Formation testing. Volume 3, Supercharge, pressure testing, and contamination models 9781119283775, 1119283779

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Formation Testing Volume 3

Scrivener Publishing 100 Cummings Center, Suite 541J Beverly, MA 01915-6106 Publishers at Scrivener Martin Scrivener ([email protected]) Phillip Carmical ([email protected])

Formation Testing Volume 3

Supercharge, Pressure Testing and Contamination Models Wilson C. Chin

This edition first published 2019 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA © 2019 Scrivener Publishing LLC For more information about Scrivener publications please visit www.scrivenerpublishing.com. All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, except as permitted by law Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.

Wiley Global Headquarters 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives, written sales materials, or promotional statements for this work The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make This work is sold with the understanding that the publisher is not engaged in rendering professional services The advice and strategies contained herein may not be suitable for your situation You should consult with a specialist where appropriate Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Library of Congress Cataloging-in-Publication Data ISBN 978-1-119-28377-5

Cover images courtesy of the author Cover design by: Kris Hackerott Set in size of 11pt and Minion Pro by Exeter Premedia Services Private Ltd., Chennai, India

Printed in the USA 10 9 8 7 6 5 4 3 2 1

Contents Preface

ix

Acknowledgements

xi

1 Formation Testing – Strategies, Capabilities and Solutions 1.1 Development Perspectives 1.2 Basic Forward and Inverse Models 1.3 Supercharge Forward and Inverse Models 1.4 Multiple Drawdown and Buildup Inverse Models 1.5 Multiphase Cleaning and Supercharge Model 1.6 System Integration and Closing Remarks 1.7 References

1 1 4 14 20 24 29 30

2 Supercharging – Forward Models and Inverse Solutions 2.1 Supercharge and Math Model Development 2.2 Supercharge Pressure and Ultimate Decay 2.3 United States Patent 7,243,537 B2 2.4 Forward and Inverse Models with Supercharging – Drawdown-Only and Drawdown-Buildup Applications and Illustrative Examples 2.4.1 General Ideas in Formation Testing Formulations 2.4.2 Mathematical Formulation 2.5 Drawdown Only Applications 2.5.1 Example DD-1, High Overbalance 2.5.2 Example DD-2, High Overbalance 2.5.3 Example DD-3, High Overbalance 2.5.4 Example DD-4, Qualitative Pressure Trends 2.5.5 Example DD-5, Qualitative Pressure Trends

31 31 34 37

v

69 69 73 78 78 84 88 92 95

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Contents 2.5.6

Example DD-6, “Drawdown-Only” Data with Multiple Inverse Scenarios for 1 md/cp Application 2.5.7 Example DD-7, “Drawdown-Only” Data with Multiple Inverse Scenarios for 0.1 md/cp Application 2.6 Drawdown – Buildup Applications 2.6.1 Example DDBU-1, Drawdown-Buildup, High Overbalance 2.6.2 Example DDBU-2, Drawdown-Buildup, High Overbalance 2.6.3 Example DDBU-3, Drawdown-Buildup, High Overbalance 2.6.4 Example DDBU-4, Drawdown-buildup, 1 md/cp Calculations 2.6.5 Example DDBU-5, Drawdown-buildup, 0.1 md/cp Calculations 2.7 Supercharged Anisotropic Flow Simulation Model 2.8 References 3 Pressure Transient Analysis – Multirate Drawdown and Buildup 3.1 Multirate Drawdown and Buildup Applications 3.1.1 Monitoring, Testing, Treatment and Retest 3.1.2 Hydrate Characterization and Production 3.2 Detailed Validations with Exact Solutions 3.2.1 Validation of PTA-App-01 Inverse Model 3.2.2 Validation of PTA-App-02 Inverse Model 3.2.3 Validation of PTA-App-03 Inverse Model 3.2.4 Validation of PTA-App-04 Inverse Model 3.2.5 Validation of PTA-App-05 Inverse Model 3.2.6 Validation of PTA-App-06 Inverse Model 3.2.7 Validation of PTA-App-07 Inverse Model 3.2.8 Validation of PTA-App-08 Inverse Model 3.2.9 Validation of PTA-App-09 Inverse Model 3.2.10 Validation of PTA-App-10 Inverse Model 3.2.11 Validation of PTA-App-11 Inverse Model 3.3 References

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102 107 107 111 114 118 122 127 131

132 133 134 138 142 144 148 156 163 172 177 182 187 192 197 202 219

Contents 4 Practical Applications and Examples 4.1 Review Objectives 4.2 Practical Applications and Examples 4.2.1 Isotropic Medium Pressure Testing 4.2.1.1 Steady-state method 4.2.1.2 Drawdown-buildup method 4.2.1.3 Drawdown only method 4.2.2 Anisotropic Media Pressure Testing (Using FT-01) 4.2.3 Supercharge Effects in Drawdown-Buildup 4.2.4 Supercharge Mechanics in Detail – Reservoir Fluid More Viscous Than Mud 4.2.5 Supercharge Mechanics in Detail – Reservoir Fluid Less Viscous Than Mud 4.2.6 Supercharge Mechanics in Detail – Reservoir Fluid Viscosity Equals Mud Viscosity 4.2.7 Perfectly Balanced Well, Mechanics in Detail – Reservoir Fluid Viscosity Equals Mud Viscosity 4.2.8 Underbalance Mechanics in Detail – Reservoir Fluid Viscosity Equals Mud Viscosity 4.2.9 Comparing Overbalance vs Underbalance Pressures for Same Reservoir and Tool Pumping Conditions 4.2.10 Consequences of Non-Performing Pump Piston 4.2.11 Batch Processing Using FT-00 4.2.12 Depth of Investigation Using FT-00 DOI Function 4.2.13 History Matching Using FT-06 Batch Mode 4.2.13.1 Operating FT-06 batch simulator 4.2.13.2 Source code documentation (color coded) 4.2.13.3 A final example – concise operational summary 4.2.14 Gas Pumping 4.2.14.1 Several field notes 4.2.14.2 Review of steady-state direct and inverse methods 4.2.14.3 Transient gas calculations 4.3 References

vii 221 221 222 222 222 226 230 232 240 250 261 264 266 268

270 284 291 296 306 308 313 328 334 334 335 338 342

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Contents

5 Best Practices and Closing Remarks 5.1 Best Practices 5.2 Recommended Reading

343 343 351

Cumulative References

352

Index

363

About the Author

369

Preface This volume, possibly the third and final of three, summarizes new and important methods and approaches for wireline and LWD/MWD formation tester pressure transient interpretation. Two prior monographs, Formation Testing: Pressure Transient and Contamination Analysis (2014) and Formation Testing: Low Mobility Pressure Transient Analysis (2015), had developed essential models and algorithms for exact forward analysis; rapid, real-time, inverse mobility and pore pressure prediction methods in low mobility environments with non-negligible flowline volume effects; steady-state and transient forward and inverse techniques; miscible, multiphase flow contamination modeling for sample quality control; fast “phasedelay” methods for realtime kh and kv prediction in tight zone applications, and other useful mathematical methods. The present book tackles two additional challenges. The first deals with high overbalance pressures, which “supercharge” the pressure fields surrounding tool probes, and addresses specific questions. How do we extrapolate mobility, compressibility and pore pressure accurately in the presence of possibly overwhelming borehole effects which have not yet dissipated? How can we use this model to obtain error bounds on the usual downhole predictions? And how does supercharging evolve dynamically, as a function of formation, borehole and mud filtration properties? Second, we consider “multiple drawdown and buildup” sequences, actually any combination of piecewise constant, positive, negative or zero flowrates, calculate their exact pressure response, and importantly, show how any three time-pressure data points taken during the final flowrate cycle can be accurately inverted to produce mobility, compressibility and pore pressure – and that’s rapidly, in the presence of flowline volume distortions and low mobility environments. Detailed theory, validations and engineering applications are given for both new models. But just as important, the present book summarizes all of our collective results obtained over the years, and highlights their application to day-to-day activities. This perspective is invaluable especially to a ix

x

Preface

researcher reflecting on his path to discovery in dealing with what appeared to be an endless array of random challenges. Out of this, a number of “best practices” were developed which focus on uses of our models, which apply to all manufacturers’ formation testing tools. It is difficult to believe that almost two decades have elapsed since the author’s first introduction to formation testing and sampling – it’s been immensely challenging and enjoyable, particularly in seeing these methods used in oil exploration – and, I suppose, an intellectual labor of love, that’s finally coasting to a satisfying ending. I think. Wilson C. Chin, Ph.D., M.I.T. Houston, Texas Email: [email protected]

Acknowledgements The author is indebted to his friends and colleagues at Halliburton Energy Services and China Oilfield Services who strongly shaped and influenced his initial approaches to formation testing. Over the past twenty years, the freedom to pursue new ideas and a continual exposure to state-of-theart technology have led to innovative interpretation and oilfield planning methods that have positively impacted our industry. This third book in our trilogy on formation testing summarizes modeling approaches that address operational concerns raised by petroleum engineers, for example, interpretation in supercharged environments, modeling contamination under overbalanced pressures, gas pumping, and applications to reservoir treatment and enhanced hydrate production. The author gratefully acknowledges the United States Department of Energy for its support through its Small Business Innovation Research (SBIR) program for Contracts DE-FG03-99ER82895, DE-FG0204ER84082, DE-FG02-04ER84083 and DE-FG02-06ER84621, and through its Research Partnership to Secure Energy for America (RPSEA) Ultra-Deepwater Program, for assistance under Contract 08121-2502-01. Such programs are essential in supporting new, high-risk ideas that may make a difference – and, certainly, to entrepreneurs dedicated to science and wanting of the chance to make the world just a bit better. Finally, thanks to Mark Proett, formation testing guru, Xiaoying Zhuang, friend and facilitator, and last but not least, Phil Carmical, Acquisitions Editor and Publisher, for their interest and continuing support over the years. Without their unwavering faith and confidence, this author’s frustrations and disappointments may have remained just that.

xi

1 Formation Testing – Strategies, Capabilities and Solutions 1.1 Development Perspectives During the mid-1990s, the present author, working with his colleague Mark Proett at Halliburton Energy Services, in Houston, focused his efforts on rapid and efficient formation tester pressure transient interpretation methods. Since the 1950s, flow rate and pressure drop data had been routinely used during sampling operations to predict “effective” or “spherical permeability” (or, more precisely, mobility) – this single-probe measurement provided reservoir characterization information complementing the retrieval and analysis of actual fluid samples. However, the interpretation made use of a steady-state formula requiring complete pressure equilibrium – that is, steady flows that, in the environment of the 1990s and beyond, possibly required hours of expensive wait times at the rigsite and increased the risk of lost tools. We were tasked with the development of more rapid methods that would “roll out” with the introduction of our new formation tester. But disruptive technology is never easy. The obvious and economic use of early time data would be contaminated by pressure distortive effects associated with flowline storage volume, a problem compounded by tight zones, heavy oils, or both. An empirical method in use at the time seemed to work well; applications to synthetic and limited field data were successful, although why, unfortunately, was anyone’s guess. But rigorous mathematics would come to the rescue. The complete initial1

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boundary value problem was formulated and laboriously solved exactly in its entirety. Closed form, analytical solutions for the “direct” or “forward problem,” in which transient pressure histories were sought given fluid, formation, tool and flow rate properties, were obtained in terms of complex complementary error functions. A special “exponential” limit of this solution was studied, which explained why our empirical method worked, and importantly, how it could be improved. This limit formed the basis for a new “inverse” model, in which permeability (mobility), pore pressure and fluid compressibility could be predicted from a limited set of pressure measurement data. Our research resulted in a number of publications and contributions, all of which were later summarized in “Advanced Permeability and Anisotropy Measurements While Testing and Sampling in Real-Time Using a Dual Probe Formation Tester,” SPE Paper No. 64650, Seventh International Oil & Gas Conference and Exhibition, Beijing, China, November 2000 (for earlier related work, refer to “Cumulative References” and “About the Author” in this book). In summary, our work led to three significant contributions – A simpler “exponential” formula was developed which allowed rapid predictions of effective spherical permeability (or mobility) in tight zones, using early time data in the presence of strong flowline volume effects. Additional by-products of this approach included pore pressure and fluid compressibility. This method forms the basis of the company’s real-time GeoTapTM logging-while drilling service operable for single and also dual probe tools. A method to predict isotropic permeability (or mobility) using phase delay measurements was also developed. Basically, the travel time for sinusoidal waves created by an oscillating pump piston source and measured at a nearby observation probe would provide the desired predictions. However, while a patent award did result from this work, the method was not economically viable since two probes were required – unlike the drawdown-buildup approach above using the exponential formula and just a single source (or pumping) probe. For dual probe tools at zero dip angle (that is, operating in vertical wells), formulas were also given for kh and kv prediction using steady pressure drops obtained at source and observation probes – these measurements, of course, may require lengthy wait times.

Formation Testing 3

In 2004, the United States Department of Energy (DOE), through its Small Business Innovation Research (SBIR) program, awarded two hundred awards nationally in areas such as plasma physics, nuclear energy, refining, waste remediation, building and ventilation, and so on. Four grants were made for fossil fuel and well logging research – two of these awards, both won by this author through his consulting firm Stratamagnetic Software, LLC, founded in 1999, related to formation tester interpretation and analysis. These grants, together with three additional DOE awards, carried stipends significant to any start-up organization and indirectly supported activities in Measurement-WhileDrilling, reservoir engineering, drilling and cementing rheology and electromagnetic logging. The freedom that the awards provided led to new methodologies which would dominate the author’s work for more than a decade. Many “loose ends” have been resolved, and over the past several years, our work has been disseminated through John Wiley & Sons; in formation testing, in three volumes, this representing our third.

Figure 1.1. Chin et al. (2014) and Chin et al. (2015). In this last volume on formation testing, we summarize new industry capabilities applicable to all manufacturers’ tools in Chapters 1. Chapter 2 highlights “supercharge” effects, where high overbalance pressures distort formation tester measurements – a new interpretation model, suitable for desktop or downhole use, is developed for early time mobility, pore pressure and compressibility prediction in the presence of flowline storage. Chapter 3 develops new inverse methods for multiple drawdown and buildup applications for reservoir characterization, formation treatment and hydrate production. Finally, Chapter 4, provides a broad range of examples for practical engineering application.

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1.2 Basic Forward and Inverse Models In this section, we discuss methods for forward and inverse analysis that employ “simple” logging techniques such as steady-state drawdown, unsteady drawdown, and drawdown-buildup. The “forward” or “direct” problem solves for the transient pressure response when fluid, formation, tool and flowrate parameters are given. On the other hand, the “inverse” or “indirect” formulation attempts to provide permeability (or, mobility), fluid compressibility and pore pressure when a limited number of time and pressure data points are given. With the exception of supercharge and multiple drawdown and buildup methods, the models discussed here are developed in detail in Chin et al. (2014) and Chin et al. (2015).

Figure 1.2.1a. FT-00 (Main Interactive) exact forward liquid simulator.

Formation Testing 5

Figure 1.2.1b. FT-00 (Batch Mode) exact forward liquid simulator. FT-00 model. Our (initial) flagship forward simulator, simply named “FT-00,” is shown in Figures 1.2.1a,b,c. The underlying math model is the exact, analytical, closed form, analytical solution solving the complete initial-boundary value problem formulation for liquids originally published in “Advanced Permeability and Anisotropy Measurements While Testing and Sampling in Real-Time Using a Dual Probe Formation Tester,” SPE Paper No. 64650, Seventh International Oil & Gas Conference and Exhibition, Beijing, China, November 2000.

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Although the solution is exact, the solution could not be used for real-time or even most desktop applications for two reasons. First, the “complex complementary error function” supplied in most scientific mathematical libraries was far too complicated for downhole use with microprocessors having limited capabilities. And second, transient pressure responses at observation probes could not be calculated for the entire range of logging applications because of very small and very large arguments. For these reasons, the “exponential model” was, and probably is currently, used, although the authors at the time were satisfied that its scientific basis had been clearly established. In the early 2000s, however, the author and other collaborators reworked the complex variables methods underlying the error function evaluation in order to render FT-00 fully functioning (details are offered in Chin et al. (2014)). As a result, the Windows program will perform dozens or more simulations per minute (in batch mode) depending on the microprocessor used, and importantly, will provide transient pressure responses at both source probe and distant observation probes. Figure 1.2.1a displays all the required inputs for the “main, interactive” mode. Standard outputs include line graphs for assumed volume flow rate versus time, source and observation probe pressure responses versus time, and finally, normalized plots showing both pressure and flow rate responses. In addition, detailed tabulations are offered to support other user applications like report generation and spreadsheet plotting. While the “main, interactive” mode is useful insofar as establishing physical intuition for the flow variables at hand, it may be less convenient in history matching applications where, for example, numerous kh, kv, or other values need to be varied systematically to match calculated pressure responses to probe measurements. As shown in Figure 1.2.1b, our FT-00 software also supports an exact “batch mode” calculator. Here, at the bottom left, a convenient setup box can be “called” to define parameter ranges and increments for physical variables of interest. Line plots and tables can be displayed during batch calculations, or more conveniently, suppressed to the very end, at which time a single large tabulation is offered to the user. In other applications, “depth of investigation” (DOI) is important in job planning and interpretation error assessment. Consider, for example, a low mobility situation – will the assumed pump rate, or the maximum mechanical rate the system is capable of, result in a measurable signal at the observation probe? Will pressure diffusion (smearing) be excessive?

Formation Testing 7

Figure 1.2.1c. FT-00 (DOI) exact forward liquid simulator. Rather than defining this quantity abstractly, as is commonplace in resistivity and electromagnetic logging, we use our ability to calculate probe responses at any distance from the source to advantage. Clicking the “DOI” button leads to the simplified menu in Figure 1.2.1c, which automatically supplies exact pressure results and plots at predetermined separation distances between zero and the “maximum probe separation” distance requested. Example calculations are offered in Chapter 4.

8 Formation Testing Volume 3 FT-01 model. It is known that numerical methods, e.g., Ansys, Comsol, and others, whether they are finite difference or finite element based, are influenced by truncation and round-off errors. In the historical context, these act as “artificial viscosities” in fluids problems. In formation testing applications hosted by Darcy’s equations, the calculated pressure response for a given inputted mobility may correspond to a different mobility whose value or even qualitative effect may be difficult to quantify. This is not acceptable for forward calculations. But the consequences are worse for the development in inverse methods because they cannot be properly validated. We noted that SPE Paper 64650 provided equations for kh and kv determination for dual probe tools, although using steady-state pressure drops in vertical wells. At the time, only Ansys synthetic data was available and applications were deferred. The book Chin et al. (2014) provides the exact, analytical, closed form solution for kh and kv determination assuming dual probe tools where steady-state, liquid assumptions are in place. However, any dip angle is permitted. The screen for “FT-01” is shown in Figure 1.2.2. The method is validated by using synthetic pressure data generated by the fully transient FT-00 code (which does not suffer from truncation or roundoff error), transferred to the first two boxes in Figure 1.2.2, and showing that predicted anisotropic permeabilities are consistent with those used in FT-00 to generate the pressure data. Example calculations appear in Chapter 4.

Figure 1.2.2. FT-01, exact inverse liquid simulator.

Formation Testing 9 FT-02 model. In our description of FT-01, our exact inverse model for liquid flows using steady pressure data, we emphasized that it was validated by running forward liquid transient simulator FT-00 until steady-state conditions were achieved in order to obtain steady pressure inputs for inverse calculations. FT-02 represents our exact inverse method for nonlinear gas flows based on exact, closed form, analytical solutions (details are offered in Chin et al. (2014)). Whereas FT-00 for liquids was constructed from simple exact solutions using linear superposition methods, an analogous forward simulator for nonlinear gas flows cannot be developed because superposition methods do not apply. Thus, a different validating forward simulator for gases was developed, in this case an exact one for steady-state nonlinear gas flows. This complementary pair of steady forward and inverse gas simulators is shown in Figure 1.2.3. The method allows simultaneous for kh and kv determination for dual probe tools using steady-state pressure drop data. It applies to all dip angles plus a range of thermodynamic effects, for instance, isothermal and adiabatic processes, and so on. We emphasize that inverse solutions need not be unique. In other words, more than a single horizontal and vertical permeability pair may be found for a given set of dual probe pressure drops. Additional logging information (outside the realm of formation tester analysis) is required to render the solution unique. Example calculations are offered in Chapter 4.

Figure 1.2.3. FT-02, exact, steady forward and inverse gas simulators.

10 Formation Testing Volume 3 FT-06 and FT-07 models. Our exact FT-00 forward simulator for liquid motions is based on closed form, analytical solutions, and its versatile flowrate capabilities are founded on general linear superposition principles. For mathematical expediency, these required “piecewise constant” rate specifications, say “ +1 cc/s for two sec, + 5 cc/s for six sec, – 10 cc/s for three sec,” and so on. In many practical applications, pumps cannot achieve such constant rates because of excessive formation resistance or mechanical issues. In fact, timewise volume flowrate functions may take the form of triangles, trapezoids or non-ideal shapes. Thus, the need for a numerically based simulator capable of handling more general volume flowrate functions is apparent.

Figure 1.2.4a. FT-06, numerical liquid and gas forward simulator.

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Figure 1.2.4b. FT-06, general flowrate functions, forward simulator. A numerical option is also required for general transient nonlinear gas flows, for which closed form analytical solutions are not available, and for which, in any event, linear superposition methods are inapplicable. Our FT-06 numerical finite difference simulator serves two combined functions. First, it solves liquid flow problems subject to arbitrarily defined flowrates, as is apparent from the flowrate schedule in Figure 1.2.4a. In fact, as shown, a numerical file read in by the user is also possible. Second, the computational engine is extended to nonlinear gas flows for a wide range of thermodynamic situations, e.g., isothermal, adiabatic or other processes of interest. Furthermore, anisotropy may be specified via “kh, kv” or “effective spherical permeability and kv/kh.” The same computational outputs as FT-00 are offered, that is, line plots for source and observation probe pressures, flowrate, and pressure-rate superposed plots versus time, plus detailed numerical tabulations. Example flowrate functions are displayed in Figure 1.2.4b. FT-06 assumes that flowline storage volume is constant for the duration of the simulation. In other applications, those focusing on hardware development efforts, the need for time-varying flowline volume simulation arises. It is known that when formations are low in mobility and flowline volumes are not small, pressure responses can be distorted or smeared. The need to dynamically “tune” flowline volume allows the field engineer to adjust the resolution in his pressure curve and

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permit more accurate interpretation using inverse prediction methods such as those offered in this book. Our FT-07 numerical simulator provides a general means to define time-varying flowline volumes, as suggested in the bottom left menu shown in Figure 1.2.4c. Examples using FT-06 are offered in Chapter 4, while applications using FT-07 are provided in Chin et al. (2015).

Figure 1.2.4c. FT-07, a FT-06 extension supporting general time-varying flowline volume.

Formation Testing 13 FT–PTA–DDBU model. Previously, we introduced two inverse models, namely FT-01 for liquids and FT-02 for gases, both requiring steady pressure drops from dual probe data. These models were based on exact, close form, analytical solutions of the respective steady Darcy formulations, and while impractical, do offer horizontal and vertical mobility predictions. In contrast, the FT-PTA-DDBU inverse model, for drawdown-buildup applications using buildup data, supports early time data usage for low mobility applications with non-negligible flowline storage effects. This model rapidly (within seconds) predicts the “effective” or “spherical mobility” kh2/3kv1/3/ where is the viscosity.

Figure 1.2.5. FT-PTA-DDBU, early time, low mobility, flowline volume non-negligible – for “drawdown only,” see Figure 1.4.4). As indicated in Figure 1.2.5, only three time-pressure data points are required, together with the time TDD1 at which drawdown ceases. Shown at the bottom right are pore pressure and mobility predictions. A “drawdown only model, using drawdown data” is also available. While both are still offered, they have been replaced by the more general inverse capabilities of the “multiple drawdown and buildup” system described later, which in addition to pore pressure and mobility, offers fluid compressibility. Note that our “multiple drawdown and buildup” options do not model supercharge due to overbalance effects, but a version of the code in Figure 1.2.5 with supercharge is available.

14 Formation Testing Volume 3 Classic inversion model. Finally, we cite for historical purposes the original single-probe model offering spherical mobility when steady pressure drops are available assuming a continuous constant flowrate fluid withdrawal. The method is based on an exact analytical solution, but the main drawback with this approach is the possibility of long waits in low mobility environments, required so that steady conditions are achievable and flowline storage effects dissipate.

Figure 1.2.6. Classic inverse model. 1.3 Supercharge Forward and Inverse Models In our prior discussion of inverse model FT-PTA-DDBU, we indicated that pore pressure, mobility and fluid compressibility were predicted from early time, single-probe, pressure transient data with nonnegligible flowline storage effects. This zero-supercharge model, for drawdown-buildup applications utilizing buildup data, is again shown in the top of Figure 1.3.1. Mathematical details are offered in the formation testing book of Chin et al. (2014), explaining both exponential function as well as “rational polynomial” implementations (the latter, used in our work, is more robust, since exponentials are prone to compiler or microprocessor quality issues). This method is extended in Chapter 2 of this book to include supercharge effects due to overbalance in the well. The screen at the bottom of Figure 1.3.1 contains one additional input box “Pover (psi)” for the over-pressure due to overbalance. Again, pore pressure, mobility and compressibility are predicted.

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Figure 1.3.1. Both software modules apply to drawdown-buildup applications using buildup data. Pore pressure, mobility and compressibility predictions, zero supercharge (top), strong supercharge or overbalance pressure (bottom).

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In addition to the supercharge inverse model shown at the bottom of Figure 1.3.1, which applies to drawdown-buildup applications using buildup data (as shown in the yellow screen), a complementary supercharge inverse model for drawdown applications using drawdown data is also available and is shown in Figure 1.3.2 with essentially identical inputs as in Figure 1.3.1, except that TDD1 (for the time when drawdown stops) is not requested. Note that all the “black DOS screen” software items shown in the figures below represent completed and fully validated algorithms, except that, as of this writing, more attractive Windows user interfaces have not been written – all of the results generated in Chapter 2 used the “black screen” interfaces below as temporary “front ends.” In addition to Model SC-DD-INVERSE-2 for inverse calculations, a complementary forward solver, which calculates transient drawdown pressure responses at the source probe when fluid and formation properties, tool characteristics, volume flowrates, pore pressure and overbalance pressure are given, is available and shown in Figure 1.3.3. In fact, the forward or direct solver in Figure 1.3.3 was run to create synthetic transient (supercharged) pressure data, which was inputted into the inverse model Figure 1.3.2. Here, inverse calculations recovered the known mobility, pore pressure and compressibility.

Figure 1.3.2. Input screen for Model SC-DD-INVERSE-2.

Figure 1.3.3. Input screen for Model SC-DD-FORWARD-3B.

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For “drawdown only” applications, a special forward simulator was also written to calculate and plot a baseline “pore pressure and flowrate given” run assuming different values of overbalance pressure. This program is shown in Figure 1.3.4a and calculated pressure responses (with automated graphics displays) are given in Figure 1.3.4b.

Figure 1.3.4a. Input screen for Model SC-DD-FORWARD-2-CREATE-TABLES-3B.

Figure 1.3.4b. Pressure trends for selected overbalance pressures.

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Now, we return to supercharge models for drawdown-buildup applications using buildup data. The software shown in Figure 1.3.5a for the inverse solver is identical to that for the bottom screen in Figure 1.3.1 except for the user interface. The forward supercharge drawdownguildup solver is shown in Figure 1.3.5b, which solves the basic “pore pressure and flowrate given” model, plus a specified overbalance pressure. Pressure transients created by this forward solver are inputted into the inverse solver. Then validation is accomplished when mobility, pore pressure and compressibility are recovered.

Figure 1.3.5a. Input screen for Model SC-DDBU-INVERSE-2.

Figure 1.3.5b. Input screen for Model SC-DDBU-FORWARD-4NOPOR.

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As of this writing, Windows user interface development is ongoing and subject to careful review. We recognize that appearance, features and menu placement strongly affect productivity. For example, forward supercharge simulators for “drawdown only” and “drawdown-buildup” applications have been integrated and our “first pass” result is shown in Figure 1.3.6. This working software version was used to generate the forward pressure transient response results given in Chapter 4.

Figure 1.3.6. Input screen for integrated forward simulator For “drawdown only” and “drawdown-buildup” applications.

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1.4 Multiple Drawdown and Buildup Inverse Models In Section 1.1, we explained how, using the exact forward solver FT-00 for liquids, we can calculate the complete pressure transient response corresponding to any multirate volume flowrate input provided “piecewise constant” rates were used (this allowed us to apply superposition methods to create exact solutions to the governing equations). Sample “curly, blue line” pressure traces are shown together with their corresponding “white box” flowrates in Figure 1.4.1.

Figure 1.4.1. Main interface, “multiple drawdown and buildup” inverse models (MDDBU).

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Figure 1.4.2. Exact steady-state inverse solver (see “center button,” main menu).

Figure 1.4.3. Inverse method, Model 2 (same as FT-PTA-DDBU). In Chapter 3, we discuss different situations when multirate pumping can be used in practice, and why an inverse calculation would be desired using (three time-pressure point) data from the last flowrate cycle. Figure 1.4.4 shows the eleven pumping scenarios supported, noting that rates (in any order) can be positive, negative or zero. The “three black dots” in each diagram indicate where three time-pressure data points are to be selected for our exact inverse analysis.

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Figure 1.4.4. Eleven transient run situations supported.

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Section 3.2 explains the exact analysis for calculating mobility, pore pressure and compressibility using three time-pressure data points from unsteady, possibly very rapid pressure responses. The models assume non-negligible flowline volume effects but do not include supercharge or overbalance pressures. Functionally, our multiple drawdown and buildup inverse model is equivalent to the simpler drawdown-buildup model FT-PTA-DDBU given previously. However, the algebra required for the underlying exact inverse solutions is extremely complicated. Detailed forward and inverse model validations are given in Chapter 3. Finally, we acknowledge what is perhaps the industry’s first use of “double drawdown” analysis (see Schlumberger Log Interpretation Principles/Applications, SMP-7017, Schlumberger Wireline & Testing, Sugarland, Texas, 1989). Figure 1.4.5 from that publication shows an important application of double drawdowns, but it is seen how both have achieved steady state and act independently. Our methodology, again, allows fully transient, interacting drawdowns and buildups, under low mobility conditions where flowline storage effects are non-neglibible. This allows us to conduct multiple independent inverse tests, or even pretests, while reducing total test time significantly – other important petroleum engineering applications are described in Chapter 3.

Figure 1.4.5. Original Schlumberger double-drawdown application.

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1.5 Multiphase Cleaning and Supercharge Model Whereas the foregoing models deal with “clean” single phase liquid and gas fluids, flows in the reservoir are often multiphase in nature and require more complicated mathematical description. The early work of Chin and Proett (2005) describes a powerful transient, compressible liquid, miscible mixing model applicable to the formation testing process: invasion and mudcake buildup, overbalance and underbalance, onset of fluid pumping and fluid redistribution within the reservoir, and so on. This model was importantly extended to high inertia applications in Chin et al. (2015) and applications to detailed well interactions, supercharging and underbalanced effects are given in Chapter 4 (simpler pressure interpretation methods are offered in Chapter 2). In our brief introduction here, we display the “main system menus” in Figure 1.5.1 while “run-time simulation menus” appropriate to a particular run appear in Figure 1.5.2. Since any simulation requires dozens of inputs, the learning curve is steep – thus, an extensive library of prior runs (i.e., bottom of Figure 1.5.1) provides access to earlier setups that may be re-run at any time – and whose inputs may be simply modified by the user. These inputs include properties related to mud, mudcake, permeability, porosity, initial, pore and well pressure, and so on, as shown in separate categorized menus in Figure 1.5.2. As we will show in Chapter 4, as we have already in Chin et al. (2015), many output options are available, from line plots, to static color plots, to dynamic color movies highlighting the mixing and pressure redistribution process. Although the math model and numerical solution are state-of-the-art, efficient coding and tightly integrated color graphics allow the software system to operate on modest Windows i5 machines rapidly and “right out of the box” – just as all of our other algorithms do. Also note that the initial run library at the bottom of Figure 1.5.1 consists of assorted runs and may be augmented by new runs saved by the user.

Formation Testing 25

Figure 1.5.1. Main system level simulation menus and options.

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Figure 1.5.2. Run-time simulation menus for specific run.

Formation Testing 27

The physical problem solved by our miscible flow simulator is readily explained. Figure 1.5.3 shows the initial invasion process (assuming overbalanced drilling) where mud penetrates the formation cylindrically. As the fluid invades and displaces or mixes with reservoir fluid, it leaves behind a growing mudcake. Figure 1.5.4 illustrates the onset of formation tester pumping. The red arrows show fluids entering the formation tester while blue denotes continuing invasion that is decreasing in rate. The withdrawn fluid, however, is not yet clean since it consists of mud and oil. If invasion eventually ceases, pumping at later times will lead to cleaner pumped fluids. To illustrate this process, screen shots typified by those in Figure 1.5.5 are offered.

Figure 1.5.3. Initial cylindrical invasion and mudcake buildup.

Figure 1.5.4. Pumpout (red) and simultaneous invasion (blue).

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Typical color plots for pressure and concentration, shown in Figure 1.5.5, appear periodically at different time intervals so that the physics in the reservoir can be monitored (for faster simulations, these screens can be turned off). Figure 1.5.5 contains two views, an early time screen shot at left and a later time view at right. Consider the left-most diagram in the left screen shot. This shows the pressure distribution in an angular section of the reservoir (red indicates high pressure in the well, while orange denotes lower pressure in the reservoir. The small “blue dot” represents low pressure at the probe where fluid is being withdrawn. The central diagram shows concentration, dirty “blue mud” at the left well interface invading clean red in situ fluid. The far right “striped” diagram indicates reservoir layer boundaries assumed for this simulation. The right side Windows screen displays analogous events at a much later time. Note how the “very colorful, dynamic” pressure plot still shows the high (red) pressure influence of the well. The nearfield is now blue (instead of orange) because pumping has continued for a significant time. The effect on concentration is more pronounced. Line plots showing viscosity and fluid contamination level (or concentration percentage) are offered as well and typical plots are shown in Chapter 4 calculations.

Figure 1.5.5. Early results (left) and later dynamics (right) times. During runs, different “snapshots” use different color scales; thus, “red” at one time denotes a different attribute value from “red” at a another time. After the simulations are completed, all color plots are collected, and color scales are normalized to the maxima and minima for the entire time collection of snapshots for “movie mode” playback. This provides a good understanding of the mixing process throughout all time, whereas the static plots provided during simulations lead to an understanding of properties as they distributed throughout in space.

Formation Testing 29

1.6 System Integration and Closing Remarks Finally, we indicate that effective productivity software provides more than individual models – an integrated platform hosting forward, inverse and multiphase software, together with manuals, papers, “best practice” tips, industry videos, marketing brochures, plus internal company and competitor information, and so on, is required. Figure 1.6.1 provides a glimpse of one such prototype that would support the needs of an active marketing organization.

Figure 1.6.1. Integrated software platform, a beginning.

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1.7 References Chin, W.C., Zhou, Y., Feng, Y., Yu, Q. and Zhao, L., Formation Testing: Pressure Transient and Contamination Analysis, John Wiley & Sons, Hoboken, New Jersey, 2014. Chin, W.C., Zhou, Y., Feng, Y. and Yu, Q., Formation Testing: Low Mobility Pressure Transient Analysis, John Wiley & Sons, Hoboken, New Jersey, 2015. Proett, M., Chin, W.C. and Mandal, B., “Advanced Permeability and Anisotropy Measurements While Testing and Sampling in Real-Time Using a Dual Probe Formation Tester,” SPE Paper 64650, Seventh International Oil & Gas Conference and Exhibition, Beijing, China, Nov. 2000. Schlumberger Staff, Log Interpretation Principles/Applications, SMP-7017, Schlumberger Wireline and Testing, Sugar Land, Texas, 1989.

2 Supercharging – Forward Models and Inverse Solutions The purposes behind formation testing pressure transient analysis are multifold, among them, pore pressure assessment for production planning and drilling safety assessment, gradient analysis for fluid identification, and identification of fluid contact interfaces and possible solid barriers. In addition, formation mobility is a key objective, with the ultimate focus being rock permeability, and fluid viscosity and compressibility. In Chapter 1, we introduced new methods to extrapolate such information from early time pressure transient measurements, and demonstrated how we ensure our inverse methods are correct. We create exact “synthetic data” using analytical solutions in FT-00, analyze resulting pressure time histories using rapid interpretation models, and predict fluid and formation quantities that agree with known FT-00 inputs. In this sense, our forward and inverse models are consistent. 2.1

Supercharge and Math Model Development In both forward FT-00 and inverse FT-01 instances, spherical flow was assumed for isotropic media (while ellipsoidal models were used for transversely isotropic formations). This was done for mathematical expediency and the practical advantages offered by simple analytical models. There is no surprise here: in all areas of mathematical physics, geometric simplifications are sought to make algebraic manipulations manageable. However, sometimes these do not apply to the physical situation, and the wide variety of drilling scenarios contain examples where simple spherical or ellipsoidal models do not apply. One important class of applications deals with formation testing when wellbore fluids inter the reservoir, known as “overbalanced drilling,” and the second, when reservoir fluids enter the well, referred to as “underbalanced drilling.” The former is used to build mudcake seals to prevent fluid lost into the formation, while the second disallows mudcake formation, thus reducing skin damage and enabling production while drilling. 31

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We can introduce three possible drilling situations using Figures 2.1, 2.2 and 2.3. In Figure 2.1, the well is drilled underbalanced and reservoir fluid enters the well along the circular cylindrical surfaces of the wellbore. At the same time, the formation tester is injecting or withdrawing fluid, so that a spherical or ellipsoidal flow in superposed on the cylindrical flow. In Figure 2.2, with overbalanced drilling, the direction of the wellbore flow is reversed. In underbalanced drilling, the well pressure is less than the pore pressure, while in overbalanced drilling, the well pressure exceeds the pore pressure.

Figure 2.1. Underbalanced drilling with reservoir outflow.

Figure 2.2. Overbalanced drilling with wellbore inflow.

Supercharging – Forward Models and Inverse Solutions 33

The geometric nature of the flow in Figures 2.1 and 2.2 precludes simple spherical flow modeling and a full cylindrical (plus spherical or ellipsoidal) flow approach is required. Fortunately, immiscible and miscible flow models are available and discussed in the author’s two prior formation testing books, namely Chapter 4 in Chin et al. (2014) and Chapters 6 and 7 in Chin et al. (2015). These books describe the physics, the underlying mathematical model and formulation, and the numerical analysis, together with detailed examples. Aside from the brief introduction given earlier in Chapter 1, we deal directly with underbalanced and overbalanced supercharge applications in Chapter 4. For these complicated flow problems, simple solutions in the sense of those underlying our FT-00 and FT-01 algorithms are not possible.

Figure 2.3. Overbalanced and underbalanced drilling applications with sealed borehole walls. In Figure 2.3, we consider a flow where mudcake seals the borehole walls against fluid influx or outflux. An overbalanced flow is assumed to have formed the mudcake, which is now thick or impermeable enough to allow at most local invasion speeds that are small compared to formation tester nozzle velocities. On the other hand, the figure would also apply to underbalanced drilling when fluid speeds leaving the formation are small compared to tester pumping speeds at the nozzle. For such cases, simpler spherical or ellipsoidal models apply.

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There are important differences from the math models used to construct forward FT-00 and inverse FT-01 simulators. These earlier methods assume Darcy’s partial differential equation, a general pumping boundary condition accounting for volume flow rate and flowline storage, a given farfield pore pressure Ppore and a uniform initial condition in which the initial pressure is identical to Ppore. These conditions, as summarized in Equations 2.1.1-2.1.4 in this chapter, typify the models in Chin et al. (2014, 2015) and earlier published papers by this author and his colleagues at other oil service companies. In this chapter, we will present a new model described in Equations 2.2.1-2.2.4. Whereas the above paragraph assumes a uniform initial condition in which the initial pressure P0 is identical to Ppore , here we will allow the initial pressure to vary spatially with P(r, t = 0) = P0 + Z/r, where a constant Z > 0 for overbalanced applications, for R > Rnozzle , while for underbalanced drilling, we take Z < 0. The “1/r” decay seen in “Z/r” is no more than an approximate description of monotonic pressure decrease away from the wellbore surface. As emphasized earlier, we assume a spherical flow (for isotropic media), which applies when mudcake sealing is effective, or if leakage does occur, it is found only farther away from the nozzle and pad arrangement. Under the circumstances, a math model with a closed form analytical solution can be developed, although a final equation must be numerically evaluated. Fortunately, the computational resources required are minimal, and downhole implementation is possible with little difficulty. 2.2 Supercharge Pressure and Ultimate Decay The older petroleum literature deals with overbalance drilling and typical overbalance pressures do not exceed 200 psi. In most applications, thick impermeable mudcakes are formed that effectively seal the formation from the effects of high wellbore pressure. The temporary nonuniformity in pressure in the sealed reservoir is a transient phenomenon that disappears quickly, say within the hour. In modern low mobility applications, however, mudcakes are thin and do not build quickly, leading to so-called “supercharged” conditions. While these too will ultimately dissipate, it is neither clear when, nor what fluid, rock and tool parameters the dissipation rates depend on. The literature offers numerous ways to predict pressure supercharge levels, but none predict the time scale of the equilibration or the effect of the supercharge on measured source probe pressures; also, inverse models are non-existent.

Supercharging – Forward Models and Inverse Solutions 35

Before developing the subject, we cite modern references on applications where supercharge can be severe. Barriol, Glaser and Pop et al. in “The Pressures of Drilling and Production,” Schlumberger Oilfield Review, Autumn 2005, state these conditions. “Subnormal pressures, or those below the normal gradient, can cause lost-circulation problems in wells drilled with liquid drilling mud. Subnormal pressure conditions frequently occur when the surface elevation of a well is much higher than the subsurface water table or sea level. This is seen when drilling in hilly or mountainous locations, but it may also occur in arid regions where the water table may be more than 1,000 ft (305 m) deep.” “Abnormally low pressures are also frequently found in depleted reservoirs. These are reservoirs whose original pressure has been reduced by production or leakage. Depletion is not unusual in mature reservoirs from which significant volumes of oil and gas have been produced without waterflooding or pressure maintenance.” “If the filtercake is totally ineffective in sealing between the formation and test probe, then wellbore pressure will be measured; if the filtercake provides a perfect seal, given sufficient time, the tester should measure the true formation pressure.” Finally, “When the difference between the measured sandface pressure and the true formation pressure is significant, the formation is usually said to be supercharged.” Just what magnitudes are considered critical? While the early literature refers to 200 psi, recent studies focus on much higher levels. Proett, Ma, Al-Musharfi and Berkane, in “Dynamic Data Analysis with New Automated Workflows for Enhanced Formation Evaluation,” SPE187040-MS, Society of Petroleum Engineers Annual Technical Conference and Exhibition, San Antonio, Texas, October 9-11, 2017, consider typical overbalance pressures in the range 500-1,000 psi in several examples. Proett, Seifert, Chin, Lysen and Sands, in “Formation Testing in the Dynamic Drilling Environment,” SPWLA 45th Annual Logging Symposium, Noordwijk, The Netherlands, June 6-9, 2004, assume overbalance pressures of 1,000 psi in their illustrative examples. The most attention-getting magnitudes are published by Halliburton and Chevron Thailand E&P, in a symposium article surveying several hundred wells. Rourke, Powell, Platt, Hall and Gardner, in “A New Hostile Environment Wireline Formation Testing Tool: A Case Study from the Gulf of Thailand,” SPWLA 47th Annual Logging Symposium, Veracruz, Mexico, June 4-7, 2006, summarize field experiences gained from over three hundred wells logged with a new formation tester. The

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key point from their paper is succinctly contained in one quote. In short, “While pressure testing in the infill development wells where depletion is often observed, it is not uncommon for the differential between hydrostatic and reservoir pressure to exceed 2000 psi.” This is also emphasized in a 2018 fact sheet, “Testing the Tight Gas Reservoir – Hostile Environment Wireline Formation Tester Reduces NPT in HPHT Boreholes,” available from Halliburton at www.halliburton.com. “Wells are drilled highly overbalanced because the differential between hydrostatic and reservoir pressure may exceed 2,000 psi.” In concluding this section, we emphasize that high values of supercharge pressure may exceed 2,000 psi, not just the “norm” cited in the older literature nearing 200 psi. Perhaps worse than the pressures themselves is the fact that supercharge effects, like high blood pressure in humans, lurks unknown and ready to do damage. Very often, operators understand that supercharge exists, with service company advisors recommending expensive wait periods ranging from thirty minutes to hours. In short, the hope is that high pressure effects will dissipate, leaving permeability, pore pressure and compressibility predictions from pressure transient analysis credible and accurate. We have emphasized supercharging and overbalanced drilling thus far, but in many applications, the opposite may be of importance. Wells are often drilled “underbalanced” to reduce formation damage, with slight production into the borehole tolerated. For such applications, the foregoing formation evaluation objectives are unchanged and just as important. If the formation tester flowfield is still spherical, the new model described briefly above still applies. In existing industry models, as well as those in Chin (2014, 2015), a uniform initial pressure P0 identical to the farfield pore pressure Ppore is assumed. In a subsequent analysis, will allow the initial pressure to vary spatially with P(r, t = 0) = P0 + Z/r, where a constant Z is introduced with Z > 0 for overbalanced applications, for R > Rnozzle , while for underbalanced drilling, we take Z < 0. The “1/r” decay seen in “Z/r” is no more than an approximate description of monotonic pressure decrease away from the wellbore surface. For applications in which spherical (or ellipsoidal) flow models are inapplicable, more complete models allowing cylindrical influx or outflux at borehole walls are required. These appear in Chin et al. (2014, 2015) and used in Chapter 4 for illustrative studies. Before developing our methods, we offer a useful patent write-up on supercharge measurement based on the work in Chin (2017) and earlier.

Supercharging – Forward Models and Inverse Solutions 37

2.3 United States Patent 7,243,537 B2 * These pages, included for informational purposes, provide describe methods for measuring formation supercharge pressure.

Figure 2.4a. United States Patent 7,243,537 B2. *

The author does not retain any financial interest in patent. Certain claims based on prior related work cited in Chin (2017). For additional information, contact your Halliburton representative.

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Figure 2.4b. United States Patent 7,243,537 B2.

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Figure 2.4c. United States Patent 7,243,537 B2.

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Figure 2.4d. United States Patent 7,243,537 B2.

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Figure 2.4e. United States Patent 7,243,537 B2.

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Figure 2.4f. United States Patent 7,243,537 B2.

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Figure 2.4g. United States Patent 7,243,537 B2.

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Figure 2.4h. United States Patent 7,243,537 B2.

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Figure 2.4i. United States Patent 7,243,537 B2.

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Figure 2.4j. United States Patent 7,243,537 B2.

Supercharging – Forward Models and Inverse Solutions 47

Figure 2.4k. United States Patent 7,243,537 B2.

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Figure 2.4l. United States Patent 7,243,537 B2.

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Figure 2.4m. United States Patent 7,243,537 B2.

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Figure 2.4n. United States Patent 7,243,537 B2.

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Figure 2.4o. United States Patent 7,243,537 B2.

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Figure 2.4p. United States Patent 7,243,537 B2.

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Figure 2.4q. United States Patent 7,243,537 B2.

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Figure 2.4r. United States Patent 7,243,537 B2.

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Figure 2.4s. United States Patent 7,243,537 B2.

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Figure 2.4t. United States Patent 7,243,537 B2.

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Figure 2.4u. United States Patent 7,243,537 B2.

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Figure 2.4v. United States Patent 7,243,537 B2.

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Figure 2.4w. United States Patent 7,243,537 B2.

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Figure 2.4x. United States Patent 7,243,537 B2.

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Figure 2.4y. United States Patent 7,243,537 B2.

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Figure 2.4z. United States Patent 7,243,537 B2.

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Figure 2.4aa. United States Patent 7,243,537 B2.

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Figure 2.4bb. United States Patent 7,243,537 B2.

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Figure 2.4cc. United States Patent 7,243,537 B2.

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Figure 2.4dd. United States Patent 7,243,537 B2.

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Figure 2.4ee. United States Patent 7,243,537 B2.

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Figure 2.4ff. United States Patent 7,243,537 B2.

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2.4

Forward and Inverse Models with Supercharging – Drawdown-Only and Drawdown-Buildup Applications and Illustrative Examples

2.4.1 General Ideas in Formation Testing Formulations In this section, we develop the general theory for forward (direct) and inverse (indirect) pressure transient analysis problems where supercharging, low mobility and flowline storage effects can be important. The approach applies to “drawdown only,” “drawdownbuildup, using buildup data,” and other forms of probe excitation. There is no requirement for the piston probe volume pumping rate to be constant. The methodology is developed for both wireline and FTWD tools but focuses on single-probe testers – the benefits for having a second observation probe include kh and kv determination but are not addressed in this section. While “supercharging” typically refers to overbalanced drilling applications, the models developed here apply to underbalanced situations as well, provided the spherical flow assumption is used, which implicitly assumes low influx or outflux velocities at the sandface compared to formation tester nozzle velocities. Conventional zero supercharge model. The formation testing book by Chin et al. (2014) provides a number of zero-supercharge forward and inverse formulations in different physical limits together with exact solutions. One particular model, quite general, is given by Equations 5.1– 5.4 in that reference, here re-numbered as, 2P(r,t)/ r2 + 2/r P/ r = ( c/k) P/ t (2.1.1)

P(r,t = 0) = P0

(2.1.2)

P(r = ,t) = P0

(2.1.3)

(2.1.4) (4 Rw2k/ ) P(Rw,t)/ r – VC P/ t = Q(t) Equations 2.1.1 – 2.1.4 represent the complete isotropic, zero skin, spherical Darcy flow formulation for compressible liquids, which was solved exactly in terms of complex complementary error functions. Note that “spherical” implies mathematical idealization. The spherical source of radius Rw will not adequately describe borehole wall curvature or the effects of tester pads – to account for these and other non-idealizations, this Rw often denotes the product of a corrective “geometric factor” (that is, “G,” determined empirically or through three-dimensional finite element analysis) and the true nozzle radius.

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We also emphasize that spherical flow assumptions implicitly assume that influx or outflux velocities at the sandface of the circular cylindrical well are small compared to typical formation tester nozzle flow speeds as suggested in Figure 2.3. Importantly, Equation 2.1.4 describes volumetric pumping in the presence of flowline storage. Existing solutions solving the above formulation include Halliburton’s GeoTapTM model, developed by this author and M.A. Proett, as well as the “rational polynomial” solution offered in Chin et al. (2014). We emphasize that, even for a given model, different numerical answers are possible depending on the compiler or microprocessor used. These differences cannot be ignored when values of the argument are very small or large, corresponding to extremes in formation mobility values. In the above, P(r,t) is the transient sand pressure with “r” and “t” being spherical radius and time, where is porosity, is viscosity, c is formation fluid compressibility, and k is the isotropic permeability. Again, Rw is the effective nozzle radius (where the subscript “w” denotes “spherical well”), V is the flowline volume, C is the compressibility of the fluid within the flowline, Q(t) is the pumping volume flow rate, and importantly, P0 is the pore pressure assumed to be equal to the initially uniform quiescent pressure. It is important to emphasize that the initial pressure P(r, t = 0) does not vary spatially nor does it characterize the higher borehole mud pressure; it is simply a constant identical to the farfield pore pressure. Thus, supercharging is not modeled by the conventional formulation – again, this is also true of the models in the formation testing books of Chin et al. (2014) and Chin et al. (2015) and in other industry methods known to the author. Supercharge extension. Now, we ask how “supercharging,” where the term will be used to represent both overbalanced and underbalanced effects, can be modeled by extending the framework of Equations 2.1.1 – 2.1.4. Our approach is straightforward. We instead solve Equations 2.1.1 – 2.1.4 with a single change, altering only the initial condition in Equation 2.1.2. In particular, we consider the model 2P(r,t)/ r2 + 2/r P/ r = ( c/k) P/ t (2.2.1) P(r,t = 0) = P0 + Z/r, Z > 0, R > Rw

(2.2.2)

P(r = ,t) = P0

(2.2.3)

(4 Rw2k/ ) P(Rw,t)/ r – VC P/ t = Q0

(2.2.4)

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where Z = Pbh – P0 may be positive (overbalanced) or negative (underbalanced) with any magnitude, Pbh being the borehole pressure just outside the reservoir sandface. In older references, “Pbh – P0” is often quoted in the 200-250 psi range. In fact, Halliburton and Chevron Thailand, as discussed previously, observe that overbalances exceeding 2,000 psi are not uncommon, particularly in infill drilling where the reservoir is depleting. For such applications, the use of inverse models based on Equations 2.1.1 – 2.1.4 will lead to incorrect pore pressures, mobilities and compressibilities. What does Equation 2.2.2 mean? In a sense, it simply describes “what the measuring source probe sees.” The constant P0 in the boundary condition in Equation 2.2.3 still represents the farfield background pore pressure, that is, the constant pressure in the undisturbed reservoir. However, the initial condition in Equation 2.2.2 recognizes the fact that pressures can exceed (or can be less than) pore pressure when the formation tester first interacts with the reservoir; the Z/r behavior describes the expected monotonic decay (or increase) away from the probe. The initial difference from farfield pore pressure is the result of initial exposure of the reservoir to the wellbore – in our spherical flow model, we assume that the well has been sufficiently sealed by mudcake to the degree that any velocity influx or outflux is small compared to velocities at the nozzle. Pressures close to the formation tester can be large because mud overpressures (or small due to underpressure) are transmitted into the rock matrix. These high (or low) pressures are trapped inside the formation near the sandface until they can be dissipated. We emphasize that overbalance effects are more significant in low permeability formations since mudcakes are substantially thinner. Also note that supercharging (as modeled through Z/r in the initial pressure description) vanishes far away as it should – in the farfield, the initial pore pressure is P0 as in conventional formulations. Supercharge pressures always decay with time leading to spatially uniform conditions, but at early times, they distort the pressure measurements needed for accurate interpretation. Our objective in developing a physical model is to simulate the transient decay of supercharge effects and to ascertain how these nonequilibrium effects influence formation tester pressure transient responses. While our suggestion of P(r,t = 0) = P0 + Z/r, r > RW, is consistent with physical intuition, there are other reasons for its choice. First, the “1/r” in Z/r is an exact solution of the spherical flow equation

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2P(r,t)/ r2 + 2/r P/ r = ( c/k) P/ t. However, it is an approximation in the sense that this “r” is not the radial coordinate “r” in the cylindrical description of flow at the sandface of the wellbore. And so, it is perhaps best that we simply view Z/r as a convenient practical description of the initial reservoir state. Working with the pressure directly through an initial condition, i.e., as through “P(r,t = 0) = P0 + Z/r,” allows us to describe global changes to the initial reservoir domain without altering at all the pumpout boundary condition, which is only physically affected by prescribed flow rate and flowline storage volume. Note that the value of Z can follow from a rough field estimate for overbalance. Before delving into the mathematics, it is useful to point out the differences between the forward supercharging models posed here and those in Chin et al. (2014) and Chin et al. (2015). In the earlier references, a cylindrical coordinate system is used to host the formulation, one allowing mudcake growth at the walls of the hole. The radial derivative P/ r (where “r” is perpendicular to the sandface) is used to model filtration at the sandface; where any pumping probes are located, P/ r is used at an isolated point location to model fluid pumpout mechanisms. Initially, properties like pressure and multiphase saturation or concentration may be uniform throughout the reservoir, or they may vary (cylindrically) radially if invasion has occurred for some time. Only when pumping commences is a spherical (or ellipsoidal) flow superposed on the cylindrical background – this pumping removes fluid mixtures leading, hopefully, to decontaminated in situ fluid with time. In other words, the prior work models mud invasion, two-phase flow development, and also, the fluid extraction process at the probe. The analysis predicts time-dependent saturations and concentrations within the reservoir. Mudcake formation may be strongly or weakly coupled to flow in the formation, depending on relative differences in permeability. Needless to say, the model is only as accurate as the required input model for cake filtration properties, one which must be measured in the laboratory. Illustrative examples are developed in Chapter 4. Unfortunately, the information and computing time and resources needed to run such general simulators is often not available, either downhole or at the surface at the rigsite. In fact, for exploration wells, little is known. Thus, the conventional model in Equation 2.1.1 – 2.1.4 is popular, requiring little knowledge of drilling conditions or downhole geology. For instance, permeabilities are isotropic, and supercharge effects are (hopefully) minimal. Most operational approaches are “wait

Supercharging – Forward Models and Inverse Solutions 73

and see” – simply wait until estimated pressure imbalances equilibrate and then apply inverse methods. But at earlier times, obvious problems arise. How much of the pressure overbalance should be subtracted from measured formation tester pressures and for how long? Should the same constant over-pressure be subtracted at every instant in time? Definitely not, but a suggestions are purely heuristic and impossible to substantiate. Because the pumping environment cannot be fully characterized, the “minimalist approach” behind Equations 2.1.1 – 2.1.4 is adopted here, with a single extension replacing Equation 2.1.2 by Equation 2.2.2. It adequately describes “what the probe sees.” The additional overpressure Pbh – P0 depends on mudcake properties which can be estimated from methods such as those taught by Proett et al. (2007) in United States Patent 7,243,537 B2, reproduced in the previously in Section 2.3, or for example, in the early formation invasion book of Chin (1995) and more recently in Chin (2017). We will not review these materials in this book and assume that methods form estimating Pbh – P0 are available. We also emphasize that the inverse models developed here can be used to estimate uncertainties in pore pressure, mobility and compressibility predictions, provided a range of over-pressure values are considered. 2.4.2 Mathematical Formulation In Chin et al. (2014), the exact solution to Equations 2.1.1 – 2.1.4 was obtained by taking Laplace transforms in time. The resulting ordinary differential equation in space was solved in terms of Bessel functions together with regularity conditions at infinity. The resulting transform in “s space” was simplified using partial fraction expansions. An exact inversion led to closed form analytical solution written in terms of the complex complementary error function, valid for all parameters and flowrate functions Q(t). The exact solution can be simplified in the low mobility limit where pressure distortions due to flowline volume effects are not insignificant. The resulting exponential pressure solution and its inversion form the basis for Halliburton’s commercially successful GeoTapTM pressure transient analysis method and also the alternative “rational polynomial expansion” in Chin et al. (2014) and Chin et al. (2015). Both methods have proven successful in field and laboratory applications, providing strong, credible support for the formulation in Equations 2.1.1 – 2.1.4. The formulation in Equations 2.2.1 – 2.2.4, however, cannot be solved using the analytical methods developed for Equations 2.1.1 –

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2.1.4. The reasons are mathematical. The ordinary differential equation for the Laplace transform now contains a right-side inhomogeneity related to the spatially-varying initial condition P(r,0) which does not lead to solutions that can be analytically inverted. In the parlance of elementary differential equations, while the “homogeneous solution” associated with complex complementary error functions is available, the “particular solution” cannot be easily expressed since it contains nontrivial combinations of Bessel functions which cannot be inverted in closed analytical form. On the other hand, a purely numerical approach, say using Stehfest integrations, is prone to inaccuracy, long computing times, and results that do not support straightforward physical interpretation. The advantages behind closed form solutions based on exponential or rational polynomials are lost, and moreover, downhole implementation is no longer possible. We were therefore led to seek a different, less conventional analysis approach. We asked, “What if it were possible to replace the Z/r in the initial condition by a simple modified flow rate in the boundary condition?” This question appears contradictory to our remarks above. We had stated that initial supercharge effects should be accommodated in an initial condition and not a boundary condition, and now, we appear to take the disputed reverse position. What we mean is the following. We retain the physical description behind Equations 2.2.1 – 2.2.4 in its entirety, but we only mathematically convert this formulation to one with uniform initial conditions and a modified boundary condition so that Laplace transform methods can be used. If this could be accomplished, the more complicated solution derived previously for general pumping rates Q(t) could be used immediately for supercharge applications with minor change. We now present the complete strategy needed to develop forward (direct) and inverse (indirect) models. Again, “forward” refers to pressure calculation once tool and formation parameters are specified, while “inverse” refers to pore pressure, mobility and compressibility prediction assuming that several (time, pressure) data points are available. In our research, we develop both models, focusing first on the forward model and using it to create synthetic pressure transient probe responses for use in evaluation the complementary inverse procedure – e.g., these responses are typically a table of (t, P(Rw,t)) values, only three sets of which are used in any particular inversion. The derivation of the inverse model is a mathematical exercise once the forward model is available,

Supercharging – Forward Models and Inverse Solutions 75

although a quite challenging exercise. We shall show that the forward supercharge model generates physically credible results, and using data from this model together with the new inverse model, we recover the generating assumptions for pore pressure, mobility and compressibility. This recovery demonstrates complete mathematical consistency between forward and inverse solvers. Our mathematical strategy is straightforward. Essentially, the analytical expression for P(Rw,t) or Pw, expressed in terms of its dependent parameters, which includes Rw, k, , VC, P0 and Pbh – P0, is first determined so that Pw(t) = F(Rw, k, , VC, P0, Pbh – P0, t), with F now being a known function. This expression is evaluated for a userselected number of time values, three in our case, so that formula representations for pressures Pw,1 at t = t1, Pw,2 at t = t2 and Pw,3 at t = t3 are available. Thus we obtain three nonlinearly coupled transcendental equations which must be solved rapidly, stably and numerically, say Pw,t1 = F(Rw, k, , VC, P0, Pbh – P0, t1)

(2.3.1)

Pw,t2 = F(Rw, k, , VC, P0, Pbh – P0, t2)

(2.3.2)

Pw,t3 = F(Rw, k, , VC, P0, Pbh – P0, t3)

(2.3.3)

The function F is complicated, and depending on the flow limit assumed, can contain polynomials, exponentials, or in the most general case, complex complementary error functions. Moreover, while it takes a simpler form in “drawdown only” applications, it is particularly cumbersome in drawdown-buildup problems where time-shifted superpositions of the foregoing functions are involved. Nonetheless, with the assistance of algebraic manipulation software, a general Fsolver inverse module can and has been developed and successfully tested. How exactly is FSolver constructed? We provide enough details for interested readers to develop their own working versions. We return to a question posed earlier, “What if it were possible to mathematically transform our physical initial condition so that Z/r effects simply appear as a modified flow rate in a slightly different boundary condition to which our prior inverse algorithms apply?” Again, this possibility underlies our entire supercharge strategy.

