honarpour - relative permeability of petroleum reservoir

Relative Permeabilityof

Petroleum Reservoirs

Authors

Mehdi HonarpourAssociate Professor of Petroleum Engineering

Department of Petroleum EngineeringMontana College of Mineral Science and Technology

Butte, Montana

Leonard KoederitzProfessor of Petroleum Engineering

Department of Petroleum EngineeringUniversity of Missouri

Rolla. Missouri

A. Herbert HarveyChairman

Department of Petroleum EngineeringUniversity of Missouri

Rolla, Missouri

@frc')CRC Press, Inc.

Boca Raton, Florida

PREFACE

In 1856 Henry P. Darcy determined that the rate of flow of water through a sand filtercould be described by the equation

h , - h .q : K A- L

where q represents the rate at which water flows downward through a vertical sand pack

with cross-sectional area A and length L; the terms h, and h, represent hydrostatic heads at

the inlet and outlet, respectively, of the sand filter, and K is a constant. Darcy's experimentswere confined to the flow of water through sand packs which were 1007o saturated with

water.Later investigators determined that Darcy's law could be modified to describe the flow

of fluids other than water, and that the proportionality constant K could be replaced by k/p, where k is a property of the porous material (permeability) and p is a property of the

fluid (viscosity). With this modification, Darcy's law may be written in a more general form

AS

k l- dz dPlu ' : * L P g o s - d s l

where

Sv

Distance in direction of flow, which is taken as positiveVolume of flux across a unit area of the porous medium in unit time alongflow path SVertical coordinate, which is taken as positive downwardDensity of the fluidGravitational accelerationPressure gradient along S at the point to which v. refers

The volumetric flux v. may be further defined as q/A, where q is the volumetric flow rate

and A is the average cross-sectional area perpendicular to the lines of flow.

It can be shown that the permeability term which appears in Darcy's law has units of

length squared. A porous material has a permeability of I D when a single-phase fluid with

a viscosity of I cP completely saturates the pore space of the medium and will flow through

it under viscous flow at the rate of I cm3/sec/cm2 cross-sectional area under a pressure

gradient of 1 atm/cm. It is important to note the requirement that the flowing fluid must

completely saturate the porous medium. Since this condition is seldom met in a hydrocarbon

reservoir, it is evident that further modification of Darcy's law is needed if the law is to be

applied to the flow of fluids in an oil or gas reservoir.A more useful form of Darcy's law can be obtained if we assurne that a rock which

contains more than one fluid has an effective permeability to each fluid phase and that the

effective permeability to each fluid is a function of its percentage saturation. The effectivepermeability of a rock to a fluid with which it is 1007.o saturated is equal to the absolute

permeability of the rock. Effective permeability to each fluid phase is considered to be

independent of the other fluid phases and the phases are considered to be immiscible.

If we define relative permeability as the ratio of effective permeability to absolute perme-

ability, Darcy's law may be restated for a system which contains three fluid phases as

tirl lows:

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dP

dS

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l 5

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. - . I

V o . : T ( 0 . , * K - * )

V* . : * (o - ' 13 - t )V o , : H ( o - r # - k )

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where the subscripts o, g, and w represent oil, gas' and water, respectively' Note that k,,,'

k.", and k,* are the relative permeabilities to the three fluid phases at the respective saturations

of the phases within the rock'

Darcy's law is the basis for almost all calculations of fluid flow within a hydrocarbon

reservoir. In order to use the law, it is necessary to determine the relative permeability of

the reservoir rock to each of the fluid phases; this determination must be made throughout

the range of fluid saturations that will be encountered. The problems involved in measuring

and predicting relative permeability have been studied by many investigators. A summary

of the major results of this research is presented in the following chapters'

l t r . ' . . \ r , tc th l t k . . , .

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THE AUTHORS

Dr. Mehdi "Matt" Honarpour is an associate professor of petroleum engineering atthe Montana College of Mineral Science and Technology, Butte, Montana. Dr. Honarpourobtained his B.S., M.S., and Ph.D. in petroleum engineering from the University of Mis-

souri-Rolla. He has authored many publications in the area of reservoir engineering and core

analysis. Dr. Honarpour has worked as reservoir engineer, research engineer, consultant,and teacher for the past 15 years. He is a member of several professional organizations,

including the Society of Petroleum Engineers of AIME, the honorary society of Sigma Xi,

Pi Epsilon Tau and Phi Kappa Phi.

Leonard F. Koederitz is a Professor of Petroleum Engineering at the University of

Missour i -Rol la. HereceivedB.S., M.S., andPh.D. degrees fromtheUniversi tyof Missour i -

Rolla. Dr. Koederitz has worked for Atlantic-Richfield and previously served as Department

Chairman at Rolla. He has authored or co-authored several technical publications and two

texts related to reservoir engineering.

A. Herbert Harvey received B.S. and M.S. degrees from Colorado School of Minesand a Ph.D. degree from the University of Oklahoma. He has authored or co-authorednumerous technical publications on topics related to the production of petroleum. Dr. Harveyis Chairman of both the Missouri Oil and Gas Council and the Petroleum EngineeringDepartment at the University of Missouri-Rolla.

ACKNOWLEDGMENT

The authors wish to acknowledge the Society of Petroleum Engineers and the AmericanPetroleum Institute for granting permission to use their publications. Special thanks are dueJ. Joseph of Flopetrol Johnston and A. Manjnath of Reservoir Inc. for their contributionsand reviews throughout the writing of this book.

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TABLE OF CONTENTS

Chapter IMeasurement of Rock Relative Permeability .

I. Introduction. . .il. Steady-State Methods .. .

A. Penn-State MethodB. Single-Sample Dynamic Method

C. Stationary Fluid MethodsD. Hassler Method.E. Hafford MethodF. Dispersed Feed Method .

II1I24455689

1 0t 2

III .IV.V .VI .

Unsteady- State MethodsCapillary Pressure MethodsCentrifuge MethodsCalculation from Field Data .

References . . . . .

Chapter 2Two-Phase Relative Permeabil ity ...... 15

I . I n t r o d u c t i o n . . . . . . . . . . 1 5

II. Rapoport and Leas .. ' 15

I I I . G a t e s , L i e t z , a n d F u l c h e r . . . . . . . . 1 6

IV. Fa t t , Dyks t ra , and Burd ine . . . . . . . 16

V. Wyl l ie, Sprangler, and Gardner. . . . . . ' . 19

VI. Timmerman, Corey, and Johnson . . . . . .20

VII. Wahl, Torcaso, and WyllieVIII. Brooks and Corey . . . .27

XIIX. Wyll ie, Gardner, and Torcaso . . . .... . .29

X. Land, Wy l l ie , Rose, P i rson, and Boatman. . . . . . . . . 30

XI. Knopp, Honarpour et al., and Hirasaki . . . . . .37

R e f e r e n c e s . . . . . . . . . . . . . 4 1

Chapter 3Factors Affecting Two-Phase Relative Permeability .... 45

I . I n t r o d u c t i o n . . . . . . . . . . 4 5

il. Two-Phase Relative Permeabil ity Curves ....45

n. Effects of Saturat ion States . . . . . .49

IV. Effects of Rock Properties .... ... 50

V. Def in i t ion and Causes of Wettabi l i ty . . . . . . . . .54

V I . D e t e r m i n a t i o n o f W e t t a b i l i t y . . . . . . . . . . . 5 8

A. Contact Angle Method ... 58

B . I m b i b i t i o n M e t h o d . . . . . . . . 6 0

C . B u r e a u o f M i n e s M e t h o d . . . . . . . 6 3

D. Cap i l la r imet r i c Method. . . . . . . . .63

E. Frac t iona lSur faceAreaMethod. . . . . .64

F. Dye Adsorp t ion Method ' . . . . . . .64

G. Drop Tes t Method. . . . . . . .64

H . M e t h o d s o f B o b e k e t a l . . . . . . . . . 6 4

I. Magnetic Relaxation Method ...64

J. Residual Saturation Methods .. .65

27

K . P e r m e a b i l i t y M e t h o d . . . . . . . . . . . 6 5

L. Connate Water-Permeabi l i ty Method . . . . . . . 66

M. Relat ive Permeabi l i ty Method . . . . . . . . 66

N. Relat ive Permeabi l i ty Summation Method . . . . . . . .61

O. Relat ive Permeabi l i ty Rat io Method . . . . . . . .67

P. Water f lood Method . . . . . . . 68

a. Capil lary Pressure Method .... . 68

R. Resist iv i ty Index Method . . . . . . . 68

VII. Factors Influencing Wettability Evaluation .. . 68

VIII. Wettability Influence on Multiphase Flow . . .72

I X . E f f e c t s o f S a t u r a t i o n H i s t o r y . . . . . . . . . . ' 7 4

X. Effects of Overburden Pressure .. ... ' .. 78

K)(I . Ef fects of Porosi ty and Permeabi l i ty . . . . . . . . .79

XII. Effects of Temperature. . .. .82

XIII. Effects of Interfacial Tension and Density . . .82

XIV. E f fec ts o f V iscos i ty . . . . ; . . . . . . . ' ' 83

XV. Effects of Init ial Wetting-Phase Saturation ... 89

XVI. Effects of an Immobile Third Phase . '. 90

XVII. Effects of Other Factors . . .92

R e f e r e n c e s . . . . . . . - . . . . . 9 7

Chapter 4Three-Phase Relative Permeability ... f 03

I . I n t r o d u c t i o n . . . . . . . . . 1 0 3

i l . D r a i n a g e R e l a t i v e P e r m e a b i l i t y . . . . . ' . 1 0 4

A. Leverett and Lewis ... ' . . 104

B. Corey, Rathjens, Henderson, and Wyllie .. 105

C . R e i d . . . 1 0 7

D . S n e l l . . . l 0 g

E. Donaldson and Dean .. . . I l0

F . S a r e m . . . . . . . 1 1 3

G. Sara f and Fat t . . . . . I 15

H . W y l l i e a n d G a r d n e r . . . . ' l l 5

m. Imbibi t ion Relat ive Permeabi l i ty . . . . . .117

A . C a u d l e , s l o b o d , a n d B r o w n s c o m b e . . . . . . . 1 1 7

B . N a a r a n d W y g a l . . . . . . . . . I 1 7

C . L a n d . . . 1 2 0

D . S c h n e i d e r a n d O w e n s . . . . . . . . . 1 2 3

E . S p r o n s e n . ' . . 1 2 3

IV. Probabil ity Models . .123

V. Exper imentalConf i rmat ion . . . . .126

U\/ I . LaboratoryApparatus. . . . .127

VII. Practical Considerations for Laboratory Tests .... ' 132

V I I I . C o m p a r i s o n o f M o d e l s . . . ' 1 3 3

R e f e r e n c e s " " ' " " " ' 1 3 4

AppendixS y m b o l s . . . . . . . . 1 3 7

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Chapter I

MEASUREMENT OF ROCK RELATIVE PERMEABILITY

I. INTRODUCTION

The relative peffneability of a rock to each fluid phase can be measured in a core sampleby either "steady-state" or "unsteady-state" methods. In the steady-state method, a fixedratio of fluids is forced through the test sample until saturation and pressure equilibria areestablished. Numerous techniques have been successfully employed to obtain a uniformsaturation. The primary concern in designing the experiment is to eliminate or reduce thesaturation gradient which is caused by capillary pressure effects at the outflow boundary ofthe core. Steady-state methods are preferred to unsteady-state methods by some investigatorsfor rocks of intermediate wettability,' although some difficulty has been reported in applyingthe Hassler steady-state method to this type of rock.2

ln the capillary pressure method, only the nonwetting phase is injected into the core duringthe test. This fluid displaces the wetting phase and the saturations of both fluids changethroughout the test. Unsteady-state techniques are now employed for most laboratory meas-urements of relative permeability.3 Some of the more commonly used laboratory methodsfor measuring relative perrneability are described below.

II. STEADY-STATE METHODS

A. Penn-State MethodThis steady-state method for measuring relative perrneability was designed by Morse et

al.a and later modified by Osoba et aI.,5 Henderson and Yuster,6 Caudle et a1.,7 and Geffenet al.8 The version of the apparatus which was described by Geffen et al., is illustrated byFigure l. In order to reduce end effects due to capillary forces, the sample to be tested ismounted between two rock samples which are similar to the test sample. This arrangementalso promotes thorough mixing of the two fluid phases before they enter the test sample.The laboratory procedure is begun by saturating the sample with one fluid phase (such aswater) and adjusting the flow rate of this phase through the sample until a predeterminedpressure gradient is obtained. Injection of a second phase (such as a gas) is then begun ata low rate and flow of the first phase is reduced slightly so that the pressure differentialacross the system remains constant. After an equilibrium condition is reached, the two flowrates are recorded and the percentage saturation of each phase within the test sample isdetermined by removing the test sample from the assernbly and weighing it. This procedureintroduces a possible source of experimental error, since a small amount of fluid may belost because of gas expansion and evaporation. One authority recommends that the core bewgighed under oil, eliminating the problem of obtaining the same amount of liquid film onthe surface of the core for each weighing.3

The estimation of water saturation by measuring electric resistivity is a faster procedurethan weighing the core. However, the accuracy of saturations obtained by a resistivitymeasurement is questionable, since resistivity can be influenced by fluid distribution as wellas fluid saturations. The four-electrode assembly which is illustrated by Figure I was usedto investigate water saturation distribution and to determine when flow equilibrium has beenattained. Other methods which have been used for in situ determination of fluid saturationin cores include measurement of electric capacitance, nuclear magnetic resonance, neutronscattering, X-ray absorption, gamma-ray absorption, volumetric balance, vacuum distilla-tion, and microwave techniques.

. l e

Relative Permeabilin of Petroleum Reservoirs

El-ectrodes

Outl-et Differential PressureTaps

Inlet

Inlet

FIGURE l. Three-section core assembly.8

After fluid saturation in the core has been determined, the Penn-State apparatus is reas-sembled, a new equilibrium condition is established at a higher flow rate for the secondphase, and fluid saturations are determined as previously described. This procedure is re-peated sequentially at higher saturations of the second phase until the complete relativepermeability curve has been established.

The Penn-State method can be used to measure relative permeability at either increasingor decreasing saturations of the wetting phase and it can be applied to both liquid-liquid andgas-liquid systems. The direction of saturation change used in the laboratory should cor-respond to field conditions. Good capillary contact between the test sample and the adjacentdownstream core is essential for accurate measurements and temperature must be heldconstant during the test. The time required for a test to reach an equilibrium condition maybe I day or more.3

B. Single-Sample Dynamic MethodThis technique for steady-state measurement of relative permeability was developed by

Richardson et al.,e Josendal et al.,ro and Loomis and Crowell.ttThe apparatus and exper-imental procedure differ from those used with the Penn-State technique primarily in thehandling of end effects. Rather than using a test sample mounted between two core samples(as illustrated by Figure 1), the two fluid phases are injected simultaneously through a singlecore. End effects are minimized by using relatively high flow rates, so the region of highwetting-phase saturation at the outlet face of the core is small. The theory which was presentedby Richardson et al. for describing the saturation distribution within the core may be de-veloped as follows. From Darcy's law, the flow of two phases through a horizontal linearsystem can be described by the equations

-dP*, : Q* , F* ,dLk*, A

tL* tl

r E C

I rr rrl

( l )

kirF .r f icFrg : f

rdt

t q rll erG

f , F :5X

and

,n Q. Fr" dL- d P n : = i ^ Q )

where the subscripts wt and n denote the wetting and nonwetting phases, respectively. Fromthe definition of capillary pressure, P", it follows that

1 . 0

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plc i r :J t r t r \ r '

3T . : ' . : t . t . t I lS

i d . - ; : J e n d

I r i , ' - . . J r - t r f -

J li. ; ., .: ' .ric rll

nr-' \ ' hcldtr\. : - mJ\

l c . l . , i * - J b )

! - : : - C\F' r - f -

D..r:. ' rn thC

C r ' : ; . . : : : l p l C r

B J . - , , . : l ' l ! l s '

f 3h " : n rsh

Jil. l-: s'ntcrj

! n - : . re ' Jc -

i z . - ' . a ( r r

5 1 0 1 5 2 0 2 5

D i s t a n c e f r o m O u t f l o w F a c e , c f f i

FIGURE 2. Comparison of saturation gradients at low flow rate.e

d P . : d P . - d P * ,

These three equations may be combined to obtain

qP. : /Q*, Fr,*, _ 9"U=\ / o

dL \ k*, kn / /

where dP"/dL is the capillary pressure gradient within the core. Since

dP. : dP. ds*,dL dS*, dL

it is evident that

(3)

(4)

(s)

(6)dS*,

dL

| /Q*, Fr*, Q"p.\ I: A \ k *

- L " / o p . r u s *

, l t

Richardson et al. concluded from experimental evidence that the nonwetting phase sat-

uration at the discharge end of the core was at the equilibrium value, (i.e., the saturation

at which the phase becomes mobile). With this boundary condition, Equation 6 can be

integrated graphically to yield the distribution of wetting phase saturation throughout the

core. If the flow rate is sufficiently high, the calculation indicates that this saturation is

virtually constant from the inlet face to a region a few centimeters from the outlet. Within

this region the wetting phase saturation increases to the equilibrium value at the outlet face.

Both calculations and experimental evidence show that the region of high wetting-phase

saturation at the discharge end of the core is larger at low flow rates than at high rates.

Figure 2 illustrates the saturation distribution for a low flow rate and Figure 3 shows the

distribution at a higher rate.

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Relative Permeability of Petroleum Reservoirs

1 . 0

\ o't I -o-o- -o--o- - :- -- : - JtT h e o r e t i c a l s a t u r a t i o n g r a d i e n t

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FIGURE 3. Comparison of saturation gradients at high flow rate.e

Although the flow rate must be high enough to control capillary pressure effects at thedischarge end of the core, excessive rates must be avoided. Problems which can occur atvery high rates include nonlaminar flow.

C. Stationary Fluid MethodsLeas et al.12 described a technique for measuring permeability to gas with the liquid phase

held stationary within the core by capillary forces. Very low gur flo* rates must be used,so the liquid is not displaced during the test. This technique was modified slightly by Osobaet al.,s who held the liquid phase stationary within the core by means of barriers which werepermeable to gas but not to the liquid. Rapoport and Leasr3 employed a similar techniqueusing semipermeable barriers which held the gas phase stationary while allowing the liquidphase to flow. Corey et al.ra extended the stationary fluid method to a three-phar. ryri..by using barriers which were permeable to water but impermeable to oil and gas. Osoba etal. observed that relative permeability to gas determined by the stationary liquid methodwas in good agreement with values measured by other techniques for some of the caseswhich were examined. However, they found that relative permeability to gas determined bythe stationary liquid technique was generally lower than by other methods in the region ofequilibrium gas saturation. This situation resulted in an equilibrium gas saturation valuewhich was higher than obtained by the other methods used (Penn-Siate, Single-SampleDynamic, and Hassler). Saraf and McCaffery consider the stationary fluid methods to beunrealistic, since all mobile fluids are not permitted to flow simultaneously during the test.2

D. Hassler MethodThis is a steady-state method for relative permeability measurement which was described

by Hasslerrs in 1944. The technique was later studied and modified by Gates and Lietz,16Brownscombe et ?1.," Osoba et al.,s and Josendal et al.ro The laboratory apparatus isillustrated by Figure 4. Semipermeable membranes are installed at each end of the Hasslertest assembly. These membranes keep the two fluid phases separated at the inlet and outletof the core, but allow both phases to flow simultaneously through the core. The pressure

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FIGURE 4. Two-phase relative permeability apparatus.r5

in each fluid phase is measured separately through a semipermeable barrier. By adjustingthe flow rate of the nonwetting phase, the pressure gradients in the two phases can be made

equal, equalizing the capillary pressures at the inlet and outlet of the core. This procedure

is designed to provide a uniform saturation throughout the length of the core, even at low

flow rates, and thus eliminate the capillary end effect. The technique works well under

conditions where the porous medium is strongly wet by one of the fluids, but some difficulty

has been reported in using the procedure under conditions of intermediate wettability.2'r8

The Hassler method is not widely used at this time, since the data can be obtained more

rapidly with other laboratory techniques.

