certain scale is a rendering of the physical pore throats assemblage is considered. Further suppose another fractal which when visualized at a certain scale is a rendering of the pore assemblage. Then these two visualizations look at the same fractal

Pore attributes of conventional reservoirs Chapter j 3

145

at different scales. The two fractals mentioned are identical. Thus, no qualitative distinction is made between a pore and a pore throat. The geometric element of this fractal, the pore space fractal, is a pore. The term “pore’ can thus mean a physical pore or a pore throat. In the rest of this discussion, the term “pore throat’ is therefore dropped. Any characterization of the physical pore space is essentially a rendering of the visualization of the pore space fractal at a given scale. It is implicitly assumed that pores in this general sense are spherical. In a rock there exists a maximum size of the pore, which is the maximum value the radius of a pore of the fractal model can assume, for a given scale of rendering the fractal. The radius of the narrowest pore throat present within the rock is the minimum value that the radius of a “pore’ of the prespace fractal can assume. The values of these two limits are, respectively, denoted as rmax and rmin. It will be seen that the condition of selfconsistency of the fractal model of pore space requires that the following condition is satisfied: rmax [1 (3.99) rmin A pore space can also be modeled as a bundle of nonjointed capillaries. In fractal models of the pore space based on this kind of visualization, the capillaries are assumed to be of right circular cross section. The unit of the fractal that models the pore space is a capillary. A capillary unit is defined through two numbers: the length or the tortuosity and the radius of cross section of the capillary. The limits of the value of the radius of cross section of a capillary, when the pore space is rendered as the visualization of the bundle of capillaries-fractal at a particular scale, are again, rmax and rmin. Here rmax and rmin mean the maximum pore radius and minimum pore throat radius available within the rock.

3.7.1 The fractal model of the pore space, based on a poreepore throat assemblage visualization of the physical pore space Let Df stand for the fractal dimension of the pore space, in this fractal model. The pore population scales on the pore radius r as:

r Df max nðr  rmax Þ ¼ (3.100) r Here, n(r  rmax) stands for the population per unit gross rock volume, of the pores whose radius ranges r to rmax. Using Eq. (3.100):   rmax Df (3.101) nðr þ dr  rmax Þ ¼ r þ dr

146 Understanding Pore Space through Log Measurements

Let n(r) stand for the number of pores whose radius lies within the interval (r, r þ dr). nðrÞ ¼ nðr  rmax Þ  nðr þ dr  rmax Þ

(3.102)

Substituting for the quantities on the RHS of Eq. (3.102) from Eqs. (3.100) and (3.101), respectively: Df

r Df  r  D  ð1þDf Þ max max f r nðrÞ ¼ dr (3.103)  yDf rmax r r þ dr Let N denote the total pore population per unit volume of rock.   rmax Df N¼ rmin

(3.104)

The probability density P(r) of the pore radius is: nðrÞ ¼ PðrÞdr N

(3.105)

Substituting for N from Eq. (3.104) into Eq. (3.105):

 Df r ð1þDf Þ PðrÞ ¼ Df rmin (3.106) R rmax Df Þðr ð1þDf Þ Þdr ¼ 1 For the fractal model to be self-consistent, rmin Df ðrmin should hold since the integrand is the probability density of the pore radius. This in turn holds if Eq. (3.99) holds. Consider a unit volume of rock. The measure of the pore volume obtained by the box counting method with a pore as the volume unit can be considered as the value of the porosity 4. Then, ln 4  

Df ¼ DE  ln

(3.107)

rmin rmax

(Yu and Li, 2001). In Eq. (3.107), DE stands for the dimensionality of the space within which the fractal (pore space fractal) is embedded. In this case, this space is the Euclidean space, with DE ¼ 3. Substituting this value for DE in Eq. (3.107) and rearranging: 1

rmin ¼ rmax 43Df

(3.108)

Eq. (3.108) conveys that more the spread in the value of the pore radius, more is the fractal dimension value.

Pore attributes of conventional reservoirs Chapter j 3

147

3.7.2 A fractal model of the pore space This model is based on a visualization of the physical pore space as a bundle of capillary tubes. A number of key results of interest follow from the fractal modeling of pore space visualized as a bundle of capillary tubes. The results in the main are presented here with a brief analysis. The detailed analysis leading to the results is found in Chapter 9. From the perspective of the fractal model of pore space as per the visualization of pore space as a bundle of capillaries, Eq. (3.103) would represent the number of capillaries of radius of cross section within (r, r þ dr), which cross unit area of a plane of cross section for a bunch of capillaries. This number density is denoted as N(r) here. Note that each capillary is of circular cross section and that the radius of cross section does not vary within a capillary. Eq. (3.106) would represent the probability density function for the radius of cross section of a capillary because the scaling of the number of capillaries on the capillary radius would be identical. The symbols rmin and rmax would, respectively, denote the minimum radius of cross section of a capillary and the maximum radius of a capillary within the bundle of capillaries that together comprise the pore space. Note that the capillaries are the conceptualized flow tubes when a pressure gradient exists. There is the implicit assumption that macroscopically the flow behavior is isotropic. Apart from its radius of cross section, a capillary is also characterized by its axial length. The axis of any capillary would necessarily have to be tortuous. The reason for this, is the fact that the fluid flow-lines inside any region of the intergranular space would always be tortuous. The axis of a capillary is considered to be a rendering of a fractal at a certain scale. If ε denotes the “measure” by which the length le of the axis of a capillary is measured, and which will henceforth, simply be called as the length of a capillary, le would scale on ε as: le ðεÞ ¼ ε1Dt LDt

(3.109)

Here, it is imagined that we are looking at fluid flow through a sample of the rock whose end faces are separated by a distance L. The minimum value that le can possibly have is L. Dt is the fractal dimensionality of the capillary axis fractal Tyler and Wheatcroft (1988). The minimum possible value of Dt is 1.0 (case of all capillaries linear), and the maximum value of Dt possible is 2.0 since this value of the fractal dimension is the fractal dimension of the fractal “space filling curve” an example being the Hilbert Fractal (Tyler and Wheatcroft, 1988). There is no loss of generality in choosing ε ¼ r. The scaling of le over r would be Yu Cheng (2002): le ðrÞ ¼ r 1Dt LDt

(3.110)

148 Understanding Pore Space through Log Measurements

The tortuosity of a capillary would therefore scale as: sðrÞ ¼

le ¼ r 1Dt LDt 1 L

(3.111)

3.7.3 Permeability from the perspective of the fractal model of the pore space NðrÞ ¼ NPðrÞdr

(3.112)

Here, N stands for the total number of capillaries that cross unit area of a cross section. Since the probability density for the radius of cross section of a capillary is represented by a capillary,

 Df r ð1þDf Þ dr NðrÞ ¼ NDf rmin (3.113)

Df N ¼ rmax for r ¼ rmin because the full range of r is (rmin, rmax). Thus, r   rmax Df (3.114) N¼ rmin We have, by the HagenePoiseuille Law (1839), the following equation for the flow rate through a capillary of radius of cross section r and length le: qðr; le Þ ¼

p 4 DP r 8h le

(3.115)

Here, DP denotes the pressure-drop across the end faces of the sample through which fluid flow is taking place. Let Q and A, respectively, denote the total discharge rate of the fluid through the sample, and the area of an end face of the sample. Then, Z rmax ANPðrÞqðr; le Þdr (3.116) Q¼ rmin

Substituting for the relevant variables in the integrand in Eq. (3.115), from Eqs. (3.106), (3.114) and (3.115), respectively, carrying out the integration and rearranging:  Z rmax  rmax Df Df  ð1þDf Þ p 4 DP r A Df rmin r dr (3.117a) Q¼ 8h le rmin rmin

Pore attributes of conventional reservoirs Chapter j 3

       Q pDP Df rmin 3þDt Df 1Dt 3þDt ¼ rmax 1  L A 8hL 3 þ Dt  Df rmax Further, in view of condition (3.99) Eq. (3.117) simplifies to:     Q 1 p Df DP 1Dt 3þDt ¼ rmax L A h 8 3 þ Dt  Df L

149

(3.117)

(3.118)

Since the second term within the last pair of braces in Eq. (3.117) < < 1. Comparison of Eq. (3.118) with Darcy Law yields: 

p Df 3þDt k¼ (3.119) L1Dt rmax 8 3 þ Dt  Df From Eq. (3.110) we have, 1Dt Dt le ðrmax Þ ¼ rmax L

(3.120)

Further it is reasonable to assume that le(rmax) would be close to L in magnitude, and: 1Dt Dt 1 rmax L y1

From Eqs. (3.119) and (3.121): 

p Df 4 k¼ L1Dt rmax 8 3 þ Dt  Df

(3.121)

(3.122)

The deduction is presented in Chapter 9.

3.7.4 Cumulative pore volume The cumulative pore volume hosted by pores of radius lying within (r,rmax) is denoted by Vp(r,rmax) here. This is also equal to the cumulated volume of capillaries whose radius of cross section lies within (r,rmax). We have, Z rmax ANPðrÞpr 2 le dr (3.123) Vp ðr; rmax Þ ¼ r

Substituting the relevant variables in the integrand in Eq. (3.123), from Eqs. (3.114) and (3.106), respectively, carrying out the integration and rearranging:       Df 1 r 3Dt Df 2 Vp ðr; rmax Þ ¼ ApL 1  rmax Dt 1 1Dt rmax 3  Dt  Df rmax L (3.124) From Eq. (3.121) it is noted that the value of the quantity within the square braces approximates to 1.0. Therefore,

150 Understanding Pore Space through Log Measurements



Df Vp ðr; rmax Þ ¼ ApL 3  Dt  Df



 1

r

rmax

3Dt Df  2 rmax

(3.125)

The pore volume per unit rock volume hosted by pores of radius lying within (r, rmax) denoted as 4(r, rmax) is:      Df r 3Dt Df 2 (3.126) 4ðr; rmax Þ ¼ p 1 rmax rmax 3  Dt  Df The total porosity is given by      Df rmin 3Dt Df 2 4 ¼ 4ðrmin ; rmax Þ ¼ p 1 rmax 3  Dt  Df rmax

(3.127)

3.7.5 Representative hydraulic tortuosity and cumulated surface area to cumulated volume of the capillaries The tortuosity of a capillary, having a radius of cross section r and length le, respectively, is denoted as s(r). Here we are considering capillaries, which connect the two parallel end faces of a sample of the rock, separated by a distance L. sðrÞ ¼

le r 1Dt LDt ¼ ¼ r 1Dt LDt 1 L L

The representative hydraulic tortuosity of the capillaries (s) is: Z rmax Z rmax

 Df sðrÞPðrÞdr ¼ r 1Dt LDt 1 Df rmin s¼ r ð1þDf Þ dr rmin

(3.128)

(3.129)

rmin

The last equality in Eq. (3.129) comes from substituting for the appropriate quantities in the integrand of the integral in the first equality, from Eqs. (3.106) and (3.128), respectively. Evaluation of the integral leads to (details in Chapter 9),        1Dt  D 1 Df rmin Dt þDf 1 L t rmin s¼ (3.130) 1 Dt þ Df  1 rmax Substituting for rmin from Eq. (3.108) into Eq. (3.130) and simplifying:       1Dt  1D D 1  3D Df rmin Dt þDf 1 s¼ (3.131) rmax t L t 4 f 1  Dt þ Df  1 rmax The quantity within the square braces in the RHS of Eq. (3.131) can be replaced with unity, in view of Eq. (3.121). We then have,

Pore attributes of conventional reservoirs Chapter j 3

151



    1D  t Df rmin Dt þDf 1 3Df s¼ 1 4 Dt þ Df  1 rmax  Further, since

rmin rmax

(3.132)

  1:   1D t Df sy 43Df Dt þ Df  1

(3.133)

The ratio of the surface area of the connected pore volume to the value of the connected pore volume is now evaluated. This ratio is denoted as Spv. This ratio is evaluated as the ratio of the cumulative surface area of all the capillaries to the cumulative volume of those capillaries, as follows. Consider a sample of the rock whose end faces are separated by a distance L: R rmax NPðrÞLsðrÞ2prdr Spv ¼ rmin (3.134) Vp Substituting for the different quantities from Eqs. (3.106), (3.114), (3.125), and (3.133), evaluating the integral and simplifying:    Dt þ Df  3 1 Spv ¼ (3.135) Dt þ Df  2 rmax The relationship between the non-wetting fluid saturation and the intrusion pressure for a drainage process, is now discussed. Consider a drainage where the initial state is that of 100% saturation of wetting fluid within the pores of a rock of porosity 4. Consider a unit volume of the rock. Let PC and Snw denote the intrusion pressure (¼ capillary pressure or pore entry pressure for the narrowest capillary drained), and the nonwetting fluid saturation, at any stage of the intrusion process. Let, r1 denote the lowest value of radius among the pores drained by the nonwetting fluid at this stage. Since we are working with a capillary bundle model of the pore space, Eq. (3.43) would apply as: r1 ¼

2s cos q PC

(3.136)

From Eq. (3.136) we have, dPC 2s cos q ¼ dr1 r12 Snw ðPC Þ ¼

4ðr1 ; rmax Þ 4

(3.137)

(3.138)

152 Understanding Pore Space through Log Measurements

Using Eq. (3.126) for the numerator of RHS of Eq. (3.138) and substituting for 4 from Eq. (3.127) in the denominator of RHS of Eq. (3.138) and simplifying: 3D Df

Snw ðPC Þ ¼

rmax t

3D Df

rmax t

3Dt Df

 r1

(3.139)

3Dt Df

 rmin

Let PB denote the core entry pressure for a sample of the rock. This pressure is the minimum injection pressure at which a filament of nonwetting fluid stretching continuously between the end faces of the sample would form, when the initial saturation state of the sample is 100% wetting fluid saturation. In the context of this analysis this pressure would correspond to the entry pressure for the nonwetting fluid to drain the widest capillary. Hence, PB ¼

2s cos q rmax

(3.140)

Let PCmax denote the intrusion pressure at which all the capillaries would be drained by the nonwetting fluid. We have, PCmax ¼

2s cos q rmin

(3.141)

Using Eqs. (3.140) and (3.141), Eq. (3.139) is stated as, Snw ðPC Þ ¼

½3ðDt þDf Þ

PC

½3ðDt þDf Þ

 PB

(3.140a)

½3ðDt þDf Þ ½3ðDt þDf Þ PCmax  PB

Let Sw ðPC Þ ¼ 1  Snw ðPC Þ

(3.141a)

Eq. (3.140a) can be stated: 1  Sw ðPC Þ ¼

½3ðDt þDf Þ

PC

½3ðDt þDf Þ PCmax

½3ðDt þDf Þ

 PB 

(3.142)

½3ðDt þDf Þ PB

The isolated porosity in rock is the capillary tube volume per unit rock volume for capillaries whose entry pressure is too high. The fractional porosity corresponding to this pore space subset is commonly referred to as irreducible saturation (Swirr). Thus, ½3ðDt þDf Þ

1  Swirr ¼ lim ½1  Sw ðPC Þ ¼ bPB PC /N

;b¼h

1 ½3ðDt þDf Þ PB

½3ðDt þDf Þ  PCmax

i

(3.143)

Pore attributes of conventional reservoirs Chapter j 3

Eq. (3.142) can be written as,

 3 D þD  ½3ðDt þDf Þ ½ ð t f Þ 1  Sw ðPC Þ ¼ b PB  PC

153

(3.144)

From Eqs. (3.143) and (3.144):  ½3ðDt þDf Þ Sw ðPC Þ  Swirr PC ¼ 1  Swirr PB

(3.145)

  1 Sw ðPC Þ  Swirr ½3ðDt þDf Þ PC ¼ PB 1  Swirr

(3.146)

Rearranging:

Eq. (3.146) is a predictor of the Capillary PressureeWetting fluid saturation curve by the fractal model of pore space. When the tortuosity of the capillaries is not high, fractal dimension Dt z 1.0. In that case, the predictor runs on basically, the fractal dimension of the pore space connected with pore size (or radius of cross section of a capillary in the capillary bundle visualization of pore space) as,   1  1 Sw  Swirr 2Df ¼ PB Sw 2Df (3.147) PC y PB 1  Swirr Here, Sw which is also known as effective wetting fluid saturation is given by   Sw Swirr 1Swirr . Assuming that the irreducible water saturation is negligible, Eq. (3.147) can be converted to a saturation height function if the height above free water level is known. The value of PB can be obtained from NMR data if the hydrocarbon fluid phase is known. This is further discussed in Chapter 9. For the methodology, the reader can also refer to Dutta and Reddy (2015), Ramamoorthy et al. 2000. Forward modeling the saturation height function for best fit to the water saturation versus height above free water level, with the water saturation computed through the petrophysical inversion of log data, leads to the estimation of the value of Df. Eq. (3.147) resembles the BrookeseCorey equation (Brookes and Corey (1964) (Eq. (3.148) below) and provides a physical meaning to the pore size heterogeneity factor l as the fractal dimension of the pore radius fractal.  1 PC ¼ PB Sw l (3.148) The pore heterogeneity factor is the quantity (2Df). There is a further insight that the similarity between Eq. (3.147) and the BrookeseCorey equation is conditional on tortuosity being not too far away

154 Understanding Pore Space through Log Measurements

from unity in value. Since increased pore size heterogeneity also leads to increased tortuosity of the flow paths, there is an upper limit to the pore heterogeneity index below which alone, BrookeseCorey equation works. The themes discussed above are discussed in greater detail in Chapter 9.

3.7.5.1 BrookeseCorey permeability and the fractal model of pore space The BrookeseCorey equation is quite popular for modeling the intrusion process in a drainage cycle where the injected fluid phase is the nonwetting fluid phase. The permeability prediction based on this equation can be seen as follows. Let dSw denote the fraction of total porosity, of the cumulated volume of capillaries whose radius of cross section lies within (r, r þ dr) and let k(r) denote the permeability contribution of this set of capillaries. Let s(r) denote the tortuosity of these capillaries. kðrÞ ¼ Z

1

k¼ 0

Z kðrÞ4dSw ¼ 0

1

r2 8ðsðrÞÞ

r2

(3.149)

8ðsðrÞÞ2 Z

4dSw ¼ 2

0

1

r2 8ðsðrÞÞ

4dSw ¼ 2

4ð2s cos qÞ2 8s2

Z

1 0

dSw P2C

(3.150)

The irreducible saturation for the rock is neglected. Substituting for PC from the BrookeseCorey Eq. (3.148) for negligible irreducible saturation, in Eq. (3.150): k¼

  Z     lþ2 lþ2 4 2s cos q 2 Sw ðPB Þ 2=l 4 2s cos q 2 l  ½S ðP l S dS ¼ ðP Þ Þ ½S w w B w Cmax w 8s2 PB 8s2 PB Sw ðPCmax Þ (3.151)

Substituting for PB from Eq. (3.140) into Eq. (3.151) and since Sw ðPB Þ[ Sw ðPCmax Þ and Sw ðPB Þy1; Eq. (3.150) can be stated as, 4 2 k ¼ 2 rmax (3.150a) 8s Substituting s from Eq. (3.133) in Eq. (3.150a) and simplifying:  2  1þ2Dt Df  1 Df 2 (3.151a) 4 3Df rmax k¼ 8 Dt þ Df  1 Using Eq. (3.135), Eq. (3.151a) can be stated also as   2    t Df 1 Df Df þ Dt  3 2 1þ2D 3Df k¼ 4 S2 pv 8 Dt þ Df  1 Dt þ Df  2

(3.152)

Pore attributes of conventional reservoirs Chapter j 3

155

Both rmax and Spv can be obtained from the NMR data. Chapter 9 details   the methodology. By plotting the parameters r2k or kS2pv against 4 the values of certain max

known functions of Dt, Df are obtained. Relation between Dt, Df as: D

Dt ¼ 4  Df þ

f log Df 1 

log 4

3  Df



(3.153)

has been derived (Wei et al., 2015). Eq. (3.153) is derived in Chapter 9. One can obtain the value of Df, the fractal dimension of pore space, expressed as the dimension of the pore radius fractal, through an analysis of permeability data as well.

3.8 Electrical formation factor from the perspective of the fractal model of the pore space As in Appendix 1, we assume that the capillary tortuosity has a low spread in value over the population of the capillaries in a rock volume. We further assume that the LHS of Eq. (3.133) has a value close to s, the representative tortuosity value of a capillary. Referring to Eq. (A1.12) of Appendix 1 and substituting for s from Eq. (3.133) into Eq. (A1.12):  2 2Dt 1þDf s2 Df F¼ ¼ 4 3Df (3.154) 4 Dt þ Df  1 Eq. (3.154) gives some physical insight into the Archie parameters a and m. Eq. (3.154) also suggests that these two parameters are not mutually independent. For the case of porous reservoirs where the pore space is largely intergranular pore space, consider s2 ys (Ghanbarian et al., 2013). In that case, Eq. (3.154) approximates to:   ðDt Df þ2Þ s Df F¼ ¼ (3.155) 4 3Df 4 Dt þ Df  1 A more detailed discussion on Formation factor is available in Chapter 9.

Appendix 1 Relation between tortuosity (s), porosity (4), and formation factor (F) Consider a sample of rock with end faces at a distance L apart and with a face area A. Consider a fluid flow as a result of a pressure differential setup between

156 Understanding Pore Space through Log Measurements

the end faces. Let N(r)dr be the number of capillaries of radius of cross section within (r,r þ dr) which cross unit area of a plane of cross section for a bunch of capillaries. Let le(r) denote the tortuous length of a capillary segment of radius r when measured between the two end faces of the sample. Let s(r) denote the tortuosity of this capillary. Let sf denote the conductivity of the fluid saturating the sample. The conductance of the N(r)dr capillaries is denoted as C(r)dr. The fractional contribution of the cumulated volume of the N(r)dr number of capillaries to the porosity of the sample is denoted as d41(r,r þ dr). We have, CðrÞdr ¼ sf

pr 2 NðrÞdr pr 2 NðrÞle ðrÞdr ¼ sf le ðrÞ ½le ðrÞ2

d41 ðr; r þ drÞ ¼

pr 2 NðrÞle ðrÞdr AL

(A1.1)

(A1.2)

Let C denote the conductance of the sample. Z rmax Z rmax Z 4 pr 2 NðrÞle ðrÞdr AL C¼ CðrÞdr ¼ sf ¼ sf d41 ðr; r þ drÞ 2 2 ðrÞ ½l ½l 0 rmin rmin e e ðrÞ (A1.3) In Eq. (A1.3) the second equality comes from substituting for C(r)dr from Eq. (A1.1) into Eq. (A1.3). The last equality of Eq. (A1.3) comes by substituting d41(r,r þ dr) from Eq. (A1.2) into Eq. (A1.3). Eq. (A1.3) can be written as: Z A 4 1 sf d41 ðr; r þ drÞ (A1.4) C¼ L 0 ½le ðrÞ=L2 It is assumed that the case where the spread in the tortuosity value of a capillary, over the total population of capillaries within the sample is low enough for the following approximation to be valid: Z 4 2 Z 4 1 1 d4 d4 ðr; r þ drÞy ðr; r þ drÞ (A1.5) 1 1 2 0 ½le ðrÞ=L 0 ½le ðrÞ=L is the common case. When the spread in the tortuosity value of a capillary over the population of capillaries within the sample is low, the following also would be valid: 2 Z 4 1 12 d41 ðr; r þ drÞ y (A1.6) s 0 ½le ðrÞ=L

Pore attributes of conventional reservoirs Chapter j 3

157

Here, the quantity 1s stands for the representative value of the reciprocal of tortuosity of a capillary, and is defined: Z 1 1 4 1 ¼ d4 ðr; r þ drÞ (A1.7) s 4 0 ½le ðrÞ=L 1 The representative tortuosity (s) of a capillary for the set of capillaries within a sample of rock, in the capillary bundle model of pore space of the rock, is defined as: s¼

11 s

(A1.8)

Thus, Eq. (A1.4) can be restated: C ¼ sf

A4 Ls2

(A1.9)

Let s denote the conductivity of the rock and let F denote the electrical formation factor: C¼s

A sf A ¼ L F L

(A1.10)

From Eqs. (A1.9) and (A1.10): s2 ¼ F4 F¼

s2 4

(A1.11) (A1.12)

References Berg, C.F., 2014. Permeability description by characteristic length tortuosity constriction and porosity. Transport in Porous Media 103, 381e400. Brooks, R.H., Corey, A.T., 1964. Hydraulic Properties of Porous Media. Hydrology Paper No. 3. Colorado State University. Coalson, E.B., Goolsby, S.M., Franklin, M.H., 1994. Subtle seals and fluid-flow barriers in carbonate rocks. In: Dolson, J.C., Hendricks, M.L., Wescott, W.A. (Eds.), Unconformity Related Hydrocarbons in Sedimentary Sequences: Rocky Mountain Association of Geologists Guidebook for Petroleum Exploration and Exploitation in Clastic and Carbonate Sediments, pp. 45e58. Coates, G.R., Dumanoir, J.L., 1974. A new approach to improved log derived permeability. The Log Analyst XV, 17e29. Dutta, R., Reddy, B., 2015. Permeability and Saturation Prediction Using NMR and Other Logs- A Clastics Case Study. GEO India Conference and Exhibition. Ghanbarian, B., Hunt, A.G., Ewing, R.P., Sahimi, M., 2013. Tortuosity in porous media: a critical review. Soil Science Society of America Journal 77, 1461e1477.

158 Understanding Pore Space through Log Measurements Hagen, G.H.L., 1839. Uber die Bewegung des Wassers in Engencylindrischen Rohren. Poggendorf’s, Annalen der Physik und Chemie 46, 423e442. Hartmann, D.J., Beaumont, E.A., Coalson, E., 1999. Chapter e Predicting reservoir quality and performance, Search and Discovery Article #40005, adapted from Chapter 9, Predicting Reservoir System Quality and Performance, by Dan J. Hartmann and Edward A. Beaumont. In: Beaumont, Edward A., Foster, Norman H. (Eds.), Exploring for Oil and Gas Traps, Treatise of Petroleum Geology, Handbook of Petroleum Geology. Johnson, D.L., et al., 1986. New pore-size parameter characterizing transport in porous media. Physical Review Letters 57, 2564e2567. Kashif, M., et al., 2019. Pore size distribution, their geometry and connectivity in deeply buried Paleogene Es1 sandstone reservoir, Nanpu Sag, East China. Petroleum Science 16, 981e1000. Muller-Huber, E., Schon, J., Borner, F., 2016. A pore body-pore throat-based capillary approach for NMR interpretation in carbonate rocks using the Coates equation. In: Paper Presented at the SPWLA 57th Annual Logging Symposium, Reykjavik, Iceland, Paper SPWLA-2016-H. Ramamoorthy, R., Boult, P.J., Neville, T.J., 2000. A novel application of nuclear magnetic resonance and formation tester data for the determination of gas saturation in the Pretty Hill Sandstone reservoirs, onshore, Otway Basin. Abs., AAPG international conference Bali October 2000. AAPG Bulletin 84, 1395e1518. Tyler, S.W., Wheatcraft, S.W., 1988. An exploration of scale dependent dispersivity in heterogeneous acquifers using concepts of fractal geometry. Water Resources Research 24, 566e578. Washburn, E.W., 1921. The dynamics of capillary flow. Physical Review 17, 273e283. Wei, W., Cai, J., Hu, X., Han, Q., 2015. An electrical conductivity model for fractal porous media. Geophysical Research Letters 42, 4833e4840. Xu, H., et al., 2019. Characterization of pore throat size distribution in tight sandstones with nuclear magnetic resonance and high-pressure mercury intrusion. Energies 12, 1528. Yu, B., Cheng, P., 2002. A fractal permeability model for bi-dispersed porous media. International Journal of Heat and Mass Transfer 45, 2983e2993. Yu, B.M., Li, J.H., 2001. Some fractal characters of porous media. Fractals 9, 365e372.

Further reading Archie, G.H., 1942. The electrical resistivity log as an aid in determining some reservoir characteristics. Transactions of the AIME 146, 54e62. Berg, R.R., 1970. Method for determining permeability from reservoir rock Properties’. Transactions of the Gulf Coast Association of Geological Societies 20, 303e317. Carman, P.C., 1937. Fluid flow through a granular bed. Transactions of the Institution of Chemical Engineers 15, 150e167. Coates, G.R., Corp, N., Peveraro, R.C.A., Hardwick, A., Roberts, D., 1991. The magnetic Resonance Imaging Log characterized by comparison with petrophysical properties and laboratory core data. In: Proceedings of the 66th Annual Technical Conference and Exhibition, Formation Evaluation and Reservoir Geology, Society of Petroleum Engineers SPE 22723. Coates, G.R., Xiao, L., Prammer, M.G., 1999. NMR Logging: Principles and Applications. Haliburton Energy Services Houston, Houston, TX, USA. Darcy, H., 1856. Fontainespubliques de la ville de Dijon, Librairie des Corps Impe´riaux des Ponts et Chausse´es et des Mines. Kostek, S., Schwartz, L.M., Johnson, D.L., 1992. Fluid permeability in porous media: comparison of electrical estimates with hydrodynamical Calculations. Physical Review B 45, 186e195.

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Li, K., 2004a. Theoretical Development of the Brooks-Corey Capillary Pressure Model from Fractal Modeling of Porous Media, pp. 1e6. SPE 89429. Li, K., 2004b. Characterization of Rock Heterogeneity Using Fractal Geometry, pp. 1e7. SPE 86975. Mao, Z.-Q., Wang, Z.X.-N., Jin, Y., Liu, X-g, Xie, B., 2012. Estimation of permeability by integrating nuclear magnetic resonance (NMR) logs with mercury injection capillary pressure (MICP) data in tight gas sands. Applied Magnetic Resonance 44, 449e468. Marschall, D., Gardener, J.S., Mardon, D., Coates, G.F., 1995. Method for correlating NMR relaxometry and mercury injection data. In: Proceedings of the Transactions of the 1995 Symposium SCA San Francisco, CA USA, Paper 9511. Schiller, L., 1933. Dre; Klassiker der Stromungslehre: Hagen, Poiseuille, Hagenbach. Akad. Verlagsgesellschaft, Leipzig, p. 1933. Seevers, D.O., 1966. A nuclear magnetic method of determining the permeability of sandstones. In: Trans. SPWLA 7th Annual Logging Symposium. Sheng, P., 1990. Effective medium theory of sedimentary rocks. Physical Review B 41. Tyler, S.W., Wheatcraft, S.W., 1990. Fractal processes in soil water retention. Water Resource Research 26, 1047e1054. Van Baaren, J.P., 1979. Quick-look permeability estimates using sidewall samples and porosity logs. In: Transactions of the 6th Annual European Logging Symposium, vol. 19. Society of Petrophysicists and Well Log Analysts. Volokitin, Y., Looyestijn, W., Slijkerman, W., Hofman, J.P., 2001. A practical approach to obtain primary drainage capillary pressure curves from NMR core and log data. Petrophysics 42 (4), 334e343. Wang, F., Yang, K., You, J., Lei, X., 2019. Analysis of pore size distribution in tight sandstone with mercury intrusion porosimetry. Results in Physics 13. Whitaker, S., 1986. Flow in porous media i: a theoretical derivation of Darcy’s law. Transport in Porous Media 1, 3e25. Wu, J.S., Yu, B.M., 2007. A fractal resistance model for flow through porous media. International Journal of Heat and Mass Transfer 50, 3925e3932. Yu, B.M., Li, H.M., 2004. A geometry model for tortuosity of flow path in porous media. Chinese Physics Letters 21, 1569e1571.

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Chapter 4

Pore space attributes of nonconventional reservoirs 4.1 CBM reservoirs 4.1.1 The components of the space occupied by fluids in coals The space occupied by fluids within coals comprises (i) cleats and fractures other than cleats, (ii) adsorption pores (methane is adsorbed and stored within these pores), and (iii) seepage pores. Through the seepage pores, diffusion and permeation of methane occur within these pores, which also support fluid transport against pressure gradient. Seepage pores are the main conduits for gas diffusion and permeation within a coal. They are the locations of water and free gas transportable within a coal. Seepage pores are also known as seepage holes. Pore classification of Hodot (Hodot, 1966) considers cleats also as a part of seepage pore system. Cleats are naturally occurring opening-mode fractures within coal seams. They form two mutually perpendicular sets of fractures, each set being perpendicular to the bedding plane. These sets are called as face cleats and butt cleats (Kendall and Briggs, 1933; Laubach et al., 1998). Face cleats are prominent and through-going fractures formed during the initial stages of coalification itself, while butt cleats are developed later. Butt cleats terminate orthogonally against face cleats (Kulander and Dean, 1993; Laubach et al., 1991; Tremain et al., 1991; Ammosov and Eramin, 1963; Busse et al., 2017; Ting, 1977). Butt cleats are considered as formed to accommodate relaxation of the stress which originally formed face cleats. This is the reason for the fact that butt cleats terminate against face cleats perpendicularly (Golab et al., 2013). In this chapter, the term “fracture’ is used to mean exclusively those fractures which are not cleats.

Understanding Pore Space through Log Measurements. https://doi.org/10.1016/B978-0-444-64169-4.00001-8 Copyright © 2022 Elsevier B.V. All rights reserved.

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162 Understanding Pore Space through Log Measurements

4.1.1.1 Hodot’s pore classification scheme As per Hodot’s pore classification scheme (Hodot, 1996), pores that occur within a coal can be divided into the following four categories: (i) micropores (size 1.0 microns size). Transition pores and micropores comprise of adsorption pores. Mesopores and macropores comprise of seepage pores. Cleats are also considered as a part of seepage pore assemblage. Gas molecules can be present in the adsorption pores either in the adsorbed or in the free state. The populations of free gas and adsorbed gas molecules occur in fact in a dynamic equilibrium. The timescale of exchange between the two populations because of gas molecules getting adsorbed or adsorbed gas molecules becoming free gas molecules is too small compared to the timescales involved even in logging techniques, e.g., in the NMR logging. Adsorbed gas relaxes rapidly through surface relaxation whereas this is not the case for free gas. However, because the timescale transition of a free gas molecule to the adsorbed state or vice versa is too small compared to the timescales involved in NMR, there is a large overlap in the T2 distributions of the adsorbed and free gas populations, which renders it difficult to differentiate the two populations on a transverse relaxation spectrum. Similar remarks also apply for the longitudinal relaxation times of the two populations mentioned above. Space occupied by fluids within coals can be as (i) micropores (1.0 micron). There are only two techniques for the characterization of the fluid-occupied volume within coals in terms of pore classes. These are (i) high-resolution electrical images and (ii) NMR logs. Out of them, high-resolution electrical images can only differentiate macropores of size exceeding hundreds of microns, cleats, and fractures. NMR can partition the fluid-occupied volume within coals into micropores, mesopores, and macropores, with cleat volume a part of the macropore class. This chapter uses the term “pore space of coal’ to mean the total fluidoccupied space (volume) within a coal body. 4.1.2 Characterization of the pore space of coals e cleats and fractures that are not cleats The minimum log suite required for a useful characterization of the pore space of coals would include conventional nuclear and resistivity suite, natural gamma ray spectrometry logs, elemental capture gamma ray spectrometry logs, NMR logs, and high-resolution electrical images.

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The characterization primarily deals with the computation of cleat aperture, noncleat fracture aperture, cleat density, the representative length of a cleat, cleat orientation cleat surface area, micropore porosity fraction, micropore surface area, and the geometric relation of the cleats with respect to the principal horizontal stress directions. Along with cleat aperture, cleat density is another important parameter concerning the characterization of CBM reservoirs.

4.1.2.1 Extraction of fractures and cleats from borehole images 4.1.2.1.1 The process for low-angle events The only logs which enable the extraction of the orientation in space of cleats and thus their geometrical relation to important directions such as the principal horizontal stress directions are the borehole image logs. Out of these, highresolution electrical image logs offer the best option of accurate characterization of the orientation in space of the cleats. Electrical images have data gaps because borehole coverage is not 100%. As a preprocessing step the gaps in the images are filled up. A tried and tested method is illustrated at Section 10.3.3 of Chapter 10 of this book. The fullborehole image is then rescaled using MSFL or LLS or medium induction data or equivalent in case MSFL data are not available. The result is a conductivity map of the borehole wall. From the original high-resolution images, information about the geometry of the surfaces, which are unlikely to be fracture/cleats, is obtained. Using this information, the set of high contrast traces/segments is filtered to retain only those potential traces (regions on the image that share connectedness and pixel intensity above a given threshold) that can be interpreted with confidence as parts of a cleat/fracture event. The regions of the conductivity map, which contain these pixels of the traces, are used to compute the cleat/fracture aperture (discussed in the next section). A process which achieves the above goals, that is well tested in the Kherroubi’s work flow (Kherroubi, 2008), is discussed at Appendix 1. The input set of traces based on which the pixels analyzed for cleat aperture depending on button current output and cleat density is discussed in “Obtaining the final cleat/fracture traces of high confidence,’ at Appendix 1. The best-fit sinusoids are now found for the group of pixels in the final conductivity map. These sinusoids carry the information of the dip and strike of the cleats/fractures. This part of the work flow is also discussed in Appendix 1. In order not to miss the high-angle as well as the low-angle fracture/cleats, the abovementioned process is carried out on original conductivity map as well as on the conductivity map rotated 90
with respect to the original conductivity map.

164 Understanding Pore Space through Log Measurements

Cleats and natural fractures are easily distinguished because cleats are fractures that commonly occur perpendicular to or at very high angles to the coal bedding. Hence these are easily identified from images and labeled. Borehole induced fractures are oriented along the present-day maximum stress direction and are easily distinguished from cleats, on images. From image analysis it is possible to differentiate primary and secondary fracture sets (face cleats and butt cleats). 4.1.2.1.2 The process for high-angle events The work flow resembles that used for extracting the low apparent dip events. The initial image in this case, however, is rotated at the start, by 90
, before the work flow is applied on it, and the same vertical structuring element used during the extraction of low apparent dip fracture segments is used.

4.1.2.2 Computation of fracture/cleat aperture From the results obtained as above, it is clear that the essential pixel location for each pixel on a fracture/cleat trace is available. This information can be used to obtain the changes in the normalized button current in the neighborhood of the fracture/cleat traces. By the term normalized button current, is meant the button current map, post the process of normalization. The process of normalization mentioned above is the process applied on the original button current data, to (i) cancel out the calibration and other instrument-related differences between the button currents and (ii) effect a scaling with respect to Rxo as given by MSFL log, for instance, so that the apparent resistivity from the button current scales to Rxo as discussed above. The true width of a cleat/fracture is much less than the button size. The typical values for cleats range 0.01e0.20 mm. When a button approaches a fracture/cleat, the button current increases when the edge of the button is close to the fracture/cleat. It continues to show an increased value over the background even after the button clears the fracture/cleat entirely. The effect of this is that a cleat with width 10 ms for the oil saturated case is also differentiated on the T2 distribution. The differentiation of pore types, even on conventional 1D inversion of echo data into a T2 distribution as per Fig. 4.13, is characteristic of shale reservoirs bearing oil. Table 4.2 brings out the ranges of the relevant parameters, which bring out the above and also can be taken as guidelines while partitioning the pore space of shale reservoirs bearing oil using the NMR data.

FIGURE 4.12 T1T2 plot of an oil resaturated shale sample. The color is a measure of the hydrogen index. The set of contours corresponding to T210 ms correspond to oil present within inorganic pores. Reproduced from Kausik et al. (2014).

Pore space attributes of nonconventional reservoirs Chapter j 4

191

FIGURE 4.13 Plots of conventional T2 distributions of native shale (in black), oil resaturated shale (in green; light gray in print version), and brine resaturated shale (in blue; dark gray in print version). Reproduced from Kausik et al. (2014).

TABLE 4.2 Ranges of T2, T1/T2 for different injected fluids as against different pore types. Injected oil: Organic pores

Injected oil: Inorganic pores

Injected water: Organic and Inorganic pores

Bitumen and Clay-bound water

T2

1 ms510 ms

> 10 ms

6 ms580 ms

< 1:5 ms

T1/T2

358

152

1:352:5

6515

Reproduced from Kausik et al. (2014).

4.2.6 The method of diffusion editing Diffusion editing deploys a special mode of acquisition for acquiring echo amplitudes for multidimensional inversion. In this case, the echo spacing between the first and second echo and the echo spacing between the second and the third echo are kept long. This echo spacing parameter is called as the long echo spacing (te,l). For the succeeding echoes, the echo spacing (te,s) is short. Diffusion attributes are embedded in the decay of the first echoes and these drive the amplitude of echoes which succeed them. The echo sequence is shown at Fig. 4.14.

192 Understanding Pore Space through Log Measurements

FIGURE 4.14 An example of diffusion editing suites deployed for multidimensional inversion of echo data. The thin black pulse is the 90
spin-flipping RF pulse and the thicker black pulses are the 180
spin-flipping RF pulses. Reproduced from Freedman et al. (2003).

The analysis given here is after Freedman et al. (2003). The analysis assumes water and oil as the fluids present within rock. In principle, the analysis is valid for water and gas being the fluids present within rock, with the appropriate inputs. The decay of the transverse magnetization M with time t, for an oilewater fluid pair assumed as present within the rock, and for the case of the echo sequence (Fig. 4.14), is given by: For t > 2te,l: "
 X ZZZ

t dDdT1 dT2 Pf ðD; T1 ; T2 Þexp M te;l ; te;s ; TW ; t ¼ T2 f ¼o;w ( ! !) 3 3 g2 G2 Dte;l g2 G2 Dte;l ad exp þ as exp exp 6 3 !#
!
 2 2 g2 G2 Dte;l te;s g2 G2 Dte;s t TW 1  exp exp 12 6 T1 (4.36) (After Freedman et al., 2003). Here, o, w, respectively, denote oil and water. The expression within the curly braces arises because the diffusion decay of transverse magnetization in an inhomogeneous magnetic field is

Pore space attributes of nonconventional reservoirs Chapter j 4

193

biexponential (Hu¨rlimann & Venkataramanan, 2002). If we assume only direct echoes, the magnetization at the end of the second echo would be: "
 ZZZ

X 2te;l M te;l ; te;s ; TW ; t ¼ 2te;l ¼ dDdT1 dT2 Pf ðD; T1 ; T2 Þexp T2 f ¼o;w ( !)#



2 2te;l g2 G2 Dte;l TW exp 1  exp T1 12 (4.37) (After Hu¨rlimann & Venkataramanan, 2002). The magnetization for t > 2te,l: Mðte;l ; te;s ; TW ; t ¼ 2te;l Þ ¼ "

!)#
( 2 2t X ZZZ g2 G2 Dte;l 2te;l e;l dDdT1 dT2 Pf ðD; T1 ; T2 Þexp exp 12 T2 f ¼o;w


! 

 2 t g2 G2 Dte;s  t  2te;l TW 1  exp exp exp ; 12 T1 T2

which simplifies to:


 ZZZ

X t dDdT1 dT2 Pf ðD; T1 ; T2 Þexp M te;l ; te;s ; TW ; t ¼ T2 f ¼o;w

(

! !)

2 2 te;l g2 G2 Dte;l g2 G2 Dte;s t TW (4.38) exp exp 1  exp 12 T1 6

(After Hu¨rlimann & Venkataramanan, 2002). Eq. (4.38) is same as Eq. (4.36) for the case ad ¼ 1, as ¼ 0. Fort2te,l: X ZZZ

dDdT1 dT2 M te;l ; te;s ; TW ; t ¼ f ¼o;w

"

!#

  2 g2 G2 Dw te;s t t TW 1  exp Pf ðD; T1 ; T2 Þexp exp T2 12 T1


(4.39)

(After Hu¨rlimann & Venkataramanan, 2002). The biexponential decay of magnetization in the transverse plane due to diffusion in an inhomogeneous magnetic field occurs since, in addition to onresonance components, contribution of stimulated echoes from the offresonance components also needs to be considered. Stimulated echoes decay at a rate which is double the rate at which direct echoes do. Therefore, the

194 Understanding Pore Space through Log Measurements

diffusion-controlled echo amplitude decay has two components. One is the conventional component coming from the direct echoes. The other component is the one that arises from the stimulated echo contribution. This contribution can be quite significant. The diffusion-related term hence comes within the curly braces in Eq. (4.36). The direct and stimulated echo coefficients are as and ad appearing in the diffusion term. The denominator of the second term within the curly braces of Eq. (4.36) is half of that of the denominator of first term within the curly braces. This is because the decay rate of stimulated echoes is double that of the direct echoes. The values of as and ad are a function of the bandwidth of the receiver of the NMR logging tool. They are found out by fitting the echoes acquired against a water sample, to the model. Eqs. (4.36) and (4.39) form the forward model used to invert echo trains to the model Pf(D,T1,T2) that comprises the hydrogen index for both the water and oil bins for the case of zero noise. The realistic forward model would also comprise noise term. The inversion is implemented by writing the forward model as a matrix equation: M ¼ LPf ðD; T1 ; T2 Þ þ ε

(4.40)

(After Hu¨rlimann & Venkataramanan, 2002). M is the column vector formed by concatenating the echo amplitudes sets of different echo trains. Pf(D,T1,T2) is the column vector of the bin weights (the hydrogen index associated with a bin is called as the bin weight of that bin). The elements of this vector are the unknowns to be determined. ε is the column vector of noise associated with the echoes. L is the matrix of the kernels which arise by virtue of the forward model mentioned above. Considering an element of M which can be the ith echo of the jth echo train, which for clarity is referred to as Mij and the noise associated as εij:

 " XXX tij t Pf ðD; T1 ; T2 Þexp Mij ¼ Pf ðD; T1 ; T2 Þexp T2 T2 D T1 T2 (

! !) ! 3 3 2 g2 G2 Dte;l g2 G2 Dte;l g2 G2 Dte;l te;s ad exp þ as exp exp 6 3 6 2 g2 G2 Dte;s tij exp 12

!#
1  exp


TWj þ εij T1

(4.41)

(After Hu¨rlimann & Venkataramanan, 2002). For tij>2te,l where tij is the time counted from the first 90
spin-flip pulse for the echo train to which the echo belongs. For tij2te,l, Mij ¼

" XXX D

T1

T2

!#
 

2 g2 G2 Dte;s tij tij TWj exp þ εij 1  exp Pf ðD; T1 ; T2 Þexp T2 12 T1 (4.42)

(After Hu¨rlimann & Venkataramanan, 2002).

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195

Eqs. (4.36) and (4.38) are the basis for building the kernel matrix L. Eqs. (4.41) and (4.42) illustrate the process how the kernels arise. The inversion of Eq. (4.40) can be implemented by SVD, or through the process of regularization. A method of regularization used is the Tikhanov Regularization. The process would involve the minimization of:

2

2 Minimize M  LPf ðD; T1 ; T2 Þ þ f Pf ðD; T1 ; T2 Þ subject to the constraint; that elements of Pf ðD; T1 ; T2 Þ are nonnegative: (4.42a) Here, k:k stands for the Euclidian norm of the matrix within the k:k. The parameter f is the regularization parameter. The retrieved model of Pf(D,T1,T2) can be used to find the porosity contribution of different pore types. This is because the oil resides mostly within the kerogen pores with some additional oil present within the inorganic pores, while water is present mainly as clay-bound water and as capillary bound water within the inorganic pores. The method discussed in the foregoing is also applied in the gas shale reservoirs. The method results in estimating the total gas hydrogen index and the total water hydrogen index components. Complexity arises since the only part of the gas is present as free gas and the remaining part is present as the adsorbed gas. While it is true that free gas mostly resides within the organic pores, unless the volume of free gas is found out, it is not possible to have estimates of the organic pore space. From the retrieved model of Pf(D,T1,T2) of gas it is possible to find out the apparent diffusion coefficient, the value of the diffusion-free T2 of the gas, and the longitudinal relaxation time of gas, all under formation conditions. There are methods of partitioning the total hydrogen index associated with gas into that associated with the free gas and with the adsorbed gas fractions, respectively, using only NMR data. A more widely used method is to estimate the adsorbed gas content at formation conditions, based on the kerogen content within the rock. The kerogen content in turn is computable through the petrophysical inversion of logs (Appendix 3). A method of partitioning the amount of gas into free and adsorbed fractions using NMR data is briefly discussed further in Appendix 3. We now discuss the method of Laplace Inversion with regularization which is used to invert echo data to T1T2 bins.

4.2.7 The method of Laplace Inversion with regularization Here the echo amplitudes of different echo trains acquired through CPMG cycles of short interecho spacing and variable wait times are considered. The echo spacing is kept short enough for diffusion decay of the transverse

196 Understanding Pore Space through Log Measurements

magnetization to be minimal. Let b(t,TW) denote the echo amplitude in terms of time and wait time. Let F(T1,T2) denote the distribution of hydrogen index over (T1,T2). We have,


 ZZ t TW dT1 dT2 FðT1 ; T2 Þexp bðt; TW Þ ¼ 1  exp þ εnoise T2 T1 (4.42b) (After Song and Kausik, 2019 and after taking into account longitudinal relaxation and noise).Here, ε stands for noise. The discrete form of equation is


j¼m X r¼n X

tr TW F T1r ; T2j exp (4.42c) 1  exp þ εik bik ¼ T2j T1k j¼1 r¼1 (After Song and Kausik, 2019 and after taking into account longitudinal relaxation and noise).Here, T1 has been divided into n number of classes and T2 has been divided into m number of classes, leading to mn number of bins. F(T1r,T2j) stands for the hydrogen index associated with bin (T1r,T2j). Concatenating the echo train vectors into a column vector S and the kernels organized as a 2D matrix denoted as K and organizing the bin-hydrogen index values with the bins indexed and ordered, as a column vector F and εik for different echo trains concatenated as a column vector ε, we can state the preceding equations as a matrix equation: S ¼ KF þ ε

(4.43)

(After Song et al., 2002). To invert Eq. (4.43), note that Eq. (4.42b) is in the form of a double Laplace Transform. The solution for F satisfying the nonnegativity constraint (every component of the solution vector should be  0) is obtained as: Minimize kS  KFk2 þ fkFk2 subject to the nonnegativity constraint (4.44) (After Song et al., 2002). Here, k:k stands for the Euclidian norm of the matrix within the k:k. f is a regularization parameter, and the above is a case of Tikhanov Regularization. Also see Song et al. (2002). Please note that the regularization parameter f is the regularization parameter relevant to Tikhanov Regularization and is different from the regularization parameter l defined elsewhere (in Appendix 1) in a different context.