76 Formation Testing Volume 3

The formulation in Equations 2.2.1 – 22..4 is posed in terms of a “regular capital P.” We motivate our approach by introducing a “bold capital P” or P(r,t) with the definition P = P – Z/r = P – Zr -1. This implies the following partial derivative relationships, Pr = Pr + Zr -2 , Prr = Prr – 2Zr –3 and Pt = Pt. Then, the prior partial differential equation Prr + 2/r Pr = ( c/k) Pt transforms accordingly as (Prr + 2Zr -3) + (2/r) (Pr – Zr –2 ) = ( c/k) Pt or Prr + 2/r Pr = ( c/k) Pt. In other words, the form of the governing equation is unchanged. Next consider the proposed physical initial condition P(r,t = 0) = P0 + Z/r. This transforms according to P + Z/r = P0 + Z/r, leaving P(r, t = 0 ) = P0 which is identical to the initial condition in Equations 2.1.1 – 2.1.4. The farfield boundary condition in Equation 2.1.3 or P(r = ,t) = P0 now transforms as P(r = ,t) = P0, P(r = ,t) + Z/r = P0 or P(r = ,t) P0 since Z/r 0 as r . Thus, it too remains identical to Equation 2.1.3. In fact, the only change associated with P = P – Z/r is found in Equation 2.1.4 or the pumpout boundary condition (4 Rw2k/ ) Pr(Rw,t) – VC Pt = Q(t). This transforms as (4 Rw2k/ ) [Pr(Rw,t) – ZRw-2] – VC Pt = Q(t), leading to (4 Rw2k/ ) Pr(Rw,t) – VC Pt = Q(t) + (4 Rw2 k/ ) ZRw-2 or, more simply, (4 Rw2k/ ) Pr(Rw,t) – VC Pt = Q(t) + 4 kZ/ . In other words, the initial-boundary value problem for P(r,t) with supercharging is mathematically identical to the zero-supercharge model for P(r,t) except that the volume flow rate Q(t) is increased by a constant value 4 kZ/ . Put simply, the effective flow rate in our bold P(r,t) formulation is now Q(t) + 4 kZ/ (note that P(r,t) is not the pressure, but an abstraction defined by P = P – Z/r). To create the formulas underlying FSolver, the change due to 4 kZ/ can be incorporated into formulas in Chin et al. (2014) for complex error functions and in terms of exponentials using the exponential solutions due to Proett and Chin. Finally, our P solution is back-converted to one in “regular capital P.” In essence, the three relationships in Equations 2.3.1, 2.3.2 and 2.3.3 are now available. FSolver further embeds a stable nonlinear transcendental equation solver that solves for the three unknowns k/ , P0 and c rapidly and accurately for all pressure-time input data. This procedure for drawdown and drawdown-buildup problems yields exact parameter solutions because we fit three data points to analytical formulas without further approximation or subjective visual curve-fitting judgements.

Supercharging – Forward Models and Inverse Solutions 77 Drawdown-only and drawdown-buildup specifics. The foregoing discussion presents the high-level strategy used to incorporate supercharge effects within the framework of a successfully implemented earlier model. Here we recapitulate the mathematics needed to develop “drawdown only” and “drawdown-buildup, using buildup data” approaches to forward and inverse model development. “Drawdown only” solutions are simple. By convention, the volume flow rate Q(t) is set to a positive constant Q0 > 0 for fluid withdrawal and a negative one for injection. For the purposes of discussion, we consider drawdown (although our comments apply to injection buildup as well). In the analyses cited above, the drawdown-only pressure solution P(Rw,t) is obtained by setting Q(t) = Q0 in the resulting analytical expression. We denote this expression by P(t)dd which is valid for all times t > 0. Let us further denote measured pressures by Pdd,1 at t = t1, Pdd,2 at t = t2 and Pdd,3 at t = t3 which are are known. When Pdd,1, Pdd,2 and Pdd,3 are equated to the analytically available solutions with times replaced by t1, t2 and t3, respectively, we obtain three nonlinearly coupled transcendental equations, namely Pdd(t1) = Pdd,1, Pdd(t2) = Pdd,2 and Pdd(t3) = Pdd,3, where the left sides refer to the analytical functions in Equations 2.3.1 – 2.3.4, which must be solved rapidly, stably and numerically. Drawdown-buildup applications are similarly addressed. Note that drawdown only problems are solved by P(Rw,t) = Pdd(t;Q0) which is valid for all times t > 0. We have added Q0 to the argument to emphasize a parametric dependence on Q0. Now suppose that the pump piston terminates its motion at time t = tdd. The drawdown solution only applies until t = tdd. Beyond this, there is no net fluid pumping and the net volume flow rate vanishes. Then, for t > tdd, the buildup solution is derived by augmenting the drawdown solution via linear superposition to give P(Rw,t) = Pdd(t;Q0) + Pdd(t– tdd; – Q0) which has the same form as the drawdown problem. The inverse procedure developed for drawdown only applications now applies to buildup data on the drawdown-buildup response curve.

78 Formation Testing Volume 3

2.5 Drawdown Only Applications Several Fortran computer programs have been written to implement our forward and inverse methods to “drawdown only” and “drawdown buildup” applications. A wide range of example calculations is offered in this section for drawdown-only problems. In each example, the forward model is used to create the pressure transient response “seen” by the formation tester probe in the presence of overbalanced pressure, with an assumed set of input parameters, e.g., permeability, viscosity, compressibility, pore pressure and so on. The forward pressure response results are given in sets of tabulated “time and pressure” numbers. Then, the inverse mode is evaluated, using three early-time data points arbitrarily selected from the above table and used to predict known pore pressure, mobility and compressibility. The inverse solver is operated in two modes, the correct supercharge mode requiring an overbalanced pressure input, and the incorrect mode – that is, the conventional inverse approach, which does not ask for overbalance inputs. The examples chosen show how supercharge can affect pore pressures (and hence, pressure gradients used for fluid, contact and barrier identification) and mobilities. Software references are given for readers wishing to duplicate or extend our published simulations. Note that authors’ comments are offered in the present Times Roman font, while computer screen results appear in Courier New font. 2.5.1 Example DD-1, High Overbalance

We first create forward data, calculating the transient pressure history measured by the probe using the overbalanced data shown immediately below. Then, in two separate inverse examples, we select three (time, pressure) data points arbitrarily, and predict pore pressure, mobility and fluid compressibility. In the first instance, we assume that we know the overbalance pressure – the predictions are excellent. In the second, we assume that it is zero, that is, we will use the incorrect inverse method that does not account for supercharging – and we demonstrate that the predictions are (as expected) not very good. The latter mode, operating without knowledge of supercharge pressure, corresponds to present industry practice.

Supercharging – Forward Models and Inverse Solutions 79 C:\FT-PTA-SC>sc-dd-forward-3B

Software reference, sc-dd-forward-3B.for. Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc):

1 1 .00001 20000 2000 1 .5 1 1000

FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Forward pressure transient predictions for drawdown-only applications, for low mobility, isotropic, supercharged flows where flowline storage is not negligible. Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc):

0.1000E+01 0.1000E+01 0.1000E-04 0.2000E+05 0.2000E+04 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0

P(psi) 22000. 21818. 21644. 21477. 21317. 21164. 21017. 20877. 20742. 20613. 20489. 20371. 20257. 20149. 20045. 19945. 19849.

(selected for inverse input)

80 Formation Testing Volume 3 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 39.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0 500.0

19758. 19670. 19586. 19505. 19428. 19354. 19284. 19216. 19151. 19089. 19029. 18972. 18917. 18865. 18814. 18766. 18720. 18676. 18634. 18593. 18554. 18517. 18482. 18447. 18176. 17999. 17883. 17808. 17759. 17727. 17669. 17668. 17668. 17668.

(selected for inverse input)

(selected for inverse input)

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

2332.0621 2000.0000 4332.0621

Supercharging – Forward Models and Inverse Solutions 81

Figure 2.5a. Pressure transient response with overbalance. In the following we use our inverse solver, take pressure data from 10, 20 and 30 seconds, entering in the input screen an overbalance pressure of 2,000 psi known from forward simulator inputs. C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for. Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1 .5 1 10 20489 20 19505 30 18865 2000

82 Formation Testing Volume 3 OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 10.0000 20489.0000 20.0000 19505.0000 30.0000 18865.0000 2000.0000

The predicted results are excellent, namely, a pore pressure of 20,003 psi instead of 20,000 psi, a mobility of 1.0164 md/cp instead of 1 md/cp, and a compressibility of 0.0100 x (cc/FloLineVol) per psi or, since the flowline volume is 1,000 cc, 0.00001 1/psi, exactly as assumed. Note that, in our calculations, we used an oil compressibility that is ten-fold larger than that of water, which results to slower transient decays. The following output summarizes inverse results. Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20003.0000 1.0164 0.0644 0.0100 x (cc/FloLineVol)

We next consider a second inverse calculation, using an incorrect overbalance pressure input that does not correspond to the value used in the forward simulation creating the pressure transient data. C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for. Here, we repeat the inverse calculation assuming an overbalance of 0 psi, that is, no overbalance – in other words, we are using the inverse method previously developed that does not account for supercharging. As expected, the results are not good. In particular, the output below shows a pore pressure of 22,003 psi instead of 20,000 psi, a mobility of 0.5463 md/cp versus an assumed value of 1 md/cp, and finally, a fluid compressibility of 0.0054 x (cc/FloLineVol)per psi instead of 0.0100 x (cc/FloLineVol)per psi. These unsatisfactory results are not surprising – they merely confirm our expected errors.

Supercharging – Forward Models and Inverse Solutions 83 Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1 .5 1 10 20489 20 19505 30 18865 0

OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 10.0000 20489.0000 20.0000 19505.0000 30.0000 18865.0000 0.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

22003.0000 0.5463 0.0346 0.0054 x (cc/FloLineVol)

84 Formation Testing Volume 3

2.5.2 Example DD-2, High Overbalance The forward model, in this example, is similar to that used in Section 2.5.2 above except that the mobility is smaller by a factor of ten. C:\FT-PTA-SC>sc-dd-forward-3B

Software reference, sc-dd-forward-3B.for. Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Plot every "NSEC" seconds:

0.1 1 .00001 20000 2000 1 .5 1 1000 50

Figure 2.5b. Pressure transient response with overbalance.

Supercharging – Forward Models and Inverse Solutions 85 FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Forward pressure transient predictions for drawdown-only applications, for low mobility, isotropic, supercharged flows where flowline storage is not negligible. Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc):

0.1000E+00 0.1000E+01 0.1000E-04 0.2000E+05 0.2000E+04 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0

P(psi) 22000. 21892. 21784. 21676. 21569. 21463. 21357. 21251. 21146. 21041. 20937. 20833. 20730. 20627. 20525. 20423. 20321. 20220. 20119. 20019. 19919. 19820. 19721. 19622. 19524. 19426. 19329. 19232. 19135. 19039. 18944. 18848. 18753. 18659.

(selected for inverse input)

(selected for inverse input)

(selected for inverse input)

86 Formation Testing Volume 3 34.0 35.0 36.0 37.0 38.0 39.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0

18565. 18471. 18378. 18285. 18193. 18101. 18009. 17114. 16256. 15434. 14647. 13893. 13170. 7420. 3674. 1235.

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

23320.6202 2000.0000 25320.6202

Following the illustrative Example DD-1 above, we perform a first inverse calculation using three (time, pressure) data points arbitrarily chosen, highlighted in red above, plus the known overbalance pressure of 2,000 psi. The predictions are excellent, in particular, a pore pressure of 20,000 psi in exact agreement with that assumed in the forward model, a mobility of 0.1040 md/cp versus an inputted 0.1 md/cp above, and a compressibility of 0.0100 x (cc/FloLineVol) per psi in agreement with the forward analysis. In the second inverse calculation, where we will assume a zero overbalance (that is, we use the older inverse model that does not account for supercharging), the results are questionable. The mobility is 0.0956 md/cp while the compressibility output is “0.0092” – these acceptable results compare with exact values of 0.1 md/cp and “0.01.” However, an incorrect pore pressure of 22,000 psi is obtained versus an assumed 20,000 psi. C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for. Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ...

Supercharging – Forward Models and Inverse Solutions 87 Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1 .5 1 10 20937 20 19919 30 18944 2000

OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 10.0000 20937.0000 20.0000 19919.0000 30.0000 18944.0000 2000.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20000.0000 0.1040 0.0646 0.0100 x (cc/FloLineVol)

C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for. Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1 0.5 1 10 20937 20 19919 30 18944 0

88 Formation Testing Volume 3 OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 10.0000 20937.0000 20.0000 19919.0000 30.0000 18944.0000 0.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

22000.0000 0.0956 0.0593 0.0092 x (cc/FloLineVol)

2.5.3 Example DD-3, High Overbalance

In the forward model below, we consider a higher mobility of 10 md/cp relative to the two prior examples. C:\FT-PTA-SC>SC-DD-FORWARD-3B

Software reference, sc-dd-forward-3B.for. Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Plot every "NSEC" seconds:

10 1 .00001 20000 2000 1 .5 1 1000 1

Supercharging – Forward Models and Inverse Solutions 89

Figure 2.5c. Pressure transient response with overbalance. FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Forward pressure transient predictions for drawdown-only applications, for low mobility, isotropic, supercharged flows where flowline storage is not negligible. Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc):

0.1000E+02 0.1000E+01 0.1000E-04 0.2000E+05 0.2000E+04 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

P(psi) 22000. 21221. 20714. 20384. 20169. 20028. 19937. 19878.

(selected for inverse input)

90 Formation Testing Volume 3 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0 500.0

19839. 19814. 19797. 19787. 19780. 19775. 19772. 19770. 19769. 19768. 19768. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767.

(selected for inverse input)

(selected for inverse input)

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

233.2062 2000.0000 2233.2062

In our first inverse calculation, we assume we know the overbalance pressure of 2,000 psi, and we predict 20,014 psi, 9.5467 md/cp and “0.0094” for compressibility – these compare well with known values of 20,000 psi, 10 md/cp and “0.01,” providing excellent results. However, in the second inverse calculation assuming an overbalance of zero, that is, using an older inverse model that does not provide for supercharging, the results are poor. We have 22,014 psi as opposed to 20,000 psi for pore pressure. The mobility is 1.0512 md/cp versus a known value of 10 md/cp – a factor of ten discrepancy, although the compressibility is correct to the number of decimal places shown.

Supercharging – Forward Models and Inverse Solutions 91 C:\FT-PTA-SC>SC-DD-INVERSE-2

Software reference, sc-dd-inverse-2.for. Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1 .5 1 5 20028 10 19797 20 19767 2000

OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 5.0000 20028.0000 10.0000 19797.0000 20.0000 19767.0000 2000.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20014.0000 9.5467 0.0606 0.0094 x (cc/FloLineVol)

C:\FT-PTA-SC>SC-DD-INVERSE-2

Software reference, sc-dd-inverse-2.for. Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor:

1 .5 1

92 Formation Testing Volume 3 1st Point Time T1 Pressure P1 2nd Point Time T2 Pressure P2 3rd Point Time T3 Pressure P3 Overbalance pressure

(sec): (psi): (sec): (psi): (sec): (psi): (psi):

5 20028 10 19797 20 19767 0

Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 5.0000 20028.0000 10.0000 19797.0000 20.0000 19767.0000 0.0000

OUTPUT SUMMARY ...

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

22014.0000 1.0512 0.0067 0.0010 x (cc/FloLineVol)

2.5.4 Example DD-4, Qualitative Pressure Trends

It is always instructive, before performing forward and inverse studies, to assess the qualitative effects of overbalance. The work reported here draws upon software reference sc-dd-forward-2-createtables-3B.for. This software automatically and conveniently produces transient pressure responses for different assumed overbalance pressures. For instance, in the following input listing, the “200 psi” is used to create pressure transient responses corresponding to overbalance pressures of 0, 200, 400, 600 and 800 psi. In this drawdown-only example, we observe that the initial pressure at t = 0 contains the effects of high overbalance, and that all pressure responses correctly tend to the same pressure at large times; in this particular example, that time is approximately 240 sec with pressures of 17,668 psi. From the graph in Figure 2.5d, it is clear that as overbalance effects disappear with time, all line graphs grow closer and closer. The rate of this convergence depends on tool, fluid and formation parameters. The effect of overbalance, as expected, is not a simple shift of the static “no overbalance” response.

Supercharging – Forward Models and Inverse Solutions 93 C:\FT-PTA-SC>sc-dd-forward-2-create-tables-3B

Software reference, sc-dd-forward-2-create-tables-3B.for. Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Plot every "NSEC" seconds:

1 1 .00001 20000 200 1 .5 1 1000 5

FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Forward pressure transient predictions for drawdown-only applications, for low mobility, isotropic, supercharged flows where flowline storage is not negligible. Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc):

0.1000E+01 0.1000E+01 0.1000E-04 0.2000E+05 0.2000E+03 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0

P0 (psi) 20000. 19902. 19808. 19718. 19632. 19550. 19471. 19395. 19323. 19253. 19187. 19123. 19062. 19003. 18947. 18894. 18842. 18793. 18746. 18700.

P1 (psi) 20200. 20094. 19992. 19894. 19801. 19711. 19626. 19543. 19465. 19389. 19317. 19248. 19182. 19118. 19057. 18999. 18943. 18889. 18838. 18789.

P2 (psi) 20400. 20285. 20175. 20070. 19969. 19873. 19780. 19692. 19607. 19525. 19447. 19373. 19301. 19233. 19167. 19104. 19044. 18986. 18931. 18878.

P3 (psi) 20600. 20477. 20359. 20246. 20138. 20034. 19935. 19840. 19749. 19661. 19578. 19497. 19421. 19347. 19277. 19209. 19144. 19082. 19023. 18966.

P4 (psi) 20800. 20669. 20543. 20422. 20306. 20196. 20089. 19988. 19890. 19797. 19708. 19622. 19540. 19462. 19386. 19314. 19245. 19179. 19115. 19055.

94 Formation Testing Volume 3 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 39.0 40.0 41.0 42.0 43.0 44.0 45.0 46.0 47.0 48.0 49.0 50.0 51.0 52.0 53.0 54.0 55.0 56.0 57.0 58.0 59.0 60.0 120.0 240.0 360.0 480.0 600.0 720.0 840.0 960.0

18657. 18616. 18576. 18538. 18501. 18466. 18433. 18401. 18370. 18340. 18312. 18285. 18259. 18234. 18211. 18188. 18166. 18145. 18125. 18106. 18088. 18070. 18053. 18037. 18021. 18007. 17992. 17979. 17966. 17953. 17941. 17930. 17919. 17908. 17898. 17888. 17879. 17870. 17862. 17854. 17846. 17682. 17668. 17668. 17668. 17668. 17668. 17668. 17668.

18742. 18697. 18654. 18612. 18573. 18535. 18498. 18463. 18430. 18398. 18367. 18338. 18310. 18283. 18257. 18232. 18209. 18186. 18164. 18143. 18124. 18104. 18086. 18069. 18052. 18036. 18020. 18005. 17991. 17978. 17965. 17952. 17940. 17929. 17918. 17907. 17897. 17888. 17878. 17870. 17861. 17683. 17668. 17668. 17668. 17668. 17668. 17668. 17668.

18827. 18778. 18732. 18687. 18644. 18603. 18564. 18526. 18490. 18456. 18423. 18391. 18361. 18332. 18304. 18277. 18251. 18227. 18204. 18181. 18160. 18139. 18119. 18100. 18082. 18065. 18048. 18032. 18017. 18002. 17988. 17975. 17962. 17949. 17938. 17926. 17915. 17905. 17895. 17886. 17876. 17684. 17668. 17668. 17668. 17668. 17668. 17668. 17668.

18912. 18859. 18809. 18762. 18716. 18672. 18630. 18589. 18550. 18513. 18478. 18444. 18411. 18380. 18350. 18322. 18294. 18268. 18243. 18219. 18195. 18173. 18152. 18132. 18112. 18094. 18076. 18059. 18042. 18027. 18012. 17997. 17983. 17970. 17957. 17945. 17934. 17922. 17912. 17902. 17892. 17685. 17668. 17668. 17668. 17668. 17668. 17668. 17668.

18996. 18941. 18887. 18836. 18787. 18740. 18695. 18652. 18611. 18571. 18533. 18497. 18462. 18429. 18397. 18366. 18337. 18309. 18282. 18256. 18231. 18208. 18185. 18163. 18143. 18123. 18104. 18085. 18068. 18051. 18035. 18020. 18005. 17991. 17977. 17964. 17952. 17940. 17928. 17917. 17907. 17686. 17668. 17668. 17668. 17668. 17668. 17668. 17668.

Note how, for the parameters chosen in this example, the effects of supercharge do not dissipate until about 15 minutes. Of course, the operational objective is not to wait for complete dissipation, but instead, to predict fluid and formation properties using early time pressure data that includes the distortive effects of supercharging.

Supercharging – Forward Models and Inverse Solutions 95

Figure 2.5d. Pressure trends for selected overbalance pressures. 2.5.5 Example DD-5, Qualitative Pressure Trends

Software reference, sc-dd-forward-2-create-tables-3B.for. Additional computations for a wider range of overbalance pressures are provided below without further comment. FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Forward pressure transient predictions for drawdown-only applications, for low mobility, isotropic, supercharged flows where flowline storage is not negligible. Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc):

0.1000E+02 0.1000E+01 0.1000E-04 0.2000E+05 0.1000E+04 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04

Transient time vs probe pressure response ...

96 Formation Testing Volume 3 T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0

P0 (psi) 20000. 19919. 19866. 19831. 19809. 19794. 19785. 19778. 19774. 19772. 19770. 19769. 19768. 19768. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767.

P1 (psi) 21000. 20570. 20290. 20107. 19989. 19911. 19861. 19828. 19807. 19793. 19784. 19778. 19774. 19771. 19770. 19769. 19768. 19768. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767.

P2 (psi) 22000. 21221. 20714. 20384. 20169. 20028. 19937. 19878. 19839. 19814. 19797. 19787. 19780. 19775. 19772. 19770. 19769. 19768. 19768. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767.

P3 (psi) 23000. 21873. 21138. 20660. 20349. 20146. 20014. 19928. 19871. 19835. 19811. 19796. 19786. 19779. 19775. 19772. 19770. 19769. 19768. 19768. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767.

P4 (psi) 24000. 22524. 21562. 20936. 20528. 20263. 20090. 19977. 19904. 19856. 19825. 19805. 19791. 19783. 19777. 19774. 19771. 19770. 19769. 19768. 19768. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767.

Figure 2.5e. Pressure trends for selected overbalance pressures.

Supercharging – Forward Models and Inverse Solutions 97 2.5.6 Example DD-6, “Drawdown-Only” Data with Multiple Inverse Scenarios for 1 md/cp Application C:\FT-PTA-SC>sc-dd-forward-3B

Software reference, sc-dd-forward-3B.for. As in prior examples, we first create the synthetic source probe pressure transient response, in this case using the following data . . . Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Plot every "NSEC" seconds:

1 1 .00001 20000 1000 1 .5 1 1000 50

FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Forward pressure transient predictions for drawdown-only applications, for low mobility, isotropic, supercharged flows where flowline storage is not negligible. Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc):

0.1000E+01 0.1000E+01 0.1000E-04 0.2000E+05 0.1000E+04 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0

P(psi) 21000. 20860. 20726. 20598. 20475. 20357. 20244. 20136. 20032. 19933. 19838. 19747.

(selected for inverse input)

98 Formation Testing Volume 3 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 39.0 40.0 41.0 42.0 43.0 44.0 45.0 46.0 47.0 48.0 49.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0 500.0

19660. 19576. 19496. 19419. 19346. 19275. 19208. 19143. 19081. 19022. 18965. 18911. 18859. 18809. 18761. 18715. 18671. 18629. 18588. 18550. 18513. 18477. 18443. 18411. 18380. 18350. 18321. 18294. 18267. 18242. 18218. 18195. 18173. 18152. 18131. 18112. 18093. 18076. 18058. 17922. 17834. 17776. 17738. 17714. 17669. 17668. 17668. 17668.

(selected for inverse input)

(selected for inverse input)

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

2332.0621 1000.0000 3332.0621

Supercharging – Forward Models and Inverse Solutions 99

Figure 2.5f. Pressure transient response with overbalance. In the first inverse application below, we select three pressure data points as shown, but assume that we know the overbalance pressure of 1,000 psi exactly. Inverse calculation #1 C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for. Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi): OUTPUT SUMMARY ...

1 .5 1 5 20357 15 19419 25 18809 1000

100 Formation Testing Volume 3 Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 5.0000 20357.0000 15.0000 19419.0000 25.0000 18809.0000 1000.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20001.0000 1.0170 0.0644 0.0100 x (cc/FloLineVol)

Predicted results for pore pressure, mobility and compressibility are excellent, agreeing with the inputs assumed in the forward analysis that created the synthetic pressure data. But what if we did not use the inverse supercharge model, or equivalently, “What if conventional inverse methods with zero overbalance were applied?” To answer this question, we simply run our inverse supercharge model with a zero input overbalance pressure. The results are shown below. Inverse calculation #2 C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for. Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi): OUTPUT SUMMARY ...

1 .5 1 5 20357 15 19419 25 18809 0.

Supercharging – Forward Models and Inverse Solutions 101 Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 5.0000 20357.0000 15.0000 19419.0000 25.0000 18809.0000 0.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

21001.0000 0.7110 0.0450 0.0070 x (cc/FloLineVol)

The answer to the foregoing question is clear – pore pressures are not correct, while mobility and compressibility depart substantially from desired values (0.7110 vs 1.0 md/cp and “0.0070” vs “0.0030” 1/psi). We might also consider the follow-up question. “What if we used the inverse supercharge model, but with an incorrect overbalance guess?” This is considered in the calculation below. The assumed overbalance is closer to the correct value rather than being set to zero. Inverse calculation #3 C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for. Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi): OUTPUT SUMMARY ...

1 .5 1 5 20357 15 19419 25 18809 500

102 Formation Testing Volume 3 Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 5.0000 20357.0000 15.0000 19419.0000 25.0000 18809.0000 500.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20501.0000 0.8369 0.0530 0.0082 x (cc/FloLineVol)

The predicted mobility and compressibility are slightly improved, but the pore pressure is still incorrect. It is, however, useful to note that predicted pressure gradient trends (from run to run) may be nonetheless useful in identifying fluid contacts. 2.5.7 Example DD-7, “Drawdown-Only” Data with Multiple Inverse Scenarios for 0.1 md/cp Application C:\FT-PTA-SC>sc-dd-forward-3B

Software reference, sc-dd-forward-3B.for. Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Plot every "NSEC" seconds: Stop - Program terminated.

0.1 1 .00001 20000 250 0.5 1 1 300 50

FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Forward pressure transient predictions for drawdown-only applications, for low mobility, isotropic, supercharged flows where flowline storage is not negligible. Fluid, formation, tool and pumping parameters ...

Supercharging – Forward Models and Inverse Solutions 103 Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc):

0.1000E+00 0.1000E+01 0.1000E-04 0.2000E+05 0.2500E+03 0.5000E+00 0.1000E+01 0.1000E+01 0.1000E+01 0.3000E+03

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 39.0 40.0

P(psi) 20250. 20079. 19912. 19750. 19593. 19440. 19292. 19147. 19007. 18871. 18738. 18609. 18484. 18363. 18245. 18130. 18018. 17910. 17804. 17702. 17602. 17506. 17412. 17320. 17231. 17145. 17061. 16980. 16901. 16824. 16749. 16676. 16606. 16537. 16470. 16405. 16342. 16281. 16222. 16164. 16108.

(selected for inverse input)

(selected for inverse input)

(selected for inverse input)

104 Formation Testing Volume 3 41.0 42.0 43.0 44.0 45.0 46.0 47.0 48.0 49.0 50.0 100.0 200.0 300.0 400.0 410.0 420.0 430.0 440.0 450.0 460.0 470.0 480.0 490.0 500.0

16053. 16000. 15948. 15898. 15850. 15802. 15756. 15711. 15668. 15626. 14519. 14190. 14171. 14170. 14170. 14170. 14170. 14170. 14170. 14170. 14170. 14170. 14170. 14170.

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

5830.1552 250.0000 6080.1552

Figure 2.5g. Pressure transient response with overbalance.

Supercharging – Forward Models and Inverse Solutions 105 Inverse calculation #1 C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for. Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

0.5 1.0 1 5 19440 15 18130 25 17145 250

OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

0.5000 1.0000 1.0000 1.0000 5.0000 19440.0000 15.0000 18130.0000 25.0000 17145.0000 250.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

19999.0000 0.1014 0.0194 0.0030 x (cc/FloLineVol)

Predicted results are excellent for pore pressure, mobility and compressibility in this low mobility calculation. Next we examine the consequences of using an inverse model that does not account for supercharge, that is, one assuming that overbalance pressure is vanishing.

106 Formation Testing Volume 3 Inverse calculation #2 C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for. Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

0.5 1.0 1 5 19440 15 18130 25 17145 0

OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

0.5000 1.0000 1.0000 1.0000 5.0000 19440.0000 15.0000 18130.0000 25.0000 17145.0000 0.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20249.0000 0.0972 0.0186 0.0029 x (cc/FloLineVol)

Predictions for mobility and compressibility are not too far from exact values, although the pore pressure is over-predicted.

Supercharging – Forward Models and Inverse Solutions 107

2.6 Drawdown – Buildup Applications 2.6.1 Example DDBU-1 – Drawdown/Buildup, High Overbalance

With a good degree of confidence from our “drawdown only” inverse supercharge model, we turn to drawdown-buildup applications. Here, our inverse model uses data from the buildup portion of the pressure transient curve. We have selected some interesting examples from numerous that we have tested. In the forward analysis below, with an overbalance of 2,000 psi, our assumed mobility is 10 md/cp and the time at which the piston stops withdrawing fluid is taken as 5 sec. In the pressure transient response, note how the expected buildup is not quite building up – in fact, the curve levels out and is almost flat. Now we turn to inverse calculations. The selected (time, pressure) data points at 5, 8 and 15 sec are 20,028 psi, 20,008 psi and 20,000 psi. In the first inverse calculation using our new supercharge model, we assumed an overbalance of 2,000 psi, and the predicted results 19,999 psi, 8.609 md/cp and “0.0090” for compressibility are excellent when compared with the known values of 20,000 psi, 10 md/cp and “0.01.” In the second inverse calculation, we assume that the overbalance is zero, using the older inverse model that does not account for supercharging. The results are not satisfactory. The pore pressure is correct at 19,999 psi versus a known value of 20,000 psi, however an unacceptable negative mobility of -71.849 md/cp and a negative compressibility are found. C:\FT-PTA-SC>SC-DDBU-FORWARD-4NOPOR

Software reference, SC-DDBU-FORWARD-4NOPOR.FOR. Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Time drawdown ends (sec):

10 1 .00001 20000 2000 1 .5 1 1000 5

g Volume 3 108 Formation Testing

Figure 2.6a. Pressure transient response with overbalance. FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc): Time drawdown ends ............. (sec):

0.1000E+02 0.1000E+01 0.1000E-04 0.2000E+05 0.2000E+04 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04 0.5000E+01

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

P(psi) 22000. 21221. 20714. 20384. 20169. 20028. 20019. 20012. 20008. 20005. 20003.

(selected for inverse input) (selected for inverse input)

g gModels and Inverse Solutions 109 Supercharging –pForward 11.0 12.0 13.0 14.0 15.0 50.0 100.0 200.0 300.0 400.0

20002. 20001. 20001. 20001. 20000. 20000. 20000. 20000. 20000. 20000.

(selected for inverse input)

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

233.2062 2000.0000 2233.2062

C:\FT-PTA-SC>SC-DDBU-INVERSE-2

Software reference, SC-DDBU-INVERSE-2.FOR. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1 .5 1 5 5 20028 8 20008 15 20000 2000

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1.0000 0.5000 1.0000 0.5000 5.0000 5.0000 20028.0000 8.0000 20008.0000 15.0000 20000.0000 2000.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

Excellent results below!

19999.000 8.609 0.0583 0.0090 x (cc/FloLineVol)

110 Formation Testing Volume 3 C:\FT-PTA-SC>SC-DDBU-INVERSE-2

Software reference, SC-DDBU-INVERSE-2.FOR. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1 .5 1 5 5 20028 8 20008 15 20000 0

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1.0000 0.5000 1.0000 0.5000 5.0000 5.0000 20028.0000 8.0000 20008.0000 15.0000 20000.0000 0.0000

Pore pressure and mobility predicted ... Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

Poor results below!

19999.000 -71.849 -0.4866 -0.0754 x (cc/FloLineVol)

Supercharging – Forward Models and Inverse Solutions 111 2.6.2 Example DDBU-2, Drawdown-Buildup, High Overbalance

Here we alter the above forward analysis inputs, to emphasize the buildup part of the pressure transient response – the buildup curve here actually shows an increasing pressure with time. C:\FT-PTA-SC>SC-DDBU-FORWARD-4BNOPOR

Software reference, SC-DDBU-FORWARD-4BNOPOR.FOR (plots more densely, every second, as opposed to Version 4). Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Time drawdown ends (sec):

10 1 .00001 20000 2000 2 .5 1 1000 5

FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc): Time drawdown ends ............. (sec):

0.1000E+02 0.1000E+01 0.1000E-04 0.2000E+05 0.2000E+04 0.2000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04 0.5000E+01

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

P(psi) 22000. 21140. 20580. 20215. 19977. 19823. 19884. 19925. 19951. 19968.

(selected for inverse input)

112 Formation Testing Volume 3 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0

19979. 19986. 19991. 19994. 19996. 19998. 19998. 19999. 19999. 20000. 20000. 20000. 20000. 20000. 20000. 20000.

(selected for inverse input)

(selected for inverse input)

Figure 2.6b. Pressure transient response with overbalance. Inverse calculation #1 C:\FT-PTA-SC>SC-DDBU-INVERSE-2

Software reference, SC-DDBU-INVERSE-2.FOR INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec):

2 .5 1 5

Supercharging – Forward Models and Inverse Solutions 113 1st Point Time T1 Pressure P1 2nd Point Time T2 Pressure P2 3rd Point Time T3 Pressure P3 Overbalance

(sec): (psi): (sec): (psi): (sec): (psi): (psi):

5 19823 10 19979 19 20000 2000 . . .

Correct overbalance used

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

2.0000 0.5000 1.0000 0.5000 5.0000 5.0000 19823.0000 10.0000 19979.0000 19.0000 20000.0000 2000.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20000.000 9.901 0.0640 0.0099 x (cc/FloLineVol)

The foregoing predictsions are excellent, with a pore pressure of 20,000 psi exactly as assumed, plus a mobility of 9.901 md/cp versus an input value of 10 md/cp, while the compressibility is “0.0099” as opposed to “0.01.” Again, this first inverse calculation assumes an overbalance pressure of 2,000 psi. In the next calculation, we take a zero value, that is, use the older inverse model which does not account for supercharging. For this analysis, the pore pressure is accurate, however, the predicted mobility is 23.403 md/cp versus an assumed 10 md/cp, while the compressibility is “0.0235” as opposed to “0.01.” Inverse calculation #2 C:\FT-PTA-SC>SC-DDBU-INVERSE-2

Software reference, SC-DDBU-INVERSE-2.FOR INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor:

2 .5 1

114 Formation Testing Volume 3 Stop time TDD1 1st Point Time T1 Pressure P1 2nd Point Time T2 Pressure P2 3rd Point Time T3 Pressure P3 Overbalance

(sec): (sec): (psi): (sec): (psi): (sec): (psi): (psi):

5 5 19823 10 19979 19 20000 0

Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

2.0000 0.5000 1.0000 0.5000 5.0000 5.0000 19823.0000 10.0000 19979.0000 19.0000 20000.0000 0.0000

OUTPUT SUMMARY ...

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20000.000 23.403 0.1513 0.0235 x (cc/FloLineVol)

2.6.3 Example DDBU-3, Drawdown-Buildup, High Overbalance

In this example, the “buildup curve” actually decreases with time. This part of the complete pressure transient response, more precisely, represents the response when the pump piston has ceased withdrawing fluid from the formation. The decrease in time is a result of high overbalance pressures. C:\FT-PTA-SC>SC-DDBU-FORWARD-4BNOPOR

Software reference, SC-DDBU-FORWARD-4BNOPOR.FOR Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Time drawdown ends (sec):

1 1 .00001 20000 2000 2 .5 1 1000 5

Supercharging – Forward Models and Inverse Solutions 115 FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc): Time drawdown ends ............. (sec):

0.1000E+01 0.1000E+01 0.1000E-04 0.2000E+05 0.2000E+04 0.2000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04 0.5000E+01

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 40.0 50.0 60.0 70.0 80.0

P(psi) 22000. 21720. 21452. 21196. 20950. 20714. 20684. 20655. 20628. 20601. 20576. 20552. 20529. 20507. 20485. 20465. 20445. 20427. 20409. 20392. 20375. 20360. 20344. 20330. 20316. 20303. 20290. 20278. 20266. 20255. 20244. 20159. 20104. 20068. 20044. 20029.

(selected for inverse input)

(selected for inverse input)

(selected for inverse input)

116 Formation Testing Volume 3 90.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0

20019. 20012. 20000. 20000. 20000. 20000. 20000. 20000. 20000. 20000. 20000.

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

4664.1242 2000.0000 6664.1242

Figure 2.6c. Pressure transient response with overbalance. For our first inverse calculation, we assume that we know the overbalance pressure of 2,000 psi. The predictions are excellent, with the pore pressure being 20,002 psi as opposed to 20,000 psi, the mobility being 1.017 md/cp versus 1 md/cp, and the compressibility taking on the value of “0.01” exactly as inputted. In the second inverse calculation, the pore pressure is accurate, however, both the mobility and compressibility take on unacceptable negative values.

Supercharging – Forward Models and Inverse Solutions 117 Inverse calculation #1 C:\FT-PTA-SC>SC-DDBU-INVERSE-2

Software reference, SC-DDBU-INVERSE-2.FOR INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

2 .5 1 5 5 20714 10 20576 20 20375 2000

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

2.0000 0.5000 1.0000 0.5000 5.0000 5.0000 20714.0000 10.0000 20576.0000 20.0000 20375.0000 2000.0000

Pore pressure and mobility predicted ... Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

Excellent results!

20002.000 1.017 0.0645 0.0100 x (cc/FloLineVol)

Inverse calculation #2 C:\FT-PTA-SC>SC-DDBU-INVERSE-2

Software reference, SC-DDBU-INVERSE-2.FOR INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi):

2 .5 1 5 5 20714

118 Formation Testing Volume 3 2nd Point Time T2 Pressure P2 3rd Point Time T3 Pressure P3 Overbalance

(sec): (psi): (sec): (psi): (psi):

10 20576 20 20375 0

Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

2.0000 0.5000 1.0000 0.5000 5.0000 5.0000 20714.0000 10.0000 20576.0000 20.0000 20375.0000 0.0000

OUTPUT SUMMARY ...

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20002.000 ... Bad negative results! -1.287 -0.0815 -0.0126 x (cc/FloLineVol)

2.6.4 Example DDBU-4, Drawdown-buildup, 1 md/cp Calculations

In this example, we illustrate forward and inverse results. The models derived above are incorporated in software references sc-ddbuforward-4NOPOR.for for forward analysis and sc-ddbu-inverse-2.for for inverse analysis. C:\FT-PTA-SC>sc-ddbu-forward-4NOPOR

Software reference, sc-ddbu-forward-4NOPOR.for. Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Time drawdown ends (sec):

1 1 .00001 10000 250 1 .5 1 1000 20

FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL

Supercharging – Forward Models and Inverse Solutions 119 Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc): Time drawdown ends ............. (sec):

0.1000E+01 0.1000E+01 0.1000E-04 0.1000E+05 0.2500E+03 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04 0.2000E+02

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0

P(psi) 10250. 10142. 10038. 9938. 9843. 9752. 9664. 9580. 9500. 9423. 9350. 9279. 9211. 9147. 9085. 9025. 8968. 8914. 8861. 8811. 8763. 8815. 8865. 8912. 8958. 9002. 9044. 9084. 9122. 9159. 9194. 9228. 9261. 9292. 9321. 9350. 9377. 9403. 9428.

(selected for inverse input)

(selected for inverse input)

120 Formation Testing Volume 3 39.0 40.0 41.0 42.0 43.0 44.0 45.0 46.0 47.0 48.0 49.0 50.0 100.0 200.0 300.0 400.0 500.0 700.0 800.0 900.0 1000.0

9452. 9475. 9497. 9518. 9539. 9558. 9577. 9594. 9611. 9628. 9643. 9658. 9960. 9999. 10000. 10000. 10000. 10000. 10000. 10000. 10000.

(selected for inverse input)

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

2332.0621 250.0000 2582.0621

In the double-column listing above, we randomly select pressures at times t = 21, 30 and 40 sec for inverse analysis later (the “21” occurs immediately after tddend = 20, the time when piston withdrawal ends – our models, again, apply to the buildup cycle of the drawdown-buildup curve. The selected points are highlighted in red, while the dynamically significant portion of the entire pressure response is plotted below.

Figure 2.6d. Pressure transient response with overbalance.

Supercharging – Forward Models and Inverse Solutions 121

We now attempt to recover the inputs used in creating the above transient pressure response, however, using only three data values (t1, Pw, #1), (t2, Pw, #2) and (t3, Pw, #3), plus nozzle, pump and overbalance inputs. C:\FT-PTA-SC>sc-ddbu-inverse-2

Software reference, sc-ddbu-inverse-2.for. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1 .5 1 20 21 8815 30 9194 40 9475 250

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1.0000 0.5000 1.0000 0.5000 20.0000 21.0000 8815.0000 30.0000 9194.0000 40.0000 9475.0000 250.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

10000.000 1.011 0.0645 0.0100 x (cc/FloLineVol)

The pore pressure is correctly predicted as 10,000 psi, while the mobility is 1.011 md/cp, for a 1% error. Note that the inverse code does not request flowline volume as an input – it predicts the product VC. However, we had assumed (in the forward analysis) that it is 1,000 cc. The “cc/FloLineVol” term is therefore 1/1,000. If we multiply 0.0100 by 1/1,000 we obtain 0.00001/psi, the correct assumed compressibility. Our methodology applies to all manufacturers’ formation testers. If the flowline volume is available from the vendor, fluid compressibility is available from our model.

122 Formation Testing Volume 3 2.6.5 Example DDBU-5, Drawdown-buildup, 0.1 md/cp Calculations

In the prior example, the forward analysis assumed a permeability of 1 md for a mobility of 1 md/cp and a commonly used overbalance pressure of 250 psi. In the illustration below, we consider a tighter formation with a mobility of 0.1 md/cp and a much higher (but less frequently encountered) overbalance of 1,000 psi. C:\FT-PTA-SC>sc-ddbu-forward-4NOPOR

Software reference, sc-ddbu-forward-4NOPOR.for. Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Time drawdown ends (sec):

0.1 1 .000003 20000 1000 1 .5 1 1000 20

FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc): Time drawdown ends ............. (sec):

0.1000E+00 0.1000E+01 0.3000E-05 0.2000E+05 0.1000E+04 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04 0.2000E+02

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

P(psi) 21000. 20655. 20315. 19979. 19648. 19323. 19001. 18684.

Supercharging – Forward Models and Inverse Solutions 123 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 39.0 40.0 41.0 42.0 43.0 44.0 45.0 46.0 47.0 48.0 49.0 50.0 51.0 52.0 53.0 54.0 55.0 56.0 57.0 58.0 59.0 60.0 70.0 80.0

18372. 18064. 17761. 17462. 17167. 16876. 16589. 16307. 16028. 15754. 15483. 15216. 14953. 15025. 15095. 15165. 15233. 15301. 15368. 15433. 15498. 15562. 15625. 15687. 15748. 15809. 15868. 15927. 15985. 16042. 16098. 16153. 16208. 16262. 16315. 16367. 16419. 16469. 16519. 16569. 16618. 16666. 16713. 16760. 16806. 16851. 16896. 16940. 16983. 17026. 17068. 17110. 17151. 17530. 17859.

(selected for inverse input)

(selectef for inverse input)

(selected for inverse input)

(selected for inverse input)

124 Formation Testing Volume 3 90.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0

18144. 18391. 19615. 19908. 19978. 19995. 19999. 20000. 20000. 20000. 20000.

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

23320.6202 1000.0000 24320.6202

The dynamically significant part of the transient pressure response is shown in Figure 2.6e.

Figure 2.6e. Pressure transient response with overbalance. Two inverse calculations are reported here. In the first illustration, we use pressures from times 21, 30 and 40 sec, noting that piston motions ceased at 20 sec. The predicted mobility and compressibility are excellent and agree with forward analysis inputs. However, the predicted pore pressure exceeds the input value by 26 psi.

Supercharging – Forward Models and Inverse Solutions 125

In the second attempt, we use pressure values from 21, 40 and 60 sec, covering a wider dynamic range for increased accuracy. Note that these values are obtained from the first 60 sec of tool operation, which is significant for the tight, overbalanced formation under consideration. Mobility and compressibility are still excellent – the pore pressure, now at 19,997 psi, is now extremely close to the input value of 20,000 psi. Inverse calculation #1 INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1 .5 1 20 21 15025 30 15625 40 16208 1000

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1.0000 0.5000 1.0000 0.5000 20.0000 21.0000 15025.0000 30.0000 15625.0000 40.0000 16208.0000 1000.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20026.000 0.100 0.0193 0.0030 x (cc/FloLineVol)

126 Formation Testing Volume 3 Inverse calculation #2 C:\FT-PTA-SC>sc-ddbu-inverse-2

Software reference, sc-ddbu-inverse-2.for. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1 .5 1 20 21 15025 40 16208 60 17151 1000

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1.0000 0.5000 1.0000 0.5000 20.0000 21.0000 15025.0000 40.0000 16208.0000 60.0000 17151.0000 1000.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

19997.000 0.101 0.0194 0.0030 x (cc/FloLineVol)

Supercharging – Forward Models and Inverse Solutions 127

2.7 Supercharged Anisotropic Flow Simulation Model In this section, we derive the math model for supercharged anisotropic (that is, transversely isotropic) media. Practical supercharge calculations can be performed for general kh, kv and dip angle following the outlined approach with the assistance of the FT-00 solver. Again, FT-00 does not by itself handle supercharge, but the algorithm here extends its capabilities in a mathematically rigorous and correct manner. Recall from Chapter 5 of Chin et al. (2014) that FT-00 covers both “no skin” and “with skin” limits, but in the zero supercharge limit. We address “zero skin” followed by the problem with skin. Slightly compressible liquids generally satisfy the dimensional formulation kv

2

2

2

2

2

2

P/ z + kh ( P/ x + P/ y ) =

c P/ t

(2.4.1)

P(x,y,z,0) = P0

(2.4.2)

P( ,t) = P0

(2.4.3)

VFR – VC P/ t = Q(t)

(2.4.4)

is Here kh and kv are horizontal and vertical permeabilities, porosity, c is reservoir fluid compressibility, is viscosity, V is flowline volume, C is flowline fluid compressibility, P is Darcy pressure, VFR is the volume flow rate contributed through the sand or matrix rock, and Q is the total volumetric flow rate. In addition, x and y are horizontal coordinates, z is the vertical coordinate, and t is time. Equation 2.4.1 is the governing partial differential equation, Equation 2.4.2 is the initial uniform pressure, Equation 2.4.3 is the farfield boundary condition for pore pressure, while Equation 2.4.4 is the zero-skin pumpout condition at the source probe with flowline volume compressibility accounted for. Physical formulation. We had shown how we can physically account for initial supercharge by modifying only the initial condition in Equation 2.4.2 – how this problem is solved is considered later in this section. Thus, we introduce the complete supercharge formulation kv

2

2

2

2

2

2

P/ z + kh ( P/ x + P/ y ) =

P(x,y,z,0) = P0 + Z/{kv

1/6

kh1/3

2

c P/ t 2

2

(2.5.1) ½

{(x + y )/kh + z /kv} }

(2.5.2)

P( ,t) = P0

(2.5.3)

VFR – VC P/ t = Q(t)

(2.5.4)

128 Formation Testing Volume 3

The modification to the initial condition consists of the single term shown, namely Z/{kv1/6 kh1/3 {(x2 + y2)/kh + z2/kv}½}, which vanishes faraway, when any of x, y or z approach infinity. Thus, at infinity, the initial pressure and the farfield pressure have identical values. We also introduced a dimensionless radius r* = {(x2 + y2)/kh + z2/kv}½ in Chin et al. (2014) for transversely isotropic media to simplify Equation 2.5.1. We then assigned a constant rw* = Rw /(kh1/3kv1/6) to characterize the “ellipsoidal well” model of the pumping source. This implies that at the well we have r* = {(x2 + y2)/kh + z2/kv}½ = Rw /(kh1/3kv1/6). To see that this is reasonable, consider the isotropic limit kh = kv = k. In this case, we have the simplification x2 + y2 + z2 = Rw2 which is exactly the locus of points for a spherical well and the anticipated solution. Initial transforms. In Chin et al. (2014), we introduced a complete set of “starred” dimensionless variables P* = (P(x,y,z,t) – P0)/Pref *

2

2

2

(2.6.1)

½

r = (x + y )/kh + z /kv}

(2.6.2)

*

t = t/tref

(2.6.3)

where Pref and tref are reference values, so that Equation 2.5.1 transforms into the simpler equation 2 *

P / r*2 + 2/r* P*/ r* =

c/tref P*/ t*

(2.7.1)

Next consider how Equation 2.5.2 transforms. This leads to a lengthy P(x,y,z,0) = P0 + Z/{kv1/6 kh1/3 {(x2 + y2)/kh + z2/kv}½} = P0 + Pref P* or Z/{kv1/6 kh1/3 {(x2 + y2)/kh + z2/kv}½} = Pref P*. Equation 2.5.2 transforms into Z/{kv1/6 kh1/3 r*} = Pref P* or P*(r*,t* = 0) = Z/( Pref kv1/6 kh1/3 r*)

(2.7.2)

which vanishes at infinity as desired. Now we examine how Equation 2.5.3 transforms. It is clear from P( ,t) = P0 = P0 + Pref P* that P*( ,t*) = 0

(2.7.3)

We lastly consider how Equation 2.5.4 transforms. Equation 5.49 in Chin et al. (2014) shows that the volume flow rate can be expressed as ½ * 2 * * VFR(t) = (4 r w Pref kv kh / ) ( p / r )w so Equation 2.5.4 becomes * 2

½

*

*

*

*

(4 r w Pref kv kh / ) ( P / r )w – VCPref/tref P / t = Q

(2.7.4)

Supercharging – Forward Models and Inverse Solutions 129

If we compare with the zero supercharge formulation in Chin et al. (2014), we find that Equations 2.7.1 – 2.7.4 above are identical except for the Z contribution in Equation 2.7.2. Again, Z represents the overpressure that the “ellipsoidal well” sees initially due to high pressure effects in the wellbore. While it correctly vanishes faraway, allowing the initial and farfield pressure to be equal, it unfortunately (for mathematical purposes, anyway) leads to a problem that is extremely difficult to solve. Change of variables. We can convert the above problem into an equivalent problem that can be solved without further mathematical simplifications. For now, let us introduce a new “curly ” defined by *

*

= P* – Z / (Pref kv1/6 kh1/3 r ) so that the above Equation 2.7.4 becomes * 2

½

*

*

*

*

(4 r w Prefkv kh/ ) ( / r )w – VCPref/tref / t = Q + 4 kv1/3kh2/3Z/ . Also we find that the farfield boundary condition in Equation 2.7.3 * becomes P*( ,t*) = 0 = * + Z / (Pref kv1/6 kh1/3 r ) which for large values of radius simplifies to *( ,t*) = 0. In addition, the initial condition in Equation 2.7.2, or P*(r*,t* = 0) = Z/( Pref kv1/6 kh1/3 r*), now transforms to * * + Z / (Pref kv1/6 kh1/3 r ) = Z/( Pref kv1/6 kh1/3 r*)* or *(r*,t* = 0) = 0. Finally, the governing parabolic partial differential equation, namely 2 * P / r*2 + 2/r* P*/ r* = c/tref P*/ t* in Equation 2.7.1, after some 2 * * * / r*2 + 2/r* / r* = c/tref / t*. Finally, in algebra, becomes summary, the boundary value problem for 2 * *

*

/ r*2 + 2/r*

*

/ r* =

c/tref

*

*

is

/ t*

(2.8.1)

* *

(r ,t = 0) = 0

(2.8.2)

*

( ,t ) = 0 * 2

½

(4 r w Prefkv kh/ ) (

(2.8.3) *

*

/ r )w – VCPref/tref

*

*

/ t =

(2.8.4a) 1/3

= Q + 4 kv kh2/3Z/ This is exactly the FT-00 zero skin boundary value problem treated in Chin et al. (2014), for a scaled pressure, except that the total flow rate Q is replaced by Q + 4 kv1/3kh2/3Z/ with a constant increase. In other words, to solve Equations 2.5.1 – 2.5.4, we first solve Equations 2.8.1 –

130 Formation Testing Volume 3

2.8.4a using the existing FT-00 engine. Once *

obtained from the earlier transform

*

is available, P* is *

= P* – Z / (Pref kv1/6 kh1/3 r ) or

*

simply P* = * + Z / (Pref kv1/6 kh1/3 r ). Solution with skin effect. The above results for Equations 2.81 – 2.84a apply to the “zero skin” case. The FT-00 formulation with skin effects is also treated in Chin et al. (2014). From Equation 5.114 in that book, the pumpout boundary condition is shown to be * 2

(4 r w Pref kv

1/2

*

*

*

*

kh / ) ( P / r )w - VCPref/tref P / t 1/2

* 2

+ {4 r w VCPref kv

2 *

kh /(ks

*

*

tref)} ( P / t r )w = Q *

*

The left term on the second line is new. Note that P*/ t = / t since the additional term in the transform does not depend on time. Thus, we find that the left term retains the same earlier form since the mixed 2 * * 2 * * derivative P*/ t r = / t r remains unchanged. It follows that the above pumpout condition in terms of Equation 2.8.4, that is, ½

* 2

(4 r w Prefkv kh/ ) ( 1/2

* 2

*

*

*

*

/ r )w – VCPref/tref

+ {4 r w VCPref kv kh /(ks

tref)}(

2

*

is a simple modification of

*

*

/ t

(2.8.4b)

*

/ t r )w = Q + 4 kv1/3kh2/3Z/

This is identical as the above pumpout equation except for the increased right side flow rate where the increase is by a constant amount. Analogously as before, we use the “with skin” solver for FT-00 to * compute and then obtain the physical pressure from P* = * + Z / (Pref *

kv1/6 kh1/3 r ). For the time being, the anisotropic “zero skin” and “with skin” formulations must be solved by hand. The schemes outlined above will be integrated in the future to support automated calculations.

Supercharging – Forward Models and Inverse Solutions 131

2.8 References Barriol, Y., Glaser, K.S., Pop, J. et al., “The Pressures of Drilling and Production,” Schlumberger Oilfield Review, Autumn 2005. Chin, W.C., Formation Invasion – with Applications to Measurement While Drilling, Time Lapse Analysis, and Formation Damage, Gulf Publishing, Houston, 1995. Chin, W.C., Quantitative Methods in Reservoir Engineering, Second Edition, Elsevier, Amsterdam, 2017. Chin, W.C., Zhou, Y., Feng, Y., Yu, Q. and Zhao, L., Formation Testing Pressure Transient and Contamination Analysis, John Wiley & Sons, Hoboken, New Jersey, 2014. Chin, W.C., Zhou, Y., Feng, Y. and Yu, Q., Formation Testing Low Mobility Pressure Transient Analysis, John Wiley & Sons, Hoboken, New Jersey, 2015. Halliburton Staff, “Testing the Tight Gas Reservoir: HostileEnvironment Wireline Formation Tester Reduces NPT in HPHT Boreholes,” available from www.halliburton.com, 2018. Proett, M.A., Chin, W.C., Beique, J.M., Hardin, J.R., Fogal, J.M., Welshans. D. and Gray, G.C., “Methods for Measuring a Formation Supercharge Pressure,” United States Patent No. 7,243,537 B2, awarded July 17, 2007. Proett, M.A., Ma., S.M., Al-Musharfi, N.M. and Berkane, M., “Dynamic Data Analysis with New Automated Workflows for Enhanced Formation Evaluation,” SPE-187040-MS, Society of Petroleum Engineers (SPE) Annual Technical Conference and Exhibition, San Antonio, Texas, October 9-11, 2017. Proett, M.A., Seifert, D., Chin, W.C., Lysen, S. and Sands, P., “Formation Testing in the Dynamic Drilling Environment,” SPWLA 45th Annual Logging Symposium, Noordwijk, The Netherlands, June 6-9, 2004. Rourke, M., Powell, B., Platt, C., Hall, K. and Gardner, A., “A New Hostile Environment Wireline Formation Testing Tool: A Case Study from the Gulf of Thailand,” SPWLA 47th Annual Logging Symposium, Veracruz, Mexico, June 4-7, 2006.

3 Pressure Transient Analysis – Multirate Drawdown and Buildup In this book and our earlier monographs Chin et al. (2014) and Chin et al. (2015), we dealt with simple drawdown and drawdown-buidup applications, where the primary drivers were constant flowrate pumping. In many real world problems, flowrates are hardly constant. For example, a pump may not be able to withdraw or inject fluid at a constant rate because formations are too resistive or because of mechanical limitations – thus, a nonconstant rate constructed from several piecewise constant rates may be required – together with the inverse model that accommodates the new pumping scheme. In several modern formation testing applications, multiple fluid withdrawal and injection sequences may be employed by design. For example, fluids may be sampled and saved in the usual manner and pressure transients may be interpreted for pore pressure, mobility and compressibility. But on the injection cycle, chemicals may be added to reduce viscosity, acids may be administered to increase permeability, additives may be introduced to induce heating or to alter hydrate slurry properties – then sampling is again repeated, together with inverse pressure transient interpretation, to determine the success of these treatments. We emphasize that, in all likelihood, the properties of the resulting fluid mixtures will vary in time – however, these will likely vary slowly so that our methods will provide qualitative assessments. There is a need for multiple drawdown and buildup forward and inverse models – those presented here represent an initial step in the correct direction and are derived in the “zero supercharge” limit. 132

Pressure Transient Analysis 133

3.1 Multirate Drawdown and Buildup Applications In the author’s earlier books Chin et al. (2014) and Chin et al. (2015), and in the present monograph, we emphasized conventional formation tester job planning and reservoir characterization applications. For example, FT-01 provides exact forward pressure simulations for liquids while FT-06 develops similar capabilities (although computationally) for nonlinear gas flows. These calculations are useful in many ways and address important questions. Are drawdowns so large that dissolved gas is released? That unconsolidated sands are further loosened? Do pressure buildups take too long? Are signals measurable at the observation probe in low mobility environments? On the other hand, FT-01 and FT-02 support inverse analyses when steady dual probe pressure drops are available, leading directly to horizontal and vertical mobility estimates. Single probe transient methods described in the earlier books and illustrated in Chapter 4 of the present book show how “effective” or “spherical permeability” (that is, kh2/3kv1/3) plus fluid compressibility can be obtained using unsteady pressure data accurately even in the presence of distortive flowline storage effects. These applications alone demonstrate the importance of simple testing schemes employing long-time, constant flow rate fluid withdrawal, short-term drawdown and also drawdown-buildup approaches. On the other hand, other production applications have been proposed by petroleum engineers that involve more complicated sequences of fluid withdrawals and injections, where the latter can involve chemical or other additive delivery to the formation. In situ changes can alter properties like viscosity and permeability or environmental variables such as pressure and temperature. These in turn affect, possibly, viscosity, permeability and local pressure over time, effects which need to be monitored over time during the course of multiple drawdown and buildup cycles. Such engineering applications motivated the complicated forward and inverse models presented in this chapter. Before discussing their development and validation, we provide summaries of two classes of problems, which we simply refer to as “monitoring and treatment” and “hydrate characterization and production.” The discussions are introductory and brief and not intended to be comprehensive. However, the references should provide a good starting point for readers interested in understanding a rapid changing field emphasizing formation tester usage and certainly the forward and inverse pressure transient algorithms developed in this chapter.