E. Hafford MethodThis steady-state technique was described by Richardson et al.e In this method the non-

wetting phase is injected directly into the sample and the wetting phase is injected througha disc which is impermeable to the nonwetting phase. The central portion of the semiperme-able disc is isolated from the remainder of the disc by a small metal sleeve, as illustrated

by Figure 5. The central portion of the disc is used to measure the pressure in the wetting

fluid at the inlet of the sample. The nonwetting fluid is injected directly into the sample andits pressure is measured through a standard pressure tap machined into the Lucite@ sur-rounding the sample. The pressure difference between the wetting and the nonwetting fluid

is a measure of the capillary pressure in the sample at the inflow end. The design of theHafford apparatus facilitates investigation of boundary effects at the influx end of the core.The outflow boundary effect is minimized by using a high flow rate.

F. Dispersed Feed MethodThis is a steady-state method for measuring relative permeability which was designed by

Richardson et al.e The technique is similar to the Hafford and single-sample dynamic meth-


G A S

IG A S P R E S S U R E G A U G E

P R E S S U R E

G A S M E T E R

O I L B U R E T T E

FIGURE 5. Hafford relative permeability apparatus.e

ods. In the dispersed feed method, the wetting fluid enters the test sample by first passingthrough a dispersing section, which is made of a porous material similar to the test sample.This material does not contain a device for measuring the input pressure of the wetting phaseas does the Hafford apparatus. The dispersing section distributes the wetting fluid so that itenters the test sample more or less uniformly over the inlet face. The nonwetting phase isintroduced into radial grooves which are machined into the outlet face of the dispersingsection, at the junction between the dispersing material and the test sample. Pressure gradientsused for the tests are high enough so the boundary effect at the outlet face of the core isnot significant.

III. UNSiuoo"-STATE METHoDS

Unsteady-state relative permeability measurements can be made more rapidly than steady-state measurements, but the mathematical analysis of the unsteady-state procedure is moredifficult. The theory developed by Buckley and Leverettre and extended by Welge2o isgenerally used for the measurement of relative permeability under unsteady-state conditions.The mathematical basis for interpretation of the test data may be summarized as follows:Leverett2r combined Darcy's law with a definition of capillary pressure in differential formto obtain

f*z

' * ; h ( * - e A p s i n o )( 7 1

r + I n . &k* Fo

where f*, is the fraction water in the outlet stream; q, is the superficial velocity of total fluidleaving the core; 0 is the angle between direction x and horizontal; and Ap is the density

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difference between displacing and displaced fluids. For the case of horizontal flow andnegligible capillary pressure, Welge2o showed that Equation 7 implies

S*.u, - S*z : f.r, Q*

where the subscript 2 denotes the outlet end of the core, S*.ou is the average water saturation;and Q* is the cumulative water injected, measured in pore volumes. Since Q* and S*.,u canbe measured experimentally, f", (fraction oil in the outlet stream) can be determined fromthe slope of a plot of Q* as a function of S*,ou. By definition

l , z : q , , / ( q , , * q * )

By combining this equation with Darcy's law, it can be shown that

If , , r : ' t l O t

I1.,/ K..,t *

tr/.,*

Since p" and pw are known, the relative permeability ratio k.o/k.* can be determined fromEquation 10. A similar expression can be derived for the case of gas displacing oil.

The work of Welge was extended by Johnson et a1.22 to obtain a technique (sometimescalled the JBN method) for calculating individual phase relative permeabilities from unsteady-state test data. The equations which were derived are

k.. :

(8)

(e)

f,,,

and

k . o : l t o o , , ,t.z ttr.

where I,, the ?elative injectivity, is defined as

( I l )

(12)

( l 3 )I , :

injectivity

initial injectivity

(q*,/Ap)

(q*,/Ap) at start of injection

A graphical technique for solving Equations 1l and 12 is illustrated in Reference L3..Relationships describing relative permeabilities in a gas-oil system may be obtained byreplacing the subscript "w" with "g" in Equations lI,12, and 13.

In designing experiments to determine relative permeability by the unsteady-state method,it is necessarv that:

The pressure gradient be large enough to minimize capillary pressure effects.The pressure differential across the core be sufficiently small compared with totaloperating pressure so that compressibility effects are insignificant.The core be homogeneous.The driving force and fluid properties be held constant during the test.2

l .2 .

3 .4 .


Laboratory equipment is available for making the unsteady-state measurements under sim-ulated reservoir conditions.2a

In addition to the JBN method, several alternative techniques for determining relativepermeability from unsteady-state test data have been proposed. Saraf and McCaffery2 de-veloped a procedure for obtaining relative permeability curves from two parameters deter-mined by least squares fit of oil recovery and pressure data. The technique is believed tobe superior to the JBN method for heterogeneous carbonate cores. Jones and Roszelle25developed a graphical technique for evaluation of individual phase relative permeabilitiesfrom displacement experimental data which are linearly scalable. Chavent et al. describeda method for determining two-phase relative permeability and capillary pressure from twosets of displacement experiments, one set conducted at a high flow rate and the other at arate representative of reservoir conditions. The theory of Welge was extended by Sarem todescribe relative permeabilities in a system containing three fluid phases. Sarem employeda simplifying assumption that the relative permeability to each phase depends only on itsown saturation, and the validity of this assumption (particularly with respect to the oil phase)has been questioned.2

Unsteady-state relative permeability measurements are frequently used to determine theratios k*/ko, ks/k", and kr/k*. The ratio k*/k" is used to predict the performance of reservoirswhich are produced by waterflood or natural water drive; kr/k" is employed to estimate theproduction which will be obtained from recovery processes where oil is displaced by gas,such as gas injection or solution gas drive. An important use of the ratio k*/k* is in theprediction of performance of natural gas storage wells, where gas is injected into an aquifier.The ratios k*/ko, kg/ko, and kr/k* are usually measured in a system which contains only thetwo fluids for which the relative permeability ratio is to be determined. It is believed thatthe connate water in the reservoir may have an influence on kg/k.,, expecially in sandstoneswhich contain hydratable clay minerals and in low permeability rock. For these types ofreservoirs it may be advisable to measure k*/k., in cores which contain an immobile watersaturation.2a

IV. CAPILLARY PRESSURE METHODS

The techniques which are used for calculating relative permeability from capillary pressuredata were developed for drainage situations, where a nonwetting phase (gas) displaces awetting phase (oil or water). Therefore use of the techniques is generally limited to gas-oilor gas-water systems, where the reservoir is produced by a drainage process. Although itis possible to calculate relative permeabilities in a water-oil system from capillary pressuredata, accuracy of this technique is uncertain; the displacement of oil by water in a water-wet rock is an imbibition process rather than a drainage process.

Although capillary pressure techniques are not usually the preferred methods for generatingrelative permeability data, the methods are useful for obtaining gas-oil or gas-water relativepermeabilities when rock samples are too small for flow tests but large enough for mercuryinjection. The techniques are also useful in rock which has such low permeability that flowtests are impractical and for instances where capillary pressure data have been measured buta sample of the rock is not available for measuring relative permeability. Another use whichhas been suggested for the capillary pressure techniques is in estimating kr/k" ratios forretrograde gas condensate reservoirs, where oil saturation increases as pressure decreases,with an initial oil saturation which may be as low as zero. The capillary pressure methodsare recommended for this situation because the conventional unsteady-state test is not de-signed for very low oil saturations.

Data obtained by mercury injection are customarily used when relative permeability isestimated by the capillary pressure technique. The core is evacuated and mercury (which is

t m c l# r } ,r;ra;t &-.rsri

kr6

rlrn I

f r *d * blr

A't|i l t r

hh

B u L J r ' l r l l T l -

D l n j : i . . r l l r c '

f - : l l - - ' . J c -

f l c " - - . . : i lCr -

J i * - . . : c i c J t o

d R . zc l l c ' : '

3 f i : . ; : ^ i l l l t s ' r

rl --.-:rhcdfE

" ' l ' . l \A t r

l r " ' - ' : r [ rf ' . : . : : ; : : l l t r

! n " ' : ' " , c J

J r - ' : : \

h ' . ' - . : . i ' '

l r c ' - ' - c i hc

O l ' - - J - - . , ' t r \

f 6 : ' " - r . t i l c

X c " : - . . ! 1 . .

A. : thcD a- -:* - iilc't

; tr . . : . .r thc

b r - , - - l l r . r l

l . . . . , : . i t .

r - ' r - \ , r '

D ' - . * - : l J l

br. : ' : ; . .urc

) ; . , . : - - ' \ J

b J ' - - : . - r r l l. 1 " ' - r r

br-, ' - i - , rc

f i - . . ^ : i S [ -

! l : ; - J : r i l n g

i l l l ' - ; . - r l l r c '

l f t " : " i ' : . u n

lq : - : l ! l t r$

ts.:. -:'.'J huttf -r-' u hrcht t : l t r r : l t ) r

E .le . :t'a:€s.

Drc :: .cthtrJr

! l : . : , ' t de -

I t rK ' - :^ . .11\ rs

;: \ ' ' i l r ;h is

9

the nonwetting phase) is injected in measured increments at increasing pressures. Approx-

imately 20 data points are obtained in a typical laboratory test designed to yield the complete

capillary pressure curve, which is required for calculating relative permeability by the meth-ods described below.

Several investigators have developed equations for estimating relative permeability from

capillary pressure data. Purcell2e presented the equations

f s * i

l, dS/pik.* , :

f l

t dS/Pi

I' ds/p!J S o i

k . n * , : f l

J, dS/pi

( l 4 )

and

( l 5 )

where the subscripts wt and nwt denote the wetting and nonwetting phases, respectively,and n has a value of 2.0. Fatt and Dykstra3o developed similar equations with n equal to3 . 0 .

A slightly different result is obtained by combining the equations developed by Burdine3lwith the work of Purcell.2e The results are

( l 6 )

( l 7 )

where S, is the total liquid saturation.

V. CENTRIFUGE METHODS

Centrifuge techniques for measuring relative permeability involve monitoring liquids pro-

duced from rock samples which were initially saturated uniformly with one or two phases.

Liquids are collected in transparent tubes connected to the rock sample holders and production

is monitored throughout the test. Mathematical techniques for deriving relative permeability

data from these measurements are described in References 26, 27, and 28.

Although the centrifuge methods have not been widely used, they do offer some advantages

over alternative techniques. The centrifuge methods are substantially faster than the steady-

state techniques and they apparently are not subject to the viscous fingering problems which

sometimes interfere with the unsteady-state measurements. On the other hand, the centrifuge

methods are subject to capillary end effect problems and they do not provide a means for

determining relative permeability to the invading phase.

O'Mera and Lease28 describe an automated centrifuge which employs a photodiode array

in conjunction with a microcomputer to image and identify liquids produced during the test.

t0 Relative Permeabiliy of Petroleum Reservoirs

C A M E R

C E N T R I F U G E

L I Q U I D P R O D U C T I O N

T R O B E

S P E E D D I S K

FIGURE 6. Automated centrifuge system.28

Stroboscopic lights are located below the rotating tubes and movement of fluid interfacesis monitored by the transmitted light. Fluid collection tubes are square in cross section,since a cylindrical tube would act as a lens and concentrate the light in a narrow band alongthe major axis of the tube. A schematic diagram of the apparatus is shown by Figure 6.

VI. CALCULATION FROM FIELD DATA

It is possible to calculate relative permeability ratios directly from field data.23In makingthe computation it is necessary to recognize that part of the gas which is produced at thesurface was dissolved within the liquid phase in the reservoir. Thus;

(produced gas) : (free gas) * (solution gas) (18)

If we consider the flow of free gas in the reservoir, Darcy's law for a radial system maybe written

SrmrLrlr

Thll. tu

rtts:rt r.

\ & t r tor fra g,;rrrrrrhrrrrrr-l t.-r

9g.fr"" :

Thc n: R r ! t n

lr*rj nr

. E ! E

h F'fr'if rttl t

:u-bil

tr r*l

t r tI tru::3 r rF F rlr}-rrf$lrI1

hor I

Fcr

- lst'

k h P . - P? . 0 9 - E - e

- w

FrB, ln (r./r*)( l 9 )

C O M P U T E R

oz

LIJ

o

ouJLIJo-a)

oU'IJJtroo

J

:

C O N T R O L L E R

S P E E D S E T P O I N T

l l

?

FIGURE 7. Calculation of gas-oil relative permeability values from production data.

Similarly, the rate of oil flow in the same system is

where r* is the well radius and r" is the radius of the external boundary of the area drained

by the well. B" and B, are the oil and gas formation volume factors, respectively. The ratio

of free gas to oil is obtained by dividing Equation 19 by Equation 20. lt we express Ro,

cumulative gas/oil ratio and R,, solution gasioil ratio, in terms of standard cubic foot per

stock tank barrel, Equation l8 implies

R o : s . 6 t s l u * ' * * .Ko ltrs be

Thus, the relative permeability ratio is given by

(20)

(22)

(2t)

k"

ko

S . : ( t - t o o , ) * , t -

s * )

_ (Ro - R.)&- ! !5 .615 B . F .

l | i ' ' : J :1Jac \

! n - . . \ e \ l l ( ) n .

I ^ - : l J r l t r 0 S

I F . . , : l 6

l : 1 ' : : : r l t ng

Dd- - . - : l t he

l \ r

| : ' , . t : : t tx?)

The oil saturation which corresponds to this relative permeability ratio may be determined

from a material balance. If we assume there is no water influx, no water production, no

fluid injection, and no gas cap, the material balance equation may be written

where minor effects such as change in reservoir pore volume have been assumed negligible.

In Equation 23 the symbol N denotes initial stock tank barrels of oil in place; No is number

of stock tank barrels of oil produced; and B", is the ratio of the oil volume at initial reservoir

conditions to oil volume at standard conditions.If total liquid saturation in the reservoir is expressed as

(23)

s , : s * + ( r - s * ) ( \ } ) ( * ) (24)

r l 9 t

then the relative permeability curve may be obtained by plotting kr/k" from Equation 22 asa function of S,- from Equation 24. Figure 7 illustrates a convenient format for tabulatingthe data. The curve is prepared by plotting column 9 as a flnction of column 6 on semilogpaper, with k/k" on the logarithmic scale. The technique is useful even if only a few high-liquid-saturation data points can be plotted. These kr/k" values can be used to verify theaccuracy of relative permeability predicted by empirical or laboratory techniques.

Poor agreement between relative permeability determined from production data and from

laboratory experiments is not uncommon. The causes of these discrepancies may include

the following:

t2 Relative Permeability of Petroleum Reservoirs

l. The core on which relative permeability is measured may not be representative of thereservoir in regard to such factors as fluid distributions, secondary porosity, etc.

2. The technique customarily used to compute relative permeability from field data doesnot allow for the pressure and saturation gradients which are present in the reservoir,nor does it allow for the fact that wells may be producing from several strata whichare at various stages of depletion.

3. The usual technique for calculating relative permeability from field data assumes thatRo at any pressure is constant throughout the oil zone. This assumption can lead tocomputational errors if gravitational effects within the reservoir are significant.

When relative permeability to water is computed from field data, a common source ofelror is the production of water from some source other than the hydrocarbon reservoir.These possible sources of extraneous water include casing leaks, fractures that extend fromthe hydrocarbon zone into an aquifer, etc.

REFERENCES

l. Gorinik, B. and Roebuck, J. F., Formation Evaluation through Extensive Use of Core Analysis, CoreLaborator ies, Inc. , Dal las, Tex. , 1979.

2. Saraf, D. N. and McCaffery, F. G., Two- and Three-Phase Relative Permeabilit ies: a Review, PetroleumRecovery Institute Report #81-8, Calgary, Alberta, Canada, 1982.

3. Mungan, N., Petroleum Consultants Ltd., personal communication, 1982.4. Morse, R. A., Terwill iger, P. L., and Yuster, S. T., Relative permeability measurements on small

samples, Oi l Gas J. , 46, 109, 1947.5. Osoba, J. S., Richardson, J. G., Kerver, J. K., Hafford, J. A., and Blair, P. M., Laboratory relative

permeability measurements, Trans. AIME, 192, 47, 1951.6. Henderson, J. H. and Yuster , S.T. , Relat ive permeabi l i ty study,World Oi l ,3,139, 1948.7. Caudle, B. H., Slobod, R. L., and Brownscombe, E. R. W., Further developments in the laboratory

determination of relative permeability, Trans. AIME, 192, 145, 1951.8. Geffen, T. M., Owens, W. W., Parrish, D. R., and Morse, R. A., Experimental investigation of factors

affecting laboratory relative permeability Teasurements, Trans. AIME, 192, 99, 1951.9. Richardson, J. G., Kerver, J. K., Hafford, J. A., and Osoba, J. S., Laboratory determination of relative

permeability, Trans. AIME, 195, 187, 1952.10. Josendal, V. A., Sandiford, B. B., and Wilson, J. W., Improved multiphase flow studies employing

radioactive tracers, Trans. AIME, 195, 65, 1952.I l. Loomis, A. G. and Crowell, D. C., Relative Permeability Studies: Gas-Oil and Water-Oil Systems, U.S.

Bureau of Mines Bulletin BarHeuillr, Okla., 1962,599.12. Leas, W. J., Jenks, L. H., and Russell, Charles D., Relative permeability to gas, Trans. AIME, 189,

65, r9s0.13. Rapoport, L. A. and Leas, W. J., Relative permeability to l iquid in l iquid-gas systems, Trans. AIME,

1 9 2 , 9 3 , l 9 5 l .14. Corey, A. T., Rathjens, C. H., Henderson, J. H., and Wyllie, M. R. J., Three-phase relative perme-

abi l i ty , J . Pet . Technol . , Nov. , 63, 1956.15. Hassler , G. L. , U.S. Patent 2,345,935, 1944.16. Gates, J. I. and Leitz, W. T., Relative permeabilities of California cores by the capillary-pressure method,

Drilling and Production Practices, American Petroleum Institute, Washington, D.C. 1950, 285.17. Brownscombe, E. R., Slobod, R. L., and Caudle, B. H., Laboratory determination of relative perrne-

ab i l i t y , O i l Gas J . ,48 ,98 , 1950 .18. Rose, W., Some problems in applying the Hassler relative permeability method, J. Pet. Technol., 8, I l6l,

1980 .

19. Buckley, S. E. and Leverett, M. C., Mechanism of fluid displacement in sands, Trans. AIME, 146,107,1942.

20 . We lge 'H .J . rAs imp l i f i edmethod fo rcomput ing recoverybygasorwa te rd r i ve ,T rans .A |ME, 195 ,91 ,1952.

21. Leverett, M. C., Capillary behavior in porous solids, Trans. AIME, 142, 152, 1941.

t l

l_1

lo

l l:.i

I r

Johrplar'cn

CridC l r f i . .SFr.-t.

Jcrl.lr.plrSlo5.irc.hfu,U r S

SPL T)O'llGacotn:Frerr,

htlF*:

Frt- |Bra- l _

l v

l r

-\

n l . r l : r c ( ) i t hg

b r t \ . i l a

bl i ; . ' l . r J t res

dlc ' : l .cn . t i r .

I . i : - : l - : l r [ 1g I

t E . . - : : t C . t h a l

D . - : i ' . c . r J to

l l l . . - : l

I|r. . -:.c tl i

Fn : - ' -< ' I \ t ) l f .

I c \ : r - . J l r r )m

r l

l E -

F.

l r -

X'r

| : ' ,

F

I .

Er

l s

It

! l t -

JI

F ' - ' . " ; : h , r J .

[ : . 'I t : . . j ' rTnC-

13

22. Johnson, E. F., Bossler, D. P., and Naumann, V. O., Calculation of relative permeability from dis-placement experiments, Trans. AIME, 216,310, 1959.

23. Crichlow, H. B., Ed., Modern Reservoir Engineering - A Simulation Approaclr, Prentice-Hall, EnglewoodCliffs, 1977, chap. 7.