Pore space attributes of nonconventional reservoirs Chapter j 4

197

4.2.8 D-T2 plots (more familiarly known as D-T2 maps) e forward models Shale oil and shale gas reservoirs come under the long-time regime from the NMR perspective. In such a regime, the diffusion length LD for the diffusionencoding time is much larger than the pore dimension (or more precisely, the characteristic length scale of the pore assemblage). The diffusion-encoding time TD is nothing but the interecho spacing in the case of the conventional CPMG cycle. The diffusion-encoding time TD is related to the long echo spacing time for the case of diffusion editing. It is to be noted that here, the diffusion length LD refers to the diffusion length for the case of unrestricted diffusion. It is also to be noted that the small dimensions of the pores within shale reservoir imply significant Knudsen diffusion rather than molecular or Fick diffusion. The diffusion-encoding time (TD) is given by: pffiffiffiffiffi2 TD ¼ te;l (4.45) pffiffiffiffiffiffi Here te;l stands for the mean value of the square roots of the long echo spacing values used during the activation. pffiffiffiffiffiffiffiffiffiffiffi LD ¼ D0 TD (4.46) The DT2 relation for water (Hu¨rlimann et al., 1994) is: 4 1 a1 ¼ pffiffiffi 9 p T2S reffw g1 ¼ 1 

D0w DNw

DNw ¼ D0w 4m1 (The value of DNw ¼ reffw ¼

(4.47)

D0w s

(4.48) ¼

D0w F4

Sgrainwater r Sgrainwater þ Sgrainoil

¼ D0w 4m1 ) (4.49)

Sxy stands for the area of the interface between phases x,y. The subscript 0 denotes unrestricted diffusion, subscript N indicates the long time limit. 4 stands for water volume. m stands for the Archie cementation exponent.
2 a1 LDw þ LLDw M DðT2 Þ ¼ g1 (4.50)
2 LDw a1 LDw þ LM þ g1 Here LDw ¼ For oil,

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi D0w TD and LM is the heterogeneity length scale of the medium.

198 Understanding Pore Space through Log Measurements

4 1 a1oil ¼ pffiffiffi 9 p T2S reffoil g1oil ¼ 1 

(4.51)

D0oil DNoil

(4.52)


 D0oil D0oil ¼ ¼ D0oil 4m1 DNoil ¼ D0oil 4m1 The value of DNoil ¼ (4.53) s F4 reffoil

Sgrainoil r Sgrainwater þ Sgrainoil

4 now stands for oil volume.

DðT2 Þ ¼ g1oil




(4.54)

2

a1oil LDoil þ
2 a1oil LDoil þ LLDoil þ g1oil M LDoil LM

(4.55)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here, LDoil ¼ D0oil TD and LM is the heterogeneity length scale of the medium. These forward models are the basis of the overlays given for the DT2 maps, and these models allow for the identification of the different fluid volumes within the shale reservoir rock. The abovementioned equations (Eqs. 4.45e4.55) are an example of nonconventional 2D NMR model. The reference for the abovementioned equations is Hu¨rlimann et al. (1994).

4.2.8.1 Gas diffusion and the role played by adsorption of gas in the modeling The gas diffusion line is modeled with the DNgas ¼ D0gas4m1 with 4 now standing for the gas volume. The value of reffgas is taken to be equal to the value of r of kerogen. The movement of the adsorbed gas molecules is through the mechanism of thermally activated hopping. This is the mechanism for the surface diffusion of the adsorbed gas molecules. When an adsorbed molecule acquires sufficient energy to overcome the surface potential barrier, it escapes from the surface and becomes a free gas molecule. The converse process is also prevalent and at any instant there is a dynamic equilibrium between the respective populations of the free gas and adsorbed gas molecules. The transition time is of the order of picoseconds for the case of shale gas reservoirs where kerogen is the adsorbent. If we consider the free gas molecules, when moleculeemolecule collisions dominate the moleculeewall collisions the diffusion is Fick diffusion. Fick

Pore space attributes of nonconventional reservoirs Chapter j 4

199

diffusion is identical to the bulk diffusion. On the other hand, when the moleculeewall collisions dominate the moleculeemolecule collisions, the diffusion is called as the Knudsen diffusion. In the case of shale gas reservoirs the pore size is much smaller than the mean free path of a gas molecule for the reservoir temperature and pressure conditions. Therefore, the restricted diffusion case of the Fick as well as the Knudsen diffusion leads to two diffusion constants. These are, respectively, Db the conventional bulk diffusion constant, and Dk, which governs Knudsen Diffusion. These two diffusion constants are, respectively, given by the equations: Db ¼

Db0 sb

(4.56)

Dk ¼

Dk0 sk

(4.57)

(After Kausik et al., 2011)

(After Kausik et al., 2011). The free gas diffusion coefficient is denoted here as Dgas. Dgas is related to Db and Dk as: 1 1 1 ¼ þ Dgas Db Dk

(4.58)

(After Kausik et al., 2011). The apparent gas diffusion coefficient coming from the echo data inversion and presented on the DT2 maps is denoted here as Deffective. Its magnitude is a function of the exchange timescale of the free gas to the adsorbed gas states, and the fractions of the free gas and adsorbed gas phases within the total gas. The diffusion coefficient for adsorbed gas is denoted as Dads, and the fraction of the number of adsorbed gas molecules of the total number of gas molecules as ε1 here. Deffective is related to Dgas and Dabs as: Deffective ¼ ð1  ε1 ÞDgas þ ε1 Dads

(4.59)

(After Kausik et al., 2011). The gas diffusion line D(T2) can be modeled as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1gas Deffective TD þ DðT2 Þ ¼ g1gas

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1gas Deffective TD þ

Here, a1gas is given by,

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 Deffective TD LM

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 Deffective TD LM

(4.60) þ g1gas

200 Understanding Pore Space through Log Measurements

4 1 a1gas ¼ pffiffiffi 9 p T2S reffgas

(4.61)

The value of reffgas is to be taken as the relaxivity of kerogen for the case of shale gas reservoirs. However, the relaxivity of kerogen is a function of the adsorbed gas amount and the internal structure at the surface according to which the adsorbed gas occurs (monomolecular layer or double molecular layer, etc.). Forward models are prepared for different values of the kerogen relaxivity. The value of ε1 is also not known per se. This value is obtained by simulating the methane gas adsorption at formation temperature and pressure in the lab using core plugs of the shale gas reservoir.

T1, T2 are relaxivity for gas shales. 4.2.8.2 Relaxation of bulk gas In bulk gas the mechanism of transverse relaxation or longitudinal relaxation of the magnetization is the spin rotation mechanism. When a molecule rotates, the rotating electron cloud generates a variable magnetic field at the location of the protons present within another molecule which is sufficiently proximal to the first molecule, and vice versa. This perturbation of the magnetic field causes relaxation (Abragam, 1961). The mechanism is known as the spin rotation mechanism. For bulk gas:   2 2 2 C þ C t I 1 sF k 1 1 ¼ ¼ kT (4.62) 2 T1 T2 3Z (Kausik et al., 2011).Here k and T are, respectively, the Boltzmann Constant and the absolute temperature, I1 is the moment of inertia of a molecule assumed to be spherical in shape, sF is the correlation time of rotation, and Ck ; Ct are, respectively, the principal components of the spin rotation tensor (Hubbard, 1963). sF is given by: sF ¼

3I1 D 4a2 kT

(4.63)

Here a is the radius of a gas molecule, and D is the diffusion coefficient of bulk gas. D¼

kT 6pah

(4.64)

(EinsteineStokes relation) where h stands for the viscosity of the gas. From Eqs. (4.63) and (4.64): sF ¼

3I1 8pha3

(4.65)

Pore space attributes of nonconventional reservoirs Chapter j 4

201

The spin rotation mechanism leading to Eq. (4.62) is impaired by increase in gas pressure. The reason for this is the fact that as the pressure increases, the frequency of collision between gas molecules increases, which disturbs the molecular rotations. Thus, relaxation takes longer time at higher temperatures than that at lower temperatures, because, as temperature increases, pressure increases.

4.2.8.3 Relaxation of gas within organic pores When the relaxation of gas within organic pores is considered, it is found that the dominant relaxation mechanism for free gas protons is no longer spin rotation, but the dipolar intermolecular interactionedriven mechanism of spin relaxation. The relaxation in this case is called as the dipolar relaxation. The partners in the interaction are the free gas molecules, and molecules/ions other than free gas molecules. Relaxation of free gas proton spins happens in the presence of the adsorbed gas molecules (gas molecules adsorbed into the surface of organic pores). A free gas molecule has a magnetic dipole moment. The magnetic field due to this dipole is a fluctuating field because of the thermal motion of the molecule. The surface of kerogen has ions and electrons strongly bound to the lattice. The thermal vibrations of the lattice perturb the electromagnetic field around the surface. The fluctuating fields due to the thermal motion of the free gas molecules and the thermal vibration of the lattice (kerogen structure) lead to the field perturbation that defines the dipolar interaction. This results in spin relaxation. It is clear that the quantitative attributes of this relaxation mechanism should depend on the rate of molecular motion. Thus, the efficiency of relaxation (and hence the relaxation time) is temperature dependent. Transverse relaxation occurs when spins get out of phase. Longitudinal relaxation occurs when the spin population goes back to the MaxwelleBoltzmann distribution, which is the equilibrium distribution. It is due to increase in the number of spins aligning with the ambient field direction. Hence, the efficiency of the kerogen surface in causing transverse relaxation is always higher than that of the kerogen surface in causing longitudinal relaxation. The efficiency of relaxation is called as relaxivity. Thus, kerogen is assigned two parameters of relaxivity, called as the longitudinal relaxivity (r1) and the transverse relaxivity (r2). The relaxation discussed above is also called as surface relaxation. 4.2.8.4 Relaxation of gas within inorganic nanopores A part of the nanopore assemblage encountered in gas shales is often present in the clay matrix. The clays have paramagnetic impurities (Loucks et al., 2009). In the case of organic pores, the strong dipolar interactions of the gas molecules and the protons of the kerogen, in the presence of the adsorbed gas

202 Understanding Pore Space through Log Measurements

molecules, lead to significant surface relaxation because of the high surface to volume ratio of the organic nanopore assemblage. The above remarks apply, both for the case of transverse relaxation and longitudinal relaxation, mentioned in the foregoing. In the case of inorganic nanopores, the strong dipolar interaction of the gas molecules with the paramagnetic impurities within the clay matrix, coupled with the high surface area to volume ratio of the inorganic nanopore assemblage, leads to strong surface relaxation of the proton spins.

4.2.8.5 Adsorption of methane gas into kerogen and the role played by this adsorption in relaxing spins The adsorption of gas results largely in a monomolecular layer of methane forming on the surface of the pores. The fraction of the total number of gas molecules in the adsorbed state averaged over time at any given pressure temperature condition follows the Langmuir isotherm (ε1). The surface relaxivity exhibited by kerogen in the presence of adsorbed methane differs for longitudinal relaxation and for the transverse relaxation. Both the transverse and the longitudinal relaxation are highly accentuated due to the high surface area to volume ratio of the kerogen pores. 1 1 rS ¼ þ 1 T1;free T1;bulk V

(4.66)

(Kausik et al., 2011). 1 T2;free

¼

1 T2;bulk

þ

r2 S ðgGTE Þ2 Deffective þ V 12

(After Kausik et al., 2011).     2 2 2 Ck2 þ Ct I1 s F Ck2 þ Ct I12 1 1 ¼ ¼ kT ¼ kT T1;bulk T2;bulk 3Z2 12pha3 Z2

(4.67)

(4.68)

(see Eqs. 4.62 and 4.65). 1  ε1 ε1 þ T1;free T1;ads

(4.69)

1 1  ε1 ε1 ¼ þ T2;effective T2;free T2;ads

(4.70)

1 T1;effective

¼

(After Kausik et al., 2011).

(After Kausik et al., 2011). The inversion of the echo amplitude data to the model distribution of hydrogen index associated with a bin (D,T2,T1) over different bins has already been discussed (at Sections 4.2.3, 4.1.2.6, 4.2.7). A useful representation of

Pore space attributes of nonconventional reservoirs Chapter j 4

203

this distribution would be a 3D cluster in 3D space with (D,T2,T1) dimensions. Here the color (palette value) of a point of the cluster being proportional to the hydrogen index of the bin. In practice, projections of this cluster on the (D,T2) plane and the (T2,T1) are presented as DT2 and T1T2 maps. Other useful representations are the following distributionsdhydrogen index over T2, hydrogen index over T1, hydrogen index over D, and T1/T2 ratio over T2. The overlays for the DT2 maps are quite important as they identify the provenance of the gas within the shale gas reservoir and allow fluid typing when the fluid type data are unavailable (generally not the case with shale reservoirs), when integrated with the following distributionsdhydrogen index over T2, hydrogen index over T1, and T1/T2 ratio over T2.

4.2.9 D-T2 maps and other plots related to the results of echo data inversiondfield examples Fig. 4.15 presents inversion for the case of a gas shale reservoir with a porosity of 8.34. The T2D maps arise by simultaneously presenting the distribution of the hydrogen index over T2, D. The peak of the distribution of hydrogen index over T2, D comes under the unconventional model gas diffusion line indicating that the gas within the reservoir is restricted gas and thus present within the kerogen pores. The diffusion distribution at the top right panel is the forward model for the gas in the kerogen pores and the match of the apparent diffusion coefficient with the peak in the diffusion distribution modeled is another representation. Through this it is seen that the gas is within organic pores. From the graphic at the left, in the middle-panel (denoted as B in Fig. 4.15) it can be seen that the clay-bound water is represented below 10 ms and gas is represented within the interval 10e100 ms. The graphic depicts the T1 and the T2 distributions in black and in red (gray in print version), respectively. From the graphic, it is seen that the T1/T2 ratio for pores hosting the identified gas is w 4.9. This indicates that the pores hosting the gas are kerogen pores because this is the typical T1/T2 ratio expected of kerogen pores. For shale reservoirs mud filtrate invasion is negligible. The fluid saturation values shown in Fig. 4.15 can be considered as virgin zone saturation of the indicated fluids.

4.2.9.1 Burst mode activation Activation has been through the diffusion editing burst mode and employing the Diffusion Editing cycle of activation for acquiring a train of echoes. The activation parameters are given in Table 4.3. If a CPMG cycle-based activation is used in the burst mode, the echo spacing and the wait time would be of the same order as in Table 4.3. We now discuss an example of the results of the inversion of NMR data for the case of an oil-bearing shale. The type of data acquired is as station data. The acquisition mode is burst mode. The parameters are typical of the

204 Understanding Pore Space through Log Measurements

FIGURE 4.15 D,T2 map for echoes acquired through DE sequence against a shale gas reservoir. D, diffusion coefficient; Dclm, logarithmic mean diffusion coefficient; DEseq, diffusion edited echo sequence; wat, water; oil error fn, oil-error function, Phi, Phiw, Phio, Phig, total porosity, wateroccupied porosity, oil-occupied porosity and gas-occupied porosity respectively; Sw, Sg, water saturation and gas saturation respectively; T2lm, logarithmic mean T2. Reproduced from Cao Minh et al. (2012).

acquisition parameters used for shale reservoirs bearing oil. The details of the acquisition parameters are available in Table 4.4. They are also displayed at the right panel of Fig. 4.16 (denoted as D in Fig. 4.16). The activation cycle deployed is the CPMG cycle. Referring to Fig. 4.16, the T2 distribution in black shown at the bottom left panel (denoted as C in Fig. 4.16) shows three peaks. The leftmost peak is identified as the irreducible water peak. The rightmost peak and the middle peak are identified as oil. While it is fairly easy to identify the rightmost peak as corresponding to oil, such identification would be difficult on the strength of

205

Pore space attributes of nonconventional reservoirs Chapter j 4

TABLE 4.3 Activation details for the example presented in Fig. 4.15. Sequence

WT (s)

TE (ms)

TEL (ms)

NECH

NRPT

1

8.2

0.6

0.6

752

1

2

2.5

0.6

0.6

602

1

3

1.2

0.6

3

602

1

4

1.2

0.6

4

602

1

5

2.5

0.6

7

602

1

6

1.2

0.6

10

602

1

7

1.2

0.6

16

602

1

8

0.8

0.6

0.6

512

2

9

0.1

0.6

0.6

144

2

10

0.03

0.6

0.6

64

2

11

0.008

0.6

0.6

16

32

NECH, number of echoes; NRPT, number of repeats in a burst; TE, short echo spacing te,s; TW: TEL: long echo spacing te,l; WT, wait time. Reproduced from Cao Minh et al. (2012).

TABLE 4.4 Activation details pertaining to the station of Fig. 4.15 as depicted at the right panel of Fig. 4.16. TW sec

TE ms

NECH

NRPT

0.55

0.2

200

1

1

1

300

1

1

2

300

1

1

4

300

1

1

6

300

1

0.08

0.2

90

20

2

0.2

1700

1

0.02

0.2

20

50

3

0.2

1700

1

0.06

0.2

20

50

206 Understanding Pore Space through Log Measurements

FIGURE 4.16 An example of oil-bearing shale. The right panel (denoted as D in the figure) shows the raw echoes and activation sequences details. The panels on the left (denoted as A, B, C in the figure) present the inversion results presented as DT2 maps and T1, T2 distributions. The data pertain to a shale reservoir bearing oil. Taken from Cao Minh et al. (2012).

the T2 distribution alone. However, note that the DT2 map at the top left panel (denoted as A in Fig. 4.16) indicates, that, the two local maxima of hydrogen index (the one centered around w30 ms T2 and the other centered around w100ms T2, respectively) indeed correspond to oil since they fall on the restricted diffusion line for oil. Referring to the bottom left panel of Fig. 4.15, note from the T1 and the T2 distribution, that the ratio T1/T2 corresponding to the peak centered around w30 ms T2 is around 5.2 while that for the peak centered around w100 ms T2 is 1.0. The surface of organic pores is hydrocarbon wetting. This results in very short T2 and the value of the T1/T2 ratio ranging 2e6. The inorganic pores are mixed wetting and the value of the T1/T2 ratio ranges 2e6 (Song et al., 2002). The following deduction can be made from the values of T1/T2 ratio data for the peak centered around w30ms T2 and that around 1.0 for peak centered around w100 ms T2. The peak centered around w30 ms corresponds to oil within organic pores while the peak centered around w100 ms T2 corresponds to oil present within inorganic water-wetting pores. Table 4.4 gives the activation parameters used, as indicated on the right panel (denoted as D in Fig. 4.16) of Fig. 4.15. In Table 4.4 the first column

Pore space attributes of nonconventional reservoirs Chapter j 4

207

stands for wait time, the second column for interecho spacing, the third column for the number of echoes, and the fourth column for the number of repeats.

4.2.10 Partitioning of total gas into free and adsorbed gas components using only NMR data The partitioning of the total methane gas-volume into free and adsorbed methane gas-volumes respectively using diffusion data from echo amplitude inversion is now discussed. The apparent gas diffusion coefficient coming from the echo data inversion, presented on the DT2 maps, is denoted here as Deffective. Its magnitude is a function of the exchange timescale of the free gas to adsorbed gas states, and the fractions of the free gas and adsorbed gas phases within the total gas. The diffusion coefficient for adsorbed gas is denoted by Dads, and the fraction of the number of adsorbed gas molecules of the total number of gas molecules is by ε1. Note the relation: Deffective ¼ ð1  ε1 ÞDgas þ ε1 Dads

(4.59)

(Kausik et al., 2011). From Eqs. (4.56)e(4.58) it can be seen that, from a given control from lab data from core plugs cut out of a shale gas reservoir, Dgas can be modeled. Deffective is retrieved from inversion of echo amplitudes. Dads is the diffusion constant associated with the diffusion of the adsorbed methane molecules. The diffusion of the molecules of the adsorbed methane falls under the category of surface diffusion. The mechanism for this is thermally activated hopping. Assuming a monomolecular layer structure for the adsorbed methane, a forward model for Dads can be possible. Such a model would be also based on data on the apparent diffusion coefficient of methane at different temperature and pressure conditions. Each set of pressure and temperature would stand for a point of dynamic equilibrium between the free and adsorbed methane gas molecular populations. With a robust model of Dads available, ε1 can be evaluated from Eq. (4.59).

4.2.10.1 Partitioning of methane into free and adsorbed methane using relaxation data from echo amplitude inversion Consider Eqs. (4.66)e(4.70). Subtracting Eq. (4.69) from Eq. (4.70):

 
 1 1 1 1 1 1    ¼ ð1  ε1 Þ þ ε1 T2;effective T1;effective T2;free T1;free T2;ads T1;ads (4.71) Subtracting Eq. (4.66) from Eq. (4.67):

208 Understanding Pore Space through Log Measurements




1

T2;free



1



T1;free


¼

1

T2;bulk



1 T1;bulk

 þ

ðr2  r1 ÞS ðgGTE Þ2 Deffective þ V 12 (4.72)

Here VS stands for the ratio of the cumulated surface area to the cumulated volume of the kerogen pore assemblage. When cores of the shale gas reservoir are available, core plugs are cut against the kerogen-rich parts of the core. The value of VS is found out by using the BET technique. Both by cumulative surface area and cumulated volume, the kerogen pore assemblage dominate the inorganic pore assemblage. Therefore, the value of VS found out for the kerogen-rich core plugs can be considered as a good estimate of the VS of the kerogen pore assemblage of the shale reservoir. This is because the VS parameter is independent of the cumulated surface area and the cumulated pore volume. Note that VS is the ratio of these two quantities, namely, the cumulated surface area and the cumulated pore volume. This makes the value of the parameter independent of the size of the sample as long as the sample size is not too small to be representative. Thus VS is a parameter with a known value.


1  1 Þ from Eq. (4.72) into (4.71): Substituting ðT2;free T1;free

1

T2;effective



1 T1;effective



  
1 1 ðr  r1 ÞS ðgGTE Þ2 Deffective þ ¼ ð1  ε1 Þ þ 2  T2;bulk T1;bulk V 12
 1 1 þ ε1  T2;ads T1;ads (4.73)

Because the adsorbed methane molecules are bound to the kerogen molecular lattice, dipolar relaxation presumably dominates spin relaxation. Since the translational motion of adsorbed methane molecules happens only through the thermal energy activated hopping, and since the lattice field would dominate the external magnetic field, the contribution of diffusion dephasing to the transverse relaxation of proton spins would be negligible in comparison with the dipolar relaxation. Therefore, Eq. (4.73) can be stated:


1

T2;effective



   
1 1 ðr  r1 ÞS ðgGTE Þ2 Deffective þ yð1  ε1 Þ þ 2  T1;effective T2;bulk T1;bulk V 12 (4.74) 1

The LHS of Eq. (4.74) is known from the NMR echo amplitudes inversion, as also the value of ðgGTE Þ2 Deffective since the value of Deffective is known from the NMR echo amplitudes inversion. A good control of r2 value can be obtained from the DT2 maps. The forward model gas diffusion line on which the inversion results (results of echo amplitudes inversion) plot gives an idea of the r2 value of kerogen, on the basis of which the diffusion line has been modeled. Thus, the values of all quantities in Eq. (4.74) are known except for

Pore space attributes of nonconventional reservoirs Chapter j 4

209

the values of r1 and ε1 . The value of r1 varies with the temperature and the pressure of the methane gas, and thus with ε1 . Forward models of r1 as a function of ε1 enable Eq. (4.74) to recast as:


1

T2;effective




   1 1 ðr  r1 ðε1 ÞÞS ðgGTE Þ2 Deffective  yð1  ε1 Þ þ 2 þ 12 T1;effective T2;bulk T1;bulk V (4.75) 1

Since Eq. (4.75) has the only unknown ε1 ; the equation can be inverted for ε1 given a known model r1 ðε1 Þ.

4.3 Characterization of fractured reservoirs We discuss reservoirs where fractures play a major role in the storage and fluid transport properties of the reservoir. The essential attributes that can be retrieved from logs are the aperture, width, height, orientation, density, storage volume, and permeability of fractures/fractured reservoir. The techniques discussed in Sections 4.1.2.1 up to Section 4.1.2.4 including Appendix 1 can also be applied for obtaining the aperture, width, orientation, and the density of fractures, but those are not discussed here. Discussion on these topics is further available in Chapter 10. Fracture height can be evaluated from image logs and the statistics of the fracture height can be generated. Fracture length is not directly available from the log data. Natural fractures can be quite long. The log-based characterization of a fractured reservoir operates at a length scale much less than the lengths of natural fractures encountered. Therefore, it is assumed that a natural fracture traverses the volume relevant to the log-based reservoir characterization. Permeability prediction based on log data for the case of fractured reservoirs is discussed in Section 10.8 of Chapter 10 and elsewhere in this book.

Appendix 1 Kherroubi’s work flow for trace extraction for low-angle events The treatment given in this Appendix mainly follows Kherroubi (2008).

Extraction of pixels which form part of a fracture The image function I(i,j) is defined as, ði; jÞ ˛ ½1; n  ½1; m/Iði; jÞ

(A1.1)

(i,j) are the pixel coordinates. Consider the following morphological operation on I(i,j) and let Iout(i,j) denote the resultant image:

210 Understanding Pore Space through Log Measurements

Iout ði; jÞ ¼ Iði; jÞ  ðO
CðIÞÞði; jÞ

(A1.2)

Here, C is the “closing’ operator and O is the “opening’ operator. These operators are defined with respect to a given structuring element B of a specific shape and size as: ðOðIÞÞði; jÞ ¼ ðD
EðIÞÞði; jÞ

(A1.3)

ðCðIÞÞði; jÞ ¼ ðE
DðIÞÞði; jÞ

(A1.4)

The operators D, E, respectively, stand for the standard image morphological operators, namely, the “dilation’ operator and the “erosion’ operator defined as below: ðDðJÞÞði; jÞ ¼ supfJði  x; i  yÞg; ðx; yÞ˛B

(A1.5)

ðEðJÞÞði; jÞ ¼ inf fJði  x; i  yÞg; ðx; yÞ˛B

(A1.6)

Here J is any binary or gray scale image in general. Let the result of application of operator C on image I be denoted by I1. The way I1 relates to I is that holes (gaps) in I whose dimensions are smaller than that of B are filled, and thus the high contrast features are accordingly linked. The meaning of the term ðO
CðIÞÞði; jÞ on the RHS of Eq. (A1.2) is that this term is the result of applying O on I1. The structuring element B is a vertical structuring element. The size of the structuring element corresponds to the vertical extent over which the image contrast measurement is deemed as important. The opening operator O applied on I1 results in the total erosion of the small isolated high contrast features present in I1. These constitute the information that is required to construct the fracture traces. This information is therefore extracted, by subtracting OðI1 Þ, which is ðO
CðIÞÞði; jÞ from the original image I. This is what is conveyed by the RHS of Eq. (A1.2). After generating the image Iout(i,j) (Refer Eq. (A1.2), the image Iout(i,j) is further modified by selecting only those pixels with intensity within the top 10% of the intensity distribution of the pixels present in the image. Let Iout(i,j) denote the resulting image. This image is a binary image. Iout forms the basis of identifying the pixels on a conductivity map of the borehole wall that belong to planar events that are not bedding planes (bedding features manifest as sets of planes at borehole scale).

Obtaining fracture traces from the extracted pixel sets Segments of the image of a fracture are called as fracture traces. The extracted pixel sets are called as traces. The image Iout(i,j) is rendered as a collection of traces that belong to geologically meaningful planar events, by applying the connected component labeling algorithm (Horn, 1986). This algorithm scans the binary image Iout(i,j) and groups the connected pixels, which share

Pore space attributes of nonconventional reservoirs Chapter j 4

211

nonzero pixel values. These groups define the traces. The algorithm also filters out the small events expressed as groups of pixels, which arise due to noise or texture artifacts. The resulting image is a binary image whose foreground is the collection of traces. Let this image be denoted as Itrace(i,j).

Finding the set of vectors (segments) that best fits a trace when placed end to end We are to compute the segments that best fit the traces and their respective orientations. A trace has a partial sinusoid geometry. A trace is approximated by a polygonal line. Such a line is composed of several segments of lines (Kherroubi, 2008) (Fig. A1.1). Each object with an arrowhead represents a segment of a polygonal line referred to above. The region (foreground) of a curved shape is referred to as “heterogeneity’ (Fig. A1.1) that contains a trace. The segments are found out as follows. We do not focus on the direction of the segment. Each segment is in fact a vector. Let the traces be indexed and consider the trace i. Let the coordinates corresponding to this trace be defined by (xi,yi). Consider a segment sj among a set of segments, of length Lj. This corresponds to a function fj

fj : x/yej ¼ aj x þ bj

(A1.7)

FIGURE A1.1 Illustration of a heterogeneity and segments in an image. Reprinted with permission from Kherroubi (2008).

212 Understanding Pore Space through Log Measurements

Let yeji denote the yej relevant to (in the context of) trace i. The penalized error Ej is defined as, !2 XX j Ej ¼ (A1.8) yei  yi þ l2 Lj ¼ Dj þ l2 Lj fj

i

Here, l is a regularization parameter, and is called as complexity penalty factor. The process of finding the required segments set which best characterizes a trace is illustrated by the following procedure (Kherroubi, 2008). Consider the trace labeled by zero. Let a fit by a segment s0 be considered, with the best-fit criterion being that D0 is minimized. The initial trace is now divided into two equal parts along the x axis. The best-fit segments s1 and s2 are found out. The energy measures of the segments s0, s1, and s2 are defined as the RHS of Eq. (A1.8) for these segments and the energy measures would be E0, E1, E2. In case E0 cosðp = 2  aÞ

(A1.9)




Here a is set to a low value of 10 to ensure that only fracture/cleat traces are selected. The above logic is generalized over depth: ci; pðzi Þ:si > cosðp = 2  aÞ Here, zi stands for the mean depth of a bedding plane.

(A1.10)

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Computing the main fracture/cleat orientation mentioned above A robust best-fit algorithm which implements the above logic is presented. This algorithm is launched over the likely fracture/cleat traces. The algorithm returns the main fracture/cleat orientation pf: X (A1.11) pf ¼ argmin rðp:si Þ i

Here, r is an M-estimator. The process that employs IRLS algorithm is described in Huber (1981) and Holland and Welsh (1977).

Obtaining the final cleat/fracture traces of high confidence Some of the traces extracted as above might belong to planar features that are not fractures/cleats. The traces that belong to planar features other than cleats are filtered out by the following process. The traces with segments orthogonal to the normal vector of the main fracture/cleat planes are selected with the criterion: ci; pf :si < cosðp = 2  bÞ

(A1.12)

The value of b is fixed interactively. A small value of b misses some cleats/ fractures while a large value of b selects planar events that are not cleats/ fractures, in addition to cleats/fractures. Eq. (A1.12) is still not sufficient for characterizing the selected traces as fracture/cleat traces (Kherroubi, 2008). The following procedure is applied on the selected traces. Consider a selected trace. A global normalized vector sg joining the extremity points of the trace is considered. The vector p orthogonal to sg minimizing the angle with the vector pf which is the main fracture/cleat orientation vector is considered:

(A1.13) pf : p ¼ pf ^sg ^sg If i is the index of the trace under discussion, the vector p is denoted as pi. pi and the depth coordinates of the global segment give the depth zi of the fracture plane. At this point all the unwanted noncleat/fracture-related traces not conforming to the pattern of spatial orientation of fractures/cleats are discriminated and hence get eliminated. A hierarchal clustering algorithm is now applied on the set of traces remaining. This algorithm, parameterized by the minimum distance between the elements in each cluster (single linkage), groups the selected traces as “clusters.” Those clusters which comprise two or more traces are classified as fractures/cleats. The rest of the clusters are discarded.

214 Understanding Pore Space through Log Measurements

The final set of traces remaining in the foreground of the image are the desired fracture/cleat traces of high confidence. These traces constitute the basis for defining the groups of pixels to compute fracture/cleat aperture and cleat density. The best-fit planes of these sets of traces are computed. The sinusoids with the traces also plotted on the input image, graphically, bring out the extracted fractures/cleats.

Appendix 2 A derivation of Eq. (4.21) Consider the simple model of identical blocks of matrix (base rock) separated by a set of intersecting orthogonal partings. These partings model the cleat system. Let the aperture of a parting be the same everywhere in the system and let it be denoted by W. Let a denote the length of a side of a cubical matrix block. A matrix block is considered as impermeable to fluid flow through it. Further, the dimension of a matrix block is assumed to be far larger than the width of a parting. The system is illustrated at Fig. A2.1.

FIGURE A2.1 A schematic of the simplest model possible for a medium intersected by orthogonal set of partings of identical aperture. The model is conceived as a repetition of the system in the three directions along the edges of the blocks of matrix. The arrow shown indicates the fluid flow direction considered. Reproduced from Reiss, (1980).

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215

Fig. A2.1 is the simplest model possible for a cleated coal. Note that the flow contribution arises from either of the two sets of partings, for the direction of flow shown. Consider a volume of the medium which contains n3 blocks of matrix enclosed in a parallelopiped of equal sides each of length (na þ nW). The volume Vpartings occupied by the partings, within this parallelopiped, is: Vpartings ¼ ðna þ nWÞ3  ðnaÞ3 x3n2 aw Porosity 4 is given by: 4¼

ðna þ nWÞ3  ðnaÞ3 ðnaÞ3

¼



 
2 nW 3 W W 1þ 1¼3 þ3 na a a

(A2.1)

(A2.2)

Since, as per the model the dimension of a matrix block is far larger than the width of a parting. Given Eq. (A2.2), and under this condition, the following approximation holds:
 W 4¼3 (A2.3) a Let DP denote the pressure drop between the fluid entry point and the fluid exit point for the system shown at Fig. A2.2. The volumetric flow rate denoted as q here through the fracture is given as per HagenePoiseuille law as, q¼

b3 l DP 12h L

(A2.4)

where h is the dynamic viscosity of the fluid.

FIGURE A2.2 Flow of an incompressible fluid through a single fracture of length L, breadth l, and width b. Arrow indicates the direction of fluid flow. Reproduced from Reiss (1980).

216 Understanding Pore Space through Log Measurements

Eq. (A2.4) can be stated also as, q¼

b2 z DP 12h L

(A2.5)

where z is the area of cross section of a fracture. Consider the system discussed where there are n matrix blocks for each side of the parallelopiped discussed in the foregoing. Referring to Fig. A2.1 it is easy to see that within this parallelopiped there are n number of horizontal fracture-like paths and n number of vertical fracture-like paths present. Length of each path is n(aþW). Area of cross section of each path is Wa. Substituting W for b, Wn(aþW) for z, and n(aþW) for L in Eq. (A2.5), and denoting the volumetric flow rate for a pressure drop of DP across the opposite faces of the parallelopiped along the direction of fluid flow, by Q we get, understanding that there are 2n flow paths present in all, Q ¼ 2n

W 2 Wnða þ WÞ DP W3 ¼ 2n DP 12h nða þ WÞ 12h

(A2.6)

Since the face area of the parallelopiped is ½nða þ WÞ2 , as per Darcy Law, and as
 k ½nða þ WÞ2 k DPyna Q¼ DP (A2.7) h nða þ WÞ h Equating the RHS of Eqs. (A2.6) and (A2.7) and simplifying, we get, W a, k¼

1 3 W 6a

(A2.8)

In the notation where b denotes the intercleat spacing, since the value of the intercleat spacing for the present case is nothing but a for the present case, Eq. (A2.8) can be stated as, k¼

1 3 W 6b

(A2.9)

Eq. (A2.9) is nothing but Eq. (4.21).

Appendix 3 Computation of kerogen volume, gas volume, and total porosity in shale gas reservoir Fig. A3.1 gives the petrophysical model of shale gas reservoirs. The nonclay inorganic mineral suite, in general, need not be confined to calcite and can include other minerals as well, such as siderite.

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217

FIGURE A3.1 Petrophysical model of shale gas reservoirs. Reproduced from Bust et al. (2014).

The log suite is assumed to include the NMR logs whose acquisition parameters, mode, and operational requirements of acquisition have been tailored to optimize data value against the shale reservoir logged, the state-ofthe-art capture neutron and inelastic neutron gamma spectrometry service, and the state-of-the-art formation bulk density logs. Consider the following response equation for the bulk density measurement: rb ¼ VK rK þ VDcl rDcl þ VInm rInm þ VCBW rw þ VFW rw þ Vgad rgad þ Vgfk rgf þ VgInm rgf (A3.1)

Here the subscripts Dcl, Inm, CBW, FW, HC denote bitumen, solid part of kerogen, dry clay, inorganic mineral composite other than dry clay, clay-bound water, pore water not bound to clay (this is also known as far water). This comprises of all water that is not clay bound (see Fig. A3.1) and hydrocarbon fluid, respectively. V stands for volume per unit bulk rock volume, and r for density. The subscripts gad, gfk, gf, respectively, mean adsorbed gas, free gas in kerogen pores, and simply free gas present in inorganic pores and kerogen pores. It is assumed that the density of free gas within the kerogen pores and that of the free gas within the inorganic pores are equal. Inorganic pores have only free gas. Let VIS ¼ VDcl þ VInm

(A3.2)

Here, the subscript IS denotes the total inorganic solids composite. Let rIS denote the density of the total inorganic solids composite. We have, rIS ðVDcl þ VInm Þ ¼ ðVDcl rDcl þ VInm rInm Þ

(A3.3)

Substituting from Eq. (A3.3) into Eq. (A3.1) we get, rb ¼ VK rK þ VDcl rIS þ VInm rIS þ VCBW rw þ VFW rw þ Vgad rgad þ Vgfk rgf þ VgInm rgf (A3.4)

218 Understanding Pore Space through Log Measurements

rIS  rb ¼ rIS  VK rK  rIS VDcl  rIS VInm  VCBW rw  VFW rw

 Vgad rgad þ Vgfk rgf þ VgInm rgf

(A3.5)

Eq. (A3.5) can be restated and simplified as:

ðrIS  rb Þ ¼ rIS VK þ VDcl þ VCBW þ VFW þ Vgad þ Vgfk þ VgInm  VK rK

 rIS VDcl  VCBW rw  VFW rw  Vgad rgad þ Vgfk rgf þ VgInm rgf (A3.6) Since, VB þ VK þ VDcl þ VInm þ VCBW þ VFW þ VHC ¼ 1

(A3.7)

Dividing Eq. (A3.6) by (rISrw) simplifying and rearranging:



rIS  rgad ðrIS  rb Þ ðrIS  rK Þ ðrIS  rDcl Þ ¼ VK þ VDcl þ VCBW þ VFW þ Vgad ðrIS  rw Þ ðrIS  rw Þ ðrIS  rw Þ ðrIS  rw Þ



rIS  rgf rIS  rgf þ VgInm þ Vgfk ðrIS  rw Þ ðrIS  rw Þ (A3.8) The following notation is introduced: ðrIS  rb Þ ðr  rK Þ ðr  rDcl Þ ; 4DK ¼ IS ; 4DDcl ¼ IS ;4 ðrIS  rw Þ ðrIS  rw Þ ðrIS  rw Þ Dgad



rIS  rgad rIS  rgf ;4 ¼ ¼ ðrIS  rw Þ Dgf ðrIS  rw Þ

4D ¼

(A3.9)

In the notation at Eq. (A3.9), Eq. (A3.8) is: 4D ¼ VK 4DK þ VDcl 4DDcl þ VCBW þ VFW þ Vgad 4Dgad þ Vgfk 4Dgf þ VgInm 4Dgf (A3.10) As already mentioned, the term kerogen denotes the solid part of the kerogen. Kerogen density is known and is an input to the computation. Dry clay density is known and it is an input to the computation. The density of hydrocarbon is an input to the computation. In shale reservoirs, invasion of filtrate is minimal. Hence the density of water means the density of pore water, which is an input to the computation. The computation assumes that the density of the pore water, which is the density of the far water, and the density of clay-bound water are equal. The state-of-the-art capture neutron and inelastic neutron gamma ray spectrometry technology delivers the value of rIS. This technology also delivers two important outputs, viz., the total organic carbon content (TOC) and the dry weight fraction of clay.

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219

From NMR echo amplitude inversion the volume of clay-bound water per MR denote the volume of clay-bound unit rock volume is computed. Let VCBW water per unit rock volume computed from the echo amplitude inversion mentioned above. The wet clay porosity (WCLP) is assumed to be known and is an input to the computation of the kerogen volume. We have, MR MR VCBW ¼ Vcl WCLP which implies Vcl ¼ VCBW =WCLP

VDcl ¼ Vcl ð1  WCLPÞ

(A3.11) (A3.12)

From Eqs. (A3.11) and (A3.12) we have, MR VDcl ¼ VCBW

ð1  WCLPÞ WCLP

(A3.13)

MR output from the echo amUsing Eq. (A3.13) computed from the VCBW plitudes inversion against every depth level. Hence, the value of VDcl is also known against each depth level. Total porosity is denoted by 4T.

4T ¼ ðVCBW þ VFW Þ þ Vgfk þ VgInm þ Vgad (A3.14)

Note that the volume of the adsorbed gas is considered as a part of the total porosity. The total hydrogen index measured by NMR is denoted by HIMR. HIMR ¼ VCBW HIW þ VFW HIW þ Vgad HIgad þ Vgfk HIgf þ VgInm HIgf

(A3.15)

In the RHS of Eq. (A3.15) HI stands for the hydrogen index of the fluid indicated by the subscript, at formation conditions. Let 4MR ¼

HIMR HIgad HIgf ;4 ¼ ;4 ¼ HIW MRgad HIW MRgf HIW

(A3.16)

Dividing Eq. (A3.15) throughout by HIW and using the notation of (A3.16): 4MR ¼ VCBW þ VFW þ Vgad 4MRgad þ Vgfk 4MRgf þ VgInm 4MRgf

(A3.17)

As already mentioned, the TOC is an output obtained from the state-of-theart capture neutron and inelastic neutron gamma ray spectrometry technology. It is defined as the weight fraction of organic carbon within a rock. Weight of organic carbon per unit volume of rock is given by TOCrb.

(A3.18) TOCrb ¼ VK CK þ Vgad Cgad þ Vgfk þ VgInm Cgf Here, C stands for the weight of organic carbon per unit volume of the substance denoted by the subscript. If WC is the weight fraction of organic carbon in the substance indicated by the subscript,

TOCrb ¼ VK WCK rK þ Vgad WCgad rgad þ Vgfk þ VgInm WCgf rgf (A3.19)

220 Understanding Pore Space through Log Measurements

4MRgad is a function of rgad and 4MRgf is a function of rgf. Eq. (A3.16) can be stated:





4MR ¼ VCBW þ VFW þ Vgad 4MRgad rgad þ Vgfk 4MRgf rgf þ VgInm 4MRgf rgf (A3.20) From Eqs. (A3.20) and (A3.10):



ð4D  4MR Þ ¼ VK 4DK þ VDcl 4DDcl þ Vgad 4Dgad  4MRgad rgad



þ Vgfk 4Dgf  4MRgf rgf þ VgInm 4Dgf  4MRgf rgf (A3.21)

rgf is modeled from the formation temperature and pressure. Accordingly, the quantity (4Dgf4MRgf(rgf)) is modeled and its value is thus known. The value of (VDcl4DDcl) is known. The quantity VgInm is determined for every depth level from the inversion results of NMR echo amplitudes because the hydrogen index of the gas within the inorganic pores is known, and the hydrogen index of the gas is also known as the gas is free gas for the case of inorganic porosity. Taking all the knowns, to the LHS, Eqs. (A3.19) and (A3.21) are stated as: TOCrb  VgInm WCgf rgf ¼ VK WCK rK þ Vgad WCgad rgad þ Vgfk WCgf rgf (A3.22) ð4D  4MR Þ  VDcl 4DDcl  VgInm 4Dgf  4MRgf rgf ¼ VK 4DK 

 þ Vgad 4Dgad  4MRgad rgad þ Vgfk 4Dgf  4MRgf rgf 

(A3.23) We know, from the inversion of the echo amplitudes, the total hydrogen index of the gas (HIKgas) hosted by the kerogen pores both as adsorbed gas and as free gas. HIKgas ¼ Vgad HIgad þ Vgfk HIgf ¼ Vgad HIgad þ Vgfk HIgf

(A3.24)

The density of the adsorbed methane (rgad) At formation pressure and temperature conditions, methane is supercritical. Hence the adsorption of methane by kerogen is supercritical adsorption. The adsorption sites on the surface of kerogen are uniformly distributed. It is reasonable to assume that adsorbed methane occurs as a monomolecular layer. These features are brought out in Fig. A3.2. The differences in the thermodynamic properties of phases I and II are small and therefore neglected. Modeling the density of adsorbed methane using Monte Carlo simulations to model the adsorption process is a useful means of estimating the density of adsorbed methane (rgad). Perez and Devegowda (2017) presented kerogen

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221

FIGURE A3.2 The left panel (denoted as A in the figure) shows the adsorbed methane, the layer of methane molecules proximal to the kerogen surface (phase I), and the methane molecules far from the kerogen surface (phase II). The right panel (denoted as B in the figure) shows the uniform distribution of adsorption centers on the surface of kerogen and some of these centers occupied by methane molecules. Reproduced from Xiong et al. (2016).

models using molecular dynamics and Monte Carlo simulations for methane adsorption. The authors demonstrate an estimation of the adsorbed methane density, using the simulations, and a three-component Langmuir Adsorption Model. Because the temperature and pressure of the system are known against every depth level it is feasible to compute the density of adsorbed methane (rgad) against every depth level. Further, models HIgad(rgad) of hydrogen index versus density are available, using which the value of HIgad against every depth level can be computed. Eq. (A3.24) can be rearranged as:   HIgf Vgad ¼ HIKgas  Vgfk (A3.25) HIgad Substituting from Eq. (A3.22) into (A3.19) and (A3.20):   HIgf TOCrb  VgInm WCgf rgf ¼ VK WCK rK þ WCgad rgad HIKgas  Vgfk HIgad þ Vgfk WCgf rgf (A3.26) 





ð4D  4MR Þ  VDcl 4DDcl  VgInm 4Dgf  4MRgf rgf ¼ VK 4DK  

 HIgf  4Dgad  4MRgad rgad þ Vgfk 4Dgf  4MRgf rgf þ HIKgas  Vgfk HIgad (A3.27) Since the value of rgad is known and so do HIgad at every depth levels, the value of 4MRgad in the RHS of Eq. (A3.27) is known against every depth level.

222 Understanding Pore Space through Log Measurements

Eqs. (A3.26) and (A3.27) are essentially two linear equations with two unknowns Vgfk,VK which can be solved. Thus, the kerogen volume per unit rock volume for a gas shale reservoir can be computed.

Computation of adsorbed methane volume per unit rock volume Let mgad denote mass of methane adsorbed per unit mass of kerogen. The value of mgad can be estimated using an appropriate adsorption isotherm. Let wgad denote the mass of adsorbed methane per unit volume of kerogen. wgad ¼ mgad rK

(A3.28)

Vgad, the volume of adsorbed gas per unit rock volume, is given by, Vgad ¼

wgad mgad VK ¼ VK r K rgad rgad

(A3.29)

Total gas volume within the kerogen pores and its partition The total gas within the kerogen pores (organic porosity) is denoted as Vgkp: Vgkp ¼ Vgfk þ Vgad

(A3.30)

Vgfk and Vgad constitute the partitioning of the total gas volume within the kerogen pores assemblage also known as organic porosity.

Total porosity Total porosity is denoted by 4T. 4T ¼ VCBW þ VFW þ Vgfk þ VgInm þ Vgad

(A3.11a)

Note that the volume of the adsorbed gas is considered as part of the total porosity. Total hydrogen index measured by NMR is denoted by HIMR. HIMR ¼ VCBW HIW þ VFW HIW þ Vgad HIgad þ Vgfk HIgf þ VgInm HIgf (A3.15a) Rearranging Eq. (A3.15), 

ðVCBW þ VFW Þ ¼ HIMR  HIgf Vgfk þ VgInm þ Vgad HIgad =HIW

(A3.31)

Substituting for (VCBW þ VFW) from Eq. (A3.31) into (A3.14), 4T ¼





1  HIMR  HIgf Vgfk þ VgInm þ Vgad HIgad þ Vgfk þ VgInm þ Vgad HIW (A3.32)

The value of the total porosity (4T) against every depth level is determined by Eq. (A3.32), since the value of each quantity in the RHS of Eq. (A3.32) against each depth level is known.

