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SOME STUDIES OF MULTIPHASE RELATIVE PERMEABILITY IN CONSOLIDATED CALIFORNIA OIL SANDS, AS DETERMINED BY THE CAPILLARY PRESSURE DISPLACEMENT METHOD
A Thesis Presented to the Faculty of The Department of Petroleum Engineering The University of Southern California
In Partial Fulfillment of the Requirements for the Degree Master of Science in Petroleum Engineering
by A. McKenzie Laurie and Harrison L* Staub June 1950
UMI Number: EP63276
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T h is thesis, w r it t e n by
A. McKenzie Laurie and Harrison L* Staub their u n d e r the g u id a n c e o f h and approved
by a l l
F a c u l t y C o m m itte e , its m em bers,
p resented to a n d accepted by the C o u n c il on G r a d u a te S tu d y a n d R e s ea rch in p a r t i a l f u l f i l l m e n t o f the re q u ire m e n ts f o r the degree o f
Master of Science in Petroleum Engineering
The authors wish to express their appreciation to the Western Gulf Oil Company, and in particular to Mr. T. H. Wallace and Mr. E. A. Galvin, for the permission to publish the contained laboratory data on oil field cores. Appreciation is also extended to Mr. W. Tempelaar-Lietz and Mr. John Gates of the Shell Oil Company, whose suggestions aided in the development of the experimental methods em ployed in this study.
TABLE OF CONTENTS CHAPTER I.
PAGE THE PROBLEM, DEFINITIONS OF TERMS USED, AND STATEMENT OF ORGANIZATION ..................
Statement of the problem .................
Importance of the s t u d y ...............
Definitions of terms used ..................
Statement of organization of the thesis. . . II.
REVIEW OF THE L I T E R A T U R E .................
Literature on fundamental fluid flow concepts
Literature on homogeneous f l o w ...........
Literature on capillarity
Literature on relative permeability measure ment by the capillary pressure displacement method
Literature on relative permeability measurement by the solution gas displacement method . .
Literature on relative permeability measurement by the dynamic displacement method
Literature on analytical methods of relative permeability determination
PAGE Evaluation of the existing literature
. . .
Principles of the apparatus.................
Description of the apparatus
Preparation of the core
Preparation of the apparatus................
Technique of operation
PRESENTATION, DISCUSSION AND GENERAL INTER PRETATION OF THE DATA
Presentation and discussion of data
Sources of error
Data on natural c o r e s ....................
General interpretation of d a t a ............
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS . . .
Tables II through I X ............................
Recommendations BIBLIOGRAPHY APPENDIX A Symbols Subscripts APPENDIX B
LIST OF TABLES TABLE
I. Laboratory data on attainment of equilibrium . .
II. Laboratory data on effective permeability of alundum core, Sample F, Run 1 III.
Laboratory data on effective permeability of
alundum core, Sample F, Run 2 IV. Laboratory data on effective permeability of alundum core, Sample F, Run 3
V. Laboratory data on relative permeability, Sample VI.
C, Run 1
Laboratory data on relative permeability, Sample
Laboratory data on relative permeability, Sample C, Run 2
Laboratory data on relative permeability, Sample
Laboratory data on relative permeability, Sample
Laboratory data on relative permeability, Sample
LIST OF FIGURES FIGURE 1.
PAG-E Schematic diagram illustrating hysteresis e f f e c t .....................................
Relative permeability apparatus
Capillary butt a s s e m b l y ....................
Technique illustration diagram ............
Technique illustration diagram ..............
Technique illustration diagram ..............
Technique illustration diagram ..............
Technique illustration diagram ..............
Effectivepermeability curves, Sample F
Capillary pressure curve of Sample F ........
Capillary pressure curve, Sample A Relative
Sample A .
. . .
. . .
permeability curves, Sample B . . . .
Sample C .
. . .
Sample D .
. . .
Sample E .
. . .
Relative permeability ratio curves
CHAPTER I THE PROBLEM, DEFINITIONS OF TERMS USED, AND STATEMENT OF ORGANIZATION Petroleum products have become a requisite in the present industrial world; at the present stage of scienti fic development, the only practical source of these products is the large and widespread accumulations of naturally oc curring liquid hydrocarbons.
It is imperative, therefore,
to determine and employ methods of production which will assure maximum recovery.
One factor which is basic to
this determination is the further study of relative permea bility phenomena. I.
Statement of the problem.
It was the purpose of this
study (1) to obtain data on the relative permeabilities of samples of California oil sand, using the capillary pressure displacement method; (2) to present a description of the ex perimental methods employed in obtaining these data; and (3) to present an interpretation and evaluation of the re sults. Importance of the study.
In response to the ever-
increasing demand for petroleum products, there has been, during the past two decades, a widespread attempt to determine
2 more effective methods of obtaining naturally occurring hydrocarbons.
Because of the increasing difficulty and ex
pense involved in discovering and developing additional pet roleum reserves, there has been a tendency to emphasize ef ficiency in the production of those reserves already accessi ble. A better understanding of the physical and chemical forces operative in the recovery of petroleum from under ground reservoirs has been sought through the medium of laboratory research.
An important phase of this research
has been the study of fluid flow through porous media, and its quantitative description in terms of relative permea bility,
Although there has been a considerable amount of
research directed toward the laboratory measurement of rela tive permeability, all the related phenomena have not been explained or defined.
Furthermore, there is a distinct
scarcity of laboratory data available for application in the solution of oil field problems.
In this study an attempt
was made to further explain relative permeability phenomena and to add to the existing supply of applicable data, II.
DEFINITIONS OF TERMS USED
The present concept of permeability
3 has evolved from the classical work of D farcy
the flow of water through a porous, saturated sand medium was investigated.
He concluded that the rate of flow of a
homogeneous fluid through a porous medium was directly pro portional to the pressure gradient and the cross-sectional area normal to the direction of flow, and inversely propor tional to the viscosity of the fluid.
constant relating the rate of flow to the other variables has become known as the “permeabilityn of the medium. In later studies, where attempts were made to apply the findings of D*arcy to flow through porou6 media contain ing binary and ternary fluid systems, it was found that the permeability constant as defined by D'arcy was not applica ble.