24. Special Core Analysis, Core Laboratories, Inc., Dallas, 1976.25. Jones, S. C. and Roszelle, W. O., Graphical techniques for determining relative permeability from

displacement experiments, J. Pet. Technol., 5, 807, 1978.26. Slobod, R. L., Chambers, A., and Prehn, W. L., Use of centrifuge for determining connate water,

residual oil, and capillary pressure curves of small core samples, Trans. AIME, 192, 127, 1952.27 . Yan Spronsen, E., Three-phase relative permeability measurements using the Centrifuge Method, Paper

SPE/DOE 10688 presented at the Third Joint Symposium, Tulsa, Okla., 1982.28. O'Mera, D. J., Jr. and Lease, W. O., Multiphase relative permeability measurements using an automated

centrifuge, Paper SPE 12128 presented at the SPE 58th Annual Technical Conference and Exhibition, SanFranc isco .1983 .

29. Purcell, W. R., Capillary pressures - their measurement using mercury and the calculation of permeabilitytherefrom, Trans. AIME, 186, 39. 1949.

30. Fatt, I. and Dyksta, H.,,Relative permeability studies, Trans. AIME, 192,41, 1951.31. Burdine, N. T., Relative Permeability Calculations from Pore Size Distribution Data, Trans. AIME, lg8,

7 t , 1 9 5 3 .

rf-

tv:

. . l f . l .

: ^ l ( ) 7 .

N t ! : - i . 9 1 .

l 5

Chapter 2

TWO-PHASE RELATIVE PERMEABILITY

I. INTRODUCTION

Direct experimental measurement to determine relative permeability of porous rock has

long been recorded in petroleum related literature. However, empirical methods for deter-

mining relative permeability are becoming more widely used, particularly with the advent

of digital reservoir simulators. The general shape of the relative permeability curves may

be approximated by the following equations: k.* : A(S*)'; k.., : B(l - S*)"'; where A,

B. n. and m are constants.Most relative permeability mathematical models may be classified under one of four

categories:Capillary models - Are based on the assumption that a porous medium consists of a

bundle of capillary tubes of various diameters with a fluid path length longer than the sample.

Capillary models ignore the interconnected nature of porous media and frequently do not

provide realistic results.Statistical models - Are also based on the modeling of porous media by a bundle of

capillary tubes with various diameters distributed randomly. The models may be described

as being divided into a large number of thin slices by planes perpendicular to the axes of

the tubes. The slices are imagined to be rearranged and reassembled randomly. Again,

statistical models have the same deficiency of not being able to model the interconnected

nature of porous media.Empirical models - Are based on proposed empirical relationships describing experi-

mentally determined relative permeabilities and in general, have provi{ed the most successful

approximations.Netwoik models - Are frequently based on the modeling of fluid flow in porous media

using a network of electric resistors as an analog computer. Network models are probably

the best tools for understanding fluid flow in porous media'r'aa

The hydrodynamic laws generally bear little use in the solution of problems concerning

single-phase fluid flow through porous media, let alone multiphase fluid flow, due to the

complexity of the porous system. One of the early attempts to relate several laboratory-

measured parameters to rock permeability was the Kozeny-Carmen equation.2 This equation

expresses the permeability of a porous material as a function of the product of the effective

path length of the flowing fluid and the mean hydraulic radius of the channels through which

the fluid flows.Purcell3 formulated an equation for the permeability of a porous system in terms of the

porosity and capillary pressure desaturation curve of that system by simply considering the

porous medium as a bundle of capillary tubes of varying sizes.

Several authorsa-r6 adapted the relations developed by Kozeny-Carmen and Purcell to the

computation of relative permeability. They all proposed models on the basis of the assumption

that a porous medium consists of a bundle of capillaries in order to apply Darcy's and

Poiseuille's equations in their derivations. They used the tortuosity concept or texture pa-

rameters to take into account the tortuous path of the flow channels as opposed to the concept

of capillary tubes. They tried to determine tortuosity empirically in order to obtain a close

approximation of experimental data.

II. RAPOPORT AND LEAS

Rapoport and Lease presented two equations for relative permeability to the wetting phase.

16 Relative Permeabilin of Petroleum Reservoirs

These equations were based on surface energy relationships and the Kozeny-Carmen equa-tion. The equations were presented as defining limits for wetting-phase relative permeability.

The maximum and minimum wetting-phase relative permeability presented by Rapoportand Leas are

k.*,(max) : ( l )

P. dSf s *

Jr*, t 'ot,['*'(tj) (T#)'

and

.['*'P. dS

k,*,(min) : (ti - j; ) ' fs- fS*,

I P . d s + | R . a sJ r ' J r

(2)

where S- represents the minimum irreducible saturation of the wetting phase from a drainagecapillary pressure curve, expressed as a fraction; S*, represents the saturation of the wettingphase for which the wetting-phase relative permeability is evaluated, expressed as a fraction;P. represents the drainage capillary pressure expressed in psi and S represents the porosityexpressed as a fraction.

III. GATES. LIETZ. AND FULCHER

Gates and Lietzs developed the following expression based on Purcell's model for wetting-phase relative permeability:

t . _K.*r -

Fulcher et al.,as have investigated the influence of capillary number (ratio of viscous tocapillary forces) on two-phase oil-water relative permeability curves.

IV. FATT, DYKSTRA, AND BURDINE

Fatt and Dykstrarr developed an expression for relative permeability following the basicmethod of Purcell for calculating the permeability of a porous medium. They considered alithology factor (a correction for deviation of the path length from the length of the porousmedium) to be a function of saturation. They assumed that the radius of the path of theconducting pores was related to the lithology factor, tr, by the equation:

ruI $

(3 )

u hcre riun -tr.r

Tlr ct \

FanE^t,rat.l

Ttrr rtrflfl

Thc 1ilrrrrd

&nJDillrd

!,! hrDrfi

crlr cr

Ffm

(4)a

\ : -ro

L7

, a I - ' - : l cL lua -

P C : - ' . . r . r l l t r .

h i i . . ' j ' p rp1

[ l . , , l : r : n J l C

J : - . . i . ' 1 t l n S

F ' - ' : ' : l ( r n -

i l i - r : ' . : , r \ l l \

i r5 : : .r hasic

srr: . . .- :ercd a

J i : ; Fr r t tusi f ; i : . , ' l t h e

Table ICALCULATION OF WETTING.PHASE RELATIVE

PERMEABILITY BASED ON THE FATT ANDDYKSTRA EQUATION

Area from 0S*, Vo P", cm Hg l/P"'], (cm Hg)-t to S*, in.2 k.*,, Vo

, l r100 4.0 0 .01569 0 4 . 5 0 . 0 1 1 080 5.0 0.0080'70 5.5 0.006060 6.0 0.0046s0 6.7 0.003340 7.s 0.002430 8.7 0 .00 1520 13.0 0.0005

' 7 .88/11.25 x 100 : 70.0 ." 5 .54111.25 x l0O : 49.2 .

n . 2 57 . 8 85 .543 .802.49t . 5 00.750.300.20

100.070.0,49.2b33 .82 2 . 11 3 . 36 . 12 . 70 .4

_ l

where r represents the radius of a pore, a and b represent material constants, and }, is afunction of saturation.

The equation for the wetting-phase relative permeability, k.*,, reported by Fatt and Dykstrais

f t* ' dst -

, J n P 2 ( l + b )

K.*, : l.r dS

Jo P2( | * b)

agreement with observed data when b :

(5)

r/r, reducingFatt and Dykstra found goodEquation 5 to

They stated that their equation fit their own data as well as the data of Gates and Lietz moreaccurately than other proposed models.

The procedure for the calculation of relative permeability from capillary pressure data isillustrated by Table I and the results are shown in Figures I and 2.

Burdine'3 reported equations for computing relative perrneability for both the wetting andnonwetting phases. His equations can be shown to reduce to a form similar to those developedby Purcell. Burdine's contribution is principally useful in handling tortuosity.

Defining the tortuosity factor for a pore as L when the porous medium is saturated withonly one fluid and using the symbol tr*, for the wetting-phase tortuosity factor when twophases are present, a tortuosity ratio can be defined as

ft*' dsJo P:

TF (6)r - r l

Ttr.*, : ;

(7)r - l )

l8 Relative Permeabilitv of Petroleum Reservoirs

9

I

| 7P o l

(cm Hg) 6

5

4

3

2

I

oo' lo 20 40 50 60 70 80S w +

FIGURE 1. Capillary pressure as a function of water saturation.

/'*' {^,*,)'�ds/(\)'�(P.)'�kr*,

/ '0r,1^;'1r.y'

fS*'

t ds/(P.)rk.*t : (tr.*.) ' r l

t ds/(p")l

In a similar fashion, the relative permeability to the nonwetting phase can

utilizing a nonwetting-phase tortuosity ratio, tr,,*,,

then

Burdine has shown that

( 9 )

be expressed

( l0)

where SThe relaphase to

where SThe e

the expr

Wyl l icomputi

( 8 )

If tr is a constant for the porous medium and tr,*t depends only on the final saturation, then

f l

I dst1e.)'^ J S * t

k .n * , : ( t r r n * , ) '

J" ds/(P.)2

S * , - S -Arwt - ( l t )

1 - S -

l9

r60

r50

r40

r30

t20

l l

roo

90

70

60

50

40

30

20

t o

o5 lo 20 30 40 50 60 70

Sw -+

il,;yul}: Reciprocal of (capillary pressure)r as a function of water

where S- represents the minimum wetting-phase saturation from a capillary-pressure curve.

The relative perrneability is assumed to approach zero at this saturation. The nonwetting

phase tortuosity can be approximated by

\ - ^ . . , . : . S n * t - - S ' ( 1 2 )r n w t l - s * - s "

where S. is the equilibrium saturation to the nonwetting phase.

The expression for the wetting phase (Equation 9) fit the data presented much better than

the expression for the nonwetting phase (Equation 10).

V. WYLLIE, SPRANGLER, AND GARDNER

Wyllie and Spran glertz reported equations similar to those presented by Burdine for

computing oil and gas relative permeability. Their equations can be expressed as follows:

I tI

Pc3 |

(Cm Hq i3

fa: : t hcn

r 9 )

f3 . r l l i ' rred

r l 0 )

r l l )

fs"

k,,,: (iil' J os"rp;

/' or",rl( l 3 )

IE


oAIa

WYLLIE ond SPANGLERGATES ond LIETZ

i l | | t l

BEREA NO.4

FIGURE 3. Reciprocal of (capillary pressure)r as a function of saturation for normalizedda ta . rT

Wetrine-

\\ 'etring.

k,* (r -r+" )' !Y_ S*,/ /' or",r3

where S- represents the lowest oil saturation at which the gas phase is discontinuous: S-: ( l - S".) .

The above equations for oil and gas relative permeabilities may be evaluated when areliable drainage capillary pressure curve of the porous medium is available, so that a plotof llP"2 as a function of oil saturation can be constructed. Obviously, reliable values of S-and So. are also needed for the oil and gas relative permeability evaluation. Figure 3 showssome examples of llP.2 vs. saturation curves.rT

Wyllie and GardnerrT developed equations for oil and gas relative permeabilities in thepresence of an ineducible water saturation, with the water considered as part of the rockmatrix:

f t ' ds *

k,.:(H), +* .s;Jr*, Pi

f ' ds*

k,, (*)' f* '6)Jr*, Pi

where Sl represents total liquid saturation. Note that these equations may be applied onlywhen the water saturation is at the irreducible level.

VI. TIMMERMAN, COREY, AND JOHNSON

Timmermanr8 suggests the following equations based on the water-oil drainage capillarypressure, for the calculation of low values of water-oil relative permeability.

( t 4 l

Corslut i l i t r ar\alunlllo

rt is fau

cquation

drarnaec

Pressuretri the ci

S . t r I

hart: tri I\3luralKr

o.

2l

Wetting-Phase Drainage Process:

k.o : S.

k.* : S*

Wetting-Phase Imbibition Process:

kro : So

Injection Curve

Injection Curve

Trap-Hysteresis Curve

Injection Curve

k.o : So (20)

Coreyre combined the work of Purcell3 and Burdiner3 into a form that has considerable

utility and is widely accepted for its simplicity. It requires limited input data (since residual

saturation is the only parameter needed to develop a set of relative permeability curves) and

it is fairly accurate for consolidated porous media with intergranular porosity. Corey's

equations are often used for calculation of relative permeability in reservoirs subject to a

drainage process or external gas drive. His method of calculation was derived from capillary

pressure concepts and the fact that for certain cases, l/P"2 is approximately a linear function

of the effective saturation over a considerable range of saturations; i.e. , llP"2 : C [(S" -

S".)/(1 - S",)] where C is a constant and S" is an oil saturation greater than S.,,. On the

basis of this observation and the findings of Burdiner3 concerning the nature of the tortuosity-

saturation function, the following expressions were derived:

fl'"H.1"LTFI

Injection Curve

Injection Curve

Injection Curve

lnjection Curve

( t7)

f[Hl"LrFl

( l 8 )gt c

b.

[l'"H 1"LTFj[[H]"Lrsl

( l e )

l - l t

J t : * ' . 1 . S , ,

lc ; . i hcn a

D l : . : : . r p lo t

ta . - . ' . , t l - S . , ,

JUr. t .ht)\\ S

i l r i : e . rn the

I 0: : : . i r (Ek

tpp. re J only

l g c . . r p r l l a r y

l - 5 )

r l 6 )(2r)

(22)

(23)\ o :k,o [ S '

- S ' * l o

L r - s * J

22 Relative Permeability of Petroleum Reservoirs

where S'- is the total liquid saturation and equal to (l - Sr); S- is the lowest oil saturation(fraction) at which the gas phase is discontinuous; and Sr* is the residual liquid saturationexpressed as a fraction.

Corey and Rathjens2o studied the effect of permeability variation in porous media on thevalue of the S- factor in Corey's equations. They confirmed that S,,, is essentially equal tounity for uniform and isotropic porous media; however, values of S,, were found to begreater than unity when there was stratification perpendicular to the direction of flow andless than unity in the presence of stratification parallel to the direction of flow. They alsoconcluded that oil relative permeabilities were less sensitive to stratification than the gasrelative permeabil it ies.

The gas-oil relative permeability equation is often used for testing, extrapolation, andsmoothing experimental data. It is also a convenient expression that may be used in computersimulation of reservoir performance.

Corey's gas-oil relative permeability ratio equation can be solved if only two points onthe k,r/k,., vs. S* curve are available. However, the algebraic solution of the k,g/k.., equationwhen two points are available is very tedious and the graphical solution that Corey offersin his original paper requires lengthy graphical construction as well as numerical computation.Johnson2r has offered a greatly simplified and useful method for determination of Corey'sconstant.

Johnson constructed three plots by assuming values of Sr*, S,,, and k.s/k.., by calculatingthe gas saturation, (1 - S,_), using Corey's equations. The calculation was carried out forvarious Sr* and S- combinations and for k.s/k,o values of l0 to 0.1, 1.0 to 0.01, and 0. Ito 0.001. Johnson's graphs may be used to plot a more complete k.g/k,,, curve based onlimited experimental data. The span of the experimental data determines which of the threefigures should be selected.

The suggested procedure for k.g/k., calculation, based on Corey's equation, is as follows:

l. Plot the experimental k.r/k," vs. S, on semilog paper with k,*/k,o on the logarithmicscale.

2. From the experimental data determine the gas saturation at k.r/k,o equal to 10.0 and0.1, 1.0 and 0.01, or 0.1 and 0.001. (The l is ted pairs of values correspond to Figures4,5, and 6 of Johnson's data, respectively, and the range of the experimental datadictates which figure is to be employed. Note that if the data do not span the entirepermeability ratio interval of 10.0 to 1.0, Figure 4 may not be employed first; insteadFigure 5 with the k,*/k.o interval of 1.0 to 0.01 or Figure 6 with the k.*/k,., interval of0.10 to 0.001 may be used f i rst . )Enter the appropriate Figure (4,5, or 6) using the gas saturations corresponding tothe pair of k.r/k.o values selected in step 2.Pick a unique S.* and S- at the intersection of the gas saturation values; interpolateif necessary.

5. Using these S.* and S- values and employing the two other figures of Johnson,determine two more gas saturation values and the k,*/k," ratio indicated on the axesof each figure.

6. Add these points to the experimental plot for obtaining the relative permeability ratioover the region of interest.

This procedure provides values of gas saturation at k.*/k.o ratios of 10.0, 1.0, 0.10, 0.01,and 0.001, which are sufficient to plot an expanded k.s/k.o curve.

It should be noted that if the data cover a wide range of permeability ratios, multipledeterminations of Sr* and S- can be made. If the calculated values differ from the exper-imental data, the discrepancy indicates that there is no single Corey curve which will fit all

t 5rq11-

rilustnl

( ' r tTr '

rrrrrahl3 .

4 .

FJ Ehc\

rk S-3tuJr

C;ttr

23

o

t l

I

o)

J

Io)

U)

l r

N :i n . . 'l n ' :f i : - '

;

lsi'

l : , "

tri

i . ' -

B;^

l . ' r . ( ) . 0 1 ,

i ' . ' : : u l t i p l e

ln : : J c\per-El'. ,,. I iit all

' . . , t \ \ \ :

. - : t r l i l c

, l n d

l : : u r c s

: - : . Ja ta

. i n t l r e

::.tc', itd' - : ' . l l o f

' . : : n g t O

' : 3 r l 3 1 s

l': fl r()Il ." r . ' t \ g s

: r ra t i o

20

S n , % k r g / k r o = 0 . 1 O

FIGURE 4. Corey equation constants.2l

the points; an average of the values for each constant should yield a better curve fit. Figure7 illustrates the graphical technique of Johnson.

Corey's equations for drainage oil and gas relative permeabilities and the gas-oil relativepermeability ratio in the simplest form are as follows:

and they are related throughI

k.o : (s".)o

k . r : ( l - S " . ) 2 x ( l - S 3 " )

k.. k.

(S*X- -

( l - S;y : I

(24)

(2s)

where So. represents the lowest oil saturation at which the gas tortuosity is infinite; S". isdefined as (S" - S",)/(l - S".).

Corey's equations in the presence of irreducible water saturation take the following form:

k,o : (s*)o

(26)

(27)

Relative Permeabilitv of Petroleum Reservoirs

q

t lo

-t

o)-g

(U

aQ

o)U)

S g , % , a t k r n / k r o = 0 . 0 1

FIGURE 5. Corey equation constants.2t

f S 1 2k,n : I t -

;---""-^ | " fl - S*)2' L ) - - ) * i J

where S- is a constant related to ( I - S*") and as a first approximation S- can be assumedto be unity. This is a good approximation, since S*" is less than 5Vo inrocks with intergranularporosity. In these equations, S* : S"/(l - S*,) and S" is the oil saturation represented asa fraction of the pore volume of the rock; S*, is the irreducible water saturation, also expressedas a fraction of the pore volume.

These equations are linked by the relationship

+ +;-q*: | (zs)(s*), ( l - s*),

Corey et al. plotted several hundred capillary pressure-saturation curves for consolidatedrocks and only a few of them met the linear relationship requirement. However, comparisonof Corey's predicted relative permeabilities with experimental values for a large number ofsamples showed close agreement, indicating that Corey's predicted relative permeabilitiesare not very sensitive to the shape of the capillary pressure curves.

Equation 24 may be employed to calculate water relative permeability if the oil saturationand the residual oil saturation are replaced by water saturation and irreducible water satu-

(28)

nrll(rtl. n

t(rr\trnaloi the p

drrtntrutl

lrtrn\ $ tl

scrB pmehqrlutcl

Cael;.ffstrlXt

hrr result

.trr-ludc

tV Ctlt1R{

-{pflxnE:Ilut3t

\ rnrlirr{rc5

n crl-gr

t h rrt

) <

0 . 9

o

o

o

J

o)

.:<

(U

Ae

oU)

(28)

crr, lrc assumed

dtl: :ntcrgranular

n r . l rc \ented as

l . ; . . , ' C r p r e S S e d

(2e)

f t ' r ; , 'nsol idated

!1c:. i trmparison

lar-ic number of

E ;\ .-rnteabi l i t ies

tx- ,r t l saturat ion

ihic u ater satu-

5 l o

S g , % , a t k r g / k r o o f O . O 0 1

FIGURE 6. Corey equation constants.2l

ration, respectively. The exponent of Corey's water relative permeability equation is ap-

proximately four for consolidated rocks, but depends somewhat on the size and arrangement

of the pores. The exponent has a value of three for rocks with perfectly uniform pore size

distribution. Several other authors have proposed similar water relative permeability equa-

tions with different exponents for other types of porous media. Values of 3.022 and 3.521

were proposed for unconsolidated sands with a single grain structure which may not be

absolutely uniform in pore size but should have a nalrow range of pore sizes.