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Perez, F., Devegowda, D., 2017. Estimation of adsorbed phase density of methane in realistic overmature kerogen models using molecular simulations for accurate gas in place calculations. Journal of Natural Gas Science and Engineering 46, 865e872. Reiss, L.H., 1980. Reservoir Engineering Aspects of Fractured Formations. Gulf Publishing Co., Houston. Ruppel, S.C., Louckes, R.M., Hammes, U., 2012. Spectrum of pore types and networks in mud rocks and a descriptive classification of matrix related mud rock pores. AAPG Bulletin 96 (6), 1071e1098. Song, Y.Q., Kausik, R., 2019. NMR applications in unconventional shale reservoirs e a new porous media research frontier. Progress in Nuclear Magnetic Resonance Spectroscopy 112e113, 17e33. Song, Y.Q., et al., 2002. T1-T2 Correlation Spectra obtained using a fast two-dimensional Laplace Inversion. Journal of Magnetic Resonance 15 (4), 261e268. Tremain, C.M., Laubach, S.E., Whitehead, N.H., 1991. Coal fracture (cleat) patterns in Upper Cretaceous Fruitland formation, San Juan Basin, Colorado and New Mexico: implications for coalbed methane exploration and development. In: Schwochow, S., Murray, D.K., Fahy, M.F. (Eds.), Coalbed Methane of Western North America, Rocky Mountain Association of Geologists, Field Conference, pp. 49e59, 1991. Ting, F.T.C., 1977. Origin and spacing of cleats in coal beds. Journal of Pressure Vessel Technology 99 (4), 624e626. Van Golf-Racht, T.D., 1982. Fundamentals of Fractured Reservoir Engineering, Volume 12. Developments in Petroleum Science, 1st Edition. Elsevier. Xiong, W., et al., 2016. Methane adsorption on shale under high temperature and high pressure reservoir condition: experiments and supercritical modelling. Adsorption Science and Technology 34 (2 e 3), 193e211. Zhang, D., et al., 2018. Petrophysical characterization of high-rank coal by nuclear magnetic resonance: a case study of the Baijiao coal reservoir, SW China. Royal Society Open Science 5 (12), 181411. Zielinski, L., Ramamoorthy, R., Cao Minh, C., Al Daghar, K., Sayed, R.H., Abdelaal, A.F., 2010. Restricted diffusion effects in saturation estimates from 2D diffusion-relaxation NMR maps. SPE 134841. Presented at Society of Petroleum Engineers Annual Technical Conference and Exhibition, Florence, Italy. https://doi.org/10.2118/134841-MS. Zodrow, E.L., et al., 2010. Medullosalean fusain trunk from the roof rocks of a coal seam: insight from FTIR and NMR. International Journal of Coal Geology 82 (5), 116e124.

Further reading Chatterji, R., Paul, S., 2012. Estimation of in Situ Stress from Cleat Orientation, for Coal Bed Methane Exploration. Grace, M., Newberry, B., Koepsell, R., 1998. Geologic application of dipmeter and borehole electrical images. Unpublished Textbook. Schlumberger Oilfield Services, Marathon, Houston, Texas. Hu¨rlimann, M.D., et al., 2009. Hydrocarbon composition from NMR- diffusion and relaxation data. Petrophysics 50 (2), 116e129. Sarkar, A., et al., 2008. Method and System for Cleat Characterization in Coal Bed Methane Wells for Completion Optimization. US Patent 7900700B2.

226 Understanding Pore Space through Log Measurements Sengupta, N., 1980. A Revision of the Geology of Jharia Coal Field with Particular Reference to Distribution of Coal Seams. Song, Y.Q., et al., 2019. NMR application in unconventional shale reservoirs e a new porous media research frontier. Progress in Nuclear Magnetic Resonance Spectroscopy 112e113, 17e33. Yu, H., et al., 2017. Application of Nuclear Magnetic resonance (NMR) logs for pore size distribution evaluation. In: Paper Presented at the SPE/AAPG/SEG Unconventional Resources Technology Conference, Paper No. URTEC e 2663389 -MS.

Chapter 5

Log measurements commonly used for finding the bulk porosity of conventional reservoirs 5.1 Pore space attributes of conventional reservoirs The pore space attributes referred to above are bulk porosity, pore types and porosity partition, pore size and pore size distribution, connectedness index of pores, pore compressibility, and shear compliance of pores, permeability, and Archie a, m, n parameters.

5.2 Measurement of bulk porosity The preferred way of bulk porosity measurement is through a combination of density neutron porosity resistivity and gamma ray measurements, augmented by volumetric photoelectric factor and acoustic slowness measurements. Another way is through the inversion of NMR echo data to bulk porosity. The way these measurements are inverted together to bulk porosity has already been covered in Chapter 2. We discuss here the techniques of measurements.

5.2.1 Measurement of formation density for bulk porosity A rock is modeled as a mixture of solids and fluids. The bulk density of a rock is therefore, rb ¼ rma ð1  4Þ þ rfl 4

(5.1)

Porosity is measured as a fraction. The porosity includes intergranular porosity, and secondary porosity such as fracture porosity, and graindissolution porosity.

Understanding Pore Space through Log Measurements. https://doi.org/10.1016/B978-0-444-64169-4.00009-2 Copyright © 2022 Elsevier B.V. All rights reserved.

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228 Understanding Pore Space through Log Measurements

The dependence of rb on 4 through a linear relationship, and the accuracy that can be achieved in the measurement of rb, makes the bulk density measurement, the key measurement for porosity. Formation bulk density measurement itself is based on the gamma backscatter technique of densitometry with single scattering events as well as multiple scattering events contribute to the photon count rates. In formation bulk density measurement, the gamma photons detected can include those, detected, after undergoing even 30 scatterings. We use the symbol re to denote the number density of electrons and not the electron density index. The term “electron density” used in this chapter refers exclusively to the “number density of electrons.”

5.2.1.1 Single scattering of gamma photons Fig. 5.1 gives the geometry of a single scattering of a gamma photon. Consider a collimated source, whose collimation angle is da. Further, consider a detector situated at a distance of d from the source. Take the volume element of the medium to be dv, which subtends a solid angle of db at the detector. Referring to Fig. 5.2, the count rate at the detector due to Compton scattering from the volume element dv is given as, dN ¼

S mðr1 þr2 Þ ds A sin b cos q e re dv dU 4pr12 r22

(5.2)

m, the linear attenuation coefficient for gamma photons, is given by, m ¼ re s

FIGURE 5.1 Single scattering events of gamma photons detected by a detector.

(5.3)

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FIGURE 5.2 Mean energy versus number of scatterings. 662 KeV source in aluminum with no photoelectric effect. Reprinted with permission from Bertozzi et al. (1981).

The Compton differential cross section is given by the KleineNishina formula (Klein and Nishina, 1928),  ds r02
1 ¼ P  P2 sin 2 4 þ P3 where P ¼ dU 2 ð1 þ Eð1 þ cos 4ÞÞ

(5.4)

where, E stands for gamma photon energy and r0 stands for the classical electron radius. ds from Eqs. (5.3) and (5.4), respectively, into Eq. (5.2) Substituting for m, dU we get, dN ¼

 Sr02 ðA sin b cos qÞ re sðr1 þr2 Þ
P  P2 sin 2 4 þ P3 dv re e 2 2 8pr1 r2

(5.4a)

230 Understanding Pore Space through Log Measurements

which; as dv ¼ r12 r22 dadbdq implies; dN ¼

(5.4b)

 Sr02 ðA sin b cos qÞ re sðr1 þr2 Þ
re e P  P2 sin 2 4 þ P3 dadbdq 8pd (5.5)

Since da is fixed, the detector counts are obtained by integrating over limits of b,q. Eq. (5.5) shows that the detector counts are driven (for a given tool geometry, and assuming no borehole effects) by the electron density, since, as per Eq. (5.5), dNfre ere sðr1 þr2 Þ

(5.5a)

5.2.1.2 Multiple-scattered photons play a big role in the formation density measurement Typical density logging tools (Fig. 5.2) have a source and an assembly of two detectors pressed against the borehole wall. The source (Cs137 emitting 662 KeV gamma photons) as well as the short-spaced detector (also called as the near detector) is collimated while the long-spaced detector (also called as the far detector) is uncollimated. Because of the above and the geometry of the arrangement, in the case of the short-spaced detector, detected photons having energy more than 100 KeV will have mostly suffered one to three scatterings during their transport. On the other hand, in case of the long-spaced detector, the detected photons have a large fraction of multiple-scattered photons. Fig. 5.2 gives the mean energy of photon versus the number of scatterings. The bottom curve is relevant for the short-spaced detector while the top curve is relevant for the long-spaced one. Density of aluminum is close to formation density normally encountered. The significance of 100 KeV being the lower energy threshold of gamma photons counted is that Compton scattering is the dominant photon transport mechanism for energy >100 KeV (see Fig. 5.3). Fig. 5.2 shows that the number of scatterings suffered by photons of energy 100 KeV, for the case of the long-spaced detector, is up to six scatterings. The general equation for count rate N Czubeck (1983) is, N ¼ arbe eðcre þpZeq Þ 3:5

(5.6)

Here, a depends on the source strength. b depends on the collimation. b expresses the sensitivity of count rate to the electron density of the medium within the active volume, which gives rise to the backscatter of the gamma photons. The more the electron density, the more is the backscatter. There is, however, an additional mechanism through which electron density of the medium controls the count rate. While gamma photons travel from the source

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FIGURE 5.3 Mass absorption coefficient versus energy for porous silica and calcite for porosity 0% and 40% (with fresh water as the saturating fluid) over the photon energy range of interest for formation density logging. Reprinted with permission from Tittman and Wahl (1965).

toward the active backscatter volume, they are scattered out of the beam, by the electrons of the medium. Further, while the backscattered photons travel from the active volume toward the detectors they are scattered out of the beam, by the electrons of the medium. These factors are ensconced in the quantity c which is a function of the Compton scattering cross section, and the geometry of the source-detector-medium system. Coming back to the quantity b, it will be shortly seen that for a singlescatter event, the value of b is 1. As the number of scatterings suffered by a photon before the photon is detected rises, the value of b reduces and tends to zero. The wider the angle of collimation for the detector, and the farther it is from the source, the greater the fraction of multiple-scattered photons, within the total number of photons detected by the detector. The quantity p of Eq. (5.6) expresses yet another aspect of the gamma photon transport, viz., the photoelectric absorption of gamma photons by the medium. The gamma source used in formation density measurement is a Cs137 which emits gamma photons of energy 662 KeV. When the gamma photons suffer multiple Compton scatterings, their energy reduces to 100 KeV/

232 Understanding Pore Space through Log Measurements

ranges below 70 KeV. The quantity Zeq is the mean atomic number of the medium. Fig. 5.3 illustrates the dependence of the probability of a gamma photon getting Compton-scattered/absorbed through photoelectric process, on the photon energy, for the case of common porous sedimentary rock media. An inspection of Fig. 5.3 brings out that the photoelectric absorption has little role to play for the photon energy >100 KeV. This is because the mass absorption coefficient for photoelectric process is less than one-tenth of the corresponding number for the Compton process. Fig. 5.3 also brings out that, since maximum possible gamma photon energy is 662 KeV, pair production has no role to play. The current technology of formation density measurement involves the usage of two detectors or sometimes three detectors. The basic measurement is of detector count rates in different energy windows. Fig. 5.3B helps visualize the two-detector formation density measurement system. As Fig. 5.4 brings out, in an open hole, there always exists a mud cake on the borehole wall, which also scatters gamma photons, in addition to the

FIGURE 5.4 The dual detector formation density measurement system. Reprinted with permission from Ellis et al. (1983).

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formation. Thus, at least two media would have to be modeled for gamma photon transport, whereas the property sought is the bulk density of one of them, which is the formation. This circumstance necessitates employing at least two detectors, and which are affected by the mud cake by different degrees, to invert the detector count rates to formation bulk density. 5.2.1.2.1 Dual detector data processing using single window (per detector) count rates In this system, formation density measurement is obtained through the inversion of detector count rates for photons which fall within the energy range around 130 Kev e 460 Kev, for the long-spaced detector and 330e460 KeV for the short-spaced detector. These ranges are only representative. The energy window whose lower energy bound exceeds 100 KeV is generally referred to as the Compton energy window because the main process which controls photon transport within this energy range is Compton Scattering. The bounds of the Compton energy window are considered in order to achieve the best tradeoff between conflicting goals. The window will have to be wide enough to collect sufficiently large number of counts and in order to mitigate the statistical fluctuation (statistical fluctuation introduces an uncertainty in the count rate which scales as p1ffiffin where n stands for the mean count rate, since the counts follow Poisson statistics). At the same time, however, the lower energy edge of the window cannot be too low since, then, photoelectric absorption events would be significant enough, to have to be accounted for, even for the simplest forward models of photon transport within the Compton energy window. In the single energy window processing technique being discussed, it is assumed that, because photoelectric events have cross section much less than Compton cross section above 35 Kev, and particularly very low for energy more than 100 KeV, the count rate is driven principally by the Compton scattering process controlling the photon transport. It is therefore assumed that 3:5 would be of little relevance for the Compton window count rate. the term dZeq Therefore, the adequate forward model for the detector count rate is N ¼ arbe ecre

(5.7)

The integral of Eq. (5.5) over the detector collimation limits yields
  kSr02 A sin b cosq
(5.8) g q; E re ere sðr1 þr2 Þ N¼ 8pd Since the detector is collimated, the angle of scattering for a singlescattered detected photon can vary only within given limits. This in turn constrains the range of energies that the detected photons could possibly have. The barred quantities in Eq. (5.8) represent the mean values over the energy range of a detected photon, referred to above. Here, k is a constant.

234 Understanding Pore Space through Log Measurements

Comparing Eqs. (5.7) and (5.8), we note that the value of b ¼ 1.0 for the case of single-scattered photon count rate. The value of b can be reasonably assumed to be zero for the case of the long-spaced detector, which is also uncollimated, but is yet more than zero for the short-spaced detector, which is collimated, and which counts gamma photons which have undergone one to three scatterings prior to reaching the detector. We will note that the value of b has been considered as zero for the case of the short-spaced detector as well. This is not the case for forward models of short-spaced detector counts which succeed the simple model introduced here. kSr 2 ðA sin b cosqÞ

The quantity a is identifiable with 0 8pd , and c with s(r1þr2) for single scattering. In case of multiple scattering, a is a function of source strength and source-detector geometry. The quantity c is identified with the path length of the multiple-scattered photon. The general form for Eq. (5.7) when mud cake is also accounted for is, N ¼ aecrae rae ¼




  1  eKhmc rmce þ eKhmc rbe

(5.9) (5.10)

where rmce, rbe, and hmc stand for mud cake electron density, formation electron density, and mud cake thickness, respectively, and K1 is the characteristic length scale of radial distance of investigation of the density measurement. rae is the apparent electron density of the composite medium within the volume probed by the photon count rate in the appropriate energy window of a detector, and N is the photon count rate of the detector in the appropriate energy window mentioned above. Eq. (5.10) is the mixing law for generating rae (Samworth, 1992). Considering the above, we write Eq. (5.9) in the notation of this analysis: 
  Khmc Þrmce þeKhmc rbe k N ¼ a 1  eKhmc rmce þ eKhmc rbe e½ð1e (5.11) We can now write the count rate equations for the long-spaced detector and the short-spaced detector, with subscripts LS, SS, respectively, denoting them, as, 
  KLS hmc Þrmce þeKLS hmc rbe kLS NLS ¼ aLS 1  eKLS hmc rmce þ eKLS hmc rbe e½ð1e (5.12) 
  KSS hmc Þrmce þeKSS hmc rbe kSS NSS ¼ aSS 1  eKSS hmc rmce þ eKSS hmc rbe e½ð1e (5.13) Fig. 5.5 gives the variation of count rates with electron density of rock, for the case of no mud cake.

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235

FIGURE 5.5 Variation of count rate with electron density. X-axis plotted as twice the electron density normalized by Avogadro Number, which is an equivalent density of the medium. Reprinted with permission, from Elkington et al. (2006).

Taking natural logarithms of Eqs. (5.12) and (5.13) we get,   
ln NLS  ln aLS ¼ ln 1  eKLS hmc rmce þ eKLS hmc rbe 
   kLS 1  eKLS hmc rmce þ eKLS h rbe   
ln NSS  ln aSS ¼ ln 1  eKSS hmc rmce þ eKSS hmc rbe 
   kLS 1  eKSS hmc rmce þ eKSS hmc rbe

(5.14)

(5.15)

The natural logarithms on the RHS of the respective Eqs. (5.14) and (5.15) are much less in magnitude in comparison with their respective arguments. The logarithms can be neglected in comparison with their arguments. We can hence write Eqs. (5.14) and (5.15), respectively, as, 
  ln NLS ¼  kLS 1  eKLS hmc rmce þ eKLS hmc rbe þ ln aLS (5.16) 
  ln NSS ¼  kSS 1  eKSS hmc rmce þ eKSS hmc rbe þ ln aSS (5.17) Expanding the exponentials as infinite series, truncating to second power of the argument, substituting in Eqs. (5.16) and (5.17), respectively, and rearranging:    ln NLS ¼  kLS rbe  ðrmce  rbe Þ ðKLS hmc Þ  ðKLS hmc Þ2 = 2 þ ln aLS (5.18)    ln NSS ¼  kSS rbe  ðrmce  rbe Þ ðKSS hmc Þ  ðKSS hmc Þ2 = 2 þ ln aSS (5.19) rbe, rmce are the number of electrons per unit volume, of the formation, and the mud cake, respectively. Let rb, rmc stand for the true densities of formation and mud cake, respectively; Ab, Amc, Zb, Zmc stand for the mean atomic weight of

236 Understanding Pore Space through Log Measurements

formation, mud cake, mean atomic number of formation, and mud cake, respectively. Let NA stand for Avogadro Number. We then have Zb r NA Ab b

(5.20)

Zmc r NA Amc mc

(5.21)

rbe ¼ rmce ¼

For the case of common minerals found in sedimentary rocks and for the case of common muds, the approximation of the ratio of the mean atomic number to the mean atomic weight being 0.5 is valid. Within the industry, the e quantity 2r NA where re is electron density is called as electron density and its be value is close to the true density. Assuming 2rNmce and 2r NA to be, respectively, A equal to rmc, rb, respectively, we write Eqs. (5.18) and (5.19) as,

ln NLS ¼ 

  NA  kLS rb  ðrmc  rb Þ ðKLS hmc Þ  ðKLS hmc Þ2 = 2 þ ln aLS 2 (5.22)

ln NSS ¼ 

  NA  kSS rb  ðrmc  rb Þ ðKSS hmc Þ  ðKSS hmc Þ2 = 2 þ ln aSS 2 (5.23)

When hmc ¼ 0, a plot of lnNLS on the y-axis versus lnNSS on the x-axis would be a straight line. This straight line is known as the spine. Selecting any point on the spine, as hmc increases from zero to different positive values, the point moves along a parabola, which closely approximates to a straight line when the value of hmc is small. Calibration of the tool involves creation of the spine, recording long-spaced and short-spaced detector counts, with the source and detectors flush with a magnesium block, and an aluminum block, respectively, whose density values are known to a high degree of accuracy. The parabolic traverses of the point, when mud cake is present, is simulated by placing materials of different density values and thicknesses and recording the short-spaced and long-spaced detector count rates. These traverses are called as ribs. The total plot generated is known as a rib and spine plot (Fig. 5.6). 5.2.1.2.1.1 Graphical method of solving the detector response equations The graphical method of solving Eqs. (5.22) and (5.23), for the count rates of the respective detectors, is as follows. It can be noted that the ribs are not differentiated by mud cake density. The reason is that the quantities KLS, KSS are independent of mud cake density. For an unknown formation, the representative point on the spine and ribs plot would fall on one of the ribs. The point of intersection of the rib with the spine gives the point where

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FIGURE 5.6 An example of a spine and ribs plot. The plot is a logelog plot. Reprinted with permission, from Wahl et al. (1964).

formation would have plotted, had there been no mud cake or tool standoff. The formation density value can be directly read off the spine, as marked on the spine. This marking has been possible because, for no mud cake case, each value of formation density results in a unique pair of the long-spaced and short-spaced count rates. 5.2.1.2.1.2 Computing density and density correction without using graph In the industry the terms rLS, rSS also called as longspaced density and short-spaced density are used. rLS is defined as the density of the formation which gives the same long-spaced detector count rate with no mud cake, as the observed long-spaced count rate against the original formation (with mud cake present). rSS is defined similarly. A quantity called as “density correction” and denoted as Dr is defined as, Dr ¼ rb  rLS

(5.24)

238 Understanding Pore Space through Log Measurements

By the definition of rLS and rSS, respectively, and using Eq. (5.10),   rLS ¼ ðrb  rmc ÞeKLS hmc þ rmc (5.25)   rSS ¼ ðrb  rmc ÞeKSS hmc þ rmc (5.26) Eliminating hmc from Eqs. (5.24) and (5.25):     K

rLS ¼ ðrb  rmc Þ

1KLS SS

KSS

ðrSS  rmc Þ

KLS KSS

þ rmc

KLS

rb ¼ ðrLS  rmc ÞKLS KSS ðrSS  rmc ÞKLS KSS þ rmc From Eqs. (5.24) and (5.28): KSS

 KLS

Dr ¼ ðrLS  rmc ÞKLS KSS ðrSS  rmc ÞKLS KSS  ðrb  rmc Þ

K 1KLS SS



(5.27) (5.28)   KLS KSS

ðrSS  rmc Þ (5.29)

To appreciate the effect of mud cake thickness hmc on the density correction, Eq. (5.25) can be written, truncating the expression for the exponential function to the second power of its argument, as,     K 2 h2 K 2 h2 rLS ¼ ðrb  rmc Þ 1  KLS hmc þ LS mc þ rmc ¼ rb  ðrb  rmc Þ KLS hmc  LS mc 2 2 (5.30)



K 2 h2 Dr ¼ ðrb  rmc Þ KLS hmc  LS mc 2

 (5.31)

When the value of hmc, the mud cake thickness, is known to a good degree of accuracy, Eq. (5.31) is very useful in modeling the density correction. Semiempirical models on the lines of Eq. (5.31) exist. Fig. 5.7 illustrates the behavior of density correction, with experimental data. From the magnesium block and aluminum block calibration we know the respective detector counts for both the blocks, with the no-mud cake case applying here. Consider the long-spaced detector counts for magnesium and aluminum. Substituting in Eq. (5.22) we have two equations for two unknowns, namely, kLS, aLS, which can be solved. Similarly, consider the shortspaced detector counts for magnesium and aluminum. Substituting in Eq. (5.23) we have two equations for two unknowns, namely, kSS, aSS, which can be solved, for the short-spaced detector case. At this point, the values of kLS, aLS, kSS, and aSS are known. Now, a mud cake simulator of known density and thickness is introduced between the tool and each of the blocks and the detector count rates are recorded. Since (i) the densities and the electron densities of the blocks and the mud cake simulator are known, as also the thickness of the mud cake simulator, and (ii) the values of kLS, aLS, kSS, and aSS are known,

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FIGURE 5.7 Behavior of density correction illustrated with experimental points (black dots). Reprinted with permission from Ellis et al. (1983).

KLS and KSS are determined, once the values of all the knowns are substituted in Eqs. (5.16) and (5.17). The apparent long-spaced electron density rLSe and the apparent shortspaced electron density rSSe are computed following these equations: rLSe ¼

1 ðln NLS  ln aLS Þ kLS

(5.30a)

rSSe ¼

1 ðln NSS  ln aSS Þ kLS

(5.31a)

rLS and rSS are next computed from rLSe and rSSe by multiplying the electron densities by 2.0. The computed values of rLS and rSS are, respectively, substituted in Eq. (5.28) to obtain the value of rb. The value of the density correction Dr is computed from Eq. (5.24). The accuracy of the results depends upon the accuracy of the value of rmc. It is not true that mud cake would be the medium between tool and the borehole wall. It can be mud also, when the tool has a standoff because of hole rugosity. By default, mud cake density is set equal in value to mud which might introduce errors. This drawback also limits the maximum cake thickness for accurate computation of formation bulk density to be feasible. The migration of fines from mud into the formation near the formation face creates an altered zone, but effect cannot be accounted for in the above method.

240 Understanding Pore Space through Log Measurements

Technology exists, wherein a third detector very close to the source is also available, and with collimation, allowing counting of only those photons which have suffered a single scattering. Because of the close spacing, the investigation zone is mostly mud cake/mud (in case of tool standoff). Integrating the third detector count rate data allows rmc to be considered as a parameter determined by the measurement. Since three detector count rates are available, they can be inverted for hmc, rb, and rmc, simultaneously. The process generates hmc and rb depth level by depth level. The advantage of this technique is that the case of microzone of altered density near the borehole wall, owing to fines migration into the formation from the drilling mud, can be accounted for and the rb value is much more realistic, as a result. Even in the case of no fines migration into the formation, the three-detector count rate inversion ensures accurate formation density computation, for higher mud cake thickness than for the case of two-detector count rate inversion. There is additional advantage in the form of user being free to apply suitable constraints on rmc and hmc to obtain robust rb, hmc, and rmc simultaneously. Finally, there is a further option of obtaining a high-resolution formation density, combining the third detector count rate data and the short-spaced detector count rate data. In the above, no correction is made for the fact that, when the borehole radius differs from the radius of curvature of the pad/mandrel surface, borehole fluid additionally comes between the pad/mandrel and
 the mud cake. Generally, pad / mandrel radius of curvature is ð0:5 7 78Þ inches and hence, the correction for the case of an 8” borehole would be negligible. Some service providers apply empirical corrections for cases where the borehole diameter is different from 8. The correction-function is usually a polynomial function of the borehole radius. 5.2.1.2.1.3 Final correction to computed bulk density to account for electron density of water not being half of the bulk density of water The assumption of the ratio of the mean atomic number to the mean atomic weight being 0.5 fails in the case of water-bearing porous rocks because hydrogen deviates from the trend. It has been observed from actual counts recording in test pits having clean porous dolomites, sandstones, and limestones that the best estimate of true rock bulk density for water saturated porous sedimentary rocks is obtained as: rbtrue ¼ 1:04704rb  0:1883

(5.32)

rbtrue computed in culmination of all the steps detailed above is considered as the true density of porous rock. This is denoted as rb in short. It is the rb that is found in Eq. (5.1).

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5.2.1.2.2 Computation of rb using multiwindow count rate inversion There are different methods of computation of formation density using multiwindow count rate inversion. Here, only the example of Litho Density Tool (Mark of Schlumberger) is illustrated. Before proceeding, note why multiwindow inversion was necessitated in the first place. The reason is twofold. To start with, as already indicated earlier, it is not possible to conceive a Compton energy window which guarantees acceptable statistics while at the same time being totally immune to the effect of photoelectric absorption events. The feasible way is then to keep the Compton window wide enough, while at the same time, having an additional window specifically for monitoring the photoelectric absorption characteristics of the formation. The count rate from this new window is then used to correct the Compton window counts for the count rate loss due to photoelectric absorption. This additional window also provides valuable information hitherto unavailable regarding the formation lithology. This, since the photoelectric absorption itself is driven by the mean atomic number of the formation, and therefore is different for different lithologies. 5.2.1.2.2.1 Photon energy spectrum versus number of scatterings Fig. 5.8 illustrates how the gamma spectrum looks like for those gamma photons which have suffered 1, 2, 3, ., 11 scatterings. From Fig. 5.8 it is seen that the gamma photons counted under Compton energy window are those that have suffered a maximum of six scatterings before they had reached the detector. The total spectrum is the sum of the individual spectra, up to six scatterings. The result is the forward model of the spectrum between 130 KeV and 430 KeV, with no reduction due to photoelectric absorption, accounted for. The capacity of the formation to absorb photons through photoelectric process is an important lithology differentiator since this capacity scales as Z3.2 where Z is average atomic number of the formation. This property is extracted from the count rate of the long-spaced detector for a low energy window below 90 KeV. The window used in the case of the Litho Density Tool, for example, 44e84 KeV approximately, is called as LITH or the Lithology-Window. Fig. 5.2A shows when this window count rate is inverted, the forward model will have to include Compton scattering process as well. At this juncture it is important to understand the controls on the photoelectric absorption process. The photoelectric absorption cross section sp is given by sp ¼ 12:1E3:15 Z 4:6

(5.33)

The unit of sp is barns/atom. Consider the LITH window. From Fig. 5.8 it is noted that the gamma photons counted in this window will have undergone on an average, 7e10 scatterings. b in Eq. (5.6) can be equated to zero.

242 Understanding Pore Space through Log Measurements

FIGURE 5.8 Energy spectra for 1 to 11 scatterings at 35 cm from a 662 KeV source in aluminum without photoelectric effect. Reprinted with permission, from Bertozzi et al. (1981).

Before proceeding further, the definition of “long-spaced detector density rLS” is recalled as “the density of a formation (with mean atomic number is equal to the mean atomic number of the actual formation for which the density is to be measured), which, with no mud cake present, would give the same counts in the Compton window of long-spaced detector as the formation with mud cake.” Similar definition applies for “short-spaced detector density rSS.” It is assumed that the formation mean atomic number is Z. Eq. (5.6) in its full form is applicable as 3:6 N ¼ aeðcre þpZ Þ

(5.34a)

and we can write for the long-spaced detector counts, where f depends on the average path length of the photons for the long-spaced detector, and g depends on the distance from the detector, at which gamma photons energy reduces

Bulk porosity of conventional reservoirs Chapter j 5

243

below around 80 KeV. This distance will be more, the more the electron density of the formation.   Compton NLS

f

¼ ae

NA Z NA A sc rLS þg A rLS sp

(5.34b) 

 N kLS rLS þg AA rLS sp

NA Z sc (5.34c) A Here, sc is the mean scattering cross section for the Compton window. sp is the mean absorption cross section where the photoelectric absorption cross R section is nonnegligible. sp is 12:1E3:15 Z 4:6 dE over the energy range within the Compton window, within which the photoelectric absorption cross section is nonnegligible, normalized by width of the Compton window. f is a function of the source-to-detector spacing. g is a function of the distance from the detector, at which the gamma photons have energy low enough, for photoelectric absorption to have nonnegligible probability of occurrence. It is to be noted that there is no mud cake assumed. We also have,   Compton ¼ ae NLS

Lith NLS ¼ ae

f

 Lith ¼ ae NLS

N kLith rLS þg0 AA rLS sp0

where; kLS ¼ f

NA Z 0 NA A sc0 rLS þg A rLS sp0

(5.35a)  where; kLith ¼ f

NA Z sc0 A

(5.35b)

Here, sc0 is the mean scattering cross section for the Lithology window and g0 is a function of the transport path length at the gamma photons whose energy falls within the lithology window in the vicinity of the detector. g0 is a function of the average distance from the detector, of the locale where gamma photon energy begins to become 150 keV, incident on the detector can, however, suffer Compton scattering inside the scintillation crystal of the detector prior to detection. The Compton scattering can degrade the photon energy to that of a value, which falls within the lithology window before getting detected and results in a spurious boosting of the lithology window counts. This is a phenomenon of the Compton background or Compton tail which occurs. We shall presently see how allowance is made for this effect on the low energy part of the spectrum.

252 Understanding Pore Space through Log Measurements

A simplified version of Eq. (5.56) given below also has a good fit with Pe Ellis et al. (1985). To arrive at this simpler version of Eq. (5.56), the third term in the denominator is ignored. Eq. (5.56) can then be restated as,    

  S ðA1 =C1 Þ A1 1 1 ¼ ¼ where C ¼ (5.57) H ð1=C1 Þ þ Pe C1 C1 C þ Pe
 From Eq. (5.57) it is noted that as P assumes very large values HS goes to zero. But in fact, a photon entering the long-spaced detector with its energy falling within the Compton energy window can also degrade, to energy within the detector crystal through multiple Compton scattering to energy falling within the Lithology window. Owing to this, extra photon counts result in the Lithology window. Hence, for a practical fit, Eq. (5.57) modifies as,  

S 1 ¼M þR (5.58) H C þ Pe   Here M stands for CA11 . M equals 3:4 C1 , since the value of A1 is 3.4. R is a constant added for the reason discussed above. The value of R is determined from calibration (see further). C1 is a known function of the average energy of the Lithology window. In the usual notation followed in this analysis Eq. (5.58) is,

1 Compton Lith NLS ¼M þR (5.59) =NLS C þ Pe Fig. 5.11 is a plot of ðHS Þ versus C þ Pe for different materials. It is seen that Eq. (5.58) is well validated by experimental data. Hence, in a field calibration, different values of Pe are simulated by introducing different materials as spacers between the density tool pad and a calibration block, and  the ratio HS is noted. This is plotted against the simulated Pe values with the respective spacers. The plot will be a straight line whose slope is M, which is  equal to 3:4 C1 . From this, the value of C1 is obtained. The value of C is ob tained as the reciprocal of the value of C1. HS is now plotted against 1/  (C þ Pe) to obtain the intercept of the resulting straight line on the HS axis, which yields the value of R.  Thus, knowing the values of C, M, and R, the Pe data for any HS can be computed inverting Eq. (5.58). 5.2.1.2.2.8 Effect of mud cake Here the effect of curvature difference between the pad/mandrel and the borehole is not considered. A mud cake of thickness h between pad/mandrel and the formation face is considered.

Bulk porosity of conventional reservoirs Chapter j 5

253

FIGURE 5.11 A plot of the ratio between the ratio of the long-spaced detector Compton energy window count rate (shown as NLS e this is same as H) to the long-spaced detector Lithology window count rate (shown as NLith e this is same as S), versus 1/(C þ Pe). Reprinted with permission from Ellis et al. (1983).

From the calibration with reference blocks, the quantities a, b, g, d, and ε are determined. Using Eqs. (5.43) and (5.48), the apparent rLS and the apparent rSS are computed. The density correction Dr is no longer given by Eq. (5.31) but is a function of the photoelectric index contrast between the formation and the mud cake, in addition to the density contrast between the mud cake and formation, and the mud cake thickness. The following approach is adopted to address the above challenge. A database is created from the empirical measurements, where the simulated formation density and photoelectric index, mud cake density and mud cake thickness, and the mud cake photoelectric index are varied using different reference blocks and artificial mud cake combinations. For the case of the litho-density tool, the count rates for each of the five windows (see Fig. 5.9), viz., LL, LU1, LU2, and SS2, for each combination, are recorded, and a total database of 900 readings is generated. The forward modeling is for Dr in

254 Understanding Pore Space through Log Measurements

terms of (rLSrSS) where rLS and the rSS are the quantities obtained by substitution of the window counts for LS window and SS1 window, in the spine equations (please refer again those equation numbers) (Fig. 5.12). The X-axis shown in Fig. 5.12 is (rLSrSS), which is indicated as (rLSrSS1) because the SS1 counts are used for short-spaced detector while substituting in the spine equation. The Y-axis is the difference between the block density and the apparent rLS and equals Dr. The dashed line in Fig. 5.12 is the locus of (rLSrSS1), when the error in rLS owing to the loss of counts due to photoelectric absorption within the LS window is compensated, and then the corrected rLS is used. The origin corresponds to the case of no mud cake and hence, any given point on the spine. Since the ribs are examples for two different blocks, the origin for the two panels is two different densities. Multiple ribs arise for the same block and thus, for the same simulated formation density. Each of these multiple ribs pertains to a given value of mud cake density, and the different points on a rib correspond to different thicknesses of the mud cake. We have, Dr ¼ Drð½rLS  rSS1 ; rmc ; Pemc Þ

(5.59a)

Here, Pe and Pemc stand for the photoelectric index of formation and the mud cake, respectively. As per Eq. (5.51), the density as well as the photoelectric index of the mud cake affect the value of Dr. One would expect that those two sets of ribs would be possible. Of these, one set would be characterized by a given value of mud cake density and the other set characterized by a given value of photoelectric index. However, only one set of ribs is seen at Fig. 5.10. The reason for this is as follows. The mud cake density and the mud cake photoelectric index are correlated since barites in the mud are the major driver of both these properties. Hence, Eq. (5.49) becomes, Dr ¼ Drð½rLS  rSS1 ; rmc Þ

(5.59b)

FIGURE 5.12 Examples of rib in two calibration blocks. Reprinted with permission from Ellis et al. (1983).

Bulk porosity of conventional reservoirs Chapter j 5

255

The effects of the photoelectric index and thickness of mud cake are illustrated in Fig. 5.13 to represent the way these parameters affect the energy spectrum of the gamma photons reaching the long-spaced detector. The forward model of ribs works well up to a maximum thickness of mud cake of 1 inch. The implementation, the algorithm of the mud cake correction evaluation can be explained graphically as follows. Against a logged depth, the values of rLS and rSS1 are known because the detector counts are known, and the spine (refer Fig. 5.7) is known from calibration. Thus, the value of (rLSrSS1) is known. The value of rmc is an input to the model. It is assumed to be equal to mud density when no other input is available. Using the values of (rLSrSS1), rmc (which uniquely fix the forward model rib) the value of Dr is computed. The bulk density corrected for mud cake is then computed as rb ¼ ðrLS þ DrÞ. The rb value computed here is still not the formation bulk density since correction will have to be applied for the effect of difference between pad/mandrel curvature and borehole radius. This is called as the “Borehole Effect.” The rb value referred to above is denoted here as rLOG. 5.2.1.2.2.9 The borehole effect The value of rb corrected for mud cake is now corrected for the presence of mud between the pad/mandrel and the mud cake. The correction is denoted here as ðrb r8} Þ.
 (5.59c) rb  r8} ¼ jðrLOG  rMUD Þ where rMUD is the density of mud (borehole fluid) and the function j is an empirical fit function experimentally determined as a part of the master calibration.

FIGURE 5.13 et al. (1985).

Effect of barite on the measured spectrum. Reprinted with permission from Ellis

256 Understanding Pore Space through Log Measurements

5.2.1.2.3 General method of simultaneous inversion of multienergy window count rates Consider the general Eq. (5.34) and write Z3.6 as Pe103.6 to get, 3:6 N ¼ aeðcre þpPe 10 Þ

(5.59d)

N ¼ aeðcre þqPe Þ

(5.59e)

Let q ¼ p103.6.

Consider the case of presence of mud cake of density, photoelectric factor, and thickness of rmc, Pemc, and hmc. Let the apparent density and apparent photoelectric factor of the medium probed by the gamma photons, which are detected in the ith window of the jth detector, be rij and Peij, respectively. Let Nij be the count rate in the ith window of the jth detector. Since the electron density is proportional to the true density: Nij ¼ aij eðcij rij þqij Peij Þ þ bij

(5.59f)

Here, aij, cij, qij, bij are inputs to the model determined through calibration. Using the model of Eq. (5.31): ! Kij2 h2mc rij ¼ ðrb  rmc Þ 1  Kij hmc þ þ rmc yrb  ðrb  rmc ÞKij hmc 2 (5.59g) L2ij h2mc Peij ¼ ðPe  Pemc Þ 1  Lij hmc þ 2

! þ Pemc yPe  ðPe  Pemc ÞLij hmc (5.59h)

Kij and Lij are inputs to the model. Their values are taken from experimental data and calibration, for all i and j. Generally, bij for any i,j is low and Nij is corrected for bij. Hence, denoting Nij is corrected for bij as Nij for any i,j and substituting for rij and Peij, respectively, from Eqs. (5.57) and (5.58) in Eq. (5.56) and rearranging: Nij ¼ aij e½ðcij rb þqij Pe Þðcij Kij ðrb rmc Þhmc Þðqij Lij ðPe Pemc Þhmc Þ

(5.59i)

Let mij ¼ cijKij, nij ¼ qijLij. Eq. (5.59) is rewritten:     Nij ¼ eðcij rb þqij Pe Þ eðrb rmc Þmij hmc eðPe Pemc Þnij hmc

(5.60)

Taking natural logarithms on both sides:   
ln Nij ¼ cij rb þ qij Pe  hmc ðrb  rmc Þmij þ ðPe  Pemc Þnij

(5.61)

Bulk porosity of conventional reservoirs Chapter j 5

257

In current technology the number of detectors is  3 and the number of energy windows is 12. Equations such as 5.61 (but not necessarily of the same form as that of Eq. 5.61), as we have seen in the case of the processing of LDT count rate data discussed above, are constructed for each window, and together comprise the forward model. Thus, up to 12 equations are possible. The number of unknowns is 5. The unknowns to be inverted from the count rates, using the forward model, are rb, Pe, rmc, and Pemc. The inversion scheme follows the least error minimization method. Here the values of each of the unknowns are updated continuously till the least squared error of the forward model, with respect to the actual measurements, reaches the global minimum. Out of the three detectors, one of them is placed very near the source and its count rate data are analyzed in a standalone mode to obtain an estimate of rmc. Also, caliper log is recorded simultaneously with the formation density log and estimate of tool standoff/hmc is available. These independent estimates serve as quality indicators or can be used as nonholonomic constraints, in the inversion as per the utility/need to increase the robustness of the solution.

Appendix 1

  S SC

Simple approach to the ratio

Eq. (5.35a) in the current notation is,  S ¼ ae

f



NA Z 0 NA A sc0 rLS þg A rLS sp0

(A1.1)

By setting sp0 to zero, we get the equation for SC as,   SC ¼ ae

f

NA Z A sc0 rLS

(A1.2)

Dividing Eq. (5.51a) by Eq. (5.51b):   S ¼e SC

g0

NA A rLS sp0

¼e

g0 d2

NA Z 3:6 A rLS Z



 12:1=2:15 1 1 where; d2 ¼  E20  E10 E202:15 E102:15 (A1.3)

Since the energy bounds of the Lithology window are 40 KeV and 80 KeV, respectively, d2 ¼ 3:92  105 . Also, Z 3:6 ¼ Pe 103:6 S 0 1:4 NA Z 0 1:4 ¼ eg 3:9210 A rLS Pe ¼ eg 3:9210 rLSe Pe SC

(A1.4)

g0 is the reflection of the representative distance x (say) from the long-spaced detector for which the energy of a photon falls < 80 KeV. We note that, the more the value of x, less the magnitude of rLSe. This is so because the distance

258 Understanding Pore Space through Log Measurements

from the source at which the energy of a photon falls below 80 KeV for aluminum of density 2.7 g/cc is 35 cm, which is comparable with the distance between source and the long-spaced detector. Sedimentary reservoir rock density is less than this. Hence, the periphery at which photons degrade in their energy to over all orientations of the ellipsoid ¼ < E0x cos2 q1 > over all values of q1 ¼ ð1 = 3ÞE0x Similarly,

   < E0 :ny ny :i > over all orientations of the ellipsoid ¼ < E0x cos2 q2 > over all values of q2 ¼ ð1 = 3ÞE0y

And < ðE0 :nz Þðnz :iÞ > over all orientations of the ellipsoid is ¼ < E0x cos2 q3 >over all values of q3 ¼ ð1 = 3ÞE0z

(A1.6)

(A1.7)

(A1.8)

Applying the averaging, on Eq. (A1.2) and substituting from Eqs. (A1.6), (A1.7), and Eq. (A1.8), we get,    εcri E0 :ny ny :i εcri ðE0 :nx Þðnx :iÞ < Eix > over all orientations of theellipsoid ¼ <  þ   εcri þ N1j εj  εcri εcri þ N2j εj  εcri εcri ðE0 :nz Þðnz :iÞ þ   > εcri þ N3j εj  εcri < Eix > over all orientations of theellipsoid ¼

E0x X εcri   3 k¼1;2;3εcri þ Nkj εj  εcri

(A1.9)

Similar equations apply for over all orientations of the ellipsoid and over all orientations of the ellipsoid. Now, the ellipsoids are randomly oriented in space. Hence, when we compute the mean field for the interior of the inclusions taken together, the mean field for interior region of an ellipsoidal inclusion, represented as , will be given by (considering the calculations above),

554 Understanding Pore Space through Log Measurements

< Ei > ¼ E0

1 X εcri   3 k¼1;2;3εcri þ Nkj εj  εcri

(A1.10)

Let the average field for the sphere be designated as . Let vj denote the cumulative volume of all the ellipsoidal inclusions (all are made of material of type j). Since dipole field of the inclusions is neglected, the field at any location in the sphere, which is exterior to inclusions, is E0. Hence, V ¼ {(V evj) P εcri E0þ vj E0 13 }/V where V is volume of sphere. If fj is the ε þN j ðε ε Þ k¼1;2;3

cri

k

j

cri

v

volumetric concentration of inclusions within the bulk medium, Vj ¼ fj. Hence,   1 X εcri < E > ¼ 1  fj E0 þ fj E0   3 k¼1;2;3εcri þ Nkj εj  εcri

(A1.11)

Let represent the averaged displacement field averaged over the volume of the sphere. Let Vi represent the spatial region inside the sphere occupied by inclusions and let Vh represent the remaining part of the sphere. Let E(r) denote electric field at a location whose position vector is r. Then, since is the averaged displacement field averaged over the volume of the sphere V, Z Z V < D > ¼ εcri EðrÞd3 r þ εj EðrÞd3 r (A1.12) where, first integral is over Vh and the second, over Vi. The above can be also written as, Z Z   3 EðrÞd3 r V < D > ¼ εcri EðrÞd r þ εj  εcri

(A1.13)

Here, the first integral is over the entire space occupied by the sphere, and the second is over Vh. Z Z  1 1 3 < D > ¼ εcri EðrÞd r þ εj  εcri EðrÞd3 r ¼ εcri < E > V V (A1.14)  Vi  εj  εcri < Ei > þ V   < D > ¼ εcri < E > þ f εj  εcri < Ei > (A1.15) We assign a macroscopic complex dielectric permittivity ε for the mixture making up the sphere, defining ε through the equation < D > ¼ε < E >

(A1.16)

Archie’s cementation exponent Chapter j 8

555

We have, from Eqs. (A1.11), (A1.12), (A1.13), and (A1.15),  < Ei >  ε ¼ εcri þ f εj  εcri

¼ εcri þ

P

εcri ε þNkj ðεj εcri Þ k¼1;2;3 cri  P fj εj  εcri  εcri 1efj E0 þ fj E0 13 ε þNkj ðεj εcri Þ k¼1;2;3 cri



ε ¼ εcri þ



E0 13

P

εcri ε þNkj ðεj εcri Þ k¼1;2;3 cri  P fj εj  εcri  εcri 1efj þ fj 13 εcri þNkj ðεj εcri Þ k¼1;2;3





1 3

  ε ¼ εcri þ fj εj  εcri

1 3 fj

ε ¼ εcri þ

(A1.17)

1 3

(A1.18)

P

εcri ε þNkj ðεj εcri Þ k¼1;2;3 cri

1 þ fj 13

P

Nkj ðεj εcri Þ

ε þNkj k¼1;2;3 cri

  P εj  εcri k¼1;2;3

ðεj εcri Þ

εcri εcri þNkj ðεj εcri Þ

  P 1  13 fj εj  εcri

Nkj ðεj εcri Þ

ε þNkj k¼1;2;3 cri

(A1.19)

(A1.20)

ðεj εcri Þ

Extending this to the case of n types of inclusions present, we get, 1 3

ε ¼ εcri þ

n 3  P    P fj εj  εcri εcri = εcri þ Nkj εj  εcri

j¼1

k¼1

n 3  P    P 1  13 fj εj  εcri Nkj = εcri þ Nkj εj  εcri j¼1

(A1.21)

k¼1

Case of aligned inclusions (identical spatial orientation of principle axes of inclusions) Consider a situation, where a strong degree of alignment of ellipsoidal inclusions is present, for the case a clastic rock. The ellipsoidal inclusions are being best modeled as oblate spheroids. Complex dielectric permittivity ε and therefore the electrical conductivity show transverse isotropy around direction of axis of symmetry of the oblate spheroids. Conveniently choosing laboratory coordinate axis z along axis of symmetry of the oblate spheroids and indexing the axis of symmetry by the number “3”, we have the following equation for εz the macroscopic permittivity exhibited by rock when the external field polarizing it is directed along the direction of, the symmetry axis.

556 Understanding Pore Space through Log Measurements n     P fj εj  εcri N3j εj  εcri

εz ¼ εcri þ

j¼1

1

n     P fj εj  εcri N3j εj  εcri

(A1.22)

j¼1

Similar equations can be written down for εx, εy. The last two (εx, εy), are equal in magnitude because depolarization factors along the respective directions are equal in magnitude. Taking the limit of frequency tending to zero we obtain sz the conductivity of rock when measurement is made along direction of axis of symmetry and sx, the conductivity of rock when measurement is made along any direction normal to the direction of axis of symmetry. The results are the eqns sx ¼ scri þ

Sxo4ðswi  scri ÞN1w ðswi  scri ÞÞ ¼ sy 1  Sxo 4ðswi  scri ÞN1w ðswi  scri ÞÞ

(A1.23)

Sxo4ðswi  scri ÞN3w ðswi  scri ÞÞ 1  Sxo 4ðswi  scri ÞN3w ðswi  scri ÞÞ

(A1.24)

sz ¼ scri þ

Appendix 2ddifferential effective medium theory for aligned inclusions case This case is relevant when we have a situation of a clastic deposition whose grains have strong alignment with respect of each other. To analyze this case, we note that ! ! 3 3 P P εε1 εε2 1 1 (1 e f) 3 þf3 ¼ 0 (see Eq. (8.63)) ðεþN 1 ðε εÞ ðεþN 2 ðε εÞ k¼1

k

1

k¼1

k

2

For the case of aligned inclusions (grains), we choose coordinate axes to be aligned with the Principal directions of the ellipse. The inclusions are assumed to be oblate spheroids as this is the shape which mimics the grain shape. Z axis of the laboratory coordinate system is chosen to be along axis of symmetry of any inclusion and X and Y axes are chosen normal to the Z axis and in any arbitrary direction in the plane normal to Z axis. The permittivity measured in a direction along X and Y axes, respectively, will be equal but different from permittivity measured in a direction along z axis. Denoting these, respectively, as εx, εy, εz, above equation gets modified to the following equations.