Empirical evidence showed that another variable ex
isted, namely, the saturation relationship between the fluids. It became apparent, therefore, that there was a need for a reconsideration of the flow relationships in order to ade quately describe the flow in these more complex systems. The parameter “effective permeability11 i*as consequently de fined as the permeability of a porous medium to one fluid of a multiphase fluid system existing in the medium.
“effective permeability11 was found to be a function of the
Numbers refer to bibliography
4 saturation relationships in the medium. It was found convenient, for correlation purposes, to descx-lbe “effective permeability" in terms of "relative permeability."
For a given porous medium, and a given satur
ation condition, the "relative permeability" has been de fined as the "effective permeability" divided by the "permea bility. " Definitions: Absolute permeability.
The absolute permeability of
a porous medium was used to mean that property which describee the capacity of the medium to transmit a homogeneous, single phase, fluid which does not react with the solids contained in the porous medium.
It should be stressed that this prop
erty is defined by the porous medium alone and is unaffected by all other quantities. Apparent gas permeability.
Apparent gas permeability
was interpreted as that permeability which is evaluated by application of D'arcy’s law to gas flow systems.
parameter is not a unique function of the porous medium, but is also a function of the gas and various thermodynamic quantities which affect the mean free path of the gas mole cules.
(See discussion of the Klinkenberg effect, Chapter
II, page 12.) Boundary effect.
The "boundary effect" was termed
that effect resulting from a discontinuity in a porous sys tem, brought about by the existence of adjacent regions of different capillary properties.
In multi-phase flow sys
tems, it is commonly manifested in the tendency for the wet ting flowing phase to accumulate at the outflow face. Capillary pressure.
Capillary pressure as it applies
to porous media was defined as the pressure difference exist ent between any two interconnected phases in the porous medium. Connate water.
Connate water was termed that water
which is contained in the intersticies of a rock as it oc curs en situ.
This water is natural to the rock and bears
no relation to "interstitial water" that may have been placed in the rock by any method whatsoever. Core.
"Core" was used to describe the samples of
oil field sand upon which laboratory experiments were per formed.
Whenever the term was used in another capacity,
it was qualified. The darcy. bility.
The darcy is the standard unit of permea
A porous medium is said to have an absolute permea
bility of one darcy if one cubic centimeter per second of a fluid of one centipoise viscosity will flow across a cross
6 sectional area of one square centimeter (measured normal to the path of flow) under a pressure gradient of one atmosphere per centimeter and conditions of viscous flow* Displacing; pres sure»
For the purpose of this study,
displacing pressure was defined as the difference between the mean pressure difference between phases and the capillary pressure within a porous medium.
It is obvious that this
quantity is characteristic of non-equilibrium systems only, its value constantly diminishing until, at equilibrium, it becomes zero. Displacement methods:
(The three following terms 5 were used as defined by Brownscombe, Slobod and Oaudle. ) Capillary pressure displacement* The capillary pres sure displacement method denoted that laboratory method in which displacement of a fluid from a porous medium is accomplished by maintaining a pressure difference between phases through the use of capillary membranes. Dynamic displacement. The dynamic displacement method was used to mean that laboratory method whereby displacement is effected by forcing a non-wetting phase through a porous medium saturated with a wetting phase. Solution gas displacement. The solution gas dis placement method was used to mean that laboratory method whereby, upon the reduction of pressure, displacement of a wetting phase from a porous medium Is accomplished by the dissolution and expansion of a gaseous phase. Displacement pressure.
Displacement pressure was
defined as that pressure necessary to force a non-wetting phase into a porous medium 100 per cent saturated with a
7 wetting phase, in effect a minimum capillary pressure. Effective permeability.
The effective permeability
of a medium was used to designate that property which describes the capacity of the medium to transmit one fluid phase of a multi-phase fluid system existing in the medium.^ The ef fective permeability is a function, though not a unique one, of the saturation relationships existent in the medium. "Effective permeability11 in itself has no meaning but must be qualified by designation of the phs.se in question; for example, effective permeability to oil. Equilibrium.
Equilibrium was used synonomously with
"steady state conditions. " The latter term was defined as those conditions obtaining in a porous medium when the fluid velocities and pressures have become independent of time. Equilibrium gas saturation.
The equilibrium gas
saturation of a system was defined after Muskat
that saturation of gas such that an infinitesimal increase in saturation xfill establish permeability to gas and a like decrea.se will render the medium impermeable to the gas phase. Equilibrium permeability.
of a medium was defined to be the effective permeability to the wetting phase, in a two-phase system, or to the oil phase, in a three-phase system, at the equilibrium gas satura-
8 tion. Interstitial water.
Interstitial water was termed
that water which is contained in the inters tides of a por ous medium. Permeability.
Throughout the report of this investi
gation, the term "permeability11 was used as a quality rather than a property —
as a qualitative measure of the ease with
which a porous medium will transmit fluid. Porous media.
Porous media was used to embrace all
those materials composed of granular bodies, regular or irregular, consolidated or unconsolidated, separated by pores of capillary dimensions some of which are intercon nected and capable of fluid transmission under certain con ditions. Relative permeability. Relative permeability was defined as the ratio of effective permeability to absolute permeability.
Since effective permeability is a function
of the saturation relationships in a heterogeneous fluid system, it is evident that relative permeability will also be a function of saturation. Relative permeability ratio.
The relative permeability
ratio for a given saturation condition has been defined as a
measure of the capacity of a porous medium to transmit one fluid as related to its ability to transmit another.
numerically equal to the ratio of the effective permeabilities to the two phases in question, which is in turn equal to the ratio of the relative permeabilities when relative permeabil ity is defined as above. Saturation.
The saturation of a given fluid in a
porous medium, unless otherwise defined, was used to mean the fraction of the total pore space of the medium which was occupied by that fluid. III.
STATEMENT OF ORGANIZATION OF THE THESIS
The remainder of the thesis has been divided into four chapters.
Chapter II has been devoted to a review of
relative permeability literature, emphasizing that work which has been more directly concerned with the measurement of relative permeability by the capillary pressure displacement method.