Corey compared the calculated values of oil and gas relative permeabilities for poorly

consolidated sands with laboratory-measured values and obtained good results. However,

his results showed some deviation at low gas saturations for consolidated sandstone. Corey

concluded that the equations are not valid when stratification, solution channels, fractures,

or extensive consolidation is present.Application of Corey's equation permits oil relative permeability to be calculated from

measurements of gas relative permeability. Since k., measurements are easily made while

k.o measurements are made with difficulty, Corey's equation is quite useful. The procedure

involves the measurement of gas relative permeability at several values of gas saturation in

an oil-gas system and then performing the following steps:

1 . Prepareanaccura tep lo to f the func t ionk . r : ( l - S" " )2 x ( l - S . " ' )byassuming

arbitrary values of So., the effective saturation, which is defined as

o n<perj-nental Data of Vlelge

Xustirated Data points


- - o o.lo o.20 0.30 0.40 0.50 0.60 0.70

S g

FIGURE 7. Example of the use of the Corey equations.rl

Prepare a tabulation of k., vs. So" for values of k,, ranging from 0.001 to 0.99 instepwise fashion.Determine values of So" for each experimental value of k., by using the above-describedtabulation.Plot these values of So. against the values of S" coffesponding to the k., values onrectangular coordinate paper. The plot should be a straight line between 50 and 807ooil saturation.Construct a straight line through the points in this range and extrapolate to S.* : 0.The value of S" at this point corresponds to S".. (See Figure 8.)Employ Equation 24, k,o : (So")o and the value of S.,. obtained in the previous stepto calculate k,o values for assumed values of S".

Corey-type equations for drainage gas-oil relative permeability (gas drive) in the presenceof connate water saturation have been suggested as follows:

ol<

o).:.

2 .

3 .

4 .

5 .

6 .

k." : ( l - S)u

k., s3(2 - s)where S represents (Sr)/(l - S*,).

Corey's equations for the drainage cycle in water-wetformations are as follows:

(30)

( 3 1 )

sandstones as well as carbonate

(32)

\\ 3l

ttfnF;

n trre

:r'tr-l l(r

Ttrf.

Cr{UJllt

rr{Tl$l

'trLrtn

Brtr

rr l,ll ttr

i t { h t u t

, l - l - s * 1 rK - . - : l - l

L l - S * , 1

S o rt

o b 20 40 60So ,

o /o

FIGURE 8. Determination of residual oil saturation based on effectiveoil saturation.

k . * : ( S * * ) o

VII. WAHL. TORCASO. AND WYLLIE

60

50

27

(33)

(341

aoo

@

ro

)70

0 t : ' 9 9 i n

O\ ( . : . . . r tbcd

. : . L IC\ ( )n

. ,nJ t l07c

\ . - 0 .

, ' J r \ t e P

I th.r ' l rc:c'nCe

(-10)

( 3 1 )

a- . . r rhonate

roo80

\r :

Wahl et al.2a suggested the use of the following equation for drainage gas-oil relative

permeability ratios based on field measurements of sandstone reservoirs:

* : +(o.o43s + o.4ss6 .l,)

(32)

where rf represents ( I - S*. - S. - Sg.)/(S,, - C); Sr. is the critical gas saturation as a

fraction of total pore space; and C is a constant equal to 0.25.

Torcaso and Wylliett compared gas-oil relative permeability ratios calculated by Corey's

equation with those obtained from Wahl et al. for various irreducible water saturations. This

comparison suggested that Corey's work was theoretically sound, since it agreed with values

obtained from field measurements by Wahl et al. (see Figure 9).t^

VIII. BROOKS AND COREY

Brooks and Corey26'27 modified Corey's original drainage capillary pressure-saturation

relationship and combined the modified equation with Burdine's equation to develop the

following expression that predicts drainage relative permeability for any pore size distribution:

9 y y ; = o ' 3


o. lo.o.03

o.olo.oo5

o.oot Lo 20 40 60

FIGURE 9' Comparison of relative permeability calculations at three irreducible watersaturations.25

for P. i Po

o.g

o,J

roo5030

to53

too.5o.3

trt e hrg

l , S .

Ttbcr alrr grr

r r th rx

itrrt it{

crFsll

\t-r lh

.rl rclrlr

IltrrtttD I

r l &r r t

k ll f . r

lrel

rb'rq

where tr, and Po are constants characteristic of the media; ), isdistribution of the media, and Po is a measure of maximum porecapillary pressure at which a continuous nonwetting phase exists).two-phase relative permeabilities are given by

and

s** : (l)^

2 + l A

k - / S * l r" r w t \ v w ,

(*, =la-- -

5 * =s :::roo80

, . lJ

(3s)

a measure of pore sizesize (minimum drainageUsing this relationship,

(36)

k .n* , : ( l - ' t * * ) ' [ t

- (S** ) (37)

where k.*, and k-*, are wetting and nonwetting phase relative permeabilities respectively.The values of tr and Po are obtained by plotting (S* - S*,)/(l - S*,) vs. capillary pr.rrur.

29

on a log-log scale and establishing a straight line with L as the slope and Po as the interceptat (S* - S* i ) / ( l - S*,) : 1.

These equations reduce to Equations 24 and 25 for \ : 2. Theoretically \ may have anyvalue greater than zero, being large for media with relative uniformity and small for mediawith wide pore size variation. The commonly encountered range for L is between two andfour for various sandstones.2t Talash28 obtained similar equations with somewhat differentexponents.

IX. WYLLIE, GARDNER, AND TORCASO

Wyllie and GardnerrT have presented the following expressions for the drainage water-oil relative permeability:

k, . : (H) 'H'

Jr*, ds*/P.'�

k,.:(5;) '$i11/' or*,1r";'

(38)

More general expressions for any wetting and nonwetting relative permeability may bewritten where

(3e)

(40)

(41)

(42)

kr*r

k.n*,

S* i

S L

Relative permeability to wetting phase (k,* and k,").Nonwetting phase relative permeability (k,r).

Irreducible water saturation.Total liquid saturation : (l - Sr).

r 35 )

I , ' l l t r C s i Z e

l ru : : . J ra inage

] ( ' - ' . - t ( r t rnshiP,

( 36) Wyllie and Gardner have also suggested the following equation for relative permeabilityto water or oil when one relative permeability is available:

k.* : (S**) ' - k,o (S**/(1 - S**)) '

( 37)

1- .J* -u t ive ly .

[an pressure

where S**, which is defined as (S* - S*,)/(1 - S*,), is the mobile wetting-phase saturationin a water-wet system.

Based on the linear relation between l/P"2 and S"/(l - S*,), they obtained a drainage waterrelative permeability equation for water-wet rocks with intergranular porosity as follows:


(43)

Togpaso and Wyllie2s suggested the following equation for calculation of gas-oilrelative permeability of water-wet sandstone, where l/P.2 is approximately a linear functionof effective saturation. Their derivation was based on the relation developed by Corey:

\ = :k.,,

( l - s*) , ( l - s* , )(44)

(s*)o

where S* represents effective oil saturation and is equal to S.,/(l - S*,). Obviously, a reliablevalue of irreducible water saturation, S*r, needs to be known to calculate the gas-oil relativepermeability ratio.

X. LAND, WYLLIE, ROSE, PIRSON, AND BOATMAN

Land2e reported that an appreciable adjustment of experimental parameters was requiredto avoid a discrepancy between experimental and calculated two-phase relative permeabil-ities. A large number of the relative permeability prediction methods are based on derivationof pore size distribution factors from the saturation and drainage capillary pressure rela-tionship. Some authors3o believe that the employment of capillary pressure relationships forthe prediction of relative permeability is not advisable, since capillary pressure is derivedfrom experiments performed under static conditions, whereas relative permeability is adynamic phenomenon. McCaffery3r in his thesis argues that the surface or capillary forcesare orders of magnitude larger than forces arising from the fluid flow and thus, predominatein controlling the microscopic distribution of the fluid phases in many oil reservoir situations.Brown's32 results from the measurement of capillary pressure under static and dynamicconditions support McCaffery's argument.

Several relative permeability prediction methods which are based on drainage capillarypressure curves assume that pore size distribution can be derived from these curves. Theseproposed models can only be applied when a strong wetting preference is known to exist.

Additionally, relative permeability calculations from capillary pressure data are developedfor a capillary drainage situation where a nonwetting phase, such as gas, displaces a wettingphase (oil in a gas-oil system, or water in a gas-water system). They are developed primarilyfor gas-oil or gas-condensate relative permeability calculations; however, water-oil relativepermeability can be calculated with a lesser certainty.

Wyllie in Frick's Petroleum Production Handbook33 suggested simple empirical gas-oiland water-oil relative permeability equations for drainage in consolidated and unconsolidatedsands as well as oolitic limestone rocks. These equations are presented in Tables 2 and3.

The oil-gas and water-oil relative permeability relations for various types of rocks presentedin Tables 2 and 3 may be used to produce k.g/k.o curves at various S*, when k., measurementsare unavailable.

It should be noted that the k,.,/k.* values obtained apply only if water is the wetting phaseand is decreasing from an initial value of unity by increasing the oil saturation. This iscontrary to what happens during natural water drive or waterflooding processes; however,Figures l0 through l4 also apply to preferentially oil-wet systems on the drainage cyclewith respect to oil if the curves were simply relabeled.

Rose6 developed a useful method of calculating a relative permeability relationship onthe basis of analysis of the physical interrelationship between the fluid flow phenomena inporous media and the static and residual saturation values. The equations for the wettingand nonwetting relative permeabilites are

k,* : (s**)o

rrblc

htr:

rEl - n5:I

r r rG

*::;!} f t l ,nr

Lf i

3 l

r J 3 )

n : : . r r -o i li nc . , : : . . ne t i t l n

b \ ( ' : e \ .

r _l_l )

n l . . . , : c l i ab le

P' rc la t i r e

{

fa i , . . rp r l la ryCu: ' , . ' . TheseD\a ' t : , . ' C f iS t .

lrc . : i \ . ' loped

I - i - . : $c t t i ng

|f*.*.: :.rrrnarily

tsr . rc ' lat ive

p r : . . . : l gas -o i l

t x - : . , t l i d a t e d

b l . ' . I and 3 .

E i . : ' : -c .ented

Dc'-: . . i rCi l9OtS

'Ec : : . n r phase

I t ; , , : : T h i s i s

I t ' . l l t t t t eVgf ,

h. t ; : : . , rc cy 'c le

i l . r : : , ' n . h i p s11

phr . : t , , l t tcna in

f : - . se t t i ng

Table 2OIL-GAS RELATIVE PERMEABILITIES (FOR

DRAINAGE CYCLE RELATIVE TO OIL)33

Type of formation k"o

Unconsolidated sand, well (S*)'

sorted

Unconsolidated sand, poorly (Sxlt :

sorted

Cemented sandstone, oolit ic (S*)'

l imestone, rocks with vugu-

lar porosity"

k.e

( l - 5 x ; r

( l - 5 x ; : ( l - 5 x ' s )

0 - s x ) , ( l - 5 x : 1

Note: In these relations the quantity Sx : S,,/(l - S*,).

Application to vugular rocks is possible only when the size of the

vugs is small by comparison with the size of the rock unit for which

the calculation is made. The unit should be at least a thousandfold

larger than a typical vug.

Table 3WATER.OIL RELATIVE PERMEABILITIES (FOR

DRAINAGE CYCLE RELATIVE TO WATER)33

Type of formation k"o

Unconsolidated sand, well (l - S**)'

sortedUnconsol idated sand, poor ly ( l - S**) ' ( l - S** ' ' ) (S**) t t

sorted

Cemented sandstone, ool i t ic ( l - S**;z ( l - 5"x: ; (S**)o

limestone

Note'. In these relations the quantity S** : (S* - S"i)/(l - S*,), where

S*, is the ineducible water saturation.

k * =l6s i (s*-s*_)t( l -s*-)

t2si(2 - 3s*.) + 3S*S*-(3S*. - 2) + S**(4 - 5S*,,)1'�

l653*,(5"*, - S"-)'( I -,lr* - S.-)

k.*

(s**)'

k - :

(4s)

(46)[253*,(2 -2rlt*- 3S.-) + 3S"*, S"-(3S n -2* 2,lr*) + S,-(l - r!*X4 -,lr* - 5S".)]'

where S* and Sn*, represent wetting and nonwetting saturations, respectively, expressed asfractions; S*- and S.- represent minimum wetting and nonwetting saturation values attainedunder dynamic flow conditions, expressed as fractions; they are the dynamic equivalents ofS*, and S". obtained from static tests. The symbol qr* represents an immobile wetting-phasesaturation expressed as a fraction. It is that part of the wetting-phase saturation which doesnot interfere with the nonwetting phase mobility and it is the maximum wetting-phasesaturation at which the nonwetting relative permeability is unity. Note that Equation 46reduces to Equation 45 for r.|l* : 0. The minimum wetting saturation, S**, depends on flowconditions and may be obtained by the Brownell and Katz3a relationship of S*- : (1/86.3)

[V(g o cos 0) dP/dx]-o264 where g is the acceleration due to gravity, o is the interfacialtension, 0 is the contact angle, k is the permeability, and dP/dx is the pressure gradient.

The principal disadvantage of Rose's method is that the residual saturation of both phasesmust be known fairlv accuratelv.


o 20 40 60 80 too

Qv r

L

FIGURE 10. Wyllie curves for water-wet cemented sandstones, oolit iclimestones, or vugular systems.rl

Pirson3s derived equations from petrophysical considerations for the wetting and non-wetting phase relative permeabilities in clean, water-wet, granular rocks for both drainageand imbibition processes. The water relative permeability for the imbibition cycle was givenAS

k.*, : (S**)"' (R.,/R,)3/2

later modified to

k,*t : (S**)t" (R"/R,)3/2

and

k.*, : (S**)t" Si

Water relative permeability for the drainage cycle was given by

k.* , : (S**) t" Si

(4e)

(s0)

oj

o).Y

r?El€

F ,.lu'l

5. rcl

::Ttrr

Ttr; r c l c

(((41)

(48)rtrt

T\

rF:

po..e.l€

33

ov

o)l<

h-i

Xlt:

rlc

. : : ' lJ non-

. l r l rnage. i . 1 . g i ven

(-17)

(48)

(s0)

i*Ylt t t Wyllie curves for poorly sorted water-wet unconsolidated

where R., represents electrical resistivity of the test core at l00%o brine saturation expressed

as ohm-meters; R, represents electrical resistivity of the test core expressed as ohm-meters;

S*, represents irreducible wetting-phase saturation; and S* represents water saturation as a

fraction of pore space.The nonwetting phase relative permeability in clean, water-wet rocks for the drainage

cycle was found to be

k**, : ( l - S**) [1 - S**r '4(R"/R,)r '4]2

or

k - * , : ( 1 - S * * ) ( l -S * * r / 4 S r /2 )2

which was later modified to

(s2)

k,n* , : ( l - S**Xl - S** t /4 Su2) t t2 (53)

The nonwetting phase relative permeability in an imbibition cycle given by

krn*, : [ t

@

st

S* - S*, l ' �l - s - , - s * J

(s 1)

(49)

(54)

where S** represents (S* -S*,)/(l - S*,) and S.*, represents the irreducible nonwettingphase saturation as a fraction of pore space. Pirson also derived equations for the wetting


and for the drainage cycle

Swi

20 40 60 80 loo

s,L

l - S . . - S * ,t',.)u - s:,.-sl,,.l'

I,OOO

roo

0.ol

o.ool

o.0ootoL

ro

oJ

o)j- o.l

FIGURE 12. Wyllie curves for well-sorted water-wet unconsolidatedcores. }

and nonwetting phase relative permeabilities in clean, oil-wet rocks for both drainase andimbibition processes:

kr., : (S.r")"' S: (5s1

where S.* is defined as (S" - S.,.)/( I - S".) and S.. represents irreducible oil saturation andis the equilvalent of of ( I - S*') for a clean, water-wet rock; S" represents total oil saturationobtained by differences from (l - S*).

The nonwetting phase relative permeability in clean, oil-wet rocks for the imbibition cyclewas found to be

So - So,(s6)

(s7)

ilktr !

:rfn3r d rrt&rI ?l4l :

qiilIlr

Fn.:ln rrrrt h'r

frk

lnSnr

be

- s ,

[ 'L

to be

l -

krr*, :

was found

krn-, : (

35

e-w

FIGURE 13. Wyl l ie curves. I

where S*, represents trapped-water saturation, which is determinable by Albert and Butault's

method.36 These investigators suggested that a capillary-pressure curve be obtained either

with a wetting fluid or with a nonwetting fluid such as mercury to obtain irreducible non-

wetting phase saturation. They also estimated that the irreducible nonwetting phase saturation

is two thirds of net pore volume made up of capillaries of radii smaller than the most common

capillary size, when the nonwetting phase displaces the wetting phase.

Pirson suggested a method to determine the in situ trapped nonwetting phase saturationby means of microresistivity logging devices, which respond to the flushed zone around a

well bore:

S n * r : 1 - ( 1 / 0 ) ( R - , / R * . , ) " t ( 5 8 )

where $ represents the porosity of the reservoir rock and R-r/R^. is the ratio of the mud-filtrate resistivity to flushed zone resistivity.

Boatman3T suggested water and gas relative permeability equations in terms of core pe-

trophysical properties obtained from laboratory data:

k,* : S**t'' (R"/R,)3'2 (se)

3.Y

ora

dr, and

t 5 -51

i l l c l - : i l ( ) n a n d

Ot . . . : i u ra t iOn

D r t . i : ' n c t c l e

( 56)

( 57 )

W e l l - - S o r t e d G r a r n s


3L

l<

oL

l<

(60)

n{

rhsnc

Th\r

rbrtr l

R*x(

tw!'rJ

rhcrflE

&lqrr

t $rr

ti-rF'ffirJ

& l,g*

I c";s

f u

1 .uFc

r frgr

C.ro

Fsc!lfr 1l

,mnlf, .,,:

* f u tHcr

rtnnt

3rF-il

* : s .

\\lm-* '

\\ - * '\ -\,L.

e-w

FIGURE 14. Wyllie curves for water-wet cemented sandstones, oolit ic l imestones, or vug-ular systems.33

20 40 60 80 roo

k , , : (1 - S* * t /4 Swt /2 ) t /2and

where

e * - S * - S * i\'w I - S*,,

Pirson et aI.38 proposed equations for oil and water relative permeabilities as follows:

k,* : (S**)"t (R"/R,)2 (61)

(60)

t l " l I t r * S l

( 6 1 )

and

k^, : (l - S*-)' (62)

where S*- represents (S* - S*,-)/(l - S*,., - S.,.); S** represents (S* - S*,*)/(l - S*,,,).Thornton5 proposed the following equation for wetting-phase relative permeability:

k.*, : Sl (PD/P.)2 (01)

where P"/P. represents the ratio of displacement pressure to drainage capillary pressure.Rose and Wyl l ieT'3e proposed a petrophysical equat ion for wett ing-phase relat ive

permeability:

k.*, : (Ir/2) (64)

where I represents resistivity index, R,/R".Jonesao proposed mathematical relationships for water-oil and water-gas relative perme-

abilities as function of S* and S*,, where S* may be determined from well logs, S*, maybe estimated from an S* - $ crossplot, and d may be determined from well logs:

k.* : (s**)'

k,-:[8H]'XI. KNOPP, HONARPOUR ET AL., AND HIRASAKI

(6s)

(66)

Knoppa' developed a correlation from 107 experimentally determined gas-oil relativepermeability ratios of Venezuelan core samples. The core samples were from consolidatedas well as poorly consolidated sandstone reservoirs of high porosity and permeability; theWelge gas-flood procedure was used for k.r/k.o determination.

A single correlation was established on the basis of the restored-state water saturation asa correlating parameter. The correlation is shown as a family of most probable k.s/k,., curvesin Figure 15.