 εx  ε1 εx  ε2    ð1  f Þ  þ f εx þ N11 ðε1  εx εx þ N12 ðε2  εx (A2.1)   ¼ 0 for the present case εx ¼ εy

Archie’s cementation exponent Chapter j 8

557

  3
3
1 X ε z  ε1 1X εz  ε2   þf   ¼0 ð1  f Þ 3 k¼1 εz þ N31 ðε1  εz 3 k¼1 εz þ N32 ðε2  εz (A2.2) Analysis on the same lines as given earlier leads to the following differential equations. Eq. (8.69) gets replaced by two differential equations (as εx ¼ εy):
 dεx 1 1  εx ðε2  εx Þ  ¼ (A2.3) df 1  f εx þ N12 ðε2  εx Þ 
dεz 1 1  εz ðε2  εz Þ  ¼ (A2.4) df 1  f εz þ N32 ðε2  εz Þ The equations, when integrated subject to the condition that εx, εz, each equal ε1 when f ¼ 0 give
 2 ðε2  εx Þ ε1 N1 ð1  f Þ ¼ (A2.5) ðε2  ε1 Þ εx ð1  f Þ ¼


 2 ðε2  εz Þ ε1 N3 2 ; N1 ¼ N22 ; εx ¼ εy ðε2  ε1 Þ εz

So, for completeness, we can write,  
 2 ε 2  ε y ε 1 N2 ð1  f Þ ¼ ðε2  ε1 Þ εy

(A2.6)

(A2.7)

generally, resistivity measurements are carried out parallel to the bedding plane because, vertical wells penetrating horizontally laid strata is the simplest case and common. For this case we have,
N12 ðε2 εx Þ ε1 ð1 f Þ ¼ ðε2 ε1 Þ εx see Eq. (A2.5) Recalling that in the model discussed earlier, the composite medium which is a mixture of cement and formation fluid is component 1, of that model, LHS of Eq. (A2.5) given above, is nothing but w4, where, 4 is pore volume per unit bulk volume (porosity) of rock. Taking the low-frequency limit we obtain, with the symbols retaining the meanings they had in the main text, 4
s1q
s ½3w  1q f ¼ where q ¼ N12 : (A2.8) ¼ w sx 2sx

558 Understanding Pore Space through Log Measurements


Rearranging; we get sx ¼

  1 3w  1 1q w sf 4q ; sy ¼ sx 2

1 Archie m parameter ¼ ¼ 1=N12 q

(A2.9) (A2.10)

Again, “rg” standing for aspect ratio of grain ellipsoids defined as ratio of the minor axis length to the major axis length of the ellipse, whose solid of revolution around minor axis is the ellipsoid representing grain shape, “e” standing for eccentricity of this ellipse, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2g  1 ð1  e2 Þ 1 2 arcsinðeÞ (A2.11) e¼ and N3 ¼ 2 1  e e rg N32 can be approximated as N32 ¼

1 1 þ 1:6rg þ 0:4rg2

(A2.12)

Friedman (2005) N12 ¼

 1 1  N32 2

1 2 ¼ Archie m parameter ¼ ¼ 1=N12 ¼  q 1  N32

(A2.12a) 2 1

!

1 1þ1:6rg þ0:4rg2



2 1 þ 1:6rg þ 0:4rgÞ ¼ mx ðsayÞ ¼2  1:6rg þ 0:4rg2

(A2.13) For measurement of resistivity in direction along the symmetry axis of the spheroidal grain shapes, the resulting conductivity is sz. This is given by the equation, 
  3w  1 1 1 (A2.14) sz ¼ wr sf 4r where r ¼ N32 2  1 1 Archie parameter m in this case ¼ ¼ 2 ¼ 1 þ 1:6rg þ 0:4rg2 ¼ mz ðsayÞ: r N3 (A2.15) For the range of grain aspect ratios encountered in case of real rocks for the transverse isotropy of conductivity as has been depicted, mx is less than mz. From measurements on real rocks by logging tools, it is seen that rock

Archie’s cementation exponent Chapter j 8

559

conductivity measured along vertical direction is always less than that measured along horizontal direction, when rock strata are horizontal. From measurements on bead packs saturated with brine, also, it is noted that conductivity of sample is less, when measurement is made, employing any methodology but where current flows are perpendicular to the oblate spheroidal grain (that is, current flows are along symmetry axis of the oblate spheroid), when compared with the case, where the methodology of measurement involves current flows in directions normal to the direction of the axis of symmetry of the oblate spheroidal grains (beads).

Appendix 3 Depolarization factors Introduction When a dielectric is embedded in another dielectric and an electric field applied, interface charges appear at both sides of the interface of the embedded dielectric with the host-dielectric. The charge distribution is net positive for one half of the interface and net negative for the remaining half of the interface. When the incident field is uniform, the field inside the host-dielectric as well as the inclusion-dielectric is uniform. Consequently, there is no bound polarization charge density anywhere. The only polarization charge is on the interface. As indicated, this distribution in space is bipolar and the inclusion behaves like a body having a dipole moment for points located outside it at distances much larger than its dimensions, and as multipole moment (dipole plus higher order multipoles related moment) for small distances from the surface. The dipole moment and its effects, alone, are considered and multipole moments, and their effects neglected. The dipole moment is responsible for fields internal to the inclusion and external to the inclusion which oppose the incident field. The opposing field is called as depolarizing field. Depolarization factors control this field by virtue of their control on the dipole moment of the inclusion. Depolarization factors of an ellipsoid in general and a spheroid in particular Consider an ellipsoid whose lengths of semi axes are ax, ay, az and whose symmetry axis is designated as principal axis z. Consider a cartesian system of coordinates, whose z axis is aligned with the principal axis z of the spheroid. Let the spheroid be made of a material having isotropic dielectric permittivity εi. Let a uniform electric field E0 be considered as existing, into which, first a medium of dielectric permittivity εh, of infinite extent is introduced, and into which medium, the dielectric spheroid is immersed. Under the influence of the external electric field, there is a rearrangement of the charge distribution inside the dielectric spheroid, as well as in the medium hosting the dielectric

560 Understanding Pore Space through Log Measurements

ellipsoid, without the overall charge of the dielectric spheroid or the host medium, deviating from zero. The rearrangement of charges, results in the net field inside the spheroid, as well as outside the spheroid, and not far away from it, deviating from E0. Let Ei be the field inside the spheroid. The additional field mentioned above, due to charge separation inside the spheroid is called the depolarization field and is denoted as Ed. Let the field outside the ellipsoid and at a distance large enough from the ellipsoid, for the field due to the ellipsoid to be negligible, be denoted as Eh Ei ¼ E0 þ Ed

(A3.1)

Eq. (A2.15) is as per the (proven) “Weak Eshelby conjecture” applied to electrostatics, for any uniform applied field E0. The field Ei is a uniform field, for ellipsoids. Let 4h be scalar potential corresponding to Eh, 4i the scalar potential corresponding to Ei and 40 the scalar potential corresponding to E0 and 4e the scalar potential corresponding to Ee the field outside the dielectric ellipsoid. The scalar fields are solutions of Laplace equations. The electric fields are the respective gradient fields of the scalar potential fields. E0 ¼  V40

(A3.2)

Eh ¼  V4h

(A3.3)

Ei ¼  V4i

(A3.4)

Ee ¼  V4e

(A3.5)

The boundary conditions to be satisfied at the ellipsoid’s surface are that i) tangential component of electric field is continuous, which is same as the condition that the electric potential should be continuous across the surface of the ellipsoid, and that ii) electric displacement be continuous across the surface of the ellipsoid (as free charge density is zero). Physically, the situation is as if an ambient field Eh was existing, inside which a dielectric ellipsoid made of a material of dielectric permittivity εi has been placed. Owing to the effect of the field, charge separation occurs. Elementary dipoles are created. Theoretically, the charge separation, makes the charge distribution, to give rise to a field, which is not just due to dipoles, but also higher order multipoles. Multipole components are neglected because the magnitude of multipole fields will be much less than that due to the dipole component when distance of point of observation from the center of gravity of the charge distribution is much larger than the spatial extent of the charge separation. Macroscopic charge density is zero everywhere in the interior of the ellipsoid, and is nonzero on the surface of the ellipsoid. Microscopically, dipole moment is nonzero at every point of the ellipsoid. The entire ellipsoid acts like a dipole, as far as field experienced at any point exterior to the ellipsoid, due to the charge separation inside the ellipsoid. The potential due to

Archie’s cementation exponent Chapter j 8

561

this ellipsoidal dipole at points external to the ellipsoid will be denoted as scattered potential field 4s The field internal, as well as to the ellipsoid due to the charge separation when added to Eh yields, respectively, Ei and Ee. Similar remarks apply to the scalar potential fields corresponding to the vector fields indicated. All the scalar potentials obey the Laplace equation. Especially V2 4 i ¼ 0

(A3.6)

Consider Eh ¼ Ehxi þ Ehyj þ Ehzk. Let 4hx, 4hy, 4hz be the scalar fields whose gradients are, respectively, Ehxi, Ehyj, and Ehzk. Let 4ix be the potential field inside the ellipsoid and Eix the internal field when Ehy ¼ Ehz ¼ 0 and Ehxi [ -V2 4i . Let 4iy be the potential field inside the ellipsoid and Eiy the internal field when Ehx ¼ Ehz ¼ 0. Let 4iz be the potential field inside the ellipsoid and Eiz the internal field when Ehx ¼ Ehy ¼ 0. The solutions for potentials and fields subject to the boundary conditions referred to above are εh 4ix ¼ 4 (A3.7) εh þ ðεi  εh ÞNx hx Eix ¼

εh Ehx I εh þ ðεi  εh ÞNx

(A3.8)

4iy ¼

εh 4 I εh þ ðεi  εh ÞNy hy

(A3.9)

εh Ehy j$I εh þ ðεi  εh ÞNy

(A3.10)

εh 4 I εh þ ðεi  εh ÞNz hz

(A3.11)

εh Ehz k$I εh þ ðεi  εh ÞNz

(A3.12)

Eiy ¼

4iz ¼ Eiz ¼

It can be seen, that the internal field is parallel to the incident field but is reduced in magnitude. This is because the charge distribution would be such as to set up an internal field opposite in direction to the incident field. This opposing field is called “Depolarizing Field.” The quantities NxNyNz are called depolarization factors. These depend only on the shape of the ellipsoid. The potential outside the ellipsoid 4ex when Ehy ¼ Ehz ¼ 0 is given by 4ex ¼ 4hx þ 4s ¼ 4hx 

ax ay az ðεi  εh Þ x Ehx εh þ ðεi  εh ÞNx 3r 3

(A3.13)

562 Understanding Pore Space through Log Measurements

It can be seen, that the scattered potential 4s is the potential due to a dipole, whose dipole moment px is given by px ¼

4p ax ay az ðεi  εh Þ Ehx 3 εh þ ðεi  εh ÞNx

(A3.14)

Thus, for points far from the center of the ellipsoid to its 
as compared 4pax ay az has a polaridimensions, the ellipsoid whose volume is given by 3 zation field Px (defined as the vector field “dipole moment per unit volume”) of

ax ay az ðεi εh Þ εh þðεi εh ÞNx Ehx i

when placed in an external field with its principle axis x

aligned with the direction of the external field. Thus, ax , the polarizability in x direction, per unit volume, is given by ax ¼

ðεi  εh Þ εh þ ðεi  εh ÞNx

(A3.15)

Px ¼ ax Ehx i

(A3.16)

Similarly, we can consider the corresponding polarizability and Polarization pertinent to the other two directions as ðεi  εh Þ εh þ ðεi  εh ÞNy

(A3.17)

Py ¼ ay Ehy j

(A3.18)

ðεi  εh Þ εh þ ðεi  εh ÞNz

(A3.19)

Pz ¼ ay Ehy k

(A3.20)

ay ¼

az ¼

It is easy to note that the depolarization factors play a crucial role in expressing the control that the shape of an inclusion has on its polarizability and internal field, when the inclusion occurs in a host is subject to an incident field, static, or propagative. The depolarization factors are given by a a a Z N ds x y z rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Nx ¼ (A3.21)       2 0 s þ a2x s þ a2y s þ a2z s þ a2x Ny ¼

a a a Z N x y z  2 0

ds ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     2 2 2 2 s þ ay s þ ax s þ ay s þ az

(A3.22)

Archie’s cementation exponent Chapter j 8

Nz ¼

a a a Z N x y z 2 0 

ds rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi      2 2 2 2 s þ ax s þ ay s þ az s þ az Nx þ Ny þ Nz ¼ 1

563

(A3.23)

(A3.24)

Whenax ¼ ay ¼ az, which is the case for a sphere, Nx ¼ Ny ¼ Nz ¼ 1/3. The closed form expressions for Nx, Ny, Nz are as below, when the ellipsoid is a prolate spheroid, with the principal axis z being the axis of symmetry. A prolate spheroid is also solid of revolution of an ellipse around an axis which contains its major axis. Let “a” denote the semimajor axis length of the ellipse. Let “b” denote the semiminor axis length of the ellipse.

  3 1 þ e2 Nz ¼ (A3.25) az þ 1 4 1  e2 In this case az ¼ a, the semimajor axis length of the ellipse referred to above. Nx ¼ Ny ¼

ð1  Nz Þ 2

(A3.26)

The closed form expressions for Nx, Ny, Nz are as below, when the ellipsoid is an oblate spheroid, with the principal axis z being the axis of symmetry. An Oblate spheroid is also solid of revolution of an ellipse around an axis which contains its minor axis.
  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 2 Nz ¼ 2 1  ð1  e Þ sin e (A3.27) e e In this case az ¼ b, the semiminor axis length of the ellipse referred to above. Nx ¼ Ny ¼

ð1  Nz Þ 2

(A3.28)

Here “e” is the eccentricity of the ellipse whose solid of revolution is the spheroid sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 b (A3.29) e¼ 1  a For spheroids rotational axis is called unique axis and the other two principal axes are called equal axes. If “r” stands for the ratio of dimension along rotational axis to dimension along an equal axis, r is given by r ¼ ab for the case of oblate spheroid. Numerical integration of RHS of Eq. (8.23) and

564 Understanding Pore Space through Log Measurements

fitting to a polynomial carried out Jones and Friedman (2000) yields the expression for Nz as given below Nz ¼ Nx ¼ Ny ¼

ð1Nz Þ 2

1 ð1 þ 1:6r þ 0:4r 2 Þ

(A3.30)

(see Eq. (A3.28))

Approaches to model “m”: in case of Shaly rocks using the Bergman spectral density representation of the effective permittivity of a binary mixture (ReferD. Stroud, D. W. Milton, B. R. De, “Analytical model for the dielectric response of brine saturated rocks”,Phys.Rev. B. vol 34, no. 8, 1986) These approaches are differentiated from other approaches, illustrated in the foregoing, in some significant ways. (i) No internal geometry of the composite including geometry of closed surfaces bounding component media is assumed. (ii) The approach derives from the Bergman’s Theorem, which is a direct consequence of the Maxwell Equations applied to propagation of electromagnetic wave through binary mixtures in the quasi-static limit. (iii) The fact that medium of rock is a disordered medium makes the spectral representation, a continuous function whose form is subject to rigorous conditions on the allowable eigenvalues, their ranges and certain bounds on the spectral function. (iv) Previously discussed approaches turn out to be specific definitions of the spectral function. (v) The pathway toward modeling the effective permittivity is through the generation, of valid forward models of the spectral function. The following analysis is after the Stroud Milton De Model (SMD Model) described at the reference cited above. The model is valid only when grain to grain contact area  the average surface area of a grain, a situation which holds true for many types of rocks. The model assumes rock to be a binary mixture of a conductive phase namely brine and nonconductive phase namely grain. This binary mixture is spatially defined as regions of space occupied by grain and brine. Here, brine includes (i) free water, (ii) capillary-bound water, and (iii) clay-bound water. This distribution is defined through 2 functions q1(r) and q2(r). Here, the function q1(r) corresponds to the distribution of the brine phase and the function q2(r) corresponds to the distribution of the grain phase. These functions are defined to take values 1 when r points to a point interior to the respective phase, and 0 otherwise. These functions therefore represent the microstructure of the spatial distribution of the two phases present. Brine is considered as phase-1 and grain is considered as the phase-2. The permittivity of brine is denoted as ε1 and that of the grain as ε2. The volume fraction of brine in the mixture is the porosity 4.

Archie’s cementation exponent Chapter j 8

565

Modeling for Archie “m” starts with modeling the mixing law. The Stroud Milton De approach does not attempt to model the geometry (microstructure) and then homogenize the composite micromodeled (that would involve modeling the low frequency behavior of double layer formed in association with the clay grains, which could make the model intricate). Instead, it aims to focus on the spectral representation of the mixing law, viewing the mixing law as an analytical function of a complex variable. This follows the line of argumentation of Bergman (1978), which notes certain properties to be exhibited by the function referred to above, for it to correspond to the conditions that any mixing law for real binary mixtures would have to honor, in the context of electromagnetic wave propagation through the mixture. Note that, the electromagnetic wave propagation is controlled by the permittivity and conductivity properties of the components of the mixture. In the quasi-static limit, the form of the mixing law, which relates the effective permittivity (ε) of the mixture with the permittivity of the components, the geometry of their arrangement in space and their volume fractions, remains unchanged, if the permittivity values of the components are replaced by ratios between their values. Hence, the mixing law is written as a relationship between two variables “f” and “s” which are defined as below, ε f ¼1  (A3.31) ε2
 ε1 1 s¼ 1  (A3.32) ε2 Note that, now, the mixing law is recast as the functional relationship between the variables “f” and “s”. This functional relationship is expressed as, “f” expressed as a single argument function of “s”. f ¼ f ðsÞ

(A3.33)

Hence once we obtain the function f(s), we will have deduced the form of the mixing law. This type of recasting is possible because the function ε defined through the equation ε ¼ εðε1 ; ε2 Þ

(A3.34)

has the property that, if l is a constant, replacing ε1,ε2 by lε1,lε2 in Eq. (A3.34) results in ε becoming lε. In other words, if ε ¼ ε(ε1,ε2), then, ε(lε1,lε2) ¼ lε(ε1,ε2) while it might appear that, in the absence of a model on the geometry at the microlevel, one would be knowing nothing about the form of the function f(s), much can in fact be learned about the function f(s), from the physical considerations and the need of mathematical self-consistency which any definition of the function f(s) has to satisfy. Stroud et al. (1986)

566 Understanding Pore Space through Log Measurements

argue that f(s), considered as an analytical function in s, which is also a rational function in s, can have only simple poles that have positive residues. To see why this must be so, consider the passage of electromagnetic wave through the medium. When an electromagnetic wave passes through the composite medium, any dissipation, which occurs within the composite medium will arise only because of the nonvanishing imaginary parts of, respectively, the components’ bulk permittivity, and the effective permittivity of the mixture. Now, when the components dissipate energy, the composite will also dissipate energy. However, dissipation is possible, only when imaginary part of permittivity is positive. Hence, Im(ε1) and Im(ε2) being positive, implies that Im(ε) is to be also positive.
1 By definition s ¼ 1  εε12 . It can be proved that If ImðsÞs0; while Imðε1 Þ > 0 and Imðε2 Þ > 0; then

Im ðsÞ < 0 (A3.35) Im fðsÞ

Hence, the change of sign of Im[f(s)]which occurs at a pole (or zero) of f(s) must always be accompanied by a change in sign of s and in such a way, that the Im[ f(s)], always has its sign opposite to the sign of Im(s) the imaginary part of its argument, namely s. Now, the condition set forth by inequality (A3.35), would certainly to be violated in the vicinity of the higher order poles of f(s) in case f(s) would have any higher order poles. This is so because, for s lying within the neighborhood of a higher order pole (of order >1), the sign of the Imaginary part of f(s) can change multiple times without sign of imaginary part of “s” changing according to the change of sign of the imaginary part of f(s). Hence, all the poles of f(s) can be only simple poles (poles of order 1). The implication of the above is that, if f(s) has a simple pole at say s ¼ si, then si necessarily lies on the real axis. Also, residue of si necessarily is positive. This is so because, if
si had an  ImðsÞ

imaginary component, or, if the real component of si was negative, Im½f ðsÞ
 ImðsÞ would be ‡ 0 near si, whereas Im½f ðsÞ would be < 0 for full range of the function f(s). Hence, if f(s) is supposed to be an analytical function, which is also a rational function: (i) Every pole of f(s) is a real pole (The interested reader may refer Appendix at the end of this analysis for a general perspective on why this should be so) (ii) Residue of every pole of f(s) is positive (iii) Since ε has to be positive whenever ε1, and ε2 are real and positive, the poles have magnitudes 0 and  1

Archie’s cementation exponent Chapter j 8

567

(iv) Further, because of (iii), f(s) 1 for s ¼ 1 Hence f(s) can be expanded as the series (referred to as the spectral expansion of f(s)) X Bn with Bn  0 (A3.36) f ðsÞ ¼ ðsn  sÞ n The reason why the expansion of f(s) as given in the RHS of Eq. (A3.36) is called as the Spectral Expansion of f(s) is because the poles and residues of f(s) happen to also be the eigenvalues, and quantities related to the eigenfunctions e which is defined at Appendix 4 that follows this of a special operator G analysis. All relevant details of this perspective on expansion of f(s) and the consequent results are present at Appendix 4. The index n of the summation in the RHS of Eq. (A3.36) takes on integer values from zero onward. sn are the poles and Bn are the residues of the poles. Because every pole of f(s) is a simple pole, the expansion of f(s) lacks terms with power >1.0 in the denominator. The above constitutes the Bergman’s Theorem (Bergman (1978). Poles of f(s) are the values of s for which f(s) becomes infinite. We note that when s equals zero, f(s) becomes infinite. Hence, the lowest valued pole of f(s) is zero. s is zero when the complex dielectric permittivity of the fluid is infinite, or when the complex dielectric permittivity of grain material is zero. Since these are extreme cases the lowest pole of zero requires a special treatment in the analysis. The residue of this pole can be shown as A, and the spectral expansion written as, A X Bn f ðsÞ ¼ þ where n  1 (A3.37) s ðsn  sÞ n The amplitudes of Bn the residues, and the location of the poles sn reflect the pore structure of the rock. Eq. (A3.37) will have finite number of terms on its RHS (discrete number of poles, or equivalently, values of s characterizing a discrete number of spectral components, if the pore structure of the rock is expressible as a combination of well-defined geometric arrangements of pores in space, with individual pores having well-defined geometrical shape, and the distribution of the poles on the real axis would be a discrete distribution. However, the pore structure of real rocks (since real rocks are media with random microgeometry) can be expressible only as a combination of an infinite number of possible pore structures intertwined with one another. In other words, the function f(s) has an infinite number of poles, all situated on the real axis and lying within the interval [0,1], and each one of them a simple pole with a positive residue. Therefore, for the case of real rocks, all the poles of f(s), excepting the pole s ¼ 0 are expected to be broadened into a branch cut along the real axis.

568 Understanding Pore Space through Log Measurements

Physically the poles would represent resonances (in s domain) (values of the argument of f(s) which make f(s) go high) and, the residues will be replaced by a density function. One can understand the terminology if one visualizes a contour integral of f(s) over s, in the complex plane. The value of the integral would be 2pi times the sum of the residues which are located interior to the contour. Since the poles are not discretely defined on the real axis, the only way forward would be to frame a function whose integral over that interval on real axis, of s, which falls interior to the contour of integration. The consequence of the reasoning detailed above, is that for the case of real rocks, since they are examples of composite systems having random spatial distribution of the constituents, the discrete summation (spectral representation of f(s)), figuring in the RHS of Eq. (A3.37) is to be expressed as a continuous integral overs, as given below Z 1 A gðs0 Þ 0 f ðsÞ ¼ þ ds (A3.38) 0 s 0 ss where gðs0 Þ ds’ represents the residue of poles lying within the interval (s0 , s0 Dds0 /). gðs0 Þ reflects the density of resonances. gðs0 Þ is also called as the “Geometric Resonance Density,” “Spectral Function”. gðs0 Þ takes only positive real values. If s, s1, s2, respectively, stand for the electrical conductivity of the binary mixture (here, the rock medium), fluid, andgrain material, respectively, then, s in the low-frequency limit, f(s) tends to 1  and s tends to
1 , s2

s

1s1 2

respectively. Therefore, in the low-frequency limit, Eq. (A3.38) can be written as,  s Z 1 1  s2 1 gðs0 Þ ¼A þ

ds0 (A3.39) 1
0 s1 s1 s 1 0 1  s2 1  s2 s 1  s2 At low frequencies, value of “s” is low and the RHS tends toA, and Eq. 1ss

2

(A3.39) reduces, in the low-frequency limit, to the equation  ¼ A, s which on rearrangement becomes, 1s1 2 A¼

ðs2  sÞ ðs2  s1 Þ

(A3.40)

s2 is the grain conductivity and is normally will be negligibly low. s1 is the conductivity of brine, and will normally be [s, the conductivity of a fully brine saturated rock.

Archie’s cementation exponent Chapter j 8

Under these conditions, Eq. (A3.40) reduces to the equation, s A¼ s1

(A3.40a)

s s1

¼ 4m. This implies that, s A ¼ ¼ 4m s1

However, as per Archie’s Law,

569

(A3.40b)

which indicates that A represents the reciprocal of electrical formation factor of the total water occupied porosity. There are two important facts to note, regarding Eq. (A3.38) I. The different mixing laws discussed are in fact, subcases of a general spectral representation of a form akin to the one discussed in the foregoing. Eq. (A3.38) explicitly expresses the general spectral representation of the complex dielectric permittivity of a binary mixture in terms of the dielectric permittivity of its components. II. The spectral representation’s form is fundamental for binary mixtures and can be seen to follow from application of Maxwell’s Equations in the behavior of binary mixtures, in the quasi-static limit when the electric field is curl free. For the interested reader, these aspects are discussed at Appendix 4. The equation given below, which is a form of Eq. (A3.38) where the first term in the RHS of Eq. (A3.38) also has been taken inside the integral, is known as the spectral representation of effective permittivity of a mixture in terms of component properties and internal geometry of arrangement of components. The equation is, Z 1 gðs0 Þ 0 f ðsÞ ¼ ds (A3.41) 0 0 ss Eq. (A3.37) which is the fundamental form, that leads to the equations succeeding it, has great significance. Firstly, the spectral representation (RHS of Eq. (A3.37)) of f(s) generally holds as long as the quasi-static approximation holds. Secondly, as brought out at Appendix 4 the discrete sum form of expansion for f(s) as expressed in the RHS of Eq. (A3.37) is, in a physical sense, the sum squares of spectral amplitudes of the eigenfunctions of a specific operator which carries all the geometry information concerning the spatial distribution of component phases of the medium. Conceptually, the eigenvalues are seen to correspond to values of “s” at which geometric resonances occur. To continue the discussion, the model of gðs0 Þ is the dispersion model of f(s) and hence dispersion model of ε. The value of s depends on value of εε12 . The dispersion of ε1 and ε2 give rise to the dispersion of f(s), through g(s”), for given zero limit frequency values of ε1 and ε2. The form of g(s”) should

570 Understanding Pore Space through Log Measurements

therefore be such that the low frequency dispersion observed in ε as a frequency dependent enhancement of permittivity over CRI predicted permittivity is adequately captured. It is to be noted that, the enhancement comes not only from the conventional Maxwell e Wagner effect and the dispersion from the Maxwell e Wagner Relaxation, but also from the equivalent effect for double layer associated with clay mineral surfaces and the associated relaxation. In order to maintain the physical perspective, it is important to bear in mind, the variation s with frequency, of measurement used. As frequency reduces, the magnitude of s0 also falls, and vice versa. The maximum value that s0 can assume is 1.0 and this happens at high frequencies. The frequency range in which the low frequency effects mentioned above are important is < 1000 MHz for a very saline brine (Resistivity of brine 0.05 U mt) saturated rock and below 100 MHz for much fresher brine saturated rock (Resistivity of brine 0.50 U mt). In both cases the value of s0 ranges 0e0.1. Fig. A1.1 illustrates the behavior of s0 which is same as that 
1  s s ¼ 1  εε12 for the full frequency range (Fig. A3.1).

1

FIGURE A3.1 The figure shows the graphs of the imaginary part of s½s ¼ ð1  εε12 Þ  versus its real part for different frequencies. The graphs are thus trajectories of the variable 1  
over the range of frequencies used in dielectric measurements, through which s s ¼ 1  εε12 the variable s is probed. Solid line pertains to the trajectory of s for a sandstone saturated with brine of conductivity 20 S/m. Dotted line pertains to the trajectory of s for a sandstone saturated with brine of conductivity 2 S/m. Reprinted from Stroud et al. (1986).

Archie’s cementation exponent Chapter j 8

571

Sum rules to be obeyed by ðsÞ It is important to note that the constraints that the spectral density function gðs0 Þ has to obey are not completely known. It is, however, known that, for the case of a medium which is nearly homogeneous, so that the deviation from homogeneity can be treated as a perturbation, a perturbation expansion of f(s) in powers of s is possible. Such an expansion (Brown, 1955; Herring, 1960), to second order in 1s is given below.
 4 4ð1  4Þ 1 þO 2 f ðsÞ ¼ þ (A3.42) 2 s 3s s R 1 0Þ 0
 f ðsÞ ¼ As þ 0 gðs ss0 ds (see Eq. (A3.38)) It is possible to expand the RHS of Eq. (A3.38) as an infinite series in 1s . When this series is truncated to a few terms the result is Z Z 1 A 1 1 0 0 1 gðs Þds þ 2 s0 gðs0 Þds0 þ ðthe sum of the remaining termsÞ: f ðsÞ ¼ þ s s 0 s 0 (A3.43) Comparing the coefficients of like powers of ð1s Þ in Eqs. (A3.42) and (A3.43) we obtain the following two sum rules (Bergman, 1982; Korringa, 1984). R1 i) R0 gðs0 Þds0 ¼ 4 e A 1 ii) 0 s0 gðs0 Þds0 ¼ 13 4(1 e 4)

Constraint on gðsÞ Since real valued and positive ε1 and ε2, together imply a real valued positive ε, a) the poles of f(s)are restricted to the interval (0,1) and b) f(s) 1 for s ¼ 1. The above statement is the stipulation that no matter what the form of f ðsÞcan be; f ð1Þ  1 (A3.44) Let us now consider Eq. (A3.38) for the case s ¼ 1. The equation for this case is, Z 1 gðs0 Þ 0 f ð1Þ ¼ A þ ds (A3.45) 0 0 1s From the inequality (A3.43) and Eq. (A3.45) we get the following constraint. Z 1 gðs0 Þ 0 ds  1 (A3.46) 0 0 1s

572 Understanding Pore Space through Log Measurements

Consider a medium R made of grain and fully saturated with brine identical in salinity to the salinity of brine saturating the pore space of the original rock. Let the microgeometrical arrangement of grain phase be identical to the microgeometrical arrangement of brine phase, of the original rock medium, and the microgeometrical arrangement of brine phase, identical to the microgeometrical arrangement of the grain phase, of the original rock medium. The medium R will be called as conjugate rock medium. Now, s ¼
1 . In the low-frequency limit, s tends to
1 . For the case of 1  εε12 s

1s1 2

original rock, since phase-1 is brine and phase-2 is grain, s is low valued. On the other hand, for the case of medium R (conjugate rock medium), since phase-1 is grain and  phase-2 is rock, s is close to 1.0. In the low frequency

s . It is reasonable to assume for real rocks, that the limit, f(s) tends to 1 s2

area of contact between neighboring grains is much less than the average cross-sectional area of a grain. Geometrically the microstructure of real rocks, therefore, is very close to the situation where grains float in brine. If this is so, then for the case of conjugate rock medium R, the geometry is that brine phase floats in grain phase as isolated droplets. In such a case, s the conductivity of the conjugate rock medium would be zero and therefore,  s ¼ 1:0 for the conjugate rock i) In the quasi-static limit, f ðsÞ ¼ 1 s2 medium, provided the area of contact between neighboring grains is much less than the average cross-sectional area of a grain, for the case of the original rock medium.  s1 ii) In the quasi-static limit, s ¼ 1 s2 ¼ 1:0 for the conjugate rock medium The implication of i) and ii) above is the following.   s1 s ¼ 1:0 also is true, When s ¼ 1 s2 ¼ 1:0, is true, then f ðsÞ ¼ 1 s2 which in turn means that when real rocks are to be modeled for complex dielectric permittivity, since in real rocks, the area of contact between neighboring grains is much less than the average cross-sectional area of a grain, it is reasonable to assume that the function f(s) has such a form that f(1) ¼ 1.0. Therefore, we can say that, For a general random binary mixture; f ð1Þ  1; for real rocks case f ð1Þ ¼ 1:0 (A3.47) Substituting for f(1) from Eq. (A3.47) into Eq. (A3.45), we get Z 1 gðs0 Þ 0 Aþ ds ¼ 1 0 0 1s

(A3.48)

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Therefore, for the case of modeling the dielectric permittivity of real rock R 1 gðs0 Þds0 media, there is little error if the constraint 0 1s0  (1 e A) (inequality (A3.46)) is modified to an equality Z 1 0 0 gðs Þds ¼ ð1  AÞ (A3.49) 0 0 1s

Further necessary condition that the spectral density function is expected to satisfy In addition to what has been discussed previously, the spectral density function should also be such that, the mixing law obtained using the spectral density function should be able to recover the, the CRI mixing law in the high frequency range. The form of spectral density function gðs0 Þ chosen in the model, for doing justice to experimental data at both low frequency range and high frequency range is b

gðs0 Þ ¼ Cðs0 Þ ð1  s0 Þ

e

(A3.50)

Here, the parameters C, b, and e cannot be arbitrary because, Eqs. (A3.44), (A3.45), (A3.49) are to be satisfied. Substituting for gðs0 Þ, in the equations R1 , it referred to above, and using the identity 0 ðs0 Þa1 ð1  s0 Þb1 ds0 ¼ GðaÞGðbÞ GðaþbÞ is seen that the equations are satisfied only if CGð1  bÞGð1 þ eÞ ¼4  A Gð2  b þ eÞ

(A3.51)

CGð2  bÞGð1 þ eÞ 4ð1  4Þ ¼ Gð3  b þ eÞ 3

(A3.52)

CGð1  bÞGðeÞ ¼1  A Gð1  b þ eÞ

(A3.53)

Dividing Eq. (A3.52) by Eq. (A3.51), and using the identity Gð1 þxÞ ¼ xGðxÞ for any x, we get ð1  bÞ 4ð1  4Þ ¼ ð2  b þ eÞ 3ð4  AÞ

(A3.54)

ð1  b þ eÞ ð1  AÞ ¼ e ð4  AÞ

(A3.55)

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Eqs. (A3.54) and (A3.55) are 2 equations in 2 unknowns. They are solved to get b ¼ 1  4ð1  4Þ=½24  Að3  4Þ

(A3.56)

e ¼ 4ð4  AÞ=½24  Að3  4Þ

(A3.57)

Substituting for b, e, respectively, into Eq. (A3.53), we get, C¼

ð1  AÞGð1  b þ eÞ Gð1  bÞGðeÞ

(A3.58)

The normal frequency range of data acquisition is below 1.1 GHz and absolute magnitude of s for this range is a prefixed value, the class is not partitioned further. Else it is partitioned, subject to a maximum population and minimum population limits. After the final partitioning is carried out, the final training class prototypes are computed again as well as their sharpness. The actual template used for scanning the training image, generate filter scores and score classes, and finally, class prototypes, is a 9  9 template and 3 multiple grids so that at the coarsest grid the template covers an area of 33  33 pixels which is large enough to see vuggy objects and their interactions. Fig. 10.7 is an illustration with using a 3  3 template.

10.3.3.3 Simulation to populate the pixels which have no data In what follows, protl(u) stands for the data in pixel defined by internal position vector u relative to the center of the prototype of class l. The simulation grid mesh size equals the template size in terms of number of pixels. Sample data are of n elements or less depending upon whether full data complement is available on the training image or not for that location of the simulation mesh. The data are relocated to the nearest grid node. This way, data are mapped to simulation grid nodes. Let n0 denote the data complement mapped to a node. The data can be of two types. It can be actual data present (hard data) or it could be data which were the result of a previous simulation and frozen as hard data. Again, the template is used to carry out the simulation. Let us denote a current node u0 visited by the template has full complement of hard data or already simulated data. This is checked by retrieving all the informed nodes which fall within the template centered at u0. An informed node can either be hard data or already simulated data. Let the number of data elements be n0 . If n0 ¼ n move to next node on the simulation path. If n0 u(2)>u(3) to emphasize the importance of hard original and frozen data. Once the training prototype (class prototype) closest to the conditioning data event is identified through the minimum value of Dl(u0), the prototype class having the maximum relevance to supplying the data to fill the unsampled locations in the template is identified. A pattern from this selected prototype class is randomly sampled and patched centered on the current simulation node, except at hard and previously simulated locations. If n0 ¼ 0 one pattern is randomly picked across all training prototypes or from the training image. This pattern is patched centered on the simulation node. This way, full-bore covering image is generated from the acquired image which has gaps which are gaps between the pads. The following flowchart in Fig. 10.9 summarizes the training pattern classification. The flowchart in Fig. 10.10 summarizes the Filtersim algorithm in brief. The data gap closure obtained by applying Filtersim is illustrated at Fig. 10.11. The full-bore image now obtained is considered as the input for the next stage of image analysis described further.

10.3.3.4 Image rescaling Before further processing the full-bore image, image is rescaled using MSFL and LSS logs. This is required in order to work with a calibrated conductivity image for the further processing steps. The processing steps now carried out on the conductivity-rescaled image are described. As above, the work flow is that described by Yamada et al. (2013). 10.3.4 Extraction of fracture segments The extraction of fracture segments draws upon image segmentation using mathematical morphology theory. A tried and tested method of extraction of fracture segments which was developed by Kherroubi (2008) is briefly described. The input to this process is the conductivity map of the borehole wall which is rendered in the form of a topographic surface. On this surface, higher elevations correspond to higher conductivity and lower elevations to lower conductivity. With respect to an average level, the lowest local conductivity pixels fall within troughs.

754 Understanding Pore Space through Log Measurements FIGURE 10.9 Flowchart summarizing training pattern classification. Reprinted with permission from Zhang (2006a, 2006b).

10.3.4.1 Extraction of low apparent dip fracture segments A vertical structuring element, called here as B, is considered. The length of B is of the order of the vertical extent over which contrast variation of pixels (or variation in apparent conductance associated with pixels) is of interest for the applications such as fracture aperture computation. The troughs smaller than the structural element get filled. The image is then browsed with a second vertical structural element longer than the first one. This has the effect of removing the lateral crests smaller in size in comparison with the vertical structural element. The residue between the resulting image from the above step and the original image is computed. This residue preserves the thin, highly conductive low apparent dip fracture segments. A threshold is now applied to keep only the highest conductivity pixels. A connected component labeling algorithm is used to identify and index the remaining pixel groups. Pixel groups with area smaller than a pre-defined value are removed, to eliminate

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FIGURE 10.10 Flowchart summarizing Filtersim. Reprinted with permission from Zhang (2006a, 2006b).

texture related features or measurement artefacts. The remnant pixel groups are finally accepted as data and a best fit sinusoid is found out for every connected component set, to obtain the dip and strike of the fractures. The input conductance image is denoted as I(i,j) where i,j stand for the pixel-coordinates. I(i,j) is closed using B and the resulting image is then opened using B. The resulting image is denoted as I1(i,j). Thus, I1 ði; jÞ ¼ ðO CðIÞÞði; jÞ

(10.22a)

Here, C is the “closing” operator and O is the “opening” operator. These operators are defined with respect to a given structuring element B of a specific shape and size as: ðOðIÞÞði; jÞ ¼ ðD EðIÞÞði; jÞ

(10.22b)

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FIGURE 10.11 Data gap closure achieved with Filtersim. Reprinted with permission from Yamada et al. (2013).

ðCðIÞÞði; jÞ ¼ ðE DðIÞÞði; jÞ

(10.22c)

The operators D,E, respectively, stand for the standard image morphological operators namely, the “dilation” operator and the “erosion” operator defined as below: ðDðJÞÞði; jÞ ¼ supfJði  x; i  yÞg; ðx; yÞ˛B

(10.22d)

ðEðJÞÞði; jÞ ¼ inf fJði  x; i  yÞg; ðx; yÞ˛B

(10.22e)

Here J is any binary or grayscale image in general. The bright pixels of I1 comprise i) areas of I which will be called as A1 that cannot fully overlap or enclose B, ii) those parts of the areas of I called as A2 which cannot have full overlap with B, while A2 encloses B. Thus, the areas of bright pixels of I1 comprise a foreground which cannot potentially be a part of fractures or bedding planes at borehole scale. The areas of bright pixels, which can potentially be a part of fractures or bedding planes at borehole scale, are therefore recovered by subtracting I1 from I. The resulting image is denoted as Iout. Iout ði; jÞ ¼ Iði; jÞ  I1 ði; jÞ ¼ Iði; jÞ  ðO CðIÞÞði; jÞ

(10.22f)

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A threshold is applied on Iout(i,j) to retain only the bright pixels whose brightness falls within the top 10 percentile, to generate a binary image, which will be denoted as IOUT(i,j). The foreground of IOUT(i,j) comprises extracted pixel sets which are called as traces. The connected component labeling algorithm Horn (1986) is now applied on IOUT(i,j). This algorithm groups the connected pixels which share nonzero pixel values and discards the rest. These groups define the traces of interest, from the perspective of fracture trace extraction. A trace is a fragment of the image expression of a planar event crossing the borehole that has geological meaning. The groups mentioned above are essentially such traces. The algorithm also filters out the small events expressed as groups of pixels, which arise due to noise or texture artifacts. The resulting image denoted as Itrace(i,j) here is a binary image whose foreground is the collection of traces which are fragments of the image expression of planar events mentioned above. These include fracture-derived traces as well as bedding derived traces, since bedding at borehole scale is essentially a collection of planes.

10.3.4.2 Spatial orientation of traces Groups of vectors, which when placed end-to-end fit best within a trace, are called as segments of that trace. Thus, each trace has a set of segments associated with it. The way these sets of segments are computed is discussed at Kherroubi (2008). The interested reader can refer to Kherroubi (2008) for the computation methodology. 10.3.4.3 Differentiating bedding planes and facture planes Bedding planes are distinguished by the fact that the vector normal to such a plane varies gently with depth. The vector function which captures this variation is obtained by interpolating the orientation of a plane, between the planes picked from routine image analysis, using linear interpolation. This information is used to filter out bedding related traces from the extracted traces from the previous steps. The process is discussed at Kherroubi (2008). The interested reader may please refer to Kherroubi (2008) for a detailed description of the process. 10.3.4.4 Main orientation of fractures A best fit algorithm Kherroubi (2008) is launched over the most likely fractures to obtain the main orientation of fractures. 10.3.4.5 Obtaining high confidence fracture traces The associated sets of segments of traces left postfiltering out of bedding related traces’ segments are subject to an algorithm which preferentially keeps only those segments that belong to fractures. This algorithm uses the

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information of the main orientation of fractures, obtained above. The traces corresponding to the final sets of segments are the high confidence fracture traces. These traces form the basis for defining the groups of pixels which are used to compute fracture aperture and fracture density. The computation of the fracture planes corresponding to these traces, and the graphical representation on the original images of the sinusoids corresponding to the fracture planes, and the extracted fracture traces also indicated, completes the process. 10.3.4.5.1 Extraction of high apparent dip fracture segments The work flow resembles the work flow used for extracting the low apparent dip fracture segments. The initial image in this case, however, is rotated at the start, by 90 degrees, before the work flow is applied on it and the same vertical structuring element used during the extraction of low apparent dip fracture segments is used. The curve which fits the final pixel groups (fracture segments) is no longer a sinusoid for this case.

10.3.5 Matrix extraction The background of the image corresponds to the geological term “matrix.” From the standpoint of micropores and mesopores, matrix includes both the pore classes. The extraction of matrix from the image proceeds through the application of grayscale reconstruction transform (Luc, 1993). The original conductivity image is taken as the mask or control. The original conductivity image is transformed to another image where the pixel conductivity values are less than the pixel conductivity values of the original image or mask. The transformed image is considered as the marker image. Now, using an appropriate structural element, geodesic dilation is carried out on the marker image, with the mask as the control image. Fig. 10.12 reflects the situation.

FIGURE 10.12 Illustration of geodesic dilation. X-axis is pixel number and Y-axis is conductivity value of the pixel. The upper solid line is the traverse for the mask and the lower solid line is the traverse for the marker. Left panel illustrates effect of one geodesic dilation. Right panel illustrates the result of successive geodesic dilations with the mask as the control. Reprinted with permission from Luc (1993).

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The graphs above show conductivity value on the y-axis and pixel number on the x-axis, for a traverse of the images. The upper solid line is the traverse for the mask and the lower solid line is the traverse for the marker. When a geodesic dilation operation is applied on the marker, the pixel value of the marker below the center of the structuring element assumes the maximum pixel value of the marker, for the pixels covered by the structuring element for that position. The effect on the marker image is that the peaks would increase in width, for any traverse and the troughs would seem to get filled. This is shown in the left side image, where the marker traverse after one geodesic dilation is shown as the curve whose interior is shaded gray. Successive geodesic dilations are applied, with the mask as the control. A final stabilized grown position is reached for the marker, as shown in the right-hand side image because, given the control curve, no further growth through geodesic dilation is possible. It can be noted that the conductive peaks of the original image or the mask, no longer figure in the stabilized dilated image’s traverse, nor do the troughs figure. Further note that, in the example, if the middle peak of the marker image has a peak value less than that of the middle peak of the mask, the pixels of the mask’s middle peak, having values greater than the peak value of the pixels of the middle peak of the marker, would not have figured in the final stabilized marker image. The procedure adopted for the removal of resistive features is symmetrical to the procedure described above, for removing the conductive features. The mask continues to be the original conductivity image. However, the marker image is obtained from the original image, by transforming the pixel values in such a manner that the marker image pixels have higher conductivity values as compared to those of the corresponding pixels of the mask. Again, using a structural element, geodesic erosion is carried out with the mask as the control (that is to say, erosion of marker image is carried out over the mask image). The geodesic erosion is carried out sequentially, till the eroded marker stabilizes, in the sense that the pixel values for any traverse of the geodetically eroded marker do not fall in value, below the value of the corresponding pixel of the mask. It is ensured that the geodesic dilation and geodesic erosion are made in such a way that preservation of features traversing in any direction as matrix is assured. As illustrated above, conductive heterogeneities are firstly removed, followed by resistive heterogeneities. The net result is that after the two procedures only matrix is left. Thus, at the end of the procedure, the image of the rock matrix is replicated in the output image, while the areas of the output image that correspond to the anomalies in the input image are indistinguishable from the image background surrounding them, in the output images. Fig. 10.13 shows the results of the procedure applied on real formation images, as reported by Yamada. et al. (2013). The left panel of the figure is the resistivity image of the borehole wall. The middle panel is the resistivity image of the matrix, which has been obtained from the conductivity image of the matrix, which in turn has been generated through the work flow described in the foregoing. The rightmost panel is the

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FIGURE 10.13 Illustration of geodesic dilation. At the left is the gap-filled conductivity image. In the middle is the extracted matrix image, and at the right is the averaged matrix conductivity. The same upper/lower limits of the color map are applied to both the color images, respectively, at the left and middle of the figure. Reprinted with permission from Yamada et al. (2013).

average conductivity of the conductivity pixel values corresponding to a segment of the image bounded by the bounding bedding surfaces as brought out at the middle panel of the figure. One meter has been shown as a scale, on the extreme left of the figure. The rightmost panel thus represents a highresolution matrix conductivity log with a higher sampling rate and finer resolution, compared to conventional logs. Matrix conductivity can be converted to matrix porosity using Archie Equation and assuming that the radial dimension of the volume of rock seen by button current is comparable with that seen by a shallow resistivity logging tool. Here we are assuming that water base mud has been used for drilling. The matrix porosity is given by, 1

4ma ¼ 4ext ðRext Ci Þm

(10.23)

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Here 4ext and Rext, respectively, stand for the rock porosity and shallow resistivity log reading against the formation from conventional logs, and m is Archie cementation exponent. The quantity Ci stands for the matrix conductivity extracted from images as discussed above.

10.3.6 The problem of computing the pore volume contribution of heterogeneities The term heterogeneity here stands for all the events which are bound by closed contours on images. These events have sizes on the images that fall within multimillimeters to decimeters scale. Their actual sizes in a rock correspond to 4e256 mm plus and fall within the “mega pore class” of Choquette and Pray (Delhomme 1992). The problem of computing the pore volume contribution of heterogeneities is twofold. First the heterogeneities will have to be extracted from the images. Then their pore volume contribution has to be evaluated. But this is not enough. It is also necessary to classify the events as connected events, events connected to leached bed boundaries or open fractures or solution channels and so on, and totally isolated heterogeneities.

10.3.6.1 The challenge of extraction of heterogeneities Heterogeneities smaller in size than the button electrodes of an imaging tool appear exaggerated in size and poor contrast with the surrounding image. When clusters of such heterogeneities such as vugs occur in close proximity to one another, too close to be resolved with the image resolution available, they merge together on the images and appear as large sized conductive heterogeneities. In general, irrespective of the size of the heterogeneity such as a vug, vis-a`-vis the button size, the point spread function of the acquisition system the vugs in the images appear to be surrounded by halos which make them look larger than they are, and make their bounding contours fuzzy (Delhomme, 1992) (The halo surrounding a vug is the visual expression on images, of the fact that a vug draws-in current in excess of that arising out of the conductivity contrast of the vug interior, with the surrounding medium. This excess current into the vug is compensated by a current deficiency around the vug). Even the minute cracks or solution channels which sometimes connect vugs, whose width is often much smaller than the electrode button size, are highlighted on the images and appear larger in width than their actual width because of same reason as mentioned above (Delhomme, 1992). The gray level value and the contrast of a heterogeneity such as a vug with its immediate surroundings on an image depends on the vug size and is hugely variable from vug to vug. This automatically renders any thresholding-based technique of possible extraction of the heterogeneities from the images, as useless. The only helpful aspect of vug behavior on images is the observation that once the vug size exceeds the button size, the inflection line of the conductivity value stays close to the physical edge of the vug, irrespective of the grayscale contrast of the vug event

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or the minor blurring observed sometimes due to the micro-standoff between the button and the borehole wall (Delhomme, 1992).