Chapter III contains a description of the experi
mental methods employed in obtaining the data, while Chapter IV includes the presentation and interpretation of these data.
The concluding Chapter V consists of a summary, con
clusions and recommendations.
CHAPTER II REVIEW OF THE LITERATURE There has been a considerable amount of literature written concerning relative permeability and related phenom ena.
The treatments of the subject have differed primarily
because various methods of laboratory measurement of relative permeability have been employed.
In this study the litera
ture has been reviewed under appropriate headings, giving emphasis to experimental methods as criteria of classifica tion. I.
LITERATURE ON FUNDAMENTAL FLUID FLOW CONCEPTS D'arcy
formally presented the concept of permeability
He studied the flow of a homogeneous fluid, water,
through a porous medium, unconsolidated sand.
of the investigation can be most concisely stated by a mathe matical expression of D'arcy's law which, for a horizontal flow system is: v =
where v is the volume flux of fluid per unit area, K is the permeability coefficient of the medium, u is the viscosity of the flowing fluid and £LE is the pressure gradient in the dL fluid in the direction of flow. D'arcy's equation, as defined above, is subject to a number of important limita tions.
The equation is only valid in the study of the
11 linear viscous flow of a single, homogeneous fluid through an inert, porous medium.
It should be further pointed out
that the requisite of viscous flow precludes the considera tion of turbulent flow and of gas flow in which slippage phenomena obtains* 30 Jamin was among the earliest investigators in estab lishing the concepts of heterogeneous flow*
the "Jamin effect, " the resistance offered by individual gas bubbles when an attempt was made to force them through the capillary constriction of a porous medium partially saturated with a wetting fluid phase.
of the microscopic character of fluid flow through porous 35 media have made extensive use of the effect. Leverett, in particular, has effectively employed the Jamin effect as a tool in the description of heterogeneous flow mechanisms. The American Petroleum Institute'*' has published Code No. 27 wherein the basic concepts of permeability have been authoritatively established and the fundamental equations of fluid flow have been presented in a usable form. noun
has further discussed these fundamentals. II.
LITERATURE ON HOMOGENEOUS FLOW was among the first to present a complete
and comprehensive study of homogeneous fluid flow.
ment furnished a sound basis on which to establish later
12 conepts of flow in more complex systems in that it effectively and collectively presented the basic principles and discussed their application# op Klinkenberg^ later studied homogeneous flow, investi gating the effect of the flowing fluid on the laboratory measurement of permeability.
Measurements were made using
four gases and nine different pure liquids.
that the permeability of a porous medium was a unique property of that medium as long as the rate of flow remained propor tional to the pressure gradient was concluded by him to be true for most fluids; but, in the case of gases, there was found to be a deviation from the normal behavior.
meability of the medium was found to be a function of the nature of the gas and was observed to vaxy linearly with the reciprocal mean pressure existing in the system#
page, closely related to the mean free path of the gas mole cules, was given as an explanation of the phenomenon. It was illustrated that if the apparent gas permea bilities were plotted against reciprocal mean pressures, the absolute permeability of the medium could be obtained by extrapolating the resulting straight line to infinite mean pressure (a zero value of reciprocal mean pressure).
permeability so obtained was shown to be a property of the medium only.
The "Klinkenberg effect" is mathematically
13 expressed by: Ka “
K (1 + 4 -
where K_ is the apparent permeability, K is the absolute Si permeability of the medium, b is a constant and P is the mean flowing pressure.
It was further shown that the effect
was less important, percentage-wise, in media of higher permeability, becoming negligible at very high permeabili ties,
Although Klinkenberg was not the first to observe gas
slippage phenomena, he established its importance in rela tion to permeability measurement of oil field porous media, correcting an erroneous assumption that had been accepted by all the earlier investigators, 57 21 Yuster and Calhoun and Yuster later extensively studied homogeneous permeability.
Their findings were in
agreement with those of Klinkenberg.
They further concluded
that, within the region of linear flow, the apparent permea bility to gas was independent of the pressure gradient. Rose
has very recently studied relative permeability
and gas slippage phenomena.
Although the cores studied were
saturated with both liquid and gas, the only flow under the conditions of the experiments was in the gas phase.
therefore deemed advisable to review this work under the heading of “homogeneous fluid flow.“ Noteworthy among the conclusions of Rose were:
(1) The corrections for slip in relative permeabilities to gas are often negligible regardless of the magnitude of these corrections, which separately control the effective and ab solute permeabilities.
(2) Relative permeability to gas
data obtained under any arbitrarily chosen mean free path conditions should equally well apply to any other condition; i. e., if the same mean free path condition obtains in the measurement of effective gas permeabilities and apparent gas permeability, the ratios of effective to apparent per meability will be representative of the true relative per meability characteristics of the medium.
The per cent
correction for slip increases with increasing gas saturation and gas permeability, since the largest average pore diameter, and therefore the smallest correction, exists at the equili brium gas saturation. Johnston and Beeson,
in an effort to demonstrate the
differences in the permeabilities of oil field sands to air, fresh water and salt water, presented data on approximately 1,200 core samples.
Ratios of air permeability to salt
water permeability were seen to vary from one to values ap proaching infinity, while the fresh water to salt water permeability ratios varied from zero to one.
It was sug
gested that the vast differences between the air permeabili ties and reservoir water permeabilities were caused by serious
15 modification in the sand structure effected by processes of cleaning and drying preparatory to measuring air permea bility,
All hydratable materials contained in the sample
would be in a hydrated condition in the reservoir; cleaning and subsequent drying would result in dehydration and conse quent volume reduction.
The disparity between salt water
and fresh water permeabilities was attributed to the effect of ion concentration of the water of hydration upon the swell ing of the interstitial clays.
In conclusion, an appeal
was made to use reservoir water as the standard fluid for measurement of permeability of oil field cores, since the permeabilities so obtained were believed to be much more representative of actual field conditions, III.
LITERATURE ON CAPILLARITY
A familiarity with capillary phenomena In general is basic to understanding heterogeneous flow in porous systems which are in themselves traversed by an intricate configuration of capillary channels.