Comparison of Knopp's correlation with experimental values is more promising when thegeometric mean of the suite of k,s/k,o curves for a given reservoir or sample group is comparedwith the corresponding most probable curves for the correlation. Knopp also suggested aprocedure for developing similar correlations for various other formations.

A comparison of Knopp's correlations with the correlation of Corey and Wahl et al. onthe basis of l5%o water saturation is shown in Figure 16.

Honarpour et al.a2 developed a set of empirical prediction equations for water-oil imbibitionrelative permeability and gas-oil drainage relative permeability from a large number ofexperimental data. Their results are presented in Tables 4 and 5. Symbols used in these twotables are defined as follows:

ku : air permeability, mdko : oil permeability, mdko(s*i) : oil permeability at irreducible water saturation, mdk., : gas relative permeability, oil and gas system, fractionk,e(so,): gas relative permeability at residual oil saturation, fractionk,o,* : oil relative permeability, water and oil system, fractionk* : water relative permeability, water and oil system, fractionk.o., : oil relative permeability, oil and gas system, fraction

Restored StateWater Saturation


roo.o

of-

.Y

o,f-

.Y

oL-

.y,- t

crb

.-

ooorb24 30 36 42 48 54 60

ssFIGURE 15. Knopp's correlation of most probable relative permeability ratios.or

Ss : gas saturation, fractionS*. : critical gas saturation, fractionS. : oil saturation, fractionS..* : residual oil saturation to gas, fractionS..* : residual oil saturation to water, fractionS* : water saturation, fractionS*, : irreducible water saturation, fraction0 : porosity, fraction

The data which were used as a basis for the study by Honarpour et al. were derived fr.moil and gas fields located in the continental U.S., Alaska, Canada, Libya, Iran, Argentina,and the United Arab Republic. Alt of the laboratory tests were made at room temperatureand atmospheric pressure' No attempt was made by the authors to group the data accordingto laboratory techniques used in measuring relative permeability, since this information wasnot available for many of the data sets. Each set of relative permeability data was classified

r s G

Sxrh r

cirri:

a5rF-!

x

@

ol-

.Y\

o)l-

.Y

39

o.ol tooo,o

o.ool roo.o

o.oool ro.o

o.ooool r.o36 42 48 54 60

s g 'FIGURE 16. Comparison of relative permeability correlations.*'

as either "carbonate" or "noncarbonate", but the information which was available was not

sufficient for more detailed lithologic characterization.In addition to the classification of data sets as "carbonate" or "noncarbonate", a further

classification was made on the basis of wettability. This rough classification was madeaccording to the following arbitrary criteria:

l. The rock was considered to be strongly water-wet if k,,, at high oil saturations in anoil-water system greatly exceeded k,o in a gas-oil system at the same saturations,provided k.* in a gas-oil system greatly exceeded k,* in an oil-water system at or nearresidual oil saturation after water-flooding.

2. The rock was considered to be oil-wet when k,o in the oil-water system was approx-imately equal to k,., in the gas-oil system, provided k,* in the gas-oil system was

approximately equal to k.* in the oil-water system.

t 8t2 3024

o/o

dc:: , ci 1'1nrn

l . \ : , l e n t i na ,

l lc:: f tratufel . r - : . . , , f d i ng

f ' I I . r l r ( )n wasra . - . . r . : i f i ed


Table 4EQUATIONS FOR THE PREDICTION OF RELATIVE PERMEABILITY IN

SANDSTONE AND CONGLOMERATE

_1. Te l

Atlerlinear nrneasunporosit''

Al l uequaliorrxks.

Thetestedin clox\a ere ul

ln u .relatrr e

ttsrTnesshrlied

Hrra'io l los:

utrrc

S .S .s-s .L.\ . .L\ -n

k... : 0.035388 (s* - s" ') - o.olo874x

( l - S * , - S , , , * )

f . , ' t " r - t " ' : '

. l " + o .sos56(S*) rn(S* - S* , ) (water -wer)L t t - S " , - S , , , " ) l

k . , . : r . 5 8 1 4 [ s * - ' s * '

l ' " ' - 0 . 5 8 6 1 7 ( s * - s " ' * ) *

l l - s " , I ( l - s * , - S , , , * )

(S" - S*,) - 1.24846( I - S*,) (S* - S*,) ( intermediately wet)

r / s ' \ - s , r .k , , , * : 0 . 7 6 0 6 7 1

\ l - s " / ' l [ , t " = - t " ' : l ' "

L ' - s, , ,*

I t ' - s*, - s, , ,*J+2.63180( l - S, , .* ) (S, , - S",*) (any wenabi l i ty)

, + ). L

t,,=- t,,,,: l . (any weuabil ity)kn' , : 0 '98372 ( l - t - , [ | - s- , - s, , r : ]

k.* : lo?2 (H) 'u,* , , , , . . , +2. i ig4*

\+ k'g's,,,r ' (anY wettabil itY)

k* : 0.002oszs \f

- 0.05r371 (s* - S*,) (i)"" (warer-wet)

k* : o 2eer. (H) - o tztot (ffi;)'-

( S * - S * . ) * 0 . 4 | 3 2 5 g ( * ) . ( i n t e r m e d i a t e | y w e t )

k.. * : 1.2624 (H:) (*) ' (any wettabl i ty)

k*,e : 0.s37s2 (jil '(ff-- ,_)' (any wettablity)

o '* : 'sossff ik 'gts, , ,* t + 8 'oo53x

r S . , - S , l r S . . . . r :-T-l= - o'o258eo {s. - S..)x

(#)'.(' - ' - t"

_ i:: - t")'

(t)"' (any wettab'ity)

(61)

(68 )

(69)

(70)

( 7 1 )

Table 5EQUATIONS FOR THE PREDICTION OF RELATIVE PERMEABILITY IN

LIMESTONE AND DOLOMITE

(72)

(13)

(74)

(75 )

A

.'1I

, i(76)

L IT \ I \

L- I1 \ I \

4r

3. The rock was considered to be of intermediate wettability when it did not clearly meet

either the water-wet or the oil-wet classification criteria.

After the data sets had been classified according to lithology and wettability, stepwise

linear regression analysis was employed to develop equations which would approximate the

measured relative permeabilities from such factors as fluid saturations, permeability, and

porosity.All water-oil system equations refer to displacement of oil by water and the oil-gas system

equations refer to drainage processes. All experimental data were measured in consolidated

rocks.The equations that were developed by Honarpour et al. have not yet been extensively

tested. However, most of the tests which have been made indicated that the equations are

in closer agreement with laboratory data than the predictions of publisfred correlations which

were used as a basis for comparison.In using empirical relationships such as those presented by Honarpour et al., any calculated

relative permeability which exceeds l�0 should be assumed equal to 1.0. If a relative

permeability value is known at any water saturation, the relative permeability curve may be

shifted to match the known data point.Hirasakia3 has suggested a relative permeability correlation for fractured reservoirs as

follows:

S * :Su - So.

l - s * - S o "

k,a : K,o (S*)"

k,, : k:" (l - S*)' (7e)

where

S* : Normalized saturation.Sd : Displacing phase saturation.So" : Immobile displacing phase saturation.So. : Residual oil saturation.k.a : Displacing phase relative permeability.ko.o : Displacing phase relative permeability at residual oil saturation.k,o : Relative permeability to oil.k".. : Relative permeability to oil at immobile displacing phase saturation.n : Exponent parameter for shape of relative permeability curves, said to be equal to

one in fractured reservoirs.

t

-..\ 'i i" ,tY REFERENCESl) rr",/J-,_trt

tdpullien, F. A. L., Ed., Porous Media: FluidTransport and Pore Stucture, Academic Press, New York,'r,l4lg.

2. Kozeny, J., Uber Kapillare Leitung des Wassersim Boden, Sitzungsber. Akad. Wiss. Wien. Math. Naturwiss.

KL., Abt . 2A, 136,2 ' l l , 1927.

3. Purcell, W. R., Capillary pressures - their measurement using mercury and the calculation of permeability

therefrom. Trans. AIME, 186.39, 1949.

(77)

(78)

r 7 3 )

r71 )

r 7 5 )

t 7 6 )


4. Rose, W. D. and Bruce, W. A., Evaluation of capillary character in petroleum reservoir rock, Trans..AIME, t86, 127, t949.

5. Thornton, o. F., valuation of relative permeability, Trans. AIME, 1g6,329, lg4g.6. Rose, W. D., Theoretical generalization leading to the evaluation of relative permeability,Trans. AIME,

186 , 1 i l , 1949 .7. Rose, W. and Wyllie, M. R. J., Theoretical description of wetting liquid relarive permeability , Trans.

AIME, 186,329, t949.8 . Ga tes ,J . I . andLe i t z ,W.J . ,Re la t i vepermeab i l i t i eso fCa l i f o rn iaco resby thecap i l l a ryp ressuremethod ,

paper presented at the API Meering, Los Angeles, california, May ll, 1950, 296.9' Rapoport, L. A. and Leas, W. J., Relative permeability to l iquid in l iquid-gas system, Trans. AIME,

1 9 2 , 9 3 , l 9 5 l .10. Wyllie' M. R. J., Interrelationship between wetting and non-wetting phase relative permeability, Trans.

A IME, 192 , 83 , 1981 .ll. Fatt, I. and Dykstra, H., Relative permeability studies, Trans. AIME, 192,249, lg5l.12. Wyllie' M. R. J. and Sprangler, M. B., Application of electrical resistivity measurements to problems

of fluid flow in porous media, Bull. AApG, 36, 359, 1952.13. Burdine, N. T., Relative permeability calculations from pore size distribution data, Trans. AIME, lgg,

7 t , 1 9 5 3 .14. Naar, J. and Henderson, J. H., An imbibition model - its application to flow behavior and the prediction

of o i l recovery, Trans. AIME, 222, 61, 1961.1 5 . N a a r , J . a n d W y g a l , R . J . , T h r e e - p h a s e i m b i b i t i o n r e l a t i v e p e r m e a b i l i t y , T r a n s . A I M E , 2 2 2 , 2 5 4 , 1 9 6 l .16. Land, C. S., Calculation of imbibition relative permeability for two- and three-phase flow from rock

properties, Soc. Pet. Eng. J., 6, 149, 1968.17. Wyllie' M. R. J. and Gardner, G. H. F., The generalized Kozeny-Carmen equation, its application to

problems of multi-phase flow in porous media, World Oit, 146, l2l, 1958.18. Timmerman, E. H. , Bd. , Prat ' t iL 'a l Resert ,o i r Engineer ing, Penwel l pubr. , r982, l0 l .19. Corey, A. T. , The interre lat ion between gas and oi l re lat ive permeabi l i t ies, Prod. Mon., 19,38, 1954.20. Corey, A. T. and Rathjens, C. H., Effect of stratif ication on relative permeability,Trons. AIME,20j,

3 5 8 , 1 9 5 6 .21. Johnson, C. E., Jr., Graphical determination of the constants in the Corey equation for gas-oil relative

permeabi l i ty rat io, J . Pet . Technol . , 10, l l l l , 1968.22. l rmay, S. , On the hydraul ic conduct iv i ty of unsaturated soi ls , Trans. AGU,35(3),463, 1954.23. Averganov, S. F., About Permeabilitl, ofSubsurfuc'e Soils in Case of Incomplete Saturation, Engineenng

Colfection, Vol. 7, 1950, cited by Polubarinova-Kochina, P, in The Theory of Ground Water Movement,Engl ish t ranslat ion by Dewiest , R. J. M., Pr inceton Univ. Press, Pr inceton. N.J. . t962.

24. Wahl, W. L., Mullins, L. D., and Elfrink, E. 8., Estimation of ultimate recovery from solution gasdrive reservoirs, Trctns. AIME, 213, 132, 1958.

25. Torcaso, M. A. and Wyllie, M. R. J., A comparison of calculated k.r/k,,, ratios with field data, J. pet.Technol., 6, 57, 1958.

26. Brooks, R. H. and Corey, A. T., Hydraulic Properties of Porous Media, Hydrology papers, No. 3,Colorado State University, Ft. Collins, Colo., 1964.

27. Brooks, R. H. and Corey, A. T., Properties of porous media affecting fluid flow, "/. Irrig. Drain. Div..6 . 6 t . 1 9 6 6 .

28. Talash, A. W., Experimental and calculated relative permeability data for systems containing tensionadditives, Paper 5810, Society of Petroleum Engineers, Dallas, Tx., 1976.

29. Land, C. S., Calculation of imbibition relative permeability for two- and three-phase flow from rockproperties, Soc. Pet. Eng. J., 6, 149, 1968.

30. Bear, J, Ed., Dynamics of Fluids in porous Media, Ersevier, Amsterdam, 1972.31. McCafferY, F. G., The Effect of Wenability of Relative Permeability and Imbibition in porous Media,

Ph.D. thesis, Universiry of Calgary, Alberta, Canada, 1973.32. Brown, H. w. , capi l lary pressure invest igat ions,Trans. AIME, 1g2,67, lg5l .33. Frick, T., Ed., Petroleum Production Handbook, Vol. 2, Society of Petroleum Engineers of AIME, Dallas,

Tx . , 1962 .25 .34. Brownell, L. E. and Katz, D., Flow of fluids through porous media, Chem. Eng. prog., 43(ll), 603,

194'7.

Pirson, S. J., Ed., Oil Reservoir Engineering, McGraw Hill, New york, 195g.Albert, P. and Butault, L., Etude des Characteristiques Capillaries du Reservoir du Cap don par LaMethode Purcel l , Pet . Ann. Combus. L iq. ,1(8) ,250, 1952.

37. Boatman, E. M., An Experimental Investigation of Some Relative Permeability-Relative ConductivityRelationships, M.S. thesis, University of Texas, Austin, 1961.

38. Pirson, S. J., Boatman, E. M., and Nettle, R. L., Prediction of relative permeability characteristics ofintergranular reservoir rocks from electrical resistivity measurements,Trans. AIME, Z3l,564. 1964.

-19 l1'rllic.

Phr '1 t - '

ro. Jones.11. Knopp

I l l r . r{1 Honerl

rc'lattr c-l-r. Hinsrl

Ga.. P."ll liopfif

t-nc l{5 tukha

pha.c r

35.36.

R.

l:

: _ -f

t -

)r

f - . .

h - . . : '

' P::.. -

I P

J \ . \ : l

I-rrt n s .

\1. \T E.

'., . 'l'runs.

. - ; r t t c thod .

.v.vE.

' . I runs .

: ' - , ' h l c r n s

t ' . 1 I I 9 t t .

. . ' : : J r c t i t l n

_ < : 1 9 6 L" , : t t rock

, , r t l ( ) n [ o

. . l e 5 - l: r / l_ 107.

r c l J l l v e

. . i : r t I l S i tS

. ' . J Pet .

. \ o 3 .

. , : D iv . ,

- tcnsion

- 'nr rock

. \ tedia,

. Da l l as ,

43

39. Wyllie, M. R. J. and Rose, W. D., Some theoretical considerations related to quantitative evaluation of

physical characteristics of reservoir rock from electrical log data, Trans. AIME, 189, 105, 1950.

40. Jones, M. A. , Waterf lood mobi l i ty contro l : a case history, J. Pet . Tec'hnol . , 9, l l5 l ,1966.

41. Knopp, C. R., Gas-oil relative permeability ratio correlation from laboratory data, J. Pet. Technol.,9,

l l t 1 , 1 9 6 5 .42. Honarpour, M. M., KoederitzrL. F., and Harvey, A. H., Empirical equations for estimating two-phase

relative permeability in consolidated rock, Trans. AIME, 2'73,2905, 1982.

43. Hirasaki, G. J., Estimation of Reservoir Parameters by History Matching Oil Displacement by Water or

Gas, Paper 4283, Society Petro leum Engineers, Dal las, Tex. , 1975.

44. Kopli.k, J. and Lasseter, T. J., Two-phase flow in random network models of porous media, Sot'. Pet.

Eng . J . , 25 , 89 , 1985 .

45. Fulcher, R. A., Ertekin, T., and Stahl, C. D., Effect of cappillary number and its constituents on two-

phase relative permeability curves, J. Pet. Technol., 2,249, 1985.

fr-,

F .

| - t -

l " : : n c c r i n g

lu' ' ,1 . t t t t t ' t t l ,

' t t

t : . I 1 t . 6 0 3 ,

L T :

u r {

'n Par La

:Juct iv i ty

: cn ' t i cs o fl e6J

h.:,.S+':

{\

Chapter 3

FACTORS AFFECTING TWO-PHASE RELATIVE PERMEABILITY

I. INTRODUCTION

The first published information concerning the simultaneous flow of multiple fluid phases

was probably by Hassler et al.r The term "relative permeability" had not yet been coined

and Hassler et al. studied only the flow characteristics of the gas phase as a function of

fluid saturation in consolidated rocks. The relative permeability concept was first postulated

by Muskat and Meres.2 Their work consisted of extending Darcy's law to two-phase systems.

For oil reservoirs, the relevant two-phase fluid combinations are water-oil and liquid-gas

(usually thought of as oil-gas). Gas-water relative permeability curves are used to describe

the performance of gas reservoirs and gas-liquid curves are used for condensate reservoirs.

II. TWO.PHASE RELATIVE PERMEABILITY CURVES

Water-oil relative permeability is usually plotted as a function of water saturation, as

shown by Figure l. At the irreducible water saturation (S*.), the water relative permeability

is zero and the oil relative permeability with respect to water is some value less than one.

At this point only oil can flow and the capability of the oil to flow is reduced by the presence

of connate water. The effect of connate water in reducing oil flow rate is illustrated sche-

matically by Figure 2.Note that data to the left of the irreducible water saturation are not useful for predicting

hydrocarbon reservoir performance, since water saturations less than S*" are not encountered.

As water saturation increases, the water relative permeability increases and the oil relative

permeability (with respect to water) decreases. A maximum water saturation is reached at

the residual oil saturation and the oil relative permeability becomes zero. Obviously, aquifer

conditions are represented by a relative permeability to water of unity, which occurs at a

water saturation of l00%o.Unfortunately, there is an alternate definition of relative permeability currently in use.

This terminology (illustrated by Figure 3) defines the oil relative permeability at irreducible

water saturation as having a value of one, and defines absolute permeability as the effective

permeability at irreducible water saturation. The effective permeabilities are identical with

both definitions of relative permeability and one set of values may be readily converted to

the other. This second definition of relative permeability (k,r) applies to both the oil and

water phases.These alternate or normalized values of relative permeability may be converted to standard

values by

k.srn : k,2 ku./kusrD

where

k.. : k"o at S*"

Also note that under this second definition of relative permeability, the water relative perme-

ability in an aquifer has a value greater than unity. Essentially, with this alternate definition,

relative permeability is normalized to the value at irreducible water saturation.

Gas-oil relative permeability and gas-liquid relative permeability are similar in concept

to water-oil relativE permeability. The preferred relative permeability values are those taken

with connate water present at the ineducible saturation value.

( l )

46 Relative Permeability of PetSoleum Reservoirs

f t r e t W a t

S w c S o r w

Svrr-+(-s o-

FIGURE l . Water-oi l re lat ive permeabi l i ty curves.

; W a t e r

FIGURE 2. Oil f low reduction due to the presence of water.

As free gas saturation increases, the oil relative permeability with respect to gas decreases;however, until the critical gas saturation (Sr") is reached, the gas relative permeability iszero. The critical gas saturation is the point at which the gas bubbles become large enoughto break through the oil and away from the rock surface. As gas saturation increases, thegas relative permeability increases and theoretically reaches a value of unity at l00%c gas.A gas-oil relative permeability curve is illustrated by Figure 4.

An experimental procedure to determine relative permeability in an unconsolidated sandwas first described by Wyckoff and Botset.3 Their work consisted of injecting a combinationof liquids and gases through a sample under steady-state conditions. Their results are shownin Figure 5, where k.. and k,, are relative permeability to oil and gas, respectively. Thefigure is typical of wetting- and nonwetting phase relative permeabilities, regardless ofwhether the system is oil- or water-wet.