10.3.7 An efficient methodology for extracting heterogeneities The work flow described in the following is as per the work flow described by Yamada. et al. (2013). In order to avoid the problem of variable image contrast for the heterogeneities with respect to their surroundings, the conductivity image is converted first into a conductivity gradient image, where the grayscale value reflects the gradient of the conductivity rather than the conductivity. Each pixel of a gradient image measures the change in intensity of that same point in the original image, in a given direction. To get the full range of direction, gradient images in the x and y directions are computed. Pixels with the largest gradient values become possible edge pixels. Starting with the gradient image, the image is made the input for the watershed transform. To understand the action of the transform, it can be imagined that each pixel of the gradient image is mapped on to each pixel of a topographic surface, where the elevation represents the gradient image pixel value (the gradient of the conductivity). This is shown at Fig. 10.14. In Fig. 10.14A conductivity image is presented with black standing for maximum conductivity, in the left. The middle panel depicts the gradient image with black indicating highest gradient value and blue (dark gray in print version) indicating the lowest gradient value. The gradient image is depicted as a topographic map, with each pixel mapping on to an elevation value, at the right panel. Highest value has the color value of black and the lowest, blue (dark gray in print version). The highest points are called as watersheds. A drop of water, falling from above, on the watershed, can roll down in any direction, toward a local minimum. The slopes joining the local minima lines to the watershed lines are called as the catchment areas. Simply put, the expression “watershed transform” denotes a labeling of the image, such that all points of a given catchment basin have the same unique label, and a special label, distinct from all the labels of the catchment basins, is assigned to all points of the watersheds. Following this topographic rendering (Meyer and Beucher, 1990), suppose a hole is bored at every local

FIGURE 10.14 The left panel displays the original image, the middle panel displays the gradient image of the original image, and the right panel displays the segmented image of the gradient image. Reprinted with permission from Yamada et al. (2013).

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minimum on the surface and the surface is immersed in a lake at a uniform speed, vertically downward. The catchments are progressively flooded, equating to recursive geodesic dilations of markers (holes or punctures referred to earlier) that progressively shrink the reliefs, on the 2D map of the middle panel of the figure. As the progressive flooding of the catchments proceeds a dam is built, along the line where the floods would merge. The points where the spreading lakes would meet each other are known as extinguishing points in mathematical morphology. The allusion to the dams is the preservation of extinguishing points by suppression of other details. These form “skeletonizing by influence zones” (SKIZ). This process is continued till all the reliefs are accounted for, by the dams. The process used follows the algorithm of Vincent and Souille (1991). The basic algorithm in pseudocode is given at Appendix 1 for the interested reader. The output of the algorithm is the segmented into catchment basins with the watersheds forming the boundaries between segments. The segmented image is a mosaic with each catchment basin, a mosaic piece. Fig. 10.15 shows the gradient image, and the topographic rendering of the gradient image, with the watersheds (SKIZ) superimposed on both the images.

10.3.7.1 Classification of the mosaic pieces The pixel values of a mosaic piece, even though it is a catchment, can be above the matrix pixel values. In that case the mosaic piece is labeled as a conductive mosaic piece. Similarly, the pixel values of a mosaic piece, even though it is a catchment, can be below the matrix pixel values. In that case the mosaic piece is labeled as a resistive mosaic piece. Thus, conductive mosaic pieces and resistive mosaic pieces are two types of mosaic pieces. Apart from type, a mosaic piece has other attributes. One important attribute of a mosaic piece is size. Other important attributes are contrast with matrix and peak value and valley value.

FIGURE 10.15 The left panel displays the segmented rendering of the gradient image of the original image. The right panel displays the segmented topographic rendering of the gradient image. Watersheds (SKIZ) superimposed on both the images express the segmentation illustrated. Reprinted with permission from Yamada et al. (2013).

764 Understanding Pore Space through Log Measurements

FIGURE 10.16 The left panel depicts the gradient image of the original image, with the crest line of the original image (in yellow; gray in print version), superimposed on it. The right panel depicts the topographic rendering (landscape image) of the gradient image of the original image, with the topographic rendering of the crest line of the original image superimposed over it. Points on the crest line so depicted have different elevations, when they have differing conductivity values associated with them. Reprinted with permission from Yamada et al. (2013).

Now, the crest lines of the original image (from which gradient image has been generated) are extracted by applying the watershed transform to the original image. The crest lines are the SKIZ. The result is shown at Fig. 10.16, on the left. Every pixel of the image, which belongs to a crest line shown in the left panel of Fig. 10.16, carries a conductivity value. Hence crest lines on a topographic rendering of the conductivity image in the left panel of Fig. 10.16 appear as curved lines, where elevation of any point on a crest line would reflect its associated conductivity pixel value. This is shown in the right panel of Fig. 10.16. Superimposing the crest lines of Fig. 10.16 on the watershed lines of Fig. 10.15 and removing the watershed lines (SKIZ) that do not intersect the crest lines, the closed contour watershed lines (SKIZ) which define the boundaries of the heterogeneities can be separated out kept. Another way this can be done is to consider reconstruction by opening, of the watershed lines (SKIZ) (which forms the mask), with a round structuring element. At this stage, the heterogeneities with closed contours as boundaries have been extracted, typed as conductive heterogeneity or resistive heterogeneity, and classified as per size peak/valley value, contrast with matrix.

10.3.7.2 Connectedness attribute of a conductive heterogeneity As already mentioned, since, every pixel of a crest line shown in the left panel of Fig. 10.16 carries a conductivity value, crest lines on a topographic rendering of the conductivity image in the left panel of Fig. 10.16 would be

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curved lines. The topographic elevation of any point on a crest line would reflect its associated conductivity pixel value. This is shown in the right panel of Fig. 10.16. A cutoff conductivity threshold is now fixed. This corresponds to a plane having a certain elevation on the topographic rendering shown in the right panel of Fig. 10.16. Now, those of the crest lines which dip below the above plane are removed and only those crest lines which are above the threshold plane are kept. The retained crest lines indicate connectivity between the conductive heterogeneities such as vugs. Thus, the attribute of connectedness is obtained from images and vugs are now classified further, as connected vugs and isolated vugs. The quantification of the connectedness is the connectedness curve. The connectedness curve is defined by the average of the differences in conductivity between matrix and crest line (zero if there is no line) at each depth level.

10.3.7.3 Final porosity partition By the term porosity partition, we mean both porosity classification and porosity quantification on to the original conductivity image. The original conductivity image is thus segmented into different mosaic pieces. Appropriate cutoff values on value and contrast attributes of mosaic pieces on a facies by facies basis are set, and neighboring mosaic pieces are then automatically merged to form a heterogeneity feature. The previously extracted fracture segments and bed boundaries are now integrated back into the system in a spatial sense. The boundaries of the heterogeneities which are connected, isolated from each other, are, respectively, marked. For ease of communication, the term “spot” will be used here to denote a heterogeneity that has a closed contour as its boundary. Using the logic of existence of pixels whose conductivity exceeds a preset threshold, and which at the same time form connecting paths between a conductive spot and a fracture, conductive spots that are connected to fractures and conductive spots that are aligned along bed boundaries are also identified and their boundaries are marked. Spots connected by crest lines to another spot are classified as connected spots. Conductive spots connected to fractures are called fracture spots. Conductive spots aligned along bed boundaries are called as bed boundary spots. Conductive spots which have no connectivity to either any other conductive spot or to a fracture or to a bed boundary are classified as isolated spots. 10.3.7.4 The mosaic image and the heterogeneity image To obtain the mosaic image, representative conductivity and conductivity contrast at mosaic piece scale are computed for conductive type mosaic pieces. Representative resistivity and resistivity contrast are computed at mosaic piece

766 Understanding Pore Space through Log Measurements

scale, for resistive type mosaic pieces. Mosaic pieces of the conductive type, whose conductivity and conductivity contrast cross pre-set thresholds, are assigned a hot color such as “red”. Mosaic pieces of the resistive type, whose resistivity and resistivity contrast cross pre-set thresholds, are assigned a cool color such as “blue”. The mosaic pieces when put together constitute the mosaic image. For the case where bed boundaries and fractures are absent, the heterogeneities mosaic image highlights the connected conductive spots and the isolated conductive spots. To create a heterogeneity image, first, each mosaic piece is considered. If the mosaic piece is a resistive type mosaic piece, every pixel of the mosaic piece whose resistivity contrast as well as resistivity exceed, respectively, preset threshold values is marked out on the mosaic piece and assigned a specific color say blue. If the mosaic piece is a conductive type mosaic piece, every pixel whose conductivity value as well as its conductivity contrast exceed, respectively, preset threshold is assigned an initial color say green (gray in print version). It is then examined if this pixel is part of a conductive spot which has connection with any other conductive spot. If yes, the pixel is assigned a color say red (light gray in print version). If the pixel is part of a conductive spot which is connected to a fracture it is assigned a color say buff (very light gray in print version). Else if the pixel if a conductive spot which is connected to a bed boundary, the pixel is assigned a color say pink (medium dark gray in print version). If none of the above is true, the previously assigned color of green (gray in print version) stands. When all the mosaic pieces are considered together, we get the Heterogeneity Image (see Fig. 10.17).

10.3.7.5 Classification of heterogeneities and quantification of the porosity associated with them The left most panel of Fig. 10.17 is the heterogeneity image. In this heterogeneity image, red (light gray in print version) color signifies the connected vugs (connected spots or vugs which are connected by crest lines as discussed earlier). Green (gray in print version) color signifies isolated spots which means isolated vug porosity. Blue (dark gray in print version) color signifies the resistive spots. The respective proportion of these events in terms of borehole wall area is quantified as the red (light gray in print version), green (gray in print version), and blue (dark gray in print version) colored curves, respectively, and presented on the first track on the right-hand panel of the figure. Cumulative area in square inches of the connected vug-sets and the resistive events have, respectively, been presented as red (light gray in print version) and blue (dark gray in print version) curves at track 2 of the right-hand panel, respectively. Track 3 shows the conductivity

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FIGURE 10.17 From left, heterogeneity image (red (light gray in print version)dconnected spots or vugs; green (gray in print version)disolated spots (isolated vugs); and blue (dark gray in print version)dresistive spots), overlay of the fractional well bore area occupied by the heterogeneities (color code is as for the heterogeneity image), overlay of the representative size of connected conductive spots (red curve; light gray in print version) and resistive spots (blue curve; dark gray in print version), overlay of the representative contrast of connected conductive spots (red curve; light gray in print version) and resistive spots (blue curve; dark gray in print version), and connectedness of the conductive spots (connectedness of the vugs). Reprinted with permission from Yamada et al. (2013).

contrast of the connected vug events with respect to the matrix conductivity, in red (light gray in print version), and the resistivity contrast of the resistive spots with respect to the matrix resistivity, in blue (dark gray in print version). Finally, the connectedness, defined by the average of the differences in conductivity between matrix and crest line (zero if there is no line) at each depth level, as mentioned above, is presented at track 4 of the panel. All these curves are the respective computed data smoothed over a 1 ft sliding window on depth. Out of this total classification conductive spots discussed above, those conductive spots that are not vugs but shale nodules are edited out as resistive spots based on visual observation and local knowledge. Similarly drilling induced fractures are edited out through visual observation.

10.3.7.6 Porosity association of the different types of spots (heterogeneities) Porosity arises from matrix, different classes of vugs, fractures, leached bed boundaries, etc. Visually, these constitute image texture. Therefore, these can be

768 Understanding Pore Space through Log Measurements

called as texture classes. Every pixel on a conductive image belongs to one of these texture classes. Using Eq. (10.23), the conductivity value associated with each pixel can be converted to a porosity value. Thus, we can imagine a porosity image made out of porosity pixels. The totality of pixels of porosity at any depth level can be classified into different texture classes. This enables one to define porosity associated with a texture class and average porosity of a texture class, as follows. At this juncture it is worthwhile to note that as an alternate work flow, heterogeneity delineation can be run directly on a porosity image. It is important to appreciate that every conductive event on an image cannot be porosity, and every fracture cannot be meaningful for understanding the transport characteristics of the formation. For example, a clay nodule can be a conductive spot but is not a vug. A drilling induced fracture event has no role to play in permeability. A clay filled stylolamination can have no role in fluid transport and so on. Based on local geological knowledge these events are edited out, prior to porosity analysis. Let “t” stand for a texture class and let I(t) denote the number of porosity pixels which fall into the texture class (matrix or fracture or bed boundary or connected vug or isolated vug or fracture vug and so on). Let 4i stand for the porosity value of a pixel i and 4imagetotal stand for total porosity from image. Let 4ðtÞcontribution denote the porosity contribution of all porosity pixels which belong to texture class t. And finally let 4ðtÞave be the average porosity which can be associated with a porosity pixel belonging to texture class t. These quantities are given by, I 1X 4 I i¼1 i

(10.24)

1 X 4 IðtÞ i˛t i

(10.25)

4imagetotal ¼

4ðtÞave ¼

4ðtÞcontribution ¼

IðtÞ4ðtÞave I

(10.26)

Here, I is like the total number of voxels each of same volume, and indexed by i with the ith subvolume having the internal porosity 4i and I(t) is like the total number of voxels each of same volume, and indexed by i with the ith subvolume having the internal porosity 4i but the voxels limited to those hosting the porosity associated with the texture class t. For the example illustrated at Fig. 10.17 there are no fractures or internal bed boundaries. The texture classes are connected vugs, isolated vugs, resistive spots, and matrix.

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FIGURE 10.18 From left, heterogeneity image, porosity distribution, porosity contribution, average porosity. Color code e Red (light gray in print version) relates to connected spots (connected vugs), green (gray in print version) relates to isolated spots (isolated vugs), blue (dark gray in print version) relates to resistive spots, and buff (very light gray in print version) relates to the matrix. Reprinted with permission from Yamada et al. (2013).

Fig. 10.18 illustrates the complete porosity partition as component values texture classwise, porosity histogram indicating the porosity partition into different texture classes, and average porosity associated with a texture class. As illustrated at Fig. 10.18, the porosity contribution of the connected vugs and the porosity contribution of the isolated vugs are now known. The contribution of the matrix to the porosity is the internal porosity of the matrix, which has been denoted as matrix porosity, earlier on in this analysis is also known from Eq. (10.7). In Fig. 10.18 red color (light gray in print version) denotes everything related to connected vugs, green color (gray in print version) denotes everything related to isolated vugs, and blue color (dark gray in print version) denotes everything related to resistive spots. Buff (very light gray in print version) color denotes everything related to matrix. Heterogeneity image is at the extreme left panel. Histogram of the porosity contribution of matrix, connected vugs, isolated vugs, and resistive spots is at the track adjoining the heterogeneity image. Bin width and vertical axis are given for the histogram displays, at the bottom of the track. Porosity contribution of the texture classes is quantitatively given at the track to the right of the track displaying the porosity distribution histograms referred to above. The pink (medium dark gray in print version) curve on this track is porosity from conventional logs.

770 Understanding Pore Space through Log Measurements

Appendix 1 The watershed algorithm of Vincent e Soille (1991).

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771

772 Understanding Pore Space through Log Measurements

10.4 Porosity partition using acoustic logs 10.4.1 Challenges of porosity partition using acoustic logs The work flow of porosity partitioning is based on inverting the elastic constants of the rock into the volume fractions of the components of the rock. Rock is modeled as a composite of a base rock and inclusions. A suitable forward model of the elastic constants of the rock is selected. The elastic constants of the components of the model are modeled in terms of the bulk solid elastic constants and the fluid elastic constants, respectively. The forward model is then built, assuming certain volume fractions of the components of the model. The volume fractions are optimized for best fit of model elastic constants of the composite, with the elastic constants of the composite, measured using acoustic shear and compressional wave slowness data. The pore volume fraction associated with the respective components are next computed using the component volume fractions and their elastic constants. Early work assumes a model of a rock as a continuum which hosts inclusions. In such a model the host has a porosity which is known as host porosity and a cumulative pore volume associated with it. Similarly, the inclusions have a cumulative pore volume associated with them which is nothing but the cumulative volume of inclusions. This is called as inclusion porosity. The porosity partition consists in partitioning the total porosity into host porosity and inclusion porosity, or as host pore volume fraction and inclusion pore volume fraction. The popular choice of the forward model of the elastic constants of the composite rock has been the Kuster and Toksoz (1974) model. The defining features of the model are single scattering from an inclusion, which precludes

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interaction between the inclusions via the wave field, and inherent asymmetry of the model with respect to the model components (host and inclusions). At the same time it assumes the applicability of the long wavelength limit (long wavelength limit implies that the wavelength of the propagating shear and compressional wave field is much larger than the dimensions of the largest heterogeneity present in the rock (dimensions of the largest sized vug)). The model assumes that the scatters are of spherical shape. This model is suitable for carbonates when the vug concentration is low, which is not always the case. Two examples (Brie et al. (1985) and Sarkar and Dutta (2018)) illustrate use of KustereToksoz model with host properties modeled using WyllieePickett relation, Krief’s relation, and Berryman’s relation respectively. For modeling the elastic constants of carbonate rocks, since the concentration of the inclusions cannot be assumed to be dilute, effective medium theories which are inherently asymmetric with respect to the rock which occurs in-between inclusions and the inclusions might not be the best choice for forward modeling rock acoustic properties. In order to avoid some of the challenges, formulations symmetric with respect to host and inclusions are used. In these models, the term “host” is used in a very limited sense. The meaning of the term “host” is limited to “rock which is present in-between inclusions.” One of the challenges of modeling fluidesaturated rocks has been the issue of the models ensuring fluid pressure equalization, between the components of the model. It is known that working with dry rock effective medium theories, porosity partition can be efficiently effected, sidestepping the aforementioned challenge. An added advantage enjoyed by such approaches is that thereby one avoids the necessity of factoring in hydrocarbon saturation. An approach is now illustrated Ramamoorthy et al. (2010), Ramakrishnan et al. (2001), which is based on a self-consistent model of Berryman Berryman (1980). A brief account of Berryman’s model is given at Appendix 2.

10.4.2 A work flow of porosity partition using acoustic logs In what follows, the term “host” is to be understood only in the limited scope of its meaning as clarified above. Also, the terms “vug” and “inclusion” are used interchangeably. In the scope of the following analysis, any spherical pore space which scatters the acoustic incident wave field falls within the ambit of the terms “vug” and “inclusion” used interchangeably, here. According to the critical porosity based forward model of Nur (Nur (.)) the shear modulus G of a rock made of solid grains of material of shear modulus GS and having intergranular porosity 4 is given by   4 G ¼ GS 1  (10.27) 4c where 4c stands for critical porosity.

774 Understanding Pore Space through Log Measurements

A modified relationship which better matches actual data has been suggested by Ramamoorthy et al. (2010)    4 4 G ¼ GS 1  1 (10.28) 4c SG Here, SG is an empirically determined constant found by fitting actual data to the model. In the case of rocks which host intragranular porosity and intergranular porosity but no vug porosity, Eq. (10.28) is modified to,      4m 4m fm fm G ¼ GS 1  1 1 1 (10.29) 4c SG 4c SG Here 4m is the intragranular porosity and fm is the intergranular pore volume per unit rock volume. Intragranular porosity falls under the category of microporosity. Intergranular porosity falls under the category of mesopore porosity. The sensitivity of G to 4m is much less than the sensitivity of G to fm. This is also in line with the general observation that shear modulus has low sensitivity to microporosity. Therefore Eq. (10.29) is modified as,    fm fm G ¼ GS 1  1 (10.30) 4c SG Eq. (10.30) is used as the forward model for the shear modulus of the host, and where fm is the pore volume per unit volume of host. The host porosity is denoted as 4host and is equal to fm which in turn represents mesopore pore volume per unit volume of host. The shear modulus of host is denoted as Ghost. We can therefore write,    4 4 Ghost ¼ GS 1  host 1  host (10.31) 4c SG In a similar manner, Khost the drained bulk modulus of host medium can be modeled as,    4host 4host Khost ¼ KS 1  1 (10.32) 4c SK Here KS stands for the bulk modulus of the grain material and SK is an empirically determined fitting constant. Let fv stand for the cumulative volume of vugs per unit rock volume. We then have, 4 ¼ ð1  fv Þ4host þ fv which implies 4host ¼

4  fv 1  fv

(10.33)

As already remarked, the only viable forward model of a vuggy rock is one that honors self-consistency criterion and assures symmetry of any effective medium

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property with respect to host and inclusions. In the context of an effective medium theory of acoustic properties of a composite medium which represents a vuggy carbonate, the best suited model is Berryman’s self-consistent forward model of a host with inclusions. This model is discussed at Appendix 2. The model is based on the Coherent Phase Approximation (CPA) scheme, which is also called as SelfConsistent (SC) scheme. The principal results which follow from Berryman’s theory (Berryman, 1980) are expressed as the following two equations, which are true in the long wavelength limit (Wavelength of the shear or compressional waves through the propagation of which, the medium properties are probed, is much larger than the dimensions of the largest heterogeneity present in the rock (dimensions of the largest sized vug)). The following equations are Eqs. (A2.4)e(A2.7) of the appendix renotated in terms of the vug fraction, dry rock moduli, since as already discussed at Appendix 2, the necessary condition for self-consistency of the model of the rock is that the medium type - *, of the appendix, is identical with the effective medium of the rock. This in turn means that the bulk elastic moduli and density of the rock, being equal to the corresponding properties of the effective medium of the rock, equal the corresponding properties of the type - * medium. Thus, Kb ¼ K  ; G ¼ m ; fv ¼ c1, Kincl ¼ K1, Khost ¼ K2 ; Ghost ¼ m2 ; Gincl ¼ m1 ;Fb ¼ F  , rincl ¼ r1 , rhost ¼ r2 12, rb ¼ r ; where Kb stands for the dry rock bulk modulus, G stands for the dry rock shear modulus, which also equals the saturated rock shear modulus, Khost stands for the “host” bulk modulus under drained condition, Ghost stands for the “host” shear modulus under drained condition, which also equals the “host” shear modulus under saturated (undrained) condition, rb stands for dry rock density, rhost stands for “host” density under drained condition, and rincl stands for inclusion material density. Finally, c1þc2 ¼ 1 which implies c2 ¼ 1fv. 1 fv 1  fv ¼ þ 4 4 Kb þ 3 G Kincl þ 3 G Khost þ 43 G

(10.34)

rb ¼ fv rincl þ ð1  fv Þrhost

(10.35)

1 fv 1  fv ¼ þ G þ Fb Gincl þ Fb Ghost þ Fb

(10.36)

with Fb given by Fb ¼

G 9Kb þ 8G 6 Kb þ 2G

(10.37)

The solution for the vug fraction, which is the cumulative vug porosity per unit rock volume, denoted by fv, is obtained from Eq. (10.36) as, fv ¼

Fb ðGhost  GÞ Ghost ðG þ Fb Þ

(10.38)

776 Understanding Pore Space through Log Measurements

Substituting for fv in Eq. (10.34) and inverting it for Kb we get, 4 Þ 3 GKhost ð1  fv Kb ¼  4 3 G þ Khost fv

(10.39)

In rocks the medium within the inclusions is a fluid and hence Gincl ¼ 0. Equipped with the above important results, the following work flow is conceived for computing fv from available data. 1. Obtain 4 the intergranular porosity from conventional logs with the parameters of the grain composite chosen appropriately, or consider the total porosity as obtained from NMR. 2. Input model parameters 4c, SG, SK, GS, KS. Values of GS, KS can be assigned using known data, in case of single mineral composition of the solid phase. In case the solid phase involves two mineral species, the mineral end point moduli are best determined using the HashineShtrikman bounds Hashin and Shtrikman (1963). When the number of mineral components exceeds 2 then Berryman extension Berryman (1995) is recommended Ramakrishnan et al. (1999). 3. Assume intragranular porosity 4m ¼ 0. 4. Start with initial value of fv ¼ 0. 5. Compute 4host using Eq. (10.24). 6. Compute Khost, Ghost using Eqs. (10.30) and (10.31), respectively. 7. Obtain G from log data of bulk density rb and shear slowness DtS as G ¼ rb ðDtS Þ2 . 8. Compute Kb using Eq. (10.30). 9. Compute Fb using Eq. (10.28). 10. Compute fv using Eq. (10.29). 11. Go to step 4 and update the value of fv and execute steps 5e11, and carry on with the iteration till the value of fv converges (magnitude of the difference between values of fv computed through two successive iteration loops falls within a preset limit). 12. Compute final 4host , Khost, Ghost, Kb. Pore volume per unit rock volume hosted by vugs is fv computed, and pore volume hosted by base rock per unit total rock volume is 4host(1fv).

What is a “vug” in the model? In the model, an inhomogeneity of spherical shape, which scatters the compressional wave field, giving rise to scattered compressional and shear wave fields, is designated in the model as a vug. Since there is no strict lower limit on the dimensions of a vug in the model, fv can exceed the vug fraction computed from image data. Since NMR thresholds on T2 to compute vug porosity are tuned using image data, fv arising from the inversion of acoustic data can exceed vug fraction computed using NMR data.

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Sensitivity of model results to the shape of the vug The general shape to which the shape of a vug is often approximated is an oblate spheroid. A measure of the shape of the spheroid is the ratio between the length of the minor axis and the length of the major axis of the ellipsoid whose solid of revolution about its minor axis is the oblate spheroid approximating the shape of a vug. The shape of the vug does have effect on the scattering of acoustic waves. The shape effect is illustrated at Fig. 10.19 (Berryman (1980)). The y-axis is computed shear modulus of solid calcite hosting oblate spheroid inclusions. In Fig. 10.19, Aspect Ratio is the ratio between the minor axis and the major axis of the ellipse, whose solid of revolution around its minor axis is the geometric shape of the pore. Aspect Ratio of 1.0 implies spherical shape. It is noted from Fig. 10.19 that sensitivity of the shear modulus to Aspect Ratio is very low when Aspect Ratio is more than 0.5. This is the normal case for vugs present in carbonate rocks. Other shapes such as cracks and short length leached solution channels and so on, if they are dominant, cause deviation between actual values of the elastic moduli of the rock and model results. Therefore, it can be concluded that the model of spherical pores is valid for a majority of rocks encountered that are normally classified as “Conventional Reservoirs”.

FIGURE 10.19 Effect of the shape of Inclusions on the Shear Modulus of the composite medium. Reprinted with permission from Berryman (1980).

778 Understanding Pore Space through Log Measurements

Appendix 2 The self-consistent theory of Berryman in the long wavelength limit The treatment given here is after Berryman (1980).

Introduction to KustereToksoz model The effective material or effective medium of a composite is a homogeneous medium which, when it replaces the composite within the same geometric boundary which the composite existed during any experiment involving acoustic wave propagation, leads to the same outcome as that of the original experiment, when that experiment is repeated with the effective material or effective medium replacing the original composite. Let the effective medium or effective material be denoted as medium E. In the following, numeral 1 denotes inclusions, numeral 2 denotes host, and * denotes the effective medium E (exclusively for this section and Section KustereToksoz estimates which follows this). The inclusions are spherical inclusions. Fig. A2.1 illustrates the model on which KustereToksoz estimates are based, where in medium 2 is a continuum hosting medium 1 in the form of inclusions.

FIGURE A2.1 Model on which KustereToksoz equations are based. Reprinted with permission from Berryman (1980).

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Please note that the dotted medium shown in the figure is the indeterminate medium, which will be referred to as matrix. Consider a scattering experiment, in which a plane compressional wave propagating along x-axis, is scattered by the region of the composite of media 1, 2, and *, bound by a spherical surface. There will be a scattered wave field inside the sphere in the form of compressional wave field and shear wave field, as well as a scattered wave field external to the sphere, which will comprise a compressional wave field and a shear wave field. The displacement vector of the incident plane compressional wave propagating along x-axis is given by, u¼b x ðikÞ1 expðkx  utÞ

(A2.1)

Here k stands for the wave number and u stands for angular frequency. b x represents unit vector along x direction. The amplitude has been shown as unity for simplicity. The factor (ik)1 is required to give u the dimensions of displacement, and to ensure that the strain tensor components are real in the limit k/0. Let us now focus on the scattered fields, and specifically on the displacement vectors of these respective scattered fields. The displacement vector of the scattered field from an inclusion indexed by i is denoted as ðus Þi . The radial component of ðus Þi is denoted as ðusr Þi and the transverse component of ðus Þi is denoted as ðust Þi . The radius of an inclusion indexed by i is denoted as ai. We then have,    s 1 1 ur i ¼ ðjkÞ1 ðkai Þ3 expðkri  utÞ B0  B1 cos q  B2 ð3 cos 2 q þ 1Þ kri 4 (A2.2)    s 1 3s ut i ¼ ðjkÞ1 ðsai Þ3 expðsri  utÞ B1 sin q þ B2 sin 2 q (A2.3) sri 4k Referring to Fig. A2.1 the medium surrounding each inclusion is the medium of type 1 or “host,” which is designated also as “matrix.” The subscript m notates matrix related quantities. Let rm, Km, and mm, respectively, denote the density, bulk modulus, and shear modulus of matrix. Then, k and s are given by, 0 112 B k ¼ u@

rm C A Km þ 43mm

(A2.4)

It is to be noted that k of Eq. (A2.4) is identical to the k of Eq. (A2.1) because the incident plane wave, when it is away from any inclusion and thus scattering effects are absent, is essentially propagating through the matrix.

780 Understanding Pore Space through Log Measurements

The shear wave number of the matrix is denoted by s which is given by,  12 r s¼u m (A2.5) mm ri is the magnitude of the position vector of the observation point relative to the center of the inclusion of type i. If x denotes the position vector of the observation point relative to the origin, and 2i denotes the position vector of the center of the inclusion relative to the origin, ri ¼ jx  2i j

(A2.6)

q stands for polar angle. B0 ðKm ; mm ; Ki Þ ¼ B1 ðrm ; ri Þ ¼ B2 ðKm ; mm ; mi Þ ¼

ðKm  Ki Þ ð3Ki þ 4mm Þ

ðrm  ri Þ ð3rm Þ

20mm ðmi  mm Þ=3 6mi ðKm þ 2mm Þ þ mm ð9Km þ 8mm Þ

(A2.7) (A2.8) (A2.9)

The KustereToksoz estimates Consider a sphere of radius a of the composite medium. The composite medium consists of a medium 1, which hosts medium 2 as spherical inclusions. Consider plane wave propagation as described above. A situation as shown in Fig. A2.1 can be imagined. When the concentration of inclusions is low, the sum of the displacement fields due to the inclusions can be considered as the displacement field due to a sphere identical to the sphere in Fig. A2.1, but whose interior is nothing but the composite medium, for locations far from the surface of the sphere. The logic behind this is noninteraction between inclusions, and the matrix itself becoming the medium 2 or “host” of the rock model. Thus, remembering the notation of * denoting the composite medium and since mh2, Eqs. (A2.2) e (A2.9), respectively, become,    s 1 1 3 1 ur 1 ¼ ðjkÞ ðka1 Þ expðkr1  utÞ B0  B1 cos q  B2 ð3 cos 2 q þ 1Þ kr1 4 (A2.10)    s 1 3s ut 1 ¼ ðjkÞ1 ðsai Þ3 expðsri  utÞ B1 sin q þ B2 sin 2 q (A2.11) sr1 4k

Permeability and electrical conductivity Chapter j 10

0

112

B k ¼ u@

r2 C A K2 þ 43m2

781

(A2.12)

It is to be noted that k of Eq. (A2.4) is identical to the k of Eq. (A2.1) because the incident plane wave, when it is away from any inclusion and thus scattering effects are absent, is essentially propagating through the medium 2. The shear wave number of the medium 2 is denoted by s which is given by,  12 r s¼u 2 (A2.13) m2 ðK2  Ki Þ ð3Ki þ 4m2 Þ

(A2.14)

ðr2  ri Þ ð3r2 Þ

(A2.15)

B0 ðK2 ; m2 ; Ki Þ ¼ B1 ðr2 ; ri Þ ¼ B2 ðK2 ; m2 ; mi Þ ¼

20m2 ðmi  m2 Þ=3 6mi ðK2 þ 2m2 Þ þ m2 ð9K2 þ 8m2 Þ

(A2.16)

The displacement field due to a sphere identical to the sphere in Fig. A2.1, but whose interior is nothing but the composite medium, is given by 

 s  1 ur ¼ ðjkÞ1 ðkaÞ3 expðkr  utÞ  kr 1    B0 ðK2 ; m2 ; K Þ  B1 ðr2 ; r Þcos q  B2 ðK2 ; m2 ; m Þð3 cos 2 q þ 1Þ (A2.17) 4

Here, a stands for the radius of the sphere. When we add the radial components of the scattered field due to all the inclusions it should be equal to ðusr Þ as per the logic given above. Similarly, we add the transverse components of the scattered field; due to all the inclusions it should be equal to ðust Þ as per the logic given above. Imposing this condition for the radial component of the displacement, we get, from Eq. (A2.1) and Eq. (A2.8), 1 ðjkÞ1 ðkaÞ3   kr 1 ¼ expðkr  utÞ B0 ðK2 ; m2 ; K  Þ  B1 ðr2 ; r Þcos q  B2 ðK2 ; m2 ; m Þð3 cos 2 q þ 1Þ 4   i¼n X 1 1 3 1 ðjkÞ ðkai Þ expðkr1  utÞ B0  B1 cos q  B2 ð3 cos 2 q þ 1Þ (A2.17a) ¼ kr1 4 i¼1

Here, n stands for the total number of inclusions. i stands for the index number of an inclusion. All inclusions are not necessarily of the same radius.

782 Understanding Pore Space through Log Measurements

It is to be noted that the very fact of summing the scattering displacements implicitly assumes independent single scattering events and absence of interaction between any two scatterers. The position vector of the observation point is x. For large distances from the center of the sphere, r1 yr  n:2i . Hence the equation above can be written, after simplification, as,

  1 a3 B0 ðK2 ; m2 ; K  Þ  B1 ðr2 ; r Þcos q  B2 ðK2 ; m2 ; m Þð3 cos 2 q þ 1Þ 4   i¼n X 1 a3i expð jkn:2i Þ B0 ðK2 ; m2 ; K1 Þ  B1 ðr2 ; r1 Þcos q  B2 ðK2 ; m2 ; m1 Þð3 cos 2 q þ 1Þ ¼ 4 i¼1 (A2.18)

Invoking the assumption of long wavelength limit We note that the only dependence of RHS of Eq. (A2.18) on k is through the phase factor expðjkn:2i Þ. In the long wavelength limit, ka  1 and for any i, kjn:2i j < ka. Consequently, the phase factor expðjkn:2i Þ can be replaced by unity in Eq. (A2.18) with no loss in accuracy. Let c1 denote the cumulative volume of inclusions per unit volume of the composite.   i¼n iP ¼n P  3 4 3 ai pa i 3   ¼ i¼1 3 (A2.19) c1 ¼ i¼1 ða Þ 4 3 3 pa Substituting from Eq. (10.19) into Eq. (10.18) replacing the phase factor with unity and since q is independent of the other quantities it is seen that the following equations have to hold for Eq. (10.18) to hold. B0 ðK2 ; m2 ; K  Þ ¼ c1 B0 ðK2 ; m2 ; K1 Þ

(A2.20)

B1 ðr2 ; r Þ ¼ c1 B1 ðr2 ; r1 Þ

(A2.21)

B2 ðK2 ; m2 ; m Þ ¼ c1 B2 ðm1 ; m2 ; K2 Þ

(A2.22)

Eqs. (10.20)e(10.22) translate to, K2  K  K2  K1 ¼ c1  3K þ 4m2 3K1 þ 4m2

(A2.23)

r2  r ¼ c1 ðr2  r1 Þ

(A2.24)

Permeability and electrical conductivity Chapter j 10

m  m2 m  m2 ¼ c1 1  m þ F2 m1 þ F1 Where F1 ¼

m2 9K1 þ 8m1 6 K1 þ 2m1

783

(A2.25) (A2.26)

Eqs. (A2.23), (A2.25) and (A2.26) constitute the KustereToksoz estimates and are known as KustereToksoz equations Kuster and Toksoz (1974). In rocks the material medium 1 which makes for inclusions is a fluid and hence m1 ¼ 0.

Self-consistent estimates in the long wavelength limit Let us start with recapping the definition of an effective medium or effective material, of a composite made up entirely of medium 1 and medium 2, as follows. Suppose we conduct an experiment whose outcomes depend on a volume of the composite. Then the effective material or effective medium of a composite is a homogeneous medium which, when it replaces the composite within the same geometric boundary which the composite existed during any experiment involving acoustic wave propagation, leads to the same outcome as that of the original experiment, when that experiment is repeated with the effective material or effective medium replacing the original composite. Let the effective medium or effective material be denoted as medium E. Consider a sphere S1 of a composite medium made up entirely of medium 1 and medium 2. Also consider a sphere S2 made up of the effective medium E of the composite, and geometrically identical to S1. Consider a case (Case 1) where the sphere S1 is embedded in the effective medium E of the composite. Consider a scattering experiment, in which a plane compressional wave propagating along x-axis, is scattered by S1. Similarly, consider another case (Case 2) where sphere S2 is embedded in the effective medium E of the composite. Consider a repeat of the scattering experiment mentioned above. It is logical to assume that the displacement at a point far away from whichever sphere is scattering the incident field would be the same, for either experiment. Now consider a case (Case 3), where the bounding surface of S1 is enclosed by a sphere S3 of radius a. Let the annular space between S1 and S3 be also filled with the composite. The displacement at a point far from S3 would still be the same as for case 2, when the scattering experiment described above is repeated for this case also. Consider a case (Case 4), where the interior of S3 of Case 3 is modified to a system where with the annular space between S1 and S3 is filled with E the effective medium of the composite, and let the modified sphere S3 be immersed in E.

784 Understanding Pore Space through Log Measurements

The displacement field in regions far from the modified S3 would still be the same as for Case 3, if the scattering experiment for Case 3 is repeated, in this case (Case 4). Let us now consider a final modification of Case 3, as follows. Consider a rearrangement of the contents of S3 of Case 3, as follows. The contents of the sphere S1 (which comprise medium 1 and medium 2 only) are now distributed as inclusions inside S3 with the volume intervening the inclusions, filled with medium E. The relative proportions of medium 1 and medium 2 in S3 are, respectively, equal to the relative proportions of medium 1 and medium 2 in S1. Let the current modified sphere S3 be embedded in E. There is no compositional change in the average properties of the current sphere S3. The effective medium of the contents of the current modified S3 continues to be identical to the effective medium of S3 of Case 3 (the case where there was S1 enclosed by the outer surface of S3 with the annular space between S1 and S3 filled with E). Therefore, if the scattering experiment described above is now conducted, with S3 embedded in E, the displacement at a point far away from S3 would be identical to that at the same point in Case 1 and Case 2. But Case 2 is nothing but only E existing. Thus, the above analysis implies that, for Case 3, in a scattering experiment the inclusions will act together, as if they, taken together, were transparent to the incident wave field. Now this will make physical sense only when the wavelength is much greater than the size of the inclusions, which is precisely what the long wavelength limit assumed means. What has been said in the preceding paragraph can be summed up as two statements given below: The sum of displacements due to the scattered field from different inclusions will be zero, individually for the radial and transverse components of the summed displacement field arising out of scattering. (Statement A) The scattered field due to the sphere S3 at points far away from the sphere will be zero, individually for its radial component as well as for its transverse component. (Statement B) The above sums up the self-consistency criterion and offers a way of obtaining the elastic moduli of the composite in terms of the elastic moduli of the constituent media. The estimates of the elastic moduli of the composite are called as self-consistent estimates. To start the calculation of elastic moduli of the composite, consider a situation, where we have a sphere S4 of radius a which contains spherical inclusions of medium 1 and medium 2, with the relative volumetric proportions of medium 1 and medium 2 within S4, being the same as the relative volumetric proportions of medium 1 and medium 2 within S1 (which is composed entirely of the original composite whose properties we are setting out to calculate). Also, let the space in-between the inclusions be filled entirely, by a medium of unknown properties, and which will be designated as type - * medium. This medium will be called as “matrix.” Now, consider the case of sphere S4 also embedded in the type - * medium (matrix). The scheme is illustrated at Fig. A2.2.

Permeability and electrical conductivity Chapter j 10

785

FIGURE A2.2 Schematic of a sphere containing inclusions of the components of a composite medium (mixture), with the space between the inclusions filled entirely, by a medium of, unknown, properties. Reprinted with permission from Berryman (1980).

Invoking statement (A) above, it is stated that only if “matrix” is identical to E i¼n   X usr i ¼ 0

(A2.27)

i¼1 i¼n   X ust i ¼ 0

(A2.28)

i¼1

It is to be noted that i is the index number of an inclusion, and it can denote an inclusion of type 1 or an inclusion of type 2. Substituting for the appropriate displacement components in Eqs. (A2.27) and (A2.28) from Eqs. (A2.2) and (A2.3), we get,   i¼n X 1 1 ðikÞ1 ðkai Þ3 expðkri  utÞ B0  B1 cos q  B2 ð3 cos 2 q þ 1Þ ¼ 0 kri 4 i¼1 (A2.29)

786 Understanding Pore Space through Log Measurements i¼n X i¼1

  1 3s ðikÞ ðsai Þ expðsri  utÞ B1 sin q þ B2 sin 2 q ¼ 0 sr1 4k 1

3

(A2.30)

Eqs. (A2.29) and (A2.30) hold only when “matrix” is identical to E. Further, Eq. (A2.9) becomes, because of Statement (B) above,   i¼n1   X   1 3 a1i exp jkn:21i B0 ðK  ; K1 m Þ  B1 ðr ; r1 Þ cos q  B2 ðm ; m1 K  Þð3 cos 2 q þ 1Þ 4 i¼1   l¼n2 X  2 3   1 ai exp jkn:22i B0 ðK  ; K2 m Þ  B1 ðr ; r2 Þ cos q  B2 ðm ; m2 ; K  Þð3 cos 2 q þ 1Þ þ 4 l¼1

0¼

(A2.31)

 1 3 ai i¼1 Let c1 ¼ j¼n  P 4 3 3 paj

(A2.32)

 2 3 al Let c2 ¼ l¼1   j¼n P 4 3 3 paj

(A2.33)

1¼n P1

4 3p

j¼1

l¼n P2

4 3p

j¼1

While writing the above equations multiple scattering has not been assumed. Thus, the above is a first-order approximation to the actual scattering phenomenon which takes place. Where a1i is the radius of an inclusion indexed by i, and made of medium 1, and a2l is the radius of an inclusion indexed by l, made of medium 2. The only dependence of RHS of Eq. (A2.31) on k is through the phase factors expðjkn:21i Þ and expðjkn:22l Þ where 2 signifies the position vector of the center of an inclusion subscripted by the index and superscripted by the medium of the inclusion.

Invoking the long wavelength limit

In the long wavelength limit, ka  1 and for any i, k n:21i < ka and k n:22l < ka. Consequently, each of the phase factors expðjkn:21i Þ and expðjkn:22l Þ can be replaced by unity in Eq. (A2.31) with no loss in accuracy. Dividing Eq.   j¼n P 4 3 (A2.31) throughout by pa and noting that q is independent of all the j 3 j¼1

other variables in Eq. (A2.31), it is seen that Eq. (A2.31) implies the following equations.

Permeability and electrical conductivity Chapter j 10

787

c1 B0 ðK  ; K1 ; m Þ þ c2 B0 ðK  ; K2 m Þ ¼ 0

(A2.34)

c1 B1 ðr ; r1 Þ þ c2 B1 ðr ; r2 Þ ¼ 0

(A2.35)

c1 B2 ðm ; m1 ; K  Þ þ c2 B2 ðm ; m2 ; K  Þ ¼ 0

(A2.36)

Using Eqs. (A2.14)e(A2.16) we write, ðK   K1 Þ ð3K1 þ 4m Þ

(A2.37)

ðr  r1 Þ ð3r Þ

(A2.38)

B0 ðK  ; K1 ; m Þ ¼ B1 ðr ; r1 Þ ¼ B2 ðm ; m1 ; K  Þ ¼

20m ðm1  m Þ=3 6m1 ðK  þ 2m Þ þ m ð9K  þ 8m Þ ðK   K2 Þ ð3K2 þ 4m Þ

(A2.40)

ðr  r2 Þ ð3r Þ

(A2.41)

B0 ðK  ; K2 ; m Þ ¼ B1 ðr ; r2 Þ ¼ B2 ðm ; m2 ; K  Þ ¼

(A2.39)

20m ðm2  m Þ=3 6m2 ðK  þ 2m Þ þ m ð9K  þ 8m Þ

(A2.42)

Eqs. (A2.37)e(A2.40) lead to the following equations. K

1 c1 c2 ¼ þ 4  4  þ 3m K 1 þ 3m K2 þ 43m r ¼ c 1 r1 þ c 2 r2

Two more equations in involving m (A2.28)e(A2.33), namely,

*

(A2.43) (A2.44)

emerge as implied by Eqs.

1 c1 c2 ¼ þ m þ F  m1 þ F  m2 þ F 

(A2.45)

With F* given by F ¼

m 9K  þ 8m 6 K  þ 2m

And the equation m ¼ 0 One considers only Eqs. (A2.34)e(A2.37). The equations are implicit equations in K*, m*.

(A2.46) (A2.47)

788 Understanding Pore Space through Log Measurements

The solutions K*, m* are the required bulk modulus and shear modulus of E the effective medium of the composite (rock) and therefore of the composite itself. In rocks the material medium 1 which makes for inclusions is a fluid and hence m1 ¼ 0.

10.5 Electrical conductivity of an unfractured composite hosting dual porosity We start the analysis by noting that the equation obeyed by ε the selfconsistent Bruggeman Complex effective permittivity of the composite (which hosts a dual porosity system in the present context) is given by n P j¼1

fj ðεj εÞ 13

3 P k¼1

½ε =ðε þ Nkj ðεj εÞÞ ¼ 0 (see Eq. (8.58) of Chapter 8).

The model of the rock hosting a dual porosity system is that of a mixture of two media, namely, medium 1 representing fluid filled vugs and medium 2 representing the base rock. We then have, f1 ðε1  εÞ

3

  1X ε = ε þ Nk1 ðε1  εÞ þ ð1  f1 Þðε2  εÞ 3 k¼1 3

  1X ε = ε þ Nk2 ðε2  εÞ ¼ 0 3 k¼1

(10.40)

Here, ε1 is the complex permittivity of component 1 which is vug, ε2 is the complex permittivity of component 2 which is the base rock, and ε is the complex permittivity of the composite. N stands for depolarization factor, superscript of N denotes the component, and subscript stands for the direction of the polarizing field. f1 is the volume fraction of component 1, within the composite. In the present context f1 is nothing but the vug fraction fv. If s stands for electrical conductivity, Eq. (10.40) in the low frequency limit of the ambient field relevant to a measurement of ε becomes, as component 1 volume fraction is vug fraction, fv ðs1  sÞ

3

  1X s = s þ Nk1 ðs1  sÞ þ ð1  fv Þðs2  sÞ 3 k¼1 3

  1X s = s þ Nk2 ðs2  sÞ ¼ 0 3 k¼1

(10.41)

The vug shapes are generally near-spherical to spheroidal, with the ratio of the length of the major axis of the generator ellipse, to the minor axis of the generator ellipse ranging 1e2.5. For this range, sensitivity of the

Permeability and electrical conductivity Chapter j 10

789

depolarization factor for a vug to the deviation of the vug geometry from a spherical geometry is low. Hence it is reasonably assumed that Nk1 ¼ 13 for k ¼ 1,2,3. Component 2 is by itself a composite made up of microporous grains and the fluids hosted by the intergranular and intragranular pore space. However, it is in fact a continuum in the macroscopic sense, which also can be reasonably assumed to have isotropic macroscopic electrical properties. The reasoning given above leads to the following way of writing Eq. (10.40) s s fv ðs1  sÞ þ ð1  fv Þðs2  sÞ ¼0 (10.42) s þ 13 ðs1  sÞ s þ 13 ðs1  sÞ Eq. (10.42) can be simplified to, fv

ðs1  sÞ ðs2  sÞ þ ð1  fv Þ ¼0 ðs1 þ 2sÞ ðs2 þ 2sÞ

(10.43)

For illustration let us assume that the vugs are water bearing fully. If sw stands for conductivity of water, Eq. (10.43) becomes, fv

ðsw  sÞ ðs2  sÞ þ ð1  fv Þ ¼0 ðsw þ 2sÞ ðs2 þ 2sÞ

(10.44)

Modeling s2 for fully water saturated component 2 case As already mentioned, component 2 is composite made up of microporous grains and the fluids hosted by the intergranular and intragranular pore space. The model requirement is that when intergranular porosity is 1.0, s2 equals sw and when intergranular porosity is 0, s2 equals sm where sm stands for the macroscopic conductivity of the material of the microporous grain. A further model requirement is that s2 does not exhibit percolation, with respect to intergranular porosity. The model of a composite, with the geometry of the arrangement of the components, as per the self-similar model of Sen et al. (1981) is assumed, since the model meets the model requirements stated above. The model referred to above builds component 2 as follows. First start with water, to which a few grains of any size but of same shape (ellipsoidal) are added. This is step 1. Now, make a paste of this mixture and then add some more grains. The situation is as if the newly added grains are coated with the composite resulting from step 1. The geometry of the building up by successive addition of grains is this. The grains added at step kþ1 are coated with the medium, which is the result of step k. This process continues till the composite attains the porosity which is equal to the intergranular porosity.

790 Understanding Pore Space through Log Measurements

Let εk be the dielectric constant, when the kth step was completed. In the (kþ1)th step let the additional grains added lead to an increased rock volume of dvk . Let the complex permittivity of the medium after the (kþ1)th step be εkþ1 . Let the volume of the composite be vk at the end of step k. Let us apply Eq. (8.58) of Chapter 8 to express εkþ1 in terms of the other relevant quantities. k It is noted that in the present context n ¼ 2, and fj ¼ vk dv þdvk . We thus have,   εkþ1  εk dvk εm  εk ¼ (10.45) Lεkþ1 þ ð1  LÞεk vk þ dvk Lεm þ ð1  LÞεk Here εm stands for the complex permittivity of the grain material which hosts intragranular porosity. L stands for the representative depolarization factor which can be assigned to a grain. This concept arises from a model choice of randomly oriented grains in space, within the rock. As another extreme one can consider grains of same shape oriented in space with their principal axes aligned, and with the electromagnetic field polarized along one of the principal directions of the grain geometric shape. In that case L would stand for the depolarization factor along the direction in which the electric field is oriented. Considering dvk as the infinitesimal dv of the grain volume increase, representing volume of water as V and volume of grains as v Eq. (10.45) can be written as   dε0 dv εm  ε0 ¼ (10.46) 0 0 0 v þ V Lεm þ ð1  LÞε0 Lðε þ dε Þ þ ð1  LÞε Here ε0 is the general complex permittivity of the composite, treated as a variable which has to be evaluated. Let j denote the grain volume per unit volume of component 2. j is a variable since we are focusing on differential changes of the different quantities during successive steps of the process which culminates in the final component 2. We have,     v dv dj and j¼ ¼ (10.47) vþV vþV 1j dv Þ from Eq. (10.47) into Eq. (10.46) and neglecting dε Substituting for ðvþV in the denominator of the LHS of Eq. (10.46) we get,   dε0 dj εm  ε0 ¼ (10.48) 1  j Lεm þ ð1  LÞε0 ε0

Let 4ig stand for the intergranular porosity of component 2. This means that the maximum possible value for j ¼ ð1 4ig Þ while the minimum values that j can have is 0.