Capillarity and its
expression in the tendency for fluids to rise in tubes of small dimensions has, of course, long been recognized. The discussion to follow has necessarily been confined to capillarity as it is related to porous media and to oil field sands in particular.
16 Since this review does not permit a detailed dis cussion of these principles, the reader is referred to Cal houn *s1^>
^>17>18,19 complete presentation of the funda
mental principles of capillarity in oil reservoirs. 53 Smith, Foote and Busang descrioed capillary rise of liquids in an ideal system composed of uniform spherical sand grains.
Their work was of particular importance since
it presented a partial explanation of the so-called hysteresis observed in the capillary behavior of liquids in granular bodies.
An analogy was drawn between the capillary system
formed by a series of interconnected pores and a series of conical tubes connected as shown in Figure 1.
Figure 1, an equation for the capillary rise was stated as:
b + (b2 + 4 A)* 2
2 S cos^ 9 ____ , d g sin 9
h is the height of capillary rise, S is the surface tension of the liquid-air system, d is the density of the liquid, g is the acceleration of gravity, and 9 is a function of the geometry of the system as defined in Figure 1.
A is a con
stant for any system, while b is a parameter determining the position above the free liquid surface of the conical cell containing the meniscus.
It was pointed out that if such a
capillary were placed in a liquid, the resulting imbibition
MEN ISC US
r2 — H
FREE N S U RF AC E
1MBIBIT ' O N
D ESATU R A TIO N
F IG URE
S C H E M A T IC DIAGRAM IL L U S T R A T IN G HYSTERESIS E F F E C T .
18 would proceed only to a minimum value of h; but if the capil lary were filled and allowed to drain, the liquid level would fall to a maximum value of h.
Use of the foregoing analogy
enhances the explanation of the variation noted in capillary pressure curves obtained by employing the two mechanisms of saturation control; (1) imbibition and (2) displacement* 51 52 Smith * further discussed the capillary rise and liquid distribution in an ideal porous medium.
three types of liquid distribution: (1) pendular distribution in which the liquid appears as single rings around the grain contacts, (2) funicular distribution which arises from the coalescence of the pendular rings, and (3) the distribution characteristic of total saturation* Hassler, Brunner and Deahl
and Leverett^' have pre
sented further discussions of fluid distribution and capil lary pressure relationships in oil field porous media.
portant among their contributions are analyses of hysteresis and of the boundary effects characteristic of flow systems in which more than one porous media is present.
has recently written a paper wherein he discusses methods of calculating the initial fluid distribution in petroleum reservoirs. Livingston'39 has reviewed the surface energy relation ships in petroleum reservoirs.
Ke discussed the effect of
19 surface active agents that act to decrease interfacial ten sions or change the nature of the sol.id surfaces.
gested that such agents may well affect the permeability to a given fluid as well as the static capillary properties of the system,
Benner and Bartell
forces in simplified systems.
later obtained data on surface They concluded that the hydro
philic rock material of petroleum reservoirs, commonly silica or limestone, may be altered in nature, or even caused to become hydrophobic, through adsorption of basic or acidic polar impurities from crude oil* Recent laboratory experiments on the capillary prop erties of porous solids have been conducted by Welge-*^ and Calhoun, Lewis and Newman, 20 IV.
LITERATURE ON RELATIVE PERMEABILITY
MEASUREMENT BY THE CAPILLARY PRESSURE DISPLACEMENT METHOD The capillary pressure method of displacing liquids from porous media was early employed by Richards^ in the study of the “capillary conductivityM of soils under vary ing degrees of water saturation.
In these studies, the
saturated soil sample was placed between semi-permeable mem branes which had previously been saturated with water. The flow of water was conducted through the membrane-soil
20 system, while the water saturation in the sample was reduced from 100 per cent by increment change of the pressure differ ence between the air and water phases within the sample.
pressure difference (and hence the degree of desaturation of the water phase) was limited by the air - water entrance pressure of the semi-permeable membranes.
A greater pressure
difference permitted the entrance of the air into the water flow system, thus preventing separate measurement of the rate of water flow.
The inflow and outflow rates of the system
were volumetrieally determined and the system was considered to be in equilibrium, for any saturation condition, when the rates became equal. The pressure differential in the water phase was mea sured by the use of pressure taps which were fitted with semipermeable water saturated plates.
The taps were inserted
into the sample face and the pressure difference between them (in the water phase) was recorded by a differential manometer system.
Using D'arcy's law, the capillary conduc
tivity of the soil sample was calculated for variable water saturation.
The capillary conductivity was noted to decrease
with decreasing water saturation. It is interesting to note that the described apparatus, although developed much earlier and used for a different purpose, is in principle essentially the same as the one employed in the present study.
Also, although the permeability
21 behavior of the soil samples was not expressed in terms of relative permeability, it io obviouo that Richards was aware of the concept of effective permeability.
The function, cap
illary conductivity, is a measure of the effective permea bility to water and its decrease with decreasing water satura tion is in agreement with current findings. pix Hassler described a method and apparatus for rela tive permeability measurement wherein he applied the prin ciples of measurement discussed by Richards to the determina tion of the relative permeability of small oil field core samples obtained by conventional coring methods.
permitted study of permeability in ternary fluid systems of oil, water and gas.
The semi-permeable membranes were, under
the conditions of the experiment, permeable to only one of the phases present in the core* Brownscombe, Slobod and Caudle
5 6 7
9 have published a
considerable amount of three-phase (oil, water and gas) relative permeability data obtained using an apparatus very similar to that described by Hassler.
Their system differed
in that two phases were flowed simultaneously through the core sample with the same pressure difference between the phases being maintained at all points in the core, in order to ensure uniform saturation and avoid end effects.
complete discussion of the work of Brownscombe et al is con tained in Chapter IV.
22 Recent investigations, wherein capillary pressure dis placement was employed in the study of relative permeabilities to gas at varying static liquid saturations, have been described by Leas, Jenks and Russell.
Important among their
conclusions were the following: (1) that the rate of gas flow had no effect on relative permeability to gas, and (2) that no definite correlation had been found between relative permeability and petrofabric factors or absolute permeability. 54
has described the apparatus used by the
investigators in the study at hand.