Figure 5 shows differently shaped relative permeability curves for the two phases. Theoil relative permeability curve is concave upward while the gas relative permeability curvehas an "S" shape. This figure also shows that the oil relative permeability at the irreducible

II

II

II

Ie r I

//

\\ o i l

0

o

,'-t R o c k )

o i f -L( R o c k \-.*-t

(or critisaturaticrelativecurve ttupward

The :reducticrapid de

47

k r e l

S w

FIGURE 3. Normalized water-oil relative permeability curves

K rcl

oD . : . . : aJ \gs :

l ] t c ' . : ^ l l r t f iS

fg r l t t ough

l fe . : .Cs . the

1 [ r r r ' , g a s .

I t t i . , : rJ :and

i l l : : t ' . l t l a t i on

S r : i . h ( )wn

i t r r c l r . T h e

tgu rJ l css o f

phr.e. . Theb r l r t r c u r v e: rr. 'c,. lucible

Swc Sorg Sg.

O - S L 4 |

t + - S G

FIGURE 4. Gas-oil relative permeability curves'

(or critical) gas saturation is less than the gas relative permeability at the ineducible oil

saturation. Leverett,s worka shows that the same general observations apply to water-oil

relative permeability data. That is, in the presence of oil, the water relative permeability

curve takes on the shape of the wetting-phase relative permeability curve or is concave

upward.The shape of the oil relative permeability curve in Figure 5 indicates that, for a small

reduction in oil saturation, there is a sizeable decrease in relative permeability to oil' This

rapid decline is due to the occupation of larger pores or flow paths by the gas phase' Figure

\\\1 Gor

IIII

o i l I,,I

l .

Relative P ermeability of t etroleum Reservoirs

f t r e l

k t sl't:.:':)'/,.

A

Nouinterstimationother rof clarof conroccur i

Relathe varto wettnomentherefcf o r a limbibiRelatirpressul

SatuAt lovwettinlpendultransman appof indi

Abapath urphasethis rephasebetwetthe n'eof sattphaseration.

Fluia porodifferefor thithe noreserv(as thedown iphase

It hito thefuncti<as wellarge cindicat

S L

FIGURE 5. Relative permeability curves for an unconsolidated sancl.r

5 also indicates a steep increase in the gas relative permeability as the gas saturation increasesabove point "A", which is the saturation at which relative permeabilities to the oil and gasphases become equal. For this unconsolidated sand, the oil relative permeability at 59Vo orlsaturation is equal to gas relative permeability at 4l%o gas saturation. The gas relativepermeability reaches nearly l00%o at a gas saturation less than l007o, which 1n.un, that partof the interconnected pore space does not significantly contribute to the gas permeability ofthe porous medium. This figure also shows that the gas relative permeability remains at ierountil the gas saturation reaches the critical gas saturation, point "B". The gas phase is notmobile at a saturation less than the critical value, but this immobile gas impedes the flowof oil and reduces oil relative permeability. As oil saturation is increased from an initialvalue of zero, the oil relative permeability remains zero until the oil forms a continu.usphase at the critical oil saturation, which is represented as point C in Figure 5. In a solution-gas-drive reservoir, often the water saturation is small and immobile. Therefore, relativepermeability values are frequently plotted against the liquid saturation rather than the wettingsaturation. Under such a condition, point "C" is the summation of the irreducible watersaturation and the residual oil saturation, as previously indicated in Figure 4.

The sum of the relative permeabilities for all phases is almost always less than unitybecause of interference among phases sharing flow channels. There are a number of reasonsfor this interference. One of these reasons is that part of the pore channels available forflow of a fluid may be reduced in size by the other fluids present in the rock. Another reasonis that immobilized droplets of one fluid may completely plug some constrictions in a porechannel through which another fluid would otherwise flow. Also, some pore channels maybecome effectively plugged by adverse capillary forces if the pressure gradient is too lowto push an interface through a constriction. A fourth reason is the trapping of a group ofglobules that are clustered together and cannot be moved, since the grain configuraiionallows fluid to flow around the trapped globules without developing a pressure gradientsufficient to move them. This is the phenomenon that has been referred to as the Jamineffect.

i th rn un i tyJ r r l t - C S s o D S

ua: l . rh lc ' for

lhcr rcason

f\ ur J pore

Bnncl \ may

I t . l r ro low

3 -rroup 6f

f, t l rr urat ion

fr ' gradient

I t hc Jam in

Nowak and Kruegers tested two cores in which the permeability to oil in the presence of

interstitial water was considerably greater than single-phase permeability to synthetic for-

mation water. Yuster6 and OdehT both found the same phenomenon based on the results of

other work. A possible explanation for the high permeability to oil is that the distribution

of clay varies within the rock and variations in water saturation cause variations in the area

of contact between water and clay minerals. Thus, increasing degrees of clay swelling may

occur at higher water saturation due to the hydration of larger amounts of clay minerals.

Relative permeability is dependent upon both the fluid saturation and the distribution of

the various fluids in the interstices of the porous network. This distribution is directly related

to wettability characteristics of the rock, which in turn give rise to capillary pressure phe-

nomena. It is well known that hysteresis exists in capillary pressure-saturation curves;

therefore, hysteresis in relative permeability-saturation curves can also be expected. Thus,

for a given wetting-phase saturation, the relative permeability measured in a rock that is

imbibing the wetting phase is not the same as that measured while the rock is draining.

Relative permeability values also may be functions of factors such as temperature, overburden

pressure, phase equil ibria,ro' etc.

III. EFFECTS OF SATURATION STATES

Saturation is a term used to describe the relative volume of fluids in a porous medium.

At low saturations of the fluid that preferentially tends to wet the grains of a rock, the

wetting phase forms doughnut-shaped rings around the grain contact points. These are called

pendular rings. The rings do not communicate with each other and pressure cannot be

transmitted from one pendular ring to another. Sometimes such a distribution may occupy

an appreciable fraction of the pore space. The amount depends upon the nature and shape

of individqal grains, distribution, as well as degree and type of cementation.

Above the critical wetting-phase saturation, the wetting phase is mobile through a tortuous

path under a pressure differential and as the wetting-phase saturation increases, the wetting-

phase relative permeability increases as well. The wetting-phase saturation distribution in

this region is called funicular and up to a point, the relative permeability to the wetting

phase is less than the relative permeability to the nonwetting phase due to the adhesion force

between the solid surface and wetting fluid, and the greater tortuosity of the flow path for

the wetting phase. The nonwetting phase moves through the larger pores within this range

of saturation, but as the saturation of the wetting phase further increases, the nonwetting

phase breaks down and forms a discontinuous phase at the critical nonwetting phase satu-

ration. This is called an insular state of nonwetting-phase saturation.

Fluid flow studies have shown that when immiscible fluids flow simultaneously through

a porous medium, each fluid follows its own flow path. This flow network changes for

different ranges of saturation and as the nonwetting phase saturation reduces, the network

for this phase breaks down and becomes discontinuous; the remaining stationary islands of

the nonwetting phase canrnt be displaced at pressure gradients encountered in hydrocarbon

reservoirs. This condition is refened to as a residual nonwetting phase saturation. Similarly,

as the wetting phase saturation decreases, the network through which this phase flows breaks

down and becomes discontinuous and immobile. This is referred to as an ineducible wetting-

phase saturation.It has been showns-rr that for strongly water-wet unconsolidated sands the permeability

to the wetting phase is dependent solely upon its own saturation, (i.e., a plot of k.* as a

function of S* has the same shape regardless of whether or not the pore space contains gas

as well as oil). However, in the petroleum related literature, some small'2''3 and some quite

large deviations are seen from these findings for consolidated rocks. Some publicationsr4'15

indicate that the nonwetting phase relative permeability depends on the wetting as well as

a v e r a g e

m i n i m u m

50

I

Relative Permeability of P,etroleum Reservoirs

0 . 5 1 . 0

S ,L

FIGURE 6. Relat ive permeabi l i ty rat ios for sands and sandstones.rs

the nonwetting phase saturation for strongly water-wet systems. In preferentially oil-wetsystems, the oil phase relative permeability is found to be strictly a function of oil saturation,r6while in water-wet rocks, the oil phase relative permeability is found to depend on bothwater and oil saturation. Donaldson and Dean'7 have pointed out that under two-phase flow,relative permeability to water was increased when oil, rather than gas was the nonaqueousphase, indicating that water relative permeability is not solely a function of water saturation.

IV. EFFECTS OF ROCK PROPERTIES

Relative permeability-saturation relations are not identical for all reservoir rocks, but mayvary from formation to formation and from one portion to another of a heterogeneousformation.

Arps and Roberts'8 have presented plots of gas-oil relative permeability ratios for 16consolidated sandstones and 25 dolomites, cherts, and limestones, all with l57o connatewater saturation. These plots are presented as Figures 6 and 7. The maximum curve inFigure 6 seems to be typical of unconsolidated sandstone, while the minimum curve appearsto be more representative of highly cemented sandstones. The average curve can be con-sidered typical of the average consolidated sandstone. The minimum curve in Figure 7,which seems to be the steepest and most unfavorable, is from a fractured chert core; at theother end of the range, no well-defined maximum case is apparent. Curve #23, adaptedfrom Bulnes and Fitting's workre representing26 samples of west Texas Permian dolomite,appears to be the best maximum curve. The curve selected as "average" on Figure 7 appearsto be typical of vugular limestones.

Bulnes and Fitting as well as Stone2o have shown that the fluid flow behavior in uniform-porosity carbonate samples is similar to fluid flow behavior in consolidated sandstones, butthe difference becomes pronounced as the rock heterogeneity increases.

oJ

o)- Y 1

.9(uTE

- l

-o(uoEoo.o. = - 0 1(uo(r

Variosandstorand thelsaturatirabil it l ' rwetting-qualitatiunconscand coneffect oin degresystem.is wider

Corerpermeatin anisosaturatirpendicuwater-olarTange(ogeneitr

Lever

5l

1 0

o-g

o). Y l

o

(!

tr

.:I

-o(uoEq,

o-o

. : . 0 1

(!

otr

. o o 1

0 . 5

s,L

FIGURE 7. Relat ive permeabi l i ty rat ios for l imestones. dolomites, andcherts.r*

Various workss''e'2r have shown that the gas-oil relative permeability of consolidatedsandstone is qualitatively similar to the gas-oil relative permeability of unconsolidated sandand there is a very close coffespondence of the two relative permeabilities to oil at high oilsaturation. It has been found that for consolidated sand, the wetting-phase relative perme-ability drops sharply and the nonwetting phase relative permeability rises steeply as thewetting-phase saturation decreases. However, Naar et aI.22 have shown that there are bothqualitative and quantitative differences between relative permeability of consolidated andunconsolidated sands. Owens and Archerrr indicated that packing as modified by cementationand consolidation affects the equilibrium saturation to the wetting phase but has a negligibleeffect on the equilibrium saturation of the nonwetting phase. Nind23 stated that an increasein degree of consolidation increases the nonwetting phase relative permeability in a gas-oilsystem. Several investigators have noted that the saturation range for a mobile fluid phaseis wider in unconsolidated rock than in consolidated rock.

Corey and Rathjens2a studied the effect of rock heterogeneity on drainage gas-oil relativepermeability. They investigated the flow parallel and perpendicular to obvious stratificationin anisotropic Berea sandstone cores and concluded that the relative permeability at a givensaturation for flow parallel to bedding was greater than the analogous value for flow per-pendicular to the bedding plane, as shown in Figures 8 and 9. Huppler2s found that thewater-oil relative permeability of composite core changes appreciably when the sections arearranged in different orders. Johnson and Sweeney'o also studied the effect of rock heter-ogeneity on the gas-oil relative permeability ratio.

Leveretta found a small but systematic change in the position of the relative permeability-

1 . 0

i a . . . , ' r l - r re t

S : l . . I . r l 1 1 t J 1 . l t '

Rn.: , ' l t bt l th'ph. : .c l low.

l l l \ : i . : i . l Ll t- 'ouS

f \ . : l ' . l f J t l O n .

L . . l L t t f f i oy

| f r ' i c n e o u s

l l r , ' . l t t r l 6

5( " . ( )nnate

ln r .u r re inln c .rppearscan hc con-n I r c u r e 7 ,C r r t C . a t t h e

J r. .rdaptedt l . i t r l1t11l1a,

rc rppears

rn unr lbrm-dr t , rnC : . bU t

m a x i m u m . \ a v e r a g e \

\ \ n r m u m

\ \\ \


FIGURE 8.sandstone. ra

coo

Relative permeability measurements

oJ

ol(

from an anlsotroprc

saturatloperimenrdistributsaturatio(sphericithe shapsystems.

[,evenwater mtis necessit can hunit volrwith largrations afluids. Tsaturatiohave larlleave linpermeab

Gorrinit is enciconcludenonwettiruniform 1the sizerelative Ihigher prefficientl

Botsetdepends rand Wylconsequeof pore s

0 1

so

FIGURE 9. Relative permeability measurements from a Berea sandstone.2a

O - k r o - p o r p o n d i c u l ! r t o b o d d i n O

O - k r o - p a r E l l o l t o b a d d l n g

O - t r g - p . r p o n d l c u l r . t o b . d d i n c

O - k r g - p ! . . l l o l t o b o d d l n g

53

T ime 1

T i m e 2

T ime 3

E otL

M W A T E R

I S A N D

FIGURE 10. The formation of residual oil by the blockingprocess.

saturation relationship due to the employment of different sizes of sand grains in his ex-periments. Botset2r confirmed Leverett's finding and concluded that the effect of grain sizedistribution was not negligible either on the relationship between relative permeability andsaturation or on the value of the equilibrium saturation. It was found that the shape"(sphericity), roundness" (angularity), and orientation2a of the grains tended to influence boththe shape of the relative permeability curve and the critical gas saturation value in gas-oilsystems.

Leveretta pointed out that the relative permeability of an unconsolidated sand to an oil-water mixture is related to the sand pore size distribution. Muskat et a1.27 suggested that itis necessary to know the pore geometry of a reservoir rock before fluid movement throughit can be analyzed. Morgan and Gordon2s found that pore geometry and surface area perunit volume influenced water-oil relative permeability curves. They have shown that rockswith large pores and correspondingly small surface areas have low irreducible water satu-rations and therefore have a relatively large amount of pore space available for the flow offluids. This condition allows high relative permeability end points to exist and allows a largesaturation change to occur during two-phase flow. Correspondingly, rocks with small poreshave larger surface areas per unit volume and they have irreducible water saturations thatleave little room for the flow of hydrocarbons. This condition creates a low initial oil relativepermeability as well as a limited saturation range for two-phase flow.

Gorring2e demonstrated that oil in a larger pore can be surrounded and blocked off whenit is encircled by smaller pores which imbibe the displacing water by capillary forces. Heconcluded that both pore size distribution and pore orientation have a direct effect onnonwetting residual equilibrium saturation, as shown by Figure l0; therefore, a perfectlyuniform packing of spheres should give a residual saturation near zero. Gorring also identifiedthe size of channels occupied by the nonwetting phase as an important factor influencingrelative permeability. Crowell et al.30 indicated that higher initial water saturation yields ahigher probability for the nonwetting phase to be in larger channels so-that it can b9 recoveredefficiently during wetting-phase imbibition.

Botset2' mentioned as early as 1939 that the relative perrneability-saturation relationdepends on the degree and the type of interconnections of the pores. Fatt,3r Dodd and Kiel,32and Wyllie33 also concluded that the relative permeability of porous media is a directconsequence of the network structure of the media. Pathak et al.3a concluded that the ratioof pore size to pore throat is a factor which controls the snapping-off of droplets of the


nonwetting phase, with a high ratio leading to a high trapped oil saturation. Other workershave investigated the possibility of describing porous media as a network of interconnectedpore bodies and pore throats.

Postdepositional alterations can form more than one type of reservoir rock from a singleoriginal rock type. Alteration may reduce pore sizes, thus causing higher irreducible watersaturation and a natrow range of saturation change during two-phase flow. The presence ofgrains such as feldspar, when partially dissolved, improves the reservoir rock quality byforming pores larger than the pores between grains not containing feldspar. This alterationcauses higher relative permeability values and a larger saturation range during two-ph4sgflow.tt Reference 35 describes alterations in pore geometry which can occur due to theintroduction of reactive fluids in the rock.

Land and Baptist36 indicated that when a reservoir sandstone contains montmorillonite ormixed-layer clay minerals containing expandable layers, the water sensitivity of the sandstoneis not necessarily a result of pore blockage due to the increased volume occupied by theswollen montmorillonite. Some sandstones containing trace amounts of clay minerals mayexhibit sensitivity to water resulting from dispersion and subsequent transportation of clayminerals to pore constrictions. Thus, permeability reduction may occur in formations thatdo not contain expandable clay minerals; however, all formations containing expandableclays are probably water-sensitive due to the ease of dispersion and expansion of this typeof clay. Permeability reduction in sands containing sodium clays is likely to be higher thanthe reduction in sands containing calcium clays.

Some rock properties that influence relative permeability variations are readily observablewith a binocular microscope or even more clearly under a scanning electron microscope.Therefore, microscopic core examination can be highly useful for evaluating relative perme-ability characteristics. Once the significant rock property variations have been identified, areservoir can be subdivided into appropriate reservoir rock types. Within each of suchreservoir rocks types, relative permeability characteristics are usually similar, varying onlyslightly for rather large changes in air permeability or median grain size.

V. DEFINITION AND CAUSES OF WETTABILITY

"Wettability" is a term used to describe the relative attraction of one fluid for a solid inthe presence of other immiscible fluids. It is the main factor responsible for the microscopicfluid distribution in porous media and it determines to a great extent the amount of residualoil saturation and the ability of a particular phase to flow. The relative affinity of a rock toa hydrocarbon in the presence of water is often described as "water-wet", "intermediate"

,or "oil-wet". Examples of formations with strongly water-wet, strongly oil-wet, and in-termediate wettability are the Spraberry formation in west Texas, the Black Bradford sandin Pennsylvania, and the Fairbank sand in south Texas, respectively.

Wettability may be represented by the contact angle formed among fluids and a flat solidsurface or the angle formed between the fluids' interface and a glass capillary tube, as shownby Figure I l. The angle is measured through the denser fluid.

The wettability of a porous medium is determined by a combination of all surface forces.A sketch is shown in Figure 12, wherein two liquids, oil and water, are in contact with asolid. The force exerted by water to spread laterally and displace oil (interfacial tensionbetween water and oil) is opposed by the resultant of the solid and liquid forces (solid-oiland solid-water interfacial tensions). This difference in opposing forces is called the adhesiontension:

A, : o.o - or* : o*o cos 0*o Q)

This relationship is referred to as the Young-Dupre equation, where A, is the adhesion

tension.tensionsmeasure(

A posisurface i:angle is

A negrthe solidsuring o,evaluateto a surfi

Underthe polarof surfac

Stegenmoleculafor the n

hcr i . ()rkers

t t . , r t l t l c -Ct€d

f )nr . r . ing le

P. t ' . lc \ \ ater

p r l . eoC€ O f

I . 1 . r . r l r t r by

i . . t l t c r a t i o n

g : , r , , -phase

f \ : . . i to thg

f r r . . ' , r t t tc or

l a . . : : : J r t t l nep : ; . : r r t he

I t l l ; : .r , . t ' l ' l i i ) '

l r ' : ' l c l a l '

!r- : : ' l t . that

c r : ' . : i t J a b l e

t t l I : 1 . : t ) P €

r h . - : . . ' r than

y , , : . c r r ab lelT l l . r r r rCOP9.

l l t r c P. 'a*a-

ide : r t r t ' i c ' d . a

l , i : , , l rUCh

I a : - , . : . t ( ) n l y

fx . , . , r l id intT l : . : , . ' tCOp iC

| , ' : : c . t dua l

o : . : r (Xk tO

! r : : : l . i ra te" ,

l c : . t nd i n -

fa":: , ' r- .1 sand

d . : : . . r t sOl id

be . .r . .hown

ItJ. e l()rceS.

f i l . : . t $ i t h a

X i . : , tc 'ns ion

f , r . t r l id -o i l

thc .tJhesion

(2)

Jrt' .'Jhesion

55

O I L - W E T

g > g o o

I N T E R M E D I A T E

g = 9 O o

W A T E R - W E T

e < g o o

FIGURE I L Wettability conditions on flat surfaces and in capillary tubes._w;,='-Kfig==

, l I \Ttrp Vian of a Drop of VJateron a Solid Surface in thePresence of Oil

Ttrree Dirrensional Sdrernatic View

FIGURE 12. Forces at a water-oil-solid interface.

tension; oso, o,*, and o*o, respectively, are solid-oil, solid-water, and water-oil interfacialtensions (usually measured in dyne/cm); 0*" is the contact angle between water and oilmeasured through the denser liquid phase (usually water).