Permeability and electrical conductivity Chapter j 10

791

Integrating Eq. (10.48) for limits for j as 0 - ð1 4ig Þ and limits εw and ε on ε0 we get,   εm  ε εw L ¼ 4ig (10.49) εm  εw ε In the low frequency limit of the electromagnetic field used to generate the permittivity measurement, Eq. (10.49) reduces to,   L sm  s2 sw ¼ 4ig (10.50) sm  sw s2 Let 4m ; mm , respectively, stand for the intragranular porosity and Archie cementation exponent for the intragranular pore space. We can then write to a reasonable degree of approximation, m sm ¼ s w 4 m m

(10.51)

4m can be calculated from mesopore volume per unit rock volume (vm (say)) and micropore volume per unit rock volume (vm (say)) as 4m ¼

4  fv  vm 14

(10.52)

While stating Eq. (10.52) it has been assumed that the microporosity is largely the intragranular porosity, and that mesopore porosity is largely intergranular porosity. Such an assumption is realistic for the case of carbonate rocks in general. 4 is the total porosity and is obtained from density-neutron logs or from NMR logs. fv is obtained as macropore porosity from image log data as has been discussed in the previous sections, or from NMR data as has been discussed in the previous sections, or from considering, acoustic data inversion results, NMR data inversion results, and the macropore porosity from processing raw image data, together to obtain the best estimate on fv. L is an unknown parameter. It is the grain depolarization factor, as discussed above, and needs input on grain shape to be modeled. In practice, L is considered as a model parameter. This parameter can be assigned a value under certain circumstances, as follows. The requirement for this assignment is that the intragranular porosity is poorly connected, resulting in the value of mm being high, or 4m is very low. In that case it can be assumed that sm  s2 ;sw . Eq. (10.50) then simplifies to,  L1  1 sw ¼ 4ig which implies s2 ¼ sw 4ig 1L (10.53) s2 To make it even clearer, we can write Eq. (10.53) above as  m s2 ¼ sw 4ig m

(10.54)

792 Understanding Pore Space through Log Measurements

Here mm stands for the Archie cementation factor for the mesopore pore space, 1 . and whose value equals 1L mm ¼

1 1L

(10.55)

It is easy to see that for vug-free rocks, the value of the Archie cementation 1 exponent is a good estimate of the value of mm. It is thus a good estimate of 1L and therefore of L. 4ig will be the mesopore porosity obtained from NMR data. On the other hand, if it is reasonable to neglect 4m for a given case, then 4ig will be close to the total porosity 4 obtained from conventional log data inversion. Reasonable estimates of the value of the Archie cementation exponent can be arrived at, through the analysis of log data acquired against selected intervals, or through the analysis of data obtained through the laboratory studies of core plugs free of vugs.

Evaluation of s the electrical conductivity of a composite hosting dual porosity The evaluation of s the electric conductivity of the composite hosting dual porosity is implemented as follows. First the conductivity of a composite denoted by superscript “0 ” representing base rock hosting isolated vugs is computed as,     fiso ðsiso  s0 Þ fiso ðs2  s0 Þ þ 1  ¼0 (10.56) 0 1  fcon ðsiso þ 2s Þ 1  fcon ðs2 þ 2s0 Þ In case inputs on the shape of grains are available from thin section microscopy or from analysis of sieved grain ensembles, the value of L can be computed (see Chapter 8 under Appendix titled “Depolarization Factors”). Else if mm is known, as discussed above, then Eq. (10.54) can be directly used to evaluate s2 and substituted in Eq. (10.56). The composite of the base rock and isolated vugs, designated with primes for its conductivity, is considered as the base rock and the conductivity of this new base rock hosting connected vugs is modeled. Connected vug pore space can be treated in the same way as fracture volume, as far as electrical conductivity modeling of a medium hosting connected vugs is considered. Consider a unit cube of the composite rock of face area unity per face. The face area corresponding to the connected vugs ¼ total volume of connected vugs/representative length of the cumulative connected vugs pore space geometry ¼ sfcon where scon is the representative tortuosity connected con with the geometry of the cumulated connected vug pore space. The representative length of the cumulated vug pore space geometry is scon since distance between a pair of opposite faces of the unit cube is unity.

Permeability and electrical conductivity Chapter j 10

793

Conductance of the base rock hosting connected vugs ¼ s0 . Hence,   1s0 þ fcon 1 s ¼ 1 s or simply, 1 sfcon 1 1 scon scon w con   fcon 0 fcon (10.57) s¼ 1 s þ 2 sw scon scon

Case of partial saturation For completeness, the case of partial saturation of pore space by water is dealt with. When a rock hosting dual porosity contains two fluids namely water and hydrocarbons within its pore space, the rock is said to be under condition of partial saturation. Two additional quantities enter the model in this case. These are, respectively, water saturation of intergranular pore space, denoted as Swig, and connected vugs fraction. Intragranular pores do not get drained by hydrocarbons as the pore throat sizes involved for these pores are very small. The forward model of the base rock electrical conductivity proceeds as above, with 4ig replaced with 4ig Swig . It is reasonable to assume that, when the base rock is under conditions of partial saturation connected, vugs volume is entirely drained by hydrocarbons, while isolated vugs volume retains its full water saturation. Given that vugs are macropores, it is reasonable to ignore pendular water saturation within a vug drained by hydrocarbons. From images it is possible to have a partition of the total vug fraction into connected vugs fraction and unconnected vugs fraction. One can reasonably assume that unconnected vugs as per imaging tool button data processing also have high entry pressure and therefore are not drained by hydrocarbons. The model of bulk rock would thus be a mixture of base rock water filled conductive inclusions and hydrocarbon filled nonconductive inclusions. Assuming that the rock hosting them is water wet, it is reasonable to assume that these nonconductive inclusions should be having a strictly spherical shape. To model the conductivity of the bulk rock one of the two types of inclusions is first considered as hosted by the base rock. The conductivity of this composite is modeled on the lines discussed above. This composite is next considered as a base rock and bulk rock is modeled as this base rock (“composite” mentioned above) hosting the other second type of inclusions. Thus, the connected and unconnected vugs are added in two steps. Order of addition of the two types of vugs does not matter and would lead to the same results. The value Swig is optimized together for least error of measured rock conductivity. The partition from images for fv into isolated vug fraction fiso and connected vug fraction fcon is known from image tool button current data processing discussed earlier in this chapter. The water saturation of vug pore space, which can also be called as water saturation of macroporosity hosted by the rock, can then be estimated as ffisov . Finally, Swig can also be equated to the water saturation of the mesoporosity hosted by the rock.

794 Understanding Pore Space through Log Measurements

10.6 Permeability of an unfractured composite hosting dual porosity Base rock permeability As has been seen especially in Chapter 9 the capillary tubes model holds an important place in permeability modeling, followed by the pores and pore throats model and the fractal model of pore space. Here we look at yet another model which has not been touched upon in Chapter 9 for modeling the permeability of base rock. The analysis below draws on Ramakrishnan et al. (2001) and Rasmus and Kenyon (1985). In the model the permeability of intragranular pore space is neglected as the micropore system (which is what intragranular pore system is) has very low permeability as compared to the intragranular pore system (which is dominantly a system of mesopores). Fluid flux and pressure fields within pore space (which follow Stokes equation, for creep flow) are assumed to follow Laplace type equations which govern electric potential and flux fields within pore space for rocks drawing similarities between the forms of ohms law of electric conduction and Darcy’s Law of fluid flow through porous media, both operate at a macroscopic scale. This enables extending Eq. (10.50) to fluid flow as, 

km  k2 km  kw

 ðmmm 1Þ m kw ¼ 4ig k2

(10.58)

(see also Eq. 10.55). Eq. (10.58) can be simplified, given that the permeability of intragranular pore space which is a micropore system has very low permeability as compared to the intragranular pore system (which is dominantly a system of mesopores) to,  m k2 ¼ kw 4ig m (10.59) Here the subscript on the LHS indicates that the base rock is the component 2 of a composite consisting of base rock hosting vugs (component 1). It is assumed that electrical tortuosity (which is the ratio of the shortest tortuous path length of ions between two points when a potential gradient exists between the two points, to the straight line distance between the two points) and hydraulic tortuosity (which is the ratio of the shortest tortuous flow line length between two points when a pressure gradient exists between the two points, to the straight line distance between the two points) are equal. The second assumption is just as electrical conduction does not per se differentiate different regions the pore space (because ionic dimensions are much smaller than pore dimensions), the same is true of fluid flow. Under these circumstances the exponent of 4ig in Eq. (10.58) is equal to that in Eq. (10.54) each of which equal the Archie cementation factor for the intergranular porosity which is present as mesopores in a rock. kw here is the permeability of intergranular

Permeability and electrical conductivity Chapter j 10

795

pore space when measured by studying the fluid flux versus pressure gradient, interior to the pore system. This would physically correspond to the permeability which can be associated with a capillary whose radius equals the representative pore throat radius of the pore system. Let this representative pore throat radius of the mesopores be denoted as Rm. The flow rate Q for a capillary tube of radius Rm and length l for a pressure drop P across its ends is given by, Q¼

pPR4m 1 pR2m P 2 R ¼ m 8l m 8ml

(10.60)

Since kw is the internal permeability (hydraulic conductivity) of macroporosity simulated by the capillary tube’s permeability, the behavior of the fluid flux against pressure gradient should follow Darcy law, Q¼

kcapillarytube pR2m ð face areaÞPðpr:differentialÞ lðlengthÞ m kw pR2m ðface areaÞPðpr:differentialÞ ¼ lðlengthÞ m

(10.61)

In stating that the capillary tube of radius equals to the representative pore throat radius it has been implicitly assumed that the dimension of the pore space which is the characteristic length scale for fluid transport through the pore space is the representative pore throat radius rather than the representative pore radius. Comparing Eqs. (10.60) and (10.61), we get, 1 kw ¼ R2m 8

(10.62)

Substituting for kw from Eq. (10.60) into Eq. (10.57) we get, 1  m 1  m k2 ¼ R2m 4ig m ¼ 2 rm2 4ig m 8 8l

(10.63)

Here l is the representative pore radius to pore throat radius ratio of the pore space. The quantity 8l1 2 is generalized to a model variable C, and Eq. (10.63) is stated as, 1  m k2 ¼ rm2 4ig m C

(10.64)

Note that the first equality of Eq. (10.61) has the form similar to power law models relating porosity, permeability, and pore throat radius developed by Winland and later improved by Pitman (Pittman, E.D.: “Relationship of Porosity and Permeability to Various Parameters Derived from Mercury Injection-Capillary Pressure Curves for sandstones AAPG Bulletin (1992)). The form of WinlandePitman Permeability Predictor is, c

k ¼ awp ð4Þbwp R35wp

(10.64a)

796 Understanding Pore Space through Log Measurements

Here, k stands for permeability, awp, bwp, cwp are constants, 4 is porosity, and R35 is the pore throat radius connected with the injection pressure at 35% mercury saturation in an MICP experiment.

Permeability of the composite rock The permeability of an intermediate composite of the base rock and isolated vugs is considered. The analogue of Eq. (10.44) relevant here is,     fiso ðkiso  k0 Þ fiso ðk2  k0 Þ þ 1  ¼0 (10.65) 1  fcon ðkiso þ 2k0 Þ 1  fcon ðk2 þ 2k0 Þ kiso here is the internal permeability for any interior volume of an isolated vug. This will be infinity, since even an infinitesimal local pressure gradient at any location would result in a finite local flux at the same location, when the location is in the interior of a vug, making the mobility associated with the vug as infinite. Eq. (10.65) simplifies to, fiso þ ð1  fv Þ

ðk2  k0 Þ ¼0 ðk2 þ 2k0 Þ

Substituting for k2 from Eq. (10.64) into Eq. (10.66) we get,    m m 1 2 0 k C rm 4ig ¼0 fiso þ ð1  fv Þ   m m 1 2 0 r þ 2k 4 ig C m

(10.66)

(10.67)

A good estimate of rm is from the logarithmic mean T2 (T2LM) of mesopores, from NMR logs as, rm ¼ arT2LM

(10.68)

Here, a stands for the representative shape factor of a mesopore. By definition, Representative Surface area/volume of a pore ¼ a /rm From Eqs. (10.64) and (10.68) a good estimate of k2 is k2 ¼ 2 mm a2 ðrT 2LM Þ ð4ig Þ Eq. (10.67) C 2 A value of around 0.35 for aC and 2 for mm is a good choice Jacob et al. (2011). From core plugs of rocks, free of vugs, laboratory studies can yield a calibration on C of Eqs. (10.64) and (10.67), once rm is determined from the same cores from laboratory NMR techniques. The value of k0 can then be inverted from Eq. (10.67). Let the composite of the original base rock and the isolated vugs be considered as the new base rock. The superscript “0 ” denotes the new base rock properties.

Permeability and electrical conductivity Chapter j 10

797

When an MICP experiment is conducted on the bulk rock, the connected vug pore space is first drained by mercury. We assume that the representative pore throat radius is close to the pore throat radius corresponding to injection pressure, when the cumulative fractional drainage of the connected vug pore space is 35% and denote this as Rcon. Let kcon be the internal permeability or hydraulic conductivity of connected vug porosity. Assuming that the electrical tortuosity and hydraulic tortuosity are close, Qcon the flow rate for a pressure difference of P across opposite faces of a unit cube of the rock can be expressed as     npPR2con scon 2 1 pPR4con fcon 2 n¼ Rcon ¼ R P (10.69) Qcon ¼ 8m scon 8ms2con 8ms2con con Here, n is the number of sets of connected vugs, which are all assumed to extend from the inlet face till the outlet face. There is no loss in generality in assuming so because the subsequent analysis does not depend on any unique distance between the inlet and outlet faces. Let Qb be the flow rate contribution of the new base rock against the pressure difference of P across opposite faces of the unit cube of the rock. Let the face area of the base rock within the two faces of the unit cube of rock across which the pressure difference of P has been applied be Ab.   (10.70) Ab ¼ 1  npR2con     k0 1  npR2con k0 fcon Qb ¼ P¼ P (10.71) 1 m m scon 1 Let Q be the total flow rate out of the output face of the unit cube.   k0 fcon fcon 2 R P (10.72) Pþ 1 Q ¼ ðQb þ Qcon Þ ¼ m scon 8ms2con con Let k denote the permeability of the rock. We have, Q¼

k 1ðface areaÞP m 1ðlength of the cubeÞ

Comparing Eqs. (10.73) and (10.74) we get,   fcon 0 fcon k¼ 1 k þ 2 R2con scon 8scon

(10.73)

(10.74)

fcon is obtained from analysis of button current data of electrical imaging tools. Rcon is obtained from MICP, and since fcon is cumulative connected vug pore volume, the injection pressure at which the mercury saturation is 35% of fcon can be noted, from which Rcon can be computed. Else, if from comparison of MICP data and NMR T2 data for long T2 the representative mean value of the pore size to pore throat size ratio can be ascertained for the same formation in

798 Understanding Pore Space through Log Measurements

other offset wells, the same can be used to obtain Rcon from the representative mean pore radius data of macropore porosity, obtained from NMR T2 data. scon can be considered as a model parameter, whose value is ascertained by fitting permeability data from other wells to the forward model for permeability for those wells, for a given formation.

10.7 Electrical conductivity of fractured rocks Fig. 10.20 illustrates the model of electrical conduction in fractured rocks. The rock modeled in the previous sections is considered as base rock. For clarity, its volume has been indicated with subscript “LS” in Fig. 10.20. The cumulative volume of fractures, within a unit cube of the fractured rock, is denoted as Vf. This is thus the “fracture fraction.” The electrical conductivity of the base rock saturated fully with water, and which has been denoted as s, is now denoted as sbase. Thus sbase is a known quantity or a fully forward modeled quantity at this stage. Let Ab, Af, respectively, denote the face areas of the base rock and cumulative area of fractures exposed on an end face. Let sf be the representative tortuosity of the fracture set. Then, since the distance between the end faces is 1.0, Af ¼

Vf sf

Ab ¼ 1  Af

(10.74a) (10.75)

Let us consider the case of the entire fracture set fully saturated with water of conductivity sw. Let I be the total current flow for a potential difference of V across the end faces. We have, srock ¼

FIGURE 10.20 Model of electric conduction in fractured rocks.

I Ab Af ¼ sbase þ sw V 1 sf

(10.76)

Permeability and electrical conductivity Chapter j 10

799

Substituting for Ab, Af, respectively, from Eqs. (10.74) and (10.75) into Eq. (10.76) we get,   Vf Vf (10.77) srock ¼ sbase 1  þ sw   2 sf sf Eq. (10.76) can also be written, substituting for Ab in Eq. (10.76) from Eq. (10.75), as,   Af srock ¼ sbase 1  Af þ sw sf

(10.78)

The extraction of fracture segments from processing of electrical imaging tool button current data has been discussed earlier. One of the outputs which can further be extracted is the estimate of the width of different fractures. These widths can be cumulated into a quantity hfrac which is the cumulative vertical thickness of the intersection of a fracture with an end face. Since the breadth of an end face here is unity, we have, Af ¼ hfrac

(10.79)

Substituting for Af from Eq. (10.79) into Eq. (10.78) we have,   hfrac srock ¼ sbase 1  hfrac þ sw sf

(10.80)

From calibrated button current data of electrical imaging tools, we can have an estimate of sbase, while sw can be considered as conductivity of mud at formation temperature. The value of srock is known from the resistivity log measurement (here “resistivity log” refers to the same resistivity log against which button currentederived resistivity has been calibrated). Hence, obtaining statistics of the back calculated sf value from data against known fully water saturated fractured zones, one can generate a representative value of sf. In general, the value of sf is close to unity.

Partial saturation case When there is partial saturation of water within the fracture volume, Eq. (1.127) becomes,   n hfrac  srock ¼ sbase 1  hfrac þ sw Swfrac frac sf

(10.81)

Here Swfrac stands for the water saturation of the fracture system and nfrac stands for the saturation exponent of the fracture system. Recalling that sbase is the s of the previous section for partial saturation case, we can represent s as s(Swig) and, using Eq. (10.57), write Eq. (10.81) as,

800 Understanding Pore Space through Log Measurements

     n hfrac  srock Swig ; Swfrac ¼ s Swig 1  hfrac þ sw Swfrac frac sf

(10.82)

and where s(Swig)

    fcon  fcon 0  s Swig ¼ 1  s Swig þ 2 sw ðSwcon Þncon scon scon

(10.83)

It has been mentioned in the previous section that s(Swig) is the forward model of base rock conductivity under conditions of partial saturation. Substituting for s(Swig) from Eq. (10.81) into Eq. (10.80) we get,       fcon  fcon 0  ncon  srock Swig ; Swfrac ¼ 1  s Swig þ 2 sw ðSwcon Þ 1  hfrac scon scon þsw

n hfrac  Swfrac frac sf

(10.84)

When rock contains low volumetric concentration of isolated vugs, we can write,    m  n s0 Swig ysw 4ig m Swig m (10.85) Substituting for s0 ðSwig Þ from Eq. (10.85) into Eq. (10.84) we get,       m  n fcon fcon srock Swig ; Swfrac ¼ 1  sw 4ig m Swig m þ 2 sw ðSwcon Þncon scon scon   n hfrac  1  hfrac þ sw Swfrac frac sf

(10.86)

Because vugs are macropores, Swcon is very low. The second term within the square braces of Eq. (10.86) can be neglected in comparison with the first term within the square braces. Eq. (10.86) further simplifies to,       m  n   n fcon hfrac  Swfrac frac srock Swig ; Swfrac y 1  sw 4ig m Swig m 1  hfrac þ sw scon sf (10.87) Here Swig, Swfrac stand for the water saturation of intergranular porosity and of fractures system, respectively, and nm, nfrac stand for the saturation exponent for the intergranular porosity which is dominantly composed of mesopores. For the case of the resistivity from a microresistivity device, the fractures would be fully flushed with the borehole fluid (mud). Denoting the microresistivity reading by sxorock and the mud filtrate saturation within intergranular porosity in the flushed zone, by Sxoig, we have,

Permeability and electrical conductivity Chapter j 10

    m  n   fcon hfrac smf 4ig m Sxoig m 1  hfrac þ smud sxorock ¼ y 1  scon sf

801

(10.88)

Eq. (10.85) can be inverted for Sxoig. From resistivity measurements downhole for unfractured rock at shallow and deep depths of investigation, and laboratory studies on cores of unfractured rock, it is possible to generate workable predictors of Swig given, Sxoig as input. Substituting the predicted value of Swig, for the quantity Swig in Eq. (10.88) can be inverted for Swfrac.

10.8 The variable cementation exponent method of computing water saturation Case of connected pore space having tortuosity unity All calculations described in the following apply to the case of the entire pore volume of rock being fully water saturated. In this method, the connected vugs and fractures are treated on the same footing. The base rock is considered as the composite of intergranular porosity, fractures/connected vugs, and grains devoid of internal porosity (which is equivalent to saying that intragranular porosity is nil). The bulk rock is built as base rock hosting nonconnected vugs. The physical model is given at Fig. 10.21 in the left half.

Pore volumes The base rock is modeled as follows in terms of volumes of the model components. A unit volume of bulk rock has Vnc units of nonconnected vug pore space, and (1Vnc) units of base rock. Connected vugs and fractures together constitute the volume Vcon per unit volume of gross rock. The pore space allocation is given at Fig. 10.21 in the right half. A unit volume of bulk rock also has Vma units of intergranular pore space.

FIGURE 10.21 Schematic of the triple porosity model of rocks with pore space allocation also depicted. Reprinted with permission from Aguilera and Aguilera (2004).

802 Understanding Pore Space through Log Measurements

The electrical tortuosity associated with the abovementioned volume Vcon is considered as 1.0. When electric current flows through the base rock, the intergranular pore space and Vcon act in parallel. sb, the conductivity of the base rock, is given by, sb ¼ sw

Vcon ð1  VconVnc Þ þ s2 ð1  Vnc Þ ð1  Vnc Þ

(10.89)

Rationale behind Equation (10.89) Note that modification of the current lines through intergranular pore space due to the presence of isolated vugs is not considered in this technique. s2 is the conductivity of the material present within the volume (1Vcon).  m s2 ¼ sw 4ig m (10.90) The bulk rock is modeled as the base rock hosting nonconnected vugs. Therefore, when an electric current flows through the bulk rock, it encounters the base rock material and the vugs (the vugs are water filled), in series. Let Rrock, Rb, R2, Rw, respectively, denote resistivity of actual rock, base rock, intergranular rock (matrix), and water. In terms of these quantities, Eq. (10.89) can be stated as, 1 1 ð1  VconVnc Þ 1 Vcon þ ¼ Rb R2 ð1  Vnc Þ Rw ð1  Vnc Þ

(10.91)

Rearranging the above equation we get, Rb ¼

R2 Rw ð1  Vnc Þ Rw ð1  VconVnc Þ þ R2 Vcon

(10.92)

Rb ¼ Rw ð4b Þmb

(10.93)

Rrock ¼ ð1  Vnc ÞRb þ Vnc Rw

(10.94)

the rationale behind Eq. (10.89) is given at Appendix 3. Eqs. (10.92) and (10.94) have been cast in the form as they appear at AlGhamdy et al. (2010). The reasoning used is noted to be parallel to the logic demonstrated at Al-Ghamdy et al. (2010). The physical model of the rock is the same as that adopted at Aguilera and Aguilera (2004). Substituting for Rb in Eq. (10.94) from Eq. (10.92) we get Rrock ¼ Vnc Rw þ

ð1  Vnc Þ2 R2 Rw Vcon R2 þ ð1  VconVnc ÞRw

(10.95)

Permeability and electrical conductivity Chapter j 10

803

Let m represent the Archie cementation exponent applicable for the rock. 4 is the total porosity. m is defined through the equation, Rrock ¼ Rw ð4Þm

(10.96)

Substituting for Rrock from Eq. (10.96) in Eq. (10.95) and dividing the resulting equation by Rw throughout, we get ð4Þm ¼ Vnc þ

ð1  Vnc Þ2 R2 ð1  Vnc Þ2 ¼ Vnc þ Vcon R2 þ ð1  VconVnc ÞRw Vcon þ ð1  VconVnc ÞðRw =R2 Þ (10.97)

Using Eq. (10.92), we can simplify Eq. (10.97) to 2 ð1Vnc Þ2 nc Þ R2 ð4Þm ¼ Vnc þ Vcon R2ð1V ¼ Vnc þ V þðð1V mm . Thus, þð1VconVnc ÞRw Þ con conVnc Þ=ð4ig Þ we have, ð4Þm ¼ Vnc þ

ð1  Vnc Þ2  mm  Vcon þ ð1  VconVnc Þ= 4ig 

(10.98)

Taking logarithms of both sides of Eq. (10.98) and rearranging, we get   ð1Vnc Þ2 log Vnc þ mm Vcon þðð1VconVnc Þ=ð4ig Þ Þ m¼ (10.99) log 4 Eq. (10.99) is identical to the equation for m of Al-Ghamdy et al. (2010). From image logs we can obtain fiso the isolated vug volume per unit volume of unfractured portion of rock. Inspecting Eq. (10.99) it can be noted that 4 can be from conventional logs including Density and Neutron logs. The quantities fiso the isolated vug volume per unit volume of unfractured portion of rock and fcon the connected vug volume per unit volume of unfractured portion of rock are obtained from processing the raw button current data of electrical imaging tools. Also, from image logs, the fracture width information as well as number of fractures present per unit thickness of formation are obtained, from which, the fracture volume per unit gross rock volume can be computed. This quantity is designated as Vf. We then have, Vcon ¼ Vf þ Vnc ¼  Vnc ¼



fcon 1  Vf

fiso 1  Vf

(10.100)

(10.101)

    fiso vug volume from NMR logs ð fv ÞNMR or from sonic datað fv Þacoustic  1  Vf (10.102)

are two inputs out of which Vnc can be chosen.

804 Understanding Pore Space through Log Measurements

mm can be evaluated using deep and shallow resistivity log readings as discussed above. If the value of mm so obtained is not reasonable, use the value of Archie cementation exponent values determined in the laboratory for core plugs free of vugs and fractures, against the formation being evaluated. Else, use a value based on area knowledge. If no data is available, the standard default value can be assumed for mm. The porosity of the base rock of the model discussed above is designated as 4b and can be evaluated as, 4b ¼

4  Vnc 1  Vnc

4ig is evaluated from 4b evaluated from Eq. (10.103), as   Vcon 4b  1Vnc   4ig ¼ Vcon 1  1Vnc

(10.103)

(10.104)

The total process of modeling m is concisely summarized below.

Level by level evaluation of the Archie cementation exponent of the formation 1. Obtain total porosity 4 from conventional logs including Density-Neutron logs. 2. Obtain the values of Vf, (fv)images, (fv)acoustic, (fv)NMR, fcon, fiso as discussed in this chapter, and as above. 3. Evaluate Vnc, Vcon, as indicated above. 4. Evaluate 4b using Eq. (10.103). Then, evaluate 4ig using Eq. (10.104). 5. Evaluate mm as discussed above. 6. Evaluate m using Eq. (10.99)

Discussion The model illustrated is called as the Triple Porosity Model. The model m computed as above agrees well with laboratory determined value of m. However, there is further scope for modification of the model. The assumption of tortuosity of connected pore volume having a value of unity is a big assumption. This implies current paths in rock constituents, being parallel to fractures. This may not always be true. Hence a modification is called for. One such modification as cited by Berg (2004) as attributed to Aguillera involves introduction of tortuosity greater than 1.0 for the connected vug and/

Permeability and electrical conductivity Chapter j 10

805

or fracture pore space. This implies that the connected vug and/or fracture pore space has an associated Archie cementation exponent greater than 1.0. This is denoted as mcon. Presence of connected pore space from fractures and/or connected vugs, having nonplanar geometry results in the conductivity of this pore space being less than the conductivity, had this pore space been of planar geometry. The gross conductivity of a unit volume of base rock which has only this pore space and no intergranular pore space (scon) of this pore space would be  mcon Vcon . This is the same conductivity, if the tortuous pore scon ¼ sw 1Vnc  mcon Vcon space referred to above is replaced by planar pore space of volume 1V nc (the Archie cementation exponent for a planar pore space case is 1.0, and  mcon 1:0 Vcon conductivity for this case would be sw 1V which is equal to scon). nc So, the electrical conductivity of a rock having tortuous fracture and/or connected vug porosity is simulated by the electric conductivity of a rock having planar fracture and/or connected vug porosity of a value as indicated above. A rock having intergranular porosity 4ig and nonplanar tortuous fracture and/or   Vcon (whose Archie cementation connected vug porosity of volume 1V nc exponent is mcon, when intergranular porosity is zero) is simulated by a rock having intergranular porosity 4ig 0 (say), and planar connected vug and/or  mcon Vcon where, fracture pore volume of magnitude 1Vnc 

mcon

4b   mcon (10.105) Vcon 1  1Vnc    mcon Vcon Vcon in Eq. 10.104 to get 40ig ) (that is, we replace 1Vnc with 1Vnc 40ig ¼

Vcon 1Vnc

m R2 ¼ Rw 40m ig

Eq. (10.91) can be written as,   Vcon 1 ¼ 1 1  Vcon is modified as þ R1w ð1V Rb R2 ð1Vnc Þ nc Þ   mcon    1 1 Vcon 1 Vcon mcon ¼ 1 þ Rb R2 Rw 1  Vnc 1  Vnc

(10.106)

(10.107)

Rearranging the above equation we get, Rb ¼

R2 Rw ð1  Vnc Þmcon Rw ½ð1  Vnc Þmcon  ðVcon Þmcon þ R2 ðVcon Þmcon

(10.108)

806 Understanding Pore Space through Log Measurements

Substituting for Rb from Eq. (10.108) into Eq. (10.94) we get Rrock ¼ ð1  Vnc Þ

 Vnc Þmcon þ Vnc Rw  ðVcon Þmcon þ R2 ðVcon Þmcon (10.109)

R2 Rw ð1 mcon

Rw ½ð1  Vnc Þ

Rw ð1  Vnc Þmcon þ1 mcon  ðVcon Þmcon þ ðVcon Þmcon R2 ½ð1  Vnc Þ

Rrock ¼ Vnc Rw þ Rw

(10.110)

Using Eqs. (10.96) and (10.106) for Rrock, R2 in Eq. (10.110) we get, Rw ð4Þm ¼ Rw Vnc þ

ðVcon Þmcon

Rw ð1  Vnc Þmcon þ1 (10.111) m þ ½ð1  Vnc Þmcon  ðVcon Þmcon =40m ig

Canceling Rw throughout, in Eq. (10.111) we get 4m ¼ Vnc þ

ðVcon Þ

mcon

ð1  Vnc Þmcon þ1 m þ ½ð1  Vnc Þmcon  ðVcon Þmcon =40m ig

" log Vnc þ m¼

(10.112)

# ð1Vnc Þmcon þ1 m ðVcon Þmcon þ½ð1Vnc Þmcon ðVcon Þmcon =40m ig

(10.113)

log 4

where 40ig is given by Eq. (10.105). Eqs. (10.112) and (10.113) are the generalization of Eqs. (10.98) and (10.99), respectively, for the case of nonplanar geometry of vug/fracture pore space. Eqs. (10.112) and (10.113) reduce to Eqs. (10.98) and (10.99), respectively, as they should, for the case mcon assuming the value of 1.0. The generalization of equations made, as referred to above, is in line with that suggested by Aguilera as cited by Berg (2004). Aguilera’s equation for m is given below. The form of the equation is as per the form given at Aguilera and Aguilera (2003) h i m log Vcon þ ð1  Vcon Þ=4m ig in our notation (10.114) m¼ log 4 The difference between the form of Eq. (10.114) and of Eq. (10.113) is not of relevance here because the porosity scaling employed while arriving at the respective equations is different. To have a better perspective on the comparison of Eq. (10.113) versus Eq. (10.114), we write Eq. (10.113) as, " log

m¼ get,

ð1Vnc Þmcon þ1 ðVcon Þmcon þ½ð1Vnc Þmcon ðVcon Þmcon =40mm ig Vnc þ ð1Vnc Þmcon þ1

log 4

# and set Vnc ¼ 0 to

Permeability and electrical conductivity Chapter j 10

m¼

h i m log ðVcon Þmcon þ ½1  ðVcon Þmcon =40m ig log 4 Where 40ig ¼

mcon 4b  Vcon mcon 1  Vcon

807

(10.115) (10.115a)

The equation comparable to Eq. (10.115) suggested by Aguilera for the case of the Archie cementation exponent of connected vug/fracture pore space allowed to have a value greater than 1.0 Berg (2004) is (again in our notation), h  0mm i  mcon mcon log Vcon þ 1  Vcon =4ig (10.116) m¼ log 4 f con where 40ig ¼ 44 14f where Vcon ¼ 4mcon 1 in Aguilera’s formulation. Comparison of Eqs. (10.114) and (10.116) brings out that the generalization of planar geometry connected vug/fracture porosity case to a general mcon nonplanar geometry entails modification of Vcon for the former case to Vcon for the latter case. Comparison of Eq. (10.116) with Eq. (10.115) brings out that the generalization, being discussed, in the analysis made above, is in line with that of Aguilera as cited at Berg (2004). The fact that 40ig in Aguilera model is not identical to the 40ig referred to in Eq. (10.115a) is not of relevance in this context. f

V mcon

Computation of water saturation For completeness, the computation of water saturation is discussed. In the variable m technique, the total water saturation (designated as Sw) defined as water volume within unit volume of rock normalized by the total porosity of the rock is calculated using Archie’s equation applied to the gross rock as, Rrock ¼ Rw 4m Sn w

(10.117)

It is a general observation that once the value of m is correctly modeled, Eq. (10.117) (Archie’s equation) gives reasonable results (Al-Ghamdy et al., 2010). The value of m varies from depth level to depth level and is computed using Eqs. (10.105) and (10.113). The value of mcon is set to 1.0 usually where fractures dominate the connected macropore pore space. When connected vugs dominate the connected macropore pore space, a value calibrated with laboratory measurements of formation factor of rock plugs dominantly vuggy with the vugs connected can be used. Else, a default value is to be used. The value of n is used based on laboratory data on the resistivity index variation with water saturation. In the absence of such data, a default value of 2.0 is used.

808 Understanding Pore Space through Log Measurements

Appendix 3 Rationale behind Equation (10.89) The gross rock model here is of a base rock within which isolated vugs are distributed. The cumulative volume of vugs is Vnc per unit volume of gross rock. The volume of base rock per unit volume of gross rock is (1Vnc). Consider a rectangular parallelepiped of face area A and length L, of gross rock. Consider a potential difference to be set up across the opposite faces. Any current line which flows from one face to the opposite face encounters base rock and the vugs in series. It is therefore assumed that, for a given net current flow I across the opposite faces of the gross rock, the total potential difference across the end faces is the same as the potential difference between the two extreme end faces of a system depicted at Fig. A3.1. In Fig. A3.1, the rectangular parallelepiped of face area A and length LB represents the base rock and the rectangular parallelepiped of face area A and length LV represents the total vug pore space of isolated vugs. The parallelepiped formed by adding the two parallelepipeds as shown in Fig. A3.1 represents the gross rock. The total volume of gross rock here is A(LB þ LV); the volume of the parallelepiped representing base rock is ALB. The volume of the parallelepiped representing the vug pore space, is ALV. Vnc is the isolated vug pore space per unit gross rock volume. We have, ALV LV ¼ ¼ Vnc AðLB þ LV Þ ðLB þ LV Þ

(A3.1)

ALB LB ¼ ¼ 1  Vnc AðLB þ LV Þ ðLB þ LV Þ

(A3.2)

When a current flow through the gross rock is set up by establishing a potential difference between the extreme end faces, the current flows through

FIGURE A3.1 Segmented representation of an intergranular rock hosting isolated vugs. The rectangular parallelepiped of face area A and length LB represents the intergranular rock (matrix). The rectangular parallelepiped of face area A and length LV represents the isolated vug pore space.

Permeability and electrical conductivity Chapter j 10

809

the base rock and vugs, which act in series. It is assumed that the average path length of current through base rock is represented by LB, and that through isolated vugs is represented by LV. Let rB, rV, and r, respectively, be the resistance of the block of base rock, the block representing cumulative isolated vug pore space and the resistance of the block formed by joining these two blocks together (as depicted in Fig. A3.1), which represents the gross rock. We have, r B ¼ Rb

LB A

(A3.3)

r V ¼ RV

LV A

(A3.4)

ðLB þ LV Þ A

(A3.5)

r ¼ Rrock

r ¼ rB þ rV

(A3.6)

Substituting for rB, rV, and r, respectively, from Eqs. (A3.3)e(A3.5) into Eq. (A3.6), we get, Rrock

ðLB þ LV Þ LV LB LV LB ¼ Rw þ Rb which implies Rrock ¼ Rw þ Rb A A A LB þ LV L B þ LV (A3.7)

V Using Eqs. (A3.1) and (A3.2), we can substitute Vnc for LBLþL and 1-Vnc for V in Eq. (A3.7), to get Rrock ¼ RwVnc þ Rb(1 Vnc) which is nothing but Eq. (10.94). Consider now the arrangement shown in Fig. A3.2). In Fig. A3.2, the rectangular parallelepiped of face area Aig and length L represents the part of the base rock which is free of fractures/connected vugs and hence hosting only intergranular porosity. The rectangular parallelepiped

LB LB þLV

FIGURE A3.2 Segmented representation of a rock hosting connected vugs and/or fractures. The rectangular parallelepiped of face area Aig and length L represents rock which is free of fractures/ connected vugs. The rectangular parallelepiped of face area Acon and length L represents the total pore space of connected vugs/fractures.

810 Understanding Pore Space through Log Measurements

of face area Acon and length L represents the total pore space of connected vugs/fractures. The parallelepiped formed by adding the two parallelepipeds side by side as shown in Fig. A3.1 represents the base rock. The total volume of base rock here is (Aig þ Acon)L; the volume of the rectangular parallelepiped representing fractures/vugs-free intergranular rock is AigL.The volume of the rectangular parallelepiped representing connected vug/fracture pore space is AconL. Vcon is the isolated vug pore space per unit Vcon is the connected vug/fracture pore space per gross rock volume, and so, ð1V nc Þ unit base rock volume. We have, Acon L Acon Vcon  ¼ ¼ ð1  Vnc Þ Aig þ Acon L Aig þ Acon

(A3.8)

Aig L Aig Vcon  ¼ ¼1 ð1  Vnc Þ Aig þ Acon Aig þ Acon L

(A3.9)

 

When a current flow through the base rock is set up by establishing a potential difference between the extreme end faces, the current flows through the intergranular rock and the connected vugs/fractures, in parallel. It is assumed that the effective area of cross section of current through the intergranular rock fraction of the base rock is represented by Aig, and that the effective area of cross section of current through the connected vugs/fractures of the base rock represented by Acon. Let cig, ccon, and cb be the conductance of the block representing the intergranular porous part of base rock, the conductance of the block representing cumulative connected vug pore space/fracture pore space, and the conductance of the block formed by joining these two blocks together (as depicted in Fig. A3.2), and which represents the base rock, respectively. We have, cig ¼ sig

Aig L

(A3.10)

ccon ¼ sw

Acon L

(A3.11)



Aig þ Acon cb ¼ s L cb ¼ cig þ ccon

 (A3.12) (A3.13)

Substituting for cig, ccon, and cb, respectively, from Eqs. (A3.10eA3.12) into Eq. (A3.13), we get,

Permeability and electrical conductivity Chapter j 10

 s

 Aig þ Acon Acon Aig ¼ sw þ sig which implies L L L Acon Aig  þ sig   s ¼ sw  Aig þ Acon Aig þ Acon

811

(A3.14)

Vcon con Using Eqs. (A3.9) and (A3.10), we can substitute ð1V for ðAigAþA and nc Þ con Þ A Vcon ig 1  ð1Vnc Þ for ðAig þAcon Þin Eq. (A3.14),  to get Vcon Vcon þ s s ¼ sw ð1V 1  which is nothing but Eq. (10.89). ig ð1Vnc Þ nc Þ

10.9 Permeability of rocks hosting connected vugs/ fractures There are multiple ways to approach the problem of permeability of a fractured rock. We confine ourselves to illustrating one approach. This analysis starts with first considering fluid flow through fractures. The fluid flow is considered as laminar flow (equivalent here to seepage flow). The fluid is considered as incompressible. The fracture plane is shown in Fig. 10.22. The flow direction is also indicated. Let DP be the pressure differential across two planes normal to the X direction, and at a distance L0 apart. Let w be the width of the fracture and a the aperture of the fracture. As indicated at Fig. 10.22, a and q are, respectively, the fracture strike azimuth relative to positive Y direction and the fracture plane angle with respect to the X Y plane. Let q be the flow rate through the fracture.   a3 w 1  cos2 a sin2 q DP q¼ (10.118) L0 12m (Miao et al. (2015). Eq. (10.118) implies permeability of a rock hosting fractures to be in general, a tensor. We shall confine ourselves to the permeability value, when measurement is made through experiments where flow is induced in a direction normal to

FIGURE 10.22 Schematic 3D representation of a fracture present within a porous medium transporting fluid. The flow direction is also indicated. Adapted from Fig. 1 of Miao et al. (2015).

812 Understanding Pore Space through Log Measurements

fracture strike, and along fracture dip. This model is useful when images establish that fracture azimuth and dip vary within narrow windows. In this case, a ¼ 0; and Eq. (10.118) becomes, q¼

a3 wðcos2 qÞ DP 12m L0 sf

(10.119a)

Eq. (10.119) is Parsons Equation (1996) when fracture tortuosity is unity. However, here we shall be following the equation given by Sarkar et al. (2004) by which, q¼

a3 w cos q DP 12m L0 sf

(10.119)

We quote verbatim from Sarkar et al. (2004) below: 5

We note a significant disagreement between our analysis and what currently exists in the major literature. Originally proposed by Parsons (1996), and thereafter adopted in standard reservoir engineering literature (e.g., van GolfRacht, 1982; Nelson 2001; etc.) the discriminating factor is cos2 q, as opposed to our proposed cos q. The present-day literature states:   3 2 h cos q DP and K ¼ h2 ðcos2 qÞ, which may cause significant difference Q ¼ 12m l 12 in estimation of these important properties, when there are large number of oriented fractures. We could not find numerical verification of these equations with NaviereStokes flow simulation. While further investigation may be required to completely resolve this discrepancy, our formulation, which is backed by numerical analysis and visualization of the true physical behavior, seems much more insightful and reliable at this point.

In their notation fracture aperture is h and the block length is l. The cumulative flow Qfrac from a set of n fractures is given by, Qfrac ¼

i¼n 3 X ai w cos q DP 12m L0 sfi i¼1

(10.120)

sfi is the tortuosity of ithfracture. This tortuosity arises because the fracture faces are not smooth, but rough, and the fractures themselves can have slight L0 sinuosity. The end-to-end straight line distance of a fracture is cos q We are considering these fractures to be present within a block of gross rock of length L0, width w, and unit height. Usually, no prior information is available on the value of sfi , while it stays close to 1.0. Hence, Eq. (10.120) is written as, Qfrac ¼

i¼n 3 X ai w cos q DP 12m L0 i¼1

(10.120a)

Let the shortest path length between the two ends of a fracture and interior to the fracture be lt.

Permeability and electrical conductivity Chapter j 10

813

l, the straight length of a fracture, is related to L0 as, l¼

L0 cos q

(10.121)

lt l

(10.122)

sf ¼

At this juncture the scope of the term “fracture” is expanded to include connected vugs. To this effect, the symbol ai will refer henceforth, to the size of connected vugs, or aperture of a fracture as the case might be, indexed by i. sf is generalized to the tortuosity of connected vug pore space/fracture and denoted as scon . lt is generalized to the length of connected vug pore space/fracture and denoted as lcon. In general, the product aw is designated as Acon. Thus, Acon ¼ aw

(10.122a)

The implicit assumption here is that the intersection of connected vug pore space/fracture with either end face of gross rock is linear and has a length, almost equal to the width w of the block. Eq. (10.122) is stated as, scon ¼

lcon l

(10.122b)

The most apt description of lcon (and thereby, sf also) would be that they are fractals. Let the fractal dimension of lt be Dt. The dimensionality of the embedding Euclidian space is 2. It is a general observation that fractures with wider apertures tend to be associated with lesser tortuosity in the sense indicated above, and vice versa. Similar remarks apply to connected vugs. The correlation of lcon, a, and l is considered as analogous to that of pores in porous media (see Chapter 9). The relation as formulated at Liu et al. (2015) for fractures will apply for the present case also, and is stated as, lcon ¼ a1Dt lDt

(10.123)

l, the straight length of a fracture/connected vug pore space, is related to L0 as, L0 cos q

(10.124)

a3 w DP 12m lcon

(10.125)

l¼ Eq. (10.119) can be written as, q¼

(since ðL0 =cos1 qÞscon is nothing but lcon). Substituting for lcon from Eq. (10.123) in Eq. (10.125) we get, q¼

wað2þDt Þ DP wðcos qÞað2þDt Þ DP ¼ t 12m lDt 12m LD 0

(10.126)

814 Understanding Pore Space through Log Measurements

The cumulative flow Q from the set of connected vugs/set of fractures present within a block of gross rock of width w, height h, and length L0 is given by, Z amax ð2þDt Þ a w cos q DP dN (10.127) Qcon ¼ t 12m LD amin 0 dN is the number of connected vug-sets/fractures whose size/aperture lies within the interval (a, aþda). Like lcon, a is a fractal. Let Da denote the fractal dimension of a. Let Nða > a1 Þ denote the number of connected vug-sets/fractures whose size/aperture exceeds the value a1, and let Nða > a1 þdaÞ denote the number of connected vug-sets/fractures whose size/aperture exceeds the value a1þda. Let amax denote the maximum value of a. We write, after Miao et al. (2015),   amax Da (10.128) Nða > a1 Þ ¼ a1   amax Da Nða > a1 þ daÞ ¼ (10.129) a1 þ da Using Eqs. (10.128) and (10.129), we can frame the following equation for dN. Da a Da  a max max  ¼ Da ðamax ÞDa aðDa þ1Þ da (10.130) dN ¼ a a þ da Substituting for dN from Eq. (10.130) in Eq. (10.127) and carrying out the integration we get,   ð2Da þDt Þ   ð2þD Þ  min t Da amax 1  aamax DP 1 w cos q Qcon ¼ (10.131) t 1 L0 12m LD ð2  Da þ Dt Þ 0 Electrical image data are useful for obtaining the distribution of Nða > a1 Þ over a1. Fitting of this distribution to Eq. (10.128) is one way to obtain an estimate of the value of Da.   ð2Da þDt Þ   ð2þD Þ  min t Da amax 1  aamax 1 DP wðcos qÞ Qcon ¼ (10.132) t 1 m L0 12LD ð2  Da þ Dt Þ 0 The value of Dt is unknown but is close to 1.0. Hence, Eq. (10.132) can be written as,   ð3Da Þ  3 min Da ðamax Þ 1  aamax 1 DP wðcos qÞ Qcon y (10.133) m L0 12 ð3  Da Þ

Permeability and electrical conductivity Chapter j 10

815

Face area of gross rock ¼ w for unit height of a block of gross rock of width w and length L0. And face area Acon of connected vugs-set/fracture is aw (see Eq. (10.122). Eq. (10.133) can be written as,   ð3Da Þ  3 min Da ðamax Þ 1  aamax 1 DP Aðcos qÞ Qcon y (10.134) m L0 12 ð3  Da Þ Let Qbase be the flow rate from base rock. Let kb be the permeability of the base rock. We have, From Darcy Law, Qbase ¼

kb DPðA  Acon Þ L0 m

(10.135)

Let k denote the permeability of the gross rock and let Q denote the total flow rate. Q¼

k DPA where Q is alsoðQcon þ Qbase Þ m L0

(10.136)

Substituting for Qcon and Qbase from Eqs. (10.134) and (10.135) in Eq. (10.136), we have,   ð3Da Þ  min Da ðamax Þ3 1  aamax k DPA 1 DP Aðcos qÞ kb DPðA  Acon Þ þ ¼ m L0 m L0 12 L0 ð4  Da Þ m (10.137) Here, A is the face area of the block of the gross medium and equals w since the height of the block is unity. Canceling out common terms on both sides of Eq. (10.137) we get,          Acon ðcosqÞ Da amin ð3Da Þ 3 k ¼ kb 1  1 ðamax Þ þ 12 A 4  Da amax (10.138)   One way of finding the value of

Acon A

is from  image  logs. In case reliable

input is absent through this channel, the value of

Acon A

can be estimated as

follows. Acon being the cumulative face area of connected vug/fracture pore space is R amax R amax ðDa þ1Þ da, where we have a given by, Acon ¼ amin awdN ¼ amin awDa aD max a used Eq. (10.130) to substitute for dN. Evaluating the integral we get,   ð1Da Þ  a wDa aDmax 1Da wDa amax 1D min a Acon ¼ 1Da ðamax amin Þ ¼ 1Da 1  aamax which is written as,

816 Understanding Pore Space through Log Measurements

     Da amax ðDa 1Þ Acon ¼ w 1 amax Da  1 amin

(10.139)

A is the face area of the block of the gross medium and equals w since the height of the block is unity. Hence, dividing both sides of Eq. (10.139) by A we get,        Acon Da amax ðDa 1Þ 1 (10.140) ¼ amax A Da  1 amin Hence, Eq. (10.138) can be stated as,          Acon ðcosqÞ Da amin ð3Da Þ 3 1 ðamax Þ þ k ¼ kb 1  12 A 3  Da amax (10.141)   where the value of

Acon A

is obtained from image logs, or from Eq. (10.140).