The work is only men
tioned here, since a complete description of the instrument and its principles of operation is contained in Chapters III and IV of this study. V.
LITERATURE ON RELATIVE PERMEABILITY MEASUREMENT BY THE SOLUTION GAS DISPLACEMENT METHOD Foremost in the studies of relative permeability em~
ploying solution gas displacement are those of Muskat, Wy45 coff, Botset and Meres, in which the flow of gas-liquid mix tures through long, unconsolidated sand packs was extensively investigated.
The liquid used was water saturated with car
bon dioxide; to effect gas saturations greater than those possible by dissolution, carbon dioxide was introduced directly into the pack from an auxiliary source.
The gas saturation
23 at any position in the flow tube was determined by conductometric methods.
The packs used were very permeable, their
absolute permeabilities varying from 16.1 to 26l darcys. They concluded that certain principal features of the relative permeability versus saturation curves were fundamental char acteristics of gas-liquid mixtures.
These principal fea
tures were: (1) a sharp drop in the liquid permeability and a very slow rise in the gas permeability for small amounts of free gas dispersed through the sand; (2) the attainment of practically complete homogene ous fluid permeability for the gas phase, even with 10 to 20 per cent liquid saturation in the core; (3 ) the property that, for a given sand, steadystate conditions cannot be maintained until the gas satura tion has increased to a certain minimal value. Wycoff and Botset, 57 in a report of a portion of the work described above, stated that ten-fold changes in flow rates and two-fold changes in surface tension showed no ap preciable influence on the observed effective permeabilites to liquid. Botset and Muokat
have applied many of the above
concepts in the study of effect of pressure reduction upon core saturation. 3 Botset^ reported on laboratory experiments in which
24 the study of flow of gas-liquid mixtures was extended to consolidated sand.
One Nichols buff sandstone core, 4,48
feet long and 4 inches in diameter, having an absolute per meability of approximately 500 millidarcys was tested. Water made conductive by addition of potassium sulphate was used as the liquid phase and carbon dioxide was used as the gas phase. methods.
Saturations were determined by conductometrie The following conclusions were among those drawn:
The mechanism of approach to equilibrium satura
tion is essentially the same as was observed in the case of unconsolidated sand. (2)
The equilibrium saturation is probably dependent
upon the cementation and grain size distribution of the sand. (3)
The cementation and grain size distribution of
the sand are also important in their effect on the relative permeability-saturation relation. (4)
Surface and interfacial tension and viscosity
must be relatively unimportant (at least over a moderate variation range), since the results obtained by flowing water and carbon dioxide through the large core agree with those obtained by flowing oil and gas through a small core. 4 It was suggested that the methods of analysis and general conclusions drawn in previous studies on unconsoli dated sand were, with certain minor adjustments, dependent upon the consolidated sand in question, equally applicable to consolidated sands.
25 LITERATURE ON RELATIVE PERMEABILITY MEASUREMENT BY THE DYNAMIC DISPLACEMENT METHOD Leverett
has reported on studies of the flow of
oil-water mixtures through unconsolidated sands.
of oil and water was introduced at a steady rate into a long vertical sand pack and the resulting flow behavior was ob served.
The following variables were considered in an at
tempt to define the effective permeability function: (1)
porosity and permeability of the sand, and
perhaps other petrofabric parameters; (2)
viscosity and density of both liquid phases;
interfacial tension between the liquids and the
contact angles between the liquids and the solid surfaces; (4)
the pressure gradient inducing flow.
He concluded that relative permeability of an uncon solidated sand to oil-water mixtures was substantially in dependent of the viscosity of either fluid, but was related to pore size distribution, displacement pressure, pressure gradient and water saturation.
The direction of the effect
of each variable was believed to be explainable by a con sideration of the Jamin effect.
Data were shown to
substantiate the two mathematical relations which follow: K
P D L) (lI“5) DP d where K„,r ^ were, respectively, the relative permeaxw and Kro * ■ “ bilities to water and oil:’ ST wr was the water saturation;7 and Pp L was a dimensionless ratio defined by Pp, the displaceDP d merit pressure, L, the length of the core, DP, the pressure difference across the core, and d,, the average pore diameter of the core.
In explanation of the mechanics of flow of oil
and water through a porous medium, he theorized that the oil flow occurred through the larger channels, each channel substantially full of oil except at low oil saturations when the oil was believed to move as discrete droplets.
case some oil was believed to remain immobile in the sand. Water was believed to flow through channels not occupied by oil, and, in the case under consideration, was believed to occur as a continuous film around each sand grain. A report of a later investigation by Leverett and Lewis'^ described a study wherein the apparatus of Leverett-^ was employed in the flow of gas-oil-water mixtures through unconsolidated sands.
The data so obtained found more wide
spread field application than others, since they were more representative of reservoir conditions; the three fluid phases inherent to most petroleum reservoirs, gas, oil and water, w^ere considered.
Although the data have since been
27 extensively applied in fluid flow calculations concerning consolidated sand reservoirs, the following restriction was clearly imposed by the authors: “....the conclusions drawn from the presort work are probably qualitatively applicable to consolidated sands*“
The following conclusions were
reached by Leverett and Lewis: (1)
The relative permeability
to water is a unique function of the water saturation and is not affected by the addition of another non-aqueous phase* (2)
The relative permeability to gas is somewhat less in a
three-phase system than it would be in a two-phase system of equal gas saturation.
The relative permeability to
oil is a function of the saturations of all three phases, oil, gas and water. The experimental data also showed that for any given oil saturation the relative permeability to oil was higher in the presence of interstitial water*
In the latter case,
the water was believed to occupy the more sharply curved portions of the pore structure, forcing the oil into the open, central portions of the pores and thus making it avail able for flow* 23 Dunlap has reported on laboratory studies of the effect of interstitial water on effective permeability to a non-wetting fluid, kerosene, desaturation of the water phase having been accomplished by dynamic displacement.