134 Formation Testing Volume 3

3.1.1 Monitoring, Testing, Treatment and Retest Consider a hypothetical formation tester that withdraws fluid for pressure testing (and possibly sampling), discharges the sample or retains it in collection vessels; then, injects fluid into the formation, with or without chemical additives, and waits; and then continues with pressure testing and sample evaluation, with the option to repeat the cycle as many times as desired. The additive might include viscosity reducers, chemicals that react with the formation to produce heat, or to change the constituency of the in-situ fluid, or to remedy formation damage – the possibilities are endless. In any event, the need to predict mobilities, local pressures and compressibilities from a minimum of data obtained during the latest cycle of withdrawal or injection tests is clear. In this chapter, we develop multiple drawdown and buildup forward models for pressure prediction – and present methods to solve the inverse problem using highly transient early time data for as many as eleven interpretation scenarios as shown in Figure 3.3.1a. We understand that applications may deal with fluids whose properties change slowly in time – thus our work represents just a beginning. The multiple flow rates in the figure need not apply to enhanced formation evaluation methods – our models can be used to simulate, for example, mechanical pumping when constant rates cannot be sustained or if oscillatory rates are desired. This section does not focus on formation tester tool design – it addresses multirate forward and inverse models we have developed. Practical applications have been published by other authors, in both “enhanced formation evaluation” and in hydrate characterization and production engineering, and we offer a limited survey of historically significant related papers, emphasizing that our discussion here is brief with no attempt made to provide a comprehensive presentation. The use of forward and reverse injection methods in drillstem and formation testing is well known in the industry and different applications are covered in the (bulleted) papers and patents below. The referenced literature discusses mechanical devices, test procedures, fluid chemical properties, hardware control devices and control diagrams. They apply to injection methods for different downhole applications, to include acidizing, fracturing, chemical reactions to release heat, and so on, and in several instances, describe both liquid and gas applications. The methods described do make use of forward simulations, which calculate pressure transients when fluid, formation and tool properties are given, and inverse calculations, which determine permeabilities, pressures and

Pressure Transient Analysis 135

mobilities when steady-state or transient measured pressures are available – our methods, because they support multiple positive and negative flow rates in any combination, simplify job planning and pressure transient interpretation. Again, other models in this book apply to both liquids and gases, and the reader should refer to them if applications to the methods below are required. The literature cited is drawn from a cross-section of offerings from oil service and operating companies. The references are not presented in any particular order. Crockett, R.K. and Cooper, R.E., “Formation Testing and Stimulation Using Modern Generation Test Tools - The TestAcidize-Test Technique,” SPE Paper 5766, Society of Petroleum Engineers of AIME, American Institute of Mining, Metallurgical and Petroleum Engineers, 1976. Paper offers a historical perspective on use of special test sequences versus simple drawdown or drawdownbuildup. Goodwin, A.R.H. and Hegeman, P.S., “Method and Apparatus for Sampling Formation Fluids,” United States Patent 7,703,317 B2, awarded April 27, 2010. Discusses “multiple retrieval processes,” including increasing the mobility of the formation fluid, plus the use of formation tester injection and heating to increase mobility. Goodwin, A.R.H. and Hegeman, P.S., “Method and Apparatus for Sampling Formation Fluids,” United States Patent 7,845,219 B2, awarded Dec. 7, 2010. Multiple retrieval processes, including increasing the mobility of the formation fluid. Also discusses use of formation tester injection and heating to increase mobility. Kuchuk, F., Ramakrishnan, T.S., Habashy, T.M., Falconer, I., Gokhan, S., Harrigan, E., Goodwin, A., Leising, L. and Mattos, F., “Instrumented Formation Tester for Injecting and Monitoring of Fluids,” United States Patent 8,191,416 B2, awarded June 5, 2012. Describes instrumented formation tester for injecting and monitoring operations for removing mudcake damage. Patent notes that formation testers typically cannot inject fluid into reservoirs and discusses how to remove mudcake for improved injection. Van Hal, R.E.G., Goodwin, A., Mullins, O.C., Hegeman, P.S., Raghuraman, B., Betancourt, S.S., Ayan, C., Vasques, R., Dubost, F.X. and Del Campo, C.S., “Formation Fluid Sampling Tools and Methods Utilizing Chemical Heating,” United States Patent 8,283,174 B2, awarded Oct. 9, 2012. Describes formation tester

136 Formation Testing Volume 3

injection of reactants which create heat energy, thus increasing mobility and decreasing sampling times. From the Abstract – “A formation fluid sampling tool is provided with reactants which are carried downhole and which are combined in order to generate heat energy which is applied to the formation adjacent the borehole. By applying heat energy to the formation, the formation fluids are heated, thereby increasing mobility, and fluid sampling is expedited.” Goodwin, A.R.H., Jones, T., Massie, K.J., Nighswander, J. and Tustin, G., “Methods and Apparatus to Change the Mobility of Formation Fluids Using Thermal and Non-Thermal Stimulation,” United States Patent Application Publication US 2010/0294493 A1, Nov. 25, 2010. Explains formation tester usage for liquids and gases, use of injection methods to support chemical reactions that create heat. Meister, M., Lee. J., Krueger, S. and Niemeyer, E., “Formation Testing Apparatus and Method for Optimizing Draw Down,” United States Patent 7,011,155 B2, awarded March 14, 2006. Patent discusses formation tester drawdown methods and means for improving subsequent tests. Blauch, M.E., McMechan, D.E., Venditto, J.J. and Tanaka, G.L., “Low Permeability Subterranean Formation Testing Methods and Apparatus,” United States Patent 5,263,360, awarded Nov. 23, 1993. Discusses formation testing for low permeability gas reservoirs, multiple fluid injection treatments. Pahmiyer, R.C. and Ringgenberg, P.D., “Methods and Apparatus for Downhole Completion Cleanup,” United States Patent 6,328,103 B1, Dec. 11, 2001. Describes downhole completion clean-up, fluid identification and monitoring. Manke, K.R., Nivens, H.W., Bianco, S., MacPhail, C.M. and Maldonado, R., “Open Hole Formation Testing,” United States Patent 6,622,554 B2, September 23, 2003. Describes open hole formation testing, forward and reverse injection methods. Gilbert, G.N., Ball, D.E., Somers, R.S., Grable, J.L. and Menezes, C.C., “Multi-Purpose Downhole Tool,” United States Patent Application Publication US 2006/0248949 A1, Nov. 9, 2006. This write-up assumes a packer geometry with a reversible pump, which pumps from and into the reservoir, determines formation pressure,

Pressure Transient Analysis 137

also introducing other fluids (e.g., acids, fracture stimulation fluids, mudcake clean-up and formation skin removal, and so on). Procedures to flush out mudcake before sampling are given. Elshahawi, H. and Hashem, M.N., “In-Situ Fluid Compatibility Testing Using a Wireline Formation Tester,” United States Patent Application Publication US 2010/0242586 A1, Sept. 30, 2010. From the Abstract – “A method for performing fluid influx tests in a wellbore (3) traversing a permeable formation (4) to test the compatibility of one or more completion fluids comprises: a) inserting a well test device (1) comprising a straddle packer assembly (6A,6B) into the wellbore (3) such that the straddle packer assembly (6A,6B) separates a well test section (13) from other sections of the well bore; b) performing a first fluid influx test during which the fluid pressure the test section (13) is reduced, pore fluid is induced to flow from the pores of the permeable formation (4) into the test section (13) and fluid influx into the test section (13) is monitored; c) injecting a completion fluid into the test section (13), thereby increasing the fluid pressure within the test section and inducing the completion fluid to flow into the pores of the surrounding formation (4); d) performing a second fluid influx test during which the fluid pressure within the test section (13) is reduced, the completion fluid and pore fluid are induced to flow into the test section, and fluid influx into the test section is monitored; e) comparing fluid influx monitoring data acquired during the first and second fluid influx tests according to step b) and d) to determine any effects of the completion fluid on the influx of formation pore fluid into the test section.” Basically, refers to packer geometries, application of 1st and 2nd influxes, comparisons to determine effects of completion fluids. Different fluids are considered, e.g., xylene injection for organic acid removal, mud acid for sandstone stimulation, hydrochloric acid for carbonate stimulation, use of fracturing fluids for permeability enhancement. Hallmark, B.J., “Well Formation Test-Treat-Test Apparatus and Method,” United States Patent 4,339,948, July 20, 1982. From the Abstract – “Discloses apparatus and method for testing, then treating, then testing the same sealed off region of earth formation within a well bore. Employs a sealing pad arrangement carried by the well tool to seal the test region to permit flow of formation fluid from the region. A fluid sample taking arrangement in the tool is adapted to

138 Formation Testing Volume 3

receive a fluid sample through the sealing pad from the test region and a pressure detector is connected to sense and indicate the build up of pressure from the fluid sample. A treating mechanism in the tool injects a treating fluid into said sealed test region of earth formation. A second fluid sample is taken through the sealing pad while the build up of pressure from the second fluid sample is indicated.” Basically, write-up for the formation tester explains “testing, then treating, then testing” procedure. Sanford, L., “Method and Apparatus for Testing and Treating Well Formations,” United States Patent 4,031,957, awarded June 28, 1977. From the Abstract – “A single trip method of testing and treating a down hole formation in which conventional drill stem testing can be carried out followed by acidizing or other treatment of the formation without the need to remove the drill stem test string, the method employing a well tool in the operating tool string which allows reverse circulation to clean out the tool string following the initial drill stem test, allows the introduction of acidizing fluid into the formation and permits a second reverse circulation following acidizing and subsequent testing, the well tool being operative without the necessity for rotation of the tool string.” Basically, explains single-trip “test, acid, test” procedure in the context of drillstem testing. 3.1.2 Hydrate Characterization and Production Energy available from presently inaccessible hydrate resources worldwide is substantial and significant engineering challenges must be addressed in the coming years. The introductory book on the subject, Hydrate Engineering, Henry L. Doherty series (Book 21, 2001), by E. Dendy Sloan, is offered through the Society of Petroleum Engineers and is recommended reading. A very important application of formation testing in hydrate field development is discussed in recent United States Patent 9291051 B2, “Reservoir pressure testing to determine hydrate composition,” awarded on March 22, 2016 to Keith C. Hester and James J. Howard, assigned to ConocoPhillips. We will quote directly from their well-written patent presentation below with special passages highlighted by the present author in blue italic font.

Pressure Transient Analysis 139 FIELD OF THE INVENTION The present invention relates to a method and system for identifying one or more characteristics within a subterranean reservoir of natural gas.

BACKGROUND OF THE INVENTION A number of hydrocarbons, especially lower boiling-point light hydrocarbons, in porous media or natural gas fluids, are known to form hydrates in conjunction with the water present under a variety of conditions—particularly at a combination of lower temperature and higher pressure. The hydrates are solid crystalline compounds which co-exist with the surrounding porous media or natural gas fluids. Any solids in produced fluids are at least a nuisance for production, handling, and transport of these fluids. It is not uncommon for solid hydrates to cause plugging and/or blockage of pipelines or transfer lines or other conduits, valves and/or safety devices and/or other equipment, resulting in shutdown, loss of production, and risk of explosion or unintended release of hydrocarbons into the environment either on-land or off-shore. Accordingly, hydrocarbon hydrates have been of substantial interest as well as concern to many industries, particularly the petroleum and natural gas industries. Natural gas hydrates are in a class of compounds known as clathrates, and are also referred to as inclusion compounds. Clathrates consist of cage structures formed between a host molecule and a guest molecule. Gas hydrates are generally composed of crystals formed by water host molecules surrounding the hydrocarbon guest molecules. The smaller or lower-boiling hydrocarbon molecules, particularly C1 (methane) to C4 hydrocarbons and their mixtures, are often the most problematic in the oil and gas industry because they form in hydrate or clathrate crystals under a wide range of production conditions. Even certain nonhydrocarbons such as carbon dioxide and hydrogen sulfide are known to form hydrates under the proper conditions. Beyond being a problem for production of hydrocarbons, hydrates are being looked at as a possible energy source. At this time the only known method for determining the composition of a hydrate found in a subterranean reservoir is to monitor the composition of gases released by the dissociation of the hydrate. This is accomplished either by sampling a hydrate-bearing core that was brought to the surface, or by collected gases in the subterranean reservoir. Preservation of hydrate-bearing cores as they are brought to the surface in coring devices is problematic as the surrounding temperatures and pressures fall outside the thermodynamic stability zones. While some hydrate remains in the core there is concern that it does not represent the composition of the original. The collection of gas samples in a borehole with the intent of bringing the sample to the surface for analysis is also difficult, especially in obtaining an uncontaminated sample. Therefore, a need exists for identifying one or more characteristics, including the composition of the actual hydrate, within the subterranean reservoir.

140 Formation Testing Volume 3 SUMMARY OF THE INVENTION In an embodiment, a method for determining one or more characteristics of a subterranean reservoir includes: (a) injecting a releasing agent into the subterranean reservoir; (b) determining an initial pressure within a subterranean reservoir; (c) reducing the pressure within the subterranean reservoir; and (d) stabilizing the pressure in the subterranean reservoir, wherein steps (c)-(d) are repeated. In another embodiment, a method for determining one or more characteristics of a subterranean reservoir includes: (a) inserting a formation testing tool into the subterranean reservoir; (b) allowing the formation testing tool to equilibrate with the subterranean reservoir; (c) injecting a releasing agent into the subterranean reservoir; (d) determining an initial pressure reduction within a subterranean reservoir, wherein the initial pressure is greater than a stability value; (e) reducing the pressure within the subterranean reservoir, wherein the pressure is incrementally reduced; and (f) stabilizing the pressure in the subterranean reservoir, wherein steps (e)-(f) are repeated. In yet another embodiment, a method for determining one of more characteristics of a subterranean reservoir, includes: (a) installing a formation testing tool into the subterranean reservoir; (b) allowing the formation testing tool to equilibrate with the subterranean reservoir; (c) injecting a releasing agent into the subterranean reservoir, wherein the releasing agent reduces the pressure within the subterranean reservoir; (d) determining an initial pressure reduction of the subterranean reservoir, wherein the initial pressure is determined by a gas hydrate stability zone of a pure methane hydrate, wherein the initial pressure is greater than a stability value; (e) reducing the pressure within the subterranean reservoir, wherein the pressure is incrementally reduced; (f) obtaining a series of pressure measurements within the subterranean reservoir, wherein the series of pressure measurements is indicative of at least one characteristic of the subterranean reservoir; and (g) stabilizing the pressure within the subterranean reservoir, wherein steps (e)-(g) are repeated. In a further embodiment, a system for determining hydrate composition including: (a) a subterranean reservoir, wherein the subterranean reservoir is a hydrate bearing subterranean reservoir; (b) a pressure reduction means for incrementally reducing the pressure within the subterranean reservoir; (c) a formation testing tool, wherein the formation testing tool is installed within the subterranean reservoir, wherein the formation testing tool is capable of evaluating the composition of released fluids and gases from the subterranean reservoir, wherein the formation testing tool is capable of evaluating the composition of liquids and gases within the subterranean reservoir; (d) a means for introducing a releasing agent into the subterranean reservoir; and (e) a means for recovering hydrocarbons from the subterranean reservoir.

End of quote.

Pressure Transient Analysis 141

While the foregoing patent addresses reservoir fluid characterization using formation testing, another production focus aims at extraction processes. Numerous references are available on “hydrate slurries,” for example, “Rheological Properties of Methane Hydrate Slurries Formed From AOT + Water + Oil Microemulsions,” by E.B. Webb, C.A. Koh and M.W. Liberatore, in Langmuir 2013, 29 (35), pp. 10997-11004, ACS Publications, American Chemical Society. From their paper, The in situ formation and flow properties of methane hydrates formed from water-in-oil microemulsions composed of water, dodecane, and aerosol OT surfactant (AOT) were studied using a unique high pressure rheometer. AOT microemulsions have high stability (order of months), well-characterized composition, and yield reproducible results compared to hydrate studies in water-in-crude oil emulsions. Viscosity increases on the order of minutes upon hydrate formation, and then decreases on the order of hours. If significant unconverted water remained after the initial formation event, then viscosity increases for a time as methane slowly dissolves and converts additional water to hydrate. In addition to transient formation measurements, yield stresses and flow curves are measured for a set of experimental conditions. Hydrate slurry viscosity and yield stress increase with increasing water volume fraction, increasing initial pressure, decreasing temperature, and decreasing formation shear rate.

End of quote. It is clear that hydrate production via slurry transport (formed within the reservoir) will depend significantly on rheological properties like viscosity, compressibility and pressure, and similarly, these properties will be important to assessing the mobility of hydrate mixtures in petroleum reservoirs as well. Information related to such evolving properties is important as transients evolve, and multiple drawdown and buildup methods that accommodate such rapidly varying environments are required to support credible interpretation models. Numerous excellent references on hydrate characterization and hydrate slurries fluids engineering are available in the literature and the reader is encouraged to study these applications and possible linkages to further formation testing development.

142 Formation Testing Volume 3

3.2 Detailed Validations with Exact Solutions Here, we demonstrate that the inverse algorithms for Models 1 – 11 in Figure 3.2.1a are correct. These are succinctly summarized by a simple convention. The white rectangular areas represent volume flow rate histories, e.g., the vertically plotted flow rate “Q” multiplied by the horizontal time “T” gives the total volume pumped for that interval. These flow rates are all “piecewise constant,” that is, they have the appearance of rectangles as shown and not triangles or trapezoids. And although they are all shown as positive, they can, in fact, generally alternate between positive and negative, with intervening zeros. Our convention is clear from Models 1 and 2. In Model 1, three “dots” are shown beneath the white rectangle. This indicates that data are taken with pumping in progress. If the flow rate is positive, we would be measuring during a “drawdown” or pressure decrease; if a negative rate, the measurement is taken during a “negative drawdown” (this is a pressure increase, however, we will not use the term “buildup” in this context). On the other hand, consider Model 2, where the three dots are shown to the right (that is, ahead in time) of the white rectangle. If the white rectangular height is positive, the pressure would have been decreasing during pumping or drawing down; then, the measurement (shown by the dots) would suggest a buildup or increase after pumping ceases. If the height is negative, the initial pressure would be increasing, the right dots now suggesting a “negative buildup” or pressure decrease. Analytical approach. Suppose a solution Pbasic = F(Q,t) represents the pressure response for a formation tester with a flowrate that is vanishing when t 0 but taking on a nonzero Q for t 0. Then, if we superpose to it the function Pbasic(– Q, t – t*), which represents a zero rate for t t* and a reversed – Q for t t*, the summed pressure response would correspond to a flowrate that is Q in 0 t t*. Now imagine that this process is repeated for different Q’s at different increasing instances in time – this is the “simple” solution process we have undertaken to construct multiple drawdown and buildup solutions, with positive, zero or negative flow rates, over non-equidistant time intervals. Of course, the algebra is nontrivial – a single solution requires pages of math derivation, given that Pbasic may be our complex complementary error function solution or its exponential or rational polynomial approximation (see Chin et al. (2014) for analytical details). In this chapter, we explain physical applications and numerical validations.

Pressure Transient Analysis 143

Figure 3.2.1a. Eleven general drawdown-buildup inverse models.

144 Formation Testing Volume 3

3.2.1

Validation of PTA-App-01 Inverse Model

Throughout all of our validations, we follow the same procedure. We create exact synthetic forward pressure data using our FT-00 simulator for liquids, which is based on closed form analytical solutions. From this solution, with all input data shown on the FT-00 screen, three data points (indicated by “black dots” on the yellow diagram and highlighted in red in tabulated results) are chosen for input into the inverse software. Then, the inverse model is run and we aim at recovering the input pore pressure, mobility and fluid compressibility. Because the mathematical equations used in developing FT-00 and each of Models 1 – 11 are completely different, our success would point to the correctness and robustness of the approach. We now begin our validation with Model 1.

Figure 3.2.1b. Model 1 function.

Pressure Transient Analysis 145

Figure 3.2.1c. FT-00 exact inputs. Our “inverse model” is an approximate one designed to predict pore pressure, spherical mobility and fluid compressibility in low mobility applications where flowline storage cannot be neglected. The method uses single-probe pressure transient data typically obtained during the first minute of logging – pressures which are highly unsteady and not steady-state. It is validated using pressure transient data created by our exact “forward model” FT-00 (immediately below) where formation and fluid properties, tool constants, reservoir conditions and pumping schedules are defined. Three time-pressure data points are arbitrarily selected (red from the output listing) and inputted into the approximate

146 Formation Testing Volume 3

inverse model. Validations performed in the “black screen, DOS window” show that predicted pore pressure, spherical mobility and fluid compressibility are consistent with those assumed in FT-00. The reader is invited to use different time and pressure inputs to evaluate the method. Different data points give slightly different predictions because the method is approximate. Times between data points should not be too close, but preferably, several seconds apart – the wider, the more likely predictions will be more accurate. The request in the “black screen,” that is, “Use decimals after integers” no longer applies to our latest software. FT-00 predictions for transient pressure appear in Courier New font.

Figure 3.2.1d. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s) 0.000E+00 0.600E+00 0.120E+01 0.180E+01 0.240E+01 0.300E+01

Rate (cc/s) 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01

Ps* (psi) 0.25000E+05 0.24803E+05 0.24609E+05 0.24419E+05 0.24233E+05 0.24051E+05

Pr* (psi) 0.25000E+05 0.25000E+05 0.24999E+05 0.24993E+05 0.24982E+05 0.24968E+05

Ps**(psi) 0.00000E+00 -0.19745E+03 -0.39088E+03 -0.58059E+03 -0.76674E+03 -0.94946E+03

Pr**(psi) 0.00000E+00 -0.53687E-02 -0.10714E+01 -0.67962E+01 -0.17797E+02 -0.32437E+02

Pr**/Ps** ----------0.27190E-04 0.27412E-02 0.11706E-01 0.23211E-01 0.34163E-01

Pressure Transient Analysis 147 0.360E+01 0.420E+01 0.480E+01 0.540E+01 0.600E+01 0.660E+01 0.720E+01 0.780E+01 0.840E+01 0.900E+01 0.960E+01 . . .

0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01

0.23871E+05 0.23695E+05 0.23522E+05 0.23352E+05 0.23185E+05 0.23022E+05 0.22861E+05 0.22703E+05 0.22547E+05 0.22395E+05 0.22245E+05

0.24951E+05 0.24933E+05 0.24915E+05 0.24898E+05 0.24880E+05 0.24863E+05 0.24847E+05 0.24831E+05 0.24816E+05 0.24802E+05 0.24788E+05

-0.11288E+04 -0.13050E+04 -0.14779E+04 -0.16477E+04 -0.18145E+04 -0.19784E+04 -0.21393E+04 -0.22974E+04 -0.24526E+04 -0.26052E+04 -0.27550E+04

-0.49118E+02 -0.66745E+02 -0.84639E+02 -0.10240E+03 -0.11979E+03 -0.13667E+03 -0.15299E+03 -0.16872E+03 -0.18384E+03 -0.19838E+03 -0.21234E+03

0.43512E-01 0.51147E-01 0.57270E-01 0.62144E-01 0.66014E-01 0.69083E-01 0.71515E-01 0.73440E-01 0.74957E-01 0.76147E-01 0.77072E-01

Figure 3.2.1e. Inverse model screen. Predicted values are as follows, (1) pore pressure is 24,981 psi (as opposed to 25,000 psi), and (2) mobility is 0.103 md/cp versus 0.100 md/cp. If the formation tester flowline volume is known, and in this case its value is 1,000 cc, then fluid compressibility is also available as (0.0031)(1/1,000) or 0.000003/psi in agreement with FT-00 assumptions. The inverse model, requiring only three data points, separated here by only 3.0 sec in a tight 0.1 md/cp formation, provides excellent results.

148 Formation Testing Volume 3

3.2.2

Validation of PTA-App-02 Inverse Model

Figure 3.2.2a. Model 2 function. Our “inverse model” is an approximate one designed to predict pore pressure, spherical mobility and fluid compressibility in low mobility applications where flowline storage cannot be neglected. The method uses single-probe pressure transient data typically obtained during the first minute of logging – pressures which are highly unsteady and not steady-state. It is validated using pressure transient data created by our exact “forward model” FT-00 (immediately below) where formation and fluid properties, tool constants, reservoir conditions and pumping schedules are defined. Three time-pressure data points are arbitrarily selected (red from the output listing) and inputted into the approximate inverse model. Validations performed in the “black screen, DOS window” show that predicted pore pressure, spherical mobility and fluid compressibility are consistent with those assumed in FT-00. The reader

Pressure Transient Analysis 149

is invited to use different time and pressure inputs to evaluate the method. Different data points will give slightly different predictions. Times should not be too close, but preferably, several seconds apart. The request in the “black screen,” that is, “Use decimals after integers” no longer applies to our latest software.

Validation No. 1

Figure 3.2.2b. FT-00 exact inputs.

150 Formation Testing Volume 3

Figure 3.2.2c. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s) 0.000E+00 0.600E+00 0.120E+01 0.180E+01 0.240E+01 0.300E+01 0.360E+01 0.420E+01 0.480E+01 0.540E+01 0.600E+01 0.660E+01 0.720E+01 0.780E+01 0.840E+01 0.900E+01 0.960E+01 0.102E+02 0.108E+02 0.114E+02 0.120E+02 0.126E+02 0.132E+02 0.138E+02 0.144E+02 0.150E+02

Rate (cc/s) 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

Ps* (psi) 0.25000E+05 0.24803E+05 0.24609E+05 0.24419E+05 0.24233E+05 0.24051E+05 0.23871E+05 0.23695E+05 0.23522E+05 0.23352E+05 0.23185E+05 0.23022E+05 0.22861E+05 0.22703E+05 0.22547E+05 0.22395E+05 0.22245E+05 0.22164E+05 0.22215E+05 0.22265E+05 0.22314E+05 0.22362E+05 0.22409E+05 0.22455E+05 0.22500E+05 0.22544E+05

Pr* (psi) 0.25000E+05 0.25000E+05 0.24999E+05 0.24993E+05 0.24982E+05 0.24968E+05 0.24951E+05 0.24933E+05 0.24915E+05 0.24898E+05 0.24880E+05 0.24863E+05 0.24847E+05 0.24831E+05 0.24816E+05 0.24802E+05 0.24788E+05 0.24774E+05 0.24761E+05 0.24751E+05 0.24747E+05 0.24748E+05 0.24753E+05 0.24759E+05 0.24767E+05 0.24775E+05

Ps**(psi) 0.00000E+00 -0.19745E+03 -0.39088E+03 -0.58059E+03 -0.76674E+03 -0.94946E+03 -0.11288E+04 -0.13050E+04 -0.14779E+04 -0.16477E+04 -0.18145E+04 -0.19784E+04 -0.21393E+04 -0.22974E+04 -0.24526E+04 -0.26052E+04 -0.27550E+04 -0.28360E+04 -0.27846E+04 -0.27345E+04 -0.26856E+04 -0.26378E+04 -0.25910E+04 -0.25452E+04 -0.25003E+04 -0.24563E+04

Pr**(psi) 0.00000E+00 -0.53687E-02 -0.10714E+01 -0.67962E+01 -0.17797E+02 -0.32437E+02 -0.49118E+02 -0.66745E+02 -0.84639E+02 -0.10240E+03 -0.11979E+03 -0.13667E+03 -0.15299E+03 -0.16872E+03 -0.18384E+03 -0.19838E+03 -0.21234E+03 -0.22574E+03 -0.23855E+03 -0.24865E+03 -0.25294E+03 -0.25196E+03 -0.24749E+03 -0.24098E+03 -0.23340E+03 -0.22535E+03

Pr**/Ps** ----------0.27190E-04 0.27412E-02 0.11706E-01 0.23211E-01 0.34163E-01 0.43512E-01 0.51147E-01 0.57270E-01 0.62144E-01 0.66014E-01 0.69083E-01 0.71515E-01 0.73440E-01 0.74957E-01 0.76147E-01 0.77072E-01 0.79600E-01 0.85667E-01 0.90929E-01 0.94182E-01 0.95517E-01 0.95516E-01 0.94680E-01 0.93349E-01 0.91741E-01

Pressure Transient Analysis 151 0.156E+02 0.162E+02 0.168E+02 0.174E+02 0.180E+02 0.186E+02 0.192E+02 0.198E+02 0.204E+02 . . .

0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

0.22587E+05 0.22629E+05 0.22671E+05 0.22711E+05 0.22751E+05 0.22790E+05 0.22829E+05 0.22867E+05 0.22904E+05

0.24783E+05 0.24791E+05 0.24799E+05 0.24806E+05 0.24814E+05 0.24821E+05 0.24827E+05 0.24834E+05 0.24840E+05

-0.24132E+04 -0.23709E+04 -0.23294E+04 -0.22887E+04 -0.22488E+04 -0.22096E+04 -0.21712E+04 -0.21335E+04 -0.20965E+04

-0.21716E+03 -0.20907E+03 -0.20118E+03 -0.19356E+03 -0.18625E+03 -0.17926E+03 -0.17260E+03 -0.16624E+03 -0.16019E+03

0.89990E-01 0.88181E-01 0.86364E-01 0.84572E-01 0.82823E-01 0.81129E-01 0.79493E-01 0.77920E-01 0.76408E-01

Figure 3.2.2d. Inverse model screen. The pore pressure is 25,003 psi compared to 25,000 psi known, and the mobility is predicted as 0.105 md/cp versus 0.100 md/cp inputted. In the present case, the flow line volume is known to be 1,000 cc so that the fluid compressibility is (0.0030)(1/1,000) or 0.000003/psi in agreement with the value assumed in FT-00. Excellent results!

152 Formation Testing Volume 3

Validation No. 2

Figure 3.2.2e. FT-00 exact inputs.

Pressure Transient Analysis 153

Figure 3.2.2f. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.500E+00

0.10000E+01

0.24835E+05

0.25000E+05 -0.16484E+03 -0.67538E-03

0.40972E-05

0.100E+01

0.10000E+01

0.24673E+05

0.25000E+05 -0.32682E+03 -0.36387E+00

0.11134E-02

0.150E+01

0.10000E+01

0.24514E+05

0.24997E+05 -0.48618E+03 -0.32158E+01

0.66145E-02

0.200E+01

0.10000E+01

0.24357E+05

0.24990E+05 -0.64303E+03 -0.99445E+01

0.15465E-01

0.250E+01

0.10000E+01

0.24203E+05

0.24980E+05 -0.79743E+03 -0.20033E+02

0.25122E-01

0.300E+01

0.10000E+01

0.24051E+05

0.24968E+05 -0.94946E+03 -0.32437E+02

0.34163E-01

0.350E+01

0.10000E+01

0.23901E+05

0.24954E+05 -0.10992E+04 -0.46248E+02

0.42075E-01

0.400E+01

0.10000E+01

0.23753E+05

0.24939E+05 -0.12466E+04 -0.60813E+02

0.48782E-01

0.450E+01

0.10000E+01

0.23608E+05

0.24924E+05 -0.13918E+04 -0.75689E+02

0.54381E-01

0.500E+01

0.00000E+00

0.23465E+05

0.24909E+05 -0.15349E+04 -0.90588E+02

0.59020E-01

0.550E+01

0.00000E+00

0.23489E+05

0.24895E+05 -0.15109E+04 -0.10532E+03

0.69709E-01

0.600E+01

0.00000E+00

0.23512E+05

0.24881E+05 -0.14877E+04 -0.11942E+03

0.80271E-01

0.650E+01

0.00000E+00

0.23535E+05

0.24869E+05 -0.14651E+04 -0.13068E+03

0.89196E-01

0.700E+01

0.00000E+00

0.23557E+05

0.24862E+05 -0.14430E+04 -0.13767E+03

0.95411E-01

154 Formation Testing Volume 3 0.750E+01

0.00000E+00

0.23579E+05

0.24859E+05 -0.14213E+04 -0.14090E+03

0.99136E-01

0.800E+01

0.00000E+00

0.23600E+05

0.24859E+05 -0.14000E+04 -0.14139E+03

0.10099E+00

0.850E+01

0.00000E+00

0.23621E+05

0.24860E+05 -0.13791E+04 -0.14006E+03

0.10156E+00

0.900E+01

0.00000E+00

0.23641E+05

0.24862E+05 -0.13586E+04 -0.13756E+03

0.10126E+00

0.950E+01

0.00000E+00

0.23662E+05

0.24866E+05 -0.13384E+04 -0.13436E+03

0.10039E+00

0.100E+02

0.00000E+00

0.23681E+05

0.24869E+05 -0.13186E+04 -0.13075E+03

0.99153E-01

0.105E+02

0.00000E+00

0.23701E+05

0.24873E+05 -0.12992E+04 -0.12692E+03

0.97694E-01

0.110E+02

0.00000E+00

0.23720E+05

0.24877E+05 -0.12800E+04 -0.12301E+03

0.96102E-01

0.115E+02

0.00000E+00

0.23739E+05

0.24881E+05 -0.12612E+04 -0.11911E+03

0.94442E-01

0.120E+02

0.00000E+00

0.23757E+05

0.24885E+05 -0.12427E+04 -0.11527E+03

0.92755E-01

0.125E+02

0.00000E+00

0.23776E+05

0.24888E+05 -0.12245E+04 -0.11151E+03

0.91070E-01

0.130E+02

0.00000E+00

0.23793E+05

0.24892E+05 -0.12066E+04 -0.10787E+03

0.89406E-01

0.135E+02

0.00000E+00

0.23811E+05

0.24896E+05 -0.11889E+04 -0.10436E+03

0.87776E-01

0.140E+02

0.00000E+00

0.23828E+05

0.24899E+05 -0.11716E+04 -0.10098E+03

0.86188E-01

0.145E+02

0.00000E+00

0.23845E+05

0.24902E+05 -0.11545E+04 -0.97723E+02

0.84645E-01

0.150E+02

0.00000E+00

0.23862E+05

0.24905E+05 -0.11377E+04 -0.94601E+02

0.83151E-01

0.155E+02

0.00000E+00

0.23879E+05

0.24908E+05 -0.11212E+04 -0.91606E+02

0.81706E-01

0.160E+02

0.00000E+00

0.23895E+05

0.24911E+05 -0.11049E+04 -0.88734E+02

0.80311E-01

0.165E+02

0.00000E+00

0.23911E+05

0.24914E+05 -0.10888E+04 -0.85980E+02

0.78964E-01

0.170E+02

0.00000E+00

0.23927E+05

0.24917E+05 -0.10731E+04 -0.83340E+02

0.77665E-01

0.175E+02

0.00000E+00

0.23942E+05

0.24919E+05 -0.10575E+04 -0.80808E+02

0.76412E-01

0.180E+02

0.00000E+00

0.23958E+05

0.24922E+05 -0.10422E+04 -0.78378E+02

0.75202E-01

0.185E+02

0.00000E+00

0.23973E+05

0.24924E+05 -0.10272E+04 -0.76046E+02

0.74035E-01

0.190E+02

0.00000E+00

0.23988E+05

0.24926E+05 -0.10123E+04 -0.73807E+02

0.72907E-01

0.195E+02

0.00000E+00

0.24002E+05

0.24928E+05 -0.99774E+03 -0.71656E+02

0.71819E-01

0.200E+02

0.00000E+00

0.24017E+05

0.24930E+05 -0.98335E+03 -0.69588E+02

0.70766E-01

0.205E+02

0.00000E+00

0.24031E+05

0.24932E+05 -0.96919E+03 -0.67600E+02

0.69748E-01

0.210E+02

0.00000E+00

0.24045E+05

0.24934E+05 -0.95524E+03 -0.65686E+02

0.68764E-01

0.215E+02

0.00000E+00

0.24058E+05

0.24936E+05 -0.94150E+03 -0.63844E+02

0.67810E-01

0.220E+02

0.00000E+00

0.24072E+05

0.24938E+05 -0.92797E+03 -0.62069E+02

0.66886E-01

0.225E+02

0.00000E+00

0.24085E+05

0.24940E+05 -0.91465E+03 -0.60358E+02

0.65991E-01

0.230E+02

0.00000E+00

0.24098E+05

0.24941E+05 -0.90152E+03 -0.58709E+02

0.65122E-01

0.235E+02

0.00000E+00

0.24111E+05

0.24943E+05 -0.88860E+03 -0.57118E+02

0.64279E-01

0.240E+02

0.00000E+00

0.24124E+05

0.24944E+05 -0.87586E+03 -0.55582E+02

0.63459E-01

0.245E+02

0.00000E+00

0.24137E+05

0.24946E+05 -0.86332E+03 -0.54098E+02

0.62663E-01

Pressure Transient Analysis 155

Figure 3.2.2g. Inverse model screen. Here, the predicted pore pressure is 24,878 psi versus 25,000 psi. The mobility is 0.127 md/cp versus 0.100 md/cp. Good qualitative agreement.

156 Formation Testing Volume 3

3.2.3

Validation of PTA-App-03 Inverse Model

Figure 3.2.3a. Model 3 function. Our “inverse model” is an approximate one designed to predict pore pressure, spherical mobility and fluid compressibility in low mobility applications where flowline storage cannot be neglected. The method uses single-probe pressure transient data typically obtained during the first minute of logging – pressures which are highly unsteady and not steady-state.. It is validated using pressure transient data created by our exact “forward model” FT-00 (immediately below) where formation and fluid properties, tool constants, reservoir conditions and pumping schedules are defined. Three time-pressure data points are arbitrarily selected (red from the output listing) and inputted into the approximate inverse model. Validations performed in the “black screen, DOS

Pressure Transient Analysis 157

window” show that predicted pore pressure, spherical mobility and fluid compressibility are consistent with those assumed in FT-00. The reader is invited to use different time and pressure inputs to evaluate the method. Different data points will give slightly different predictions. Times should not be too close, but preferably, several seconds apart. The request in the “black screen,” that is, “Use decimals after integers” no longer applies to our latest software.

Validation No. 1

Figure 3.2.3b. FT-00 exact inputs.

158 Formation Testing Volume 3

Figure 3.2.3c. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s) 0.000E+00 0.600E+00 0.120E+01 0.180E+01 0.240E+01 0.300E+01 0.360E+01 0.420E+01 0.480E+01 0.540E+01 0.600E+01 0.660E+01 0.720E+01 0.780E+01 0.840E+01 0.900E+01 0.960E+01 0.102E+02 0.108E+02 0.114E+02 0.120E+02 0.126E+02 0.132E+02 0.138E+02 0.144E+02 0.150E+02

Rate (cc/s) 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.30000E+01 0.30000E+01 0.30000E+01 0.30000E+01 0.30000E+01 0.30000E+01 0.30000E+01 0.30000E+01 0.30000E+01

Ps* (psi) 0.25000E+05 0.24803E+05 0.24609E+05 0.24419E+05 0.24233E+05 0.24051E+05 0.23871E+05 0.23695E+05 0.23522E+05 0.23352E+05 0.23185E+05 0.23022E+05 0.22861E+05 0.22703E+05 0.22547E+05 0.22395E+05 0.22245E+05 0.21965E+05 0.21428E+05 0.20902E+05 0.20385E+05 0.19878E+05 0.19380E+05 0.18891E+05 0.18411E+05 0.17939E+05

Pr* (psi) 0.25000E+05 0.25000E+05 0.24999E+05 0.24993E+05 0.24982E+05 0.24968E+05 0.24951E+05 0.24933E+05 0.24915E+05 0.24898E+05 0.24880E+05 0.24863E+05 0.24847E+05 0.24831E+05 0.24816E+05 0.24802E+05 0.24788E+05 0.24774E+05 0.24761E+05 0.24744E+05 0.24717E+05 0.24681E+05 0.24639E+05 0.24594E+05 0.24548E+05 0.24503E+05

Ps**(psi) 0.00000E+00 -0.19745E+03 -0.39088E+03 -0.58059E+03 -0.76674E+03 -0.94946E+03 -0.11288E+04 -0.13050E+04 -0.14779E+04 -0.16477E+04 -0.18145E+04 -0.19784E+04 -0.21393E+04 -0.22974E+04 -0.24526E+04 -0.26052E+04 -0.27550E+04 -0.30349E+04 -0.35716E+04 -0.40981E+04 -0.46147E+04 -0.51219E+04 -0.56199E+04 -0.61089E+04 -0.65892E+04 -0.70609E+04

Pr**(psi) 0.00000E+00 -0.53687E-02 -0.10714E+01 -0.67962E+01 -0.17797E+02 -0.32437E+02 -0.49118E+02 -0.66745E+02 -0.84639E+02 -0.10240E+03 -0.11979E+03 -0.13667E+03 -0.15299E+03 -0.16872E+03 -0.18384E+03 -0.19838E+03 -0.21234E+03 -0.22574E+03 -0.23877E+03 -0.25568E+03 -0.28277E+03 -0.31904E+03 -0.36099E+03 -0.40577E+03 -0.45152E+03 -0.49711E+03

Pr**/Ps** ----------0.27190E-04 0.27412E-02 0.11706E-01 0.23211E-01 0.34163E-01 0.43512E-01 0.51147E-01 0.57270E-01 0.62144E-01 0.66014E-01 0.69083E-01 0.71515E-01 0.73440E-01 0.74957E-01 0.76147E-01 0.77072E-01 0.74382E-01 0.66851E-01 0.62390E-01 0.61276E-01 0.62289E-01 0.64234E-01 0.66422E-01 0.68524E-01 0.70403E-01

Pressure Transient Analysis 159 0.156E+02 0.162E+02 0.168E+02 0.174E+02 0.180E+02 0.186E+02 0.192E+02 0.198E+02 . . .

0.30000E+01 0.30000E+01 0.30000E+01 0.30000E+01 0.30000E+01 0.30000E+01 0.30000E+01 0.30000E+01

0.17476E+05 0.17021E+05 0.16574E+05 0.16134E+05 0.15703E+05 0.15279E+05 0.14862E+05 0.14453E+05

0.24458E+05 0.24415E+05 0.24372E+05 0.24332E+05 0.24292E+05 0.24254E+05 0.24218E+05 0.24183E+05

-0.75242E+04 -0.79793E+04 -0.84264E+04 -0.88656E+04 -0.92971E+04 -0.97210E+04 -0.10138E+05 -0.10547E+05

-0.54189E+03 -0.58549E+03 -0.62771E+03 -0.66847E+03 -0.70773E+03 -0.74552E+03 -0.78187E+03 -0.81683E+03

0.72019E-01 0.73376E-01 0.74493E-01 0.75400E-01 0.76124E-01 0.76692E-01 0.77127E-01 0.77449E-01

Figure 3.2.3d. Inverse model screen. Very good results are obtained above. A pore pressure of 24,979 psi is found versus a 25,000 psi known value, and the mobility is 0.106 md/cp versus 0.100 inputted. With a known flowline volume of 1,000 cc from FT-00, we predict 0.0000030/psi consistently with FT-00. Now let us pursue a second validation, where Q2 is now injecting as opposed to withdrawing fluid.

160 Formation Testing Volume 3

Validation No. 2

Figure 3.2.3e. FT-00 exact inputs.

Pressure Transient Analysis 161

Figure 3.2.3f. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s) 0.000E+00 0.600E+00 0.120E+01 0.180E+01 0.240E+01 0.300E+01 0.360E+01 0.420E+01 0.480E+01 0.540E+01 0.600E+01 0.660E+01 0.720E+01 0.780E+01 0.840E+01 0.900E+01 0.960E+01 0.102E+02 0.108E+02 0.114E+02 0.120E+02 0.126E+02 0.132E+02 0.138E+02 0.144E+02 0.150E+02

Rate (cc/s) 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 -0.30000E+01 -0.30000E+01 -0.30000E+01 -0.30000E+01 -0.30000E+01 -0.30000E+01 -0.30000E+01 -0.30000E+01 -0.30000E+01

Ps* (psi) 0.25000E+05 0.24803E+05 0.24609E+05 0.24419E+05 0.24233E+05 0.24051E+05 0.23871E+05 0.23695E+05 0.23522E+05 0.23352E+05 0.23185E+05 0.23022E+05 0.22861E+05 0.22703E+05 0.22547E+05 0.22395E+05 0.22245E+05 0.22363E+05 0.23002E+05 0.23629E+05 0.24243E+05 0.24846E+05 0.25438E+05 0.26018E+05 0.26589E+05 0.27148E+05

Pr* (psi) 0.25000E+05 0.25000E+05 0.24999E+05 0.24993E+05 0.24982E+05 0.24968E+05 0.24951E+05 0.24933E+05 0.24915E+05 0.24898E+05 0.24880E+05 0.24863E+05 0.24847E+05 0.24831E+05 0.24816E+05 0.24802E+05 0.24788E+05 0.24774E+05 0.24762E+05 0.24758E+05 0.24777E+05 0.24815E+05 0.24866E+05 0.24924E+05 0.24985E+05 0.25046E+05

Ps**(psi) 0.00000E+00 -0.19745E+03 -0.39088E+03 -0.58059E+03 -0.76674E+03 -0.94946E+03 -0.11288E+04 -0.13050E+04 -0.14779E+04 -0.16477E+04 -0.18145E+04 -0.19784E+04 -0.21393E+04 -0.22974E+04 -0.24526E+04 -0.26052E+04 -0.27550E+04 -0.26370E+04 -0.19975E+04 -0.13710E+04 -0.75656E+03 -0.15376E+03 0.43782E+03 0.10185E+04 0.15886E+04 0.21483E+04

Pr**(psi) Pr**/Ps** 0.00000E+00 -----------0.53687E-02 0.27190E-04 -0.10714E+01 0.27412E-02 -0.67962E+01 0.11706E-01 -0.17797E+02 0.23211E-01 -0.32437E+02 0.34163E-01 -0.49118E+02 0.43512E-01 -0.66745E+02 0.51147E-01 -0.84639E+02 0.57270E-01 -0.10240E+03 0.62144E-01 -0.11979E+03 0.66014E-01 -0.13667E+03 0.69083E-01 -0.15299E+03 0.71515E-01 -0.16872E+03 0.73440E-01 -0.18384E+03 0.74957E-01 -0.19838E+03 0.76147E-01 -0.21234E+03 0.77072E-01 -0.22574E+03 0.85605E-01 -0.23832E+03 0.11931E+00 -0.24162E+03 0.17624E+00 -0.22311E+03 0.29490E+00 -0.18487E+03 0.12024E+01 -0.13398E+03 -0.30602E+00 -0.76195E+02 -0.74812E-01 -0.15291E+02 -0.96260E-02 0.46418E+02 0.21607E-01

162 Formation Testing Volume 3 0.156E+02 0.162E+02 0.168E+02 0.174E+02 0.180E+02 0.186E+02 0.192E+02 0.198E+02 . . .

-0.30000E+01 -0.30000E+01 -0.30000E+01 -0.30000E+01 -0.30000E+01 -0.30000E+01 -0.30000E+01 -0.30000E+01

0.27698E+05 0.28238E+05 0.28768E+05 0.29288E+05 0.29800E+05 0.30302E+05 0.30795E+05 0.31280E+05

0.25108E+05 0.25167E+05 0.25225E+05 0.25281E+05 0.25335E+05 0.25387E+05 0.25437E+05 0.25484E+05

0.26979E+04 0.32376E+04 0.37676E+04 0.42882E+04 0.47995E+04 0.53018E+04 0.57951E+04 0.62798E+04

0.10756E+03 0.16736E+03 0.22536E+03 0.28134E+03 0.33523E+03 0.38699E+03 0.43668E+03 0.48435E+03

0.39870E-01 0.51692E-01 0.59814E-01 0.65609E-01 0.69846E-01 0.72993E-01 0.75353E-01 0.77129E-01

Figure 3.2.3g. Inverse model screen. Very good predictions are obtained. We obtain the pore pressure 25,006 psi versus a 25,000 psi known. The mobility is 0.108 md/cp versus 0.100 md/cp inputted. We know the flowline volume to be 1,000 cc. Thus, the fluid compressibility is 0.000003/psi consistently with inputted value in FT-00.

Pressure Transient Analysis 163

3.2.4

Validation of PTA-App-04 Inverse Model

Figure 3.2.4a. Model 4 function. Our “inverse model” is an approximate one designed to predict pore pressure, spherical mobility and fluid compressibility in low mobility applications where flowline storage cannot be neglected. The method uses single-probe pressure transient data typically obtained during the first minute of logging – pressures which are highly unsteady and not steady-state. It is validated using pressure transient data created by our exact “forward model” FT-00 (immediately below) where formation and fluid properties, tool constants, reservoir conditions and pumping schedules are defined. Three time-pressure data points are arbitrarily selected (red from the output listing) and inputted into the approximate inverse model. Validations performed in the “black screen, DOS

164 Formation Testing Volume 3

window” show that predicted pore pressure, spherical mobility and fluid compressibility are consistent with those assumed in FT-00. The reader is invited to use different time and pressure inputs to evaluate the method. Different data points will give slightly different predictions. Times should not be too close, but preferably, several seconds apart. The request in the “black screen,” that is, “Use decimals after integers” no longer applies to our latest software.

Validation No. 1

Figure 3.2.4b. FT-00 exact inputs.

Pressure Transient Analysis 165

Figure 3.2.4c. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.800E+00

0.10000E+01

0.24738E+05

0.25000E+05 -0.26235E+03 -0.73748E-01

0.28110E-03

0.160E+01

0.10000E+01

0.24482E+05

0.24996E+05 -0.51775E+03 -0.42500E+01

0.82087E-02

0.240E+01

0.10000E+01

0.24233E+05

0.24982E+05 -0.76674E+03 -0.17797E+02

0.23211E-01

0.320E+01

0.10000E+01

0.23990E+05

0.24962E+05 -0.10096E+04 -0.37835E+02

0.37474E-01

0.400E+01

0.10000E+01

0.23753E+05

0.24939E+05 -0.12466E+04 -0.60813E+02

0.48782E-01

0.480E+01

0.10000E+01

0.23522E+05

0.24915E+05 -0.14779E+04 -0.84639E+02

0.57270E-01

0.560E+01

0.10000E+01

0.23296E+05

0.24892E+05 -0.17037E+04 -0.10824E+03

0.63534E-01

0.640E+01

0.10000E+01

0.23076E+05

0.24869E+05 -0.19241E+04 -0.13110E+03

0.68138E-01

0.720E+01

0.10000E+01

0.22861E+05

0.24847E+05 -0.21393E+04 -0.15299E+03

0.71515E-01

0.800E+01

0.10000E+01

0.22651E+05

0.24826E+05 -0.23494E+04 -0.17383E+03

0.73986E-01

0.880E+01

0.10000E+01

0.22445E+05

0.24806E+05 -0.25546E+04 -0.19360E+03

0.75782E-01

0.960E+01

0.10000E+01

0.22245E+05

0.24788E+05 -0.27550E+04 -0.21234E+03

0.77072E-01

0.104E+02

0.20000E+01

0.21917E+05

0.24770E+05 -0.30829E+04 -0.23009E+03

0.74635E-01

0.112E+02

0.20000E+01

0.21467E+05

0.24752E+05 -0.35328E+04 -0.24799E+03

0.70197E-01

0.120E+02

0.20000E+01

0.21028E+05

0.24727E+05 -0.39717E+04 -0.27283E+03

0.68693E-01

166 Formation Testing Volume 3 0.128E+02

0.20000E+01

0.20600E+05

0.24695E+05 -0.44000E+04 -0.30529E+03

0.69385E-01

0.136E+02

0.20000E+01

0.20182E+05

0.24658E+05 -0.48181E+04 -0.34154E+03

0.70888E-01

0.144E+02

0.20000E+01

0.19774E+05

0.24621E+05 -0.52262E+04 -0.37881E+03

0.72483E-01

0.152E+02

0.20000E+01

0.19375E+05

0.24584E+05 -0.56248E+04 -0.41564E+03

0.73894E-01

0.160E+02

0.20000E+01

0.18986E+05

0.24549E+05 -0.60140E+04 -0.45132E+03

0.75044E-01

0.168E+02

0.20000E+01

0.18606E+05

0.24514E+05 -0.63941E+04 -0.48553E+03

0.75935E-01

0.176E+02

0.20000E+01

0.18235E+05

0.24482E+05 -0.67653E+04 -0.51818E+03

0.76593E-01

0.184E+02

0.20000E+01

0.17872E+05

0.24451E+05 -0.71279E+04 -0.54924E+03

0.77055E-01

0.192E+02

0.20000E+01

0.17518E+05

0.24421E+05 -0.74821E+04 -0.57878E+03

0.77356E-01

0.200E+02

0.00000E+00

0.17172E+05

0.24393E+05 -0.78280E+04 -0.60686E+03

0.77523E-01

0.208E+02

0.00000E+00

0.17359E+05

0.24367E+05 -0.76413E+04 -0.63340E+03

0.82892E-01

0.216E+02

0.00000E+00

0.17539E+05

0.24350E+05 -0.74607E+04 -0.65044E+03

0.87183E-01

0.224E+02

0.00000E+00

0.17715E+05

0.24352E+05 -0.72852E+04 -0.64752E+03

0.88881E-01

0.232E+02

0.00000E+00

0.17885E+05

0.24370E+05 -0.71146E+04 -0.63047E+03

0.88617E-01

0.240E+02

0.00000E+00

0.18051E+05

0.24394E+05 -0.69485E+04 -0.60647E+03

0.87280E-01

0.248E+02

0.00000E+00

0.18213E+05

0.24420E+05 -0.67868E+04 -0.57976E+03

0.85425E-01

0.256E+02

0.00000E+00

0.18371E+05

0.24447E+05 -0.66291E+04 -0.55255E+03

0.83351E-01

0.264E+02

0.00000E+00

0.18524E+05

0.24474E+05 -0.64755E+04 -0.52592E+03

0.81217E-01

0.272E+02

0.00000E+00

0.18674E+05

0.24500E+05 -0.63257E+04 -0.50041E+03

0.79107E-01

0.280E+02

0.00000E+00

0.18820E+05

0.24524E+05 -0.61797E+04 -0.47621E+03

0.77060E-01

0.288E+02

0.00000E+00

0.18963E+05

0.24547E+05 -0.60373E+04 -0.45339E+03

0.75099E-01

0.296E+02

0.00000E+00

0.19102E+05

0.24568E+05 -0.58983E+04 -0.43193E+03

0.73228E-01

0.304E+02

0.00000E+00

0.19237E+05

0.24588E+05 -0.57628E+04 -0.41176E+03

0.71451E-01

0.312E+02

0.00000E+00

0.19369E+05

0.24607E+05 -0.56305E+04 -0.39280E+03

0.69763E-01

0.320E+02

0.00000E+00

0.19498E+05

0.24625E+05 -0.55015E+04 -0.37499E+03

0.68160E-01

0.328E+02

0.00000E+00

0.19624E+05

0.24642E+05 -0.53756E+04 -0.35822E+03

0.66638E-01

0.336E+02

0.00000E+00

0.19747E+05

0.24658E+05 -0.52527E+04 -0.34242E+03

0.65190E-01

0.344E+02

0.00000E+00

0.19867E+05

0.24672E+05 -0.51327E+04 -0.32753E+03

0.63812E-01

0.352E+02

0.00000E+00

0.19984E+05

0.24687E+05 -0.50156E+04 -0.31346E+03

0.62497E-01

0.360E+02

0.00000E+00

0.20099E+05

0.24700E+05 -0.49013E+04 -0.30017E+03

0.61242E-01

0.368E+02

0.00000E+00

0.20210E+05

0.24712E+05 -0.47898E+04 -0.28758E+03

0.60041E-01

0.376E+02

0.00000E+00

0.20319E+05

0.24724E+05 -0.46808E+04 -0.27565E+03

0.58890E-01

0.384E+02

0.00000E+00

0.20426E+05

0.24736E+05 -0.45745E+04 -0.26434E+03

0.57786E-01

0.392E+02

0.00000E+00

0.20529E+05

0.24746E+05 -0.44706E+04 -0.25360E+03

0.56725E-01

Note that there is no flow in the selected time period.

Pressure Transient Analysis 167

Figure 3.2.4d. Inverse model screen. The predicted pore pressure is 24,756 psi versus a known value of 25,000 psi. The mobility is 0.112 md/cp versus 0.100 md/cp. Flowline volume is 1,000 cc and leads to a fluid compressibility of 0.000003/psi consistently with FT-00. Next, consider another example, with slightly higher permeability, although formation is still “tight.”

168 Formation Testing Volume 3

Validation No. 2

Figure 3.2.4e. FT-00 exact inputs.