Alternate simpler way of estimating Qcon Extending the scope of Eq. (10.120) to connected vug/fracture pore space, and assuming a value of unity for the tortuosity of the vug/fracture pore space, we restate Eq. (10.120) as, Qcon ¼

i¼n 3 X ai w cos q DP 12m L0 i¼1

(10.142)

iP ¼n 3 iP ¼n ai w cos q DP a3i We evaluate 12m L0 processing image tool data for evaluating i¼1 i¼1 iP ¼n and write, a3i ¼ na3 where a3 represents the mean value of the cubed size of i¼1

the vug/fracture events. Since the height of the block is one unit, n stands for the vertical density of occurrence of the vug/fracture events. We then have, Qcon ¼ na3

w cos q DP 12m L0

(10.143)

The cumulative face area of connected vug/fractures (Acon) is evaluated as, iP ¼n Acon ¼ ai w. Acon is expressed as, i¼1 ! i¼n 1X ai w ¼ wna (10.144) Acon ¼ n n i¼1 n Acon ¼ A

1 n

iP ¼n i¼1

w

! ai w ¼ na

(10.144a)

where n is vertical density of occurrence of connected vug/fracture events.

Permeability and electrical conductivity Chapter j 10

817

Here the pointed brackets indicate mean value of the quantity within them which is evaluated from image tool data processing. Using Eq. (10.144), we write Eq. (10.143) as, Qcon ¼

na3 cos q DP Acon na 12m L0

Using Eqs. (10.135), (10.136), and (10.145) we write     3  Acon na cos q DP Acon k ¼ kb 1  þ A na 12m L0 A

(10.145)

(10.146)

Permeability kb of the base rock The model for the base rock is as intergranular rock containing isolated vugs. The cumulative volume of vugs is Vnc per unit volume of base rock. 4b is the pore volume per unit volume of base rock. 4ig is the intergranular porosity. 4b ¼ Vnc þ ð1  Vnc Þ4ig

(10.146a)

In analogy with Eq. (10.65) the equation satisfied by kb is seen to be Vnc

ðkiso  kb Þ ðk2  kb Þ þ ð1  Vnc Þ ¼0 ðkiso þ 2kb Þ ðk2 þ 2kb Þ

(10.147)

where kiso is internal permeability of a vug, and k2 is permeability of intergranular rock, and which is given by Eq. (10.64). Recalling that Vcon is the connected vug/fracture pore volume per unit gross rock volume, and Viso is the isolated vug volume per unit gross rock volume, Viso Vnc ¼ 1V and hence, Eq. (10.147) becomes, con     Viso ðkiso  kb Þ Viso ðk2  kb Þ þ 1 ¼0 (10.148) 1  Vcon ðkiso þ 2kb Þ 1  Vcon ðk2 þ 2kb Þ kiso here is the internal permeability for any interior volume of an isolated vug. This will be infinity, since even an infinitesimal local pressure gradient at any location would result in a finite local flux at the same location, when the location is in the interior of a vug, making the mobility associated with the vug as infinite. Hence, Eq. (10.148) is stated as,     Viso Viso ðk2  kb Þ ¼0 (10.149) þ 1 1  Vcon 1  Vcon ðk2 þ 2kb Þ Substituting for k2 from Eq. (10.64) into Eq. (10.149) we get,    mm 1 2      Crm 4ig  kb Viso Viso ¼0  þ 1  mm 1  Vcon 1  Vcon 1 2 4 r þ 2k b ig C m

(10.150)

818 Understanding Pore Space through Log Measurements

A good estimate of rm is from the logarithmic mean T2 (T2LM) of mesopores, from NMR logs in accordance with Eq. (10.68).

Permeability of gross rock Using Eq. (10.141), we can state that the permeability k of gross rock is given as,          Acon ðcosqÞ Da amin ð3Da Þ k ¼ kb 1  1 ðamax Þ3 þ 12 A 3  Da amax (10.151)   where kb is the solution of Eq. (10.150). AAcon is computed from image data, or from Eq. (10.140). Another choice, if strong image tool data complement is available, is,     3  Acon a cos q Acon k ¼ kb 1  þ (10.152) A a 12m A   Substituting for AAcon from Eq. (10.144a) in Eq. (10.152) we get,  3  a cos q ðnaÞ (10.153) k ¼ kb ð1  naÞ þ a 12m where n is the number of connected vug/fracture events per unit vertical distance exposed on a vertical face of the gross rock and where kb is the solution of Eq. (10.150). Before concluding the analysis of permeability of heterogeneous porous rocks, a brief account of the Bray e Smith method is given below.

10.10 Bray e Smith method of computing permeability Connected vugs arise in different contexts within carbonate rocks. They can be multiple vugs which form chains, with neighboring vugs in the chain, touching endwise. They can be vugs which are connected by fractures or vugs connected by open stylolaminations. They can be vugs connected by bed boundaries which have developed hydraulic conductivity, and so on. The one trait common to the pore space, in all these different settings, is the fact that we have connected macroporosity, meaning thereby that the characteristic length scale exceeds 2.5 microns. In the BrayeSmith model, individual permeability is ascribed to different pore classes, with TimureCoates model used as a template. This way, a forward model is constructed based on pore classes. This approach neglects the role of nontouching vugs in permeability attenuation, and hence is best suited for modeling permeability of rocks which do not have a significant fraction of isolated vug porosity.

Permeability and electrical conductivity Chapter j 10

819

The predicted permeability from BrayeSmith model is denoted here as kBraySmith. Total porosity from NMR logs is denoted as 4NMR . Irreducible fluid volume per unit rock volume is denoted as VBF. The pore volume per unit rock volume associated with an NMR T2 value falling within the range (T2il,T2iR) where the quantities within the brackets are the left and right edge T2 values which define a particular bin i is denoted as 4NMR ðT2i Þ which is also called as the NMR porosity for T2-Bin “i”. kBraySmith is then given by, ( " #)S i¼N X   4NMR ðT2i Þ P wfi (10.154) kBraySmith ¼ ð4NMR Þ VBF i¼1 " 4NMR ¼

i¼N X

# 4NMR ðT2i Þ

(10.155)

i¼1

Here, N stands for the number of bins into which the T2 distribution has been partitioned, wfi is the weight factor for the bin porosity for the bin i, and P and S are constants, which are model variables. VBF is obtained based on laboratory determined T2 cutoff for cutting off microporosity. When N is equal to 2, and the bins span microporosity and meso plus macroporosity, respectively, and wf1 is set to 0, (kBraySmith) is TimureCoates permeability, for P ¼ 8and S ¼ 2. For a practical use of the predictor the bin margins are so selected that the bins span microporosity, mesoporosity, and macroporosity, respectively. Keeping the values P ¼ 8 and S ¼ 2, the weight factor for microporosity bin porosity is set to a low value, and the weight factor for mesoporosity bin porosity is set to a value which calibrates the permeability predicted to actual permeability for rock section dominated by mesopore porosity (intergranular porosity). Once this calibration is performed, the weight factor for macropore porosity bin porosity is adjusted for best fit of the predictor for the rock having macropore porosity as well. Final adjustments can be on P,S for generating the forward model on permeability having the best fit with actual permeability.

10.11 Another approach to permeability modeling which also relies on NMR log data Consider the TimureCoates permeability predictor   FFV 2 k ¼ C44 BFV

(10.156)

Here, 4 is total porosity, BFV is bound fluid volume or pore volume which cannot be drained, and FFV is free fluid volume which is the difference between the total pore volume and the bound fluid volume. In a rock, since BFV

820 Understanding Pore Space through Log Measurements

is bound fluid volume or pore volume which cannot be drained, this pore volume cannot participate in fluid transport when a pressure gradient exists inside the rock. It is known that when tortuosity decreases, the exponent of porosity reduces to values less than 4. In a more general sense, the exponent of porosity decreases when the branching of pore network reduces in intensity. For the case of connected vugs the exponent is near 2.0. It is logical therefore, to consider that when a permeability predictor is considered for rocks whose connected pore space is mainly, connected vug pore space, the value of the exponent of the porosity is likely to be near 2.0. Rocks which do have such pore fabric as indicated above are common. These usually are carbonates which had intragranular (micropore), intergranular (mesopore), as well as grain dissolution (macropore) porosity, when at the stage of postneogenesis and early diagenesis but had the micropore and mesopore porosity mostly cemented out/be as disconnected pores, as diagenesis progressed. For predicting the permeability of such a rock, where macropore porosity is the main participator in fluid transport, it is reasonable to consider, BFV ¼ ðVmicro þ Vmeso Þ

(10.157)

FFV ¼ 4  ðVmicro þ Vmeso Þ

(10.158)

 kvug ¼ Cvug 42

4  ðVmicro þ Vmeso Þ ðVmicro þ Vmeso Þ

4 ¼ ðVmacro þ Vmicro þ Vmeso Þ

2 (10.159) (10.160)

Here, kvug is the permeability predictor for a rock as described above, and Vmicro, Vmeso, Vmacro are, respectively, the micropore pore volume, mesopore pore volume, and macropore pore volume per unit rock volume. Cvug is a constant. Using Eq. (10.160), Eq. (10.159) is stated as,  2 Vmacro 2 kvug ¼ Cvug 4 (10.161) Vmicro þ Vmeso Eq. (10.161) can be written as,  2  2 Vmacro vmacro Vmacro 2 2 ¼ Cvug 4 where vmacro ¼ kvug ¼ Cvug 4 4  Vmacro 1  vmacro 4 (10.162) An equation of the type of Eq. (10.162) is described by Jacob et al. (2011). The permeability of a rock whose pore space comprises intragranular and intergranular pores has permeability arising from intergranular pore space alone. We denote the permeability of such a rock by kintg. kintg is nothing but k2 of Eq. (10.67). We can therefore state Eq. (10.67) in the present context as,

Permeability and electrical conductivity Chapter j 10

kintg ¼

 m a2 ðrT2LM Þ2 4ig m C

821

(10.162a) 44

If 4 is total porosity of such a rock as above, 4ig ¼ 14 m where 4m is m intragranular pore volume per unit rock volume. Ignoring 4m as usually 4m  4, we can replace 4ig by 4 in Eq. (10.162) and write, kintg ¼

a2 ðrT2LM Þ2 ð4Þmm C

(10.163)

An equation of the type of Eq. (10.162) is described by Jacob et al. (2011), 2 who use the value of 0.35 for aC and 2 for mm. T2LM refers to logarithmic mean T 2. Hence, for the case of rocks where macropores are the dominant fluid transport pores, permeability is modeled as per equation,  2 vmacro kvug ¼ Cvug 42 (10.164) 1  vmacro Not only vugs touching each other, but vugs connected by fractures/open stylolites and the fractures/open stylolites themselves are also included in macropore porosity. For the case of rocks where mesopores are the dominant fluid transport pores, and which are largely devoid of macroporosity, permeability is modeled as, k¼

a2 ðrT2LM Þ2 ð4Þmm C

(10.165)

The preferred values are 0.35 for aC and 2 for mm. The right values are found out by fitting a straight line to logk versus log 4, and obtaining the intercept and slope of the fit, respectively. In the context of a rock containing micropores, mesopores, as well as macropores, the bulk rock permeability is modeled as follows. The permeability of a rock having total porosity equal to the total porosity of bulk rock but having no micropore porosity and no macropore porosity can be considered as the base or unperturbed permeability value. The pore structure is now perturbed, without changing the total pore volume, to a structure where some of the pores are compounded to form macropores. The base unperturbed 2 permeability is given by kbase ¼ aC ðrT2LM Þ2 ð4Þmm where T2LM corresponds to the mesopore class. The perturbation model (which results in the perturbed permeability, which is the bulk rock permeability k) is,  2 vmacro (10.166) k ¼ kbase 1  vmacro 2

The permeability predictor of Jacob et al. (2011) has the form of Eq. 2 (10.166). Specifically, since kbase ¼ aC ðrT2LM Þ2 ð4Þmm the model prediction of bulk rock permeability k is,

822 Understanding Pore Space through Log Measurements



a2 ðrT2LM Þ2 ð4Þmm k¼ C



vmacro 1  vmacro

2 (10.166a)

Eq. (10.166) is the permeability model of Jacob et al. (2011) for rocks hosting macropore porosity in addition to micropore porosity and mesopore porosity. This equation reduces to that for intergranular rock when macropore pore volume is zero, as it should, and has a fast rise with increasing macropore pore volume, justifying empirical findings.

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823

Machado, V., et al., 2011. Carbonate petrophysics in wells drilled with oil-base mud. In: SPWLA 52nd Annual Logging Symposium, Colorado Springs, Colorado. Marzouk, I., Takezaki, H., Miwa, M., 1995. Geologic controls on wettability of carbonate reservoirs, Abu Dhabi, U.A.E. In: SPE paper 29883 presented at SPE Middle East Oil Show, Bahrain, March, pp. 11e14. Marzouk, I., Takezaki, H., Suzuki, M. 1998. New classification of carbonate rocks for reservoir characterization. In: SPE paper 49475 presented at the 8th ADIPEC (Abu Dhabi International Petroleum Exhibition and Conference), November 1998, Abu Dhabi. Meyer, F., Beucher, S., 1990. Morphological segmentation. Journal of Visual Communication and Image Representation 1 (1), 21e46. Miao, T., et al., 2015. Fractal analysis of permeability of dual-porosity media embedded with random fractures. International Journal of Heat and Mass Transfer 88. Nelson, R.A., 2001. Geological Analysis of Naturally Fractured Reservoirs, second Edition. Gulf Professional Publishing. Parsons, R.W., 1996. Permeability of idealized fractured rock. Society of Petroleum Engineers Journal 6 (2), 126e136. Rasmus, J.C., Kenyon, W.E., 1985. An improved petrophysical evaluation of Oomoldic Lansing Kansas City formation utilizing conductivity and dielectric measurements. In: Paper V, 26th Annual Logging Symposium Transactions, Society of Professional Well Log Analysts. Ramakrishnan, T.S., et al., 1999. Forward models for nuclear magnetic resonance in carbonate rocks. The Log Analyst 40 (4), 260e270. Ramakrishnan, T.S., et al., 1998. A petrophysical and petrographic study of carbonate cores of the thamama formation. In: SPE Paper 49502, Presented at the 9th ADIPEC Abu Dhabi. Ramakrishnan, T.S., et al., 2001. In: A model-based interpretation methodology for evaluating carbonate reservoirs. SPE Society of Petroleum Engineers. Paper SPE 71704. Ramamoorthy, R., et al., 2010. A new workflow for petrophysical and textural evaluation of carbonate reservoirs. Petrophysics 51 (1), 17e31. Sarkar, S., Dutta, R., March 2018. Characterization of a dual porosity reservoir integrating advanced acoustic log data, electrical image data with conventional petrophysical analysis e a case study. In: Presented at the SEG Middle East Workshop, Bahrain. Sarkar, S., Toksoz, M.N., Burns, D.R., 2004. Fluid flow modeling in fractures. Massachusetts Institute of Technology, Earth Resources Laboratory. http://hdl.handle.net/1721.1/68616. Sen, P.N., Scala, C., Cohen, M., 1981. A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads. Geophysics 46, 781e795. Smith, H., Nelson, K., Hall, A., 2014. Comparison of image logs to nuclear magnetic resonance logs. In: AAPG International Conference and Exhibition. van Golf-Racht, T.D., 1982. Fundamentals of Fractured Reservoir Engineering, Developments in Petroleum Science, no. 12. Elsevier Scientific Publishing Company, Netherlands. Vincent, L., Soille, P., 1991. Watersheds in digital spaces: an efficient algorithm based on immersion simulations. IEEE Transactions on Pattern Analysis and Machine Intelligence 13 (6), 583e598. Yamada, T., et al., 2013. Revisiting porosity analysis from electrical borehole images: integration of advanced texture and porosity analysis. In: 54th Annual Logging Symposium. Zhang, T., 2006a. Filter-based Training Image Pattern Classification for Spatial Pattern Simulation (Ph.D. dissertation). Stanford University, Stanford, California, USA. Zhang, T., 2006b. Multiple-point Pattern Simulation Using Filter Scores: The Filtersim Algorithm and Theory. SCRF.

824 Understanding Pore Space through Log Measurements

Further reading Berryman, J.G., 1992. Single-scattering approximations for coefficients in Biot’s equations of poro elasticity. Journal of the Acoustical Society of America 91 (2). Bruggeman, D.A.G., 1935. Calculation of various physical constants of heterogeneous substances, part I: dielectric constant and conductivity of mixtures and isotropic substances. Annals of Physics 24 (5), 636e679. Carr, H.Y., Purcell, E.M., 1954. Effects od diffusion on free precession in nuclear magnetic resonance experiments. Physical Review 94 (8). Choquette, Pray, 1970. Geologic nomenclature and classification of porosity in sedimentary carbonates. AAPG Bulletin 54, 207e250. Christian Espina, C., et al., 2013. Nuclear magnetic resonance permeability response in oil production conditions. In: AAPG International Conference and Exhibition, Columbia. Dunham, R.J., 1962. Classification of carbonate rocks according to depositional texture. In: Ham, W.E. (Ed.), Classification of Carbonate Rocks: American Association of Petroleum Geologists Memoir, pp. 108e121. Frank, S., et al., 2005. Carbonate rock typing using NMR data: a case study from Al Shaheen field, Offshore Qatar. In: IPTC-10889 Presented at the International Petroleum Technology Conference, Doha, Qatar. Gomaa, N., et al., 2006. Case study of permeability, vug quantification and rock typing, in a complex carbonate. In: SPE 102888,Transactions of the SPE Annual Technical Conference and Exhibition, San Antonio, TX, U.S.A. Hassall, J.K., et al., 2004. Comparison of permeability predictors from NMR, formation image and other logs in a carbonate reservoir. In: SPE 88683, presented at 12th ADIPEC. Jennings, J.W., Lucia, F.J., 2001. Predicting permeability from well logs in carbonates with a link to geology for inter well permeability mapping. In: SPE 71336. Li, B., 2001. Characterizing Secondary Porosity and Its Connectivity in Carbonate Reservoirs Using Electrical Borehole Images: CSPG Annual Convention. Lucia, F.J., 1983. Petrophysical parameters estimated from visual descriptions of carbonate rocks: a field classification of carbonate pore space. Journal of Petroleum Technology 35 (3), 629e637. Lucia, F.J., 1995. Rock-fabric/petrophysical classification of carbonate pore space for reservoir characterization. AAPG Bulletin 79 (9), 1275e1300. Newberry, B.M., Grace, L.M., Stief, D.D., 1996. Analysis of carbonate dual porosity system from borehole electrical images. In: SPE Paper 35158-MS Presented at the Permian Basin Oil and Gas Recovery Conference. Rasmus, J.C., 1983. A variable cementation exponent, M, for fractured carbonates. The Log Analyst 13e23. Sen, P.N., 1984. Grain shape effects on the electrical and dielectric properties of rocks. Geophysics 49, 586e587. Smith, C.H., Bray, J., Ramakrishna, 2013. Utilization of magnetic resonance bin distribution to develop specific permeability. In: Rocky Mountain Geology and Energy Conference, Denver Colorado. Zhang, T., Hurley, N.F., Zhao, W., 2009. Numerical modeling of heterogeneous carbonates and multiscale dynamics. In: SPWLA 50th Annual Logging Symposium, Paper 50700.

Index ‘Note: Page numbers followed by “f ” indicate figures and “t” indicate tables.’

A Absolute pore body radius, 120 Absolute pore throat radius, 120 Acoustic body wave slowness data, 292 dipole excitation, 316e320 measurement, 315e320 monopole excitation, 315e316 Acoustic fluid mobility impedance, 722 Acoustic logging tools, 306 Acoustic logs, 456e465 computation of cleat density from, 165e166 Stoneley reflection coefficient, transmission coefficient, and energy loss, 165e166 Acoustic slowness data inversion of acoustic slowness data to pore aspect ratio in sandstones, 457e463 DEM model, 460e462 evaluation of representative grain-shear modulus, 463 spherical pores, 462e463 to pore aspect ratio in carbonates, 471e472 Acoustic wave measurement of porosity using acoustic wave slowness data, 292e315 slowness, 478 velocity, 456e457 Acquisition of spectra, 402 Activated hopping mechanism, 207 Active dispersion frequency band, 321e322 Adsorbed gas, 382 Adsorbed methane molecules, 208 Adsorption gas diffusion and role played by adsorption of gas in modeling, 198e200 of methane gas into kerogen and role by adsorption in relaxing spins, 202e203 pores, 6, 161e162 process, 220e221 Aguilera model, 807

Aguilera’s equation, 806 Airy frequency, 336 Airy Phase Frequency, 321, 330 Aligned inclusions, 555e556 differential effective medium theory for, 556e559 Alpha processing method, 108e109 Alpha Processing of Neutron tool data, 109 Aluminum, 395 Angular frequency, 351 Apparent compressional grain modulus of elasticity, 298 Apparent hydrogen index, 284 Arbitrary permittivity, 536e537 Archie “m” factor, 503, 592e594 approaches used in modeling, 501 based on differential effective medium theory, 530e551 illustration of modeling “m” using differential effective medium theory, 539 salient features of Sheng’s model for sedimentary rocks, 540e547 Sheng’s model and Archie’s “a”, 548e549 Sheng’s model and Archie’s “m”, 549e551 symmetric formulation of effective medium of composite, 530e538 inversion of Archie “m” from singlefrequency dielectric data, 503e511 inversion of multifrequency dielectric data, 511e522 real data of conductivity vs. textural model, 517f real data of permittivity vs. textural model, 516f rock model, 515f through NMR data analysis, 589e592 Sheng’s model and, 549e551

825

826 Index Archie cementation application of Maxwell’s Equations applied to binary mixtures, 581e585 approach for computing “m” from grain attributes obtained through multifrequency dielectric data inversion, 503e551 using multifrequency dielectric data and using Archie’s equation, 501e502 using single-frequency dielectric data and using Archie’s equation, 501 approaches to model “m”, 564e581 graphs of the imaginary part, 570f approaches used in modeling Archie’s m parameter, 501 Archie “m” through NMR data analysis, 589e592 constraint on g(sʹ), 571e573 depolarization factors, 559e564 differential effective medium theory for aligned inclusions, 556e559 essence of analysis made, and importance, 587e589 estimate “m” through fractal model of pore space, 602e607 exponent, 63, 803e804 of connected vug/fracture pore space, 807 level by level evaluation of, 804 prediction of value of Archie cementation exponent “m” using effective medium theories, 500e501 fact of every pole of f(s), 586e587 fact of positive residues of poles of f(s) arises naturally in analysis, 586 factor, 574e575, 673 logarithmic mixing law for effective permittivity of mixture, 594e602 modified Maxwell Garnett mixing law, 551e555 percolation theories and Archie “m” factor, 592e594 poles of (s) are real numbers, 585e586 prediction of value of Archie cementation exponent “m” using effective medium theories, 500e501 spectral density function is expected to satisfy, 573e581 sum rules to be obeyed by (sʹ), 571 Archie Parameter, 62 Archie tortuosity parameter, 495 Archie’s “a”, 548e549

Archie’s a, m parameters, 710 Archie’s Equation, 495e496, 673, 679, 760, 807 approach for computing “m” using multifrequency dielectric data, 501e502 approach for computing “m” using singlefrequency dielectric data, 501 Archie’s formula, 509 Archie’s Law, 495e496, 498, 500, 509, 525, 528, 569, 593e594, 649, 663, 666, 692 Array data, 525e526 Array of receivers, 525e526 Array receiver signal processing using semblance, illustration of, 322 Aspect ratio, 12e13, 455 pore aspect ratio, grain aspect ratio, and dielectric logs, 482e486 Attributes of pore space, 117e123 characteristic length scale of pore space, 121 concept of pore class, 120 Geometry of pore body, 119 hydraulic radius measure of pore space, 121e122 pore shape, pore size, and pore throat size, 118 pore shape factor, 122 pores as building blocks of pore space, 118 Size of pore, 119e120 surface area to volume ratio, 120e121 Average density of solid part of formation, computation of, 398e399 Average fluid velocity field, 614, 715 to changes in driving pressure, 614e616 Average permeability field, 631e632

B Backpropagation method, 104, 319 Backscattered gamma photons. See Singlescattered gamma photons Barites, 285 Base rock, 801 permeability kb of, 817 Bed boundary, 765 Bedding planes, 757 resistivity along and normal to formation, 79e86 Bergman theorem, 567, 588 Berg’s equation, 655e656, 678

Index Berg’s model, 656 Berg’s permeability model, 656 Berryman’s model, 773 Berryman’s scattering theory, 294 Biexponential decay of magnetization, 193e194 Bin porosity, 376 total porosity and, 377e379 Binary mixture of matrix, 600 Bins, 376 Bins span meso plus macroporosity, 819 Bins span microporosity, 819 Biot-Willis Coefficient, 723 Biot’s pore space modulus, 723 Biot’s slow compressional wave, 720e721 Bitumen, 177e178, 385e387 hosts poresystem, 3 pores in shale gas reservoirs, 9 morphology of pyrobitumen hosting porosity, 11f pores in solid bitumen, 11f pyrobitumen pores in shale gas reservoir rock, 12f Bloch’s equations with diffusion for case of static magnetic field varying spatially, 361e364 Blocky reservoirs, evaluation of, 76 Boltzmann distribution, 349 Boltzmann transport equations, 278e279 Boltzmann’s distribution, 187 Bond percolation, 592 Borehole Acoustic Wave Propagation, 317e318 Borehole effect, 255 and factoring-in, 277e278 standard conditions, 277e278 Borehole electric images, 414e415 Borehole fluid density, 336 slowness, 336 Borehole wall, high-resolution electrical images of, 415e421 Bound fluid, 183 index, 655 volume, 171, 653e655 in Coates equation, 668e670 Bound water, 389 Bound-stones, 1 Box counting method, 146, 602 BrayeSmith model, 818 of computing permeability, 818e819 Brine, 16, 564

827

Brine compressional modulus method, 308 Brookes-Corey Permeability Model, 154e155, 606, 682e683 Brookes-Corey relation, 22 BrookeseCorey equation, 153e154 Brooks-Corey equations, 694e699 permeability from, 699e701 Brooks-Corey pore heterogeneity index, 698 Brownian motion, 369, 738 Bruggeman effective permittivity, 537, 539 Bruggeman mixing, 589 law, 532e533, 536e537, 546 Bruggeman theory, 540 BruggemanneLandauer Equation, 537 Bruggeman’s equation, 535 Bruggeman’s model, 537e538, 541 Bulk density, 34 computation of total porosity from, 86e87 of rock, 227 of wet clay and silt and value of silt index of shale, 76e78 Bulk gas, relaxation of, 200e201 Bulk material conductivity, 675 Bulk medium, 554 Bulk modulus, 476e477 forward model of bulk modulus of water saturated rock, 469e486 differential effective medium model, 472e481 grain modulus, 469e472 pore aspect ratio, grain aspect ratio, and dielectric logs, 482e486 Bulk permittivity, 519 of matrix, 600 Bulk porosity of conventional reservoirs acoustic body wave slowness measurement, 315e320 diffusion length, 271e273 dispersion model used in quadrupole waveform data inversion, 336e339 dispersive STC, 325e329 illustration of array receiver signal processing using semblance, 322 importance of frequency of bandwidth of excitation, 321e322 inversion method, 339 mean squared displacement of fast neutrons, 285e288 measurement of bulk porosity, 227e257 measurement of formation density for bulk porosity, 227e257

828 Index Bulk porosity (Continued ) measurement of porosity using acoustic wave slowness data, 292e315 measurement of slowing down length and diffusion length of neutrons for bulk porosity, 258e271 multiple-scattered photons role in formation density measurement, 230e257 neutron migration length, 273e285 effect of noise, 329e332 nondispersive STC or simply STC, 322e324 pore space attributes of conventional reservoirs, 227 quadrupole excitation, 332 salient features of quadrupole excitation, 332e333 simple approach to ratio, 257e258 single scattering of gamma photons, 228e230 solution of Eqs, 288e292 STC and DSTC, 322 tool modes interference or with borehole modes, 333e336 estimation of bulk porosity of laminated formations using deterministic approach, 31e54 clay/shale within clastic rocks, 44e45 computation of effective porosity using shallow resistivity, gamma ray, formation bulk density, and neutron porosity data, 38e40 computation of effective porosity using shallow resistivity, neutron capture gamma ray spectrometry, formation bulk density, and neutron porosity data, 32e38 pore volume not shared with shale, per unit rock volume, 40e44 ThomaseStieber approach, 45e54 total porosity, 40e44 wet clay content of shale, 39e40 Bulk relaxation, 367e369 Bulk rock, 801 permeability, 821e822 shear modulus, 478 Bulk volume, 630 Bundle of capillaries, 118, 135e136, 147 model of intergranular pore space, 121 Bundle of capillary-tubes, 681 model of pore space, 621e626

Bundled matchstick model, 175 Burst acquisition mode, 187e188 Butt cleats, 161 Button conductivity matrix (BCM), 420 Button current map, 417 process flow for creating borehole images from, 418e421 BCM, 420 data equalization, 418 EMEX correction, 418 magnetic declination correction, 419e420 normalization, 420e421 scaled button conductivity data, 421 speed and depth corrections, 419 Button current matrix, 417 Button porosity histogram, 26

C Calibrated button conductivity matrix (CBCM), 421 Calibration-based approach, 462 Capillary PressureeWetting fluid saturation curve, 153 Capillary/capillaries, 617e622, 685 bound water, 653e654 bundle model of pore space, 649 using log measurements, 710 model of pore space, 647e648 modeling constriction of, 623e624 pore space attributes influence permeability, 620e621 pressure, 650, 668e669, 703 curve, 657 saturation behavior vs., 621 radius fractal, 693 radius of, 622 representative hydraulic tortuosity and cumulated surface area to cumulated volume of, 150e155 tubes model, 616, 625e626 bundle for modeling flow of incompressible fluid through porous rocks, 20 geometry, 622e623 of pore space, 621, 681, 684 unit, 145 Capture gamma counts spectrum, 392 rays relative yield of, 392e395 spectra, 399e403

Index yield, 391e392 Carbon, 259 fraction, 386 Carbonate(s), 456e457, 471 reservoirs, 123 rocks, 471, 476e477, 479, 735 MICP data in, 736 NMR data in, 736 pore space of, 4 porosity characterization, 473 Cartesian coordinate system, 626 Catchment areas, 762e763 basins, 762e763 Cation exchange capacity (CEC), 59e60, 92, 578 CBCM. See Calibrated button conductivity matrix (CBCM) CBM reservoirs, 161e177. See also Conventional reservoirs. See also Nonconventional reservoirs characterization of pore space of coals using NMR data, 170e172 interpore-class connectivity, 170e171 partitioning of pore space of coals as bound fluid and free fluid volumes using NMR data, 171e172 partitioning of pore space of coals using NMR data by pore size, 170 characterizing pore space of CBM reservoirs using image data, 403e413 cleat/fracture aperture and volume, and matrix porosity of, 403e413 components of space occupied by fluids in coals, 161e162 characterization of pore space of coals, 162e169 cleat orientation and direction of maximum principal horizontal stress, 169 cleat volume per unit rock volume, 166e168 computation of cleat density from acoustic logs, 165e166 computation of cleat density from images, 165 computation of fracture/cleat aperture, 164e165 Hodot’s pore classification scheme, 162 noncleat fracture volume per unit rock volume, 168e169 NMR and porosity of, 381e385

829

cleat volume per unit rock volume, 385 coal pores, 381e382 porosity available within rock for holding free gas, 385 T2 relaxation spectra of coals, 382e384 total porosity and gas volume, 385 permeability of coal, 173e177 permeability prediction for CBM reservoirs using image data, 434e439 CCA. See Charge coupled amplifier (CCA) CEC. See Cation exchange capacity (CEC) Cesium-137 gamma source, 95e96 Charge coupled amplifier (CCA), 402 Circular cylindrical pipe, 663 Clastic rocks, 44e45 Clastic-laminated rock, 31 Clay, 673e674 bound water, 574e575 clay-bound water, 581 volume per unit, 390 clay/shale manifest within clastic rocks, 44e45 Clayey rocks, applicability of model to, 527 Claystone, 45 Cleat density (CD), 163, 165e166 computation from acoustic logs, 165e166 from images, 165 Cleat(s), 161, 164 cleat/fracture aperture and volume, and matrix porosity of CBM reservoirs, 403e413 cleat volume per unit rock volume, 408e410 estimation of, 405e410, 407f evaluation of matrix pore volume, 411e413 extraction of fracture segments from image data, 405e406 fracture volume per unit rock volume, 410 cleat/fracture permeability, 403e404 in coals, 17 extraction of cleats from borehole images, 163e164 length, 408 system, 173 volume, 408 per unit rock volume, 166e168, 385, 408e410, 435 Clusters, 213

830 Index Coal(s) measurement of permeability of coal using log data, 173e175 modeling permeability of coal using log data, 175e177 using NMR data characterization of pore space of, 170e172 partitioning of pore space of using NMR data by pore size, 170 permeability of, 173e177 pores, 381e382 space, 4e6 T2 relaxation spectra of, 382e384 Coates equation bound fluid volume in, 668e670 for permeability, 652e655 Coherence peak, 104 Coherent Phase Approximation scheme (CPA scheme), 530, 774e775 Coherent Potential Approximation, 537 Collar mode. See Tool quadrupole mode Color component intensities, 420 Common factor, 449 weights, 450 Complex dielectric component, 673 Complex dielectric permittivity, 555e556 data, 523e524 Complex permittivity, 501e502 coming from unconnected pores, 528 Complex Refractive Index model, 504 Complexity penalty factor, 212 Composite rock, permeability of, 795e797 Compressional slowness, 307e308 equation for, 56 model-based inversion of compressional slowness to porosity for sandstones, 298e301 Compressional waves, 317e320 evaluating from shear wave velocity and compressional wave velocity, 478e479 inversion of compressional wave slowness log data to pore aspect ratio, 465 porosity dependence of compressional wave propagation speed for rocks pores, 298e308 propagation velocity, 309e310 slowness, 293 Compton energy window, 233 Compton process, 232

Compton scattering process, 230, 233, 242e243, 251 Conductive heterogeneity, connectedness attribute of, 764e765 Conductive spots, 765 Conductivity, 538 of base rock, 802 formation factor, 498e499 map of borehole wall, 404 Conjugate rock medium, 572 Connected macropore pore space, 807e808 Connected pore space, 805 tortuosity unity, 801 Connected vugs, 796e797, 807e808, 818 connected vug/fracture pore space, 816 permeability of rocks hosting, 811e818 pore space, 792 Connectedness curve, 764e765 connecting porosity and compressional slowness in rocks, 308e310 Connectivity, 498 Conventional density logs, alpha processing for improving vertical resolution of, 109 Conventional formation density log, 105 Conventional logs, 803e804 Conventional neutron porosity logs, alpha processing for improving vertical resolution of, 108e109 Conventional nuclear suite, 162 Conventional reservoirs, 123, 777. See also Nonconventional reservoirs. See also CBM reservoirs attributes of pore space, 117e123 computation of CPT(R) from NMR data, 131e133 linear conversion work flow, 131e132 nonlinear conversion work flow, 132e133 distribution of incremental porosity over pore radius, 124e130 electrical formation factor from perspective of fractal model of pore space, 155 fractal attributes of pore space, 144e155 cumulative pore volume, 149e150 fractal model of pore space, 147e148 fractal model of pore space, based on a poreepore throat assemblage visualization of physical pore space, 145e146 permeability from perspective of fractal model of pore space, 148e149

Index representative hydraulic tortuosity and cumulated surface area to cumulated volume of capillaries, 150e155 pore shape factor through integrating NMR and MICP, 134e135 frequency distribution of pore radius, 134e135 pore space attributes of, 227 pore space of intergranular rocks, 117 relation between tortuosity, porosity, and formation factor, 155e157 simple visualization of constriction and effect on gross permeability of pore space, 135e144 model prediction of permeability, 141e142 TimureCoates permeability predictor from perspective of constriction, 143e144 Conventional resistivity suite, 162 Conventional water-wet reservoirs, 179e180 Converse process, 198 Coordinate system, 285 Cost function, 388 Count rate data to porosity, 277e278 Covariance matrix, 97 CPA scheme. See Coherent Phase Approximation scheme (CPA scheme) CPMG cycle, 203e204, 358, 377 CPMG cycle-based activation, 203 Cramer’s rule, 490 CRI equation, 521 CRI medium, 482e483, 515 CRI mixing law, 484, 504e506, 515, 522, 551, 573, 594 Critical percolation threshold, 593 Critical porosity, 294 Cross plot porosity, 507e508 Crystalline quartz, 302 Cumulated surface area to cumulated volume of capillaries, representative hydraulic tortuosity and, 150e155 Cumulated vug pore space geometry, 792 Cumulative connected vug pore space/ fracture pore space, 810 Cumulative face area of connected vug/ fractures, 816 pore space, 815 Cumulative pore volume, 149e150, 689, 772 Cumulative surface area of connected pore space, 691 Cumulative variance method, 450e451

831

Cumulative volume of connected pore space, 691 Cutoff frequency, 336 Cyan curve, 181 Cylindrical geometry, 427 Cylindrical pores, 119e120 geometry, 739e740

D D-T2 plots, 189, 197e200 adsorption of methane gas into kerogen and role played by this adsorption in relaxing spins, 202e203 D-T2 maps and plots related to results of echo data inversion, 203e207 activation detail, 205t burst mode activation, 203e207 partitioning of methane into free and adsorbed methane using relaxation data from echo amplitude inversion, 207e209 gas diffusion and role played by adsorption of gas in modeling, 198e200 partitioning of total gas into free and adsorbed gas components using only NMR data, 207e209 relaxation of bulk gas, 200e201 gas within inorganic nanopores, 201e202 gas within organic pores, 201 Damped least squares, LevenbergeMarquardt method of, 72e73 Darcy Law, 141, 616, 618e619, 630, 635, 646, 663, 688, 718 Darcy velocity, 617e618, 635, 637 Darcy’s Law, 719, 794e795 Data acquisition, 417 dataebased porosity partition, 745e746 equalization, 418 inversion method, 339 preconditioning of, 747 Data gap, 747e753 image rescaling, 753 score class prototype, 751e752 simulation to populate pixels have no data, 752e753 training pattern classification, 748e751, 754f default filters, 749f electrical image, 750f

832 Index Deconvolution technique, 105e108 Degree of connectivity between pores, 18 180 degrees tipping of spins, 357 Delayed gamma photons, 399e400 DEM model. See Differential effective medium model (DEM model) Density, 803e804 correction, 237 neutron cross plot porosity, 503e504 porosity of clay, 34 Departure curves, 281 Depolarization factors, 519e520, 559e564, 788 of ellipsoid in general and spheroid in particular, 559e564 of grain, 673e674 Depolarization field, 559e561 Depth corrections, 419 Deuterium-tritium reaction (D-T reaction), 390 Diagonal matrix, 626e627 Dielectric logs, 482e486 data, 580e581 Dielectric mixing law, 504 Dielectric permittivity, 559e560, 672e673 dispersion model of, 525 Klein and Swift model for dispersion of, 486e487 work flow for generation of dispersion model of, 522e524 Dielectric spheroid, 559e560 Differential effective medium model (DEM model), 298, 457e458, 460e462, 539, 541e543, 549, 589 Differential volume, 629 Diffusion coefficient, 265 dimension, 182e183 editing method, 191e195 length, 271e273 process, 265 Diffusion dephasing time constant (T2D), 367 Diffusion limited regime. See Slow diffusion limit Diffusion-encoding time (TD), 197 Diffusive coupling, 23 Diffusivity, 722e723 Dilation’ operator, 210, 756 Dip fracture segments, 757e758 Dip picking, 747 Dipolar relaxation, 201 Dipole excitation, 316e320, 333e334

recorded dipole waveforms for case of fast formation, 318e320 recorded waveforms time evolution of wave field for slow formation case, 317e318 Dipole moment, 532, 559e561 Dipole receiver, 320 axis of, 320 Dipole transmitter, 315e316 Dipole waveforms for case of fast formation, 318e320 Dirichlet type boundaries, 713e714 Dispersed shale, 45e46 Dispersion analysis, 319 Dispersion model, 325e326, 336, 510 of dielectric permittivity, 525 for rock conductivity, 524e527 work flow for generation of dispersion model of dielectric permittivity, 522e524 Dispersion of dielectric permittivity of brines, Klein and Swift model for, 486e487 Dispersive STC (DSTC), 320, 322, 325e329, 339 Dissolution pores, 12e13 Distance function, 752 Distribution space, 184 Divergence theorem, 630 Dolomitization, 2e3 Dominant grainsize, 666 Double Laplace Transform, 196 Drainage process, 151 Dry clay density, 218 hydrogen index of dry clay and silt, 78e79 permittivity, 527 to water trend, 76 Dry frame, 459 case of, 467e468 compressional modulus, 298 moduli forward models, 481 with no compliant pores and stiff pores spherical in shape, 468e469 Dry rock elastic moduli, 306 Poisson ratio, 298 Dry weight clay fraction, 389 total carbon content, 386

Index DSTC. See Dispersive STC (DSTC) Dual detector data processing using single window count rates, 233e240 Dual porosity, unfractured composite hosting, 787e793 Dual Water Equation, 58e59, 90 Dual Water Model, 63e64

E Eagleford Shale, 190 Echo(es), 373 amplitude, 183e184 temporal decay of, 360e361 vector representation of quantum states, 360f data D-T2 maps and plots related to results of echo data inversion, 203e207 estimation of total porosity directly from, 379e380 spacing, 358e359 Effective electron density, 32 Effective medium, 530 model, 472e481 aspect ratio of intergranular pores, 479e481 evaluating from shear wave velocity, 478 evaluating from shear wave velocity and compressional wave velocity, 478e479 of R model, 522 theory for aligned inclusions, 556e559 Effective permeability factor, 631 and permeability, relation between, 632 Effective porosity estimation, 389e390 vcbw, 389e390 Elastic moduli, 293 Electric conductivity, 673 Electric displacement field, 583 Electric fields, 560, 584e585 Electric formation factor log, 676, 707 Electrical conduction, 639 Electrical conductivity, 620e621 of fractured rocks, 797e801 partial saturation case, 799e801 of unfractured composite hosting dual porosity, 787e793 case of partial saturation, 792e793 evaluation of, 792 modeling for fully water saturated component 2 case, 789e791

833

Electrical formation factor, 649, 692e693 from perspective of fractal model of pore space, 155 Electrical image data, 814 Electrical tortuosity, 22e23, 642e643, 663, 794, 796e797, 802 Electro-osmotic coupling coefficient, 665 Electrolyte, 676 Electromagnetic field, 790 Electromagnetic wave, 528, 523e524 Electron density, 228, 236 Element carbon, 259 Elemental capture gamma ray spectrometry logs, 162 Elemental concentration in rocks estimation of, 390e403 inelastic gamma spectrometry, 395e398 neutron capture gamma spectrometry, 390e395 Elemental standard spectra, 431 Elemental stream tubes, 629e630, 632 permeability factor field, 633e637 decomposition of permeability into macroscopic pore space attributes, 636e637 explicit representation of permeability factor of streamline in some of attributes, 634e636 Elementary stream tube, 633 Ellipsoidal grains, 546e547 Ellipsoids, 519, 536e537, 545, 563 representing grain shape, 558 EMEX correction, 418 Empirical relations, 296e297 Entropy principle, multidimensional inversion of NMR echo data using maximum, 183e188 Environment slowness parameter, 339 Epithermal counts ratio, 278 Epithermal flux, 274e275 Epithermal neutron detector counts, Group 1 neutron flux and, 276 Equations of “spine”, 248 Equivalent tool model, 336 Erosion’ operator, 210, 756 Eshelby’s tensor, 488e489 Euclidian dimensionality, 602, 605, 685, 693 Evanescent wave, 317e318 Excess conductance, 164e165 Excitation bandwidth, importance of frequency of, 321e322

834 Index Extraction of fractures and cleats from borehole images, 163e164 Extrusion, 728 cycle, 728e729

F F-dimensional space, 750e751 Face cleats, 161, 169 Factor analysis, 448, 473 Facture planes, 757 Fast diffusion, 426 limit, 368e369 Fast formations, 316 FDM. See Finite Difference Method (FDM) Fick diffusion, 198e199 Field induction decay, 357 Field perturbation, 482e483 Filter, 748 score vector, 748e749 Filtersim, 748 Finite Difference Method (FDM), 336e338 model, 336 waveform dispersion, 336 First moment equation, 589 Fit-for-processing spectrum, 392, 395 Flexural mode, 316 Flexural wave, 318, 320, 325e326 Fluid flux per unit, 635e636 Fluid pressure, 277e278 Fluid salinity, 580e581 Fluid transport model, 625e626 Fluid-filled porous media certain aspects of relaxation of magnetization within, 423e427 cylindrical geometry, 427 spherical geometry, 427 Fluid-phase interaction, 174 Flushed zone, 32 equation for flushed zone conductivity, 58e59 forward model of flushed zone resistivity, 502 total porosity of rock within, 57e60 Fora cylindrical conduit, 638e639 Formation bulk density, computation of effective porosity using, 32e40 Formation factor, relation between tortuosity, porosity and, 155e157 Formation water, salinity of, 92 Forward model, 65e66 of ribs, 255 Fourier transform, 165e166, 321

of Stoneley Wave’s, 721 Fractal attributes of pore space, 144e155 Fractal dimension, 147 of connected pore space and influence on permeability, 682e688 for lengths of capillaries using log measurements, 710 of pore radius fractal and Archie’s a, m parameters, 710 Fractal models, 605e606, 678, 694e696 impact of, 699e701 of pore space, 21e22, 147e148, 154e155 electrical formation factor from perspective of, 155 permeability from perspective of, 148e149 Fracture segments differentiating bedding planes and facture planes, 757 extraction, 753e758 of fracture segments from image data, 405e406 of low apparent dip fracture segments, 754e757 high confidence fracture traces, 757e758 extraction of high apparent dip fracture segments, 757e758 main orientation of, 757 spatial orientation of traces, 757 Fracture(s), 16e17, 161, 812 aperture and volume, and matrix porosity of CBM reservoirs, 403e413 delineation of fractures within shale reservoirs using image data, 414 extraction of fractures from borehole images, 163e164 fraction, 797e798 fractures/open stylolites, 821 plane, 811 porosity, 227 spots, 765 traces, 210e211 volume per unit rock volume, 410 Fractured reservoirs, characterization of, 209 Fractured rocks, electrical conductivity of, 797e801 Free fluid index, 655 porosity, 183 volume, 655 Frequency of bandwidth of excitation, importance of, 321e322

Index

G Gamma activity of laminated formation, 50 Gamma energy, 391e392 Gamma photons, 391e392 single scattering of, 228e230 Gamma ray computation of effective porosity using, 38e40 data, 109e110 enhancement of vertical resolution of gamma ray logs, 109e111 Gamma ray spectra using standard spectra, 431 Gas diffusion line, 199e200 and role played by adsorption of gas in modeling, 198e200 Gas molecules, 162 Gas volume, total porosity and, 385 Gassmann’s equation, 306, 471, 477 Gaussian curve, 453e455 GausseNewton Method, 99 General Purpose Inclinometry Tool (GPIT), 416e417 Generic treatment of transport of incompressible fluid through medium, 20 Geodesic dilations, 759 Geodesic erosion, 759 Geometric Resonance Density function, 568, 588 Geometric tortuosity, 22e23 Gradient-less magnetic field, 362 Grain aspect ratio, 482e486 Grain bulk material, 483 Grain compressional modulus, 298 Grain ellipsoids, 558 Grain modulus, 469e472 inversion of acoustic slowness data to pore aspect ratio in carbonates, 471e472 Grain shapes, 485 Grain size and grain size distribution from NMR data, 656e660 porosity bins end points of porosity bins, 658t Grain solid, addition of cracks to, 473 Grain surface relaxivity, 367e368 Grain-dissolution pores, 12e13, 25e26 Grain-dissolution porosity, 227 Grain-mineral modulus, 309 Grain-supported rocks, 1 Grain-wetting fluid, 16

835

Grainosity, 526e527 Granular rocks, 16 pore space, 1e3 of granular rocks not falling within categories, 3 intercrystalline pore space, 3 Green’s Functions, 583 of Laplacian, 712 Gross attributes, 259 Gross permeability of pore space, simple visualization of constriction and effect on, 135e144 Gross rock, 807 model, 808 permeability of, 818 Guided mode, 317e318 slowness data, 292

H HagenePoiseuille Law, 148, 215 Hamiltonian operator, 346 Hanai-Bruggeman-Sen equation, 672 HashineShtrikman bounds, 776 Heterogeneities, 211, 355 challenge of extraction of, 761 computing pore volume contribution of, 760e761 efficient methodology for extracting heterogeneities, 761e769 classification of heterogeneities and quantification of porosity, 766e767 classification of mosaic pieces, 763e764 connectedness attribute of conductive heterogeneity, 764e765 final porosity partition, 765 mosaic image and, 765e766 porosity association of different types of spots, 768e769 Hierarchal clustering algorithm, 213 High apparent dip segments, extraction of, 406 High atomic number nuclei, 259 High-angle events, process for, 164 High-resolution bulk density, 94e99 data, 93e94 electric(al) images, 23, 162 data acquisition, 417 generation of high-resolution electrical images of borehole wall, 415e421 position of sensor in space, 416e417