28 Large, unconsolidated sand packs having 3> ^*5 and 8 darcys absolute permeability were used.
There was no evidence that
the relation between relative permeability to kerosene and water saturation was a function of absolute permeability. In the sands studied, it was noted that when the water satur ation was increased above approximately 15 per cent of the pore volume, the effective permeability to kerosene decreased rapidly, A method of measuring two-phase relative permeabilitysaturation relationships in small consolidated core samples 4l has been described by Morse, Terwilliger and Yuster , Henderson and Yuster
shows a similarity of shape over the sat uration range common to both curves. Figure 9 shows the relative permeability curves for Core 3.
This core contained ^9 per cent interstitial water
and had an absolute permeability of 18.6 millidarcys.
graph is characterized by the rapid decline of the relative permeability to oil curve at very low gas saturations.
RELATIVE -PERMEABILITY TO GAS
r~ C A P IL L A R Y
(p s i)
RELATIVE :PERMEABILITY TO GAS
-(A -g —
RELATIVE (PERMEABILITY TO
The equilibrium gas saturation is approximately 10 per cent, and the relative permeability at that point is approximately 5 per cent.
The relative permeability to oil curve approaches
zero at approximately 68 per cent liquid saturation. Relative permeability curves for Core C are shown in Figure 10.
This graph shox^s the variation of the curves
for two different interstitial water contents.
increase in interstitial water content reduced the permea bility to oil approximately 20 per cent.
The gas flow out
let was found to be partially plugged during Run 1, result ing in an apparent high equilibrium gas saturation.
true equilibrium gas saturation, as determined by a rerun, was approximately 10 per cent.
The relative permeability
to oil curves of the two runs tend to merge as they approach zero permeability to oil.
The absolute permeability of the
core was ^0 millidarcys, and the interstitial water in Run 1 was 4A per cent, while that in Run 2 was 52 per cent. The relative permeability curves of Core D are pre sented in Figure 11.
The absolute permeability of the core
was 152 millidarcys and the interstitial water content was 26 per cent.
The graph is characterized by the rapid in
crease in gas permeability above equilibrium gas saturation, approximately 17 per cent, thus indicating that soon after equilibrium gas saturation was attained, a number of the
i.O K4 =40.0mdj
■ffr 0 9
P o ro s ity P 17i8 %
Run | I 08
sw= A a -
'$w=> 52 %
Kro -j •
Kro - 75
210 : -3 0
RELATIVE !.....J :
PERMEABILITY sam ple Ci
RELATIVE • P E R M E A B ILIT Y
• RELATIVE " P E R M E A B ILIT Y
71 large pore channels existing in the core had been made avail able to gas flow by the kerosene dcsaturation, which had been progressing from the ends toward the middle of the core* Figure 12 shows the relative permeability curves of Core E, the least permeable core tested with the apparatus employed in this investigation.
The absolute permeability
of the core was 11.8 millidarcys, and the water satura tion during the run was 45 per cent.
The plot of the data
is characterized by the unusual slope of the gas curve and very low equilibrium gas saturation. over the entire saturation range.
The curve is very flat
This latter behavior,
though not completely understood, was probably the result of an abnormally non-uniform pore size distribution, and may conceivably have been the result of small scale permea bility stratification within the actual core. The relative permeability ratios for cores A, B, C, and E were plotted versus gas saturation in Figure 13* Because of the rather poor definition of the relative per meability to gas curve, no relative permeability ratios were calculated for Core D.
The cores, w’hich were from the
same reservoir, appear to possess common characteristics of permeability ratio. well defined range.
The data are seen to fall within a
The data obtained from Core E appear
to be more characteristic of permeability ratios as calcu lated from the usual pool performance, in that they show a
= 11.8 m d
Ne l a T ive
p e r m e a b il it y
Sw = 45% Porosity *17.
: 10 20 30 40 5 TOTAL LIQUID S A fU R A ION ,i % PORE
tO 20 30 40;
0 \> O L U M E
ad ‘ 80 ‘ too
-OIL-. S A T U R A T IO N ,
% NET; . VOID
PERMEABILITY SAMPLE E
CORE A a - CORE • S 0 CORE; C 1 CORE E
:: 40 : 60 PER CENT GAS SATURATION
74 lower equilibrium gas saturation and a reduction in slope at higher gas saturation values. The tabulated laboratory data, on the natural cores are found in Tables V through IX in Appendix B. II.
GENERAL INTERPRETATION OF DATA
Earlier data obtained by Muskat, Wyckoff, Botset 45 and Meres have shown that, in relative permeability studies by solution gas displacement, the existence of steady-state conditions at low gas saturations was questionable.
the capillary pressure displacement method, however, it is believed that steady-state conditions are established in a relatively short time at low gas saturations.
ment mechanism is such that a transient condition exists only while the displacing pressure is operative.
be emphasized, however, that 11steady-state conditions11 implies no consideration of the static spatial distribution of the phases within the core, A limited supply of data on the effect of time on the attainment of steady-state conditions is shown in Table I, page 62.
Repeated measurements of effective permeability to
oil were made at a single low value of gas saturation.
inspection of the data suggested that equilibrium was attained rapidly, in approximately 45 minutes.
75 in measurement were believed to be representative of the sensitivity of the apparatus and the magnitude of opera tional errors* The equilibrium gas saturation for the five natural cores studied varied from 5 to 18 per cent.
of the magnitude of this value on pore size distribution pre cludes any general conclusion concerning positive correla tion of absolute permeability and equilibrium gas saturation* Points of inflection in the relative permeability curves were interpreted, softer Brownscombe, Slobod and Caudle, 5 *6 *71 to be the reflection of uneven pore size dis tribution.
The effect is shown in the relative permeability
to gas curve for Core B, Figure 9> page 6?, wherein the curve is particularly well defined by the data. The values of effective permeability to gas were not corrected for the Klinkenberg effect.
Over the range of
gas saturations Investigated, the Klinkenberg correction should be negligible because of the relatively large diameter of the gas flow channels* Of the data obtained in this study, the relative permeability ratios shown in Figure 13, page 73, are believed to be the most accurate.
Any experimental inaccuracies common
to the relative permeabilities to both gas and oil were can celed.