Pressure Transient Analysis 169

Figure 3.2.4f. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.800E+00

0.10000E+01

0.24765E+05

0.24983E+05 -0.23494E+03 -0.17383E+02

0.73986E-01

0.160E+01

0.10000E+01

0.24580E+05

0.24967E+05 -0.41994E+03 -0.33153E+02

0.78947E-01

0.240E+01

0.10000E+01

0.24434E+05

0.24957E+05 -0.56650E+03 -0.42874E+02

0.75683E-01

0.320E+01

0.10000E+01

0.24317E+05

0.24951E+05 -0.68291E+03 -0.49246E+02

0.72112E-01

0.400E+01

0.10000E+01

0.24224E+05

0.24946E+05 -0.77557E+03 -0.53560E+02

0.69058E-01

0.480E+01

0.10000E+01

0.24151E+05

0.24943E+05 -0.84946E+03 -0.56524E+02

0.66541E-01

0.560E+01

0.10000E+01

0.24092E+05

0.24941E+05 -0.90845E+03 -0.58568E+02

0.64470E-01

0.640E+01

0.10000E+01

0.24044E+05

0.24940E+05 -0.95563E+03 -0.59971E+02

0.62756E-01

0.720E+01

0.10000E+01

0.24007E+05

0.24939E+05 -0.99342E+03 -0.60922E+02

0.61326E-01

0.800E+01

0.10000E+01

0.23976E+05

0.24938E+05 -0.10237E+04 -0.61551E+02

0.60124E-01

0.880E+01

0.10000E+01

0.23952E+05

0.24938E+05 -0.10481E+04 -0.61951E+02

0.59108E-01

0.960E+01

0.10000E+01

0.23932E+05

0.24938E+05 -0.10677E+04 -0.62186E+02

0.58244E-01

0.104E+02

0.20000E+01

0.23792E+05

0.24932E+05 -0.12081E+04 -0.68387E+02

0.56605E-01

0.112E+02

0.20000E+01

0.23571E+05

0.24911E+05 -0.14291E+04 -0.88632E+02

0.62019E-01

0.120E+02

0.20000E+01

0.23396E+05

0.24899E+05 -0.16040E+04 -0.10088E+03

0.62889E-01

170 Formation Testing Volume 3 0.128E+02

0.20000E+01

0.23257E+05

0.24891E+05 -0.17430E+04 -0.10865E+03

0.62332E-01

0.136E+02

0.20000E+01

0.23146E+05

0.24886E+05 -0.18537E+04 -0.11379E+03

0.61387E-01

0.144E+02

0.20000E+01

0.23058E+05

0.24883E+05 -0.19419E+04 -0.11726E+03

0.60384E-01

0.152E+02

0.20000E+01

0.22987E+05

0.24880E+05 -0.20125E+04 -0.11962E+03

0.59437E-01

0.160E+02

0.20000E+01

0.22931E+05

0.24879E+05 -0.20690E+04 -0.12120E+03

0.58579E-01

0.168E+02

0.20000E+01

0.22886E+05

0.24878E+05 -0.21143E+04 -0.12224E+03

0.57818E-01

0.176E+02

0.20000E+01

0.22849E+05

0.24877E+05 -0.21507E+04 -0.12291E+03

0.57149E-01

0.184E+02

0.20000E+01

0.22820E+05

0.24876E+05 -0.21800E+04 -0.12355E+03

0.56675E-01

0.192E+02

0.20000E+01

0.22796E+05

0.24876E+05 -0.22036E+04 -0.12417E+03

0.56350E-01

0.200E+02

0.00000E+00

0.22777E+05

0.24875E+05 -0.22226E+04 -0.12452E+03

0.56022E-01

0.208E+02

0.00000E+00

0.23232E+05

0.24910E+05 -0.17682E+04 -0.89901E+02

0.50843E-01

0.216E+02

0.00000E+00

0.23589E+05

0.24942E+05 -0.14107E+04 -0.58373E+02

0.41378E-01

0.224E+02

0.00000E+00

0.23872E+05

0.24961E+05 -0.11278E+04 -0.38851E+02

0.34447E-01

0.232E+02

0.00000E+00

0.24097E+05

0.24974E+05 -0.90335E+03 -0.25963E+02

0.28741E-01

0.240E+02

0.00000E+00

0.24275E+05

0.24983E+05 -0.72486E+03 -0.17151E+02

0.23661E-01

0.248E+02

0.00000E+00

0.24417E+05

0.24989E+05 -0.58271E+03 -0.11014E+02

0.18901E-01

0.256E+02

0.00000E+00

0.24531E+05

0.24993E+05 -0.46936E+03 -0.67038E+01

0.14283E-01

0.264E+02

0.00000E+00

0.24621E+05

0.24996E+05 -0.37884E+03 -0.36702E+01

0.96881E-02

0.272E+02

0.00000E+00

0.24694E+05

0.24998E+05 -0.30647E+03 -0.15424E+01

0.50329E-02

0.280E+02

0.00000E+00

0.24751E+05

0.25000E+05 -0.24852E+03 -0.62990E-01

0.25346E-03

0.288E+02

0.00000E+00

0.24798E+05

0.25000E+05 -0.20208E+03

0.00000E+00

0.00000E+00

0.296E+02

0.00000E+00

0.24835E+05

0.25000E+05 -0.16479E+03

0.00000E+00

0.00000E+00

0.304E+02

0.00000E+00

0.24865E+05

0.25000E+05 -0.13482E+03

0.00000E+00

0.00000E+00

0.312E+02

0.00000E+00

0.24889E+05

0.25000E+05 -0.11069E+03

0.00000E+00

0.00000E+00

0.320E+02

0.00000E+00

0.24909E+05

0.25000E+05 -0.91233E+02

0.00000E+00

0.00000E+00

0.328E+02

0.00000E+00

0.24924E+05

0.25000E+05 -0.75517E+02

0.00000E+00

0.00000E+00

0.336E+02

0.00000E+00

0.24937E+05

0.25000E+05 -0.62799E+02

0.00000E+00

0.00000E+00

0.344E+02

0.00000E+00

0.24948E+05

0.25000E+05 -0.52486E+02

0.00000E+00

0.00000E+00

0.352E+02

0.00000E+00

0.24956E+05

0.25000E+05 -0.44106E+02

0.00000E+00

0.00000E+00

0.360E+02

0.00000E+00

0.24963E+05

0.25000E+05 -0.37279E+02

0.00000E+00

0.00000E+00

0.368E+02

0.00000E+00

0.24968E+05

0.25000E+05 -0.31704E+02

0.00000E+00

0.00000E+00

0.376E+02

0.00000E+00

0.24973E+05

0.25000E+05 -0.27139E+02

0.00000E+00

0.00000E+00

0.384E+02

0.00000E+00

0.24977E+05

0.25000E+05 -0.23388E+02

0.00000E+00

0.00000E+00

0.392E+02

0.00000E+00

0.24980E+05

0.25000E+05 -0.20297E+02

0.00000E+00

0.00000E+00

No flow is seen in selected time period.

Pressure Transient Analysis 171

Figure 3.2.4g. Inverse model screen. The pore pressure is 24,992 psi vs a known value of 25,000 psi. The mobility is 1.159 md/cp vs 1.000 md/cp. Both predictions are very good. The flowline volume of 1,000 cc leads to a compressiblity of 0.0000037/psi versus a value of 0.000003/psi.

172 Formation Testing Volume 3

3.2.5

Validation of PTA-App-05 Inverse Model

Figure 3.2.5a. Model 5 function. Our “inverse model” is an approximate one designed to predict pore pressure, spherical mobility and fluid compressibility in low mobility applications where flowline storage cannot be neglected. The method uses single-probe pressure transient data typically obtained during the first minute of logging – pressures which are highly unsteady and not steady-state. It is validated using pressure transient data created by our exact “forward model” FT-00 (immediately below) where formation and fluid properties, tool constants, reservoir conditions and pumping schedules are defined. Three time-pressure data points are arbitrarily selected (red from the output listing) and inputted into the approximate inverse model. Validations performed in the “black screen, DOS

Pressure Transient Analysis 173

window” show that predicted pore pressure, spherical mobility and fluid compressibility are consistent with those assumed in FT-00. The reader is invited to use different time and pressure inputs to evaluate the method. Different data points will give slightly different predictions. Times should not be too close, but preferably, several seconds apart. The request in the “black screen,” that is, “Use decimals after integers” no longer applies to our latest software.

Figure 3.2.5b. FT-00 exact inputs.

174 Formation Testing Volume 3

Figure 3.2.5c. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.800E+00

0.10000E+01

0.24765E+05

0.24983E+05 -0.23494E+03 -0.17383E+02

0.73986E-01

0.160E+01

0.10000E+01

0.24580E+05

0.24967E+05 -0.41994E+03 -0.33153E+02

0.78947E-01

0.240E+01

0.10000E+01

0.24434E+05

0.24957E+05 -0.56650E+03 -0.42874E+02

0.75683E-01

0.320E+01

0.10000E+01

0.24317E+05

0.24951E+05 -0.68291E+03 -0.49246E+02

0.72112E-01

0.400E+01

0.10000E+01

0.24224E+05

0.24946E+05 -0.77557E+03 -0.53560E+02

0.69058E-01

0.480E+01

0.10000E+01

0.24151E+05

0.24943E+05 -0.84946E+03 -0.56524E+02

0.66541E-01

0.560E+01

0.10000E+01

0.24092E+05

0.24941E+05 -0.90845E+03 -0.58568E+02

0.64470E-01

0.640E+01

0.10000E+01

0.24044E+05

0.24940E+05 -0.95563E+03 -0.59971E+02

0.62756E-01

0.720E+01

0.10000E+01

0.24007E+05

0.24939E+05 -0.99342E+03 -0.60922E+02

0.61326E-01

0.800E+01

0.10000E+01

0.23976E+05

0.24938E+05 -0.10237E+04 -0.61551E+02

0.60124E-01

0.880E+01

0.10000E+01

0.23952E+05

0.24938E+05 -0.10481E+04 -0.61951E+02

0.59108E-01

0.960E+01

0.10000E+01

0.23932E+05

0.24938E+05 -0.10677E+04 -0.62186E+02

0.58244E-01

0.104E+02

0.20000E+01

0.23792E+05

0.24932E+05 -0.12081E+04 -0.68387E+02

0.56605E-01

0.112E+02

0.20000E+01

0.23571E+05

0.24911E+05 -0.14291E+04 -0.88632E+02

0.62019E-01

0.120E+02

0.20000E+01

0.23396E+05

0.24899E+05 -0.16040E+04 -0.10088E+03

0.62889E-01

Pressure Transient Analysis 175 0.128E+02

0.20000E+01

0.23257E+05

0.24891E+05 -0.17430E+04 -0.10865E+03

0.62332E-01

0.136E+02

0.20000E+01

0.23146E+05

0.24886E+05 -0.18537E+04 -0.11379E+03

0.61387E-01

0.144E+02

0.20000E+01

0.23058E+05

0.24883E+05 -0.19419E+04 -0.11726E+03

0.60384E-01

0.152E+02

0.20000E+01

0.22987E+05

0.24880E+05 -0.20125E+04 -0.11962E+03

0.59437E-01

0.160E+02

0.20000E+01

0.22931E+05

0.24879E+05 -0.20690E+04 -0.12120E+03

0.58579E-01

0.168E+02

0.20000E+01

0.22886E+05

0.24878E+05 -0.21143E+04 -0.12224E+03

0.57818E-01

0.176E+02

0.20000E+01

0.22849E+05

0.24877E+05 -0.21507E+04 -0.12291E+03

0.57149E-01

0.184E+02

0.20000E+01

0.22820E+05

0.24876E+05 -0.21800E+04 -0.12355E+03

0.56675E-01

0.192E+02

0.20000E+01

0.22796E+05

0.24876E+05 -0.22036E+04 -0.12417E+03

0.56350E-01

0.200E+02

0.30000E+01

0.22777E+05

0.24875E+05 -0.22226E+04 -0.12452E+03

0.56022E-01

0.208E+02

0.30000E+01

0.22527E+05

0.24858E+05 -0.24730E+04 -0.14205E+03

0.57439E-01

0.216E+02

0.30000E+01

0.22329E+05

0.24842E+05 -0.26706E+04 -0.15783E+03

0.59101E-01

0.224E+02

0.30000E+01

0.22173E+05

0.24833E+05 -0.28273E+04 -0.16747E+03

0.59233E-01

0.232E+02

0.30000E+01

0.22048E+05

0.24826E+05 -0.29521E+04 -0.17370E+03

0.58840E-01

0.240E+02

0.30000E+01

0.21948E+05

0.24822E+05 -0.30516E+04 -0.17783E+03

0.58275E-01

0.248E+02

0.30000E+01

0.21869E+05

0.24819E+05 -0.31311E+04 -0.18059E+03

0.57675E-01

0.256E+02

0.30000E+01

0.21805E+05

0.24818E+05 -0.31947E+04 -0.18241E+03

0.57097E-01

0.264E+02

0.30000E+01

0.21754E+05

0.24816E+05 -0.32457E+04 -0.18358E+03

0.56562E-01

0.272E+02

0.30000E+01

0.21713E+05

0.24816E+05 -0.32867E+04 -0.18431E+03

0.56077E-01

0.280E+02

0.30000E+01

0.21680E+05

0.24815E+05 -0.33197E+04 -0.18472E+03

0.55642E-01

0.288E+02

0.30000E+01

0.21654E+05

0.24814E+05 -0.33463E+04 -0.18585E+03

0.55539E-01

0.296E+02

0.30000E+01

0.21632E+05

0.24813E+05 -0.33678E+04 -0.18656E+03

0.55394E-01

0.304E+02

0.00000E+00

0.21989E+05

0.24831E+05 -0.30113E+04 -0.16867E+03

0.56014E-01

0.312E+02

0.00000E+00

0.22599E+05

0.24892E+05 -0.24008E+04 -0.10817E+03

0.45053E-01

0.320E+02

0.00000E+00

0.23081E+05

0.24929E+05 -0.19186E+04 -0.71316E+02

0.37171E-01

0.328E+02

0.00000E+00

0.23464E+05

0.24952E+05 -0.15363E+04 -0.47664E+02

0.31025E-01

0.336E+02

0.00000E+00

0.23767E+05

0.24968E+05 -0.12325E+04 -0.31729E+02

0.25742E-01

0.344E+02

0.00000E+00

0.24009E+05

0.24979E+05 -0.99074E+03 -0.20715E+02

0.20909E-01

0.352E+02

0.00000E+00

0.24202E+05

0.24987E+05 -0.79799E+03 -0.13005E+02

0.16298E-01

0.360E+02

0.00000E+00

0.24356E+05

0.24992E+05 -0.64413E+03 -0.75808E+01

0.11769E-01

0.368E+02

0.00000E+00

0.24479E+05

0.24996E+05 -0.52114E+03 -0.37670E+01

0.72284E-02

0.376E+02

0.00000E+00

0.24577E+05

0.24999E+05 -0.42270E+03 -0.11017E+01

0.26064E-02

0.384E+02

0.00000E+00

0.24656E+05

0.25000E+05 -0.34380E+03

0.00000E+00

0.00000E+00

0.392E+02

0.00000E+00

0.24720E+05

0.25000E+05 -0.28049E+03

0.00000E+00

0.00000E+00

176 Formation Testing Volume 3

Figure 3.2.5d. Inverse model screen. Excellent results are obtained. A pore pressure of 24,998 psi is predicted versus a known value of 25,000 psi. The predicted mobility is 1.027 md/cp versus 1.000 md/cp. Since the flow line volume is 1,000 cc, the fluid compressibility is 0.0000031/psi versus 0.0000030/psi as assumed in FT-00.

Pressure Transient Analysis 177

3.2.6

Validation of PTA-App-06 Inverse Model

Figure 3.2.6a. Model 6 function. Our “inverse model” is an approximate one designed to predict pore pressure, spherical mobility and fluid compressibility in low mobility applications where flowline storage cannot be neglected. The method uses single-probe pressure transient data typically obtained during the first minute of logging – pressures which are highly unsteady and not steady-state. It is validated using pressure transient data created by our exact “forward model” FT-00 (immediately below) where formation and fluid properties, tool constants, reservoir conditions and pumping

178 Formation Testing Volume 3

schedules are defined. Three time-pressure data points are arbitrarily selected (red from the output listing) and inputted into the approximate inverse model. Validations performed in the “black screen, DOS window” show that predicted pore pressure, spherical mobility and fluid compressibility are consistent with those assumed in FT-00. The reader is invited to use different time and pressure inputs to evaluate the method. Different data points will give slightly different predictions. Times should not be too close, but preferably, several seconds apart. The request in the “black screen,” that is, “Use decimals after integers” no longer applies to our latest software.

Figure 3.2.6b. FT-00 exact inputs.

Pressure Transient Analysis 179

Figure 3.2.6c. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.100E+01

0.10000E+01

0.24715E+05

0.24978E+05 -0.28535E+03 -0.22133E+02

0.77566E-01

0.200E+01

0.10000E+01

0.24503E+05

0.24961E+05 -0.49745E+03 -0.38552E+02

0.77499E-01

0.300E+01

0.10000E+01

0.24344E+05

0.24952E+05 -0.65626E+03 -0.47883E+02

0.72963E-01

0.400E+01

0.10000E+01

0.24224E+05

0.24946E+05 -0.77557E+03 -0.53560E+02

0.69058E-01

0.500E+01

0.10000E+01

0.24135E+05

0.24943E+05 -0.86547E+03 -0.57108E+02

0.65985E-01

0.600E+01

0.10000E+01

0.24067E+05

0.24941E+05 -0.93335E+03 -0.59336E+02

0.63573E-01

0.700E+01

0.10000E+01

0.24015E+05

0.24939E+05 -0.98474E+03 -0.60719E+02

0.61660E-01

0.800E+01

0.10000E+01

0.23976E+05

0.24938E+05 -0.10237E+04 -0.61551E+02

0.60124E-01

0.900E+01

0.10000E+01

0.23947E+05

0.24938E+05 -0.10534E+04 -0.62023E+02

0.58879E-01

0.100E+02

0.20000E+01

0.23924E+05

0.24938E+05 -0.10760E+04 -0.62258E+02

0.57860E-01

0.110E+02

0.20000E+01

0.23621E+05

0.24916E+05 -0.13787E+04 -0.84473E+02

0.61272E-01

0.120E+02

0.20000E+01

0.23396E+05

0.24899E+05 -0.16040E+04 -0.10088E+03

0.62889E-01

0.130E+02

0.20000E+01

0.23227E+05

0.24890E+05 -0.17731E+04 -0.11013E+03

0.62113E-01

0.140E+02

0.20000E+01

0.23100E+05

0.24884E+05 -0.19003E+04 -0.11570E+03

0.60883E-01

0.150E+02

0.20000E+01

0.23004E+05

0.24881E+05 -0.19963E+04 -0.11911E+03

0.59666E-01

180 Formation Testing Volume 3 0.160E+02

0.20000E+01

0.22931E+05

0.24879E+05 -0.20690E+04 -0.12120E+03

0.58579E-01

0.170E+02

0.20000E+01

0.22876E+05

0.24878E+05 -0.21241E+04 -0.12244E+03

0.57642E-01

0.180E+02

0.20000E+01

0.22834E+05

0.24877E+05 -0.21661E+04 -0.12313E+03

0.56846E-01

0.190E+02

0.20000E+01

0.22802E+05

0.24876E+05 -0.21981E+04 -0.12405E+03

0.56432E-01

0.200E+02

0.30000E+01

0.22777E+05

0.24875E+05 -0.22226E+04 -0.12452E+03

0.56022E-01

0.210E+02

0.30000E+01

0.22473E+05

0.24853E+05 -0.25268E+04 -0.14681E+03

0.58102E-01

0.220E+02

0.30000E+01

0.22247E+05

0.24837E+05 -0.27534E+04 -0.16320E+03

0.59271E-01

0.230E+02

0.30000E+01

0.22076E+05

0.24828E+05 -0.29235E+04 -0.17238E+03

0.58963E-01

0.240E+02

0.30000E+01

0.21948E+05

0.24822E+05 -0.30516E+04 -0.17783E+03

0.58275E-01

0.250E+02

0.30000E+01

0.21852E+05

0.24819E+05 -0.31483E+04 -0.18112E+03

0.57527E-01

0.260E+02

0.30000E+01

0.21778E+05

0.24817E+05 -0.32216E+04 -0.18306E+03

0.56823E-01

0.270E+02

0.30000E+01

0.21723E+05

0.24816E+05 -0.32773E+04 -0.18416E+03

0.56193E-01

0.280E+02

0.30000E+01

0.21680E+05

0.24815E+05 -0.33197E+04 -0.18472E+03

0.55642E-01

0.290E+02

0.30000E+01

0.21648E+05

0.24814E+05 -0.33521E+04 -0.18607E+03

0.55507E-01

0.300E+02

0.30000E+01

0.21623E+05

0.24813E+05 -0.33770E+04 -0.18677E+03

0.55307E-01

0.310E+02

0.00000E+00

0.22460E+05

0.24879E+05 -0.25401E+04 -0.12062E+03

0.47486E-01

0.320E+02

0.00000E+00

0.23081E+05

0.24929E+05 -0.19186E+04 -0.71316E+02

0.37171E-01

0.330E+02

0.00000E+00

0.23546E+05

0.24957E+05 -0.14537E+04 -0.43095E+02

0.29645E-01

0.340E+02

0.00000E+00

0.23895E+05

0.24974E+05 -0.11048E+04 -0.25726E+02

0.23286E-01

0.350E+02

0.00000E+00

0.24158E+05

0.24985E+05 -0.84218E+03 -0.14686E+02

0.17438E-01

0.360E+02

0.00000E+00

0.24356E+05

0.24992E+05 -0.64413E+03 -0.75808E+01

0.11769E-01

0.370E+02

0.00000E+00

0.24506E+05

0.24997E+05 -0.49444E+03 -0.30076E+01

0.60828E-02

0.380E+02

0.00000E+00

0.24619E+05

0.25000E+05 -0.38108E+03 -0.94492E-01

0.24796E-03

0.390E+02

0.00000E+00

0.24705E+05

0.25000E+05 -0.29504E+03

0.00000E+00

0.00000E+00

0.400E+02

0.00000E+00

0.24770E+05

0.25000E+05 -0.22959E+03

0.00000E+00

0.00000E+00

0.410E+02

0.00000E+00

0.24820E+05

0.25000E+05 -0.17970E+03

0.00000E+00

0.00000E+00

0.420E+02

0.00000E+00

0.24858E+05

0.25000E+05 -0.14156E+03

0.00000E+00

0.00000E+00

0.430E+02

0.00000E+00

0.24888E+05

0.25000E+05 -0.11232E+03

0.00000E+00

0.00000E+00

0.440E+02

0.00000E+00

0.24910E+05

0.25000E+05 -0.89835E+02

0.00000E+00

0.00000E+00

0.450E+02

0.00000E+00

0.24928E+05

0.25000E+05 -0.72488E+02

0.00000E+00

0.00000E+00

0.460E+02

0.00000E+00

0.24941E+05

0.25000E+05 -0.59053E+02

0.00000E+00

0.00000E+00

0.470E+02

0.00000E+00

0.24951E+05

0.25000E+05 -0.48602E+02

0.00000E+00

0.00000E+00

0.480E+02

0.00000E+00

0.24960E+05

0.25000E+05 -0.40435E+02

0.00000E+00

0.00000E+00

0.490E+02

0.00000E+00

0.24966E+05

0.25000E+05 -0.34019E+02

0.00000E+00

0.00000E+00

Pressure Transient Analysis 181

Figure 3.2.6d. Inverse model screen. Pore pressure is predicted as 24,966 psi versus a known value of 25,000 psi. The mobility is 1.043 md/cp versus 1.000 md/cp. A flowline volume of 1,000 cc implies a prediction of 0.0000031/psi versus an assumed 0.000003/psi. The results are very acceptable.

182 Formation Testing Volume 3

3.2.7

Validation of PTA-App-07 Inverse Model

Figure 3.2.7a. Model 7 function. Our “inverse model” is an approximate one designed to predict pore pressure, spherical mobility and fluid compressibility in low mobility applications where flowline storage cannot be neglected. The method uses single-probe pressure transient data typically obtained during the first minute of logging – pressures which are highly unsteady and not steady-state. It is validated using pressure transient data created by our exact “forward model” FT-00 (immediately below) where formation and fluid properties, tool constants, reservoir conditions and pumping schedules are defined. Three time-pressure data points are arbitrarily

Pressure Transient Analysis 183

selected (red from the output listing) and inputted into the approximate inverse model. Validations performed in the “black screen, DOS window” show that predicted pore pressure, spherical mobility and fluid compressibility are consistent with those assumed in FT-00. The reader is invited to use different time and pressure inputs to evaluate the method. Different data points will give slightly different predictions. Times should not be too close, but preferably, several seconds apart. The request in the “black screen,” that is, “Use decimals after integers” no longer applies to our latest software.

Figure 3.2.7b. FT-00 exact inputs.

184 Formation Testing Volume 3

Figure 3.2.7c. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.100E+01

0.10000E+01

0.24715E+05

0.24978E+05 -0.28535E+03 -0.22133E+02

0.77566E-01

0.200E+01

0.10000E+01

0.24503E+05

0.24961E+05 -0.49745E+03 -0.38552E+02

0.77499E-01

0.300E+01

0.10000E+01

0.24344E+05

0.24952E+05 -0.65626E+03 -0.47883E+02

0.72963E-01

0.400E+01

0.10000E+01

0.24224E+05

0.24946E+05 -0.77557E+03 -0.53560E+02

0.69058E-01

0.500E+01

0.10000E+01

0.24135E+05

0.24943E+05 -0.86547E+03 -0.57108E+02

0.65985E-01

0.600E+01

0.10000E+01

0.24067E+05

0.24941E+05 -0.93335E+03 -0.59336E+02

0.63573E-01

0.700E+01

0.10000E+01

0.24015E+05

0.24939E+05 -0.98474E+03 -0.60719E+02

0.61660E-01

0.800E+01

0.10000E+01

0.23976E+05

0.24938E+05 -0.10237E+04 -0.61551E+02

0.60124E-01

0.900E+01

0.10000E+01

0.23947E+05

0.24938E+05 -0.10534E+04 -0.62023E+02

0.58879E-01

0.100E+02

0.20000E+01

0.23924E+05

0.24938E+05 -0.10760E+04 -0.62258E+02

0.57860E-01

0.110E+02

0.20000E+01

0.23621E+05

0.24916E+05 -0.13787E+04 -0.84473E+02

0.61272E-01

0.120E+02

0.20000E+01

0.23396E+05

0.24899E+05 -0.16040E+04 -0.10088E+03

0.62889E-01

0.130E+02

0.20000E+01

0.23227E+05

0.24890E+05 -0.17731E+04 -0.11013E+03

0.62113E-01

0.140E+02

0.20000E+01

0.23100E+05

0.24884E+05 -0.19003E+04 -0.11570E+03

0.60883E-01

0.150E+02

0.20000E+01

0.23004E+05

0.24881E+05 -0.19963E+04 -0.11911E+03

0.59666E-01

Pressure Transient Analysis 185 0.160E+02

0.20000E+01

0.22931E+05

0.24879E+05 -0.20690E+04 -0.12120E+03

0.58579E-01

0.170E+02

0.20000E+01

0.22876E+05

0.24878E+05 -0.21241E+04 -0.12244E+03

0.57642E-01

0.180E+02

0.20000E+01

0.22834E+05

0.24877E+05 -0.21661E+04 -0.12313E+03

0.56846E-01

0.190E+02

0.20000E+01

0.22802E+05

0.24876E+05 -0.21981E+04 -0.12405E+03

0.56432E-01

0.200E+02

0.30000E+01

0.22777E+05

0.24875E+05 -0.22226E+04 -0.12452E+03

0.56022E-01

0.210E+02

0.30000E+01

0.22473E+05

0.24853E+05 -0.25268E+04 -0.14681E+03

0.58102E-01

0.220E+02

0.30000E+01

0.22247E+05

0.24837E+05 -0.27534E+04 -0.16320E+03

0.59271E-01

0.230E+02

0.30000E+01

0.22076E+05

0.24828E+05 -0.29235E+04 -0.17238E+03

0.58963E-01

0.240E+02

0.30000E+01

0.21948E+05

0.24822E+05 -0.30516E+04 -0.17783E+03

0.58275E-01

0.250E+02

0.30000E+01

0.21852E+05

0.24819E+05 -0.31483E+04 -0.18112E+03

0.57527E-01

0.260E+02

0.30000E+01

0.21778E+05

0.24817E+05 -0.32216E+04 -0.18306E+03

0.56823E-01

0.270E+02

0.30000E+01

0.21723E+05

0.24816E+05 -0.32773E+04 -0.18416E+03

0.56193E-01

0.280E+02

0.30000E+01

0.21680E+05

0.24815E+05 -0.33197E+04 -0.18472E+03

0.55642E-01

0.290E+02

0.30000E+01

0.21648E+05

0.24814E+05 -0.33521E+04 -0.18607E+03

0.55507E-01

0.300E+02

0.30000E+01

0.21623E+05

0.24813E+05 -0.33770E+04 -0.18677E+03

0.55307E-01

0.310E+02

0.40000E+01

0.21318E+05

0.24791E+05 -0.36815E+04 -0.20915E+03

0.56812E-01

0.320E+02

0.40000E+01

0.21092E+05

0.24774E+05 -0.39084E+04 -0.22553E+03

0.57702E-01

0.330E+02

0.40000E+01

0.20921E+05

0.24765E+05 -0.40788E+04 -0.23463E+03

0.57524E-01

0.340E+02

0.40000E+01

0.20793E+05

0.24760E+05 -0.42071E+04 -0.23997E+03

0.57039E-01

0.350E+02

0.40000E+01

0.20696E+05

0.24757E+05 -0.43040E+04 -0.24312E+03

0.56486E-01

0.360E+02

0.40000E+01

0.20622E+05

0.24755E+05 -0.43775E+04 -0.24493E+03

0.55950E-01

0.370E+02

0.40000E+01

0.20567E+05

0.24754E+05 -0.44334E+04 -0.24588E+03

0.55461E-01

0.380E+02

0.40000E+01

0.20524E+05

0.24754E+05 -0.44760E+04 -0.24630E+03

0.55026E-01

0.390E+02

0.40000E+01

0.20491E+05

0.24752E+05 -0.45086E+04 -0.24809E+03

0.55026E-01

0.400E+02

0.00000E+00

0.20466E+05

0.24751E+05 -0.45336E+04 -0.24903E+03

0.54930E-01

0.410E+02

0.00000E+00

0.21588E+05

0.24839E+05 -0.34115E+04 -0.16083E+03

0.47142E-01

0.420E+02

0.00000E+00

0.22422E+05

0.24905E+05 -0.25781E+04 -0.95088E+02

0.36884E-01

0.430E+02

0.00000E+00

0.23045E+05

0.24943E+05 -0.19545E+04 -0.57460E+02

0.29399E-01

0.440E+02

0.00000E+00

0.23514E+05

0.24966E+05 -0.14864E+04 -0.34302E+02

0.23077E-01

0.450E+02

0.00000E+00

0.23866E+05

0.24980E+05 -0.11340E+04 -0.19581E+02

0.17267E-01

0.460E+02

0.00000E+00

0.24132E+05

0.24990E+05 -0.86818E+03 -0.10108E+02

0.11642E-01

0.470E+02

0.00000E+00

0.24333E+05

0.24996E+05 -0.66722E+03 -0.40101E+01

0.60102E-02

0.480E+02

0.00000E+00

0.24485E+05

0.25000E+05 -0.51498E+03 -0.12598E+00

0.24463E-03

0.490E+02

0.00000E+00

0.24601E+05

0.25000E+05 -0.39938E+03

0.00000E+00

0.00000E+00

186 Formation Testing Volume 3

Figure 3.2.7d. Inverse model screen. Excellent results are obtained. Predictions are a pore pressure of 24,982 psi versus 25,000 psi, and a mobility of 1.029 md/cp versus 1.000 md/cp. For our flowline volume of 1,000 cc, the fluid compressibility is obtained as 0.0000031/psi versus 0.000003/psi as assumed in FT-00.

Pressure Transient Analysis 187

3.2.8

Validation of PTA-App-08 Inverse Model

Figure 3.2.8a. Model 8 function. Our “inverse model” is an approximate one designed to predict pore pressure, spherical mobility and fluid compressibility in low mobility applications where flowline storage cannot be neglected. The method uses single-probe pressure transient data typically obtained during the first minute of logging – pressures which are highly unsteady and not steady-state. It is validated using pressure transient data created by our exact “forward model” FT-00 (immediately below) where formation and fluid properties, tool constants, reservoir conditions and pumping schedules are defined. Three time-pressure data points are arbitrarily selected (red from the output listing) and inputted into the approximate inverse model. Validations performed in the “black screen, DOS

188 Formation Testing Volume 3

window” show that predicted pore pressure, spherical mobility and fluid compressibility are consistent with those assumed in FT-00. The reader is invited to use different time and pressure inputs to evaluate the method. Different data points will give slightly different predictions. Times should not be too close, but preferably, several seconds apart. The request in the “black screen,” that is, “Use decimals after integers” no longer applies to our latest software.

Figure 3.2.8b. FT-00 exact inputs.

Pressure Transient Analysis 189

Figure 3.2.8c. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.120E+01

0.10000E+01

0.24667E+05

0.24974E+05 -0.33287E+03 -0.26288E+02

0.78976E-01

0.240E+01

0.10000E+01

0.24434E+05

0.24957E+05 -0.56650E+03 -0.42874E+02

0.75683E-01

0.360E+01

0.10000E+01

0.24268E+05

0.24948E+05 -0.73187E+03 -0.51607E+02

0.70514E-01

0.480E+01

0.10000E+01

0.24151E+05

0.24943E+05 -0.84946E+03 -0.56524E+02

0.66541E-01

0.600E+01

0.10000E+01

0.24067E+05

0.24941E+05 -0.93335E+03 -0.59336E+02

0.63573E-01

0.720E+01

0.10000E+01

0.24007E+05

0.24939E+05 -0.99342E+03 -0.60922E+02

0.61326E-01

0.840E+01

0.10000E+01

0.23963E+05

0.24938E+05 -0.10366E+04 -0.61775E+02

0.59595E-01

0.960E+01

0.10000E+01

0.23932E+05

0.24938E+05 -0.10677E+04 -0.62186E+02

0.58244E-01

0.108E+02

0.20000E+01

0.23675E+05

0.24920E+05 -0.13252E+04 -0.79716E+02

0.60156E-01

0.120E+02

0.20000E+01

0.23396E+05

0.24899E+05 -0.16040E+04 -0.10088E+03

0.62889E-01

0.132E+02

0.20000E+01

0.23199E+05

0.24889E+05 -0.18015E+04 -0.11147E+03

0.61879E-01

0.144E+02

0.20000E+01

0.23058E+05

0.24883E+05 -0.19419E+04 -0.11726E+03

0.60384E-01

0.156E+02

0.20000E+01

0.22958E+05

0.24880E+05 -0.20423E+04 -0.12049E+03

0.58996E-01

0.168E+02

0.20000E+01

0.22886E+05

0.24878E+05 -0.21143E+04 -0.12224E+03

0.57818E-01

0.180E+02

0.20000E+01

0.22834E+05

0.24877E+05 -0.21661E+04 -0.12313E+03

0.56846E-01

190 Formation Testing Volume 3 0.192E+02

0.20000E+01

0.22796E+05

0.24876E+05 -0.22036E+04 -0.12417E+03

0.56350E-01

0.204E+02

0.30000E+01

0.22645E+05

0.24869E+05 -0.23554E+04 -0.13069E+03

0.55486E-01

0.216E+02

0.30000E+01

0.22329E+05

0.24842E+05 -0.26706E+04 -0.15783E+03

0.59101E-01

0.228E+02

0.30000E+01

0.22107E+05

0.24829E+05 -0.28933E+04 -0.17091E+03

0.59072E-01

0.240E+02

0.30000E+01

0.21948E+05

0.24822E+05 -0.30516E+04 -0.17783E+03

0.58275E-01

0.252E+02

0.30000E+01

0.21835E+05

0.24818E+05 -0.31647E+04 -0.18159E+03

0.57381E-01

0.264E+02

0.30000E+01

0.21754E+05

0.24816E+05 -0.32457E+04 -0.18358E+03

0.56562E-01

0.276E+02

0.30000E+01

0.21696E+05

0.24815E+05 -0.33041E+04 -0.18455E+03

0.55853E-01

0.288E+02

0.30000E+01

0.21654E+05

0.24814E+05 -0.33463E+04 -0.18585E+03

0.55539E-01

0.300E+02

0.40000E+01

0.21623E+05

0.24813E+05 -0.33770E+04 -0.18677E+03

0.55307E-01

0.312E+02

0.40000E+01

0.21268E+05

0.24787E+05 -0.37323E+04 -0.21332E+03

0.57155E-01

0.324E+02

0.40000E+01

0.21018E+05

0.24770E+05 -0.39824E+04 -0.22977E+03

0.57696E-01

0.336E+02

0.40000E+01

0.20840E+05

0.24762E+05 -0.41600E+04 -0.23816E+03

0.57249E-01

0.348E+02

0.40000E+01

0.20713E+05

0.24757E+05 -0.42867E+04 -0.24262E+03

0.56597E-01

0.360E+02

0.40000E+01

0.20622E+05

0.24755E+05 -0.43775E+04 -0.24493E+03

0.55950E-01

0.372E+02

0.40000E+01

0.20557E+05

0.24754E+05 -0.44429E+04 -0.24600E+03

0.55370E-01

0.384E+02

0.40000E+01

0.20510E+05

0.24753E+05 -0.44901E+04 -0.24710E+03

0.55032E-01

0.396E+02

0.40000E+01

0.20476E+05

0.24751E+05 -0.45244E+04 -0.24874E+03

0.54979E-01

0.408E+02

0.00000E+00

0.21390E+05

0.24820E+05 -0.36097E+04 -0.17980E+03

0.49811E-01

0.420E+02

0.00000E+00

0.22422E+05

0.24905E+05 -0.25781E+04 -0.95088E+02

0.36884E-01

0.432E+02

0.00000E+00

0.23150E+05

0.24948E+05 -0.18499E+04 -0.51926E+02

0.28070E-01

0.444E+02

0.00000E+00

0.23667E+05

0.24972E+05 -0.13334E+04 -0.27620E+02

0.20715E-01

0.456E+02

0.00000E+00

0.24034E+05

0.24987E+05 -0.96566E+03 -0.13408E+02

0.13884E-01

0.468E+02

0.00000E+00

0.24297E+05

0.24995E+05 -0.70307E+03 -0.50226E+01

0.71439E-02

0.480E+02

0.00000E+00

0.24485E+05

0.25000E+05 -0.51497E+03 -0.12597E+00

0.24461E-03

0.492E+02

0.00000E+00

0.24620E+05

0.25000E+05 -0.37982E+03

0.00000E+00

0.00000E+00

0.504E+02

0.00000E+00

0.24718E+05

0.25000E+05 -0.28239E+03

0.00000E+00

0.00000E+00

0.516E+02

0.00000E+00

0.24788E+05

0.25000E+05 -0.21190E+03

0.00000E+00

0.00000E+00

0.528E+02

0.00000E+00

0.24839E+05

0.25000E+05 -0.16069E+03

0.00000E+00

0.00000E+00

0.540E+02

0.00000E+00

0.24877E+05

0.25000E+05 -0.12333E+03

0.00000E+00

0.00000E+00

0.552E+02

0.00000E+00

0.24904E+05

0.25000E+05 -0.95923E+02

0.00000E+00

0.00000E+00

0.564E+02

0.00000E+00

0.24924E+05

0.25000E+05 -0.75706E+02

0.00000E+00

0.00000E+00

0.576E+02

0.00000E+00

0.24939E+05

0.25000E+05 -0.60694E+02

0.00000E+00

0.00000E+00

0.588E+02

0.00000E+00

0.24951E+05

0.25000E+05 -0.49462E+02

0.00000E+00

0.00000E+00

Pressure Transient Analysis 191

Figure 3.2.8d. Inverse model screen. A pore pressure of 24,947 psi is predicted versus a known value of 25,000 psi. The predicted mobility is 1.042 md/cp versus 1.000 md/cp. A fluid compressibility 0.0000031/psi is consistent with that assumed in the exact forward pressure transient simulator FT-00.

192 Formation Testing Volume 3

3.2.9

Validation of PTA-App-09 Inverse Model

Figure 3.2.9a. Model 9 function. Our “inverse model” is an approximate one designed to predict pore pressure, spherical mobility and fluid compressibility in low mobility applications where flowline storage cannot be neglected. The method uses single-probe pressure transient data typically obtained during the first minute of logging – pressures which are highly unsteady and not steady-state. It is validated using pressure transient data created by our exact “forward model” FT-00 (immediately below) where formation and fluid properties, tool constants, reservoir conditions and pumping schedules are defined. Three time-pressure data points are arbitrarily selected (red from the output listing) and inputted into the approximate inverse model. Validations performed in the “black screen, DOS

Pressure Transient Analysis 193

window” show that predicted pore pressure, spherical mobility and fluid compressibility are consistent with those assumed in FT-00. The reader is invited to use different time and pressure inputs to evaluate the method. Different data points will give slightly different predictions. Times should not be too close, but preferably, several seconds apart. The request in the “black screen,” that is, “Use decimals after integers” no longer applies to our latest software.

Figure 3.2.9b. FT-00 exact inputs.

194 Formation Testing Volume 3

Figure 3.2.9c. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.120E+01

0.10000E+01

0.24667E+05

0.24974E+05 -0.33287E+03 -0.26288E+02

0.78976E-01

0.240E+01

0.10000E+01

0.24434E+05

0.24957E+05 -0.56650E+03 -0.42874E+02

0.75683E-01

0.360E+01

0.10000E+01

0.24268E+05

0.24948E+05 -0.73187E+03 -0.51607E+02

0.70514E-01

0.480E+01

0.10000E+01

0.24151E+05

0.24943E+05 -0.84946E+03 -0.56524E+02

0.66541E-01

0.600E+01

0.10000E+01

0.24067E+05

0.24941E+05 -0.93335E+03 -0.59336E+02

0.63573E-01

0.720E+01

0.10000E+01

0.24007E+05

0.24939E+05 -0.99342E+03 -0.60922E+02

0.61326E-01

0.840E+01

0.10000E+01

0.23963E+05

0.24938E+05 -0.10366E+04 -0.61775E+02

0.59595E-01

0.960E+01

0.10000E+01

0.23932E+05

0.24938E+05 -0.10677E+04 -0.62186E+02

0.58244E-01

0.108E+02

0.20000E+01

0.23675E+05

0.24920E+05 -0.13252E+04 -0.79716E+02

0.60156E-01

0.120E+02

0.20000E+01

0.23396E+05

0.24899E+05 -0.16040E+04 -0.10088E+03

0.62889E-01

0.132E+02

0.20000E+01

0.23199E+05

0.24889E+05 -0.18015E+04 -0.11147E+03

0.61879E-01

0.144E+02

0.20000E+01

0.23058E+05

0.24883E+05 -0.19419E+04 -0.11726E+03

0.60384E-01

0.156E+02

0.20000E+01

0.22958E+05

0.24880E+05 -0.20423E+04 -0.12049E+03

0.58996E-01

0.168E+02

0.20000E+01

0.22886E+05

0.24878E+05 -0.21143E+04 -0.12224E+03

0.57818E-01

0.180E+02

0.20000E+01

0.22834E+05

0.24877E+05 -0.21661E+04 -0.12313E+03

0.56846E-01

Pressure Transient Analysis 195 0.192E+02

0.20000E+01

0.22796E+05

0.24876E+05 -0.22036E+04 -0.12417E+03

0.56350E-01

0.204E+02

0.30000E+01

0.22645E+05

0.24869E+05 -0.23554E+04 -0.13069E+03

0.55486E-01

0.216E+02

0.30000E+01

0.22329E+05

0.24842E+05 -0.26706E+04 -0.15783E+03

0.59101E-01

0.228E+02

0.30000E+01

0.22107E+05

0.24829E+05 -0.28933E+04 -0.17091E+03

0.59072E-01

0.240E+02

0.30000E+01

0.21948E+05

0.24822E+05 -0.30516E+04 -0.17783E+03

0.58275E-01

0.252E+02

0.30000E+01

0.21835E+05

0.24818E+05 -0.31647E+04 -0.18159E+03

0.57381E-01

0.264E+02

0.30000E+01

0.21754E+05

0.24816E+05 -0.32457E+04 -0.18358E+03

0.56562E-01

0.276E+02

0.30000E+01

0.21696E+05

0.24815E+05 -0.33041E+04 -0.18455E+03

0.55853E-01

0.288E+02

0.30000E+01

0.21654E+05

0.24814E+05 -0.33463E+04 -0.18585E+03

0.55539E-01

0.300E+02

0.40000E+01

0.21623E+05

0.24813E+05 -0.33770E+04 -0.18677E+03

0.55307E-01

0.312E+02

0.40000E+01

0.21268E+05

0.24787E+05 -0.37323E+04 -0.21332E+03

0.57155E-01

0.324E+02

0.40000E+01

0.21018E+05

0.24770E+05 -0.39824E+04 -0.22977E+03

0.57696E-01

0.336E+02

0.40000E+01

0.20840E+05

0.24762E+05 -0.41600E+04 -0.23816E+03

0.57249E-01

0.348E+02

0.40000E+01

0.20713E+05

0.24757E+05 -0.42867E+04 -0.24262E+03

0.56597E-01

0.360E+02

0.40000E+01

0.20622E+05

0.24755E+05 -0.43775E+04 -0.24493E+03

0.55950E-01

0.372E+02

0.40000E+01

0.20557E+05

0.24754E+05 -0.44429E+04 -0.24600E+03

0.55370E-01

0.384E+02

0.40000E+01

0.20510E+05

0.24753E+05 -0.44901E+04 -0.24710E+03

0.55032E-01

0.396E+02

0.40000E+01

0.20476E+05

0.24751E+05 -0.45244E+04 -0.24874E+03

0.54979E-01

0.408E+02

0.50000E+01

0.20216E+05

0.24733E+05 -0.47844E+04 -0.26671E+03

0.55747E-01

0.420E+02

0.50000E+01

0.19935E+05

0.24712E+05 -0.50653E+04 -0.28785E+03

0.56827E-01

0.432E+02

0.50000E+01

0.19736E+05

0.24702E+05 -0.52645E+04 -0.29816E+03

0.56636E-01

0.444E+02

0.50000E+01

0.19594E+05

0.24696E+05 -0.54064E+04 -0.30352E+03

0.56141E-01

0.456E+02

0.50000E+01

0.19492E+05

0.24694E+05 -0.55079E+04 -0.30625E+03

0.55601E-01

0.468E+02

0.50000E+01

0.19419E+05

0.24693E+05 -0.55809E+04 -0.30749E+03

0.55097E-01

0.480E+02

0.50000E+01

0.19366E+05

0.24692E+05 -0.56336E+04 -0.30788E+03

0.54651E-01

0.492E+02

0.50000E+01

0.19328E+05

0.24690E+05 -0.56719E+04 -0.31043E+03

0.54731E-01

0.504E+02

0.00000E+00

0.19923E+05

0.24719E+05 -0.50765E+04 -0.28112E+03

0.55377E-01

0.516E+02

0.00000E+00

0.21379E+05

0.24854E+05 -0.36206E+04 -0.14593E+03

0.40306E-01

0.528E+02

0.00000E+00

0.22405E+05

0.24921E+05 -0.25954E+04 -0.79441E+02

0.30609E-01

0.540E+02

0.00000E+00

0.23131E+05

0.24957E+05 -0.18690E+04 -0.42877E+02

0.22941E-01

0.552E+02

0.00000E+00

0.23648E+05

0.24978E+05 -0.13524E+04 -0.21676E+02

0.16027E-01

0.564E+02

0.00000E+00

0.24016E+05

0.24991E+05 -0.98379E+03 -0.91756E+01

0.93268E-02

0.576E+02

0.00000E+00

0.24280E+05

0.24998E+05 -0.71996E+03 -0.18361E+01

0.25503E-02

0.588E+02

0.00000E+00

0.24469E+05

0.25000E+05 -0.53053E+03

0.00000E+00

0.00000E+00

196 Formation Testing Volume 3

Figure 3.2.9d. Inverse model screen. A pore pressure of 24,993 psi is predicted versus the known value of 25,000 psi. A mobility of 1.025 md/cp is obtained versus the exact value of 1.000 md/cp. Fluid compressibility is also acurate, taking on a value of 0.0000031/psi as opposed to 0.0000030/psi.

Pressure Transient Analysis 197

3.2.10

Validation of PTA-App-10 Inverse Model

Figure 3.2.10a. Model 10 function. Our “inverse model” is an approximate one designed to predict pore pressure, spherical mobility and fluid compressibility in low mobility applications where flowline storage cannot be neglected. The method uses single-probe pressure transient data typically obtained during the first minute of logging – pressures which are highly unsteady and not steady-state. It is validated using pressure transient data created by our exact “forward model” FT-00 (immediately below) where formation and fluid properties, tool constants, reservoir conditions and pumping schedules are defined. Three time-pressure data points are arbitrarily selected (red from the output listing) and inputted into the approximate inverse model. Validations performed in the “black screen, DOS window” show that predicted pore pressure, spherical mobility and fluid

198 Formation Testing Volume 3

compressibility are consistent with those assumed in FT-00. The reader is invited to use different time and pressure inputs to evaluate the method. Different data points will give slightly different predictions. Times should not be too close, but preferably, several seconds apart. The request in the “black screen,” that is, “Use decimals after integers” no longer applies to our latest software.

Figure 3.2.10b. FT-00 exact inputs.

Pressure Transient Analysis 199

Figure 3.2.10c. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.120E+01

0.10000E+01

0.24667E+05

0.24974E+05 -0.33287E+03 -0.26288E+02

0.78976E-01

0.240E+01

0.10000E+01

0.24434E+05

0.24957E+05 -0.56650E+03 -0.42874E+02

0.75683E-01

0.360E+01

0.10000E+01

0.24268E+05

0.24948E+05 -0.73187E+03 -0.51607E+02

0.70514E-01

0.480E+01

0.10000E+01

0.24151E+05

0.24943E+05 -0.84946E+03 -0.56524E+02

0.66541E-01

0.600E+01

0.10000E+01

0.24067E+05

0.24941E+05 -0.93335E+03 -0.59336E+02

0.63573E-01

0.720E+01

0.10000E+01

0.24007E+05

0.24939E+05 -0.99342E+03 -0.60922E+02

0.61326E-01

0.840E+01

0.10000E+01

0.23963E+05

0.24938E+05 -0.10366E+04 -0.61775E+02

0.59595E-01

0.960E+01

0.10000E+01

0.23932E+05

0.24938E+05 -0.10677E+04 -0.62186E+02

0.58244E-01

0.108E+02

0.20000E+01

0.23675E+05

0.24920E+05 -0.13252E+04 -0.79716E+02

0.60156E-01

0.120E+02

0.20000E+01

0.23396E+05

0.24899E+05 -0.16040E+04 -0.10088E+03

0.62889E-01

0.132E+02

0.20000E+01

0.23199E+05

0.24889E+05 -0.18015E+04 -0.11147E+03

0.61879E-01

0.144E+02

0.20000E+01

0.23058E+05

0.24883E+05 -0.19419E+04 -0.11726E+03

0.60384E-01

0.156E+02

0.20000E+01

0.22958E+05

0.24880E+05 -0.20423E+04 -0.12049E+03

0.58996E-01

0.168E+02

0.20000E+01

0.22886E+05

0.24878E+05 -0.21143E+04 -0.12224E+03

0.57818E-01

0.180E+02

0.20000E+01

0.22834E+05

0.24877E+05 -0.21661E+04 -0.12313E+03

0.56846E-01

200 Formation Testing Volume 3 0.192E+02

0.20000E+01

0.22796E+05

0.24876E+05 -0.22036E+04 -0.12417E+03

0.56350E-01

0.204E+02

0.30000E+01

0.22645E+05

0.24869E+05 -0.23554E+04 -0.13069E+03

0.55486E-01

0.216E+02

0.30000E+01

0.22329E+05

0.24842E+05 -0.26706E+04 -0.15783E+03

0.59101E-01

0.228E+02

0.30000E+01

0.22107E+05

0.24829E+05 -0.28933E+04 -0.17091E+03

0.59072E-01

0.240E+02

0.30000E+01

0.21948E+05

0.24822E+05 -0.30516E+04 -0.17783E+03

0.58275E-01

0.252E+02

0.30000E+01

0.21835E+05

0.24818E+05 -0.31647E+04 -0.18159E+03

0.57381E-01

0.264E+02

0.30000E+01

0.21754E+05

0.24816E+05 -0.32457E+04 -0.18358E+03

0.56562E-01

0.276E+02

0.30000E+01

0.21696E+05

0.24815E+05 -0.33041E+04 -0.18455E+03

0.55853E-01

0.288E+02

0.30000E+01

0.21654E+05

0.24814E+05 -0.33463E+04 -0.18585E+03

0.55539E-01

0.300E+02

0.40000E+01

0.21623E+05

0.24813E+05 -0.33770E+04 -0.18677E+03

0.55307E-01

0.312E+02

0.40000E+01

0.21268E+05

0.24787E+05 -0.37323E+04 -0.21332E+03

0.57155E-01

0.324E+02

0.40000E+01

0.21018E+05

0.24770E+05 -0.39824E+04 -0.22977E+03

0.57696E-01

0.336E+02

0.40000E+01

0.20840E+05

0.24762E+05 -0.41600E+04 -0.23816E+03

0.57249E-01

0.348E+02

0.40000E+01

0.20713E+05

0.24757E+05 -0.42867E+04 -0.24262E+03

0.56597E-01

0.360E+02

0.40000E+01

0.20622E+05

0.24755E+05 -0.43775E+04 -0.24493E+03

0.55950E-01

0.372E+02

0.40000E+01

0.20557E+05

0.24754E+05 -0.44429E+04 -0.24600E+03

0.55370E-01

0.384E+02

0.40000E+01

0.20510E+05

0.24753E+05 -0.44901E+04 -0.24710E+03

0.55032E-01

0.396E+02

0.40000E+01

0.20476E+05

0.24751E+05 -0.45244E+04 -0.24874E+03

0.54979E-01

0.408E+02

0.50000E+01

0.20216E+05

0.24733E+05 -0.47844E+04 -0.26671E+03

0.55747E-01

0.420E+02

0.50000E+01

0.19935E+05

0.24712E+05 -0.50653E+04 -0.28785E+03

0.56827E-01

0.432E+02

0.50000E+01

0.19736E+05

0.24702E+05 -0.52645E+04 -0.29816E+03

0.56636E-01

0.444E+02

0.50000E+01

0.19594E+05

0.24696E+05 -0.54064E+04 -0.30352E+03

0.56141E-01

0.456E+02

0.50000E+01

0.19492E+05

0.24694E+05 -0.55079E+04 -0.30625E+03

0.55601E-01

0.468E+02

0.50000E+01

0.19419E+05

0.24693E+05 -0.55809E+04 -0.30749E+03

0.55097E-01

0.480E+02

0.50000E+01

0.19366E+05

0.24692E+05 -0.56336E+04 -0.30788E+03

0.54651E-01

0.492E+02

0.50000E+01

0.19328E+05

0.24690E+05 -0.56719E+04 -0.31043E+03

0.54731E-01

0.504E+02

0.00000E+00

0.19923E+05

0.24719E+05 -0.50765E+04 -0.28112E+03

0.55377E-01

0.516E+02

0.00000E+00

0.21379E+05

0.24854E+05 -0.36206E+04 -0.14593E+03

0.40306E-01

0.528E+02

0.00000E+00

0.22405E+05

0.24921E+05 -0.25954E+04 -0.79441E+02

0.30609E-01

0.540E+02

0.00000E+00

0.23131E+05

0.24957E+05 -0.18690E+04 -0.42877E+02

0.22941E-01

0.552E+02

0.00000E+00

0.23648E+05

0.24978E+05 -0.13524E+04 -0.21676E+02

0.16027E-01

0.564E+02

0.00000E+00

0.24016E+05

0.24991E+05 -0.98379E+03 -0.91756E+01

0.93268E-02

0.576E+02

0.00000E+00

0.24280E+05

0.24998E+05 -0.71996E+03 -0.18361E+01

0.25503E-02

0.588E+02

0.00000E+00

0.24469E+05

0.25000E+05 -0.53053E+03

0.00000E+00

0.00000E+00

Pressure Transient Analysis 201

Figure 3.2.10d. Inverse model screen. Here, a pore pressure of 24,926 psi is predicted versus a known 25,000 psi. The mobility is 1.040 md/cp versus 1.000 md/cp, while fluid compressibility is obtained as 0.0000031/psi. Good qualitative agreement is seen.

202 Formation Testing Volume 3

3.2.11

Validation of PTA-App-11 Inverse Model

Figure 3.2.11a. Model 11 function. Our “inverse model” is an approximate one designed to predict pore pressure, spherical mobility and fluid compressibility in low mobility applications where flowline storage cannot be neglected. The method uses single-probe pressure transient data typically obtained during the first minute of logging – pressures which are highly unsteady and not steady-state. It is validated using pressure transient data created by our exact “forward model” FT-00 (immediately below) where formation and fluid properties, tool constants, reservoir conditions and pumping schedules are defined. Three time-pressure data points are arbitrarily selected (red from the output listing) and inputted into the approximate inverse model. Validations performed in the “black screen, DOS window” show that predicted pore pressure, spherical mobility and fluid compressibility are consistent with those assumed in FT-00. The reader

Pressure Transient Analysis 203

is invited to use different time and pressure inputs to evaluate the method. Different data points will give slightly different predictions. Times should not be too close, but preferably, several seconds apart. The request in the “black screen,” that is, “Use decimals after integers” no longer applies to our latest software.

Validation No. 1

Figure 3.2.11b. FT-00 exact inputs.

204 Formation Testing Volume 3

Figure 3.2.11c. Source probe pressure and pumpout schedule. Flow rate diagram shows accelerating withdrawal of piston. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.140E+01

0.10000E+01

0.24622E+05

0.24970E+05 -0.37768E+03 -0.29935E+02

0.79261E-01

0.280E+01

0.10000E+01

0.24372E+05

0.24954E+05 -0.62804E+03 -0.46378E+02

0.73847E-01

0.420E+01

0.10000E+01

0.24204E+05

0.24946E+05 -0.79564E+03 -0.54407E+02

0.68382E-01

0.560E+01

0.10000E+01

0.24092E+05

0.24941E+05 -0.90845E+03 -0.58568E+02

0.64470E-01

0.700E+01

0.10000E+01

0.24015E+05

0.24939E+05 -0.98474E+03 -0.60719E+02

0.61660E-01

0.840E+01

0.10000E+01

0.23963E+05

0.24938E+05 -0.10366E+04 -0.61775E+02

0.59595E-01

0.980E+01

0.10000E+01

0.23928E+05

0.24938E+05 -0.10720E+04 -0.62225E+02

0.58048E-01

0.112E+02

0.20000E+01

0.23571E+05

0.24911E+05 -0.14291E+04 -0.88632E+02

0.62019E-01

0.126E+02

0.20000E+01

0.23289E+05

0.24893E+05 -0.17112E+04 -0.10700E+03

0.62530E-01

0.140E+02

0.20000E+01

0.23100E+05

0.24884E+05 -0.19003E+04 -0.11570E+03

0.60883E-01

0.154E+02

0.20000E+01

0.22972E+05

0.24880E+05 -0.20278E+04 -0.12007E+03

0.59213E-01

Pressure Transient Analysis 205 0.168E+02

0.20000E+01

0.22886E+05

0.24878E+05 -0.21143E+04 -0.12224E+03

0.57818E-01

0.182E+02

0.20000E+01

0.22827E+05

0.24877E+05 -0.21732E+04 -0.12334E+03

0.56754E-01

0.196E+02

0.20000E+01

0.22786E+05

0.24876E+05 -0.22136E+04 -0.12437E+03

0.56185E-01

0.210E+02

0.30000E+01

0.22473E+05

0.24853E+05 -0.25268E+04 -0.14681E+03

0.58102E-01

0.224E+02

0.30000E+01

0.22173E+05

0.24833E+05 -0.28273E+04 -0.16747E+03

0.59233E-01

0.238E+02

0.30000E+01

0.21971E+05

0.24823E+05 -0.30288E+04 -0.17695E+03

0.58423E-01

0.252E+02

0.30000E+01

0.21835E+05

0.24818E+05 -0.31647E+04 -0.18159E+03

0.57381E-01

0.266E+02

0.30000E+01

0.21743E+05

0.24816E+05 -0.32568E+04 -0.18380E+03

0.56436E-01

0.280E+02

0.30000E+01

0.21680E+05

0.24815E+05 -0.33197E+04 -0.18472E+03

0.55642E-01

0.294E+02

0.30000E+01

0.21637E+05

0.24814E+05 -0.33629E+04 -0.18642E+03

0.55434E-01

0.308E+02

0.40000E+01

0.21372E+05

0.24796E+05 -0.36277E+04 -0.20438E+03

0.56340E-01

0.322E+02

0.40000E+01

0.21054E+05

0.24772E+05 -0.39465E+04 -0.22777E+03

0.57714E-01

0.336E+02

0.40000E+01

0.20840E+05

0.24762E+05 -0.41600E+04 -0.23816E+03

0.57249E-01

0.350E+02

0.40000E+01

0.20696E+05

0.24757E+05 -0.43040E+04 -0.24312E+03

0.56486E-01

0.364E+02

0.40000E+01

0.20598E+05

0.24755E+05 -0.44017E+04 -0.24539E+03

0.55749E-01

0.378E+02

0.40000E+01

0.20532E+05

0.24754E+05 -0.44684E+04 -0.24625E+03

0.55109E-01

0.392E+02

0.40000E+01

0.20486E+05

0.24752E+05 -0.45141E+04 -0.24834E+03

0.55014E-01

0.406E+02

0.50000E+01

0.20273E+05

0.24739E+05 -0.47273E+04 -0.26127E+03

0.55268E-01

0.420E+02

0.50000E+01

0.19935E+05

0.24712E+05 -0.50653E+04 -0.28785E+03

0.56827E-01

0.434E+02

0.50000E+01

0.19708E+05

0.24701E+05 -0.52915E+04 -0.29931E+03

0.56563E-01

0.448E+02

0.50000E+01

0.19556E+05

0.24695E+05 -0.54440E+04 -0.30465E+03

0.55960E-01

0.462E+02

0.50000E+01

0.19453E+05

0.24693E+05 -0.55474E+04 -0.30701E+03

0.55343E-01

0.476E+02

0.50000E+01

0.19382E+05

0.24692E+05 -0.56179E+04 -0.30782E+03

0.54793E-01

0.490E+02

0.50000E+01

0.19334E+05

0.24690E+05 -0.56663E+04 -0.31011E+03

0.54729E-01

0.504E+02

0.60000E+01

0.19176E+05

0.24682E+05 -0.58245E+04 -0.31761E+03

0.54530E-01

0.518E+02

0.60000E+01

0.18817E+05

0.24652E+05 -0.61830E+04 -0.34767E+03

0.56231E-01

0.532E+02

0.60000E+01

0.18577E+05

0.24640E+05 -0.64226E+04 -0.36038E+03

0.56112E-01

0.546E+02

0.60000E+01

0.18416E+05

0.24634E+05 -0.65840E+04 -0.36617E+03

0.55616E-01

0.560E+02

0.60000E+01

0.18307E+05

0.24631E+05 -0.66933E+04 -0.36865E+03

0.55078E-01

0.574E+02

0.60000E+01

0.18232E+05

0.24631E+05 -0.67679E+04 -0.36943E+03

0.54586E-01

0.588E+02

0.60000E+01

0.18181E+05

0.24628E+05 -0.68191E+04 -0.37170E+03

0.54509E-01

0.602E+02

0.00000E+00

0.18531E+05

0.24632E+05 -0.64686E+04 -0.36774E+03

0.56850E-01

0.616E+02

0.00000E+00

0.20641E+05

0.24825E+05 -0.43595E+04 -0.17512E+03

0.40170E-01

0.630E+02

0.00000E+00

0.22041E+05

0.24914E+05 -0.29590E+04 -0.86190E+02

0.29128E-01

0.644E+02

0.00000E+00

0.22979E+05

0.24959E+05 -0.20214E+04 -0.41430E+02

0.20496E-01

0.658E+02

0.00000E+00

0.23610E+05

0.24982E+05 -0.13905E+04 -0.17528E+02

0.12606E-01

0.672E+02

0.00000E+00

0.24036E+05

0.24995E+05 -0.96406E+03 -0.46272E+01

0.47998E-02

0.686E+02

0.00000E+00

0.24325E+05

0.25000E+05 -0.67466E+03

0.00000E+00

0.00000E+00

206 Formation Testing Volume 3

Figure 3.2.11d. Inverse model screen. Pore pressure here is predicted as 25,006 psi versus a known value of 25,000 psi. The mobility is 1.023 md/cp versus 1.000 md/cp and the fluid compressibility is obtained as 0.0000031/psi. Good results.