836 Index High-resolution (Continued ) process flow for creating borehole images from button current maps, 418e421 sensors, 415 inversion, 99e100 model, 99e100 output of, 100 NMR data, 75, 111e112 photoelectric factor, 94e99 resistivity imaging tool data into pore size heterogeneity, 24e26 Histogram equalization, 420e421, 439 Hodot’s pore classification scheme, 162 Holding free gas, porosity available within rock for, 385 Homogeneous isotropic medium for group 1, neutron flux distribution for point source in, 263 Homogeneous isotropic medium for group 2, neutron flux distribution for point source in, 269e271 Homogenized permeability density, 632 Horizontal permeability, 68e69 Host porosity, 772, 774 “Host” bulk modulus, 775 “Host” shear modulus, 775 Hydraulic cleat aperture, 177 Hydraulic conductance, 636 Hydraulic conductivity, 818 Hydraulic constriction factor, 621, 635e637, 641, 667e668 of connected pore space, 640e641 Hydraulic pore radius, 636e637, 641 Hydraulic radius, 638e642, 663 hydraulic constriction factor of connected pore space, 640e641 measure of pore space, 121e122 microscopic streamline attributes and macroscopic pore space attributes in porous medium, 640 in permeability modeling, 638 Hydraulic tortuosity, 22e23, 618, 636e637, 641e643, 663, 690e691, 710, 796e797 Hydrocarbon(s), 526e527, 668e669 atoms, 355, 361 density, 387 thermal maturity of, 388 fluid, 129e130 hydrocarbon-corrected bulk density, 48 hydrocarbon-corrected density porosity, 34

index, 183, 259, 281, 376, 380, 422 of dry clay and silt, 78e79 of fluids, 378 of matrix, 34e35 of wet clay and silt and value of silt index of shale, 76e78 property model, 387 property model, 430 volume of, 66 Hydrogen nuclei, 353, 378 nucleus, 345

I Identity matrix, 488 Imbibition capillary pressure, 728e729 process, 728 saturation of nonwetting fluid, 728 Inclusion, 773 porosity, 772 Incompressible fluid through porous rocks capillary tube bundle for modeling flow of, 20 generic treatment of transport of, 20 Incremental porosity over pore radius computation of CP(r) using NMR data, 124e125 distribution, 124e130 of incremental porosity over pore throat radius, 125 hard data on distribution of pore throat size over incremental porosity, 127e128 obtaining CPT(R) from log data, 129e130 obtaining CPT(R) from mercury injection data, 128e129 ratio of pore size to pore throat size, 125e126 Indeterminate medium, 778 Individual probability density, 667 Inelastic gamma spectrometry, 395e398. See also Neutron capture gamma spectrometry acquisition of inelastic and capture gamma ray spectra, 399e403 transforming pulse height spectrum to energy spectrum fit for spectral decomposition, 402e403 computation of average density of solid part of formation, 398e399 inelastic gamma yield, 396

Index relative yield of inelastic gamma rays of element, 397e398 Inelastic gammas, 395 spectrum, 395 Inelastic gate, 400 Inelastic neutron gamma ray spectrometry technology, 218 Injection pressure, 694 value, 701 Injection process, 133 Inorganic granular rocks, 1 Inorganic nanopores, relaxation of gas within, 201e202 Input bin volume data set, principal components of, 450e451 Intercollision displacement vector, 285 Intercrystalline pores, 3 space, 3 Interecho spacing, 366 Intergranular pore(s), 6e7, 12e13 aspect ratio of, 479e481 space, 2, 6, 121, 792e793, 802 Intergranular porosity, 227, 774 Intergranular rocks, pore space of, 117 Intermediate diffusion, 426 Internal fluid flux, 635 pressure, 630e631 Internal porosity of matrix, 769 Internal pressure field, 614 Internal velocity of flow, 617 Interparticle pore system, 178 Interpore connectivity, 16 Interpore diffusion, 741e743 Interpore-class connectivity, 170e171 degree of connectivity between mesopores and macropore assemblages, 170 degree of connectivity between micropore and mesopore assemblages, 171 Intragranular pores, 471 space, 2 Intragranular porosity, 743e746, 791 Intraparticle pore system, 178 Intrinsic macroscopic resistance of porous medium, 619 Intrinsic transverse relaxation time (T2), 367 Inverse Fourier transform, 327 Inversion of log data to gross attributes of pore space alpha processing, 108e109

837

bulk density and hydrogen index of wet clay and silt and value of silt index of shale, 76e78 case of oil base mud or SOBM, 89 case of water base mud, 88 computation of total porosity from bulk density and magnetic resonance logs, 86e87 computation of value of Cbw, 91 deconvolution technique, 105e108 dual water equation and input parameters, 90e93 enhancement of vertical resolution of gamma ray logs, 109e111 estimation of bulk porosity of laminated formations using deterministic approach, 31e54 evaluation of blocky reservoirs, 76 evaluation of microporosity, 75 high-resolution bulk density and photoelectric factor, 94e99 high-resolution data, 93e94 high-resolution model, 99e100 high-resolution NMR data, 111e112 hydrogen index of dry clay and silt, 78e79 improved-resolution acoustic slowness, 100e105 improved-resolution density and neutron porosity logs from conventional 2-detector tool data, 105e108 for improving vertical resolution of conventional density logs, 109 for improving vertical resolution of conventional neutron porosity logs, 108e109 input parameters, 91 inversion process, 82e86 output of high-resolution model, 100 resistivity along and normal to formation bedding, 79e86 S, T plane and S, T plots, 103e105 signal processing for subarrays, 102e103 source of input parameters, 92e93 stochastic inversion of log data for laminated formation, 55e75 method, 71e72, 339, 503e504, 511 presentation of results of, 187e188 process, 82e86, 175, 307, 336 Irreducible saturation, 152 Irreducible water, 653e654

838 Index Irreducible wetting fluid saturation of rock, 16 Isolated conductive anomalies, removal of, 411e412, 416f Isolated pores, 16, 737 Isolated resistive anomalies, removal of, 412 Isolated vugs, 795, 800, 808e809, 817 Isotropic Darcy-Medium, 619 Isotropic medium, 626e628

K Kaiser method, 450e451 Kerogen, 177e178, 385e386 density, 218, 387 kerogen-hosted organic pore system, 178 pores, 3 photographs of different samples from Bakken shale, 12f in shale gas reservoirs, 9 space, 3 property model, 387, 429 thermal maturity of, 388 “Killing regime”, 368 Kinematics of neutron transport, 258e271 viscosity of fluid, 614 Klein model, 484 for dispersion of dielectric permittivity of brines, 486e487 KleineNishina formula, 229 KleineSwift dispersion, 509 KleineSwift model, 513e514 Knudsen diffusion, 198e199 KozenyeCarman equation, 142e143, 638, 641e663 conventional form of, 643e646 equations for permeability prediction from log measurements, 646e651 fitting-constant C in, 680e681 to Van Baaren’s equation, 663e664 KozenyeCarman relation, 20 Krief equation, 295 KustereToksoz equations, 471e472, 782 KustereToksoz estimates, 780e781 KustereToksoz model, 457e458, 772e773, 777e779

L Laboratory coordinate system, 556e557 Laminar flow, 661 Laminar shale, 45e46 Laminated formation

estimation of bulk porosity of laminated formations using deterministic approach, 31e54 gamma activity of, 50 stochastic inversion of log data for, 55e75 Laminated shale volume using tensor resistivity data, computation of representative resistivity of, 54 Langmuir isotherm, 202 Laplace equations, 560e561 Laplace inversion, 195 with regularization method, 195e196 Larmor frequency, 351, 353e354, 357e358, 367 Late Capture’ gates, 400 Lateral quadrupole, 332 Law of Diffusion, 265 Least square error minimization process, 133 Length scale driven approaches to permeability, 621e628 Lenz’s law, 373 LevenbergeMarquardt method, 72e73, 403 Lichtenecker’s logarithmic mixing law, 504, 506, 589 LichtenekereRother mixture law, 313, 315 Linear behavior, 619 Linear conversion, 131 work flow, 131e132 Linear equations, 221e222 Liquid-filled volumes, 720e721 LITH, 241 Lithium fluoride crystals, 275e276 Litho Density Tool, 241, 246 Lithology-Window, 241 ratio of count rate in Compton window, 248 Local fluid pressure field, 718e719 Local fluid velocity field, 614, 715, 718e719 in driving pressure, 614e616 Local permeability density, 630, 632 Local pressure field in driving pressure, 614e616 Local velocity, 617, 629 Log data measurement of permeability of coal using, 173e175 modeling permeability of coal using, 175e177 stochastic inversion of log data for laminated formation, 55e75 use of, 22e26

Index degree of connectivity between two pore size classes using NMR log data, 23e24 insights from high-resolution resistivity imaging tool data into pore size heterogeneity, 24e26 Log measurements, 680 effective permeability factor and permeability using, 680 fractal dimension for lengths of capillaries using, 710 log measurements-based prediction, 680 pore space attributes, and, 678e679 rock attributes control permeability, and inversion from, 677e678 Logarithmic mean, 377e378, 648e649, 658e659 pore radius, 123 Logarithmic mean T2 of mesopores (T2LM), 817 Logarithmic mixing law for effective permittivity of mixture, 594e602 behavior of mixing law in zero-frequency limit, 601e602 charge distribution within medium, 598e601 Logging process, 180e181 tools, 293 Logs and pore space, 16e21 capillary tube bundle for modeling flow of incompressible fluid through porous rocks, 20 generic treatment of transport of incompressible fluid through medium, 20 modeling orientation of pores in space, 20e21 pore attributes used in petrophysical models of pore networks, 17e18 pore throat models, 18e20 Long wavelength limit Berryman in, 777e779 invoking, 786e787 invoking assumption of, 781e782 self-consistent estimates in, 782e786 Long wavelength limit, Berryman in, 777e779 Long-spaced density, 237 Long-spaced detector, 95e96 density, 242 Longitudinal relaxation

839

of magnetization, 200 time, 350 Longitudinal relaxivity, 201 Loop signal downhole ratio (LOOPDH), 376 Low apparent dip fracture segments, extraction of, 754e757 Low apparent dip segments, extraction of, 405e406 Low-angle events, process for, 163e164 Low-frequency forward model, 721 Low-frequency limit, 557e558, 576 of dispersion model, 502 Low-resolution inversion, 99e100 LWD collar quadrupole wave, 334 Lyophobic substance, 179e180

M Macropores, 14, 162, 737e738, 740e741 Macroscopic charge density, 560e561 Macroscopic complex dielectric permittivity, 554 Macroscopic counterpart of streamline attribute, 640e641 Macroscopic dielectric permittivity, 672 Macroscopic fluid velocity, 626 Macroscopic hydraulic conductance, 637 Macroscopic permeability, 637 factor, 633 field scalar, 628 Macroscopic permittivity, 532, 536e537, 555e556 Macroscopic pore space attributes related in porous medium, 640 Magnetic declination correction, 419e420 Magnetic resonance logs, computation of total porosity from, 86e87 Magnetization, 373 Magnetization loop (mLOOP), 376 Master calibration signal amplitude, 422 Matrix, 1, 449e452, 616, 779, 784 conductivity, 413, 760 equation, 194 extraction, 758e760 geodesic dilation, 758f models, 743 pore volume, 413 evaluation of, 411e413 matrix conductivity and matrix pore volume, 413 removal of isolated conductive anomalies, 411e412

840 Index Matrix (Continued ) removal of isolated resistive anomalies, 412 porosity of CBM reservoirs, 403e413 Maximum Entropy Principle (MEP), 185 application of, 187e188 Maxwell equations, 506e507 Maxwell Garnett equation, 533 Maxwell Garnett Mixing law, 483, 514e515, 518e519, 551e555, 589 aligned inclusions, 555e556 Maxwell Velocity Distribution, 261 MaxwelleBoltzmann distribution, 201 Maxwell’s Equations, 82, 594, 601 applied to binary mixtures, 581e585 MaxwelleWagner Relaxation Time, 513, 569e570, 599e600 Mean capture cross section, 391e392 Mean pore radius, 20 Mean pore throat radius, 20 Mean squared displacement of fast neutrons, 285e288 Measured length-scale, 603 Medium modeling approach, 499e500 Medium permeability, 634 Medium permittivity, 532 Medium theory, 512 modeling “m” using differential effective, 539 Membrane impedance, 721e722 MEP. See Maximum Entropy Principle (MEP) Mercury injection capillary pressure data, 624 process, 127e128 Mercury Injection-Capillary Pressure Curves, 795 Mesopores, 14, 162, 179, 737e738 gas, 385 volume per unit rock volume, 790e791 Meyer-Stowe theory, 660 Microcracks, 12e13 Microelectrodes, 747 Microgeometry, 499 Micropermeability, 632 Micropore(s), 14, 162, 179, 381, 737e738 dimension, 411 system, 793 volume per unit rock volume, 790e791 Microporosity, 385 evaluation of, 75 Microscopic hydraulic conductance, 638, 635e636

Microscopic models, 482e483 Microscopic permeability, 631e632 density, 630 Microscopic streamline attributes in porous medium, 640 Mixing Law, 502e503, 512, 514e515, 565, 600 of Bruggeman, 536 in zero-frequency limit, 601e602 Modal grainsize, 666 Model formalism, 473e474 Model prediction of permeability, 141e142 Model-based inversion of acoustic slowness to porosity using XueWhite scheme, 302e304 of compressional slowness to porosity for sandstones, 298e301 Model-based porosity partition, 743e746 Modeling exponent, 297 Modeling orientation of pores in space, 20e21 Modeling permeability of coal using log data, 175e177 Modeling process, 804 Modes of order, 424 Monopole excitation, 315e316 Monopole transmitter, 315 Monte Carlo simulation, 248 Mosaic pieces, 765e766 classification of, 763e764 MPS. See Multipoint statistics (MPS) MSFL tool, 510 Mud cake, effect of, 252e255 Mud solids, 285 Mud wave, 316 Mud-supported rocks, 1, 3 Multidimensional inversion of NMR echo data using maximum entropy principle, 183e188 application of MEP, 187e188 Multifrequency dielectric data approach for computing “m” from grain attributes obtained through, 503e551 applicability of model to clayey rocks, 527 contribution to complex permittivity, coming from unconnected pores, 528 dispersion model for rock conductivity, 524e527 estimation of Archie “m” based on differential effective medium theory, 530e551

Index inversion of Archie “m” from multifrequency dielectric data, 511e522 inversion of Archie “m” from singlefrequency dielectric data, 503e511 work flow for generation of dispersion model of dielectric permittivity ε, 522e524 approach for computing “m” using, 501e502 Multiple-scattered photons in formation density measurement, 230e257 computation of rb using multiwindow count rate inversion, 241e255 borehole effect, 255 equations of “spine”, 248 effect of mud cake, 252e255 photoelectric index, 246e247 photon energy spectrum vs. number of scatterings, 241e246 quantity, 248e252 ratio of count rate in lithology window to that in compton window, 248 dual detector data processing using single window count rates, 233e240 computing density and density correction without using graph, 237e240 final correction to computed bulk density to account for electron density of water not being half of bulk density of water, 240 graphical method of solving detector response equations, 236e237 general method of simultaneous inversion of multienergy window count rates, 256e257 Multipoint statistics (MPS), 405 algorithm, 405 technique, 747e748 Multivariate inversion, 73

N Nanopores, 178 Natural fractures, 164 Natural gamma ray spectrometry logs, 162 NaviereStokes Equation, 614, 616, 618e619, 711, 714, 718e719 Net Capture Spectrum (NCS), 401 Net Inelastic Spectrum (NIS), 400 Neumann type boundaries, 713e714

841

surface, 714 Neutron capture gamma spectrometry, 390e395. See also Inelastic gamma spectrometry capture gamma yield, 391e392 gamma spectra, 391t relative yield of capture gamma rays of element, 392e395 Neutron migration length, 273e285 effect of borehole and factoring-in of borehole effect, 277e278 count rate data are inverted to porosity, 278e285 apparent water-filled limestone porosity for standard conditions, 281e285 neutron detector count rates, 274e277 group 1 neutron flux and epithermal neutron detector counts, 276 group 2 neutron flux and thermal neutron detector counts, 276e277 neutron detectors, 274 Neutron(s), 395, 399 capture, 259 computation of effective porosity using Neutron capture gamma ray spectrometry, 32e38 cross section, 391e392 detectors, 274 count rates, 274e277 diffusion equations, 271 energy, 391e393, 397 flux distribution for point source in homogeneous isotropic medium for group 1, 263 for point source in homogeneous isotropic medium for group 2, 269e271 kinematics of neutron transport, 258e271 apparent hydrogen index, 259 slowing down length, diffusion length, and migration length, 260 two-group model of neutron transport, 260e266 logs, 803e804 measurement of slowing down length and diffusion length of neutrons for bulk porosity, 258e271 porosity, 42e43 computation of effective porosity using neutron porosity data, 32e40 of wet clay, 35 pulse, 395

842 Index Neutron(s) (Continued ) slowing down density of, 264 slowing down length, 266e271 tools, 284 transport kinematics of, 258e271 two-group model of, 260e266 Noise channel, 372 Nom dimensional space, 451 Noncleat fracture volume per unit rock volume, 168e169 Nonconventional reservoirs. See also Conventional reservoirs. See also CBM reservoirs characterization of fractured reservoirs, 209 computation of adsorbed methane volume per unit rock volume, 222 computation of kerogen volume, gas volume, and total porosity in shale gas reservoir, 216e222 computing main fracture/cleat orientation mentioned above, 213 extraction of pixels which form part of fracture, 209e210 finding set of vectors that best fits trace when placed end to end, 211e212 Kherroubi’s work flow for trace extraction for low-angle events, 209 obtaining final cleat/fracture traces of high confidence, 213e214 obtaining fracture traces from the extracted pixel sets, 210e211 shale reservoirs, 177e209 D-T2 plots, 197e200 differentiation of pore classes for, 179e183 method of diffusion editing, 191e195 method of Laplace inversion with regularization, 195e196 multidimensional inversion of NMR echo data using maximum entropy principle, 183e188 pore size encountered within, 178e179 porosity partition, 188e190 presentation of results of inversion, 188 total porosity, 222 Nondispersive STC, 322e324 Nonlinear conversion, 131 work flow, 132e133 Nonlinear Weighted Regularized LeastSquares Method, 66e67

Nonsedimentary rocks, pore space of, 13 Normal mode, 425e426 Normalization process, 164, 420e421 nth echo, amplitude of, 366e367 nth flipping pulse, 364 nth spin flip, 364e365 Nuclear magnetic resonance (NMR), 735 data, 52e54, 447e448 analysis, 677 average grain size and grain size distribution from, 656e660 in carbonate rocks, 736 characterization of pore space of coals using, 170e172 computation of CP(r) using, 124e125 estimation of bound fluid volume in Coates equation from, 668e670 partitioning of pore space of coals using NMR data by pore size, 170 echo amplitude inversion, 208e209, 219 data using maximum entropy principle, multidimensional inversion of, 183e188 sequence, 624 estimation of total porosity directly from echo data, 379e380 log data, 24e25, 679 estimation of rmax from, 703 permeability modeling approach relies on, 819e822 porosity using density and, 427e429 logging, 178e179 logs, 162, 817 measurement of total porosity using, 345e381 obtaining total porosity using formation density and NMR data, 380e381 pore classification using NMR T2 distribution, 738e746 limits of validity of T2 threshold-based porosity partition, 741e743 model-based porosity partition for case, 743e746 relation between pore sizes and NMR T2, 738e741 vug porosity, 746 pore size distribution from, 447e455 pore space attributes and relaxation of transverse magnetization, 367e377 porosity, 379 porosity calibration for, 422e423

Index and porosity of CBM reservoirs, 381e385 response equation, 64 spin relaxation mechanism, 123 theory, 345e367 perturbation, 351e353 perturbation in classical picture of perturbation, 353e367 spin distribution in magnetic field, 352f total porosity and bin porosities, 377e379 transverse relaxation, 177 time distribution, 130 transverse relaxation micropores and mesopores, 738e739 Null matrix, 450

O Oblate spheroids, 563 Oblate-spheroidal pores, 465e467 OBM, 739 “Ohmic sense”, 528 Oil, 16 case of oil base mud, 89 shale reservoirs bearing, 189e190 Oolitic rocks, 1 Optical dielectric constant, 484 Optimal frequency band, 336 Order of mode, 425e426 Organic granular rocks, 1 Organic matter, distribution of, 414e415 Organic pores, 3 relaxation of gas within, 201 in shale oil reservoirs, 9e11 in shale reservoir rocks, 8 Organic porosity, 222 Orthogonal factor rotation methods, 452 Oscillating quadrupole, 332 Oxides closure condition, 394

P Parallel plate capacitor, 594e595 Parsons Equation, 811 Partial saturation case, 792e793, 799e801 Particular Integral, 291 Peano curves, 686 Pendular water, 650 Penny cracks, 18 Percentile deviation, 656 Percolation theory, 293e294 Permeability, 3, 613, 619, 642, 821 BrayeSmith method of computing, 818e819

843

from Brooks-Corey equation, 699e701 of coal, 173e177 measurement of permeability of coal using log data, 173e175 modeling permeability of coal using log data, 175e177 permeability from Stoneley full waveform inversion, 173e175 Coates equation for, 652e655 density, 631 and effective permeability factor, relation between, 632 equations for permeability prediction from log measurements, 646e651 SDR equation for permeability, 646e650 Timur equation for permeability, 650e651 Timur’s permeability equation, 651e662 factor, 634 of medium, 632 and permeability using log measurements, 680 of streamline in some of attributes, 634e636 to fluid flow, 20 fractal dimension of connected pore space and influence on, 682e688 of gross rock, 818 integral representation of, 631 kb of base rock, 817 into macroscopic pore space attributes, 636e637 model prediction of, 141e142 modeling approach on NMR log data, 819e822 based on NMR, 177 hydraulic radius in, 638 from perspective of fractal model of pore space, 148e149 predictor as continuous log, 706 of rocks, 821 alternate simpler way of estimating Qcon, 816e817 hosting connected vugs/fractures, 811e818 permeability kb of base rock, 817 permeability of gross rock, 818 surface area, or, representative pore dimension or characteristic length scale driven approaches to, 621e628

844 Index Permeability (Continued ) “bundle of capillary tubes” model of pore space, 621e626 forward model of permeability in terms of pore space attributes, 626e627 stream tube model, 628 tensor field, 616 of unfractured composite hosting dual porosity, 793e797 base rock permeability, 793e795 permeability of composite rock, 795e797 Permeability-Field Tensor, 616 Permittivity, 531, 535, 569e570 Perturbation, 351e353 in classical picture, 353e367 acquisition of spin echoes, 358e360 amplitude of nth echol, 366e367 Bloch’s equations with diffusion for case of static magnetic field varying spatially, 361e364 dephasing of spins, 355e357 rephasing of spins, 357e358 effect of spatial variation of B0 field, 361 effect of spin flips on phase, 364e366 temporal decay of echo amplitude, 360e361 vectors representing spin quantum states, 356f field, 357e358 model, 821 purpose of applying, 353 Petrophysical inversion, 188e189 Petrophysical models, 44e45, 385e386 pore attributes used in petrophysical models of pore networks, 17e18 Phase alternating pairs (PAP), 111e112 Photoelectric absorption process, 241 Photoelectric index, 246e247 Photoelectric process, 231e232, 241 Photon energy spectrum, 241e246 Ping Sheng models, 548 Pixel weight, 748 Planar geometry, 426 Planar pore space, 805 Point plot, 658 Poissieulle-Hagen Law, 687 Poissiuelles equation, 661 Poisson Equation, 614 Poisson ratio, 301 of solid phase of rock, 304 Polarized components, 353e354

Polyhedral shapes, 667 Pore size(s), 118, 455, 682, 735e738 degree of connectivity between two pore size classes using NMR log data, 23e24 distribution, 447e455, 658 from NMR, 447e455 pore shapes from logs, 455e465 work flow for extracting NMR T2 signatures, 454f encountered within shale reservoirs, 178e179 heterogeneity factor, 153 heterogeneity index, 707 computation of electric formation factor log, 707 from logs, 706e707 micropores, mesopores, and macropores, 737e738 and NMR T2, relation between, 738e741 image logs for calibration of NMR T2 cutoff, 741f porosity partitioning scheme, 742f partitioning of pore space of coals using NMR data by, 170 pore-size-to-throat-size ratio, 15 Pore space, 1, 145, 622, 638e639, 662e663, 685, 694e696, 705, 710e711 attributes, 117e123, 620, 680 calculation of Cs, 678e679 conventional reservoirs, 227 data, 620e621 influence permeability, 620e621 and log measurements, 678e679 permeability in terms of, 626e627 of carbonate rocks, 4 characteristic length scale of, 121 of coals, 4e6 of coals, 162 characterization of, 162e169 extraction of fractures and cleats from borehole images, 163e164 using NMR data by pore size, partitioning of, 170 using NMR data characterization of, 170e172 process for high-angle events, 164 process for low-angle events, 163e164 depiction of pore space in model, 628e629 stream tubes, 628e629 electrical formation factor from perspective of fractal model of, 155

Index estimate “m” through fractal model of, 602e607 fractal attributes of, 144e155 fractal model of, 21e22, 118, 147e148 of granular rocks, 1e3 hydraulic constriction factor of connected, 640e641 and influence on permeability, 682e688 by inorganic pores, 6 of intergranular rocks, 117 logs and, 16e21 modeling, 620e621 models, 621 and role played by “hydraulic radius”, 638 of nonsedimentary rocks, 13 permeability from perspective of fractal model of, 148e149 of pores, 696e699 pores and pore throats model of, 682 precise definition of term “representative tortuosity of, 640 of shale reservoir rocks, 6e11 simple model, 616e617 tight sedimentary rocks, 8 as union of subspaces modeled capillary tubes, 694e696 use of log data, 22e26 Pore(s), 117 addition of, 475 aspect ratio, 456e465, 482e486 attributes, 17 used in petrophysical models of pore networks, 17e18 channel, 117, 136 class, 120 bound fluid and free fluid porosity, 183 differentiation of pore classes for shale reservoirs, 179e183 ranges of T2 for case of oil-bearing shale reservoir, 180t classification by size, 13e14 classification of carbonate rock porosity, 14t classification using NMR T2 distribution, 738e746 connectivity, 16, 499 dimension of pore radius fractal from logs, 707e709 estimation of Df from f, 709 estimation of Df from rmax, and k, 708e709

845

estimation of Df from rmax, and NMR, 707e708 estimation of Df from rmax, rmin, 707 estimation of Df from l, 707 dimensions, 677 facies, 447e465 case of dry frame, 467e468 case of dry frame with no compliant pores and stiff pores spherical in shape, 468e469 forward model of bulk modulus of water saturated rock, 469e486 Klein and Swift model for dispersion of dielectric permittivity of brines, 486e487 pore size distribution, 447e455 Wu’s tensor, 487e491 modeling orientation of pores in space, 20e21 morphology, 17e18 networks, 499 pore attributes used in petrophysical models of, 17e18 and pore throats, 14e16 pore-arrangement of macro level volume segment, 628 pore-network connectivity, 498e499 pore-structure, 620 radius fractal, 710 shapes, 118, 485, 498e499 factor, 122, 738e739 inversion of acoustic slowness data to pore aspect ratio in sandstones, 457e463 inversion of compressional wave slowness log data to pore aspect ratio, 465 inversion of shear wave slowness log data to pore aspect ratio, 463e464 from logs, 455e465 pore aspect ratio and acoustic logs, 456e465 surface volume, 677 systems, 21, 627, 649 throats, 15, 117e118, 145, 682, 696e699 models, 18e20 pores and, 14e16 size, 118, 682, 705, 729 use of log data, 22e26 volume, 117e118, 630, 802e804 per unit bulk volume, 497 and porosity, 688e689

846 Index Pore(s) (Continued ) of rock, 648 water, 518 Porosity, 497, 523e524, 710 association of different types of spots, 768e769 of base rock, 804 calibration for NMR, 422e423 computation of porosity using density and NMR log data, 427e429 count rate data, 277e278 data, 24e25 dependence of compressional wave propagation speed for rocks pores, 298e308 empirical relations connecting porosity and compressional slowness in rocks, 308e310 evaluation of, 372e376 input data, 372e376 heterogeneities and quantification of, 766e767 measurement using acoustic wave slowness data, 292e315 slowness porosity relations, 293e296 model-based inversion of acoustic slowness to porosity using XueWhite scheme, 302e304 model-based inversion of compressional slowness to porosity for sandstones, 298e301 NMR and porosity of CBM reservoirs, 381e385 partition, 188e190, 384, 765 using acoustic logs, 772e777 challenges of, 772e773 sensitivity of model results to shape of vug, 777 shale reservoirs bearing oil, 189e190 vug in model, 776 work flow of, 773e777 relation between tortuosity, formation factor and, 155e157 within rock for holding free gas, 385 of shale gas/shale oil reservoirs, 385e390 Porous media, capillary tubes models of pore space, succeed in predicting fluid transport properties of, 681 Porous medium, 498e499, 602, 628, 677 microscopic streamline attributes and macroscopic pore space attributes in, 640

Porous rocks capillary tube bundle for modeling flow of incompressible fluid through, 20 generic treatment of transport of incompressible fluid through medium such as, 20 Ports, 117 Postdepositional dolomitization process, 3 Postspeed correction, 416e417 Potential field, 583 Predictor, 143 Principal direction, 451 Probability density, 684, 697 of fractional number, 644 function for pore size, 602 of pore radius, 135, 146 Probability distribution, 262 Prolate spheroid, 563 Prompt gammas, 399e400 Psuedo Rayleigh wave, 316 Pulse height spectrum to energy spectrum fit for spectral decomposition, 402e403 Pulsed neutron generator system, 399 Pyrobitumen, 9

Q Quadrupole excitation, 332 salient features of, 332e333 Quadrupole transmitter, 315 Quality control of inversion, 377 Quasi-spherical grains, 677 Quasi-static characteristic length scale, 639 Quasi-Static Field assumption, 594 Quasi-static limit, 565, 581

R Random array, 593 RaoeCramer bound, 330, 332 RaoeCramer frequency, 330 RaymereHunteGardner equation to rocks containing clay, extending, 310e311 RaymereHunteGardner relation, 309e310 for shear wave speed, 311e313 Realistic permittivity dispersion model of water, 513e514 Recorded waveforms time evolution of wave field for slow formation case, 317e318 Recording point, 416e417 Regularization matrix, 374e375

Index method, 195 parameter, 374e375 Relaxation time, 370 constant, 370 Relaxivity, 201 Representative current conduction path, 497 Representative grain-shear modulus, evaluation of, 463, 469 Representative hydraulic tortuosity and cumulated surface area to cumulated volume of capillaries, 150e155 BrookeseCorey permeability and fractal model of pore space, 154e155 Representative pore dimension to permeability, 621e628 Representative tortuosity of pore space, 640 Reservoir rocks, 372 Resistivity along and normal to formation bedding, 79e86 data, 24e25, 747 imaging tools, 404 log, 799 Resonance condition, 353 Resonance factor, 352 Restricted diffusion, 199, 203e204 RGPZ equation, 661e662, 665e667 bundle of capillary tubes models of pore space, 681 derivation of, 670e677 effective permeability factor and permeability using log measurements, 680 estimation of rmax, rmin and g1, 701e703 estimation of rmax from NMR log data, 701e703 fractal dimension of connected pore space and influence on permeability, 682e688 Kozeny-Carman equation, 680e681 permeability from Brooks-Corey equation, 699e701 pore space attributes, and log measurements, 678e679 pores and pore throats model of pore space, 682 relation of results from fractal model, 694e699 relations, 688e694 pore volume and porosity, 688e689

847

ratio of cumulative surface area of connected pore space, to cumulative volume of connected pore space, 691 relationship between Df, Dt and, and between Df, F, and ɸ, 691e694 representative hydraulic tortuosity, 690e691 rock attributes, 677e678 Ribs, 236 plot, 236 RMS. See Root mean square (RMS) Rock(s), 1, 117, 626, 772 attributes, 677e678 bulk modulus, 471 dispersion model for rock conductivity, 524e527 elemental concentration in, 390e403 empirical relations connecting porosity and compressional slowness in, 308e310 estimation of elemental concentration in, 390e403 model, 483 multimodal pore systems and fractures approach to permeability modeling relies on NMR log data, 819e822 BrayeSmith method of computing permeability, 818e819 electrical conductivity of fractured rocks, 797e801 electrical conductivity of unfractured composite hosting dual porosity, 787e793 KustereToksoz estimates, 780e781 long wavelength limit, 781e782, 786e787 permeability of rocks hosting connected vugs/fractures, 811e818 permeability of unfractured composite hosting dual porosity, 793e797 pore classification using NMR T2 distribution, 738e746 pore classification using well bore images, 747e769 pore size nomenclature, 735e738 porosity partition using acoustic logs, 772e777 self-consistent estimates in long wavelength limit, 782e786 self-consistent theory of Berryman in long wavelength limit, 777e779

848 Index Rock(s) (Continued ) variable cementation exponent method of computing water saturation, 801e808 permeability, 793e795 total porosity of, 57 Rock’s ohmic conductivity model, 524 Root mean square (RMS), 372 Root mean squared displacement, 273 Rotating coordinate system, 355, 357e358 Roughness factor, 677 Rugosity of cleat-walls, 4e6

S Salinity value (S), 578 Sand computation of representative resistivity of, 54 equation for sand conductivity in uninvaded zone, 60e65 Sandstones, 309 inversion of acoustic slowness data to pore aspect ratio in, 457e463 model-based inversion of compressional slowness to porosity for, 298e301 SBKG, 401 SC. See Self-consistency (SC) Scaled button conductivity data, 421 Scales, 420 Scaling factor, 614e615, 658 Scattered field, 482e483, 534 Schlumberger-Doll Research Equation for permeability prediction (SDR equation for permeability), 646e650 Schrodinger equation, 350, 352 ScintillatorePhotomultiplier system, 402 Score class, 750e751 prototype, 750e752 Scree plot method, 450e451 Screw wave, 332e333 SEAR, 401 Sedimentary grains, 1 Seepage pores, 381e382 Segments of trace, 757 Seleznev’s model, 521 Self-consistency (SC), 536, 774e775 approximation, 530 Bruggeman complex effective permittivity, 787 effective complex dielectric permittivity, 672

estimates in long wavelength limit, 782e786 method of modeling, 671e672 permittivity, 537 theory of Berryman in long wavelength limit, 777e779 KustereToksoz model, 777e779 Semblance, 339 array receiver signal processing using, 322 Semiempirical relations, 296e297 Sensors, 415 Shale grains, 45 oil reservoirs, 177, 197 pores, 413e414 wet clay content of, 39e40 Shale gas porosity of shale gas/shale oil reservoirs, 385e390 estimation of effective porosity, 389e390 reservoirs, 177, 197 Shale reservoirs, 177e209 bearing oil, 189e190 characterizing pore space of shale reservoirs using image data, 413e415 borehole electric images and distribution of organic matter, 414e415 delineation of fractures within, 414 shale pores, 413e414 rocks bitumen pores in, 9 kerogen pores in, 9 organic pores in, 8e11 pore space of, 6e11 SEM photograph organic pores within organic-rich shale with honeycomb pattern, 9f SEM photographs examples of inorganic pores found within organic-rich shales, 10f Shallow resistivity, computation of effective porosity using, 32e40 Shape factor, 453 Sharpness index, 751e752 Shear slowness dispersion, 328e329 Shear wave(s), 319e320 porosity dependence of shear wave propagation speed for rocks pores interconnected, 293e296 critical porosity for different rock groups, 294t

Index relation like RaymereHunteGardner relation for shear wave speed, 311e313 slowness, 293 inversion of shear wave slowness log data to pore aspect ratio, 463e464 velocity and compressional wave velocity, 478e479 evaluating from, 478 Sheng’s model, 539 and Archie’s “a”, 548e549 and Archie’s “m”, 549e551 for sedimentary rocks, 540e547 Short-spaced density, 237 Short-spaced detector, 95e96 density, 242 Siderite, 216 Signal channel, 372 Signal-to-noise ratio, 100 Signatures-set of Gaussian functions, 452e453 Siliciclastic rocks, 462, 738e739 Silt index, 39 bulk density of silt index value of shale, 76e78 hydrogen index of silt index value of shale, 76e78 of shale, 78 Single scattering of gamma photons, 228e230 Single-frequency dielectric data approach for computing “m” using, 501 inversion of Archie “m” from, 503e511 Single-scattered gamma photons, 94e95 Singular value decomposition (SVD), 185 Skeletonizing by influence zones (SKIZ), 762e764 SLAT, 401 Slit-shaped pores, 119e120 Slow diffusion, 426 limit, 368 Slow killing regime. See Fast diffusion limit Slowing down density of neutrons, 264 Slowing down length, 260e266, 269, 273 of neutrons, 266 Slowness equation, 173e174 Slowness porosity relations, 293e296 elastic moduli and wave propagation speeds, 293 empirical and semiempirical relations, 296e297

849

empirical relations connecting porosity and compressional slowness in rocks, 308e310 RaymereHunteGardner relation, 309e310 extending RaymereHunteGardner equation to rocks containing clay, 310e311 generalization of Wyllie’s relation, 314e315 porosity dependence of compressional wave propagation speed for rocks pores, 298e308 case of spherical pores, 302 evaluation of exponents p and q, 304e308 fluid properties and grain properties input, 308 model-based inversion of acoustic slowness to porosity using XueWhite scheme, 302e304 model-based inversion of compressional slowness to porosity for sandstones, 298e301 modeling of p and q, 301 porosity dependence of shear wave propagation speed for rocks pores interconnected, 293e296 relation like RaymereHunteGardner relation for shear wave speed, 311e313 Wyllie’s relation, 313e314 Slowness Time Coherence (STC), 319e320, 322 SlownesseTime plane (S-T plane), 320 SOBM, 89 Software-based approaches, 284 Solid bitumen, 9 Solid-phase interaction, 174 Solids regime, 526e527 Solution set, 74 Solution vector, 376 Solvent extraction process, 177e178 Sonic data, multishot processing of, 100e105 Space-filling curves, 686 Spatial eigenfunctions, 424 Specific surface conductance, 674, 676 Spectral backpropagation, 320 Spectral decomposition, 431e434 Spectral density function, 573 Spectral expansion of f(s), 567, 586e587 Spectral function, 568

850 Index Spectral representation of effective permittivity, 569 Spectral stripping, 392, 398 Speed corrections, 416e417, 419 Spherical coordinates system, 288 Spherical geometry, 427 Spherical pores, 119e120, 462e463 case of, 302 Spherical volume (V1), 536e537, 551e552 Spin echoes, 358e360 Spin(s), 367 180 degrees tipping of, 357 acquisition of spin echoes, 358e360 dephasing of, 355e357, 356f echo, 357 effect flips on phase, 364e366 flipping pulses, 358e359 rephasing of, 357e358 rotation mechanism, 200 state, 355 Spine, 236 equations of, 248 plot, 236 Spinelattice relaxation, 361 Spinespin relaxation, 361 Spots, 765 porosity association of different types of, 768e769 Static normalization, 420 STAU, 401 STC, 322 Stochastic inversion of log data for laminated formation, 55e75 additional outputs computed, 68 basic response equations that lead to forward model, 55e65 equation for sand conductivity in uninvaded zone, 60e65 total porosity of rock, 57 total porosity of rock within flushed zone, 57e60 essential constraints, 66e68 forward model, 65e66 horizontal permeability and vertical permeability, 68e69 usage of high-resolution data, 69e75 solution when constraints are present, 72e73 solution when constraints are present, 74e75 Stokes Equation, 626 Stokes theorem, 629e630

Stoneley energy loss, 165e166 Stoneley full waveform inversion, permeability from, 173e175 Stoneley mobility, 706, 720e727 Stoneley permeability, 726e727 Stoneley reflection coefficient, 165e166 Stoneley transmission coefficient, 165e166 Stoneley wave, 316 Stoneley waveforms, 173 Stoneley wavepacket, 324 Stream tubes, 622, 627e628, 656 Streaming-potential coupling coefficient, 665 Streamline in attributes, 634e636 Stress-concentration tensor, 488 Strong killing regime. See Slow diffusion limit Stroud Milton De Model (SMD Model), 564 Structural dispersed shale volume fractions in formation, computation of, 50e52 Structural laminated shale volume fractions in formation, computation of, 50e52 Structural shale, 45e47 simple analysis ignoring, 49e50 gamma activity of laminated formation, 50 Surface area to permeability, 621e628 Surface area to volume ratio (Spv), 120e121 Surface conductivity of grain, 675 Surface fluids, 370 Surface limited regime. See Fast diffusion limit Surface relaxation, 201, 367e369 Surface relaxivity, 371, 624e625 Surface wave, 332e333 Swift model, 484 for dispersion of dielectric permittivity of brines, 486e487

T T2 relaxation spectra of coals, 382e384 Tensor, 626 computation of representative resistivity of sand and laminated shale volume using tensor resistivity data, 54 field, 616, 719e720 Texture classes, 768e769 porosity contribution of, 769 Thermal counts ratio, 278 Thermal maturity of kerogen and hydrocarbon density, 388 Thermal neutron capture, 270

Index energy spectrum, 261e262 group, 262 group 2 neutron flux and thermal neutron detector counts, 276e277 ThomaseStieber approach, 45e54 analysis considering structural shale, 50e52 computation of structural laminated and dispersed shale volume fractions in formation, 50e52 computation of representative resistivity of sand and laminated shale volume using tensor resistivity data, 54 evaluation of 4max, 47e48 using NMR data, 52e54 simple analysis ignoring structural shale, 49e50 Three-dimensional space (3D space), 629 Threshold capillary pressure, 653e654 Threshold pressure, 16 Threshold-based porosity partition, 741e743 Tight rocks, pore structure of, 18 Tight sedimentary rocks, pore space of, 8 Tikhanov Regularization Scheme, 72, 195 Time instant, 365 Time zero, 366, 422 Timur equation for permeability, 650e651 Timur-Coates Equation, 655 TimureCoates model, 818 TimureCoates permeability, 819 predictor from perspective of constriction, 143e144 Timur’s permeability equation, 651e662 Berg’s equation, 655e656 Coates equation for permeability, 652e655 computation of average grain size and grain size distribution from NMR data, 656e660 relation between representative or effective pore and grain dimension, 661e662 TOC. See Total organic content (TOC) Tool quadrupole mode, 339 Tool screw wave. See LWD collar quadrupole wave Tortuosity, 497e498, 603, 635, 662, 711 case of connected pore space having, 801 of cleat-walls, 4e6 relation between porosity, formation factor and, 155e157 Tortuous pathways, 496e497 Tortuous pore space, 805 Total carbon content, 386

851

Total flow rate, 662 Total organic content (TOC), 177, 218 Total porosity, 390, 473, 485e486 and bin porosities, 377e379 computation of total porosity from bulk density, 86e87 computation of total porosity from magnetic resonance logs, 86e87 equation for, 52 estimation of total porosity directly from echo data, 379e380 obtaining total porosity using formation density and NMR data, 380e381 of rock, 57 within flushed zone, 57e60 Total porosity and gas volume, 385 Total vug pore space of isolated vugs, 808 Traces, 210e211 components, 490 spatial orientation of, 757 Transition pores, 14, 162, 381 Transmission coefficient, 166 Transport model, 260 Transverse magnetization pore space attributes and relaxation of, 367e377 evaluation of porosity, 372e376 obtaining CPBj from mj(T2j), 376 quality control of inversion, 377 relaxation of transverse magnetization of pore saturated with grain-wetting fluid, 370e372 surface relaxation and bulk relaxation, 367e369 relaxation of transverse magnetization of pore saturated with grain-wetting fluid, 370e372 Transverse relaxation of magnetization, 200 time, 624 Transverse relaxivity, 201 Triple porosity model, 804 Two-exponential model, 745 Two-group model of neutron transport, 260e266 energy partitioning defining groups, 261e262 thermal neutron energy spectrum, 261e262 neutron flux distribution for point source in homogeneous isotropic medium for group 1, 263

852 Index Two-group model of neutron transport (Continued ) two-group model of neutron transport, inelastic collisions of neutrons are ignored, 1 in, 263 neutron flux distribution for point source in homogeneous isotropic medium for group 2, 269e271 removal cross section for group 1, 263e266 slowing down length of neutrons, 266 physical meaning of term “neutron slowing down length”, 266e271

U Unconventional reservoirs, pore space of certain aspects of relaxation of magnetization within fluid-filled porous media, 423e427 characterizing pore space of CBM reservoirs using image data, 403e413 characterizing pore space of shale reservoirs using image data, 413e415 computation of porosity using density and NMR log data, 427e429 decomposition of acquired gamma ray spectra using standard spectra, 431 estimation of elemental concentration in rocks, 390e403 generation of high-resolution electrical images of borehole wall, 415e421 histogram equalization, 439 kerogen property model and hydrocarbon property model, 429e430 measurement of total porosity using nuclear magnetic resonance, 345e381 NMR and porosity of CBM reservoirs, 381e385 permeability prediction for CBM reservoirs using image data, 434e439 porosity calibration for NMR, 422e423 porosity of shale gas/shale oil reservoirs, 385e390 spectral decomposition, 431e434 Unfractured composite hosting dual porosity electrical conductivity of, 787e793 permeability of, 793e797 Unimodal pore system depiction of pore space in model, 628e629 elemental stream tube, 629e630 flow through capillary, 617e621

integral representation of macroscopic, and concept of microscopic permeability density or local permeability density, 630 average permeability field, 631 concept of hydraulic radius, 638e641 derivation of RGPZ equation, 670e677 elemental stream tube permeability factor field, 633e637 estimation of bound fluid volume in Coates equation, 668e670 estimation of rmax from NMR log data, 703 estimation of g1 g1, 703e711, 728e730 estimation of dimension of pore radius fractal from logs, 707e709 estimation of pore size heterogeneity index l from logs, 706e707 estimation of s hydraulic tortuosity, 710e711 fractal dimension for lengths of capillaries using log measurements, 710 fractal dimension of pore radius fractal and Archie’s a, m parameters, 710 generation of valid permeability predictor, 706 significance of parameter Df, 709 fluid transport through perfectly rigid framework of grains, 711e720 local fluid velocity field and local fluid pressure field, 718e719 tensor field, 719e720 generalized Kozeny-Carman equation, 641e662 hydraulic constriction factor, 667e668 integral representation of average permeability field and relation between permeability and effective permeability factor, 632 integral representation of permeability, 631 Kozeny-Carman equation, 662e663 to Van Baaren’s equation, 663e664 QDP expressed as integral involving internal fluid pressure and velocity fields, 630e631 RGPZ equation, 665e667 simplest possible pore space model and role played by “hydraulic radius”, 638 Stoneley wave slowness and Stoneley mobility, 720e727 response of local pressure field, local fluid velocity field, and average fluid

Index velocity field to changes in driving pressure, 614e616 simple model of pore space presented, 616e617 surface area, or, representative pore dimension or characteristic length scale driven approaches to permeability, 621e628 Uninvaded zone, equation for sand conductivity in, 60e65 Unit dry weight, 389

V Validation, 502 Van Baaren’s equation, 661e664 Variable cementation exponent method of computing water saturation, 801e808 case of connected pore space having tortuosity unity, 801 computation of water saturation, 807e808 level by level evaluation of Archie cementation exponent of formation, 804 pore volumes, 802e804 Varimax rotation method, 452 Velocity fields, 630e631 of fluid, 638 Vertical permeability, 68e69 Vertical resolution of conventional density logs, alpha processing for improving, 109 Vertical resolution of conventional neutron porosity logs, alpha processing for improving, 108e109 VoigteReuss mean, 304 VoigteReusseHill average model (VRH average model), 469 Volume fraction of wet clay, 304 Volume ratio, 591 surface area to, 120e121 Volumes closure equation, 386 Vug, sensitivity of model results to shape of, 777 Vug model, 773, 776, 800, 821 Vug pore space, 479, 792e793 Vug porosity, 746 Vug/fracture pore space, nonplanar geometry of, 806

853

W Wall impedance. See Acoustic fluid mobility impedance Water base mud, case of, 88 Water salinity, 523 Water saturated rock, 353 forward model of bulk modulus of, 469e486 Water saturation computation of, 807e808 of fracture system, 799 of macroporosity, 792e793 variable cementation exponent method of computing, 801e808 Water-bearing rocks, 314 Water-filled limestone porosity, 284 Water-wet pores, 739 Water-wet rocks, 509e510 hydrocarbons, 523e524 Watersheds, 762e763 transform, 762e763 Waves, 316 propagation speeds, 293 Weak Eshelby conjecture, 560 Weak killing regime. See Fast diffusion limit Weighted Least Squared Error Minimization (WLS), 433 Well bore images closing data gap, 747e753 dip picking, 747 efficient methodology for extracting heterogeneities, 761e769 extraction of fracture segments, 753e758 matrix extraction, 758e760 pore classification using, 747e769 preconditioning of data, 747 problem of computing pore volume contribution of heterogeneities, 760e761 Wet clay bulk density of wet clay and silt, 76e78 content of shale, 39e40 hydrogen index of wet clay and silt, 76e78 Wet clay porosity (WCLP), 219 Wetting phase, 657 WFCF model, 451e452 WhittakereSlattery Theorem, 718 WinlandePitman Permeability Predictor, 795 Wireline acquisition dipole excitation, 332 Wu’s Tensor, 301, 487e491 Wyllie’s relation, 313e314 generalization of, 314e315

854 Index

X Xu’s White scheme, 297 model-based inversion of acoustic slowness to porosity using, 302e304

Z Zero ohmic conductivity, 547

Zero-error forward model, 67 Zero-frequency case, 595 limit mixing law in, 601e602 Zero-mean ordered set of values of random variable, 449