These data are more applicable in the common types
76 of reservoir studies, particularly as they pertain to material balance and secondary recovery calculations* As has been previously pointed out by Brownscombe £ rp
et al,*'1 *( the field application of relative permeability data requires a consideration of the producing mechanism operative in the reservoir*
The present investigators be
lieve that the data obtained by the capillary pressure dis placement method is most applicable to problems of gravity drainage and fluid accumulation in petroleum reservoirs, and that in no case should data obtained by this method be applied to solution gas drive reservoirs in stages of de pletion where the gas saturation is below the equilibrium value*
If it may be assumed that data obtained above the
equilibrium gas saturation by the solution gas displacement method are applicable to solution gas drive reservoirs, it is the opinion of the present investigators that data ob tained by the use of capillary pressure displacement are also applicable at these saturations.
The latter interpretation
presupposes that a similarity between the two methods exists for gas saturations greater than equilibrium gas saturation:.. At the present stage of development of laboratory methods for relative permeability studies, the capillary pressure displacement method is the most valid for applica tion in conventional core analysis using small core samples#
The other method applicable to measurement of small samples obtainable by routine coring methods is that proposed by the 28.^1 Pennsylvania State College group. The method of measure ment of the pressure drop and the probable existence of dis continuities in the saturation profile at the outflow and inflow faces of the sample detract from the validity of the latter method. In this study, the use of interstitial water as a third phase has added greatly to the significance of the data obtained.
The data represent the first publication
of any appreciable amount of three-phase data on small con solidated cores of a permeability range representative of reservoir sands.
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS I.
A statement of the problem and a validation of the importance of the study have been presented.
in the study which were capable of variable interpretation were explicitly defined at the outset. The fundamental concepts of permeability and capillary phenomena were presented to establish the validity of the basic principles of the apparatus and method employed. The related studies by previous investigators were re viewed and summarized, in an effort to reveal the inter dependence, similarities and disparities of the various studies. To facilitate future laboratory investigations, an extensive description of the experimental methods employed in this study was presented. Data evincing the degree of reproducibility of the experimental results were presented and evaluated.
obtained from the study of California oil sand cores were presented and interpreted. II.
The following conclusions were drawn from the study:
The capillary pressure displacement method is
the most valid for application 1n conventional core analysis using small core samples* 2.
The displacement mechanism employed in this study
obviates the justifiable application of these data to solu tion gas drive reservoirs when the gas saturation is below the equilibrium value* 3.
The equilibrium gas saturations observed in this
study appeared to bear no relationship to the absolute per meability of the samples. h.
The relative permeability ratios obtained are
the most accurate and the most applicable to field problems. 5.
The degree of accuracy of the capillary pressure
displacement method is less, for a given core, at lower values of gas saturation. III.
The limited scope of the study has prevented the investigation of numerous problems which have arisen.
of these problems, along with recommended modifications of the experimental method, are enumerated: 1.
Additional research should be directed toward
the development of more expeditious and economical methods
80 of obtaining applicable relative permeability data for the solution of reservoir problems, 2.
The effect of the convergence of the liquid flux
at the middle of the core, as discussed in Chapter IV, page 62, should be investigated, 3.
The method of continuous two-phase flow through
the core should be adopted in future investigations, unless insurmountable obstacles arise, h.
The use of a pressure sleeve type core holder
should be adopted in order to facilitate the experimental technique, 5,
(This modification is currently in process.) The ends of the core should be faced with a dia
mond saw, to ensure that the faces are smooth and normal to the axis of the core, thereby improving the capillary con tact. 6,
The fact that capillary pressure and relative
permeability data are obtained simultaneously by this method affords an excellent opportunity for future investigators to obtain a correlation between experimental data and values of relative permeability obtained by the analytical methods discussed in Chapter II, page 29. 7,
The authors recommend the use of absolute permea
bility (as determined by the extrapolation of air permeability to infinite mean pressure) as the base for the calculation of
relative permeabilities to all phases*
This parameter has
shown itself to be highly reproducible between measurements and between laboratories.
Since effective permeability ratios
are among the most important data sought, and since they are calculated as the quotients of the relative permeabilities, it follows that a common base should be used for the relative permeability values. 8.
Laboratory relative permeability data should be
compared with x^rell performance data, whenever possible.
A. P. I. Code No. 27, “Standard Procedure for Determining Permeability of Porous Media,“ American Petroleum Institute, Division of Production, April, 19^2.
Benner, F. C., and F. E. Bartell, “The Effect of Polar Impurities upon Capillary and Surface Phenomena In Petroleum Production,“ A. P. I. Drilling and Produetlon Practice, 341-348, 1941.
3. Botset, H. G., “Flow of Gas-Liquid Mixtures Through Con solidated Sands, “ A. I.. M. E. Transactions, I3S: 91105, 1940. 4
Botset, H. G., and M. Muskat, “Effect of Pressure Reduc tion upon Core Saturation, “ A. I.. M. E. Transactions, 132: 172-83, 1939.
Brownscombe, E. R. , R. L. Slobod, and B. H. Caudle, "Relative Permeability of Cores Desaturated by Capil lary Pressure Method,“ presented at the meeting of the Pacific Coast District, Division of Production, Ameri can Petroleum Institute, at Los Angeles, California, May 12-13, 1949.
, “Laboratory Determination of Relative Permeability Part I,“ The Oil and Gas Journal, 48: 68, February 9, 1950.
7_________, “Laboratory Determination of Relative Permeability, Part II,“ The Oil and Gas Journal, 48: 98, February 16, 1950. 8
Calhoun, John C., Jr., "Permeability — D'arcy^ Law," The Oil and Gas Journal, 46: 71, January 1 , 1948.
9_________ , "Effective and Relative Permeability,“ and Gas Journal, 46: 107, January 22, 1948. 10_________, “Relative Permeability Ratio, " Journal, 46: 107, February 12, 1948. 11 12
The Oil and Gas
_____ , "Effect of Connate Water on Relative Permeability Ratio,“ The Oil and Gas Journal, 46: 135, February 26, 1948. _______, “Relative Permeability in Various Types of Media,“ The Oil and Gas Journal, 46: 149, February 26, 1948.