Pressure Transient Analysis 207

Validation No. 2

Figure 3.2.11e. FT-00 exact inputs.

208 Formation Testing Volume 3

Figure 3.2.11f. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi)

NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.800E+00

0.10000E+01

0.24765E+05

0.24983E+05 -0.23494E+03 -0.17383E+02

0.73986E-01

0.160E+01

0.10000E+01

0.24580E+05

0.24967E+05 -0.41994E+03 -0.33153E+02

0.78947E-01

0.240E+01

0.10000E+01

0.24434E+05

0.24957E+05 -0.56650E+03 -0.42874E+02

0.75683E-01

0.320E+01

0.10000E+01

0.24317E+05

0.24951E+05 -0.68291E+03 -0.49246E+02

0.72112E-01

0.400E+01

0.10000E+01

0.24224E+05

0.24946E+05 -0.77557E+03 -0.53560E+02

0.69058E-01

0.480E+01

0.10000E+01

0.24151E+05

0.24943E+05 -0.84946E+03 -0.56524E+02

0.66541E-01

0.560E+01 -0.10000E+01

0.24454E+05

0.24965E+05 -0.54554E+03 -0.34611E+02

0.63443E-01

0.640E+01 -0.10000E+01

0.24800E+05

0.25000E+05 -0.20028E+03 -0.10084E+00

0.50352E-03

0.720E+01 -0.10000E+01

0.25073E+05

0.25021E+05

0.72520E+02

0.20736E+02

0.28594E+00

0.800E+01 -0.10000E+01

0.25289E+05

0.25034E+05

0.28878E+03

0.34214E+02

0.11848E+00

0.880E+01 -0.10000E+01

0.25461E+05

0.25043E+05

0.46060E+03

0.43308E+02

0.94026E-01

0.960E+01 -0.10000E+01

0.25597E+05

0.25050E+05

0.59735E+03

0.49579E+02

0.82998E-01

0.104E+02

0.25457E+05

0.25042E+05

0.45703E+03

0.41784E+02

0.91425E-01

0.10000E+01

Pressure Transient Analysis 209 0.112E+02

0.10000E+01

0.25128E+05

0.25004E+05

0.44179E+01

0.34617E-01

0.120E+02

0.10000E+01

0.24868E+05

0.24982E+05 -0.13201E+03 -0.17991E+02

0.12762E+03

0.13629E+00

0.128E+02

0.10000E+01

0.24662E+05

0.24968E+05 -0.33750E+03 -0.32187E+02

0.95367E-01

0.136E+02

0.10000E+01

0.24499E+05

0.24958E+05 -0.50056E+03 -0.41661E+02

0.83229E-01

0.144E+02

0.10000E+01

0.24370E+05

0.24952E+05 -0.63019E+03 -0.48218E+02

0.76514E-01

0.152E+02 -0.10000E+01

0.24395E+05

0.24949E+05 -0.60481E+03 -0.51008E+02

0.84337E-01

0.160E+02 -0.10000E+01

0.24755E+05

0.24988E+05 -0.24503E+03 -0.12066E+02

0.49242E-01

0.168E+02 -0.10000E+01

0.25038E+05

0.25013E+05

0.38188E+02

0.13361E+02

0.34987E+00

0.176E+02 -0.10000E+01

0.25262E+05

0.25029E+05

0.26228E+03

0.29175E+02

0.11123E+00

0.184E+02 -0.10000E+01

0.25440E+05

0.25040E+05

0.44008E+03

0.39622E+02

0.90035E-01

0.192E+02 -0.10000E+01

0.25581E+05

0.25047E+05

0.58142E+03

0.46753E+02

0.80412E-01

0.200E+02

0.10000E+01

0.25694E+05

0.25052E+05

0.69398E+03

0.51958E+02

0.74870E-01

0.208E+02

0.10000E+01

0.25314E+05

0.25021E+05

0.31386E+03

0.20842E+02

0.66406E-01

0.216E+02

0.10000E+01

0.25016E+05

0.24992E+05

0.15550E+02 -0.81551E+01 -0.52444E+00

0.224E+02

0.10000E+01

0.24780E+05

0.24974E+05 -0.22022E+03 -0.25837E+02

0.11732E+00

0.232E+02

0.10000E+01

0.24593E+05

0.24963E+05 -0.40713E+03 -0.37380E+02

0.91815E-01

0.240E+02

0.10000E+01

0.24444E+05

0.24955E+05 -0.55562E+03 -0.45210E+02

0.81368E-01

0.248E+02

0.10000E+01

0.24326E+05

0.24949E+05 -0.67379E+03 -0.50822E+02

0.75427E-01

0.76177E-01

0.256E+02 -0.10000E+01

0.24595E+05

0.24969E+05 -0.40507E+03 -0.30857E+02

0.264E+02 -0.10000E+01

0.24912E+05

0.25002E+05 -0.87805E+02

0.22711E+01 -0.25865E-01

0.272E+02 -0.10000E+01

0.25163E+05

0.25022E+05

0.16270E+03

0.22127E+02

0.13600E+00

0.280E+02 -0.10000E+01

0.25361E+05

0.25035E+05

0.36118E+03

0.34911E+02

0.96657E-01

0.288E+02 -0.10000E+01

0.25519E+05

0.25044E+05

0.51881E+03

0.43518E+02

0.83879E-01

0.296E+02 -0.10000E+01

0.25644E+05

0.25050E+05

0.64422E+03

0.49579E+02

0.76959E-01

0.304E+02

0.00000E+00

0.25619E+05

0.25048E+05

0.61949E+03

0.47865E+02

0.77265E-01

0.312E+02

0.00000E+00

0.25491E+05

0.25031E+05

0.49104E+03

0.30706E+02

0.62534E-01

0.320E+02

0.00000E+00

0.25390E+05

0.25021E+05

0.39018E+03

0.20561E+02

0.52698E-01

0.328E+02

0.00000E+00

0.25311E+05

0.25014E+05

0.31059E+03

0.14192E+02

0.45693E-01

0.336E+02

0.00000E+00

0.25248E+05

0.25010E+05

0.24762E+03

0.99459E+01

0.40166E-01

0.344E+02

0.00000E+00

0.25198E+05

0.25007E+05

0.19769E+03

0.69607E+01

0.35209E-01

0.352E+02

0.00000E+00

0.25158E+05

0.25005E+05

0.15805E+03

0.46440E+01

0.29383E-01

0.360E+02

0.00000E+00

0.25127E+05

0.25003E+05

0.12652E+03

0.30039E+01

0.23742E-01

0.368E+02

0.00000E+00

0.25101E+05

0.25002E+05

0.10143E+03

0.18393E+01

0.18135E-01

0.376E+02

0.00000E+00

0.25081E+05

0.25001E+05

0.81423E+02

0.10135E+01

0.12447E-01

0.384E+02

0.00000E+00

0.25065E+05

0.25000E+05

0.65462E+02

0.43109E+00

0.65853E-02

0.392E+02

0.00000E+00

0.25053E+05

0.25000E+05

0.52713E+02

0.24554E-01

0.46582E-03

210 Formation Testing Volume 3

Figure 3.2.11g. Inverse model screen. The pore pressure is predicted exactly as 25,000 psi, and the mobility is 1.056 md/cp versus an input value of 1.000 md/cp. Very good. In the next validation, we will reduce the input mobility by a factor of 10.

Pressure Transient Analysis 211

Validation No. 3

Figure 3.2.11h. FT-00 exact inputs.

212 Formation Testing Volume 3

Figure 3.2.11i. Source probe pressure and pumpout schedule. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.800E+00

0.10000E+01

0.24738E+05

0.25000E+05 -0.26235E+03 -0.73748E-01

0.28110E-03

0.160E+01

0.10000E+01

0.24482E+05

0.24996E+05 -0.51775E+03 -0.42500E+01

0.82087E-02

0.240E+01

0.10000E+01

0.24233E+05

0.24982E+05 -0.76674E+03 -0.17797E+02

0.23211E-01

0.320E+01

0.10000E+01

0.23990E+05

0.24962E+05 -0.10096E+04 -0.37835E+02

0.37474E-01

0.400E+01

0.10000E+01

0.23753E+05

0.24939E+05 -0.12466E+04 -0.60813E+02

0.48782E-01

0.480E+01

0.10000E+01

0.23522E+05

0.24915E+05 -0.14779E+04 -0.84639E+02

0.57270E-01

0.560E+01 -0.10000E+01

0.23691E+05

0.24892E+05 -0.13088E+04 -0.10823E+03

0.82697E-01

0.640E+01 -0.10000E+01

0.23985E+05

0.24874E+05 -0.10151E+04 -0.12642E+03

0.12454E+00

0.720E+01 -0.10000E+01

0.24271E+05

0.24874E+05 -0.72914E+03 -0.12572E+03

0.17243E+00

0.800E+01 -0.10000E+01

0.24549E+05

0.24891E+05 -0.45051E+03 -0.10895E+03

0.24184E+00

0.880E+01 -0.10000E+01

0.24821E+05

0.24916E+05 -0.17883E+03 -0.83739E+02

0.46825E+00

0.960E+01 -0.10000E+01

0.25086E+05

0.24945E+05

0.86167E+02 -0.54988E+02 -0.63816E+00

0.104E+02

0.10000E+01

0.25080E+05

0.24975E+05

0.80491E+02 -0.25297E+02 -0.31429E+00

0.112E+02

0.10000E+01

0.24815E+05

0.25002E+05 -0.18472E+03

0.18834E+01 -0.10196E-01

0.120E+02

0.10000E+01

0.24557E+05

0.25012E+05 -0.44276E+03

0.12464E+02 -0.28150E-01

Pressure Transient Analysis 213 0.128E+02

0.10000E+01

0.24306E+05

0.25005E+05 -0.69418E+03

0.136E+02

0.10000E+01

0.24061E+05

0.24987E+05 -0.93932E+03 -0.13155E+02

0.48804E+01 -0.70305E-02 0.14005E-01

0.144E+02

0.10000E+01

0.23822E+05

0.24964E+05 -0.11784E+04 -0.36026E+02

0.30571E-01

0.152E+02 -0.10000E+01

0.23721E+05

0.24939E+05 -0.12791E+04 -0.60666E+02

0.47429E-01

0.160E+02 -0.10000E+01

0.24014E+05

0.24915E+05 -0.98576E+03 -0.84778E+02

0.86003E-01

0.168E+02 -0.10000E+01

0.24300E+05

0.24904E+05 -0.70047E+03 -0.96174E+02

0.13730E+00

0.176E+02 -0.10000E+01

0.24577E+05

0.24912E+05 -0.42253E+03 -0.88359E+02

0.20912E+00

0.184E+02 -0.10000E+01

0.24848E+05

0.24932E+05 -0.15156E+03 -0.68480E+02

0.45183E+00

0.192E+02 -0.10000E+01

0.25113E+05

0.24957E+05

0.11273E+03 -0.42881E+02 -0.38039E+00

0.200E+02

0.10000E+01

0.25371E+05

0.24985E+05

0.37058E+03 -0.15145E+02 -0.40868E-01

0.208E+02

0.10000E+01

0.25098E+05

0.25013E+05

0.97500E+02

0.216E+02

0.10000E+01

0.24832E+05

0.25032E+05 -0.16770E+03

0.31797E+02 -0.18960E+00

0.224E+02

0.10000E+01

0.24574E+05

0.25031E+05 -0.42595E+03

0.31063E+02 -0.72926E-01

0.232E+02

0.10000E+01

0.24322E+05

0.25016E+05 -0.67766E+03

0.16114E+02 -0.23779E-01

0.240E+02

0.10000E+01

0.24077E+05

0.24994E+05 -0.92313E+03 -0.59986E+01

0.64981E-02

0.248E+02

0.10000E+01

0.23837E+05

0.24969E+05 -0.11626E+04 -0.31075E+02

0.26729E-01

0.12739E+02

0.13065E+00

0.256E+02 -0.10000E+01

0.23999E+05

0.24943E+05 -0.10013E+04 -0.56909E+02

0.56833E-01

0.264E+02 -0.10000E+01

0.24285E+05

0.24922E+05 -0.71526E+03 -0.77744E+02

0.10869E+00

0.272E+02 -0.10000E+01

0.24563E+05

0.24920E+05 -0.43673E+03 -0.79808E+02

0.18274E+00

0.280E+02 -0.10000E+01

0.24835E+05

0.24934E+05 -0.16527E+03 -0.65745E+02

0.39781E+00

0.288E+02 -0.10000E+01

0.25099E+05

0.24957E+05

0.99457E+02 -0.43118E+02 -0.43354E+00

0.296E+02 -0.10000E+01

0.25358E+05

0.24983E+05

0.35770E+03 -0.16798E+02 -0.46960E-01

0.304E+02

0.00000E+00

0.25478E+05

0.25011E+05

0.47758E+03

0.10628E+02

0.22254E-01

0.312E+02

0.00000E+00

0.25465E+05

0.25037E+05

0.46475E+03

0.36780E+02

0.79140E-01

0.320E+02

0.00000E+00

0.25453E+05

0.25054E+05

0.45265E+03

0.54290E+02

0.11994E+00

0.328E+02

0.00000E+00

0.25441E+05

0.25062E+05

0.44109E+03

0.62226E+02

0.14107E+00

0.336E+02

0.00000E+00

0.25430E+05

0.25064E+05

0.42998E+03

0.64385E+02

0.14974E+00

0.344E+02

0.00000E+00

0.25419E+05

0.25064E+05

0.41927E+03

0.63563E+02

0.15160E+00

0.352E+02

0.00000E+00

0.25409E+05

0.25061E+05

0.40890E+03

0.61304E+02

0.14992E+00

0.360E+02

0.00000E+00

0.25399E+05

0.25058E+05

0.39887E+03

0.58416E+02

0.14645E+00

0.368E+02

0.00000E+00

0.25389E+05

0.25055E+05

0.38914E+03

0.55319E+02

0.14216E+00

0.376E+02

0.00000E+00

0.25380E+05

0.25052E+05

0.37969E+03

0.52226E+02

0.13755E+00

0.384E+02

0.00000E+00

0.25371E+05

0.25049E+05

0.37052E+03

0.49244E+02

0.13291E+00

0.392E+02

0.00000E+00

0.25362E+05

0.25046E+05

0.36160E+03

0.46423E+02

0.12838E+00

214 Formation Testing Volume 3

Figure 3.2.11j. Inverse model screen. The pore pressure is 24,983 psi versus a value of 25,000 psi assumed. The mobility is 0.111 md/cp versus 0.100 md/cp inputted. Very good agreement is obtained. Next, we reduce inputted mobility 10 times.

Pressure Transient Analysis 215

Validition No. 4

Figure 3.2.11k. FT-00 exact inputs.

216 Formation Testing Volume 3

Figure 3.2.11l. Source probe pressure and pumpout schedule.

DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.800E+00

0.10000E+01

0.24734E+05

0.25000E+05 -0.26595E+03 -0.19886E-29

0.74773E-32

0.160E+01

0.10000E+01

0.24469E+05

0.25000E+05 -0.53094E+03 -0.39905E-13

0.75159E-16

0.240E+01

0.10000E+01

0.24205E+05

0.25000E+05 -0.79511E+03 -0.11820E-07

0.14866E-10

0.320E+01

0.10000E+01

0.23941E+05

0.25000E+05 -0.10585E+04 -0.67074E-05

0.63367E-08

0.400E+01

0.10000E+01

0.23679E+05

0.25000E+05 -0.13211E+04 -0.30878E-03

0.23372E-06

0.480E+01

0.10000E+01

0.23417E+05

0.25000E+05 -0.15830E+04 -0.40315E-02

0.25467E-05

0.560E+01 -0.10000E+01

0.23555E+05

0.25000E+05 -0.14451E+04 -0.25557E-01

0.17685E-04

0.640E+01 -0.10000E+01

0.23825E+05

0.25000E+05 -0.11751E+04 -0.10299E+00

0.87642E-04

0.720E+01 -0.10000E+01

0.24094E+05

0.25000E+05 -0.90616E+03 -0.30654E+00

0.33828E-03

0.800E+01 -0.10000E+01

0.24362E+05

0.24999E+05 -0.63807E+03 -0.73748E+00

0.11558E-02

0.880E+01 -0.10000E+01

0.24629E+05

0.24998E+05 -0.37082E+03 -0.15188E+01

0.40958E-02

0.960E+01 -0.10000E+01

0.24896E+05

0.24997E+05 -0.10437E+03 -0.27794E+01

0.26631E-01

Pressure Transient Analysis 217 0.104E+02

0.10000E+01

0.24895E+05

0.24995E+05 -0.10491E+03 -0.46284E+01

0.44117E-01

0.112E+02

0.10000E+01

0.24629E+05

0.24993E+05 -0.37088E+03 -0.71220E+01

0.19203E-01

0.120E+02

0.10000E+01

0.24364E+05

0.24990E+05 -0.63586E+03 -0.10237E+02

0.16099E-01

0.128E+02

0.10000E+01

0.24100E+05

0.24986E+05 -0.89996E+03 -0.13865E+02

0.15406E-01

0.136E+02

0.10000E+01

0.23837E+05

0.24982E+05 -0.11633E+04 -0.17828E+02

0.15326E-01

0.144E+02

0.10000E+01

0.23574E+05

0.24978E+05 -0.14258E+04 -0.21903E+02

0.15362E-01

0.152E+02 -0.10000E+01

0.23446E+05

0.24974E+05 -0.15543E+04 -0.25862E+02

0.16639E-01

0.160E+02 -0.10000E+01

0.23716E+05

0.24970E+05 -0.12840E+04 -0.29517E+02

0.22988E-01

0.168E+02 -0.10000E+01

0.23985E+05

0.24967E+05 -0.10147E+04 -0.32756E+02

0.32280E-01

0.176E+02 -0.10000E+01

0.24254E+05

0.24964E+05 -0.74639E+03 -0.35562E+02

0.47645E-01

0.184E+02 -0.10000E+01

0.24521E+05

0.24962E+05 -0.47888E+03 -0.38007E+02

0.79366E-01

0.192E+02 -0.10000E+01

0.24788E+05

0.24960E+05 -0.21219E+03 -0.40228E+02

0.18959E+00

0.200E+02

0.10000E+01

0.25054E+05

0.24958E+05

0.208E+02

0.10000E+01

0.24787E+05

0.24955E+05 -0.21301E+03 -0.44651E+02

0.20962E+00

0.216E+02

0.10000E+01

0.24521E+05

0.24953E+05 -0.47859E+03 -0.47099E+02

0.98411E-01

0.224E+02

0.10000E+01

0.24257E+05

0.24950E+05 -0.74325E+03 -0.49748E+02

0.66933E-01

0.232E+02

0.10000E+01

0.23993E+05

0.24947E+05 -0.10071E+04 -0.52526E+02

0.52158E-01

0.240E+02

0.10000E+01

0.23730E+05

0.24945E+05 -0.12701E+04 -0.55298E+02

0.43539E-01

0.248E+02

0.10000E+01

0.23468E+05

0.24942E+05 -0.15323E+04 -0.57892E+02

0.37780E-01

0.53727E+02 -0.42392E+02 -0.78902E+00

0.256E+02 -0.10000E+01

0.23605E+05

0.24940E+05 -0.13947E+04 -0.60146E+02

0.43125E-01

0.264E+02 -0.10000E+01

0.23875E+05

0.24938E+05 -0.11250E+04 -0.61949E+02

0.55064E-01

0.272E+02 -0.10000E+01

0.24144E+05

0.24937E+05 -0.85636E+03 -0.63273E+02

0.73886E-01

0.280E+02 -0.10000E+01

0.24411E+05

0.24936E+05 -0.58854E+03 -0.64175E+02

0.10904E+00

0.288E+02 -0.10000E+01

0.24678E+05

0.24935E+05 -0.32154E+03 -0.64787E+02

0.20149E+00

0.296E+02 -0.10000E+01

0.24945E+05

0.24935E+05 -0.55337E+02 -0.65283E+02

0.11797E+01

0.304E+02

0.00000E+00

0.25077E+05

0.24934E+05

0.76995E+02 -0.65842E+02 -0.85515E+00

0.312E+02

0.00000E+00

0.25076E+05

0.24933E+05

0.76244E+02 -0.66604E+02 -0.87356E+00

0.320E+02

0.00000E+00

0.25076E+05

0.24932E+05

0.75624E+02 -0.67629E+02 -0.89428E+00

0.328E+02

0.00000E+00

0.25075E+05

0.24931E+05

0.75079E+02 -0.68891E+02 -0.91758E+00

0.336E+02

0.00000E+00

0.25075E+05

0.24930E+05

0.74595E+02 -0.70286E+02 -0.94224E+00

0.344E+02

0.00000E+00

0.25074E+05

0.24928E+05

0.74140E+02 -0.71658E+02 -0.96652E+00

0.352E+02

0.00000E+00

0.25074E+05

0.24927E+05

0.73725E+02 -0.72825E+02 -0.98780E+00

0.360E+02

0.00000E+00

0.25073E+05

0.24926E+05

0.73326E+02 -0.73623E+02 -0.10040E+01

0.368E+02

0.00000E+00

0.25073E+05

0.24926E+05

0.72960E+02 -0.73922E+02 -0.10132E+01

0.376E+02

0.00000E+00

0.25073E+05

0.24926E+05

0.72599E+02 -0.73652E+02 -0.10145E+01

0.384E+02

0.00000E+00

0.25072E+05

0.24927E+05

0.72261E+02 -0.72794E+02 -0.10074E+01

0.392E+02

0.00000E+00

0.25072E+05

0.24929E+05

0.71929E+02 -0.71375E+02 -0.99229E+00

218 Formation Testing Volume 3

Figure 3.2.11m. Inverse model screen. The predicted pore pressure is 24,968 psi versus 25,000 psi which is excellent. Here the mobility is 0.017 md/cp versus 0.010 md/cp, in an accepted accuracy range given this extremely low value.

Pressure Transient Analysis 219

3.3 References Blauch, M.E., McMechan, D.E., Venditto, J.J. and Tanaka, G.L., “Low Permeability Subterranean Formation Testing Methods and Apparatus,” United States Patent 5,263,360, awarded Nov. 23, 1993. Discusses formation testing for low permeability gas reservoirs, multiple fluid injection treatments. Chin, W.C., Zhou, Y., Feng, Y., Yu, Q. and Zhao, L., Formation Testing: Pressure Transient and Contamination Analysis, John Wiley & Sons, Hoboken, New Jersey, 2014. Chin, W.C., Zhou, Y., Feng, Y. and Yu, Q., Formation Testing: Low Mobility Pressure Transient Analysis, John Wiley & Sons, Hoboken, New Jersey, 2015. Crockett, R.K. and Cooper, R.E., “Formation Testing and Stimulation Using Modern Generation Test Tools - The TestAcidize-Test Technique,” SPE Paper 5766, Society of Petroleum Engineers of AIME, American Institute of Mining, Metallurgical and Petroleum Engineers, 1976. Dendy Sloan, E., Hydrate Engineering, Henry L. Doherty series (Book 21), Society of Petroleum Engineers, Richardson, Texas, 2001. Elshahawi, H. and Hashem, M.N., “In-Situ Fluid Compatibility Testing Using a Wireline Formation Tester,” United States Patent Application Publication US 2010/0242586 A1, Sept. 30, 2010. Gilbert, G.N., Ball, D.E., Somers, R.S., Grable, J.L. and Menezes, C.C., “Multi-Purpose Downhole Tool,” United States Patent Application Publication US 2006/0248949 A1, Nov. 9, 2006. Goodwin, A.R.H. and Hegeman, P.S., “Method and Apparatus for Sampling Formation Fluids,” United States Patent 7,703,317 B2, awarded April 27, 2010. Goodwin, A.R.H. and Hegeman, P.S., “Method and Apparatus for Sampling Formation Fluids,” United States Patent 7,845,219 B2, awarded Dec. 7, 2010. Goodwin, A.R.H., Jones, T., Massie, K.J., Nighswander, J. and Tustin, G., “Methods and Apparatus to Change the Mobility of

220 Formation Testing Volume 3

Formation Fluids Using Thermal and Non-Thermal Stimulation,” United States Patent Application Publication US 2010/0294493 A1, Nov. 25, 2010. Hallmark, B.J., “Well Formation Test-Treat-Test Apparatus and Method,” United States Patent 4,339,948, July 20, 1982. Hester, K.C. and Howard, J.J., “Reservoir Pressure Testing to Determine Hydrate Composition,” United States Patent 9291051B2, issued March 22, 2016. Kuchuk, F., Ramakrishnan, T.S., Habashy, T.M., Falconer, I., Gokhan, S., Harrigan, E., Goodwin, A., Leising, L. and Mattos, F., “Instrumented Formation Tester for Injecting and Monitoring of Fluids,” United States Patent 8,191,416 B2, awarded June 5, 2012. Manke, K.R., Nivens, H.W., Bianco, S., MacPhail, C.M. and Maldonado, R., “Open Hole Formation Testing,” United States Patent 6,622,554 B2, September 23, 2003. Meister, M., Lee. J., Krueger, S. and Niemeyer, E., “Formation Testing Apparatus and Method for Optimizing Draw Down,” United States Patent 7,011,155 B2, awarded March 14, 2006. Pahmiyer, R.C. and Ringgenberg, P.D., “Methods and Apparatus for Downhole Completion Cleanup,” United States Patent 6,328,103 B1, Dec. 11, 2001. Sanford, L., “Method and Apparatus for Testing and Treating Well Formations,” United States Patent 4,031,957, awarded June 28, 1977. Van Hal, R.E.G., Goodwin, A., Mullins, O.C., Hegeman, P.S., Raghuraman, B., Betancourt, S.S., Ayan, C., Vasques, R., Dubost, F.X. and Del Campo, C.S., “Formation Fluid Sampling Tools and Methods Utilizing Chemical Heating,” United States Patent 8,283,174 B2, awarded Oct. 9, 2012. Webb, E.B., Koh, C.A. and Liberatore, M.W., “Rheological Properties of Methane Hydrate Slurries Formed From AOT + Water + Oil Microemulsions,” Langmuir, 2013, 29 (35), ACS Publications, American Chemical Society, pp. 10997-11004.

4 Practical Applications and Examples 4.1 Review Objectives Our earlier book publications summarized extensive developments in several areas of formation tester pressure transient interpretation and multiphase contamination modeling, gave detailed mathematical derivations and validations, and presented numerous calculated results showing how different inputs individually and collectively affect pressure response. Chapters 2 and 3 of this book address two additional new areas, namely, supercharge forward and inverse modeling, and inverse methods for general multiple drawdown and buildup sequences.

Figure 4.1. Earlier 2014 and 2015 research monographs. In the present chapter, we develop a well-rounded suite of practical applications examples that show how all of our methods can be used in situations important to petroleum engineering. We specifically omit new “phase delay” methods developed in our 2015 book, because, while rigorous, commercial hardware is not yet available to take advantage of the new interpretation and reservoir characterization approaches. 221

222 Formation Testing Volume 3

4.2

Practical Applications and Examples

4.2.1

Isotropic Medium Pressure Testing

In this example, we will consider different methods to perform pressure testing in isotropic media. Again, we will generate exact synthetic pressure data using our forward pressure simulator FT-00. From isolated pressure data points taken at different instants in time, we will attempt to recover input properties such as permeability and compressibility. 4.2.1.1

Steady-state method

Figure 4.2.1a. FT-00 data inputs.

Practical Applications and Examples 223

Our exact FT-00 forward simulator is run above with the input data shown. The calculated time pressure response appears in Figure 4.2.1b and shows that steady-state is reached somewhat before 100 sec. Tabulated pressure values follow immediately thereafter.

Figure 4.2.1b. Source probe pressure response. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice

224 Formation Testing Volume 3 Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.600E+01

0.10000E+01

0.20108E+05

0.24597E+05 -0.48924E+04 -0.40274E+03

0.82320E-01

0.120E+02

0.10000E+01

0.17393E+05

0.24393E+05 -0.76067E+04 -0.60735E+03

0.79844E-01

0.180E+02

0.10000E+01

0.15847E+05

0.24314E+05 -0.91528E+04 -0.68591E+03

0.74940E-01

0.240E+02

0.10000E+01

0.14951E+05

0.24282E+05 -0.10049E+05 -0.71762E+03

0.71416E-01

0.300E+02

0.10000E+01

0.14424E+05

0.24270E+05 -0.10576E+05 -0.73025E+03

0.69047E-01

0.360E+02

0.10000E+01

0.14107E+05

0.24265E+05 -0.10893E+05 -0.73493E+03

0.67470E-01

0.420E+02

0.10000E+01

0.13913E+05

0.24264E+05 -0.11087E+05 -0.73638E+03

0.66421E-01

0.480E+02

0.10000E+01

0.13792E+05

0.24263E+05 -0.11208E+05 -0.73662E+03

0.65722E-01

0.540E+02

0.10000E+01

0.13713E+05

0.24263E+05 -0.11287E+05 -0.73654E+03

0.65255E-01

0.600E+02

0.10000E+01

0.13660E+05

0.24264E+05 -0.11340E+05 -0.73645E+03

0.64945E-01

0.660E+02

0.10000E+01

0.13624E+05

0.24264E+05 -0.11376E+05 -0.73649E+03

0.64740E-01

0.720E+02

0.10000E+01

0.13598E+05

0.24263E+05 -0.11402E+05 -0.73666E+03

0.64605E-01

0.780E+02

0.10000E+01

0.13578E+05

0.24263E+05 -0.11422E+05 -0.73695E+03

0.64519E-01

0.840E+02

0.10000E+01

0.13563E+05

0.24263E+05 -0.11437E+05 -0.73733E+03

0.64466E-01

0.900E+02

0.10000E+01

0.13550E+05

0.24262E+05 -0.11450E+05 -0.73778E+03

0.64436E-01

0.960E+02

0.10000E+01

0.13540E+05

0.24262E+05 -0.11460E+05 -0.73828E+03

0.64422E-01

0.102E+03

0.10000E+01

0.13531E+05

0.24261E+05 -0.11469E+05 -0.73880E+03

0.64419E-01

0.108E+03

0.10000E+01

0.13524E+05

0.24261E+05 -0.11476E+05 -0.73934E+03

0.64424E-01

0.114E+03

0.10000E+01

0.13517E+05

0.24260E+05 -0.11483E+05 -0.73989E+03

0.64435E-01

0.120E+03

0.10000E+01

0.13511E+05

0.24260E+05 -0.11489E+05 -0.74043E+03

0.64449E-01

0.126E+03

0.10000E+01

0.13506E+05

0.24259E+05 -0.11494E+05 -0.74097E+03

0.64466E-01

0.132E+03

0.10000E+01

0.13501E+05

0.24259E+05 -0.11499E+05 -0.74149E+03

0.64485E-01

0.138E+03

0.10000E+01

0.13497E+05

0.24258E+05 -0.11503E+05 -0.74201E+03

0.64506E-01

0.144E+03

0.10000E+01

0.13493E+05

0.24257E+05 -0.11507E+05 -0.74251E+03

0.64527E-01

0.150E+03

0.10000E+01

0.13489E+05

0.24257E+05 -0.11511E+05 -0.74299E+03

0.64548E-01

0.156E+03

0.10000E+01

0.13486E+05

0.24257E+05 -0.11514E+05 -0.74347E+03

0.64570E-01

0.162E+03

0.10000E+01

0.13483E+05

0.24256E+05 -0.11517E+05 -0.74393E+03

0.64591E-01

0.168E+03

0.10000E+01

0.13480E+05

0.24256E+05 -0.11520E+05 -0.74437E+03

0.64613E-01

0.174E+03

0.10000E+01

0.13477E+05

0.24255E+05 -0.11523E+05 -0.74480E+03

0.64635E-01

0.180E+03

0.10000E+01

0.13474E+05

0.24255E+05 -0.11526E+05 -0.74521E+03

0.64656E-01

0.186E+03

0.10000E+01

0.13472E+05

0.24254E+05 -0.11528E+05 -0.74562E+03

0.64676E-01

0.192E+03

0.10000E+01

0.13469E+05

0.24254E+05 -0.11531E+05 -0.74600E+03

0.64697E-01

0.198E+03

0.10000E+01

0.13467E+05

0.24254E+05 -0.11533E+05 -0.74638E+03

0.64717E-01

0.204E+03

0.10000E+01

0.13465E+05

0.24253E+05 -0.11535E+05 -0.74675E+03

0.64737E-01

0.210E+03

0.10000E+01

0.13463E+05

0.24253E+05 -0.11537E+05 -0.74710E+03

0.64756E-01

0.216E+03

0.10000E+01

0.13461E+05

0.24253E+05 -0.11539E+05 -0.74744E+03

0.64775E-01

0.222E+03

0.10000E+01

0.13459E+05

0.24252E+05 -0.11541E+05 -0.74777E+03

0.64793E-01

0.228E+03

0.10000E+01

0.13457E+05

0.24252E+05 -0.11543E+05 -0.74810E+03

0.64811E-01

0.234E+03

0.10000E+01

0.13456E+05

0.24252E+05 -0.11544E+05 -0.74841E+03

0.64829E-01

0.240E+03

0.10000E+01

0.13454E+05

0.24251E+05 -0.11546E+05 -0.74871E+03

0.64846E-01

0.246E+03

0.10000E+01

0.13452E+05

0.24251E+05 -0.11548E+05 -0.74901E+03

0.64863E-01

0.252E+03

0.10000E+01

0.13451E+05

0.24251E+05 -0.11549E+05 -0.74929E+03

0.64879E-01

0.258E+03

0.10000E+01

0.13450E+05

0.24250E+05 -0.11550E+05 -0.74957E+03

0.64895E-01

0.264E+03

0.10000E+01

0.13448E+05

0.24250E+05 -0.11552E+05 -0.74984E+03

0.64911E-01

0.270E+03

0.10000E+01

0.13447E+05

0.24250E+05 -0.11553E+05 -0.75010E+03

0.64926E-01

0.276E+03

0.10000E+01

0.13446E+05

0.24250E+05 -0.11554E+05 -0.75036E+03

0.64941E-01

0.282E+03

0.10000E+01

0.13444E+05

0.24249E+05 -0.11556E+05 -0.75061E+03

0.64956E-01

0.288E+03

0.10000E+01

0.13443E+05

0.24249E+05 -0.11557E+05 -0.75085E+03

0.64970E-01

0.294E+03

0.10000E+01

0.13442E+05

0.24249E+05 -0.11558E+05 -0.75108E+03

0.64984E-01

Practical Applications and Examples 225

Our intention is to use a steady flow model, applying data obtained from the steady portion of the response curve. While we know where this behavior is from Figure 4.2.1b, we emphasize that this may not be so straightforward in field practice. For instance, in a very low mobility environment, pressure variations in time are slow and may be mistaken as steady-state. For now, we are confident that we are using actual steady values highlighted in red above, and so, steady pressure drops can be calculated, for example, as 30 sec., 25,000 – 14,424 = 10,576 psi 60 sec., 25,000 – 13,660 = 11,340 psi (used below) 120 sec, 25,000 – 13,511 = 11,489

We attempt to predict spherical mobility using our exact steady-state solver, which only requires geometric probe inputs, volume flow rate and pressure drop.

Figure 4.2.1c. Steady-state inverse solver (see “button,” Figure 4.2.1f). We obtain a predicted spherical mobility of 0.104 md/cp, which compares well with the 0.1 md/(1.0 cp) or 0.10 md/cp inferred from FT00 inputs. If we had used earlier inputs at t = 30 sec, we would have obtained a slightly worse 0.112 md/cp, whereas 120 sec values yields a slight improvement of 0.103 md/cp. Note that steady conditions were attained at approximately 100 sec and we had assumed a volume flowrate of 1 cc/sec. This means that at 100 sec, we would have collected 100 cc of fluid, which may be considered an excessive volume for our formation testing tool. At 30 sec, we would have collected 30 cc, still on the high side; of course, we could have lowered the flowrate, but in practice a lower flowrate will reduce the depth of investigation.

226 Formation Testing Volume 3 4.2.1.2 Drawdown-buildup method

How else could we have predicted mobility without encountering possible volume constraints? Previously, we used steady-state drawdown with large-time data. Now we attempt a ten second drawdown, followed by a pressure build-up and use build-up data. The FT-00 assumptions are shown below together with the calculated transient response.

Figure 4.2.1d. FT-00 data inputs.

Practical Applications and Examples 227

Figure 4.2.1e. Source probe pressure response. Time (s) 0.000E+00 0.333E+01 0.667E+01 0.100E+02 0.133E+02 0.167E+02 0.200E+02 0.233E+02 0.267E+02 0.300E+02 0.333E+02 0.367E+02 0.400E+02 0.433E+02 0.467E+02 0.500E+02 0.533E+02 0.567E+02 0.600E+02 0.633E+02 0.667E+02 0.700E+02 0.733E+02 0.767E+02 0.800E+02 0.833E+02 0.867E+02 0.900E+02

Rate (cc/s) 0.10000E+01 0.10000E+01 0.10000E+01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

Ps* (psi) 0.25000E+05 0.21921E+05 0.19724E+05 0.18132E+05 0.20051E+05 0.21398E+05 0.22361E+05 0.23057E+05 0.23562E+05 0.23930E+05 0.24200E+05 0.24398E+05 0.24544E+05 0.24653E+05 0.24733E+05 0.24794E+05 0.24839E+05 0.24873E+05 0.24899E+05 0.24918E+05 0.24933E+05 0.24945E+05 0.24954E+05 0.24961E+05 0.24967E+05 0.24971E+05 0.24975E+05 0.24978E+05

Pr* (psi) 0.25000E+05 0.24783E+05 0.24563E+05 0.24440E+05 0.24586E+05 0.24763E+05 0.24860E+05 0.24916E+05 0.24949E+05 0.24970E+05 0.24982E+05 0.24990E+05 0.24994E+05 0.24997E+05 0.24999E+05 0.24999E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.24999E+05 0.24999E+05 0.24999E+05

Ps**(psi) 0.00000E+00 -0.30790E+04 -0.52765E+04 -0.68675E+04 -0.49489E+04 -0.36024E+04 -0.26387E+04 -0.19431E+04 -0.14382E+04 -0.10699E+04 -0.80025E+03 -0.60199E+03 -0.45567E+03 -0.34727E+03 -0.26664E+03 -0.20642E+03 -0.16123E+03 -0.12715E+03 -0.10133E+03 -0.81650E+02 -0.66557E+02 -0.54905E+02 -0.45845E+02 -0.38746E+02 -0.33138E+02 -0.28668E+02 -0.25073E+02 -0.22154E+02

Pr**(psi) 0.00000E+00 -0.21706E+03 -0.43702E+03 -0.56012E+03 -0.41444E+03 -0.23704E+03 -0.13966E+03 -0.83844E+02 -0.50646E+02 -0.30475E+02 -0.18095E+02 -0.10486E+02 -0.58407E+01 -0.30496E+01 -0.14201E+01 -0.51520E+00 -0.57608E-01 0.00000E+00 0.00000E+00 0.00000E+00 -0.93336E-02 -0.13231E+00 -0.25707E+00 -0.37469E+00 -0.48059E+00 -0.57288E+00 -0.65128E+00 -0.71642E+00

Pr**/Ps** ----------0.70497E-01 0.82825E-01 0.81560E-01 0.83745E-01 0.65801E-01 0.52928E-01 0.43149E-01 0.35215E-01 0.28483E-01 0.22611E-01 0.17419E-01 0.12818E-01 0.87815E-02 0.53257E-02 0.24959E-02 0.35731E-03 0.00000E+00 0.00000E+00 0.00000E+00 0.14024E-03 0.24098E-02 0.56074E-02 0.96703E-02 0.14503E-01 0.19983E-01 0.25975E-01 0.32338E-01

228 Formation Testing Volume 3 0.933E+02 0.967E+02 0.100E+03 0.103E+03 0.107E+03 0.110E+03 0.113E+03 0.117E+03 0.120E+03 0.123E+03 0.127E+03 0.130E+03 0.133E+03 0.137E+03 0.140E+03 0.143E+03 0.147E+03 0.150E+03 0.153E+03 0.157E+03 0.160E+03 0.163E+03 0.167E+03 0.170E+03 0.173E+03 0.177E+03 0.180E+03 0.183E+03 0.187E+03 0.190E+03 0.193E+03 0.197E+03 0.200E+03 0.203E+03 0.207E+03 0.210E+03 0.213E+03 0.217E+03 0.220E+03 0.223E+03 0.227E+03 0.230E+03 0.233E+03 0.237E+03 0.240E+03 0.243E+03 0.247E+03 0.250E+03 0.253E+03 0.257E+03 0.260E+03 0.263E+03 0.267E+03 0.270E+03 0.273E+03 0.277E+03 0.280E+03 0.283E+03 0.287E+03 0.290E+03 0.293E+03 0.297E+03

0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

0.24980E+05 0.24982E+05 0.24984E+05 0.24985E+05 0.24986E+05 0.24987E+05 0.24988E+05 0.24989E+05 0.24990E+05 0.24990E+05 0.24991E+05 0.24991E+05 0.24992E+05 0.24992E+05 0.24993E+05 0.24993E+05 0.24993E+05 0.24994E+05 0.24994E+05 0.24994E+05 0.24994E+05 0.24995E+05 0.24995E+05 0.24995E+05 0.24995E+05 0.24995E+05 0.24995E+05 0.24996E+05 0.24996E+05 0.24996E+05 0.24996E+05 0.24996E+05 0.24996E+05 0.24996E+05 0.24996E+05 0.24997E+05 0.24997E+05 0.24997E+05 0.24997E+05 0.24997E+05 0.24997E+05 0.24997E+05 0.24997E+05 0.24997E+05 0.24997E+05 0.24997E+05 0.24997E+05 0.24997E+05 0.24998E+05 0.24998E+05 0.24998E+05 0.24998E+05 0.24998E+05 0.24998E+05 0.24998E+05 0.24998E+05 0.24998E+05 0.24998E+05 0.24998E+05 0.24998E+05 0.24998E+05 0.24998E+05

0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.24999E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05 0.25000E+05

-0.19761E+02 -0.17780E+02 -0.16123E+02 -0.14724E+02 -0.13532E+02 -0.12506E+02 -0.11616E+02 -0.10837E+02 -0.10151E+02 -0.95412E+01 -0.89963E+01 -0.85063E+01 -0.80633E+01 -0.76607E+01 -0.72932E+01 -0.69563E+01 -0.66463E+01 -0.63600E+01 -0.60949E+01 -0.58486E+01 -0.56192E+01 -0.54051E+01 -0.52047E+01 -0.50169E+01 -0.48404E+01 -0.46744E+01 -0.45179E+01 -0.43701E+01 -0.42304E+01 -0.40981E+01 -0.39727E+01 -0.38538E+01 -0.37407E+01 -0.36331E+01 -0.35306E+01 -0.34330E+01 -0.33398E+01 -0.32508E+01 -0.31658E+01 -0.30844E+01 -0.30065E+01 -0.29318E+01 -0.28602E+01 -0.27915E+01 -0.27256E+01 -0.26622E+01 -0.26012E+01 -0.25426E+01 -0.24861E+01 -0.24317E+01 -0.23793E+01 -0.23288E+01 -0.22800E+01 -0.22329E+01 -0.21874E+01 -0.21435E+01 -0.21010E+01 -0.20599E+01 -0.20201E+01 -0.19816E+01 -0.19443E+01 -0.19082E+01

-0.76941E+00 -0.81156E+00 -0.84421E+00 -0.86868E+00 -0.88616E+00 -0.89774E+00 -0.90438E+00 -0.90688E+00 -0.90598E+00 -0.90228E+00 -0.89629E+00 -0.88844E+00 -0.87912E+00 -0.86863E+00 -0.85722E+00 -0.84510E+00 -0.83247E+00 -0.81945E+00 -0.80619E+00 -0.79278E+00 -0.77929E+00 -0.76582E+00 -0.75240E+00 -0.73908E+00 -0.72591E+00 -0.71290E+00 -0.70009E+00 -0.68749E+00 -0.67511E+00 -0.66297E+00 -0.65108E+00 -0.63943E+00 -0.62803E+00 -0.61688E+00 -0.60598E+00 -0.59534E+00 -0.58494E+00 -0.57479E+00 -0.56487E+00 -0.55520E+00 -0.54576E+00 -0.53654E+00 -0.52755E+00 -0.51878E+00 -0.51022E+00 -0.50187E+00 -0.49372E+00 -0.48576E+00 -0.47800E+00 -0.47042E+00 -0.46303E+00 -0.45581E+00 -0.44877E+00 -0.44189E+00 -0.43517E+00 -0.42861E+00 -0.42220E+00 -0.41594E+00 -0.40982E+00 -0.40384E+00 -0.39801E+00 -0.39230E+00

0.38935E-01 0.45645E-01 0.52360E-01 0.58997E-01 0.65487E-01 0.71784E-01 0.77855E-01 0.83681E-01 0.89252E-01 0.94566E-01 0.99628E-01 0.10444E+00 0.10903E+00 0.11339E+00 0.11754E+00 0.12149E+00 0.12525E+00 0.12884E+00 0.13227E+00 0.13555E+00 0.13868E+00 0.14168E+00 0.14456E+00 0.14732E+00 0.14997E+00 0.15251E+00 0.15496E+00 0.15732E+00 0.15959E+00 0.16177E+00 0.16389E+00 0.16592E+00 0.16789E+00 0.16979E+00 0.17164E+00 0.17342E+00 0.17514E+00 0.17681E+00 0.17843E+00 0.18000E+00 0.18153E+00 0.18301E+00 0.18445E+00 0.18584E+00 0.18720E+00 0.18852E+00 0.18980E+00 0.19105E+00 0.19227E+00 0.19345E+00 0.19461E+00 0.19573E+00 0.19683E+00 0.19790E+00 0.19894E+00 0.19996E+00 0.20095E+00 0.20192E+00 0.20287E+00 0.20380E+00 0.20470E+00 0.20559E+00

Practical Applications and Examples 229

We now run Model 2 from Chapter 3, and use 10, 30 and 60 sec data (note that our host solver “Multiple-DDBU-Solver.exe” contains eleven inverse models). From Figure 4.2.1f, we obtain a prediction of 0.106 md/cp which compares well with the 0.1 md/cp inferred from Figure 4.2.1d. There is an added benefit in using the transient (and not steady-state) part of the pressure response curve – this data contains compressibility effects which predominate at early time and dissipate at late time. The last line of Figure 4.2.1f provides one further prediction, namely, a prediction for fluid compressibility in the form Compressibility (1/psi):

0.0009

(cc/FloLineVol).

This means that, if we knew the flowline volume associated with the formation testing tool used, and in this case it is given in the FT-00 input screen of Figure 4.2.1d as 300 cc, then the compressibility is 0.0009 (1/300) or 0.000003/psi – exactly the value assumed on the left side of Figure 4.2.1d! Recapitulating, we obtain both spherical mobility and compressibility by not using the steady-state method, but interpreting pressure transient results instead with our algorithm.

Figure 4.2.1f. Inverse method, Model 2.

230 Formation Testing Volume 3

In many field situations, petroleum engineers tend to focus primarily on mobility, or the quotient “mobility = permeability/viscosity” since this is the net quantity that controls flow resistance. In a first test performed early on during drilling, the mobility is measured and the filtrate viscosity used to provide rock permeability. If a second test is performed later, once reservoir fluid occupies the test volume, the newer value of mobility combined with the earlier rock permeability will yield reservoir fluid viscosity. This viscosity is just one indicator of fluid type – a second, the compressibility, which is also available from our analysis method, allows us to characterize the reservoir fluid more exactly. In the steady-state method, it suffices to use one data point, the steady pressure drop, to provide answers – of course, performing the calculation several times with different values will ensure measurement accuracy and validate that true steady conditions have been reached. We do emphasize that our transient methods are not without error. The FT00 algorithm, while formally an exact analytical model, utilizes complex complementary error functions which in turn use numerical algorithms with sixteen digit accuracy for most (but not all) parameters. Our inversion algorithms typically use exponential functions and rational polynomial approximations that are not as precise. Additionally, values may be computed differently depending on the microprocessor used and the algorithm employed. Thus, our forward modeling and inverse validations are often not exact, although they are very good. Furthermore, we only provide five digits in our tabulated output – which, while not entirely adequate, is more than possible with typical sensor gauges. In using our transient inverse procedures, typically three (time, pressure) data points are required for input data. It is best to use widely separated points in time, so that the entire dynamic character of the pressure response is captured. Taking data at a close 11, 12 and 13 seconds, for example, will guarantee an unacceptable inversion due to the combined effects of truncation errors in computations and round-off errors in displays. 4.2.1.3

Drawdown only method

We had successfully predicted spherical mobility from a steadystate model requiring relatively long wait times, and additionally, successfully employed a drawdown-buildup procedure (using buildup data) that provided both accurate mobility and fluid compressibility. Now we ask, “Can we use data from the drawdown portion of the

Practical Applications and Examples 231

previous pressure response curve, that is, from the first ten seconds of data?” To do this, we use Model 1 and the first several lines of pressure data in the tabulation immediately following Figure 4.2.1e.

Figure 4.2.1g. Inverse method, Model 1. The screen in Figure 4.2.1g shows that, using very early 10 sec data, we have predicted a spherical mobility of 0.1089 md/cp very well (as compared to an exact value of 0.1 md/cp) as well as an exact compressibility of 0.0009/300 or 0.000003/psi as before. Because of the short 10 sec duration of the entire drawdown process, the data used was taken across the entire time interval to capture as much of the dynamical behavior as possible. Note that there are times when buildup data are preferred over drawdown data, and vice-versa. For pro’s and con’s related to these questions, the reader should consult the “best practices” comments provided in Chapter 5 of this book and elsewhere.

232 Formation Testing Volume 3

4.2.2

Anisotropic Media Pressure Testing (Using FT-01)

In Section 4.2.1, we introduced three methods for mobility prediction in isotropic media, namely, algorithms using steady-state, transient drawdown and transient buildup pressures. The first offers spherical mobility only, while the latter two additionally predict fluid compressibility. We showed how mobilities and compressibilities can be obtained from pressure data.

Figure 4.2.2a. Anisotropic simulation at 45 deg dip.

Practical Applications and Examples 233

When horizontal and vertical permeabilities are identical, as in the prior examples, the dip angle of the formation testing tool is not relevant. However, most sedimentary layers are characterized by significant anisotropy, often with kh >> kv where “h” and “v” respectively denote “horizontal” and “vertical.” In this example, we address the prediction of anisotropic mobilities. As before, we will use FT-00 to create synthetic pressure data, and again, emphasize that the data so created are exact in a mathematical sense. The assumed conditions are shown in Figure 4.2.2a while the calculated responses for source and observation probe are given in Figure 4.2.2b (probe separations between source and observation pressure sensors are entered in the FT-00 input screen).

Figure 4.2.2b. Source and observation probe responses – note that source probe responses equilibrate much more rapidly than those at the observation probe (the latter may require significant times). DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice

234 Formation Testing Volume 3 Time (s) 0.000E+00 0.400E+02 0.800E+02 0.120E+03 0.160E+03 0.200E+03 0.240E+03 0.280E+03 0.320E+03 0.360E+03 0.400E+03 0.440E+03 0.480E+03 0.520E+03 0.560E+03 0.600E+03 0.640E+03 0.680E+03 0.720E+03 0.760E+03 0.800E+03 0.840E+03 0.880E+03 0.920E+03 0.960E+03 0.100E+04 0.104E+04 0.108E+04 0.112E+04 0.116E+04 0.120E+04 0.124E+04 0.128E+04 0.132E+04 0.136E+04 0.140E+04 0.144E+04 0.148E+04 0.152E+04 0.156E+04 0.160E+04 0.164E+04 0.168E+04 0.172E+04 0.176E+04 0.180E+04 0.184E+04 0.188E+04 0.192E+04 0.196E+04

Rate (cc/s) 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01

Ps* (psi) 0.25000E+05 0.22516E+05 0.22507E+05 0.22504E+05 0.22501E+05 0.22500E+05 0.22499E+05 0.22498E+05 0.22497E+05 0.22497E+05 0.22496E+05 0.22496E+05 0.22496E+05 0.22495E+05 0.22495E+05 0.22495E+05 0.22495E+05 0.22494E+05 0.22494E+05 0.22494E+05 0.22494E+05 0.22494E+05 0.22494E+05 0.22493E+05 0.22493E+05 0.22493E+05 0.22493E+05 0.22493E+05 0.22493E+05 0.22493E+05 0.22493E+05 0.22493E+05 0.22493E+05 0.22493E+05 0.22492E+05 0.22492E+05 0.22492E+05 0.22492E+05 0.22492E+05 0.22492E+05 0.22492E+05 0.22492E+05 0.22492E+05 0.22492E+05 0.22492E+05 0.22492E+05 0.22492E+05 0.22492E+05 0.22492E+05 0.22492E+05

Pr* (psi) 0.25000E+05 0.24925E+05 0.24916E+05 0.24913E+05 0.24910E+05 0.24909E+05 0.24908E+05 0.24907E+05 0.24906E+05 0.24905E+05 0.24905E+05 0.24904E+05 0.24904E+05 0.24904E+05 0.24903E+05 0.24903E+05 0.24903E+05 0.24903E+05 0.24902E+05 0.24902E+05 0.24902E+05 0.24902E+05 0.24902E+05 0.24902E+05 0.24901E+05 0.24901E+05 0.24901E+05 0.24901E+05 0.24901E+05 0.24901E+05 0.24901E+05 0.24901E+05 0.24901E+05 0.24900E+05 0.24900E+05 0.24900E+05 0.24900E+05 0.24900E+05 0.24900E+05 0.24900E+05 0.24900E+05 0.24900E+05 0.24900E+05 0.24900E+05 0.24900E+05 0.24900E+05 0.24900E+05 0.24900E+05 0.24900E+05 0.24900E+05

Ps**(psi) 0.00000E+00 -0.24837E+04 -0.24927E+04 -0.24964E+04 -0.24986E+04 -0.25001E+04 -0.25011E+04 -0.25020E+04 -0.25026E+04 -0.25032E+04 -0.25036E+04 -0.25040E+04 -0.25044E+04 -0.25047E+04 -0.25050E+04 -0.25052E+04 -0.25054E+04 -0.25056E+04 -0.25058E+04 -0.25060E+04 -0.25061E+04 -0.25063E+04 -0.25064E+04 -0.25066E+04 -0.25067E+04 -0.25068E+04 -0.25069E+04 -0.25070E+04 -0.25071E+04 -0.25072E+04 -0.25072E+04 -0.25073E+04 -0.25074E+04 -0.25075E+04 -0.25075E+04 -0.25076E+04 -0.25077E+04 -0.25077E+04 -0.25078E+04 -0.25079E+04 -0.25079E+04 -0.25080E+04 -0.25080E+04 -0.25081E+04 -0.25081E+04 -0.25081E+04 -0.25082E+04 -0.25082E+04 -0.25083E+04 -0.25083E+04

Pr**(psi) 0.00000E+00 -0.75413E+02 -0.83622E+02 -0.87404E+02 -0.89691E+02 -0.91263E+02 -0.92429E+02 -0.93338E+02 -0.94073E+02 -0.94682E+02 -0.95198E+02 -0.95643E+02 -0.96031E+02 -0.96374E+02 -0.96680E+02 -0.96954E+02 -0.97203E+02 -0.97429E+02 -0.97636E+02 -0.97827E+02 -0.98004E+02 -0.98167E+02 -0.98320E+02 -0.98462E+02 -0.98596E+02 -0.98721E+02 -0.98839E+02 -0.98950E+02 -0.99056E+02 -0.99156E+02 -0.99251E+02 -0.99341E+02 -0.99427E+02 -0.99509E+02 -0.99588E+02 -0.99663E+02 -0.99735E+02 -0.99804E+02 -0.99870E+02 -0.99934E+02 -0.99995E+02 -0.10005E+03 -0.10011E+03 -0.10017E+03 -0.10022E+03 -0.10027E+03 -0.10032E+03 -0.10037E+03 -0.10041E+03 -0.10046E+03

Pr**/Ps** ----------0.30363E-01 0.33547E-01 0.35012E-01 0.35897E-01 0.36504E-01 0.36955E-01 0.37306E-01 0.37590E-01 0.37825E-01 0.38024E-01 0.38195E-01 0.38345E-01 0.38477E-01 0.38595E-01 0.38701E-01 0.38797E-01 0.38884E-01 0.38964E-01 0.39037E-01 0.39105E-01 0.39168E-01 0.39227E-01 0.39282E-01 0.39333E-01 0.39382E-01 0.39427E-01 0.39470E-01 0.39511E-01 0.39549E-01 0.39586E-01 0.39620E-01 0.39653E-01 0.39685E-01 0.39715E-01 0.39744E-01 0.39772E-01 0.39798E-01 0.39824E-01 0.39848E-01 0.39872E-01 0.39895E-01 0.39917E-01 0.39938E-01 0.39958E-01 0.39978E-01 0.39997E-01 0.40016E-01 0.40033E-01 0.40051E-01

For instance, the source and observation probe pressure drops (see last line of above listing, highlighted in red) relative to a known pore pressure can be calculated as follows, 22492 – 25000 = – 2508 24900 – 25000 = – 100

We are interested in steady pressure drops at both probes because the inverse method demonstrated next is developed for (possibly long) steady-state data. Methods for anisotropic inverse mobility predictions requiring much shorter times, say a minute of less, are possible with the “phase delay” tools and methods under development (see Chin et al. (2015) but not discussed in this book.

Practical Applications and Examples 235

Figure 4.2.2c. FT-01 inverse, steady-state, anisotropic solver for liquids with general dip angle.

Figure 4.2.2d. Solution printout for FT-01 (45 deg dip).

236 Formation Testing Volume 3 INVERSE KH AND KV LIQUID SOLVER (ZERO SKIN) Copyright (2005), Wilson C. Chin, Ph.D., M.I.T. All rights reserved. Input parameters ... Dip angle ..................... (deg): Source probe delta-p .......... (psi): Observation probe delta-p ..... (psi): Effective source probe radius .. (cm): Probe separation ............... (cm): Volume flow rate ............. (cc/s): Liquid viscosity ............... (cp):

0.4500E+02 -0.2508E+04 -0.1000E+03 0.1000E+01 0.1500E+02 0.1000E+01 0.1000E+01

Source probe delta-p is source probe pressure minus pore pressure .. negative for fluid withdrawal from formation (positive flow rate). It is positive for fluid injection into formation (negative flow rate). Similar definition for observation probe delta-p. Possible solutions ... Tentative permeabilities (md) ... Complex KH root # 1: -1.14 + 0.00 i, KV: Complex KH root # 2: 1.06 + 0.00 i, KV: Complex KH root # 3: 0.08 + 0.00 i, KV:

0.08 0.09 14.85

KH above strictly valid -- if real part is positive and imaginary part is zero ... sometimes imaginary part is allowed, if small compared to positive real part. KH with negative real part is never correct. CAUTION: KV above is computed using real part of KH even if KH has nonzero imaginary part .... Careful! Exact conclusions below ... Following based on strict adherence to requirements that real(KH)>0 and imag(KH)=0 ..... mathematically correct KH and KV pairs, if shown. Root No. 1: Kh = 1.06 md, Kv = 0.09 md Root No. 2: Kh = 0.08 md, Kv = 14.85 md Multiple permeabilities found, more log data needed.