A positive value of adhesion tension means the contact angle is less than 90' and the solidsurface is preferentially water-wet . A zero value of adhesion tension indicates that the contactangle is equal to 90'; this is intermediate wettability.

A negative value of adhesion tension means the contact angle is greater than 90' and thatthe solid surface is preferentially oil wet. There is no practical laboratory method for mea-suring trso or o.*. However, o*o and cos 0 are measurable quantities which can be used toevaluate the wettability of a solid surface. A fluid is referred to as wetting or nonwettingto a surface depending on whether the contact angle is less than or greater than 90".

Understanding the causes of wettability requires a study of the chemistry of the fluids,the polarity and molecular weight of reservoir hydrocarbon compounds, and the occurrenceof surface chemical processes at the solid-fluid interfaces.

Stegemeier and Jensen3T experimentally found that the contact angles vary directly withmolecular weight for liquids with similar chemical structures. Figure l3 shows this variationfor the normal paraffin series compounds.

n - c t 6n-c l 4

n-c l 2

n - C ^t'

n-c6

I

I


5 0

1 5 0 2 0 0

M O L E C U L A R W E I G H T

FIGURE 13. Contact angle as a funct ion of molecular weight . r i

fsooctanefsooctane + Isoqrinoline Naphthenic

5.7E Isoquinoline

Calcite Sr-rrface

FIGURE 14. Interfacial contact angles.38

Benner and Bartell38 examined various multi-liquid systems in contact with silica andcalcite surfaces. Figure l4 illustrates some of the findings of this study. It was reported bythese investigators that when water and iso-octane are used, the silica and calcite surfacesare preferentially wet by water; but when water and naphthenic acid are used, water wetthe silica but oil wet the calcite surface. The experiment of Benner and Bartell illustratedthe effects of chemical as well as fluid composition of phases on wettability of a porousmedium. Contact angles as low as 30o and as high as l58o were observed when variouschemicals were employed in the study.

Salathiel3e discovered that the wettability of mineral surfaces may be altered not only byadsorbed monolayers of surface-active polar compounds, but also by much thicker layersof deposited organic materials. Several other workers have reported the formation of stablefilms on solid surfaces when the surfaces stand in contact with certain crude oils. Reisbergand Doshelo described the deposition on glass or quartz surfaces of highly stable andappreciably thick films of strongly oil-wet material from Ventura crude oil.

Early experimenters thought that all oil-bearing formations were strongly water-wet be-

4 0ooo

o)

r,u 30J

oz

F

o< z vFzo

cause althermonstudies :of crudereadilr rstudies I

One cor deposbeen elirgeochencrude oideasphalin solutirrock. Irquartz iuthe resul

DespirpreSente(particulaand theWoodbirreservoirMungan'formatiorwet wherfilm leftf i lm wil lcan thenSuggeSteleither oilwet reselwith airlrthe aquer

Authorreservoirfractionathe fractithe waterof capill:while the"spotted

dition inGimaludithat the rto the rorinsular npreferentisome pon

The erchemistrlcompounminerals

1 0

1 0 05 0

83o -

Silica Surface

dt l : . r l r ca and|s :s3rr1sd [yi l r - r lc .UrfaCeShJ. ri tter wetle l l r l lustrated

' , ' r r porousrr hcn various

)d n,' l only bythr; Icr layersllt, 'n of stableor l . Reisberglr . table and

S.r lcr-w€t be-

i7

cause an aqueous phase was always the fluid initially in contact with reservoir rock; fur-thermore, silica and carbonates are normally water-wet in their clean state. Subsequentstudies suggested that many oil reservoirs are not strongly water-wet and that the presenceof crude oils containing natural surface-active agents, such as asphaltic or wax type materialreadily adsorbable by solid-liquid interfaces, can render the solid surface oil-wet.ar Otherstudies provide evidence that reservoir rock wetting preference may cover a broad spectrum.

One criticism of the idea of reservoir rock surfaces becoming modified by the adsorptionor deposition of polar organic material from the oil phase is that such materials should havebeen eliminated during migration from the source rock to the reservoir. On the other hand,geochemists are now finding substantial evidence of various alteration processes which affectcrude oils subsequent to their accumulation in reservoirs. [n a discussion of natural gasdeasphalting, Evans et al.a2 suggested a reasonable hypothesis that the more gas a crude hasin solution the more of its heavy ends have come out of solution, plating out on the reservoirrock. It may be noted in this respect that Salathiel's strongly oil-wet film deposition onquartz and porous rocks from a mixture of evacuated crude oil and heptane was also probablythe result of a deasphalting process.

Despite uncertainty as to the causes of reservoir wettability, much evidence has beenpresented in recent years to suggest that many oil reservoirs are not strongly water-wet. Inparticular, there are the many brine/crude oil contact-angle measurements of Treiber et aI.62and the conclusions of Salathiel with regard to the apparent wetting characteristics of theWoodbine reservoir in the East Texas Field. Nuttinga3 as early as 1934 indicated that somereservoir rocks are oil-wet. Leach et al.aa described a reservoir believed to be oil-wet.Munganas studied fresh carefully preserved cores from a reservoir and concluded that theformation was oil-wet. Schmida6 has shown that strongly water-wet cores became less water-wet when equilibrated with some crudes. Kusakov et al.a1 studied the thickness of a waterfilm left on a quartz surface under crude oil drops and found that for two of the crudes, thefilm will rupture, bringing the crude oil into direct contact with the quartz surface; the surfacecan then be described as water-wet at some spots and oil-wet at others. Also, Craigassuggested that most formations are of intermediate wettability with no strong preference foreither oil or water. There is recent evidence to suggest that water may not always completelywet reservoir rock in gas-water flow following solvent injection. Soil scientists concernedwith airlwater/soil systems have reported situations in which there is incomplete wetting bythe aqueous phase.ae

Authors such as Holbrook and Bernard,so and Fatt and Klikoffs' assumed that wetting ofreservoir solids was heterogeneous rather than uniform. Holbrook and Bernard measuredfractional wettability by dye adsorption. Brown and Fatts2 defined fractional wettability asthe fraction of surface area in contact with water. This may not be a constant value sincethe water and oil saturations change as a reservoir is produced. Schmida6 showed by meansof capillary pressure-saturation data, that in preserved cores the fine pores were water-wetwhile the large pores were much less water-wet. This type of wetting is often referred to as"spotted", "dalmation", or "fractional". That heterogeneous wettability is a normal con-dition in oil sands has also been suggested by Salathiel,3e Iwankow,s3 Brown and Fatt,s2Gimaludinov,s4 and McGhee and Crocker.ss Several of these investigators have suggestedthat the wetting phase completely occupies the smaller pores of a reservoir rock in additionto the rock surface of the larger pores, while the nonwetting phase primarily occupies theinsular regions of the larger pores. Evidence suggests that some oil reservoirs are partlypreferentially water-wet and partly preferentially oil-wet. Such a condition could arise ifsome pores are lined with one type of mineral and other pores are lined with another mineral.

The existence of different minerals in porous media can create differences in surfacechemistry of the grains, so all grain surfaces do not have the same affinity for surface activecompounds. For instance, a tertiary sand reservoir in Alaska contains quartz and sideriteminerals which are strongly water-wet and calcite which is strongly oil-wet. The overall

58 R e lat iv e P e rme ab i I i ty of Pe tro le um Re'rervoi rs

necedi-ng

l^laterdi spLaced

by oi1

|st t i .Advancmg

I

II

t

FIGURE 15. Advancing and receding contact angles in capillary tubes.

rock system is water-wet, probably due to the presence of quartz and siderite surfaces in

the main flow channels. The presence of anhydrite or gypsum in the flow channels of some

carbonate rock may alter its wettability. These minerals are found to create a strongly water-

wet system, while many carbonate rocks are probably oil-wet under reservoir conditions'

Heavy metal sulfides are known to render a surface oil-wet when they are present in the

flow channels of Porous media'

Wagner and Leachs6 stated that in some oil reservoirs the rock surface is covered by a

firmly attached bituminous or other organic coating. Such surfaces would be preferentially

oil-wet in the presence of oil and water, regardless of oil and water composition. Boneau

and Clampitt.tt- reported that the oil-wet character of the North Burbank reservoir is due to

a coating of chamosite clay which covers approximately 77o of the quartz surface'

VI. DETERMINATION OF WETTABILITY

The wettability of a rock can be either evaluated experimentally or estimated qualitatively'

There is no satisfactory method to determine in si/a reservoir wettability. However' labo-

ratory-measured wettability has been used to evaluate in situ wettability. Many of the widely

used experimental methods of wettability evaluation utilize either the reservoir rock or the

reservoir fluids, but not both. Therefore, a laboratory wettability evaluation should be related

to actual reservoir conditions using a great deal of caution.

A. Contact Angle MethodThe contact angle method is used by a number of laboratories; the technique has received

considerable attention in the literature as a quantitative method of wettability measurement'

The method consists of measuring the contact angle 0 that a drop of pure liquid resting on

a smooth, flat, incompressible, nonporous, homogeneous solid forms when immersed in

another fluid. In most iractical situations, the contact angle formed between the solid surface

and the water-oil interface is found to exhibit two limiting values rather than a single

equilibrium value. The value of the contact angle when water is brought into contact with

oil on a solid surface previously in contact with oil is called the "advancing contact angle"'

The value of contact angle when oil is brought into contact with water on a solid surface

previously in contact with water is called the "receding contact angle".

Figure l5 shows a comparison of advancing and receding contact angles in a capillary

tube. The fact that advancing and receding contact angles are not equal is referred to as

contact angle hysteresis and it is usually attributed to surface heterogeniety and roughness,

as well as the presence of surface-active materialsss and rate of fluid movement. As the

surface roughness of a rock increases, the contact angle will further increase, provided the

contact angle measured on the smooth surface of the rock is above 90o; however, if the

contact angle measured on a smooth surface is less than 90o, the increase in surface roughness

would further decrease the angle. The smooth surface contact angle is found to increase in

advancing and decrease in receding, on the rough surface over most of the 0 to l80o contact

angle range.tnSurface-active materials in the fluids may cause adsorption processes at the solid-fluid

interfaces which give rise to appreciable contact angle hysteresis even with a smooth,

homogeneous solid. Motion of thl three-phase line of contact increases contact angle hys-

teresis as the rate of movement increases'

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FIGURE 16. Schematrc measurement of contact angles.5oB .ur l rces in

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Influence of aging on laboratory-measured contact angle.18

Advancing and receding contact angles can be shown in a capillary tube for oil displacingwater (receding angle) and water displacing oil (advancing angle). The procedure to determinethe contact angle using a contact angle cell is described by Wagner and Leachs6 and isillustrated schematically by Figure 16. Briefly, samples of polishea, Rat plates of the mineralwhich is the main constituent of the reservoir rock are immersed in a sample of formationwater. A drop of reservoir oil is held between the two flat samples of the mineral and thetwo plates are moved horizontally so that the water advances on the surface of the plateinitially covered by oil. The contact angle formed between the interface and the newly water-occupied surface of the mineral is a measure of the water advancing contact angle. Theadvancing contact angle is the one that is customarily measured and often reported withoutbeing identified as advancing.

The contact angle measured in the laboraotry is often influenced by aging. It has beenshown that contact angle increases with age of the oil-solid interface until an Lquilibrium isreached. This may require several days and it is one of the disadvantages of the contactangle method.a8 Figure l7 shows this effect.

Reliable wettability measurement requires that both the reservoir rock and the fluids befree from contaminants. Uncontaminated reservoir rocks can probably be obtained if thecores are recovered with coring fluid containing no surface-active additives or with reservoiroil that has not been exposed to oxygen. It has been reported that exposure of cores to aircould result in alteration from water-wet to intermediate wettability. Uncontaminated res-ervoir water and oil are easier to obtain than unaltered reservoir rock. Since contact anglemeasurement can be done without a sample of (uncontaminated) reservoir rock, it has becomea widely used method for determining wettability.

Zismanm and other investigators studied contact angles under controlled conditions and

l .

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3 .

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5 .


expressed varying opinions concerning the method's usefulness. Melrose and Brandner6rbelieved that the contact angles provides the only direct and clear specification of thewettability property characteristic of a given oil-water-rock system. Treiber et a|.62 foundthat the water-advancing contact angles correlate well with other wettability indicators whilewater-receding angles do not.

Brown and Fatts2 questioned the ability of the contact angle method to provide a reliablescale for determining wettability and suggested that the concept of a contact angle repre-sentation of wettability of reservoir rock be abandoned and that this method be replacedwith a "fractional surface area" method. Morrow et al.63 also observed that several factorscast doubt on the utility of the contact angle method. Mungans described some of thelimitations and pitfalls of contact angle measurement as follows:

The mineral chosen for the contact angle measurement is the principal constituent ofthe reservoir rock. For the purpose of contact angle measurement, silica or quartz isused to represent a sandstone; calcite is used to represent a carbonate or reef reservoir.Laboratory measurement of contact angle or mineral surfaces may not simulate truereservoir contact angle.The contact angle at the water/oil displacement front is "advancing" while at theleading edge of the oil bank it is "receding". These values sometimes differ by asmuch as 50o. This variation can be on the same order of magnitude as the laboratory-measured contact angle.Contact angle measurement should be done when the solid surface and a fluid remainin contact for an adequate time before the second fluid is introduced over the surface.This is referred to as pre-equilibrium time and it is of different length for each crudeoil-water system. Without adequate pre-equilibrium time, a stable contact angle is notreached. In some cases it has been reported that a stable contact angle is never obtainedif the solid surface comes into contact with some types of crude oils. Contact anglemeasurement is frequently time consuming.Contact angle measurement should be performed with actual reservoir fluids, sincethey are in equilibrium and solubility effects are negligible; otherwise, the fluids mustbe equilibrated with one another so that the solubility effects become negligible.Contact angle measurement preferably should be done with bottom-hole fluid samples;however, because of the time and expenses involved, flow line samples are often used.Fluid samples taken from the storage or treating facilities are not reliable, due to thepossible accumulation of asphaltenes. When produced water is not available, syntheticbrine is commonly used.Contact angle measurements should be made under controlled conditions so that theoxidation of crude oil can be prevented.Contact angle measurement requires extreme care to assure cleanliness and inertnessof the apparatus.

B. Imbibition MethodAn imbibition test is a reliable technique of wettability determination provided unaltered

reservoir fluids are available. The method consists of the measurement of rate of flow of awetting fluid spontaneously imbibed into a core and replacing a nonwetting fluid by theaction of capillary forces alone.

Imbibition tests may be performed at standard conditions or at reservoir conditions. Figures18, 19, and 20 illustrate equipment that is used for conducting the tests at ambient conditions.The imbibition test at standard conditions may be performed as follows:

l. A cylindrical plug of reservoir rock I to I tlrin. in diameter is cut with water as acoolant in the cutting process.

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FIGURE 18. Imbibi t ion cel l .

2. The sample is placed under water in a beaker and evacuated to remove trapped gas.3. The sample is flushed with water to reduce the oil saturation to residual level.4. The core plug is placed in an imbibition cell under oil and oil imbibition is monitored.5. The drained water is measured; it is equal to the amount of imbibed oil. Sufficient

time should be allowed for the system to reach equilibrium; this may take several daysdepending on the permeability of the plug.

6. The plug is then saturated with oil to reduce the remaining water to the ineduciblelevel.

7 . The sample is placed in an imbibition cell under water and water imbibition is monitoredby the amount of oil being drained. The fluid that imbibes into the sample (oil orwater) is the wetting phase.

The imbibition test under reservoir conditions is more complex. Irreducible water satu-ration is established by flushing the core with live oil and the imbibition tests are made atreservoir pressure and temperature.

C a p i l l a r y


t o W a t e r R e s e r v o i r)7

FIGURE 19. Imbibi t ion cel l

Rubber S topper - - - - -+

FIGURE 20. Imbibition cell.

Amott6s developed a quantitative technique for defining the degree of water-wetness ofcores. He expressed the degree of water wetness by a water index, which he defined as theratio of the volume of water spontaneously imbibed into a core to the total volume of oildisplaced by a water drive (forced displacement of oil by water). Similarly, an oil indexwas defined at the ratio of the volume of oil spontaneously imbibed to total water displaced

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by an oil drive (forced displacement of water by oil). Amott's test consists of the followingsteps:

l �2 .3 .

Flush theImmerseImmersedisplacedMeasureImmerse20 hr.Measure

reservoir sample with water to reduce the oil saturation to its residual level.the sample in water and evacuate to remove gas.the sample in kerosene (or reservoir oil) and measure the volume of waterby imbibition of oil after 20 hr.

the volume of water displaced when the sample is centrifuged under oil.the sample in water and measure the volume of oil displaced by water after

the volume of oil displaced when the sample is centrifuged under water.

4 .5 .

6 .

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Oil index is the ratio of the volume of fluid measured in step 3 to the volume of fluidmeasured in step 4. Water index is the ratio of fluid volume from step 5 to fluid volumefrom step 6.

The preferential wettability of a rock is determined by the magnitude of these two indexes,i.e., strong wettability is indicated by values approaching one and a weak preference inindicated by values approaching zero. A water index of one indicates a strongly water-wetsurface while an oil index of one indicates a strongly oil-wet surface. Values between thesetwo extremes or a value near zero for both ratios cover the range of intermediate wettability.

Amott's test of wettability of porous media received high marks from Raza et aI.66, althoughMoore and Slobad,67 Bobek et aI.,68 Kil lens et al.,6e and RichardsonTo have indicated thatthe imbibition rate cannot be entirely attributed to the wettability of the core, but that it isalso influenced by rock porosity, permeability, pore structure, and pore size distribution, aswell as viscosity and interfacial tension of the fluids involved in the experiment. Donaldsonet al.7' tried to eliminate extraneous effects from the wettability measurement by comparingthe volumes of fluids imbibed into preserved reservoir cores with the volumes of fluidsimbibed in the same cores after extraction and resaturation. Although the use of the samecore would appear to offer identical pore size distributions, the change in fluid distributionscaused by the cleaning process may have offset the advantage gained.

MunganT2 reported the use of an imbibition test to evaluate the wettability of native-statecores. Emery et a1.73 used an imbibition test after incubation of cores for up to 1,000 hrwith gas-saturated oil under pressure; water was the first phase to contact the rock in thetest. Kyte et al.7a described imbibition tests conducted at reservoir temperature and pressure.

C. Bureau of Mines MethodThe U.S. Bureau of Mines method of wettability determination of a porous rock, commonly

referred to as the "Centrifuge Method", is based on the assumption that an elemental areaof the internal surface of the porous medium is either wettable or nonwettable by one of thefluids involved. The problem is one of determining the fraction of the internal surface wettedby each fluid. A method of measuring wettability based on the above theory was suggestedby Gatenby and MarsdenTs and was later developed by Donaldson.Tr These investigatorsmade use of the areas obtained from the drainage and imbibition cycles of the capillarypressure curve to produce a numerical representation of wettability. The Bureau of Minesmethod is quite rapid and it can be employed with reservoir fluids.

D. Capillarimetric MethodJohansen and DunningT6 recognized the importance of the liquid used in determining

wettability of a rock-liquid-brine system and suggested the use of a capillarimeter whichjoins the two liquid phases, oil and water, through a small diameter glass capillary tube,with a capillary pressure across the interface joining the two phases. Adhesion tension or


displacement energy, was calculated from the difference in height of the two liquids in thetwo arms of the capillarimeter, the difference in densities, and the acceleration due to gravity.The instrument is capable of measuring interfacial forces with either an advancing or recedinginterface. Major limitations of this method are the exclusion of reservoir rock as a factorinfluencing wettability and lack of provision to prevent oil from oxidizing.