_____ , "Capillary Forces — Surface Tension, " The Oil and G-as Journal, 46: 137> March 18, 1948,
_____ , "Capillary Forces — Wettability, 11 The Oil and G-as Journal, 46: 203, March 23, 1948*
______, "Capillary Pressure — The Straight Capillary," The Oil and Gas Journal, 46: 99> April 8, 1948.
_____ , "Capillary Pressure — Unconsolidated Sand," The Oil and Gas Journal, 46; 119, April 22, 1948,
______ , "Use of Capillary Pressure Curves — Original Field Distribution," The Oil and Gas Journal, 46: 113, May 6, 1948.
______ , "Laboratory Determination of Capillary Pressure Curve, " The Oil arid Gas Journal, 46: 427, May 13, 1948.
______ , "Correlation of Connate Water xvith Permeability," The Oil and Gas Journal, 46: 241, May 20, 19^8.
Calhoun, John C. Jr., Maurice Lewis and R. C. Newman, "Experiments on the Capillary Properties of Porous Solids," Journal of Petroleum Technology, 1: 189-96, July, 1949.
Calhoun, John C. Jr., and S. T. Yuster, "A Study of the Flow of Homogeneous Fluids Through Ideal Porous Media,"A. P. I. Drilling and Production Practice, 335-55, 194o.
D'arcy, Henry, Les Fontaines publiques de la vllle de Pi .ion, Victor Dalmont, Paris, I856.
Dunlap, Eldon H., "Influence of Connate Water on Permea bility of Sands to Oil, " A. I_. M. E. Transactions, 127: 213-25, 1938.
Hassler, G. L., "Method and Apparatus for Permeability Measurement," U. S. Patent~No. 2 ,343,933.
Hassler, Gerald L., "The Measurement of Permeability of Reservoir Rocks and its Application," Science of Petroleum, 1: 198-209, 1938*
Hassler, G. L., E. Brunner and T. J. Deahl, "The Role of Capillarity in Oil Production," A. 1. M. E. Tran sactions , 133: 135-74, 1944,
Hassler, Gerald L., Raymond R. Rice and Erwin H. Leeman, “Investigations on the Recovery of Oil from Sandstones by Gas Drive,“ A. I. M. E. Transactions, 118: 11637, 1936.
Henderson, J. H., and S. T. Yuster, "Relative Permea bility Studies," Producers Monthly, 12: 13-20, Janu ary, 1948.
Holgrem, C. R., “Some Results of Gas and Water Drives on a Long Gore," A. I. M. E. Transactions, 179: IO3-I8, 19^9
Jamin, J. C., “comptes rendus," 5°: 172, I860,
Johnston, Norris, and C. M. Beeson, "Water Permeabilities of Reservoir Sands, “ A. I_. M. E. Transactions, 160: 43-55, 1945.
Johnston, Norris, and N. van Wingen, “Reservoir Fluid Flow Research, “ A, P. Im Drilling and Production Prac tice, 201-07, 19^5.
Klinkenberg, L. J., “The Permeability of Porous Media to Liquids and Gases, “ A. P. I_. Drilling and Produc tion Practice, 200-13, 1941.
Krumbein, W. C., and G. D. Monk, "Permeability as a Function of the Size Parameters of Unconsolidated Sands," A. I. M. E. Transactions, 151: 153-63, 19^3*
Leas, W. J., L. H. Jenks and Charles D. Russell, "Rela tive Permeability to Gas," Journal of Petroleum Tech nology, 2: 65-72, March, 1950*
Leverett, M. C., "Floxir of Oil-Water Mixtures through Un consolidated Sands," A. I.* M* E * Transactions, 132: 149-71, 1939.
_____ _, “Capillary Behavior in Porous Solids, " A. I.. M. E. Transactions, 1^2: 151-69, 1941.
Leverett, M. C,, and W. B. Lewis, “Steady Flow of GasOil-Water Mixtures through Unconsolidated Sands,“ A. M. E. Transactions, 142: 107-16, 1941.
Livingston, H. K. , “Surfs.ce Energy Relationships in Petroleum Reservoirs, “ A. I. M. E. Transactions, 151: 147-52, 1943.
Moore, T. V., "A Review of the Principles of Oil — Reservoir Performance," A. P. _I. Drilling: and Produc tion Practice, 97-104, 1940.
Morse, R. A., P. L. Terwilliger and S. T. Yuster, “Rela tive Permeability Measurements on Small Core Samples," The Oil and G-as Journal, 46: 109-25, August 23, 1947*
Muskat, Morris, Flow of Homogeneous Fluids, New York: McG-raw-Hill Book Company, Inc. , 1937*
______ , Physical Principles of Oil Production. New York: McG-raw-Hill Book Comoany, Inc., 19^9* Chapters IV, VII, and X.
(as determined by extraction at the end of the run) Pore volume = 1.88 cc Ke = 9.0 x 10b Qu W
TABLE IX LABORATORY DATA Oil RELATIVE PERMEABILITY SAMPLE X EFFECTIVE PERMEABILITIES
SATURATIONS m Total Liquid-::/ :o Pore Volume
■6 Ne t Void
100.0 99.5 99.5 98.4 97.8 96.8 95.7 95.2 94.0 93.0 91.3 88.6 83.3 78.4 75.6 73.0 68.1
(i) -.a .
RELATIVE PERM EA 3ILITIES Oil
T6T Gas (4J/11.8
100 99.2 99.2 97.3 96.0 94.2 92.2 91.3 89.1 87.3 84.2 79.3 69.6 60.8 55.6 50.9 42.0
6.08 6.04 6.00 5.30 4.86 4.62 4.09 3.78 3.19 2.20 1.34 0.70 0.375 0.124 0.0 + 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.052 0.113 0.456 0.490 0.544 — 0.633 0.696 0.737 0.897
51.5 51.2 50. 9 44. 9 41.2 39.2 34.7 32.0 27.0 18.7 11.4 5.9 3.2 1.1 0.0 + 0.0 0.0
0.0 0.0 0. 0 0. 0 0.0 0.0 0.0 0.4 1.0 3.9 4.2 5.0 —
5.4 5.9 6.3 7.6
Absolute permeability * 11.8 millidarcys -::-Y/ater saturation * 45 (as determined by extraction at the end of the run) Pore volume = 1.85 cc = 1.25 x 10' DP