The general theory behind the inverse method is given in Chin et al. (2014), where it is shown that anisotropic mobility solutions are not unique. The last two lines above (in red) show one pair with kh = 1.06 md and kv = 0.09 md (correctly corresponding to our assumed 1 and 0.1 values). Also shown is an unrealistic mathematical solution with kv >> kh. When multiple permeability pairs are found, additional logging information (or physical intuition) is required to correctly determine the physically acceptable answer. *Note added in proof: Numerical output formats were changed to exponential “E” notation, e.g., 1.06 now reads 0.106E+01 in future output.

Practical Applications and Examples 237

Now, let us reconsider the foregoing problem, but with a 90 deg dip angle – that is, a horizontal well logging situation. The assumptions for forward pressure generation are given in Figure 4.2.2e below.

Figure 4.2.2e. FT-00 forward anisotropic run at 90 deg dip.

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Figure 4.2.2f. Source and observation probe pressure response. The source probe response for this 90 deg run is the same as before, that is, as for a 45 deg dip angle – however, observation probe pressure drops will differ with dip angle. The corresponding FT-01 run is shown in Figure 4.2.2g, where we use a dip angle of 89 deg instead of 90 deg to avoid internal code divisions by zero.

Figure 4.2.2g. FT-01 inverse calculation.

Practical Applications and Examples 239 INVERSE KH AND KV LIQUID SOLVER (ZERO SKIN) Copyright (2005), Wilson C. Chin, Ph.D., M.I.T. All rights reserved. Input parameters ... Dip angle ..................... (deg): Source probe delta-p .......... (psi): Observation probe delta-p ..... (psi): Effective source probe radius .. (cm): Probe separation ............... (cm): Volume flow rate ............. (cc/s): Liquid viscosity ............... (cp):

0.8900E+02 -0.2508E+04 -0.2470E+03 0.1000E+01 0.1500E+02 0.1000E+01 0.1000E+01

Source probe delta-p is source probe pressure minus pore pressure .. negative for fluid withdrawal from formation (positive flow rate). It is positive for fluid injection into formation (negative flow rate). Similar definition for observation probe delta-p. Possible solutions ... Tentative permeabilities (md) ... Complex KH root # 1: -.185E+02 + 0.000E+00 i, KV: 0.293E-03 Complex KH root # 2: 0.175E+02 + 0.000E+00 i, KV: 0.328E-03 Complex KH root # 3: 0.102E+01 + 0.000E+00 i, KV: 0.971E-01 KH above strictly valid -- if real part is positive and imaginary part is zero ... sometimes imaginary part is allowed, if small compared to positive real part. KH with negative real part is never correct. CAUTION: KV above is computed using real part of KH even if KH has nonzero imaginary part .... Careful! Exact conclusions below ... Following based on strict adherence to requirements that real(KH)>0 and imag(KH)=0 ..... mathematically correct KH and KV pairs, if shown. Root No. 1: Kh = 0.175E+02 md, Kv = 0.328E-03 md Root No. 2: Kh = 0.102E+01 md, Kv = 0.971E-01 md Multiple permeabilities found, more log data needed.

Two mathematical roots are found, with the second solution “Root No. 2” being closest to the assumed permeabilities of 1 and 0.1 md. In summary, we indicate that the method is very accurate and based on exact, closed form analytical solutions. However, dual probe measurements are required and wait times may be long if permeabilities are low.

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4.2.3

Supercharge Effects in Drawdown-Buildup

A forward simulator similar to FT-00 is available for isotropic supercharged applications, SC-DDBU-FORWARD-4NOPOR.EXE. We emphasize that FT-00 handles anisotropic media without supercharge – the supercharge simulator does not and applies to isotropic problems exclusively. We will discuss forward and inverse examples with and without supercharge in this section. Let us return to an example in Section 4.2.2 and use data from the drawdown-buildup input screen, in particular, Figure 4.2.1d. That prior screen is shown at left and the (simple black) input screen for the present application is shown at the right of Figure 4.2.3a below. Note that, in the black screen, the value for overbalance pressure is requested – for reference, we consider a zero value here. Inputs are identical for both input screens.

Figure 4.2.3a. Zero supercharge runs, FT-00 (left) and SC-DDBU-FORWARD-4NOPOR (right). Clicking after the last entry on the black screen produces the transient history below, FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Fluid, formation, tool and pumping parameters ...

Practical Applications and Examples 241 Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc): Time drawdown ends ............. (sec):

0.1000E+00 0.1000E+01 0.3000E-05 0.2500E+05 0.0000E+00 0.1000E+01 0.1000E+01 0.1000E+01 0.1000E+01 0.3000E+03 0.1000E+02

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 39.0

P(psi) 25000. 23940. 22977. 22101. 21304. 20581. 19922. 19324. 18780. 18286. 17836. 18487. 19079. 19617. 20107. 20551. 20956. 21323. 21657. 21961. 22237. 22489. 22717. 22924. 23113. 23285. 23440. 23582. 23711. 23828. 23935. 24032. 24120. 24200. 24272. 24338. 24399. 24453. 24503. 24548.

Note, pressures start increasing below.

242 Formation Testing Volume 3 40.0 41.0 42.0 43.0 44.0 45.0 46.0 47.0 48.0 49.0 50.0 51.0 52.0 53.0 54.0 55.0 56.0 57.0 58.0 59.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0

24589. 24627. 24660. 24691. 24719. 24745. 24768. 24789. 24808. 24826. 24842. 24856. 24869. 24881. 24892. 24902. 24911. 24919. 24926. 24933. 24939. 24976. 24991. 24996. 24999. 25000. 25000. 25000. 25000. 25000. 25000. 25000. 25000. 25000.

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

11660.3103 0.0000 11660.3103

Practical Applications and Examples 243

Figure 4.2.3b. Source probe response without supercharge. The above “no supercharge” results correctly show the initial reservoir pressure of 25,000 psi decreasing for 10 sec before increasing, we emphasize, to a final value of 25,000 psi identical to the initial pressure. Next, let us repeat the calculation with all parameters identical except that we now assume an overbalance pressure of 2,000 psi. Calculated results are shown immediately following Figure 4.2.3c.

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Figure 4.2.3c. Forward simulation with supercharge pressure. FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc): Time drawdown ends ............. (sec):

0.1000E+00 0.1000E+01 0.3000E-05 0.2500E+05 0.2000E+04 0.1000E+01 0.1000E+01 0.1000E+01 0.1000E+01 0.3000E+03 0.1000E+02

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0

P(psi) 27000. 25758. 24630. 23604. 22671. 21823. 21052. 20351. 19713. 19134. 18607. 19188.

Fluid withdrawal stops here.

Practical Applications and Examples 245 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 39.0 40.0 41.0 42.0 43.0 44.0 45.0 46.0 47.0 48.0 49.0 50.0 51.0 52.0 53.0 54.0 55.0 56.0 57.0 58.0 59.0 60.0 70.0 80.0 90.0 100.0 110.0 120.0

19717. 20197. 20633. 21030. 21391. 21719. 22017. 22288. 22535. 22759. 22963. 23148. 23316. 23469. 23608. 23735. 23850. 23954. 24049. 24136. 24214. 24286. 24351. 24410. 24463. 24512. 24556. 24597. 24633. 24667. 24697. 24725. 24750. 24772. 24793. 24812. 24829. 24845. 24859. 24871. 24883. 24894. 24903. 24912. 24920. 24927. 24934. 24940. 24945. 24979. 24992. 24997. 24999. 25000. 25000.

246 Formation Testing Volume 3 130.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0

25000. 25000. 25000. 25000. 25000. 25000. 25000. 25000. 25000. 25000. 25000. 25000. 25000. 25000. 25000. 25000.

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

11660.3103 2000.0000 13660.3103

Figure 4.2.3d. Source probe response with supercharge pressure.

Practical Applications and Examples 247

Figure 4.2.3d correctly shows an initial supercharged pressure of 27,000 psi (due to an assumed 25,000 psi reservoir value plus a 2,000 psi overbalance from the wellbore contribution) decreasing until t = 10 sec, at which point a pressure buildup is initiated, with asymptotic values achieved at the expected pore pressure 25,000 psi (not 27,000 psi). So far, we have run two forward transient pressure simulations, one without and the second with supercharge. Both yield expected physical behavior. The supercharge results shows how large time buildup will lead to true reservoir pore pressures because, over the scale of the simulation, the effects of overbalance pressure will have dissipated. The problem in an actual logging situation is, “How long is the wait time until supercharge dissipation has removed the excess pressure?” At the present time, oil service companies have not addressed this issue – some suggest a thirty minute wait, others more. However, we have treated this problem rigorously in this book and next we consider some practical questions. Zero supercharge inversion. Now, we will introduce our inverse simulators. Figure 4.2.3e below shows the host Windows user interface for Model 2 from our multiple drawdown-buildup system, noting that this system does not model supercharge effects. In the background are forward (zero supercharge) effects previously computed. For this inverse calculator, zero supercharge is consistently assumed. Entering the required inputs and clicking “Find” leads to the pore pressure of 24,997 psi and the mobility of 0.1 md/cp shown at the bottom right answer boxes – which are, as known from our forward simulation screen inputs, correct.

248 Formation Testing Volume 3

Figure 4.2.3e. Zero supercharge inversion, successful results. Non-zero supercharge inversion. Next let us repeat the inversion using supercharged pressure transient input data, with our upgraded inverse method with supercharge accounted for. The bottom right of Figure 4.2.3g shows that very accurate predictions of pore pressure and mobility can be obtained in the presence of very high overbalance pressures. Moreover, they are obtained without significant wait times, thus reducing logging costs and risks of lost tools. The algorithm underlying Figure 4.2.3g is compact, requiring minimal computer storage and processing resources – it can be programmed to run downhole in real-time. The algorithm does require an estimate for overbalance pressure. But even if such an estimate is not readily available, a high numerical input can be used to provide error estimates or upper and lower bounds on predicted quantities. Such estimates are not currently possible with formation testing interpretation tools.

Practical Applications and Examples 249

Figure 4.2.3f. Inversion with supercharge simulator, with POVER (psi) input box requesting overbalance pressure in psi units – question marks (bottom right) indicate where predictions appear.

Figure 4.2.3g. Pore pressure and mobility predictions successful in presence of strong supercharge or overbalance.

250 Formation Testing Volume 3

Predicted results, again, are excellent, particularly because early time data is used for a low mobility example with strong flowline storage effects. Our inversion algorithm gives 24,999 psi versus 25,000 psi pore pressure, 0.1013 md/cp versus 0.1 md/cp mobility, plus the correct fluid compressibility. The bottom right box shows that the compressibility is 0.0009 cc/FloLineVol) = 0.0009 cc/300 cc) or 0.000003 /psi where we have used our knowledge of the flowline volume from forward inputs. Note that data at times 11, 30 and 40 sec were used on both inverse calculations. 4.2.4

Supercharge Mechanics in Detail – Reservoir Fluid More Viscous Than Mud

Whereas Section 4.2.4 deals with “simple” forward and inverse pressure transient analysis with and without supercharge, the present section focuses on the dynamical process or underlying mechanics leading to supercharge. In other words, we will try to understand what is happening in the reservoir. The basic physics and underlying math models were covered in the recent book of Chin et al. (2015) which derives a miscible multiphase formulation valid for high rate pumping speeds – this significantly improves upon an earlier method the author developed (see Chin and Proett (2005)). The 2015 book also describes the updated software interface in detail. We will only briefly review key items. The main menu options are given below.

(continued next page)

Practical Applications and Examples 251

Figure 4.2.4a. Main simulation menus and options. The “main menus” shown in Figure 4.2.4a provide access to all high level functions needed to operate the simulator. Once the user decides on a particular formation or job, whether it is new or previously modeled (see last library menu in Figure 4.2.4a), a set of submenus appears allowing the user to alter physical formation, fluid or tool properties. These are shown in Figure 4.2.4b in menus segregated by physical function, e.g., “mudcake properties,” “boundary and initial conditions,” “layer properties,” and so on. Hence, run changes are easily performed.

252 Formation Testing Volume 3

Figure 4.2.4b. Submenus used to segregate different physical properties. While the simulator is operating, a status line appears on screen, indicating Integrating, invading (elapsed time, . . . min, . . . % done)

while the high pressure wellbore mud is invading the formation and creating mudcake. Once the formation tester tool is “turned on,” the following status line appears, indicating Integrating, pumping (elapsed time, . . .

min, . . . %done)

Thus, the user will know exactly when the pumping process begins its complex interaction with fluid invasion from the well.

Practical Applications and Examples 253

Figure 4.2.4c. Submenus used to segregate different physical properties. Color plots for pressure and concentration, shown starting with Figure 4.2.4d, periodically appear on screen at different time intervals so that the physics in the reservoir can be monitored (for faster simulations, these screens can be turned off). The left-most screen shows the pressure distribution in a section of the axisymmetric reservoir (red indicates high pressure in the well, while blue denotes lower pressure in the reservoir). The central figure gives the concentration (blue at the left indicates mud, while red at the right represents clean oil). The far right figure is a static diagram indicating where layer interfaces have been assumed. Different “time shots” use different color mapping scales during simulations. Thus, “red” at one time denotes a different attribute value from “red” at a another time. After the simulations are completed, all color plots are collected, and scale mappings are normalized to the maxima and minima for the entire collection of snapshots for “movie mode” playback. This provides a good understanding of the mixing process throughout all time, whereas the static plots provided during simulations lead to an understanding of properties as they vary in space.

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Figure 4.2.4d. Time 0.17 min results, invasion in progress, no pumping.

Figure 4.2.4e. Time 1.00 min results, invasion in progress, no pumping.

Practical Applications and Examples 255

Figure 4.2.4f. Time 3.00 min results, invasion in progress, no pumping.

Figure 4.2.4g. Time 3.17 min results, pumping has started.

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Figure 4.2.4h. Time 6.00 min results, pumping has started.

Figure 4.2.4i. Time 7.33 min results, pumping has started.

Practical Applications and Examples 257

Figure 4.2.4j. Time 8.50 min results, pumping has started.

Figure 4.2.4k. Time 11.00 min results, pumping has ended (see upper right of Figure 4.2.4c showing “end time” of t = 11 min).

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Figure 4.2.4l. Source probe pressure response (see explanation).

Figure 4.2.4m. Oil concentration history (see explanation).

Practical Applications and Examples 259

Figure 4.2.4n. Viscosity variation due to miscible mixing.

Figure 4.2.4o. Source probe volume flow rate.

260 Formation Testing Volume 3

Figure 4.2.4p. Source probe cumulative volume production. The source probe pressure response is given in Figure 4.2.4l, showing the required drawdown followed by a buildup. From Figure 4.2.4c, note how the well pressure of 25,000 psi is higher than the pore (and also initial) pressure of 22,000 psi in the reservoir – there is significant overbalance or supercharge. The pressures in Figure 4.2.4l correctly fall between the two limits and have not quite reached asymptotic pore pressure yet. Figure 4.2.4m shows the reservoir fluid concentration as it changes in time. Refer to the top right of Figure 4.2.4c for simulation parameters. This indicates that “time before pumping” is 0.05 hr or 3 min, and that the pumpout time is 4 min. Figure 4.2.4m correctly shows how reservoir fluid concentration is falling during the first three minutes due to invasion. During the next four minutes, the oil is becoming cleaner as the removed fluid interacts less with the invading mud. Finally, after seven minutes, pumping has stopped, and the invasion process continues as it did initially – the concentration at the probe falls as it is getting dirtier. Figures 4.2.4n,o,p describe the consequences of miscible mixing and production and are self-explanatory. In this Section 4.2.4, we dealt with supercharge, with “reservoir fluid more viscous than mud.” In Section 4.2.5, it is less, while in Section 4.2.6, it is equal. Results are self-explanatory, but the reader should note the shape of the source probe pressure response curves.

Practical Applications and Examples 261

4.2.5

Supercharge Mechanics in Detail – Reservoir Fluid Less Viscous Than Mud

Figure 4.2.5a. Run conditions.

262 Formation Testing Volume 3

Figure 4.2.5b. Run conditions.

Figure 4.2.5c. Source probe pressure.

Practical Applications and Examples 263

Figure 4.2.5d. Source probe concentration.

Figure 4.2.5e. Source probe viscosity.

264 Formation Testing Volume 3

4.2.6

Supercharge Mechanics in Detail – Reservoir Fluid Viscosity Equals Mud Viscosity

Figure 4.2.6a. Run conditions.

Practical Applications and Examples 265

Figure 4.2.6b. Run conditions.

Figure 4.2.6c. Source probe pressure.

266 Formation Testing Volume 3

4.2.7

Perfectly Balanced Well, Mechanics in Detail – Reservoir Fluid Viscosity Equals Mud Viscosity

The present example does not deal with overbalanced or supercharged reservoirs. Instead, it considers a “perfectly balanced well” in which the well pressure is identically equal to the initial or reservoir pore pressure. This study is undertaken for academic “what if” purposes.

Figure 4.2.7a. Run conditions.

Practical Applications and Examples 267

Figure 4.2.7b. Run conditions.

Figure 4.2.7c. Source probe pressure.

268 Formation Testing Volume 3

4.2.8

Underbalance Mechanics in Detail – Reservoir Fluid Viscosity Equals Mud Viscosity

This example deals with an underbalanced situation, where the well pressure is less than the reservoir pressure by 500 psi. Both mud and reservoir fluid viscosities are identical. The source probe pressure response curve appears “normal.”

Figure 4.2.8a. Run conditions.

Practical Applications and Examples 269

Figure 4.2.8b. Run conditions.

Figure 4.2.8c. Source probe pressure increases in time.

270 Formation Testing Volume 3

4.2.9 Comparing Overbalance vs Underbalance Pressures for Same Reservoir and Tool Pumping Conditions In prior run examples, spatial color plots for pressures and saturations were provided during simulations, at specific instances in time. At each instant, magnitudes of pressures (or saturations) in each plot were normalized to the maximum found for that spatial plot at that instant in time. Holding time fixed provides a good understanding of spatial dynamics by taking advantage of variations over a wide range of colors. However, these individual displays could not compare very well events at widely separated instances in time since their color normalizations are different. A good example consists of plots from “while invading, without pumping” and “pumping commences.” As soon as the pump piston retracts, large under-pressures are found, so that the renormalization completely differs from that during the more passive “invasion only” period. Thus, a “movie playback mode” where all frames are subject to the same normalization is essential for users in comparing the detailed physics between two problems.

Figure 4.2.9a. Overbalance simulation, assumptions.

Practical Applications and Examples 271

Figure 4.2.9b. Overbalance simulation, assumptions (continued). In this section, we perform two separate simulations, one overbalanced and the second underbalanced, to be described in the following paragraph. We do not plot any color screens until the simulations are completed in time. Then, all frames are normalized to the maximum found for the entire time duration of the simulation. This process provides a better sense of the physics. For instance, by watching the evolution of the pressure field in time, we can monitor high pressures (very red) displacing lower pressures (somewhat red) as time progresses and also observe the front motion in an overbalanced situation. At the same time, fluid withdrawal can be observed by watching lower blue pressures. On the other hand, in an underbalanced drilling application, we can watch high pressure (very red) fluid stay as high pressure because mud has not been able to invade the formation. Note that both display options are built into the miscible flow software. The user can examine pressure and saturation plots at specified times as simulations progress. Again, a “movie playback” mode is available once the entire simulation is completed, in which, again, all frames are normalized by the same constant to display all results on the same basis. Our display scheme provides only twelve colors, so resolution limits do exist.

272 Formation Testing Volume 3 Overbalance simulation. The assumptions for the present overbalance simulation assumes a well pressure of 25,000 psi and a pore pressure of 22,000 psi, as shown in Figures 4.2.9a and 4.2.9b. The source probe pressure response appears in Figure 4.2.9c. Free of “wiggles,” the plot indicates that results were stably computed. Spatial pressure and saturation plots, for different times each labeled by a frame number, appear in our Figures 4.2.9d – 4.2.9k.

Figure 4.2.9c. Overbalance simulation, source probe response. Spatial color plots are provided for pressure and concentration that convey a sense of the dynamics. Figures starting with Figure 4.2.9d, for instance, periodically appear on screen at different time intervals so that the physics in the reservoir can be monitored (for faster simulations, these screens can be turned off). The left-most rectangular plot shows the pressure distribution in a section of the axisymmetric reservoir (red indicates high pressure in the well, while lighter orange denotes lower pressure in the reservoir to the right). The central rectangular gives the concentration or saturation (yellow at the left indicates invading mud, while red at the right represents clean oil). The far right rectangular is a static diagram indicating where layer interfaces have been assumed. This is useful in assessing the effects of lithology on invasion, which is in turn important in helping interpret resistivity logs.

Practical Applications and Examples 273

We focus on spatial pressures in our discussion. Frame 0 shows the results of initial invasion, with high pressures at the sandface. In Frames 1 and 3, fluid withdrawal starts, with the probe situated in the “very red” high pressure zone. Frame 13 shows the “very red” zone narrowing, but nonetheless, the “orange” at the center of the figure shows that high pressures relative to pore values still exist. Even in Frame 23, slight supercharge is still evident. Immediately afterward, pumping terminates, and in Frames 28-47, higher pressures are returning to the near sandface.

Figure 4.2.9d. Overbalance simulation – Frame 0.

274 Formation Testing Volume 3

Figure 4.2.9e. Overbalance simulation – Frame 1.

Figure 4.2.9f. Overbalance simulation – Frame 5.

Practical Applications and Examples 275

Figure 4.2.9g. Overbalance simulation – Frame 8.

Figure 4.2.9h. Overbalance simulation – Frame 13.

276 Formation Testing Volume 3

Figure 4.2.9i. Overbalance simulation – Frame 23.

Figure 4.2.9i. Overbalance simulation – Frame 28.

Practical Applications and Examples 277

Figure 4.2.9k. Overbalance simulation – Frame 47.

278 Formation Testing Volume 3 Severe Underbalance. We run exactly the same input parameters as before, except that the well pressure is reduced from an overbalance of 25,000 psi (relative to a reservoir pressure of 22,000 psi) to a severe underbalance of 20,000 psi. This situation is not likely to be encountered in practice, but our purpose here is to demonstrate correct capturing of the physics. We provide results at the same frame numbers as before, in order to allow comparisons at identical times.

Figure 4.2.9l. Underbalance simulation, assumptions. We now describe spatial pressure results, which should be contrasted with those comments in the previous overbalance description. From Figures 4.2.9e and 4.2.9f, the higher or “redder” pressures appear at the left of the screen near the sandface of the wellbore. This is the correct result for overbalance drilling. In contrast, for Figures 4.2.9n to 4.2.9u, the “redder” shading appears at the right, showing that higher reservoir pressure dominates. A slight variation in background pressure is inferred from the “lighter orange” at the left, reflecting expected lowering of pressures as fluid moves into the well.

Practical Applications and Examples 279

Figure 4.2.9m. Underbalance simulation, source probe response.

Figure 4.2.9n. Underbalance simulation – Frame 0.

280 Formation Testing Volume 3

Figure 4.2.9o. Underbalance simulation – Frame 1.

Figure 4.2.9p. Underbalance simulation – Frame 5.

Practical Applications and Examples 281

Figure 4.2.9q. Underbalance simulation – Frame 8.

Figure 4.2.9r. Underbalance simulation – Frame 13.

282 Formation Testing Volume 3

Figure 4.2.9s. Underbalance simulation – Frame 23.

Figure 4.2.9t. Underbalance simulation – Frame 28.

Practical Applications and Examples 283

Figure 4.2.9u. Underbalance simulation – Frame 47.

284 Formation Testing Volume 3

4.2.10 Consequences of Non-Performing Pump Piston In many applications and idealized analyses, the pump piston is assumed to withdraw reservoir fluid at a constant flow rate (thus reducing pressure) and terminate after a given time to initiate a buildup. Often, the assumption for constant rate is not realistic for various reasons, for example, mechanical issues, overly resistive formation, and so on. In this section, we will consider a 0.05 md/cp formation with a target flow rate of 1 cc/sec that could not be met. The formation tester tool actually ramps up slowly, overshoots to 1.5 cc/s, before ramping down and terminating pumping. The pumpout schedule is displayed in Figure 4.2.10a, which shows an average pump rate of about 1 cc/sec, while the simulation assumptions are stated in Figure 4.2.10b. Source and observation probe pressure responses are shown in Figure 4.2.10c. Tabulated results are shown immediately thereafter, with data used for buildup analysis highlighted in red font. We emphasize that calculated results are exact.

Figure 4.2.10a. Pumpout schedule for drawdown-buildup action.

Practical Applications and Examples 285

Figure 4.2.10b. FT-00 simulation assumptions for drawdown-buildup.

286 Formation Testing Volume 3

Figure 4.2.10c. Source and observation probe pressure responses. FORMATION TESTER PRESSURE TRANSIENT ANALYSIS Exact analytical solution for Darcy ellipsoidal flow (homogeneous anisotropic media with flowline storage and skin) using complex complementary error function. Specific limit considered ...... Isotropic flow model with storage and no skin Developer, Wilson C. Chin, Ph.D., MIT. Copyright (C) 2004-2007, StrataMagnetic Software, LLC. All rights reserved. Email, [email protected]. FLUID AND FORMATION PARAMETERS Formation kh permeability ....... (md): Formation kv permeability ....... (md): Formation spherical permeability (md): Porosity ................... (decimal): Viscosity ....................... (cp): Pore fluid compressibility ... (1/psi): Pore pressure .................. (psi): Skin factor .......... (dimensionless): Dip angle ................. (0-90 deg):

0.5000E-01 0.5000E-01 0.5000E-01 0.1500E+00 0.1000E+01 0.3000E-05 0.2500E+05 0.0000E+00 0.9000E+02

TOOL PROPERTIES Flowline volume ................. (cc): Probe radius .................... (cm): Probe separation ................ (cm): Pad geometric factor . (dimensionless): Flowline fluid compressibility (1/psi):

0.3000E+03 0.1000E+01 0.1500E+02 0.1000E+01 0.3000E-05

Practical Applications and Examples 287 PUMPING SCHEDULE AND SIMULATION PARAMETERS Schedule #1, Flow rate ........ (cc/s): Beginning time .... (sec): Ending time ....... (sec): Schedule #2, Flow rate ........ (cc/s): Beginning time .... (sec): Ending time ....... (sec): Schedule #3, Flow rate ........ (cc/s): Beginning time .... (sec): Ending time ....... (sec): Schedule #4, Flow rate ........ (cc/s): Beginning time .... (sec): Ending time ....... (sec): Schedule #5, Flow rate ........ (cc/s): Beginning time .... (sec): Ending time ....... (sec): Schedule #6, Flow rate ........ (cc/s): Beginning time .... (sec): Ending time ....... (sec): Total simulation time .......... (sec):

0.5000E+00 0.0000E+00 0.1000E+02 0.1000E+01 0.1000E+02 0.2000E+02 0.1500E+01 0.2000E+02 0.3000E+02 0.1000E+01 0.3000E+02 0.4000E+02 0.5000E+00 0.4000E+02 0.5000E+02 Drawdown ends 0.0000E+00 Buildup begins 0.5000E+02 0.6000E+02 0.3000E+03

DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.50000E+00

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.333E+01

0.50000E+00

0.23326E+05

0.24937E+05 -0.16743E+04 -0.62796E+02

0.37506E-01

0.667E+01

0.50000E+00

0.21921E+05

0.24783E+05 -0.30790E+04 -0.21706E+03

0.70497E-01

0.100E+02

0.10000E+01

0.20733E+05

0.24657E+05 -0.42674E+04 -0.34306E+03

0.80391E-01

0.133E+02

0.10000E+01

0.18049E+05

0.24500E+05 -0.69507E+04 -0.49982E+03

0.71908E-01

0.167E+02

0.10000E+01

0.15786E+05

0.24276E+05 -0.92142E+04 -0.72423E+03

0.78599E-01

0.200E+02

0.15000E+01

0.13865E+05

0.24097E+05 -0.11135E+05 -0.90318E+03

0.81112E-01

0.233E+02

0.15000E+01

0.10556E+05

0.23900E+05 -0.14444E+05 -0.11003E+04

0.76179E-01

0.267E+02

0.15000E+01

0.77579E+04

0.23644E+05 -0.17242E+05 -0.13557E+04

0.78629E-01

0.300E+02

0.15000E+01

0.53790E+04

0.23441E+05 -0.19621E+05 -0.15587E+04

0.79438E-01

0.333E+02

0.10000E+01

0.50260E+04

0.23351E+05 -0.19974E+05 -0.16488E+04

0.82546E-01

0.367E+02

0.10000E+01

0.46996E+04

0.23390E+05 -0.20300E+05 -0.16101E+04

0.79316E-01

0.400E+02

0.50000E+00

0.44076E+04

0.23428E+05 -0.20592E+05 -0.15723E+04

0.76354E-01

0.433E+02

0.50000E+00

0.58230E+04

0.23517E+05 -0.19177E+05 -0.14833E+04

0.77347E-01

0.467E+02

0.50000E+00

0.69991E+04

0.23689E+05 -0.18001E+05 -0.13111E+04

0.72838E-01

0.500E+02

0.00000E+00

0.79860E+04

0.23827E+05 -0.17014E+05 -0.11727E+04

0.68926E-01

0.533E+02

0.00000E+00

0.10492E+05

0.23993E+05 -0.14508E+05 -0.10070E+04

0.69410E-01

0.567E+02

0.00000E+00

0.12599E+05

0.24224E+05 -0.12401E+05 -0.77599E+03

0.62575E-01

0.600E+02

0.00000E+00

0.14382E+05

0.24408E+05 -0.10618E+05 -0.59201E+03

0.55755E-01

288 Formation Testing Volume 3 0.633E+02

0.00000E+00

0.15895E+05

0.24546E+05 -0.91052E+04 -0.45375E+03

0.49835E-01

0.667E+02

0.00000E+00

0.17181E+05

0.24651E+05 -0.78188E+04 -0.34944E+03

0.44693E-01

0.700E+02

0.00000E+00

0.18277E+05

0.24730E+05 -0.67232E+04 -0.26997E+03

0.40155E-01

0.733E+02

0.00000E+00

0.19211E+05

0.24791E+05 -0.57890E+04 -0.20889E+03

0.36085E-01

0.767E+02

0.00000E+00

0.20009E+05

0.24838E+05 -0.49912E+04 -0.16165E+03

0.32386E-01

0.800E+02

0.00000E+00

0.20691E+05

0.24875E+05 -0.43093E+04 -0.12492E+03

0.28989E-01

0.833E+02

0.00000E+00

0.21274E+05

0.24904E+05 -0.37258E+04 -0.96295E+02

0.25846E-01

0.867E+02

0.00000E+00

0.21774E+05

0.24926E+05 -0.32259E+04 -0.73938E+02

0.22920E-01

0.900E+02

0.00000E+00

0.22203E+05

0.24944E+05 -0.27972E+04 -0.56469E+02

0.20188E-01

0.933E+02

0.00000E+00

0.22571E+05

0.24957E+05 -0.24293E+04 -0.42830E+02

0.17631E-01

0.967E+02

0.00000E+00

0.22887E+05

0.24968E+05 -0.21132E+04 -0.32200E+02

0.15238E-01

0.100E+03

0.00000E+00

0.23159E+05

0.24976E+05 -0.18413E+04 -0.23942E+02

0.13003E-01

0.103E+03

0.00000E+00

0.23393E+05

0.24982E+05 -0.16072E+04 -0.17590E+02

0.10945E-01

0.107E+03

0.00000E+00

0.23595E+05

0.24987E+05 -0.14055E+04 -0.12719E+02

0.90493E-02

0.110E+03

0.00000E+00

0.23769E+05

0.24991E+05 -0.12315E+04 -0.89927E+01

0.73024E-02

0.113E+03

0.00000E+00

0.23919E+05

0.24994E+05 -0.10812E+04 -0.62422E+01

0.57735E-02

0.117E+03

0.00000E+00

0.24049E+05

0.24996E+05 -0.95125E+03 -0.42104E+01

0.44261E-02

0.120E+03

0.00000E+00

0.24161E+05

0.24997E+05 -0.83879E+03 -0.26973E+01

0.32157E-02

0.140E+03

0.00000E+00

0.24585E+05

0.25000E+05 -0.41521E+03 -0.18076E+00

0.43533E-03

0.160E+03

0.00000E+00

0.24773E+05

0.24999E+05 -0.22740E+03 -0.10623E+01

0.46717E-02

0.180E+03

0.00000E+00

0.24861E+05

0.24998E+05 -0.13887E+03 -0.24055E+01

0.17322E-01

0.200E+03

0.00000E+00

0.24906E+05

0.24997E+05 -0.93876E+02 -0.33464E+01

0.35647E-01

0.220E+03

0.00000E+00

0.24931E+05

0.24996E+05 -0.68956E+02 -0.38446E+01

0.55754E-01

0.240E+03

0.00000E+00

0.24946E+05

0.24996E+05 -0.53874E+02 -0.40402E+01

0.74994E-01

0.260E+03

0.00000E+00

0.24956E+05

0.24996E+05 -0.43967E+02 -0.40521E+01

0.92163E-01

0.280E+03

0.00000E+00

0.24963E+05

0.24996E+05 -0.36997E+02 -0.39602E+01

0.10704E+00

0.290E+03

0.00000E+00

0.24966E+05

0.24996E+05 -0.34233E+02 -0.38915E+01

0.11368E+00

The inverse solver for “drawdown-buildup applications using buildup data” is shown in Figure 4.2.10d with “question marks” occupying the pore pressure and mobility prediction boxes at the lower right corner. Pressure data from t = 50 sec to 120 sec are used for this low mobility problem. Clicking “Find” populates these two prediction boxes. The pore pressure of 24,797 psi is not quite correct, as it does not agree with the 25,000 psi input used in FT-00 to create the synthetic pressure transient response – nor is it equal to the large time asymptotic source probe pressure response of 25,000 psi (assuming one could wait long enough to measure this value) seen in the left side of Figure 4.2.10c. Mobility, of course, is not a direct measurement furnished by the formation tester. In Figure 4.2.10e, the prediction of 0.06 md/cp is satisfactory in comparison to the known 0.05 md/cp from FT-00 input, especially given our use of a very approximate flowrate average of 1 cc/sec (computed from three readings, namely, 0.5, 1.0 and 1.5 cc/sec). In summary, the use of an approximate mean flow rate in the inversion in this case does yield results that may be acceptable in practice.

Practical Applications and Examples 289

Figure 4.2.10d. Starting screen for inverse drawdown-buildup analysis.

Figure 4.2.10e. Starting screen for inverse drawdown-buildup analysis.

290 Formation Testing Volume 3

Since the inverse drawdown-buildup solver in Figures 4.2.10d and 4.2.10e was constructed, Model 2 in the multiple drawdown-buildup work of Chapter 2 in this book was developed, as shown in Figure 4.2.10f. This inverse procedure predicts the same pore pressure and mobility as before, but also yields fluid compressibility by analyzing data from the early time pressure response. For the present case, the result is reported (to only two significant digits) as Compressibility (1/psi):

0.0011 x (cc/FloLineVol)

This means that, if the tester flowline volume were known, an explicit value for compressibility could be calculated. In our study, we know from Figure 4.2.10b that the flowline volume is 300 cc. Thus, 0.0011/300 yields 0.0000036/psi, in good agreement with the 0.000003/psi assumed in Figure 4.2.10b.

Figure 4.2.10f. Model No. 2 with enhanced compressibility prediction.

Practical Applications and Examples 291

4.2.11 Batch Processing Using FT-00 While our exact forward pressure simulator FT-00 has been designed to support user-friendly interactive use, there are applications, especially in history matching, when dozens, hundreds or even thousands of simulations must be performed. FT-00 is based on a closed form analytical solution published in Chin et al. (2014), expressed in terms of complex complementary error functions, which can be evaluated extremely rapidly. Typically, a hundred practical simulation will require no more than a few seconds if interactive graphical presentations are suppressed. Here we describe the “numbers only, batch processing” mode embedded in FT-00 software.

Figure 4.2.11a. FT-00 input screen.

292 Formation Testing Volume 3

Figure 4.2.11b. Loop parameter setup menu. The information box appearing in Figure 4.2.11a indicates that “looped” calculations are available – these are only available for the upper left “Fluid and Formation Parameters” menu. If, instead of numbers, a question mark ? is placed in a text entry box, clicking “Simulate” will cause FT-00 to execute a series of runs in which the parameter related to the ? is systematically varied. If a double question mark ?? is placed in a second box, simulations related to a double doloop are executed. The left menu in Figure 4.2.11b appears when the “Set Range” box is clicked. This defines upper and lower ranges for the parameters shown are specified in looped calculations. Once user values are “Saved,” these values are retained for future ? and ?? single and double do-loop calculations.

Practical Applications and Examples 293

Figure 4.2.11c. Sample calculation varying kh and kv.

Figure 4.2.11d. Graphics suppression menu. A sample looped calculation is shown in Figure 4.2.11c. Upon clicking “Simulate,” the submenu in Figure 4.2.11d appears. The user is permitted to view all line graphs for all runs, or he may view none at all. Batch runs may be terminated at any time by clicking “Exit job.” The next three screens show output headers for Simulations 1, 99 and 100.

294 Formation Testing Volume 3 Simulation No. 1 FORMATION TESTER PRESSURE TRANSIENT ANALYSIS Exact analytical solution for Darcy ellipsoidal flow (homogeneous anisotropic media with flowline storage and skin) using complex complementary error function.

. . . Simulation No. 99 FORMATION TESTER PRESSURE TRANSIENT ANALYSIS Exact analytical solution for Darcy ellipsoidal flow (homogeneous anisotropic media with flowline storage and skin) using complex complementary error function. Specific limit considered .... Anisotropic flow model with storage and no skin Developer, Wilson C. Chin, Ph.D., MIT. Copyright (C) 2004-2007, StrataMagnetic Software, LLC. All rights reserved. Email, [email protected]. FLUID AND FORMATION PARAMETERS Formation kh permeability ....... (md): Formation kv permeability ....... (md): Formation spherical permeability (md): Porosity ................... (decimal): Viscosity ....................... (cp): Pore fluid compressibility ... (1/psi): Pore pressure .................. (psi): Skin factor .......... (dimensionless): Dip angle ................. (0-90 deg):

0.5000E+03 0.8900E+02 0.2813E+03 0.1500E+00 0.1000E+01 0.1000E-04 0.2500E+05 0.0000E+00 0.0000E+00

TOOL PROPERTIES Flowline volume ................. (cc): Probe radius .................... (cm): Probe separation ................ (cm): Pad geometric factor . (dimensionless): Flowline fluid compressibility (1/psi):

0.3000E+03 0.5000E+00 0.1500E+02 0.1000E+01 0.1000E-04

PUMPING SCHEDULE AND SIMULATION PARAMETERS Schedule #1, Flow rate ........ (cc/s): 0.1000E+01 Beginning time .... (sec): 0.0000E+00 Ending time ....... (sec): 0.1000E+01

Practical Applications and Examples 295 Simulation No. 100 FORMATION TESTER PRESSURE TRANSIENT ANALYSIS Exact analytical solution for Darcy ellipsoidal flow (homogeneous anisotropic media with flowline storage and skin) using complex complementary error function. Specific limit considered .... Anisotropic flow model with storage and no skin Developer, Wilson C. Chin, Ph.D., MIT. Copyright (C) 2004-2007, StrataMagnetic Software, LLC. All rights reserved. Email, [email protected]. FLUID AND FORMATION PARAMETERS Formation kh permeability ....... (md): Formation kv permeability ....... (md): Formation spherical permeability (md): Porosity ................... (decimal): Viscosity ....................... (cp): Pore fluid compressibility ... (1/psi): Pore pressure .................. (psi): Skin factor .......... (dimensionless): Dip angle ................. (0-90 deg):

0.5000E+03 0.1000E+03 0.2924E+03 0.1500E+00 0.1000E+01 0.1000E-04 0.2500E+05 0.0000E+00 0.0000E+00

TOOL PROPERTIES Flowline volume ................. (cc): Probe radius .................... (cm): Probe separation ................ (cm): Pad geometric factor . (dimensionless): Flowline fluid compressibility (1/psi):

0.3000E+03 0.5000E+00 0.1500E+02 0.1000E+01 0.1000E-04

PUMPING SCHEDULE AND SIMULATION PARAMETERS . . .

296 Formation Testing Volume 3

4.2.12 Depth of Investigation Using FT-00 DOI Function In formation tester job planning, one often knows vaguely what the mobility might be, but only in an order-of-magnitude sense. To determine its value more accurately, the engineer must select a flow rate such that the pressure response is measurable at the observation probe location. The pressure at this probe will depend on all the eight fluid and formation parameters shown in the FT-00 menu, as well as all tool constants in the tool properties menu. Various questions are raised. What can be seen? Is the pressure signal measurable? Which flow rate and pumping duration are optimal? These issues are addressed by clicking on the “Run” button in the “Depth of Investigation” box at the bottom right of the FT-00 menu.

Figure 4.2.12a. FT-00 screen with “Depth of Investigation” option.

Practical Applications and Examples 297

Figure 4.2.12b. “Depth of investigation” menu. Clicking on the “Run” button closes the FT-00 menu and opens the “Depth of Investigation” simulator shown in Figure 4.2.12b with its much simpler menu. Once flow rate parameters are entered at the top right, and the “Tool Properties” menu is completed with the “Max probe distance (cm)” input box filled in with the maximum observation probe distance required, simulations are automatically performed, with pressure transient responses at different locations displayed and tabulated on completion of the runs. The software plots pump schedule, source probe

298 Formation Testing Volume 3

response, followed by observation probe response at different distances from source probe, in the present example, every 10 cm until 150 cm is reached, for entire 30 sec duration selected. Rather than giving a formal definition of “depth of investigation,” or calculating its value from an abstract formula, we have taken the “seeing is believing” approach to DOI analysis – if a clear pressure profile can be seen, the source signal is obviously usable.

Figure 4.2.12c. Volume flow rate.

Practical Applications and Examples 299

Figure 4.2.12d. Source probe response.

Figure 4.2.12e. Observation probe response at 10 cm.

300 Formation Testing Volume 3

Figure 4.2.12f. Observation probe response at 50 cm.

Figure 4.2.12g. Observation probe response at 100 cm.

Practical Applications and Examples 301

Figure 4.2.12h. Observation probe response at 150 cm. Tabulated results are also offered at the end of on-screen pressure displays, which are useful for detailed analysis. For example, the user may wish to determine how accurate his use of approximate inverse models is for the formation properties at hand. FORMATION TESTER PRESSURE TRANSIENT ANALYSIS Exact analytical solution for Darcy ellipsoidal flow (homogeneous anisotropic media with flowline storage and skin) using complex complementary error function. Specific limit considered ...... Isotropic flow model with storage and no skin Developer, Wilson C. Chin, Ph.D., MIT. Copyright (C) 2004, StrataMagnetic Software, LLC. All rights reserved. Email, [email protected]. FLUID AND FORMATION PARAMETERS Formation kh permeability ....... (md): Formation kv permeability ....... (md): Formation spherical permeability (md): Porosity ................... (decimal): Viscosity ....................... (cp): Pore fluid compressibility ... (1/psi): Pore pressure .................. (psi): Skin factor .......... (dimensionless): Dip angle ................. (0-90 deg):

0.1000E+01 0.1000E+01 0.1000E+01 0.1500E+00 0.1000E+01 0.3000E-05 0.2000E+05 0.0000E+00 0.0000E+00

TOOL PROPERTIES Flowline volume ................. (cc): Probe radius .................... (cm):

0.3000E+03 0.1000E+01

302 Formation Testing Volume 3 Probe separation ................ (cm): Pad geometric factor . (dimensionless): Flowline fluid compressibility (1/psi):

0.1000E+02 FIRST LOCATION 0.1000E+01 0.3000E-05

PUMPING SCHEDULE AND SIMULATION PARAMETERS Schedule #1, Flow rate ........ (cc/s): 0.1000E+02 Beginning time .... (sec): 0.0000E+00 Ending time ....... (sec): 0.1000E+02 Schedule #2, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.1400E+06 Ending time ....... (sec): 0.1500E+06 Schedule #3, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.1600E+06 Ending time ....... (sec): 0.1700E+06 Schedule #4, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.1800E+06 Ending time ....... (sec): 0.1900E+06 Schedule #5, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.2000E+06 Ending time ....... (sec): 0.2100E+06 Schedule #6, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.2200E+06 Ending time ....... (sec): 0.2300E+06 Total simulation time .......... (sec): 0.3000E+02 DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s) 0.000E+00 0.600E+00 0.120E+01 0.180E+01 0.240E+01 0.300E+01 0.360E+01 0.420E+01 0.480E+01 0.540E+01 0.600E+01 0.660E+01 0.720E+01 0.780E+01 0.840E+01 0.900E+01 0.960E+01 0.102E+02 0.108E+02 0.114E+02 0.120E+02 0.126E+02 0.132E+02 0.138E+02 0.144E+02 0.150E+02 0.156E+02 0.162E+02 0.168E+02 0.174E+02 0.180E+02 0.186E+02 0.192E+02 0.198E+02 0.204E+02 0.210E+02 0.216E+02 0.222E+02 0.228E+02 0.234E+02

Rate (cc/s) 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

Ps* (psi) 0.20000E+05 0.15108E+05 0.12393E+05 0.10847E+05 0.99515E+04 0.94238E+04 0.91074E+04 0.89135E+04 0.87917E+04 0.87130E+04 0.86604E+04 0.86239E+04 0.85976E+04 0.85779E+04 0.85625E+04 0.85502E+04 0.85400E+04 0.10506E+05 0.14498E+05 0.16737E+05 0.18018E+05 0.18762E+05 0.19201E+05 0.19465E+05 0.19626E+05 0.19728E+05 0.19793E+05 0.19836E+05 0.19865E+05 0.19886E+05 0.19902E+05 0.19913E+05 0.19923E+05 0.19930E+05 0.19936E+05 0.19941E+05 0.19946E+05 0.19949E+05 0.19953E+05 0.19956E+05

Pr* (psi) 0.20000E+05 0.19219E+05 0.18934E+05 0.18818E+05 0.18770E+05 0.18752E+05 0.18748E+05 0.18749E+05 0.18753E+05 0.18757E+05 0.18762E+05 0.18766E+05 0.18770E+05 0.18773E+05 0.18776E+05 0.18779E+05 0.18781E+05 0.19102E+05 0.19692E+05 0.19903E+05 0.19991E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05

Ps**(psi) 0.00000E+00 -0.48924E+04 -0.76067E+04 -0.91528E+04 -0.10049E+05 -0.10576E+05 -0.10893E+05 -0.11087E+05 -0.11208E+05 -0.11287E+05 -0.11340E+05 -0.11376E+05 -0.11402E+05 -0.11422E+05 -0.11437E+05 -0.11450E+05 -0.11460E+05 -0.94940E+04 -0.55018E+04 -0.32630E+04 -0.19823E+04 -0.12381E+04 -0.79891E+03 -0.53517E+03 -0.37366E+03 -0.27247E+03 -0.20737E+03 -0.16421E+03 -0.13463E+03 -0.11366E+03 -0.98248E+02 -0.86539E+02 -0.77361E+02 -0.69967E+02 -0.63868E+02 -0.58736E+02 -0.54346E+02 -0.50539E+02 -0.47201E+02 -0.44245E+02

Pr**(psi) 0.00000E+00 -0.78085E+03 -0.10660E+04 -0.11818E+04 -0.12298E+04 -0.12479E+04 -0.12524E+04 -0.12510E+04 -0.12472E+04 -0.12427E+04 -0.12382E+04 -0.12340E+04 -0.12303E+04 -0.12269E+04 -0.12240E+04 -0.12214E+04 -0.12190E+04 -0.89819E+03 -0.30783E+03 -0.97313E+02 -0.91263E+01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

Pr**/Ps** ----------0.15960E+00 0.14014E+00 0.12912E+00 0.12239E+00 0.11799E+00 0.11498E+00 0.11284E+00 0.11128E+00 0.11010E+00 0.10919E+00 0.10847E+00 0.10789E+00 0.10742E+00 0.10701E+00 0.10667E+00 0.10637E+00 0.94606E-01 0.55950E-01 0.29823E-01 0.46039E-02 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

Practical Applications and Examples 303 0.240E+02 0.246E+02 0.252E+02 0.258E+02 0.264E+02 0.270E+02 0.276E+02 0.282E+02 0.288E+02 0.294E+02

0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

0.19958E+05 0.19961E+05 0.19963E+05 0.19965E+05 0.19967E+05 0.19968E+05 0.19970E+05 0.19971E+05 0.19972E+05 0.19974E+05

0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05

-0.41609E+02 -0.39241E+02 -0.37102E+02 -0.35160E+02 -0.33389E+02 -0.31768E+02 -0.30278E+02 -0.28905E+02 -0.27636E+02 -0.26460E+02

0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

FORMATION TESTER PRESSURE TRANSIENT ANALYSIS Exact analytical solution for Darcy ellipsoidal flow (homogeneous anisotropic media with flowline storage and skin) using complex complementary error function. Specific limit considered ...... Isotropic flow model with storage and no skin Developer, Wilson C. Chin, Ph.D., MIT. Copyright (C) 2004, StrataMagnetic Software, LLC. All rights reserved. Email, [email protected]. FLUID AND FORMATION PARAMETERS Formation kh permeability ....... (md): Formation kv permeability ....... (md): Formation spherical permeability (md): Porosity ................... (decimal): Viscosity ....................... (cp): Pore fluid compressibility ... (1/psi): Pore pressure .................. (psi): Skin factor .......... (dimensionless): Dip angle ................. (0-90 deg):

0.1000E+01 0.1000E+01 0.1000E+01 0.1500E+00 0.1000E+01 0.3000E-05 0.2000E+05 0.0000E+00 0.0000E+00

TOOL PROPERTIES Flowline volume ................. (cc): Probe radius .................... (cm): Probe separation ................ (cm): Pad geometric factor . (dimensionless): Flowline fluid compressibility (1/psi):

0.3000E+03 0.1000E+01 0.2000E+02 SECOND LOCATION 0.1000E+01 0.3000E-05

PUMPING SCHEDULE AND SIMULATION PARAMETERS Schedule #1, Flow rate ........ (cc/s): 0.1000E+02 Beginning time .... (sec): 0.0000E+00 Ending time ....... (sec): 0.1000E+02 Schedule #2, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.1400E+06 Ending time ....... (sec): 0.1500E+06 Schedule #3, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.1600E+06 Ending time ....... (sec): 0.1700E+06 Schedule #4, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.1800E+06 Ending time ....... (sec): 0.1900E+06 Schedule #5, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.2000E+06 Ending time ....... (sec): 0.2100E+06 Schedule #6, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.2200E+06 Ending time ....... (sec): 0.2300E+06 Total simulation time .......... (sec): 0.3000E+02 DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice

0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

304 Formation Testing Volume 3 Time (s) 0.000E+00 0.600E+00 0.120E+01 0.180E+01 0.240E+01 0.300E+01 0.360E+01

Rate (cc/s) 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02

Ps* (psi) 0.20000E+05 0.15108E+05 0.12393E+05 0.10847E+05 0.99515E+04 0.94238E+04 0.91074E+04

Pr* (psi) 0.20000E+05 0.19836E+05 0.19702E+05 0.19645E+05 0.19618E+05 0.19604E+05 0.19595E+05

Ps**(psi) 0.00000E+00 -0.48924E+04 -0.76067E+04 -0.91528E+04 -0.10049E+05 -0.10576E+05 -0.10893E+05

Pr**(psi) 0.00000E+00 -0.16387E+03 -0.29801E+03 -0.35510E+03 -0.38179E+03 -0.39619E+03 -0.40543E+03

Pr**/Ps** ----------0.33496E-01 0.39178E-01 0.38797E-01 0.37994E-01 0.37461E-01 0.37220E-01

. .

Tabulated output is printed for different observation probe spacings, every 10 cm this run. Pressures are given for entire time duration of simulation. . . .

FORMATION TESTER PRESSURE TRANSIENT ANALYSIS Exact analytical solution for Darcy ellipsoidal flow (homogeneous anisotropic media with flowline storage and skin) using complex complementary error function. Specific limit considered ...... Isotropic flow model with storage and no skin Developer, Wilson C. Chin, Ph.D., MIT. Copyright (C) 2004, StrataMagnetic Software, LLC. All rights reserved. Email, [email protected]. FLUID AND FORMATION PARAMETERS Formation kh permeability ....... (md): Formation kv permeability ....... (md): Formation spherical permeability (md): Porosity ................... (decimal): Viscosity ....................... (cp): Pore fluid compressibility ... (1/psi): Pore pressure .................. (psi): Skin factor .......... (dimensionless): Dip angle ................. (0-90 deg):

0.1000E+01 0.1000E+01 0.1000E+01 0.1500E+00 0.1000E+01 0.3000E-05 0.2000E+05 0.0000E+00 0.0000E+00

TOOL PROPERTIES Flowline volume ................. (cc): Probe radius .................... (cm): Probe separation ................ (cm): Pad geometric factor . (dimensionless): Flowline fluid compressibility (1/psi):

0.3000E+03 0.1000E+01 0.1500E+03 LAST LOCATION 0.1000E+01 0.3000E-05

PUMPING SCHEDULE AND SIMULATION PARAMETERS Schedule #1, Flow rate ........ (cc/s): 0.1000E+02 Beginning time .... (sec): 0.0000E+00 Ending time ....... (sec): 0.1000E+02 Schedule #2, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.1400E+06 Ending time ....... (sec): 0.1500E+06 Schedule #3, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.1600E+06 Ending time ....... (sec): 0.1700E+06 Schedule #4, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.1800E+06 Ending time ....... (sec): 0.1900E+06 Schedule #5, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.2000E+06 Ending time ....... (sec): 0.2100E+06 Schedule #6, Flow rate ........ (cc/s): 0.0000E+00 Beginning time .... (sec): 0.2200E+06 Ending time ....... (sec): 0.2300E+06 Total simulation time .......... (sec): 0.3000E+02

Practical Applications and Examples 305 DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s) 0.000E+00 0.600E+00 0.120E+01 0.180E+01 0.240E+01 0.300E+01 0.360E+01 0.420E+01 0.480E+01 . . . . .

Rate (cc/s) 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02 0.10000E+02

Ps* (psi) 0.20000E+05 0.15108E+05 0.12393E+05 0.10847E+05 0.99515E+04 0.94238E+04 0.91074E+04 0.89135E+04 0.87917E+04

Pr* (psi) 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05

Ps**(psi) 0.00000E+00 -0.48924E+04 -0.76067E+04 -0.91528E+04 -0.10049E+05 -0.10576E+05 -0.10893E+05 -0.11087E+05 -0.11208E+05

Pr**(psi) 0.00000E+00 -0.17364E-25 -0.41337E-12 -0.12654E-07 -0.22749E-05 -0.52086E-04 -0.42476E-03 -0.19189E-02 -0.59915E-02

Pr**/Ps** ----------0.35492E-29 0.54343E-16 0.13825E-11 0.22639E-09 0.49248E-08 0.38995E-07 0.17309E-06 0.53456E-06

306 Formation Testing Volume 3

4.2.13 History Matching Using FT-06 Batch Mode Two general formation testing forward simulators are available for pressure transient response calculations, namely, FT-00 and FT-06, and their access screens are shown in Figure 4.2.13a. We emphasize that FT00 applies to liquids only, and then, only when volume flow rates are “piecewise constant” in time. That is, they must be prescribed in the form “1 cc/s for t = 0 to 3 sec,” “-3 cc/sec for t = 3 to 5 sec,” and so on – in other words, slow ramp-ups or ramp-downs, periodic oscillations, triangular or trapezoidal rate versus time functions and the like are not permitted. With these restrictions in place, however, computed pressure transient responses at source and observation problems are exact since the solver evaluates analytical solutions based on the complex complementary error function formulation. This makes FT-00 useful in creating synthetic data for the purpose of evaluating inverse methods – there are no round-off or truncation errors associated with numerical finite difference or finite element solutions.

Figure 4.2.13a. FT-00 (left) and FT-06 (right) input screens.

Practical Applications and Examples 307

On the other hand, sometimes “piecewise constant” flow rates cannot be used in practice, for example, because of mechanical issues or overly resistive formations. Or perhaps, if (nonlinear) gases are being pumped since the FT-00 simulator is written for compressible liquids. The more flexible FT-06 simulator was written for broader applications. It will model both liquids and gases (under general thermodynamic assumptions) and it will also allow arbitrary flow rate functions in time. Pressure responses are obtained directly from high-order finite difference integration and superposition methods (which do not apply to nonlinear gases) are not used. For example, a flow rate function such as that in Figure 4.2.13b may be defined from the “Pumping Schedule” menu at the right, or a custom function may be read in from a user-supplied file.

Figure 4.2.13b. General flow rate functions supported for liquid and gas flows.

308 Formation Testing Volume 3

4.2.13.1 Operating FT-06 batch simulator This section explains (i) why the FT-06 “batch mode” is used, (ii) how to run the batch simulator and interpret results, and (iii) the steps needed to modify Fortran source code for a user’s own specialized purposes. Suppose experimental observation probe pressure transient results for a wide range of time values have been collected and stored in a file called “batch-data.dat.” For the purposes of this write-up, we assume the user is confident that all the numbers shown in the menu in Figure 4.2.13c are correct, except for the values for Kh and Kv. We wish to find the combination of (Kh,Kv) that is correct and consistent with experiment. The FT-06 batch simulator will assume a particular set of (Kh,Kv) values. It then calculates the observation probe pressure at numerous instances in time and stores all transient results in a file. The calculated results in this file (which contains results associated with incorrect permeability values) are compared to “batch-data.dat” and an “error” value is determined. Then, FT-06 repeats the procedure with a slightly different assumption for (Kh,Kv) and calculates another error value. This procedure is repeated hundreds or thousands of times, and the result with the smallest error is assumed to correspond to the correct choices for Kh and Kv and is presented to the user. It’s that simple. In the above, we stated that all of the numbers shown in the menu are assumed to be correct except Kh and Kv. To run the batch solver, we first need to operate “FT-06 (Interactive)” with ANY assumptions for Kh and Kv. We have chosen “1” and “1” for Figure 4.2.13c, but in another application, we may use “7” and “3.” The choice does not matter, since our operation of the interactive version of FT-06 first only serves to create special background files needed to run the batch solver later – it is simply a pre-processing routine. That is, the Kh and Kv values in the menu of Figure 4.2.13c are ignored later in the batch solver, while the batch simulator creates its own comprehensive test sets for (Kh,Kv). After the interactive solver has been operated by clicking “Solve,” we next run the batch solver to initiate batch processing.

Practical Applications and Examples 309

Figure 4.2.13c. User experimental data. Two means are available to operate the batch simulator. In the first, we run the executable file “ELIPS-14-BATCH-4.EXE” directly from the MS-DOS command line. On the other hand, for users with an integrated version of our formation testing job planning software, the “Run Batch Mode” can be selected from the FTSim system user interface (an example is shown in Figure 4.2.13d).

310 Formation Testing Volume 3

Figure 4.2.13d. Example client software system screen. Once the batch solver is initiated, the following status notes appear on the black MS-DOS screen. Note that hundreds, even thousands of FT-06 simulations can be performed in a minute since graphical results are not displayed during the simulations. Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

1., 1., 1., 1., 1., 1., 1., 1., 2., 2., 2., 2., 2., 2., 2., 2., 3., 3., 3., 3., 3., 3., 3., 3., 4., 4., 4., 4., 4., 4., 4., 4., 5., 5., 5., 5.,

KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

1. 2. 3. 4. 5. 6. 7. 8. 1. 2. 3. 4. 5. 6. 7. 8. 1. 2. 3. 4. 5. 6. 7. 8. 1. 2. 3. 4. 5. 6. 7. 8. 1. 2. 3. 4.

Practical Applications and Examples 311 Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating Evaluating

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD KHMD

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

5., 5., 5., 5., 6., 6., 6., 6., 6., 6., 6., 6., 7., 7., 7., 7., 7., 7., 7., 7., 8., 8., 8., 8., 8., 8., 8., 8., 9., 9., 9., 9., 9., 9., 9., 9.,

KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD KVMD

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

5. 6. 7. 8. 1. 2. 3. 4. 5. 6. 7. 8. 1. 2. 3. 4. 5. 6. 7. 8. 1. 2. 3. 4. 5. 6. 7. 8. 1. 2. 3. 4. 5. 6. 7. 8.

Close SCRATCH.DAT to continue ...

Note how different values of Kh and Kv have been considered above. Each line refers to a completed FT-06 time simulation. When all required simulations are finished, the following summary appears, with an Error value at the far right column. Batch file computations for FT-06 ... Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md)

= = = = = = = = = = = = = =

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2.000 2.000 2.000 2.000 2.000 2.000

Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md)

= = = = = = = = = = = = = =

1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000

Error Error Error Error Error Error Error Error Error Error Error Error Error Error

= = = = = = = = = = = = = =

5951.430 7458.258 6965.629 6078.306 5181.186 4385.321 3702.576 3128.267 2825.995 5358.436 5919.768 5681.864 5216.870 4618.719

312 Formation Testing Volume 3 Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md)

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

2.000 2.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 5.000 5.000 5.000 5.000 5.000 5.000 5.000 5.000 6.000 6.000 6.000 6.000 6.000 6.000 6.000 6.000 7.000 7.000 7.000 7.000 7.000 7.000 7.000 7.000 8.000 8.000 8.000 8.000 8.000 8.000 8.000 8.000 9.000 9.000 9.000 9.000 9.000 9.000 9.000 9.000

Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md)

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000

Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

4064.451 3512.589 1198.071 2791.451 3686.459 3849.362 3623.159 3309.531 2958.865 2593.101 747.051 1200.283 1883.090 2148.817 2159.537 2022.090 1816.262 1602.816 936.549 367.095 714.027 983.375 1086.729 1068.738 993.543 905.998 1491.641 104.719 163.851 315.262 434.059 489.755 493.716 498.274 2245.641 218.389 0.000 57.302 157.455 241.572 311.261 374.573 3084.639 598.314 136.857 96.826 169.835 268.677 373.857 482.759 4008.590 1143.483 492.366 364.955 404.968 508.574 634.391 774.725

Practical Applications and Examples 313

Note how, for Kh = 7 md and Kv = 3 md, the Error is 0. Thus, these two values of permeability have created a simulation that exactly fits the experimental data “batch-data.dat.” It follows that Kh = 7 md and Kv = 3 md represent the anisotropic permeabilities of the tested formation. The “Source Code Changes (Detailed)” button in Figure 4.2.13d opens a document that describes numerous ways our FT-06 batch simulator can be customized. For example, different ranges of Kh and Kv are possible, and other definitions of “Error” may be used, and other types of well logging constraints can be added, in simple Fortran source code changes. The document explains how the Fortran source code can be modified and is reproduced in the next section – it is easy to read and color-coded to assist the programmer or engineer in implementing his changes. Finally, while we have shown how Kh and Kv can be varied simultaneously in a two-level nested loop, it is also possible to vary Kh, Kv, porosity, viscosity . . . that is, any number of variables. All computed results are stored for later viewing. FT-06 can be operated hundreds, even thousands of times, per minute – it is extremely fast and accurate. It is not possible to anticipate all conceivable user applications – thus, the Fortran source code for FT-06 is available to interested parties. 4.2.13.2 Source code documentation (color coded) Here we describe in detail the Fortran source code ELIPS-14BATCH-4.FOR developed for batch runs of FT-06. The numerical fluid flow algorithms for both batch and interactive codes are identical. Modifications to FT-06 are introduced so that multiple simulations can be performed quickly and the end results are compared against an experimental pressure data file or other file. Source code is printed in COURIER font for clarity – RED color is used to indicate changes to the original FT-06 source code developed for interactive runs. Our comments are given in blue Times New Roman font. These comments show how the capabilities of the batch code can be extended. This document will help users customize or extend this batch run program for their own specialized uses – entry points for graphics are clearly identified from reading the highly commented source code. Very often, the formation tester is used in rock with unknown properties and, for example, it is desired to determine values of KH and KV, assuming that all other parameters are known. The flow rate time function is defined by the engineer, and it may consist of constant rate pumping, triangular or trapezoidal rate pumping, fluid withdrawal or

314 Formation Testing Volume 3

injection, rates from a user-defined file, and so on. The numerical algorithm in FT-06 will compute the time-dependent pressure response at the observation probe, which will be compared to the measured pressure data PDATA( ) obtained in the lab or field test. This batch software computes the observation probe pressure responses POBS( ) for many values of KH and KV, and compares this response to the experimental file PDATA( ). When the error between PDATA and POBS is minimized, we have identified the correct KH and KV. In the code below, KH is fixed, and KV is varied throughout a range of values. Then, the next KH is considered, and so on. The code is therefore illustrated using only two variables that need to be systematically changed, so we use a two-level “do-loop.” If, for example, five variables need to be systematically changed, e.g., KH, KV, viscosity, compressibility and gas constant, this source code should be easily modified to show a five-level do-loop. The explanations for the source code below make the changes simple to complete for the programmer. C$DEBUG C ELIPS-14-Batch-4.FOR C

Please disregard next three lines. We do not have experimental data so we created our “experimental data” from a separate run of FT-06 and stored that data in Batch-Data.dat. Please see the Appendix for the assumptions used in creating Batch-Data.dat. C Batch-Data-ELIPS-14.FOR used to define BATCH-DATA.DAT C "exact experimental" data file. Never use this code again after C BATCH-DATA.DAT is created. C C ELIPS-14.FOR ... Add geometric factor and initial r grid size C ELIPS-13.FOR ... Perms now specified thru (kh,kv) or (ks,kv/kh) C ELIPS-12.FOR ... Skin input now read from Windows menu. C ELIPS-11.FOR ... Add skin effect modeling to boundary condition C ELIPS-10.FOR C Anisotropic infinite medium without layers, transient flow model C assuming ellipsoidal source for linear liquids and nonlinear gases. C ------------------------------------------------------------C Compile with Compaq Fortran V6.5 without Quickwin. Instructions: C From command line, C:\> FL32 ELIPS-*.FOR [RET]. Run ELIPS-* [RET]. C -------------------------------------------------------------

Practical Applications and Examples 315 C

PROGRAM MAIN INTERFACE TO FUNCTION SYSTEM[C] (STRING) INTEGER*2 SYSTEM CHARACTER*1 STRING[REFERENCE] END INTEGER*2 SYSTEM

Delete the password system used in FT-06.FOR. Do not want to use it for the batch code. C C . . . C

PASSWORD SYSTEM

CALL ELIPS C C FOR BATCH CALCULATIONS, THE ELIPS.DAT FILE IS TOO LARGE AND C THE GRAPHS WOULD ONLY REFER TO LAST RUN. SO SKIP THE FOLLOWING C WHICH ARE INTENDED FOR SINGLE INTERACTIVE RUNS. C GO TO 998 C I=SYSTEM('NOTEPAD.EXE ELIPS.DAT'C) IF(I.EQ.-1) STOP 'Could not run COMMAND.COM ...' I=SYSTEM('CLS'C) IF(I.EQ.-1) STOP 'Could not run COMMAND.COM ...' C C Line plot, pump rate (cc/s) vs time (sec) I=SYSTEM('LPLOT1.EXE'C) IF(I.EQ.-1) STOP 'Could not run COMMAND.COM ...' I=SYSTEM('CLS'C) IF(I.EQ.-1) STOP 'Could not run COMMAND.COM ...' C C Line plot, source pressure (psi) vs time (sec) I=SYSTEM('LPLOT.EXE'C) IF(I.EQ.-1) STOP 'Could not run COMMAND.COM ...' I=SYSTEM('CLS'C) IF(I.EQ.-1) STOP 'Could not run COMMAND.COM ...' C C Line plot, observation probe pressure (psi) vs time (sec) I=SYSTEM('LPLOT3.EXE'C) IF(I.EQ.-1) STOP 'Could not run COMMAND.COM ...' I=SYSTEM('CLS'C) IF(I.EQ.-1) STOP 'Could not run COMMAND.COM ...' C

The file SCRATCH.DAT contains comparisons of PDATA( ) and POBS( ) for the various values of KH and KV. At the end of each simulation, the cumulative “pressure difference squared” error for that simulation is given. If you look at all the simulations, the simulation with the lowest error identifies the correct KH and KV. The

316 Formation Testing Volume 3

SCRATCH.DAT file automatically opens when the calculations are done. Note that on a Windows i5 computer, it requires about one (1) second to run forty (40) simulations. Thus, many simulations can be run very quickly, and higher level do-loops are practical. 998

I=SYSTEM('NOTEPAD.EXE SCRATCH.DAT'C) IF(I.EQ.-1) STOP 'Could not run COMMAND.COM ...' I=SYSTEM('CLS'C) IF(I.EQ.-1) STOP 'Could not run COMMAND.COM ...' 999 STOP END C ------------------------------------------------------------SUBROUTINE ELIPS COMMON T0,T1M,T1P,T2M,T2P,T3M,T3P,T4M,T4P,T5M,T5P,T6M,T6P,T7M,T7P, 1T8,Q0,Q1M,Q1P,Q2M,Q2P,Q3M,Q3P,Q4M,Q4P,Q5M,Q5P,Q6M,Q6P,Q7M,Q7P,Q8, 2QREAD,READF C There are 3600 seconds in one hour. Input flowrate function given C at 0.01 sec interval, one hour pumping permitted, for 360000 steps. DIMENSION QREAD(360000) C Note, 100 spatial meshes, 10000 time values ... DIMENSION RSTAR(100),P(100,10000) DIMENSION AA(100),BB(100),CC(100),VV(100),WW(100) DIMENSION TPLOT(10000),QPLOT(10000),PPLOT(10000),POBS(10000) C Pwell for use when skin is nonzero Dimension Pwell(10000)

The array PDATA( ) contains the experimental observation probe data obtained from a laboratory test or a field test. Since we have no empirical results, we created “experimental results” using FT-06 from the interactive mode. We read in the pressure data later from the file BATCH-DATA.DAT. C to C

PDATA is observation probe pressure data that we are trying

match computed results to Dimension PDATA(10000) CHARACTER*1 STRING DIMENSION STRING(15) REAL KV,KH,KVMD,KHMD,KS,KSMD 1234 FORMAT(15A1)

C C Create dummy array to pass to Function Q in case we do not read it DO 1111 NQREAD=1,360000 QREAD(NQREAD) = 0. 1111 CONTINUE C C General information, output file for users ...

Practical Applications and Examples 317 OPEN(UNIT=4,FILE='ELIPS.DAT',STATUS='UNKNOWN') C C Open "experimental data" BATCH-DATA.DAT file containing observation probe pressure C results from an engineering experiment. We will compare computed results to this data. C This file has 248 data points in 248 lines. This number and the exact times must C be matched to the FT-06 calculation underway. Remind programmers of this!

Explanation - in the BATCH-DATA.DAT file, the times are reported at every 0.1 second and there are 248 pressure data points. The simulator must be consistent with this, or the comparisons will be meaningless. Look at the FT-06 interactive menu to see how input data files are set up. C OPEN(UNIT=17,FILE='Batch-Data.dat',STATUS='UNKNOWN') DO 8888 NKOUNT=1,248 READ(17,3330) TDUMMY,PDATA(NKOUNT) C WRITE(18,3330)TDUMMY,PDUMMY C We have verified this is reading and writing correctly 3330 FORMAT(1X,F10.5,F20.3) 8888 CONTINUE CLOSE(17,STATUS='KEEP') C C We will compare experimental PDATA and computed POBS data in SCRATCH file OPEN(UNIT=18,FILE='SCRATCH.dat',STATUS='UNKNOWN') WRITE(18,3336) 3336 FORMAT(' Batch file computations for FT-06 ...') WRITE(18,3335) C C Read inputs from Visual Basic interface ... OPEN(UNIT=13,FILE='FDMSPH-G.DAT',STATUS='UNKNOWN') DO 2005 NKT=1,52 READ(13,1234) (STRING(NN),NN=1,15) CALL READRL(STRING,TEMPOR) C If "spectype = 0" below, set Value1 = kvmd amd Value2 = khmd C If "spectype = 1" below, set Value1 = kv/kh amd Value2 = ksmd IF(NKT.EQ. 1) VALUE1 = TEMPOR IF(NKT.EQ. 2) VALUE2 = TEMPOR IF(NKT.EQ. 3) PHI = TEMPOR IF(NKT.EQ. 4) VISCX = TEMPOR C For gases (QX=0), EM is dimensionless thermodynamic exponent C For liquids (QX=1), EM is fluid compressibility (1/psi) IF(NKT.EQ. 5) EM = TEMPOR IF(NKT.EQ. 6) PORPSI = TEMPOR IF(NKT.EQ. 7) RWELLX = TEMPOR IF(NKT.EQ. 8) DISTX = TEMPOR IF(NKT.EQ. 9) DIPDEG = TEMPOR IF(NKT.EQ.10) FLVLCC = TEMPOR IF(NKT.EQ.11) FLCMPX = TEMPOR

318 Formation Testing Volume 3 C QX is fluid type indicator used in matrix coefficient definition C logic. QX = 0 for gases, QX = 1 for liquids. IF(NKT.EQ.12) QX = TEMPOR IF(NKT.EQ.13) RRATE = TEMPOR IF(NKT.EQ.14) DTSEC = TEMPOR IF(NKT.EQ.15) TMXSEC = TEMPOR IF(NKT.EQ.16) T0 = TEMPOR IF(NKT.EQ.17) T1M = TEMPOR IF(NKT.EQ.18) T1P = TEMPOR IF(NKT.EQ.19) T2M = TEMPOR IF(NKT.EQ.20) T2P = TEMPOR IF(NKT.EQ.21) T3M = TEMPOR IF(NKT.EQ.22) T3P = TEMPOR IF(NKT.EQ.23) T4M = TEMPOR IF(NKT.EQ.24) T4P = TEMPOR IF(NKT.EQ.25) T5M = TEMPOR IF(NKT.EQ.26) T5P = TEMPOR IF(NKT.EQ.27) T6M = TEMPOR IF(NKT.EQ.28) T6P = TEMPOR IF(NKT.EQ.29) T7M = TEMPOR IF(NKT.EQ.30) T7P = TEMPOR IF(NKT.EQ.31) T8 = TEMPOR IF(NKT.EQ.32) Q0 = TEMPOR IF(NKT.EQ.33) Q1M = TEMPOR IF(NKT.EQ.34) Q1P = TEMPOR IF(NKT.EQ.35) Q2M = TEMPOR IF(NKT.EQ.36) Q2P = TEMPOR IF(NKT.EQ.37) Q3M = TEMPOR IF(NKT.EQ.38) Q3P = TEMPOR IF(NKT.EQ.39) Q4M = TEMPOR IF(NKT.EQ.40) Q4P = TEMPOR IF(NKT.EQ.41) Q5M = TEMPOR IF(NKT.EQ.42) Q5P = TEMPOR IF(NKT.EQ.43) Q6M = TEMPOR IF(NKT.EQ.44) Q6P = TEMPOR IF(NKT.EQ.45) Q7M = TEMPOR IF(NKT.EQ.46) Q7P = TEMPOR IF(NKT.EQ.47) Q8 = TEMPOR IF(NKT.EQ.48) READF = TEMPOR IF(NKT.EQ.49) SKIN = TEMPOR IF(NKT.EQ.50) SPECTY = TEMPOR IF(NKT.EQ.51) RPCT = TEMPOR IF(NKT.EQ.52) GEOMF = TEMPOR 2005 CONTINUE CLOSE(13,STATUS='KEEP') C C SPECTY (permeability specification type) now known as 0 or 1 C If "spectype = 0", set Value1 = kvmd amd Value2 = khmd C If "spectype = 1", set Value1 = kv/kh amd Value2 = ksmd IF(SPECTY.EQ.0.) THEN KVMD = VALUE1 KHMD = VALUE2 ANISO = KVMD/KHMD KSMD = KVMD**(1./3.)*KHMD**(2./3.)

Practical Applications and Examples 319 ENDIF C IF(SPECTY.EQ.1.) THEN ANISO = VALUE1 KSMD = VALUE2 KHMD = KSMD/ANISO**(1./3.) KVMD = ANISO*KHMD ENDIF C C BATCH CODE NOTE C At this point, all inputs have been read in from Visual Basic interface, and C KH and KV have been defined. This batch code will ignore KH and KV and redefine C ranges of these parameters internally. These will be placed in a double do-loop C and computations (and comparisons with experimental data) will be done. C Note that 775 and 776 are outside the time integration loop. 775 KHMD = 1. 776 KVMD = 1. 777 JUNK = 0 MAXKV = 8 MAXKH = 9 C Initialize error to zero ERROR = 0. C C READF = 0 means read flowrates from VB schedule C READF = 1 means read flowrates from file C Read volume flow rate from file if required, overwrite dummy values C Units read in are CC/S as function of time, in increments of 0.01 sec. IF(READF.EQ.1.) THEN OPEN(UNIT=23,FILE='QWRITE.DAT',STATUS='UNKNOWN') DO 1115 NQREAD=1,360000 READ(23,1114) QREAD(NQREAD) 1114 FORMAT(1X,F11.5) 1115 CONTINUE CLOSE(23,STATUS='KEEP') ENDIF C C NOMENCLATURE AND INPUTS C KVMD .... Vertical permeability (md) C KHMD .... Horizontal permeability (md) C KSMD .... Spherical permeability (md) C ANISO ... Anisotropy or Kv/Kh dimensionless ratio C PHI ..... Porosity (decimal) C VISCX ... Viscosity (cp) C EM ...... If gas, this is thermodynamic exponent "m" (dimensionless) C If liquid, this is fluid compressibility (1/psi) C PORPSI .. Pore pressure (psi) C RWELLX .. Nozzle effective radius (cm) C RPCT .... Initial r grid expressed as % of spherical radius

320 Formation Testing Volume 3 C GEOMF ... Geometric factor for spherical source (dimless) C DISTX ... Probe separation (cm) C DIPDEG .. Dip angle (deg) C FLVLCC .. Flowline volume (cc) C FLCMPX .. Flowline fluid compressibility (1/psi) C QX ...... Fluid type indicator used in Fortran logic to C select matrix coefficients for liquids versus gases C T* ...... Input time points (sec) C Q* ...... Input volume flow rate (cc/sec), + means fluid withdrawal C from rock, - is fluid injection into rock. Note, our pump C schedule allows both continuous and discontinuous functions C of time C RRATE ... Radial grid expansion rate (dimensionless > 1) C SKIN .... Dimensionless skin factor, greater or equal to zero C DTSEC ... Time step (sec) C TMXSEC .. Maximum simulation time (sec) C RSTAR ... Dimensionless r coordinate C P ....... Pressure array (lbf/ft^2) depends on RSTAR and C dimensional time (sec) C 1 FORMAT(' ') WRITE(4,2500) WRITE(4,2501) WRITE(4,2502) WRITE(4,1) WRITE(4,2503) WRITE(4,1) 2500 FORMAT(' Transient integration of ellipsoidal source equation ') 2501 FORMAT(' in anisotropic homogeneous medium for linear liquid and') 2502 FORMAT(' nonlinear gas Darcy flows ... ') 2503 FORMAT(' Copyright (C), 2006, by Wilson C. Chin, Ph.D., M.I.T. ') IF(QX.EQ.1.0) WRITE(4,2900) IF(QX.EQ.0.0) WRITE(4,2905) 2900 FORMAT(' INPUT PARAMETERS, LIQUID FLOW ANALYSIS ...') 2905 FORMAT(' INPUT PARAMETERS, GAS FLOW ANALYSIS ...') WRITE(4,1) WRITE(4,3001) KVMD WRITE(4,3002) KHMD WRITE(4,2920) KSMD WRITE(4,2921) ANISO WRITE(4,3003) PHI WRITE(4,3004) VISCX IF(QX.EQ.1.0) WRITE(4,3100) EM IF(QX.EQ.0.0) WRITE(4,3101) EM 3100 FORMAT(' Liquid compressibility ....... (1/psi): ',E11.4) 3101 FORMAT(' Gas thermodynamic exponent m (dimless): ',E11.4) WRITE(4,3005) SKIN WRITE(4,3006) PORPSI

Practical Applications and Examples 321

3001 3002 2920 2921 3003 3004 3005 3006 3007 3008 3009 3010 3011 3013 3014 3015 3016 3017

WRITE(4,3007) RWELLX WRITE(4,3016) RPCT WRITE(4,3017) GEOMF WRITE(4,3008) DISTX WRITE(4,3009) DIPDEG WRITE(4,3010) FLVLCC WRITE(4,3011) FLCMPX WRITE(4,3013) RRATE WRITE(4,3014) DTSEC WRITE(4,3015) TMXSEC WRITE(4,1) FORMAT(' Vertical permeability ........... (md): FORMAT(' Horizontal permeability ......... (md): FORMAT(' Spherical permeability .......... (md): FORMAT(' Anisotropy ratio kv/kh ..... (dimless): FORMAT(' Porosity ................... (decimal): FORMAT(' Fluid viscosity ................. (cp): FORMAT(' Skin factor .......... (dimensionless): FORMAT(' Pore pressure .................. (psi): FORMAT(' Effective nozzle radius ......... (cm): FORMAT(' Probe separation ................ (cm): FORMAT(' Dip angle ...................... (deg): FORMAT(' Flowline volume ..................(cc): FORMAT(' Flowline fluid compressibility (1/psi): FORMAT(' Grid expansion rate ........ (dimless): FORMAT(' Time step ...................... (sec): FORMAT(' Maximum simulation time ........ (sec): FORMAT(' Initial r grid (% of spherical radius): FORMAT(' Source geometric factor .... (dimless):

',E11.4) ',E11.4) ',E11.4) ',E11.4) ',E11.4) ',E11.4) ',E11.4) ',E11.4) ',E11.4) ',E11.4) ',E11.4) ',E11.4) ',E11.4) ',E11.4) ',E11.4) ',E11.4) ',E11.4) ',E11.4)

C WRITE(4,3200) WRITE(4,1) WRITE(4,3201) WRITE(4,3202) WRITE(4,1) C IF(READF.EQ.0.) THEN WRITE(4,3205) Q0 ,T0 ,Q1M,T1M WRITE(4,3205) Q1P,T1P,Q2M,T2M WRITE(4,3205) Q2P,T2P,Q3M,T3M WRITE(4,3205) Q3P,T3P,Q4M,T4M WRITE(4,3205) Q4P,T4P,Q5M,T5M WRITE(4,3205) Q5P,T5P,Q6M,T6M WRITE(4,3205) Q6P,T6P,Q7M,T7M WRITE(4,3205) Q7P,T7P,Q8 ,T8 ENDIF C IF(READF.EQ.1.) THEN WRITE(4,3206) WRITE(4,3207) 3206 FORMAT(' Volume flow rate read from file. Refer to output file') 3207 FORMAT(' for detailed tabulation of "cc/s" vs time (sec).') ENDIF C

322 Formation Testing Volume 3 WRITE(4,1) C 3200 FORMAT(' PUMPING SCHEDULE') 3201 FORMAT(' Positive flow rate denotes fluid withdrawal from rock,') 3202 FORMAT(' while negative corresponds to injection ... ') 3205 FORMAT(1X,E11.3,' cc/s at',E11.3,' sec, changes to',E11.3,' cc/s 1 at',E11.3,' sec') C C UNITS CONVERSIONS ("LBF, FT, SEC" STANDARD UNITS USED) PI = 3.1415926 KV = KVMD*0.00000000001/(12.*12.*2.54*2.54) KH = KHMD*0.00000000001/(12.*12.*2.54*2.54) VISC = VISCX*0.0000211 PPORE = PORPSI*144. RWELL = RWELLX/(12.*2.54) DIST = DISTX/(12.*2.54) DIPRAD = (PI/180.)*DIPDEG FLVOL = FLVLCC/(12.*12.*12.*2.54*2.54*2.54) FLCOMP = FLCMPX/144. C C CONVENIENT CONSTANTS NMAX = TMXSEC/DTSEC IMAX = 100 IMAXM1 = IMAX-1 BCCOEF = 4.*PI*GEOMF*RWELL**2*KV**(1./6.)*KH**(1./3.)/VISC VC = FLVOL*FLCOMP SIN2 = SIN(DIPRAD)**2 COS2 = COS(DIPRAD)**2 C C Gas compressibility, typically 0.001 1/psi C C Interpretation of old user "initial radial grid" DRCM is C ambiguous. In isotropic case, x^2 + y^2 + z^2 = Rwell^2, C initial radial grid DR (expressed as a fraction of Rwell) C makes sense. In the anisotropic case, DR/(KV**(1/6)*KH**(1/3)) C used earlier is somewhat arbitrary. Better not to use it, just C use the initial dim-less radial grid increment as a % of RstarW. C C RSTAR(1) is dimensionless ellipsoidal effective radius RSTAR(1) = GEOMF*RWELL/(KV**(1./6.)*KH**(1/3.)) C C DRSTAR is dimensionless initial grid, assumed 10% of RSTAR(1) C DRSTAR = 0.1*RSTAR(1) C NOTE - Above 0.1 chosen for fast computational speed, however, C 0.01 value leads to better agreement with exact analytical solution C Now, let % be specified by user instead ... DRSTAR = (RPCT/100.)*RSTAR(1)

Practical Applications and Examples 323 RSTAR(2) = RSTAR(1) + DRSTAR C C C

More constants, modeling skin effects (see 6/19/2006 notes) SKIN = 5. - now read in from Windows interface BETA = GEOMF*RWELL*VC*SKIN/(KV**(1./6.)*KH**(1./3.)) BETAXX = BETA/((RSTAR(2)-RSTAR(1))*DTSEC)

C WRITE(4,39) 39 FORMAT(' Dimensionless, RSTAR mesh (see theory for definitions) .. 1.') WRITE(4,1) WRITE(4,40) RSTAR(1) WRITE(4,41) RSTAR(2) 40 FORMAT(' Rstar(1) = ',F12.4) 41 FORMAT(' Rstar(2) = ',F12.4) DO 50 I = 3,IMAX C RRATE is expansion rate assumed by user RSTAR(I) = RSTAR(I-1) + RRATE*(RSTAR(I-1)-RSTAR(I-2)) C Constant mesh below C RSTAR(I) = RSTAR(I-1) + DRSTAR WRITE(4,45) I,RSTAR(I) 45 FORMAT(1X,I7,F18.4) 50 CONTINUE WRITE(4,1) C C ROBS is dimensionless distance at observation probe C DIST is probe separation, DIPDEG is dip angle for SIN2 and COS2 ROBS = DIST*SQRT(SIN2/KH + COS2/KV) DO 60 I=1,IMAXM1 IF(ROBS.GE.RSTAR(I).AND.ROBS.LT.RSTAR(I+1)) GO TO 65 60 CONTINUE 65 ISMALL = I ILARGE = I+1 C C Initialize the pressure field to pore pressure ... DO 100 I=1,IMAX P(I,1) = PPORE 100 CONTINUE C C TPLOT is time plotting array in seconds, QPLOT is volume flow C rate plotting array in cc/sec. Q( ) is read in from Fortran C function statement Q(T,...) TPLOT(1) = 0. QPLOT(1) = Q(0.) DO 333 N=2,NMAX DO 200 I=2,IMAXM1

Proprietary source code removed. . . . DO 300

I=1,IMAX

324 Formation Testing Volume 3 300

P(I,N) = VV(I) CONTINUE

C C When skin is nonzero, P refers to pressure inside reservoir at the C sandface. There is additional pressure drop through skin layer. The C pressure inside the flowline (outside the reservoir) is Pwell, which C adjusts sandface pressure by additional pressure drop thru skin. C IF(SKIN.NE.0.) THEN PWELL(1) = PPORE DPDR = (P(2,N)-P(1,N))/(RSTAR(2)-RSTAR(1)) PWELL(N) = P(1,N) GEOMF*RWELL*SKIN*DPDR/(KV**(1./6.)*KH**(1./3.)) ENDIF C If skin = 0, p(1,n) itself is sandface pressure and flowline pressure C and Pwell need not be used. C C Printed pressure output below in PSI, given at each time step IF(N.EQ.2) THEN WRITE(4,310) WRITE(4,311) WRITE(4,312) WRITE(4,1) 310 FORMAT(' Pressures P1, P2, ..., Pimax at successive time steps tab 1ulated next.') c c311 FORMAT(' P1 (left column) is formation tester pressure, Pimax (far c 1 right) represents farfield infinity.') c312 FORMAT(' Top row is t = 0, each new row describes pressure at late c 1r time step.') c 311 FORMAT(' T(sec), Q(cc/s), 1st and 2nd columns. P1(psi), next, is 1formation tester pressure, Pimax (far right) is farfield infinity. 2') 312 FORMAT(' Top row is t = 0 sec, each new row describes pressure at 1later time step.') ENDIF C C Write solutions in space for each time level ... IF(SKIN.EQ.0.) THEN IF(N.EQ.2) WRITE(4,330) tplot(n-1),Qplot(N1)/0.0000353147,(P(I,1) 1/144.,I=1,IMAX) WRITE(4,330) tplot(n),Qplot(N)/0.0000353147,(P(I,N)/144.,

Practical Applications and Examples 325 1I=1,IMAX) ENDIF C But ... IF(SKIN.NE.0.) THEN C First i=1 node treated differently using skin "p = p - () dpdr" formula IF(N.EQ.2) WRITE(4,330) tplot(n-1),Qplot(N-1)/0.0000353147, 1PWELL(1)/144.,(P(I,1)/144.,I=2,IMAX) WRITE(4,330) tplot(n),Qplot(N)/0.0000353147,PWELL(N)/144., 1(P(I,N)/144.,I=2,IMAX) ENDIF C 330 FORMAT(1X,102F10.2) POBS(N) = P(ISMALL,N) + 1(ROBS - RSTAR(ISMALL))*(P(ILARGE,N)-P(ISMALL,N))/ 2(RSTAR(ILARGE)- RSTAR(ISMALL)) POBS(N) = POBS(N)/144.

In the code below, we define the cumulative “pressure difference squared” error for the simulation. ERROR = ERROR + (POBS(N)-PDATA(N-1))**2.

It is important to understand that the above definition of error is just a mathematical abstraction and is assumed here for simplicity. There may be other definitions of error that the engineer wants to use or other ways of defining the correct set of KH and KV. If there are other relationships that the engineer wants to use in addition to the above or to replace the above, he should do that in this location of the source code. When the results of all the simulations are available, there may be another piece of code that decides what is best. C C

Below is comparing arrays properly, indexing correct ... WRITE(18,3331) TPLOT(N),POBS(N),PDATA(N-1),ERROR 3331 FORMAT(1X,F10.5,3F20.3)

C 333 CONTINUE C C End of time integration C C Tabulate final error between computed and inputted pressure files. C ERROR above represents summed cumulative error over all time points. WRITE(*,3332) KHMD,KVMD 3332 FORMAT(' Evaluating ... KHMD = ',F5.0,', KVMD = ',F5.0) IF(KHMD.EQ.MAXKH.AND.KVMD.EQ.MAXKV) WRITE(*,3335) IF(KHMD.EQ.MAXKH.AND.KVMD.EQ.MAXKV) WRITE(*,3334) 3334 FORMAT(' Close SCRATCH.DAT to continue ...') 3335 FORMAT('') WRITE(18,3333) KHMD,KVMD,ERROR 3333 FORMAT(' Kh(md) = ',F10.3,' Kv(md) = ',F10.3,' Error = ',F15.3)

326 Formation Testing Volume 3 C

335

NWARN = 0 DO 335 I=1,IMAX DO 335 N=1,NMAX IF(P(I,N).LT.0.0) NWARN = 1 CONTINUE IF(NWARN.EQ.1) THEN WRITE(4,1) WRITE(4,337) WRITE(4,338) FORMAT(' WARNING: Negative P detected, desired flowrate may

337 not be 1 physically achievable for the permeabilities assumed.') 338 FORMAT(' To obtain positive pressures, e.g., increase pore pressur 1e, increase permeabilities, change flowrates, etc.') ENDIF C

The code below shows how the KH and KV calculations are “nested” in a two-level do-loop. If we wanted to vary five parameters, e.g., KH, KV, porosity, viscosity, compressibility, we would replace the code below by its five-level generalization. C

Define the order in which calculations are done KVMD = KVMD + 1. IF(KVMD.GT.MAXKV) KHMD = KHMD + 1. IF(KVMD.GT.MAXKV.AND.KHMD.LE.MAXKH) GO TO 776 IF(KHMD.GT.MAXKH) GO TO 9999 GO TO 777

Note that the line color plots below are not used in batch mode. C C C C C

+++++++++++++++++++ Line Plot No. 1 + +++++++++++++++++++

C 340 346

348

351

DO 340 N=1,NMAX Source pressure in psi IF(SKIN.EQ.0.) PPLOT(N) = P(1,N)/144. IF(SKIN.NE.0.) PPLOT(N) = PWELL(N)/144. CONTINUE OPEN(UNIT=14,FILE='LPLOT.DAT',STATUS='UNKNOWN') WRITE(14,346) FORMAT(' Source Probe Pressure') CLOSE(14,STATUS='KEEP') OPEN(UNIT=14,FILE='XNAME.DAT',STATUS='UNKNOWN') WRITE(14,348) FORMAT(' Time (sec)') CLOSE(14,STATUS='KEEP') OPEN(UNIT=14,FILE='YNAME.DAT',STATUS='UNKNOWN') WRITE(14,351) FORMAT(' Pressure (psi)')

Practical Applications and Examples 327

352 C 354 358 C C C C C C

CLOSE(14,STATUS='KEEP') NBRPLT = 1 OPEN(UNIT=14,FILE='MYFILE.DAT',STATUS='UNKNOWN') WRITE(14,352) NBRPLT,NMAX FORMAT(' ARRAY',2I6) Values just below are time values on horizontal axis WRITE(14,354) (TPLOT(NPOINT),NPOINT=1,NMAX) FORMAT(1X,10000F13.3) WRITE(14,358) (PPLOT(NPOINT),NPOINT=1,NMAX) FORMAT(1X,10000F13.3) CLOSE(14,STATUS='KEEP') +++++++++++++++++++ Line Plot No. 2 + +++++++++++++++++++

366

368

370

372 C 374 C 376 C C C C C C

Plot volume flow rate ... OPEN(UNIT=14,FILE='LPLOT1.DAT',STATUS='UNKNOWN') WRITE(14,366) FORMAT(' Volume Flow Rate') CLOSE(14,STATUS='KEEP') OPEN(UNIT=14,FILE='XNAME1.DAT',STATUS='UNKNOWN') WRITE(14,368) FORMAT(' Time (sec)') CLOSE(14,STATUS='KEEP') OPEN(UNIT=14,FILE='YNAME1.DAT',STATUS='UNKNOWN') WRITE(14,370) FORMAT(' Flow Rate (cc/sec)') CLOSE(14,STATUS='KEEP') NBRPLT = 1 OPEN(UNIT=14,FILE='MYFILE1.DAT',STATUS='UNKNOWN') WRITE(14,372) NBRPLT,NMAX FORMAT(' ARRAY',2I6) Values just below are time values on horizontal axis WRITE(14,374) (TPLOT(NPOINT),NPOINT=1,NMAX) FORMAT(1X,10000F13.3) QPLOT in cubic ft/sec, convert to cc/sec ... WRITE(14,376) (QPLOT(NPOINT)/0.0000353147,NPOINT=1,NMAX) FORMAT(1X,10000F13.3) CLOSE(14,STATUS='KEEP') +++++++++++++++++++ Line Plot No. 3 + +++++++++++++++++++

466

468

Plot observation probe pressure (psi) ... OPEN(UNIT=14,FILE='LPLOT3.DAT',STATUS='UNKNOWN') WRITE(14,466) FORMAT(' Observation Probe Pressure') CLOSE(14,STATUS='KEEP') OPEN(UNIT=14,FILE='XNAME3.DAT',STATUS='UNKNOWN') WRITE(14,468) FORMAT(' Time (sec)') CLOSE(14,STATUS='KEEP') OPEN(UNIT=14,FILE='YNAME3.DAT',STATUS='UNKNOWN')

328 Formation Testing Volume 3 470

WRITE(14,470) FORMAT(' Pressure (psi)') CLOSE(14,STATUS='KEEP') NBRPLT = 1

. . . C CLOSE(4,status='keep') CLOSE(17,STATUS='KEEP') CLOSE(18,STATUS='KEEP') 9999 JUNK = 0 RETURN END C ------------------------------------------------------------C END FORTRAN SOURCE CODE C -------------------------------------------------------------

4.2.13.3 A final example – concise operational summary

As explained earlier, this batch program is used primarily for “history matching,” that is, numerous values of KH and KV are assumed, and simulations are performed. The observation probe pressure response in time POBS( ) is compared with the experimental result PDATA( ), and for each simulation, a total “pressure difference squared” error is computed. When we examine all the simulations, the simulation with the lowest value of error is the one associated with the correct KH and KV. Now, again, we do not have experimental data to work with, so we created our own synthetic experimental data. We did this by running FT06 in the interactive mode using the following assumptions in Figure 4.2.13e –

Practical Applications and Examples 329

Figure 4.2.13e. Sample set up, parameters defined. Note that the above assumptions correspond to the flow rate and observation probe pressure data shown below in Figures 4.2.13f and 4.2.13g,

330 Formation Testing Volume 3

Figure 4.2.13f. Volume flow rate.

Figure 4.2.13g. Observation probe pressure response.

Practical Applications and Examples 331

The observation probe pressure data (in psi units) are stored in BATCH-DATA.DAT which should not be changed for code development purposes. In the final code developed by the programmer, of course, the BATCH-DATA.DAT will be the actual experimental data corresponding to the inputted flow rate function. Our BATCHDATA.DAT file was created from the assumptions Kh = 7 and Kv = 3 as seen from the input screen above (these “7” and “3” numbers are not used by the batch processing solver, which generates its own sets of test Kh and Kv internally). There are 248 lines in Batch-Data.dat. The contents look like: 0.10000 0.20000 0.30000 0.40000 0.50000 0.60000 0.70000 0.80000 0.90000 1.00000 . . . 24.30000 24.40000 24.50000 24.60000 24.70000 24.80000

1999.975 1999.927 1999.862 1999.785 1999.697 1999.603 1999.503 1999.399 1999.292 1999.181

1986.749 1986.837 1986.920 1986.996 1987.069 1987.138

Running the batch solver. First the input data needs to be set up. To do this efficiently, run the interactive FT-06 solver first. Ignore the solutions as they are not needed. Running this interactive solver creates the basic input file that will be read by ELIPS-14-BATCH-4.EXE. The ELIPS-14-BATCH-4.EXE program will keep all the inputs EXCEPT KH and KV. Internally, it will vary KH and KV according to the doloops in the code. If the programmer does not want to run the interactive solver first, he can replace the GREEN source code directly using values of the other input parameters. However, running the interactive solver is recommended because it will plot the flow rate function that is created by the built-in menu or it will read the flow rate file defined by the engineer. It is just a convenience tool. Now, run the ELIPS-14-BATCH-4.EXE program. The batch solver runs rapidly. On an i5 Windows computer, 40 – 50 simulations can be completed in one second. In the source code,

332 Formation Testing Volume 3

we increment KH by 1 until KH = 9, and increment KV by 1 until KV = 8. Thus, there are 72 calculations, requiring less than two seconds. If the programmer wants to use different incrementing strategies, he can change the source code appropriately. When all the computations are completed, the SCRATCH.DAT file automatically opens. For the above source code, the file is Batch file computations for FT-06 ... Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md)

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 5.000 5.000 5.000 5.000 5.000 5.000 5.000 5.000 6.000 6.000 6.000 6.000 6.000 6.000 6.000 6.000 7.000

Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md)

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000

Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

5951.430 7458.258 6965.629 6078.306 5181.186 4385.321 3702.576 3128.267 2825.995 5358.436 5919.768 5681.864 5216.870 4618.719 4064.451 3512.589 1198.071 2791.451 3686.459 3849.362 3623.159 3309.531 2958.865 2593.101 747.051 1200.283 1883.090 2148.817 2159.537 2022.090 1816.262 1602.816 936.549 367.095 714.027 983.375 1086.729 1068.738 993.543 905.998 1491.641 104.719 163.851 315.262 434.059 489.755 493.716 498.274 2245.641

Practical Applications and Examples 333 Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md) Kh(md)

= = = = = = = = = = = = = = = = = = = = = = =

7.000 7.000 7.000 7.000 7.000 7.000 7.000 8.000 8.000 8.000 8.000 8.000 8.000 8.000 8.000 9.000 9.000 9.000 9.000 9.000 9.000 9.000 9.000

Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md) Kv(md)

= = = = = = = = = = = = = = = = = = = = = = =

2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000

Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error Error

= = = = = = = = = = = = = = = = = = = = = = =

218.389 0.000 57.302 157.455 241.572 311.261 374.573 3084.639 598.314 136.857 96.826 169.835 268.677 373.857 482.759 4008.590 1143.483 492.366 364.955 404.968 508.574 634.391 774.725

Note how the errors (third column at far right) for each of the simulations are quite large, except for the zero error corresponding to KH = 7 and KV = 3. In other words, we have performed many simulations in which KH and KV were changed. By examining all the simulation results and identifying that with the lowest error, we found the KH and KV used in creating the “experimental data” file PDATA( ) in BATCHDATA.DAT. In other words, we have successfully concluded the history matching exercise. Again, in practice, the engineer will probably use his own definition of error, select different incrementing schemes for KH and KV depending on the formations he is logging, add his own physical constraints and exclusions, and so on. We have proved, in constructing this batch solver from the interactive solver, that all our programming logic is correct, so that the source code in this document serves as a good starting point for other software development.

334 Formation Testing Volume 3

4.2.14 Gas Pumping Formation tester sampling and pressure transient analysis are often associated with liquid applications, but these techniques also prove vital to gas operations. Gases are categorized as “dry” or “wet.” Although several definitions are available, we will simply regard dry natural gas as almost completely methane – the higher the methane concentration within the gas, the drier it is. A wet gas is any gas that contains a small amount of liquid, e.g., a humid gas or a gas saturated with liquid vapor, a multiphase flow with a significant volume of gas, and so on. In developing models for pressure transient interpretation, it is not so much “dry” versus “wet” that matters – the viscosity of the gas and its thermodynamic state are the essential parameters. In addition, from mathematical and physical perspectives, gas flows are nonlinear while those associated with liquids are linear. The latter allow for simpler modeling since superposition methods can be employed, while for nonlinear flows, less general methods must be developed on a case-bycase basis. In this book, and in those due to Chin et al. (2014) and Chin et al. (2015), FT-00 and FT-01 were developed for liquids, while FT-02 and FT-06 were developed for gases (FT-06 also contains a liquid simulation option). 4.2.14.1 Several field notes

Dry gas sampling, and even retrograde condensate gas samples which are more challenging, have been successfully performed using conventional formation testers. Dry gas samples are recoverable, although some are wet. Condensate gases are always wet, however, it is desirable to obtain them as dry gases downhole and reconstitute at the surface in the lab. Some gas sampling in client applications have been observed with “high contamination,” but in certain cases, the laboratory reported contamination by weight while the actual contamination was very small when determined by volume (< 5%). Various formation testing tools can measure the fluid compressibility by drawing the sample into the pretest chamber and performing PVT tests while sampling. Normally this requires pumping to be stopped during PVT analysis, although recent tools do allow PVT testing while pumping. Finally,

fluid compressibility and bubble point have been used on a limited basis to monitor pumping and contamination. One tool offers an acoustic cell to monitor compressibility but possibly only for liquids.

Practical Applications and Examples 335 4.2.14.2 Review of steady-state direct and inverse methods

In this section, we briefly review steady-state direct and inverse methods for gases. The work of Chin et al. (2014) derives exact, closed form, analytical solutions which form the basis of the direct forward solver in Figure 4.2.14a and the inverse solver in Figure 4.2.14b. This section shows that these two modules are physically and mathematically consistent with each other. In Section 4.2.14.3, we show how the transient pressure relates to the steady results described here.

Figure 4.2.14a. Direct steady-state exact solution. Let us calculate the source and observation probe pressure responses for the input parameters shown in Figure 4.2.14a. Clicking “Solve” yields the following report – DIRECT GAS SOLVER FOR PRESSURES (ZERO SKIN) Copyright (2005), Wilson C. Chin, Ph.D., M.I.T. All rights reserved. Input parameters ... Dip angle ..................... (deg): KV permeability ................ (md): KH permeability ................ (md): Pore pressure ................. (psi): Effective source probe radius .. (cm): Probe separation ............... (cm): Volume flow rate ............. (cc/s): Gas viscosity .................. (cp): Gas exponent m ...... (dimensionless):

0.4500E+02 0.9305E-02 0.8287E-01 0.1000E+05 0.1000E+01 0.1500E+02 0.1000E+02 0.1000E-01 0.1000E+01

336 Formation Testing Volume 3 Newton-Raphson convergence history ... Iteration

Source Psi

Pore Psi

1

0.7083E+04

0.1000E+05

2

0.7485E+04

0.1000E+05

3

0.7500E+04

0.1000E+05

4

0.7500E+04

0.1000E+05

5

0.7500E+04

0.1000E+05

6

0.7500E+04

0.1000E+05

7

0.7500E+04

0.1000E+05

8

0.7500E+04

0.1000E+05

9

0.7500E+04

0.1000E+05

10

0.7500E+04

0.1000E+05

Observation probe (psi):

0.9905E+04

Thus, for isothermal (m=1) gases, the exact, closed form, analytical solution yields . . . Source probe (psi): Observation probe (psi):

0.7500E+04 0.9905E+04

We next use these pressures as inputs to the inverse algorithm in Figure 4.2.14b and attempt to recover the horizontal and vertical permeabilities. We have –

Figure 4.2.14b. Inverse steady-state exact solution.

Practical Applications and Examples 337 INVERSE KH AND KV GAS SOLVER (ZERO SKIN) Copyright (2005), Wilson C. Chin, Ph.D., M.I.T. All rights reserved. Input parameters ... Dip angle ..................... (deg): Source probe pressure ......... (psi): Observation probe pressure .... (psi): Pore pressure ................. (psi): Effective source probe radius .. (cm): Probe separation ............... (cm): Volume flow rate ............. (cc/s): Gas viscosity .................. (cp): Gas exponent m ...... (dimensionless):

0.4500E+02 0.7500E+04 0.9905E+04 0.1000E+05 0.1000E+01 0.1500E+02 0.1000E+02 0.1000E-01 0.1000E+01

Possible solutions ... COMPLEX KH ROOT # 1: -0.8720E-01 0.0000E+00 COMPLEX KH ROOT # 2: 0.8720E-01 0.0000E+00 COMPLEX KH ROOT # 3: -0.1717E-06 0.0000E+00 Root No. 1: Kh = 0.8720E-01 md, Kv = 0.8402E-02 md One realistic permeability root found.

The solution Root No. 1:

Kh = 0.8720E-01 md, Kv = 0.8402E-02 md

is consistent with the known values of kh = 0.08287 md and kv = 0.009305 md (the exact math models are solved numerically and involve approximations). In the next section, we describe one transient situation which would lead to the steady-state reported here.

338 Formation Testing Volume 3 4.2.14.3 Transient gas calculations

In Section 4.2.14.2, we demonstrated how direct and inverse calculations for a nonlinear gas flow example are consistent, using our model FT-02. The two software modules were developed from exact, analytical, closed form solutions obtained by solving the steady-state gas algebraic flow equations. We emphasize that these steady equations are completely different from those for transient flow, which satisfy an unsteady partial differential equation system – these are solved in FT-06 below for the same parameters as assumed in Figure 4.2.14a.

Figure 4.2.14c. Transient forward approximate numerical solution.

Practical Applications and Examples 339

Figure 4.2.14d. Source probe pressure response.

Figure 4.2.14e. Observation probe pressure response. It is clear from Figures 4.2.14d,e that while source and observation probe responses are calculated stably (that is, no “wiggles”) and yield expected horizontal asymptotes, the converged pressures are not quite the 7,500 psi and 9,905 psi obtained on an exact basis. We did attempt improved results by changing mesh expansion rates, initial mesh sizes, and so on, but a perfect match to both numbers proved difficult. The reader is therefore cautioned that numerical methods are only approximate, even the high-order accurate schemes underlying FT-06. For this reason, the exact analytical models in FT-00, FT-01 and also our early time inverse methods (all for liquids) is invaluable for both job planning and interpretation.

340 Formation Testing Volume 3 Using inapplicable formulas. The author does not support using

mathematical formulas that are inappropriate to the physics, but then again, sometimes the engineer has no choice – but on occasion, discoveries are made and additional research ideas are motivated. One formula that is derived in Chin et al. (2014) in the context of liquid flows, which satisfy a linear parabolic differential equation very different from the nonlinear one for gas, is the straight line equation “Pw(t) = P0 – Qt/(VC)” applicable to very early time drawdown and buildup. This equation, when inverted in the form C = Qt/{V(P0 – Pw)}, allows us to predict fluid compressibility C in the flowline. Here Q is the volume flow rate, V is the flowline volume, t is time and (P0 – Pw) is the pressure drop. In Figure 4.2.14f below, we show the first thirty seconds of the source probe pressure response, not quite a straight line, as calculated from the prior FT-06 simulation. Application of the formula yields C = Qt/{V(P0 – Pw)} = (10 cc/s) (30 s)/{100 cc (10,000 psi – 8,100 psi) or 0.0016/psi which is the same order-of-magnitude as the 0.001/psi assumed in the forward simulation for FT-06 – not exactly a bad result.

Figure 4.2.14f. Source probe response – very early time.

Practical Applications and Examples 341

A second curiosity item is permeability prediction using pressure drop data in an algorithm designed for liquids, that is, FT-01. This, of course, is inappropriate use of the software. The required pressure drops at source and observation probes are obtained from Source probe pressure drop = 10,000 – 7,500 = 2,500 psi Observation probe pressure drop = 10,000 – 9,905 = 95 psi

Figure 4.2.14g. Inverse method for liquid flows. Running the FT-01 software leads to the following output – INVERSE KH AND KV LIQUID SOLVER (ZERO SKIN) Copyright (2005), Wilson C. Chin, Ph.D., M.I.T. All rights reserved. Input parameters ... Dip angle ..................... (deg): Source probe delta-p .......... (psi): Observation probe delta-p ..... (psi): Effective source probe radius .. (cm): Probe separation ............... (cm): Volume flow rate ............. (cc/s): Liquid viscosity ............... (cp):

0.4500E+02 -0.2500E+04 -0.9500E+02 0.1000E+01 0.1500E+02 0.1000E+02 0.1000E-01

Source probe delta-p is source probe pressure minus pore pressure .. negative for fluid withdrawal from formation (positive flow rate). It is positive for fluid injection into formation (negative flow rate). Similar definition for observation probe delta-p. Possible solutions ...

342 Formation Testing Volume 3 Tentative permeabilities (md) ... Complex KH root # 1: -.116E+00 + 0.000E+00 i, KV: 0.758E-02 Complex KH root # 2: 0.116E+00 + 0.000E+00 i, KV: 0.758E-02 Complex KH root # 3: -.228E-06 + 0.000E+00 i, KV: 0.196E+10 KH above strictly valid -- if real part is positive and imaginary part is zero ... sometimes imaginary part is allowed, if small compared to positive real part. KH with negative real part is never correct. CAUTION: KV above is computed using real part of KH even if KH has nonzero imaginary part .... Careful! Exact conclusions below ... Following based on strict adherence to requirements that real(KH)>0 and imag(KH)=0 ..... mathematically correct KH and KV pairs, if shown. Root No. 1: Kh = 0.116E+00 md, Kv = 0.758E-02 md One realistic permeability root found.

Thus, the (incorrect) liquid flow assumption leads to approximately kh = 0.12 md and kv = 0.0076 md, which while differemt, are not too far from the 0.083 md and 0.0093 md assumed in FT-06 in Figure 4.2.14c. It is worth noting that the author expected more pessimistic predictions for both compressibility and permeability, but these limited successes may suggest that a simpler linear model for nonlinear gas flow might be developed with much greater potential for flexibility and portability. 4.3 References Chin, W.C., Formation Invasion – with Applications to Measurement While Drilling, Time Lapse Analysis, and Formation Damage, Gulf Publishing, Houston, 1995. Chin, W.C., Quantitative Methods in Reservoir Engineering, Second Edition, Elsevier, Amsterdam, 2017. Chin, W.C., Zhou, Y., Feng, Y., Yu, Q. and Zhao, L., Formation Testing: Pressure Transient and Contamination Analysis, John Wiley & Sons, Hoboken, New Jersey, 2014. Chin, W.C., Zhou, Y., Feng, Y. and Yu, Q., Formation Testing: Low Mobility Pressure Transient Analysis, John Wiley & Sons, Hoboken, New Jersey, 2015.

5 Best Practices and Closing Remarks In this final chapter, we offer comments on “best practices” designed to take maximum advantage of the formation evaluation methods we have developed. These are subjective: there are no “absolutes” and no single rule will apply to all situations. But it comes as no surprise that from qualitative observations such as those below, many drawn from experienced engineers in well logging, will emerge future quantifiable well logging methods – and several have already motivated some interesting new areas for research and development. 5.1 Best Practices The following general “rules of thumb” have proven important to all manufacturers’ tools. In addition, there are specific procedures that apply to particular hardware designs, e.g., maintenance procedures, testing sequences, and so on, that obviously cannot be covered here. Finally, a list of publications for “recommended reading” is offered at the end of this chapter to cover important aspects of formation testing not addressed in this book and more specialized aspects of fluid mechanics. The rules below are not presented in any particular order of importance and all are deemed useful in field application. Collect pressure transient data with mud pumps off since pump pulsations will decrease data quality and accuracy of pore pressure and mobility predictions. Other views have been published. They note that different frequency ranges apply to pump noise relative to those for formation tester pump rates and that frequency filtering will improve the signal. It is also possible that two-transducer differential detection methods in MWD telemetry can be deployed downhole on dual probe tools. All of these ideas require further study. 343

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In low-mobility zones, measure how rapidly (or slowly) recorded pressures are changing. The changes must be sufficiently large in order to be meaningful, e.g., better than gauge resolution, large pressure differences, and so on. Typically, ten pressure measurements may be taken along the buildup curve, with each measurement separated by seconds to minutes, depending on the mobility, as suggested by one vendor. The test is terminated when a “ test” (described later) indicates that collected results truly describe the formation. In high-mobility zones, the pressure in the formation reaches steadystate very rapidly and the pressure gauge may not act quickly enough to record transient pressure values needed for interpretation. In a buildup test, the measured steady value is the pore pressure; fortunately, this value together with one additional pressure-time dataset (in addition to the pressure value obtained when drawdown stops) suffices to determine mobility. In high-mobility zones, the pressure drop during drawdown may be small and not readily detectable by the pressure gauge. To make the pressure drop detectable, the flow rate may be increased. For example, for liquids, doubling the flow rate will double the pressure drop; the buildup pressure is correspondingly detectable. This linearity applies only to liquids. For gas pumping, no general conclusions are available. Measurement errors are different in type. For example, slopes at the bottom of drawdown curves that differ from ideal calculated shapes may indicate phase change effects. The presence of wiggles can suggest the presence of loose sand in the nozzles. Very often, the effects of supercharge are unknown and long wait times are suggested to allow overpressures to dissipate. We suggest use of our supercharge models to estimate errors in pore pressure and mobility prediction. Use a high overbalance pressure to be conservative. In principle, pore pressure and mobility can be obtained from both drawdown and buildup curves. However, there are important differences. The time duration of the drawdown cycle is limited by the length and speed of the pump piston. Often, drawdown occurs quickly and there is not enough pressure transient data needed for predictions – this limits depth of investigation. In addition, a quick

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drawdown may release dissolved gas from the petroleum liquid, thus leading to a two-phase immiscible flow with very different permeability characteristics. Also, drawdown occurs at the very beginning of testing when the influence of the formation tester packer sealing and mud invasion through the cake are most pronounced. For these reasons, interpretation using drawdown data is not suggested. Alternatively, once pump piston motion ceases, the buildup cycle can literally take as long as is required to obtain a lengthy set of pressure transient data. Gas that was previously liberated can now recombine with the petroleum liquid to form a single-phase flow again. In addition, since the buildup test is allowed to run a significantly longer time, it provides a deeper reading of the formation. The disadvantages for long duration data acquisition are cost and increased possibility of stuck and lost tools. The models used in forward and inverse analysis do not include practical effects such as sand being trapped in the tool and probe cleanup – the pressure transient characteristics associated with these events depend on the mechanical design of the formation tester and should be measured in the laboratory and documented in field operations manuals. Responses vary with specific tool designs. The single-phase flow interpretation models used obviously will not apply when liberated gas is found in the reservoir. Here, relative permeability and capillary pressure effects will be important, but since this type of information can only be obtained from detailed laboratory analysis, interpretation of immiscible flow pressure response is not possible downhole in real-time. Single-phase flow interpretation models can be approximately used to study miscible multiphase flow. During the pumping process, the fluid sampled at first is very likely the mud filtrate, and as pumping continues, this is continuously replaced by formation fluid. Thus, there is a continuous time-varying change of viscosity during the pumping process. Many modern formation testers are designed with sensors that measure viscosity. For example, if FT-PTA-DDBU is used to predict mobility, and a viscosity value is available from separate measurement, then the spherical permeability can be calculated. The time-varying value of viscosity itself can be used to infer the value of formation fluid viscosity using different mixing laws available in the literature.

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Pressure data collected by the formation tester is not perfect and subject to scatter. Also, as noted previously, it is not possible to transmit the complete dataset to the surface using presently slow MWD mud pulse telemetry methods. Thus, downhole curve-fitting is used. In this approach, a math model of the pressure transient behavior is assumed with adjustable parameters A and B related to the pore pressure and mobility. This limited but smoothed dataset is usually transmitted to the surface. For FTWD model FT-PTADDBU, only three pressure-time datasets are needed to predict the analogous A and B. One strategy to determine the A and B valid for all the measured data is to repeat the FT-PTA-DDBU procedure, which is extremely fast, for different combinations of test points, say, t1, t2, t3 and t1, t5, t8 and t2, t6, t11 and so on. Then, the average values of A and B can be calculated, assuming that “bad” low and high values have been discarded. The A, B and P0 so obtained can be transmitted to the surface using MWD methods and the pressure transient curve can be replicated at the surface for customer reports. This type of averaging ensures data integrity. A special data integrity check is also possible prior to sending A and B values to the surface. Suppose that a representative A, B and P0 have been obtained from curve-fitting. Then, for each measurement time t1, t2, t3 . . . t10, t11, tnmax, we can use the host equation for pressure, e.g., an exponential model or one based on rational polynomials, to calculate the corresponding (italicized) predictions P1,2,3, …, nmax. If we denote measured pressure values by the symbol P1,2,3, … , nmax without italics, then we can obviously construct the positive quantity (Pi – Pi)2/Pi2 where the summation is taken over n = 1 . . . nmax data points. This quantity is both positive and dimensionless. If the interpreted curve fits all of the data points, the summation vanishes identically to become zero, indicating a perfect fit. Obviously, there is some arbitrariness to the definition. One can might also use (Pi – Pi)2/Pi2 and this appropriate. If the summation is close to zero, this indicates a good data fit, and also, that the measured data satisfy a physically useful model. This use of the summation is known as “chi-square” fitting and is employed in various interpretation schemes. If this error measure is computed periodically, tests can be terminated when a good fit is indicated.

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Real-time tool software should also monitor for pressures that are not changing with time, e.g., when steady-state is achieved rapidly in high mobility problems. Additional testing is not needed and will increase costs. The sum (Pi – Pi)2/Pi2 can be computed periodically during the testing. For instance, it is not necessary to wait until ten pressure measurements are obtained before its computation. If five measurements lead to a value that is close to zero, the (tight zone) test can be terminated economically. It is not necessary for measurements to have reached steady-state be they are terminated. Extrapolations for pore pressure and spherical mobility can be accurately obtained. When testing in heavy oils, high pump flow rates are not suggested for two reasons. First, because mobilities are low, the drawdown pressure drops can be very large and gas can be released from the petroleum liquid. Second, the high viscosity of the liquid may entrain loose sand in the formation, and plug the formation testing tool. Immiscible multiphase effects should be avoided. When performing a drawdown in liquids, do not pump below the “bubble point,” since gas may be released. Similarly, when pumping gas, “dew point” considerations are important so that liquids do not precipitate out and introduce permeability changes associated with two-phase flow near the well. For single-probe FTWD applications, FT-PTA-DDBU will only predict the mobility associated with the spherical permeability ks. If detailed information related to kh and kv are required, one technique requires the use of dual-probe measurements. These methods are described in the paper by Chin and Zhuang, “Formation Tester Inverse Permeability Interpretation for Liquids in Anisotropic Media with Flowline Storage and Skin at Arbitrary Dip,” 48th Annual SPWLA Meeting, Austin, TX, June 3-6, 2007. Alternatively, a more detailed single-probe math model can be used, this being FT-03, also described in the SPWLA paper. In this approach, we recognize that the piston source is not really an idealized point source, but is really a nozzle mounted on one side of the drill collar. Here, multiple pressure measurements are obtained by rotating the drill collar every

348 Formation Testing Volume 3

90 deg. The data are interpreted using FT-03. One service company uses this method, but notes that it is often difficult to control torsional twisting of the drillstring and to obtain good angular measurements. Also, the interpretation model is complicated and requires longer computing times. The compressibility of the formation fluid can be obtained from very early time pressure transient measurements. Because the data used is obtained early on when the tester pads may not have completely sealed filtrate flow, accuracy is not guaranteed for all physical quantities. While pore pressure and mobility predictions are demonstrated to be very good, it is common experience that the compressibility can be over-predicted. One must be careful in using calculated compressibilities to identify oil versus gas. Service companies are secretive about the methods used in pore pressure and mobility prediction. This is so even in published patents, which disclose few details about exact procedures used. One company appears to use a method that does not consider flowline storage and is therefore restricted to large times or high mobilities. Another uses what we term “material balance” analysis, which this author believes to be approximate at best. Mobilities predicted using the exponential model and also our rational polynomial methodology apply to operationally important situations where we have early time data or low mobility data, where flowline storage is important and forms part of the model. When the data is late-time or mobilities are high, this model should not be used. Instead, the user should apply spherical Horner plots for probes in thick layers, with suggestions to use standard “cylindrical flow” or “radial flow” Horner plots for thinner layer applications. Horner plot analysis is no longer patent-protected and numerous papers discussing the technology are available. Pre-tests using small pump rates are intended to test for repeatability, e.g., ensuring that the formation tester pad is sealed properly against the formation. Thus, two pre-tests with identical pumping conditions may not yield identical pressure results; this is particularly true since the first test will be more likely influenced by events near the well. For this reason, it is preferable to perform interpretations on later pre-tests only.

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Pre-tests are also used, together with pressure transient interpretation, to provide an idea about pore pressure and mobility in preparation for a large flowrate test designed to collect greater volumes of fluid. The predictions from pre-tests are used to estimate the increased pressure drops from the large flowrate test and to possibly avoid mechanical problems and gas liberation from the formation fluid. We have demonstrated an interpretation method for pretest sequences using FT-PTA-DDBU in the prior section. We have noted that the nozzle radius Rw is an “effective radius” and not the true radius. What does this mean? We recall that the simpler approximate model used for downhole processing assumes spherical symmetry for isotropic media and ellipsoidal flow for transversely isotropic formations. The actual source is modeled as a spherical or ellipsoidal well. In reality, there are borehole effects such as borehole radius, the geometry of the formation tester pad, and so on, that influence the measurements. These effects are understood by all the oil service companies. They construct detailed three-dimensional finite difference or finite element models to account for these effects and then choose an “effective radius” different from the actual radius to model these three-dimensional effects. Generally, the effective radius may be 20-50% greater than the actual radius. Its value should be determined for each tool that is designed. Preferably, the determination should also be performed experimentally in an oil well setting. A “geometric factor” (not to be confused with that used in induction logging) also serves this purpose. Single-probe FTWD tools can offer only the spherical permeability ks at best. Determination of kh and kv is often required. This requires dual-probe tools. At the present time, all the service companies assume vertical wells in their software models. However, the inverse models describe in Chin and Zhuang’s paper “Formation Tester Inverse Permeability Interpretation for Liquids in Anisotropic Media with Flowline Storage and Skin at Arbitrary Dip,” 48th Annual SPWLA Meeting, Austin, TX, June 3-6, 2007 apply to arbitrary dip angles and are also based on exact analytical solutions. These methods are also summarized in this book. At high dip angles and in horizontal wells, dual-probe tools can be conveyed by coiled tubing or pipe. In addition, this paper explains how short-time “phase delay” and “pulse” methods can be used to obtain horizontal

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and vertical permeabilities in low-mobility applications. These methods have been tested by Halliburton and Shell, and possibly, other companies. They are explained in Chin et al. (2014) and Chin et al. (2015). Gas pumping has attracted strong interest with the importance of unconventional and shale gas resources worldwide. As noted in Chapter 8, when the fluid pumped is entirely gaseous, simulators must be numerically intensive (and not analytically based) and simple models cannot be developed for downhole use. Fortunately, this does not mean that pore pressure and mobility prediction are not possible for gas pumping. In fact, for gas applications, pore pressure and mobility prediction are possible and can be performed accurately. This is true because, in gas reservoirs, the well is initially drilled using drilling mud, a liquid that has invaded into the formation adjacent to the well sandface. When a pretest, or a test with a smaller flow rate, is performed, the fluid that is first withdrawn is a liquid, whose dynamics can be modeled using FT-00 and FT-PTA-DDBU. Thus, gas reservoirs can be characterized using the (liquid) software. For steady pressures obtained using dual probe tools, however, both horizontal and vertical permeabilities can be obtained analytically at any dip angle.

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5.2 Recommended Reading An Internet web search on “formation testing, best practices” will yield a number of useful references. The author has found the following very useful. Cribbs, B., “Practical Wellbore Formation Test Interpretation,” Search and Discovery Article No. 12009 (2009), adapted from a presentation at AAPG Geoscience Technology Workshop, “Geological Aspects of Estimating Resources and Reserves,” Houston, Texas, September 9-11, 2009. Hashem, M., Elshahawi, H. and Ugueto, G., “A Decade of Formation Testing – Do’s and Don’t’s and Tricks of the Trade,” SPWLA 45th Annual Logging Symposium, Noordwijk, The Netherlands, June 6-9, 2004. Lal, M.K., Tran, V.H and Drennan, L.E., “Challenges and Values of Formation Testing in Tight Sand in Monterey Formation Using Modular Dynamic Tester (MDT),” Search and Discovery Article No. 80463 (2015), adapted from a presentation given at Pacific Section AAPG, SEG and SEPM Joint Technical Conference, Oxnard, California, May 3-5, 2015. Weinheber, P.J., Boratko, E.C., Contreiras, K.D., Van-Dunem, F., Spaeth, R.L., Dussan, E.B., Rueda, M.A. and Gisolf, A., “Best Practices for Formation Testing in Low-Permeability Reservoirs,” SPE-115825-MS, SPE Annual Technical Conference and Exhibition, Society of Petroleum Engineers, Denver, Colorado, September 2124, 2008.

Cumulative References Abramowitz, M., and Stegun, I. A., Handbook of Mathematical Functions, Dover Publications, New York, 1970. Adeyemi, O.S. and Edwards, M.D., “Matrix Selection for Conventional and Nonconventional Testing and Sampling in Gas Reservoirs,” SPE Paper No. 96972, 2005 SPE Annual Technical Conference and Exhibition, Dallas, Texas, Oct. 9-12, 2005. Akkurt, R., Bowcock, M., Davies, J., Del Campo, C., Hill, B., Joshi, S., Kundu, D., Kumar, S., O’Keefe, M., Samir, M., Tarvin, J., Weinheber, P., Williams, S., Zeybek, M., “Focusing on Downhole Fluid Sampling and Analysis,” Oilfield Review, Winter 2006/2007. Al-Mohsin, A., Tariq, S. and Haq, S.A., "Application of Wireline Formation Tester (Openhole and Cased-Hole) Sampling Techniques for Estimation of Nonhydrocarbon Gas Content of Khuff Reservoir Fluids in the North Field, Qatar," IPTC Paper No. 10622, International Petroleum Technology Conference, Doha, Qatar, Nov. 21-23, 2005. Andrews, R.J., Beck, G., Castelijns, K., Chen, A., Cribbs, M.E., Fadnes, F.H., Irvine-Fortescue, J., Williams, S., Hashem, M., Jamaluddin, A., Kurkjian, A., Sass, B., Mullins, O.C., Rylander, E., and van Dusen, A., “Quantifying Contamination Using Color of Crude and Condensate,” Oilfield Review, Autumn 2001. 352

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Angeles, R., Torres-Verdin, C., Lee, H.J., Alpak, F.O. and Sheng, J., “Estimation of Permeability and Permeability Anisotropy from Straddle Packer Formation Tester Measurements Based on the Physics of Two-Phase Immiscible Flow and Invasion,” SPE Journal, Sept. 2007, pp. 339-354. Ayan, C., Hafez, H., Hurst, S., Kuchuk, F., O’Callaghan, A., Peffer, J., Pop, J., and Zeybek, M., “Characterizing Permeability with Formation Testers,” Oilfield Review, Autumn 2001. Aziz, K. and Settari, A., Petroleum Reservoir Simulation, Applied Science Publishers, London, 1979. Barriol, Y., Glaser, K.S., Pop, J. et al., “The Pressures of Drilling and Production,” Schlumberger Oilfield Review, Autumn 2005. Bassiouni, Z. and Yildiz, T., “Interpretation of Wireline Formation Tester (WFT) Data in Tight Gas Sands,” Journal of Canadian Petroleum Technology, June 1999, Vol. 38, No. 6, pp. 29-36. Blauch, M.E., McMechan, D.E., Venditto, J.J. and Tanaka, G.L., “Low Permeability Subterranean Formation Testing Methods and Apparatus,” United States Patent 5,263,360, awarded Nov. 23, 1993. Bowles, D., “Bridging the Gap Between Wireline Formation, Drill Stem Testing,” Offshore Magazine, June 1, 2004. Brigham, W.E., Peden, J.M., Ng, K.F., and O’Neill, N., “The Analysis of Spherical Flow with Wellbore Storage,” SPE Paper No. 9294, 55th Annual Fall Technical Conference and Exhibition of the Society of Petroleum Engineers of AIME, Dallas, Texas, Sept. 2124, 1980. Carnahan, B., Luther, H.A., and Wilkes, J.O., Applied Numerical Methods, John Wiley, New York, 1969. Carslaw, H.S., and Jaeger, J.C., Conduction of Heat in Solids, Oxford University Press, London, 1946. Chin, W.C., Modern Reservoir Flow and Well Transient Analysis, Gulf Publishing, Houston, 1993.

354

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Chin, W.C., Formation Invasion, with Applications to Measurement-While-Drilling, Time Lapse Analysis, and Formation Damage, Gulf Publishing, Houston, 1995. Chin, W.C., Computational Rheology for Pipeline and Annular Flow, Butterworth-Heinemann, Reed Elsevier, Boston, 2001. Chin, W.C., Quantitative Methods in Reservoir Engineering, First Edition, Elsevier Science, Amsterdam, 2002. Chin, W.C., Managed Pressure Drilling, Elsevier Science, Amsterdam, 2012. Chin, W.C., Quantitative Methods in Reservoir Engineering, Second Edition, Elsevier Science, Amsterdam, 2017. Chin,W.C., “FTWD-Processing-Algorithms-V56a,” Stratamagnetic Software Internal Report, July 2012. Chin, W.C., “Formation Tester Flow Analysis in Anisotropic Media With Flowline Storage and Skin at Arbitrary Dip,” Well Logging Technology Journal, Xi’an, China, Feb. 2013. Chin, W.C. and Proett, M.A., “Formation Evaluation Using Phase Shift Periodic Pressure Pulse Testing,” United States Patent No. 5,672,819 issued Sept. 30, 1997. Chin, W.C. and Proett, M.A., “Formation Tester Immiscible and Miscible Flow Modeling for Job Planning Applications,” SPWLA 46th Annual Logging Symposium, New Orleans, Louisiana, June 2629, 2005. Chin, W.C., Suresh, A., Holbrook, P., Affleck, L. and Robertson, H., “Formation Evaluation Using Repeated MWD Logging Measurements,” SPWLA 27th Annual Logging Symposium, Houston, Texas, June 9-13, 1986. Chin, W.C., Zhou, Y., Feng, Y., Yu, Q. and Zhao, L., Formation Testing: Pressure Transient and Contamination Analysis, John Wiley & Sons, Hoboken, New Jersey, 2014. Chin, W.C., Zhou, Y., Feng, Y. and Yu, Q., Formation Testing: Low Mobility Pressure Transient Analysis, John Wiley & Sons, Hoboken, New Jersey, 2015.

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Chin, W.C. and Zhuang, X., “Formation Tester Inverse Permeability Interpretation for Liquids in Anisotropic Media with Flowline Storage and Skin at Arbitrary Dip,” SPWLA 48th Annual Logging Symposium, Austin, Texas, June 3-6, 2007. Chow, W.K., Ho, C.M., and Fong, N.K., “Evaluation of the Finite Control Volume Method in Simulating Thermal Fire Resistance of Building Elements,” Building Simulation, Fifth International IBPSA Conference, Sept. 8-10, 1997, Prague, Czech Republic, Vol. 1, p. 273. Cole, J.D., Perturbation Methods in Applied Mathematics, Blaisdell Publishing, Waltham, Massachusetts, 1968. Cribbs, B., “Practical Wellbore Formation Test Interpretation,” Search and Discovery Article #12009 (2009), adapted from a presentation at AAPG Geoscience Technology Workshop, “Geological Aspects of Estimating Resources and Reserves,” Houston, Texas, September 9-11, 2009. Crockett, R.K. and Cooper, R.E., “Formation Stimulation Using Modern Generation Test Tools Acidize-Test Technique,” SPE Paper 5766, Society Engineers of AIME, American Institute of Mining, and Petroleum Engineers, 1976.

Testing and - The Testof Petroleum Metallurgical

Crombie, A., Halford, F., Hashem, M., McNeil, R., Thomas, E.C., Melbourne, G. and Mullins, O.C., “Innovations in Wireline Fluid Sampling,” Oilfield Review, Autumn 1998. Doll, Henri-Georges, “Methods and Apparatus for Determining Hydraulic Characteristics of Formations Traversed by a Borehole,” U. S. Patent No. 2,747,401, issued May 29, 1956. Dussan, E.B., Auzerais, F.M. and Kenyon, W.E., “Apparatus for Determining Horizontal and/or Vertical Permeability of an Earth Formation,” United States Patent No. 5,279,153 awarded Jan. 18, 1994. Elshahawi, H. and Hashem, M.N., “In-Situ Fluid Compatibility Testing Using a Wireline Formation Tester,” United States Patent Application Publication US 2010/0242586 A1, Sept. 30, 2010.

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Index Complex complementary error function, 6, 73, 142, 286, 294-295, 301, 303-304, 306 Complex variables, 6 Compressibility, 2-4, 13-16, 18, 23, 31, 36, 70, 73-75, 78-79, 82-93, 95, 97, 100-103, 105-111, 113-119, 121-122, 124-127, 132-133, 141, 144-148, 151, 156-157, 162-164, 167, 172-173, 176-178, 182-183, 186-188, 191-193, 196-198, 201202, 206, 222, 229-232, 241, 244, 250, 286, 290, 294-295, 301-304, 314, 317, 319-322, 326, 334, 340, 342, 348 Concentration, 28, 72, 253, 258, 260, 263, 272, 334 Constant rate pumping, 313 Contamination, 28, 30, 131, 219, 221, 334, 342 Cylindrical flow, 32, 348

A Acidizing, 132-138. 219 Adiabatic, 9, 11 Anisotropy, 2, 5, 11, 30, 233, 319, 321

B Barriers, 31 Batch, 5-6, 291, 293, 306, 308-311, 313-317, 319, 326, 328, 331-333 Best practices, 231, 343, 345, 347, 349, 351 Bubble point, 334, 347 Buildup, 2-4, 13-16, 18-20, 23-24, 27, 69, 75-78, 107, 111, 114, 118, 120, 122, 132-135, 137, 139, 141143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187, 189, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 226, 230232, 240, 247, 260, 284-285, 287290, 340, 344-345

D Darcy flow, 69 Depth of investigation, 6, 225, 296298, 344 Dew point, 347 Diffusion, 6 Dimensionless variables, 128 Dip, 2, 8-9, 127, 232-233, 235-239, 286, 294-295, 301, 303-304, 320321, 323, 335, 337, 341, 347, 349350

C Cake, 72, 345 Capillary pressure, 345 Chemical, 133-136, 141, 220 Clean-up, 136-137 Color, 24, 28, 253, 270-272, 313, 326 Completion, 136-137, 220, 297 363

364 Index Dip angle, 2, 8, 127, 233, 235-239, 286, 294-295, 301, 303-304, 320321, 323, 335, 337, 341, 350 Do-loop, 292, 314, 319, 326 DOI, 6-7, 296, 298 Double-drawdown, 23 Drawdown, 2-4, 13-20, 23, 69, 7579, 81, 83, 85-87, 89, 91-93, 95, 97, 99-102, 105-108, 111, 114-115, 118-120, 122, 132-137, 139, 141143, 145-147, 149-151, 153, 155, 157-159, 161, 163, 165, 167, 169, 171, 173-175, 177, 179, 181, 183185, 187, 189, 191, 193-195, 197, 199, 201, 203-205, 207-209, 211213, 215-217, 219, 221, 223, 226, 230-233, 240-241, 244, 247, 260, 284-285, 287-290, 302-303, 305, 340, 344-345, 347 Drawdown-buildup, 2, 4, 13-16, 18-19, 23, 69, 75-77, 107, 111, 114, 118, 120, 122, 133, 143, 226, 230, 240, 247, 284-285, 288-290 Drilling mud, 35, 350 Drillstem, 134, 138 Drillstem testing, 138 Dual-probe, 347, 349

E Effective radius, 79, 82-83, 85, 8789, 91-93, 95, 97, 100-103, 105106, 108-111, 113-115, 117-119, 121-122, 125-126, 241, 244, 319, 322, 349 Electromagnetic logging, 3, 7 Ellipsoidal flow, 32, 286, 294-295, 301, 303-304, 349 Ellipsoidal source, 314, 320

Equilibration, 34 Error function, 6, 73, 142, 286, 294-295, 301, 303-304, 306 Experiment, 308, 317 Experimental results, 316 Explicit, 290 Exponential model, 6, 346, 348

F Filtercake, 35 Filtrate, 230, 345, 348 Finite difference, 8, 11, 306-307, 349 Finite element, 8, 69, 306, 349 Flowline storage, 1, 3, 11, 13-14, 23, 34, 69-70, 72, 79, 81, 83, 8587, 89, 91, 93, 95, 97, 99-102, 105106, 133, 145, 148, 156, 163, 172, 177, 182, 187, 192, 197, 202, 250, 286, 294-295, 301, 303-304, 347349 Fortran, 78, 308, 313-314, 320, 323, 328 Forward model, 74, 78, 84, 86, 88, 145, 148, 156, 163, 172, 177, 182, 187, 192, 197, 202 FTWD, 69, 346-347, 349

G Gas exponent, 335, 337 Gas flow, 320, 338, 342 Gas pumping, 334, 344, 350 Geometric factor, 69, 79, 81-85, 87-89, 91-93, 95, 97, 99-103, 105115, 117-119, 121-122, 125-126,

Index 365 241, 244, 286, 294-295, 302-304, 314, 320-321, 349

H Horizontal permeability, 319, 321 Horizontal well, 237 Horner, 348 Hydrate, 3, 132-134, 138-141, 219220 Hydrates, 139, 141

I Immiscible flow, 345 Injection, 77, 132, 134-137, 219, 236, 239, 314, 320, 322, 341 Invasion, 24, 27, 33, 72-73, 131, 252, 254-255, 260, 270, 272-273, 342, 345 Inverse, 2-4, 8-9, 12-14, 16, 18, 2021, 23, 29, 31, 33-35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73-75, 7783, 85-87, 89-93, 95, 97-103, 105113, 115-121, 123-127, 129, 131134, 142-148, 151, 155-156, 159, 162-163, 167, 171-172, 176-178, 181-183, 186-187, 191-192, 196197, 201-202, 206, 210, 214, 218, 221, 225, 229-231, 234-236, 238240, 247-248, 250, 288-290, 301, 306, 335-339, 341, 345, 347, 349 Inverse model, 9, 13-14, 16, 23, 7475, 77, 81, 83, 86-87, 90-91, 99101, 105-107, 109-110, 112-113, 117, 121, 125-126, 132, 144-148, 151, 155-156, 159, 162-163, 167, 171-172, 176-178, 181-183, 186-

187, 191-192, 196-197, 201-202, 206, 210, 214, 218 Inverse problem, 134 Isothermal, 9, 11, 336 Isotropic, 2, 31, 34, 69-70, 72, 79, 81, 83, 85-87, 89, 91, 93, 95, 97, 99-102, 105-106, 127-128, 222, 232, 240, 286, 301, 303-304, 322, 349

J Job planning, 6, 133, 135, 296, 309, 339

L Laplace transform, 74 Linear liquid, 320 Liquid, 4-5, 7-11, 24, 35, 79, 8485, 88-89, 93, 95, 97, 102-103, 107-108, 111, 114-115, 118-119, 122, 134, 236, 239, 241, 244, 307, 319-320, 334, 340-342, 345, 347, 350 Liquids, 5, 9, 13, 20, 69, 127, 133, 135-136, 140, 144, 235, 306-307, 314, 317-318, 320, 334, 339, 341, 344, 347, 349 Low mobility, 6, 13-14, 23, 30, 34, 69, 73, 79, 81, 83, 85-87, 89, 91, 93, 95, 97, 99-102, 105-106, 131, 133, 145, 148, 156, 163, 172, 177, 182, 187, 192, 197, 202, 219, 225, 250, 288, 342, 348

M Measurement While Drilling, 131, 342 Microemulsions, 141, 220

366 Index Miscible flow, 27, 33, 271 Mobility, 1-4, 6, 8, 11, 13-16, 18, 23, 30-31, 34, 69-70, 73-75, 78-79, 81-93, 95, 97, 99-102, 105-107, 109-110, 113-114, 116-118, 121122, 124-126, 131-133, 135-136, 141, 144-148, 151, 155-157, 159, 162-164, 167, 171-173, 176-178, 181-183, 186-188, 191-193, 196197, 201-202, 206, 210, 214, 218219, 225-226, 229-232, 234, 236, 247-250, 288, 290, 296, 342-350 Movie, 28, 253, 270-271 Mud filtrate, 345 Mudcake, 24, 27, 31, 33-34, 71-73, 135, 137, 251-252 Mudcake buildup, 24, 27 Multiphase flow, 334, 345 Multiple drawdown, 3-4, 13, 20, 23, 132-135, 137, 139, 141-143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187, 189, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 247, 290 Multiple drawdown and buildup, 34, 13, 20, 23, 132-135, 137, 139, 141-143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187, 189, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221 Multirate pumping, 21 MWD, 343, 346

N Newton-Raphson, 336 Nonlinear gas, 9, 11, 133, 320, 338, 342

O Observation probe, 2, 6, 11, 69, 133, 233-234, 236, 238-239, 284, 286, 296-301, 304, 308, 314-317, 323, 327-331, 335-337, 339, 341 Observation probes, 2, 6, 341 Overbalance, 3, 13-18, 23-24, 3435, 71-73, 78-93, 95-122, 124-126, 240-244, 246-249, 260, 270-278, 344 Overbalanced, 27, 31-34, 36, 6971, 78, 125, 266, 271

P Packer, 136-137, 345 Partial differential equation, 34, 76, 127, 129, 338 Permeability, 1-2, 4-5, 9, 11, 24, 30-31, 36, 70-72, 78-79, 84-85, 8889, 93, 95, 97, 102-103, 107-108, 111, 114-115, 118-119, 122, 132133, 136-137, 167, 219, 222, 230, 236, 241, 244, 286, 294-295, 301, 303-304, 308, 313, 318-319, 321, 335, 337, 341-342, 345, 347, 349, 351 Phase delay, 2, 221, 234, 349 Piecewise constant, 10, 20, 132, 142, 306-307 Playback, 28, 253, 270-271 Point source, 347

Index 367 Pore pressure, 2-4, 13-18, 23, 3132, 34, 36, 70-71, 73-75, 78-79, 8293, 95, 97, 100-103, 105-111, 113119, 121-122, 124-127, 132, 144148, 151, 155-157, 159, 162-164, 167, 171-173, 176-178, 181-183, 186-188, 191-193, 196-197, 201202, 206, 210, 214, 218, 234, 236, 239, 241, 244, 247-250, 260, 266, 272, 286, 288, 290, 294-295, 301, 303-304, 319, 321, 323, 335, 337, 341, 343-344, 346-350 Pressure profile, 298 Pressure transient, 1, 14, 19-20, 3031, 36, 69, 71, 73-74, 78-79, 81-82, 84-85, 89, 92-93, 95, 97, 99, 102, 104, 107-108, 111-112, 114-116, 118, 120, 122, 124, 131-133, 135, 145, 148, 156, 163, 172, 177, 182, 187, 191-192, 197, 202, 219, 221, 229, 240, 244, 248, 250, 286, 288, 294-295, 297, 301, 303-304, 306, 308, 334, 342-346, 348-349 Pumpout, 27, 72, 76, 127, 130, 146, 150, 153, 158, 161, 165, 169, 174, 179, 184, 189, 194, 199, 204, 208, 212, 216, 260, 284 PVT, 334

R Rational polynomial, 14, 70, 73, 142, 230, 348 Relative permeability, 345 Reservoir characterization, 1, 3, 133, 221

S Saturation, 72, 271-272 Sealing, 34-35, 137-138, 345 Single-phase flow, 345 Skin, 31, 69, 127, 129-130, 137, 236, 239, 286, 294-295, 301, 303304, 314, 316, 318, 320-321, 323326, 335, 337, 341, 347, 349 Skin damage, 31 Skin effect, 130, 314 Smearing, 6 Software, 3, 6, 15-16, 18-19, 24, 29, 75, 78-79, 81-82, 84, 86-88, 9193, 95, 97, 99-102, 105-107, 109114, 117-118, 121-122, 126, 144, 146, 149, 157, 164, 173, 178, 183, 188, 193, 198, 203, 250, 271, 286, 291, 294-295, 297, 301, 303-304, 309-310, 314, 333, 338, 341, 347, 349-350 Software reference, 79, 81-82, 84, 86-88, 91-93, 95, 97, 99-102, 105107, 109-114, 117-118, 121-122, 126 Source code, 308, 313-314, 323, 325, 328, 331-333 Source probe, 6, 16, 34, 71, 97, 127, 146, 150, 153, 158, 161, 165, 169, 174, 179, 184, 189, 194, 199, 204, 208, 212, 216, 223, 227, 233, 236, 238-239, 243, 246, 258-260, 262-263, 265, 267-269, 272, 279, 288, 297-299, 326, 335-337, 339341 Spherical flow, 31, 33-34, 69-71

368 Index Spherical permeability, 1-2, 11, 133, 286, 294-295, 301, 303-304, 319, 321, 345, 347, 349 SPWLA, 35, 131, 347, 349, 351 Steady-state, 1, 4, 8-9, 21, 135, 145, 148, 156, 163, 172, 177, 182, 187, 192, 197, 202, 222-223, 225226, 229-230, 232, 234-235, 335338, 347 Storage, 1, 3, 11, 13-14, 23, 34, 6970, 72, 79, 81, 83, 85-87, 89, 91, 93, 95, 97, 99-102, 105-106, 133, 145, 148, 156, 163, 172, 177, 182, 187, 192, 197, 202, 248, 250, 286, 294-295, 301, 303-304, 347-349 Straddle packer, 137 Supercharging, 24, 31, 33, 35-37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69-73, 7579, 81-83, 85-87, 89-91, 93-95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131 Superposition, 9-11, 20, 77, 307, 334

T Thermal, 136, 220 Thermodynamic, 9, 11, 139, 307, 317, 319-320, 334 Three-dimensional, 69, 349 Tight zone, 347 Time integration, 319, 325 Total flow rate, 129 Transient, 1-2, 4, 6, 8-9, 11, 14, 16, 19-20, 22-24, 30-31, 34, 36, 69-71, 73-74, 78-79, 81-82, 84-85, 89, 9293, 95, 97, 99, 102-104, 107-108,

111-112, 114-116, 118-122, 124, 131-135, 141, 145-146, 148, 156, 163, 172, 177, 182, 187, 191-192, 197, 202, 219, 221, 226, 229-230, 232, 240-241, 244, 247-248, 250, 286, 288, 294-295, 297, 301, 303304, 306, 308, 314, 320, 334-335, 337-338, 342-346, 348-349 Transversely isotropic, 31, 127128, 349 Two-phase flow, 72, 347

U Underbalance, 24, 268, 270, 278283 Underbalanced, 24, 31-34, 36, 6971, 268, 271 User interface, 18-19, 247, 309

V Viscosity, 13, 28, 31, 70, 78-79, 84-85, 88-89, 93, 95, 97, 102-103, 107-108, 111, 114-115, 118-119, 122, 127, 132-134, 141, 230, 236, 239, 241, 244, 259, 263-264, 266, 268, 286, 294-295, 301, 303-304, 313-314, 319, 321, 326, 334-335, 337, 341, 345, 347

W Windows, 6, 16, 19, 24, 28, 247, 314, 316, 323, 331 Wireline, 23, 30, 35-36, 69, 131, 137, 219 Withdrawal, 14, 77, 120, 132-134, 204, 236, 239, 244, 271, 273, 313, 320, 322, 341

About the Author Wilson C. Chin earned his Ph.D. from the Massachusetts Institute of Technology and his M.Sc. from the California Institute of Technology. He has authored more than twenty books with John Wiley & Sons and Elsevier Scientific Publishing. Also, Mr. Chin is recipient of five prestigious research awards with the United States Department of Energy and holds about four dozen patents related to inventions in the energy geosciences, sensor design and advanced signal processing. Mr. Chin’s professional interests include fluid mechanics, electromagnetics, wave propagation and computational methods. Applications areas cover reservoir engineering, formation testing, Measurement-While-Drilling design, signal processing, software engineering, and drilling and cementing technology. Before forming Stratamagnetic Software, LLC in January 2000, he had been affiliated with Boeing as Research Aerodynamicist and Pratt & Whitney Aircraft as Turbomachinery Manager. His oilfield experience includes management roles with Schlumberger, British Petroleum and Halliburton. Since then, Mr. Chin has been awarded four prestigious Small Business Innovation Research Grants and a major RPSEA contract award from the United States Department of Energy. He has served on the Board of Directors of Harris County Education Foundation; as Senior Advisor for “Education for the Energy Industry,” spearheaded by Houston’s Aldine Independent School District and Rice University in Houston; and as Visiting Professor for leading petroleum engineering universities in China. Mr. Chin has developed several well regarded software systems for petroleum engineering and exploration interpretation used internationally by major operators. He may be contacted by email at [email protected] or by phone at (832) 483-6899 in the United States. Company updates are conveniently posted on the Internet at www.stratamagnetic.com.

369

370 About the Author

BOOK PUBLICATIONS Over twenty original oil and gas books, describing personal research in reservoir engineering, electromagnetics, formation evaluation, Measurement-While-Drilling, sensor design, managed pressure drilling, and drilling and cementing rheology, namely, Formation Testing: Supercharge, Pressure Contamination Models (John Wiley & Sons, 2019)

Testing

and

Modern Aerodynamic Methods for Direct and Inverse Applications (John Wiley, 2019) Measurement While Drilling Signal Analysis, Optimization and Design, 2nd Edition (John Wiley, 2018) Modern Borehole Analytics – Annular Flow, Hole Cleaning and Pressure Control (John Wiley, 2017) Quantitative Methods in Reservoir Engineering, 2nd Edition – with New Topics in Formation Testing and Multilateral Well Flow Analysis (Elsevier, 2017) Managed Pressure Drilling: Modeling, Strategy and Planning (Chinese Language, Elsevier, 2016)

Resistivity Modeling: Propagation, Laterolog and Micro-Pad Analysis (John Wiley, 2016) Reservoir Engineering in Modern Oilfields: Vertical, Deviated, Horizontal and Multilateral Well Systems (John Wiley, 2016) Formation Testing: Low Mobility Pressure Transient Analysis (with CNOOC, John Wiley, 2015) Wave Propagation in Drilling, Well Logging and Reservoir Applications (John Wiley, 2014) Measurement While Drilling Signal Analysis, Optimization and Design (with CNPC, John Wiley, 2014) Electromagnetic Well Logging: Models for Interpretation and Tool Design (John Wiley, 2014)

MWD/LWD

About the Author 371

Formation Testing Pressure Transient and Contamination Analysis (with CNOOC, John Wiley, 2014) Managed Pressure Drilling: Modeling, Strategy and Planning (Elsevier, 2012) Quantitative Methods in Reservoir Engineering (Elsevier, 2002) Computational Rheology for Pipeline and Annular Flow (Elsevier, 2001) Formation Invasion, with Applications to Measurement While Drilling, Time Lapse Analysis and Formation Damage(Gulf Publishing, 1995) Wave Propagation in Petroleum Engineering, with Applications to Drillstring Vibrations, Measurement-While-Drilling, Swab-Surge and Geophysics (Gulf Publishing, 1994) Modern Reservoir Flow and Well Transient Analysis (Gulf Publishing, 1993) Borehole Flow Modeling in Horizontal, Deviated and Vertical Wells (Gulf Publishing, 1992)

372 About the Author

UNITED STATES PATENTS U.S. Patent No. 9,850,754, “High Speed Telemetry Signal Processing,” with Y. Jiang, Dec. 26, 2017 U.S. Patent No. 7,243,537, “Methods for Measuring a Formation Supercharge Pressure,” with M. Proett, J. Beique, J. Hardin, J. Fogal, D. Welshans, and G. Gray, July 17, 2007 U.S. Patent No. 7,224,162, “System and Methods for Upscaling Petrophysical Data,” with M. Proett, J. Fogal, and P. Aadireddy, May 29, 2007 U.S. Patent No. 7,082,078, “Magneto-Rheological Fluid Controlled Mud Pulser,” with M. Fripp and N. Skinner, July 25, 2006 U.S. Patent No. 7,059,179, “Multi-Probe Pressure Transient Analysis for Determination of Horizontal Permeability, Anisotropy and Skin in an Earth Formation,” with M. Proett, June 13, 2006 U.S. Patent No. 6,327,538, “Method and Apparatus for Evaluating Stoneley Waves, and for Determining Formation and Parameters in Response Thereto,” Dec. 4, 2001 U.S. Patent No. 5,969,638, “Multiple Transducer MWD Surface Signal Processing,” Oct. 19, 1999 U.S. Patent No. 5,831,177, “Fluid Driven Siren Flowmeter,” with J. Anders, M. Proett, and M. Waid, Nov. 3, 1998 U.S. Patent No. 5,787,052, “Snap Action Rotary Pulser,” with W. Gardner, July 28, 1998 U.S. Patent No. 5,740,126, “Turbosiren Signal Generator for Measurement While Drilling Systems,” with T. Ritter, April 14, 1998 U.S. Patent No. 5,703,286, “Method of Formation Testing,” with M. Proett and C. Chen, Dec. 30, 1997 U.S. Patent No. 5,672,819, “Formation Evaluation Using Phase Shift Periodic Pressure Pulse Testing,” with M. Proett, Sept. 30, 1997 U.S. Patent No. 5,644,076, “Wireline Formation Tester Supercharge Correction Method,” with M. Proett and M. Waid, July 1, 1997 U.S. Patent No. 5,586,083, “Turbosiren Signal Generator for Measurement While Drilling Systems,” with T. Ritter, Dec. 17, 1996

About the Author 373

U.S. Patent No. 5,583,827, “Measurement-While-Drilling System and Method,” Dec. 10, 1996 U.S. Patent No. 5,535,177, “MWD Surface Signal Detector Having Enhanced Acoustic Detection Means,” with K. Hamlin, July 9, 1996 U. S. Patent No. 5,515,336, “ MWD Surface Signal Detector Having Bypass Loop Acoustic Detection Means,” with W. Gardner and M. Waid, May 7, 1996 U. S. Patent No. 5,459,697, “MWD Surface Signal Detector Having Enhanced Acoustic Detection Means,” with K. Hamlin, Oct. 17, 1995 U. S. Patent No. 4,785,300, “Pressure Pulse Generator,” with J. Trevino, Nov. 15, 1988

RECENTLY FILED PATENT APPLICATIONS (2015-18) Formation Evaluation Using Phase Shift Periodic Pressure Pulse Testing in Anisotropic Media, June 2015 Multiple Drawdown Pressure Transient Analysis for Low Mobility Formation Testing, June 2015 High Signal Strength Mud Siren for MWD Telemetry, with K. Iftikhar, Jan. 2016 High Speed Telemetry Signal Processing, with Y. Jiang, May 2016 (issued Dec. 2017) Array Formation Tester for Rapid Anisotropic Permeability Prediction in Low Mobility Reservoirs, May 2017 Self-Spinning, Power-Generating, High Signal Strength, Sandwich Siren for MWD Telemetry, with S. Saleh and S. Brazil, May 2017 Methods and Systems for Downhole Sensing and Communications in Gas Lift Wells, with V. Shah, S. Brazil and R. Lusted, May 2017 Turbosiren Flowmeter for Open Hole Reservoir Applications, June 2017

374 About the Author

INTERNATIONAL AND DOMESTIC PATENTS Update in progress.

CA2556937, 9/21/2010, Methods for measuring a formation supercharge pressure BRPI0508357, 7/24/2007, Método para determinar a pressão de supercarga em uma formação interceptada por um furo de sondagem US7243537, 7/17/2007, Methods supercharge pressure US7224162, 5/29/2007, System petrophysical data

for and

measuring methods

a

formation

for

upscaling

CA2156224, 10/17/2006, MWD surface signal detector having bypass loop acoustic detection means CA2156223, 8/1/2006, MWD surface signal detector having bypass loop acoustic detection means US7082078, 7/25/2006, Magneto-rheological fluid controlled mud pulser US7059179, 6/13/2006, Multi-probe pressure transient analysis for determination of horizontal permeability, anisotropy and skin in an earth formation WO2005084332, 9/15/2005, Methods for measuring a formation supercharge pressure WO2005036338, 4/21/2005, System and methods for upscaling petrophysical data WO2005017301, 2/24/2005, Electroactive fluid controlled mud pulser US06327538, 12/04/2001, Method and apparatus for evaluating Stoneley waves, and for determining formation parameters in response thereto EP00936477A3, 12/13/2000, Evaluating Stoneley waves and formation parameters

About the Author 375

EP1053488, 11/22/2000, Multiple transducer MWD surface signal processing NO20003826A, 9/26/2000, Behandling av signal fra multippel MWD-transducer p overflaten NO20003826A0, 7/26/2000, Behandling av signal fra multippel MWD-transducer p overflaten EP00747571B1, 2/02/2000, Downhole pressure pulse generator EP0950795, 10/20/1999, Tool for and method of geological formation evaluation testing US5969638, 10/19/1999, Multiple transducer MWD surface signal processing NO00990872A, 10/18/1999, Verktoey og geologisk formasjonsevaluering og testing

fremgangsmte

for

EP936477A2, 8/18/1999, Evaluating Stoneley waves and formation parameters NO00990615A, 8/18/1999, Evaluering av Stoneley-boelger og formasjonsparametre WO09938032, 7/29/1999, Multiple transducer MWD surface signal processing NO00990872A0, 2/24/1999, Verkt y og geologisk formasjonsevaluering og testing

fremgangsmte

for

NO00990615A0, 2/09/1999, Evaluering av Stoneley-boelger og formasjonsparametre US05831177, 11/03/1998, Fluid driven siren flowmeter US05787052, 7/28/1998, Snap action rotary pulser US05740126, 4/14/1998, Turbosiren measurement while drilling systems

signal

generator

US05703286, 12/30/1997, Method of formation testing EP0747571, 12/11/1996, Downhole pressure pulse generator

for

376 About the Author

US05672819, 9/30/1997, Formation evaluation using phase shift periodic pressure pulse testing EP00697499A3, 7/30/1997, Apparatus for detecting an acoustic signal in drilling mud US05644076, 7/01/1997, Wireline formation tester supercharge correction method US05586083,12/17/1996, Turbosiren measurement while drilling systems

signal

generator

for

EP00747571A2, 12/11/1996, Downhole pressure pulse generator US05583827, 12/10/1996, Measurement-while-drilling system and method US05535177, 7/09/1996, MWD surface signal detector having enhanced acoustic detection means US05515336, 5/07/1996, MWD surface signal detector having bypass loop acoustic detection means EP00697498, 2/21/1996, Apparatus for detecting pressure pulses in a drilling fluid supply EP00697499A2, 2/21/1996, Apparatus for detecting an acoustic signal in drilling mud EP00697498A2, 2/21/1996, Apparatus for detecting pressure pulses in a drilling fluid supply NO00953224A, 2/19/1996, Anordning til paavisning av trykkpulser i en ledning for tilfoersel av borevaeske NO00953223A, 2/19/1996, Overflate-signaldetektor for maaling I loepet av boringen med forsterket akustisk detektorinnretning CA02156224AA, 2/18/1996, MWD surface signal detector having bypass loop acoustic detection means CA02156223AA, 2/18/1996, MWD surface signal detector having enhanced acoustic detection means US05459697, 10/17/1995, MWD surface signal detector having enhanced acoustic detection means

About the Author 377

NO00953224A0, 8/16/1995, Anordning for aa detektere trykkpulser I en borefluid NO00953223A0, 8/16/1995,Overflate-signaldetektor for maaling I loepet av boringen med forsterket akustisk detektorinnretning US4785300, 11/15/1988, Pressure pulse generator CA1228909, 11/03/1987, Pressure pulse generator BRPI8405278, 8/27/1985, Gerador de pulsos de pressao EP0140788, 5/8/1985, Pressure pulse generator NO00844026A, 4/25/1985, Trykkpulsgenerator

378 About the Author

JOURNAL ARTICLES AND CONFERENCE PUBLICATIONS Single-authored unless noted otherwise. Copies available upon request.

PETROLEUM ENGINEERING AND ENERGY GEOSCIENCES Exact Three-Dimensional Electromagnetic Model: MWD/LWD Anisotropic Prediction for Rh and Rv, Journal of Sustainable Energy Engineering, 2015 High-Data-Rate Measurement-While-Drilling System for Very Deep Wells, with Y. Su, L. Sheng, L. Li, H. Bian and R. Shi, Well Logging Technology Journal, Xi’an, China, Dec. 2014 Strategies in High-Speed MWD Mud Pulse Telemetry, with Y. Su, L. Sheng, L. Li, H. Bian, R. Shi and X. Zhuang, Journal of Sustainable Energy Engineering, Dec. 2014 Formation Testing: New Methods for Rapid Mobility and Pore Pressure Prediction, with Y. Zhou, Z. Hao, Y. Feng and Q. Yu, Paper OTC-24890-MS, 2014 Offshore Technology Conference Asia (OTC Asia), Kuala Lumpur, Malaysia, Mar. 25-28, 2014 Formation Testing: New Methods for Rapid Mobility and Pore Pressure Prediction, with Y. Zhou, L. Zhao, Y. Feng and Q. Yu, Paper 17214, 7th International Petroleum Technology Conference (IPTC), Doha, Qatar, Jan. 19-22, 2014 Formation Tester Flow Analysis in Anisotropic Media With Flowline Storage and Skin at Arbitrary Dip, Well Logging Technology Journal, Xi’an, China, Feb. 2013 Advances in Swab-Surge Modeling for Managed Pressure Drilling, with X. Zhuang, Paper OTC-21115-PP, 2011 Offshore Technology Conference, Houston, TX, May 2-5, 2011 Effect of Rotation on Flowrate and Pressure Gradient in Eccentric Holes, with X. Zhuang, Paper AADE-11-NTCE-45, AADE 2011 National Technical Conference and Exhibition, Houston, TX, April 12-14, 2011 Advances in Swab-Surge Modeling for Managed Pressure Drilling, with X. Zhuang, Paper AADE-11-NTCE-46, AADE 2011 National Technical Conference and Exhibition, Houston, TX, April 12-14, 2011

About the Author 379

Transient, Multiphase, Three-Dimensional Pumping Models for Cementing and Drilling, with X. Zhuang, Paper AADE-11-NTCE72, AADE 2011 National Technical Conference and Exhibition, Houston, TX, April 12-14, 2011 Comprehensive Annular Flow Models for Drilling and Completions, with X. Zhuang, Paper AADE-11-NTCE-73, AADE 2011 National Technical Conference and Exhibition, Houston, TX, April 12-14, 2011 High-Data-Rate Measurement-While-Drilling System for Very Deep Wells, with Y. Su, L. Sheng, L. Li, H. Bian and R. Shi, Paper AADE-11-NTCE-74, AADE 2011 National Technical Conference and Exhibition, Houston, TX, April 12-14, 2011 Flow Simulation Methods for Managed Pressure Drilling and Cementing, Drilling and Completing Trouble Zones Conference, Galveston, TX, Oct. 19-21, 2010 Modeling and Simulation of Managed Pressure Drilling for Improved Design, Risk Assessment, Training and Operations, RPSEA Ultra-Deepwater Technology Conference, Houston, TX, June 22-23, 2010 Exact Non-Newtonian Flow Analysis of Yield Stress Fluids in Highly Eccentric Borehole Annuli with Pipe or Casing Translation and Rotation, with X. Zhuang, SPE Paper 131234-PP, CPS/SPE International Oil & Gas Conference and Exhibition, Beijing, China, June 8-10, 2010 Displacement of Viscoplastic Fluids in Eccentric Annuli: Numerical Simulation and Experimental Validation, with T. Deawwanich, J.C. Liew, Q.D. Nguyen, M. Savery and N. Tonmukayakul, Chemeca 2008 Conference, Newcastle, Australia, Sept. 28 - Oct. 1, 2008 Laminar Displacement of Viscoplastic Fluids in Eccentric Annuli – Numerical Simulation and Experimental Validations, with M. Savery, P. Tonmukayakul, T. Deawwanich, J. Liew and Q. Dzuy Nguyen, XXII International Congress of Theoretical and Applied Mechanics (ICTAM 2008), Adelaide, Australia, Aug. 24-29, 2008 Flow Visualization and Numerical Simulation of Viscoplastic Fluid Displacements in Eccentric Annuli, with Q.D. Nguyen, T.

380 About the Author

Deawwanich, N. Tonmukayakul and M.R. Savery, XVth International Congress on Rheology (ICR 2008), Society of Rheology 80th Annual Meeting, Monterey, CA, Aug. 3-8, 2008 Modeling Cement Placement Using a New Three-Dimensional Flow Simulator, with M. Savery, AADE 2008 Fluids Technology Conference, Houston, April 8-9, 2008 Formation Tester Inverse Permeability Interpretation for Liquids in Anisotropic Media with Flowline Storage and Skin at Arbitrary Dip, with X. Zhuang, 48th Annual SPWLA Meeting, Austin, TX, June 3-6, 2007 Modeling Fluid Interfaces During Cementing Using a ThreeDimensional Mud Displacement Simulator, with M. Savery and R. Darbe, OTC Paper 18513, 2007 Offshore Technology Conference (OTC), Houston, TX, April 30 – May 3, 2007 Formation Tester Immiscible and Miscible Flow Modeling for Job Planning Applications, with M. Proett, 46th Annual SPWLA Meeting, New Orleans, LA, June 26-29, 2005 MWD Siren Pulser Fluid Mechanics, Petrophysics, Journal of the Society of Petrophysicists and Well Log Analysts (SPWLA), Vol. 45, No. 4, July – August 2004, pp. 363-379 Formation Testing in the Dynamic Drilling Environment, with M. Proett, D. Seifert, S. Lysen, and P. Sands, SPWLA 45th Annual Logging Symposium, Noordwijk, The Netherlands, June 6-9, 2004 Job Planning Simulators for Immiscible and Miscible Flow, SPWLA Spring Topical Conference, Formation Testing: Applications and Practices, Taos, NM, Mar. 28 – Apr. 1, 2004 Mudcake Growth, Invasion, and Dynamic Coupling with Reservoir Flow: Experiment and Theory, SPWLA Spring Topical Conference, Formation Testing: Applications and Practices, Taos, NM, Mar. 28 – Apr. 1, 2004 Sample Quality Prediction with Integrated Oil and Water-based Mud Invasion Modeling, with M. Proett, D. Belanger, M. Manohar, and J. Wu, PetroMin Magazine, Sept. 2003 Improved Rheology Model and Hydraulics Analysis for Tomorrow’s Wellbore Fluid Applications, with R. Morgan, T. Becker, and J.

About the Author 381

Griffith, SPE Paper 82415, SPE Production and Operations Symposium, Oklahoma City, OK, Mar. 2003 Sample Quality Prediction with Integrated Oil and Water-based Mud Invasion Modeling, with M. Proett, D. Belanger, M. Manohar, and J. Wu, SPE Paper 77964, SPE Asia Pacific Oil & Gas Conference and Exhibition (APOGCE), Melbourne, Australia, Oct. 2002 Multiple Factors That Influence Wireline Formation Tester Pressure Measurements and Fluid Contacts Estimates, with M. Proett, M. Manohar, R. Sigal, and J. Wu, SPE Paper 71566, SPE Annual Technical Conference and Exhibition, New Orleans, LA, Oct. 2001 Comprehensive Look at Factors Influencing Wireline Formation Tester Pressure Measurements and Fluid Contacts, with M. Proett, M. Manohar, and R. Sigal, 42nd SPWLA Annual Symposium, Society of Professional Well Log Analysts, Houston, TX, June 2001 New Wireline Formation Testing Tool With Advanced Sampling Technology, with M. Proett, G. Gilbert, and M. Monroe, SPE Reservoir Evaluation and Engineering, April 2001 Advanced Permeability and Anisotropy Measurements While Testing and Sampling in Real-Time Using a Dual Probe Formation Tester, with M. Proett, SPE Paper 64650, Seventh International Oil & Gas Conference and Exhibition, Beijing, China, Nov. 2000 Modern Flow Assurance Methods, Part III: Coupled Velocity and Temperature Fields in Bundled Pipelines, Offshore, Nov. 2000 Modern Flow Assurance Methods, Part II: Detailed Physical Properties and Engineering Application, Offshore, Oct. 2000 Modern Flow Assurance Methods, Part I: Clogged Pipelines, Wax Deposition, and Hydrate Plugs, Offshore, Sept. 2000 Advanced Permeability and Anisotropy Measurements While Testing and Sampling in Real Time Using a Dual Probe Formation Tester, with M. Proett, SPE Paper 62919, 2000 SPE Annual Technical Conference and Exhibition, Dallas, TX, Oct. 2000 General Three-Dimensional Electromagnetic Model for Nondipolar Transmitters in Layered Anisotropic Media With Dip, Well Logging Technology Journal, Xi’an, China, Aug. 2000

382 About the Author

New Dual Probe Wireline Formation Testing and Sampling Tool Enables Real-Time Permeability and Anisotropy Measurements, with M. Proett, 41st SPWLA Annual Symposium, Society of Professional Well Log Analysts, Dallas, TX, June 2000 Clogged Pipe, Non-Newtonian Fluid, and Coupled Solids Deposition Flow Modeling, Final Technical Report, Brown & Root Energy Services, Houston, TX, May 2000 Irregular Grid Generation and Rapid 3D Color Display Algorithm, Final Technical Report, DOE Grant No. DE-FG03-99ER82895, United States Department of Energy, May 2000 New Dual-Probe Wireline Formation Testing and Sampling Tool Enables Real-Time Permeability and Anisotropy Measurements, with M. Proett, M. Manohar, G. Gilbert, and M. Monroe, SPE Paper 59701, SPE Permian Basin Oil & Gas Recovery Conference, Midland, TX, March 2000. Also presented at SPE Rocky Mountain Regional Low Permeability Reservoirs Symposium and Exhibition, Denver, CO, March 2000 New Wireline Formation Testing Tool with Advanced Sampling Technology, with M. Proett, G. Gilbert, and M. Monroe, SPE Paper 56711, 1999 SPE Annual Technical Conference and Exhibition, Houston, TX Exact Spherical Flow Solution with Storage for Early-Time Test Interpretation, with M. Proett, SPE Journal of Petroleum Technology, Nov. 1998 New Exact Spherical Flow Solution with Storage and Skin for EarlyTime Interpretation, with Applications to Wireline Formation and Early-Evaluation Drillstem Testing, with M. Proett, SPE Paper 49140, 1998 SPE Annual Technical Conference and Exhibition, New Orleans, LA, Sept. 1998 New Exact Spherical Flow Solution for Early-Time Well Test Interpretation with Applications to Wireline Formation Testing and Early-Evaluation Drillstem Testing, with M. Proett, SPE Paper 39915, SPE Rocky Mountain Regional Meeting/Low Permeability Reservoirs Symposium, April 1998 New Exact Spherical Flow Solution for Early-Time Well Test Interpretation with Applications to Wireline Formation Testing and

About the Author 383

Early-Evaluation Drillstem Testing, with M. Proett, SPE Paper 39768, SPE Permian Basin Oil and Gas Recovery Conference, Midland, TX, March 1998 Pressure Test Validity Shown by Comparing Field Tests and Simulations, Part II: Formation Pressure Test Tools, with N. Skinner, M. Proett, P. Ringgenberg, and R. Aadireddy, Oil & Gas Journal, Jan. 12, 1998 Testing System Combines Advantages of Wireline and Drillstem Testers, Part I: Formation Pressure Test Tools, with N. Skinner, M. Proett, P. Ringgenberg, and R. Aadireddy, Oil & Gas Journal, Jan. 5, 1998 New Early Formation Pressure System Field Test Results and Advances in Early Time Pressure Buildup Analysis, with N. Skinner, M. Proett, P. Ringgenberg, K. Manke, H. Smith, and R. Aadireddy, SPE Paper 38648, 1997 SPE Annual Technical Conference and Exhibition, San Antonio, TX, Oct. 1997 Permeability Prediction from Formation Tester “Phase Delay” and “Sonic Pulse” Analysis, with M. Proett, GRI – SPWLA Research Forum on Permeability Logging, Houston, TX, Feb. 1997 Permeability Prediction from Stoneley Waveform Data, GRI – SPWLA Research Forum on Permeability Logging, Houston, TX, Feb. 1997 Supercharge Pressure Compensation Using a New Wireline Testing Method and Newly Developed Early Time Spherical Flow Model, with M. Proett, SPE Paper 36524, 1996 Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Denver, CO, Oct. 1996 Supercharge Pressure Compensation with New Wireline Formation Testing Method, with M. Proett, 1996 Annual Meeting of the Society of Professional Well Log Analysts, New Orleans, LA, June 1996 Supercharge Pressure Compensation with New Wireline Formation Testing Method, with M. Proett, SPE Paper 35178, Permian Basin Oil and Gas Recovery Conference, Midland, TX, March 1996 Modeling Complex Horizontal Wells in Heterogeneous Formations, Offshore, Oct. 1993

384 About the Author

Eccentric Annular Flow Modeling for Highly Deviated Boreholes, Offshore, Aug. 1993 Model Offers Insight Into Spotting Fluid Performance, Offshore, Feb. 1991 Annular Flow Model Explains Conoco’s Borehole Cleaning Success, Offshore, Oct. 1990 Exact Cuttings Transport Correlations Developed for High Angle Wells, Offshore, May 1990 Advances in Annular Borehole Flow Modeling, Offshore, Feb. 1990 Comparative Studies in Dual Porosity Reservoir Simulation, BP Exploration Reservoir Technology Report No. H090.0005, Houston, TX, Jan. 1990 Automating the Acquisition Planning Process at the Johnson Space Center (using Artificial Intelligence Methods), with J. Golej, MITRE Report MTR-88D00065, NASA/JSC Contract No. NAS9-18057, Sept. 1988 Simulating Horizontal Well Fracture Flow, Offshore, Aug. 1988 Why Drill Strings Fail at the Neutral Point, Petroleum Engineer International, May 1988 Fatal Tubular Bending Motions Difficult to Detect Uphole, Offshore, April 1988 Formation Evaluation Using Repeated MWD Logging Measurements, with A. Suresh, P. Holbrook, L. Affleck, and H. Robertson, SPWLA 27th Annual Logging Symposium, Houston, TX, June 9-13, 1986

About the Author 385

HIGH SPEED AERODYNAMICS, JET ENGINE AND TURBOMACHINERY DESIGN Selected publications from MIT Aerophysics Laboratory, Boeing Airplane Company, Pratt & Whitney Aircraft (United Technologies) and NASA Johnson Space Center in ASME J. Applied Mechanics, AIAA Journal, J. Aircraft and J. Hydronautics (classified Naval Missile Center and United States Air Force technical reports and designs not included).

Inviscid Steady Flow Past Turbofan Mixer Nozzles, ASME J. Applied Mechanics, Dec. 1984 Comment on “PAN AIR Applications Integration,” J. Aircraft, Nov. 1984

to Aero-Propulsion

Thin Airfoil Theory for Planar Inviscid Shear Flow, ASME J. Applied Mechanics, Mar. 1984 Engine Power Simulation for Transonic Flow-Through Nacelles, AIAA Journal, Oct. 1983 An Axisymmetric Nacelle and Turboprop Inlet Analysis Including Power Simulation, with D. Golden and T. Barber, J. Aircraft, June 1983 Superpotential Solution for Jet Engine External Potential and Internal Rotational Flow Interaction, ASME J. Applied Mechanics, June 1983 An Axisymmetric Nacelle and Turboprop Inlet Analysis with FlowThrough and Power Simulation Capabilities, with D. Golden and T. Barber, AIAA Paper 82-0256, AIAA 20th Aerospace Sciences Meeting, Orlando, FL, Jan. 1982 Direct Approach to Aerodynamic Design Problems, ASME J. Applied Mechanics, Dec. 1981 Harmonic Analysis of Unsteady Transonic Flow, AIAA Journal, Feb. 1981 Kinematic Barrier for Gravity Waves on Variable Currents, J. Hydronautics, Jan. 1981 Optimal Coordinates for Squire’s Jet, AIAA Journal, Jan. 1981 Transonic Nacelle Inlet Analyses, with W. Presz, D. Ives, D. Paris and D. Golden, NASA Lewis Workshop on Application of Advanced Computational Methods, Nov. 1980

386 About the Author

Wave Focusing and Hydraulic Jump Formation, J. Hydronautics, July 1980 Inviscid Parallel Flow Stability with Mean Profile Distortion, J. Hydronautics, July 1980 Class of Shockfree Airfoils Producing the Same Surface Pressure, J. Aircraft, April 1980 Effect of Dissipation and Dispersion on Slowly Varying Wavetrains, AIAA Journal, Feb. 1980 Airfoil Design in Subcritical and Supercritical Flow, with D. Rizzetta, ASME J. Applied Mechanics, Dec. 1979 Kinematic Wave Approach to Hydraulic Jumps with Waves, J. Hydronautics, Oct. 1979 Effect of Frequency in Unsteady Transonic Flow, with D. Rizzetta, AIAA Journal, July 1979 Stability of Inviscid Shear Flow Over Flexible Membranes, AIAA Journal, June 1979 Some Exact Solutions to Guderley’s Equation, AIAA Journal, April 1979 On the Design of Thin Subsonic Airfoils, ASME J. Applied Mechanics, Mar. 1979 Algorithm for Inviscid Flow Using the Viscous Transonic Equation, AIAA Journal, Aug. 1978 Type-Independent Solutions for Mixed Compressible Flows, AIAA Journal, Aug. 1978 Similar Solutions for Unsteady Transonic Flow, AIAA Journal, June 1978 Nonlinear Formulation for Low-Frequency Transonic Flow, AIAA Journal, June 1978 Supersonic Wave Drag of Planar Singularity Distributions, AIAA Journal, May 1978 Some Singular Aspects of Three-Dimensional Transonic Flow, AIAA Journal, Mar. 1978

About the Author 387

Pseudo-Transonic Equation with a Diffusion Term, AIAA Journal, Jan. 1978 Goethert’s Rule with an Improved Boundary Condition, AIAA Journal, Oct. 1977 Numerical Solution for Viscous Transonic Flow, AIAA Journal, Sept. 1977 Supersonic Wave Drag for Nonplanar Singularity Distributions, AIAA Journal, June 1977

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