E. Fractional Surface Area MethodThis method, developed by Brown and Fatt,s2 uses mixtures of untreated sand and sand

rendered oil-wet by organosilane vapors to obtain wetting conditions ranging from completelywater-wet to completely oil-wet.

Wettability is represented by the fraction of solid surface made artificially oil-wet. Al-though use of the method to evaluate field behavior is not in evidence, the concept of afractionally wet surface has been presented in the work of other writers.3e

F. Dye Adsorption MethodThis method, developed by Holbrook and Bernard,-50 is based upon the ability of reservoir

rock to adsorb a dye such as methylene blue from aqueous solution, while rock surface areascovered by contaminants from the oil phase remain unaffected. The test is based on acomparison of the adsorption capacity of the test sample with that of an adjacent sampleextracted by chloroform and methanol. This method makes assumptions similar to those ofBrown and Fatts2 in their "fractional surface area" method.

G. Drop Test MethodThis method is often used to confirm rock wettability. The procedure involves placing

drops of oil and water on the surface of a fresh break in the core. The fluid that imbibes isthe wetting phase while the fluid that forms a ball and does not wet the surface is nonwetting.The drop test is a qualitative determination and is sometimes misleading.

H. Methods of Bobek et al.Bobek et aI.68 proposed a laboratory test to ascertain preferential wettability in a qualitative

fashion. The technique consists of determining which fluid will displace the other from arock sample by imbibition. The results of this imbibition test are compared with those of areference imbibition test on the same core sample after it has been heated to 400'F for 24hr to remove any organic materials and to make it more water-wet. The assignment ofqualitative wettability designations is based on the relative amounts and rates of imbibitionin the two tests.

In the same paper a method for estimating the wettability of unconsolidated material isdiscussed. A thin layer of the unconsolidated sand is spread on a microscope slide. The oilcontent of the sand is increased by adding a clear refined oil. Droplets of water are thenplaced on the surface of the sand grains and the fluid movement is observed. If the sand iswater-wet, the added water will displace oil from the surfaces of the sand grains and theoil will form spherical droplets, indicating that oil is the nonwetting phase. A similarprocedure is used to test for oil wettability.

I. Magnetic Relaxation MethodA nuclear magnetic relaxation technique was suggesteds2 for determining the portions of

the rock surface area that are preferentially water-wet or oil-wet. A rock sample is firstexposed to a strong magnetic field, then to a much weaker field. The magnetic relaxationrate - that is, the rate at which the initially imposed magnetism is lost - is then measured.In sandpacks containing known mixtures of oil-wet and water-wet sand grains, a linearrelationship was observed between the relaxation rate and the fraction of the surface area

that isa testi lpetrolrroutin(

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oil sata nati\foundones.be conthe coscriber(See Iprefento thea rockfluid scurve

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FIGURE 21. Interstit ial water saturation for sand mixtures.sr

that is oil-wet. Though the authors reported no studies using natural cores, they proposeda testing procedure. Their technique requires specialized equipment not normally found inpetroleum laboratories and there are no indications in the literature that the method has foundroutine use.

J. Residual Saturation MethodsMcGhee et al.,ss Lorenz et al.7e and Reznik et al.80 reported a correlation between residual

oil saturation and wettability. Treiber et al.62 reported that the connate water saturation ina native core can sometimes be used as an indication of formation wetting preference. Theyfound that oil-wet formation have much lower connate water saturations than the water-wetones. In addition, the connate water saturation in a strongly oil-wet reservoir was found tobe constant regardless of the sample permeability, while in reservoirs of other wettabilitiesthe connate water saturation decreased with increase in permeability. Iwankow53 also de-scribed the effect of heterogenous sand wettability in terms of a fraction of drifilmed sand.(See Figure 21.) Drifilm is a solution commonly used in the laboratory to make sandspreferentially oil-wet. Coley et al.8r were not successsful in using the ratio of the wettingto the nonwetting residual saturation from relative permeability-saturation relationships asa rock preferential wettability indicator; however, they found that the volume of mobilefluid shown by the spread between the residual saturation values of a relative permeabilitycurve appears to decrease as the oil wettability increases.

K. Permeability MethodThe determination of wettability of a sample from permeability data is accomplished by

comparing the ratio of water permeability at residual oil saturation with the oil permeabilityat connate water saturation. If this ratio is less than 0.3, the sample is considered to bewater-wet, while a value near unity indicates that the sample is oil-wet.82 The relationshipbetween absolute permeability and connate water saturation has been frequently mentionedin the petroleum literature and the relationship between connate water saturation and rockwettability has been discussed. Rocks with low connate water saturation are considered tobe weakly water-wet to oil-wet, while rocks with high connate water saturation are normallydesignated as water-wet.

( W a t e r - w e t )

S w

Relative Permeability of P etroleum Reservoirs

f t re l

k r e l

S w

FIGURE 22. Schematic wettability effecrs on relativepermeability curves.

L. Connate Water-Permeability MethodA correlation of absolute permeability as a function of water saturation in cores cut with

oil-base mud has been used for qualitative identification of core wettability.6s Water saturationis measured in freshly cut cores and absolute permeability is determined after extraction anddrying. A plot of water saturation as a function of absolute permeability to air is prepared.The curve will have a gentle slope over a large saturation interval for water-wet systems,while it will exhibit a nearly vertical slope over a narrow saturation range for oil-wet systems.This technique is applicable primarily to thick hydrocarbon reservoirs with sufficient variationin permeability and water saturation so the required plot can be prepared.

M. Relative Permeability MethodFor a given water saturation, the water relative permeability of a water-wet rock is lower

than that of a comparable oil-wet rock. For the systems studied by Owens and Archern itwas found that an increase in oil wetness (at constant water saturation) produced an increasein k,* and a decrease in k,.,. Treiber et aI.62 concluded that water-wet consolidated porousmedia normally have a water relative permeability less than l5Vo at residual oil saturation,while oil-wet porous media show a 50Vo or higher relative permeability to water at flood-out.

Craigas offers the following heuristic guidelines, which are illustrated by Figure 22:

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In a water-wet rock, residual oil globules in the large flow channels block the easy flowof water and cause a low water relative permeability; however, the oil in an oil-wet systemoccupies smaller flow channels and coats the walls of the larger ones, causing a minimumdisturbance to water flow and a higher water relative permeability.tt This is why an oil-wetreservoir will waterflood poorly, with early water breakthrough, rapid increase in water cut,and high residual oil saturation.

The water-oil relative permeability relationship of native-state cores under steady-stateconditions is one of the best indicators of the rock wettability preference. Keelan82 pointedout that a sharp drop in oil relative permeability over a small saturation change accompaniedby a rapid rise in relative permeability to water, to a terminal value in excess of one thirdthe initial oil relative permeability, often indicates oil wetness. Careful sample examinationis essential in using this technique, for heterogeneous or cracked samples yield relativepermeability data similar to the data obtained from oil-wet cores.

Water relative permeability curves in water-oil systems show good agreement with theoil relative permeability curve obtained during gas-oil relative permeability tests in a stronglywater-wet core.62'63'84 This effect does not exist under any other wetting condition. In astrongly water-wet core, the water relative permeability curve of a water-oil system alsoshows good agreement with the water relative permeability of a gas-water system in thepresence of residual oil saturation. This agreement will occur, even though the direction ofthe change in saturation may not be the same in the two systems. In the same manner, instrongly oil-wet cores, the gas relative permeability of a gas-water system is comparable tothe gas relative permeability of a gas-water system in the presence of residual oil saturation.8a

The point of intersection of the water and oil relative permeability curves has beensuggested as an indication of rock wettability. Owens and Archerrr have shown that therelative permeability intersection point moves toward higher values of water saturation andlower values of relative permeability in a water-oil system as the sample wettability ischanged from oil-wet to water-wet. As illustrated by Figure 22, a relative permeabilityintersection point on the left of 507o water saturation indicates that the system is oil-wet,while an intersection to the right of this saturation suggests that the system is water-wet.

N. Relative Permeability Summation MethodThe summation of relative permeabilities to the water and oil phase at fixed saturations

also gives some insight into the immiscible flow processes. McCafferyse noted a trend inthe minimum values of the sum of relative permeabilities of samples according to theirpreferential wettabilities.

O. Relative Permeability Ratio MethodIf the ratio of displacing to displaced phase relative permeability is plotted as a function

of the displacing-phase saturation, the shape of the plot is related to preferential wettabilityof the rock.66 It has been shown that the water-oil relative permeability ratio shifts to ahigher value as the rock becomes more oil-wet; furthermore, a semilog plot of water-oil andgas-oil relative permeability indicates that the gas-oil relative permeability ratio curve movesfrom under to over the water-oil relative permeability ratio curve as the rock becomespreferentially water-wet.sr The water-oil relative permeability ratio curves of rock withvarious degrees of intermediate wettability are found to be practically the same in the presence


of constant initial water saturation.85 Imbibition water-oil relative permeability ratio curvesin the absence of initial water saturations show higher values of residual oil saturation asthe cores become more oil-wet.8s Steady-state relative permeability measurements shouldbe used for determination of wettability. Unsteady-state methods may not allow equilibriumto occur during the flow test; therefore, they may indicate more oil wettness than actuallyexists.

P. Waterflood MethodSeveral attempts to find a single correlation of wettability with waterflood oil recovery

for different porous media have failed, even though the tests were carried out under astandard set of conditions.6s However, the waterflood performance of a native-state coreunder carefully controlled laboratory conditions has been used as an indication of rockpreferential wettability. It is found that in a strongly water-wet system, a large fraction ofthe oil is produced prior to water breakthrough and very little additional oil is recoveredafter breakthrough. For the test to be reliable, an equilibrium wetting condition must prevailprior to the passage of the flood front through the core.

Q. Capillary Pressure MethodBoth displacement pressure and the ratio of drainage to imbibition displacement pressure

have been proposed as qualitative indicators of preferential wettability of porous media. Anincrease in displacement pressure or in the ratio of drainage to imbibition displacementpressure signifies a tendency of the core to become more oil-wet. The above technique isapplicable when oil-water capillary tests are made on native-state cores. However, mostcapillary pressure tests are either of the mercury injection or air-brine type, which providelittle information concerning wettability.8l

R. Resistivity Index MethodFormation resistivity obtained from electric logs can be used as a qualitative technique

for wettability identification. Resistivity index is defined as the ratio of true formationresistivity to resistivity of the formation when 1007a saturated with formation water. A highvalue of resistivity index indicates a low water saturation or a discontinuous water phase,which characterize an oil-wet system. A knowledge of the water saturation in the rock mayyield sufficient information to make a judgement about rock wettability.

There is considerable uncertainty concerning the nature of the wettability characteristicsof reservoir rocks in situ. Tests of wettability made on cores taken from reservoirs are notnecessarily valid indicators of subsurface conditions, since the coring process itself mayalter wettability. Cores cut in oil-base mud, for example, are often rendered entirely orpartially preferentially oil-wet. Therefore special precautions must be observed during bothcoring and transporting to minimize the danger of altering the true wettability of the rock.In the absence of convincing evidence to the contrary (for example, abnormally high resis-tivity index) the assumption of preferential water wettability has been frequently used.86

VII. FACTORS INFLUENCING WETTABILITY EVALUATION

It has been suggested that four factors may influence the results of experimental deter-mination of rock wettability.8T One of these factors is core recovery and preservation. Inthe process of core recovery from a reservoir, heavy hydrocarbon components of crude oilbecome less soluble as the oil loses its associated solution gas (as a result of pressurereduction). The heavy hydrocarbon components can precipitate on the rock grains, leadingto less water-wet or even oil-wet core behavior.8s-m Drilling fluid containing surface-activematerials may drastically change a core wettability, but it has been shown that bentonite

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and as:

of this

A rhiStainle:In sprteni th telidatedhehar rrgrain sr\l'el .

-{ '

| [ . r l l r ) C U f V e S

sJtur . r t i ( )n as

Ilc'r t r should

r c q u r l i b r i u m

lh. rn . rc tua l ly

t t l , rc 'COVefY

r \u t undef a

t c - . t . t t d COfe

I t , ' : t t r i rock

F : : . r j t i on o f

l r l i , t ) \ ' € f e d

I r I l . . . i p rc 'va i l

l l r ' ! l l [ tr t 'Ssgfe

l . : : tc t l ia . An

d r - l l . r cemen t

t c . hn ique i s

n r c . c r . m o s trh : . : : prov ide

n c : e ; hn ique

1g :, ' r t t tat ion

rat . r . \ h ighr* . r tdF phase,

lhc r,rc[ 63y

har.re tcr ist ics

[ t r ] ! ' r a fe nOt

g r l l . c l i f f i o !

d cn t r re ly o rI J.rrrng bothI t , l thc- rOCk.

I nrgh resis-; l r u . c d . E 6

IO\

mc'ntal deter-

5('r\ ation. In

I tr t crude oi l

I tr t Plattuaat?rn: . lead ing

Urtrcr '-act ive

hat hcntonite

69

and carboxymethyl cellulose have no observable effect on rock wettability when they areused in the coring fluid.7a Weathering and contamination of cores during preservation andstorage are also found to influence core wettabilities.er Strongly water-wet cores may becomeless water-wet as a result of air exposure, while cores with intermediate wettability showno significant change.6s Oil-wet cores also may become water-wet upon exposure to air.72It has been suggested that alteration due to air exposure can be minimized and native-statewettability can be restored by incubation of the core in reservoir oil for two weeks at reservoirtemperature.2s

Crude oil is probably the best coring fluid for preserving wettability and maintainingnative interstitial water saturation;e2 however, use of the wetting phase as a coring fluid maypreserve the rock properties properly.2s NaCl brine containing CaCO, powder with no otheradditives is considered a good fluid for cutting cores.e3 Care must be taken to avoid con-tamination of the coring fluid with air, sediments, etc. The use of crude oil as a coring fluidis likely to introduce a fire hazard into the coring operation, especially if a high API gravityoi l is used.

Native state wettability of cores is obviously the most desirable condition, and the besttechnique for obtaining cores in this condition is by employing a pressure core barrel. Themethod allows cores to be cut and retrieved at reservoir pressure. At the surface, the coresare frozen, cut into sections, and sent to the laboratory.ea Although early attempts at pressurecoring met with l imited success, recent developments indicate a success ratio of 80 to90Va.

Cores that have been cleaned, dried, and restored to some saturation and wettabilitycondition are known as "restored state" cores.a8 This technique has been employed formany years and it is an established procedure; unfortunately, quite frequently, the cores arenot restored to their native state and the use of these cores invalidates results obtained usingsophisticated measurement techniques. Put very simply, restored state cores are not.

Factors that influence the core wettability evaluation include the laboratory core cleaningand preparation procedure. Mungane2 states that the cleaning procedure neither changes thepore size distribution nor the quantity of kaolinite and illite in the core. He concludes thatthe change in fluid flow behavior is basically due to wettability alteration. Salathiel3e reasonsthat the extraction of a core with strong solvents dissolves the strongly oil-wet surface coatingof heavy organic molecules and thereby alters fluid displacement behavior of many fresh orpreserved cores, as shown in Figure 23.28

Jenning'se5 results show a small but measurable change in the water-oil relative perme-ability ratio curve after toluene extraction of a variety of core samples from oil-bearingsandstones and limestones. The changes are not thought to be caused by significant changesin wettability. The results of Richardson et al.er show a higher rate of imbibition and a lowerineducible water saturation when East Texas Woodbine cores are extracted by hexane andmethanol. Morgan and Gordon's28 results show that the effect of cleaning procedure on corewettability may be minimized if reservoir fluids are used as testing fluids. Richardson etal.et believe a change in fluid flow behavior occurs as a result of repeated flooding of EastTexas Woodbine cores. This change appears as a decrease in irreducible water saturationand as an increase in residual oil saturation. Further work is necessary for better understandingof this problem.

A third category of factors that influence core wettability evaluation is the testing condition.Stainless steel wettability can be altered by pressure increase in a methane-water system.e6In spite of decrease in interfacial forces, the oil-water-solid system became more water-wetwith temperature increases in a clean unconsolidated Houston sand and a natural unconso-lidated California oil sand.eT One explanation for the effect of temperature on displacementbehavior is that polar components of the crude oil may not be adsorbed as readily on thegrain surfaces of a rock at elevated temperature, so the flow behavior becomes more water-rA'g1.7+'9a


air

4 k t o

permeab i f i t y : - 229 md.

o F r e s hA E x t r a c t e d

k r *

- : i - '

sw

il:|':?"il,,'t:Hiil:iJT11',:' permeabi'itv data rrom the same

A fourth category of factors that influence the core wettability evaluation is the type offluid used in the test. Carbonates are very sensitive to nitrogeneous surfactant compoundscontaining sulfur and oxygen.ar Sandstones containing large percentages of silica possessacid type surfaces.38'er Crude oil containing normal paraffins are inert and inactive withregard to the surfaces of porous media, while napthene and aromatics are more active withporous surfaces. Heterocyclics and asphaltenes containing oxygen, nitrogen, sulfur, andmetallic atoms are active with regard to the acid or basic sites. Reisberg and Doschelo haveindicated that different crude oils probably have different proportions of these compoundswhich are believed to be responsible for the wettability characteristics of surfaces.

The critical gas saturation decreases and that for oil increases with increasing concentra-tions of polar substances.ee Furthermore, increasing the concentration of polar compoundsin oil causes the cumulative water production to increase and cumulative oil production todecrease in laboratory tests.

Oxidation of crude oil frequently appears to modify the wettability of porous media. Thedegree of modification depends on the amount of oxidizable polar compounds in contactwith air and wettability may even be reversed.e8 Morgan and Gordon28 and Cuiece8 haveinvestigated the effect of fluids and laboratory handling on relative permeability. Mungane2saturated an extracted core with reservoir fluid and let it sit at reservoir temperature for 6

days. He discovered that the measured relative permeability values were identical to thoseof freshly preserved cores; but when he used purified fluids in place of reservoir fluids amore water-wet condition in the core was developed, as indicated in Figure 24.

The initial fluid saturation in a core,s salinity alteration,eo water alkalinity and hardness,ee

as well as the aging processe' can influence the preferential wettability of a core. Wagner

and Leach-'6 have shown that the wettability of an oil- or intermediately wet sample of

sandstone or carbonate can be changed to a more water-wet condition by the addition ofchemicals such as hydrochloric acid, sodium hydroxide, and sodium chloride. They inves-

6 0 ' 5

j

l lgi lc ' t

lreale(pH vrel-fect

Braanglerep|()rtrand 1-rnxksamine,t C H - ldtxlec;approrcausedsuch atou ardnoled\r'ettab

7l

s t l e t r pe o f

I . , ' l r r p t t undsp l r . . r p t r ssess

in . : . l r r c w i t h

E . r . l r r c w i t h

r . . . i i l u r . a n d

Drs le I-:" have

B . , ' l l lP t )U f ldS

l r e '

D S . , , n c e n t r a -

J . , \ l l l P ( ) U D d S

Jr ' \ l . le t lon to

I i : r r J ra . The

d. .r ' , eontact

Cu re e ' ' have

l ) \ lungane2

pr.rturc for 6

I t .J l to those

; r t r l r t lu ids a

hl

d h . r rdness,ee

;rrrc \\'agner

Et . . i lnple of

r .rJ. l i t ion of

. I hcr inves-

0 . 9

Sw o . 7

FIGURE 24. Effect of fluid and laboratory handling on relative permeability."l

tigated the influence of water pH on wettability of a quartz sample and used a n-octylaminetreated synthetic oil to produce an oil-wet quartz surface. Their results indicated that lowerpH solutions tend to produce water-wet surfaces under controlled salinity conditions. Thiseffect is shown in Figure 25.

Bradleyroo has shown that a basic 57o NaCl solution spontaneously decreases the contactangle of oil-wet cores and as a result increases the amount of imbibition. These effects werereported to be most pronounced on cores of intermediate wettability. Morrow et al. ,63 Wagnerand Leach,s6 and McCaffery and Munganror have shown that wettability of typical reservoirrocks can be easily changed to any desired degree by adding polar compounds such asamines or carboxylic acids. Bradleyrm found that carboxylic acids such as stearic acid CH.,(CHr)16 COOH at concentrations greater than 10-6 moll( altered the wettability of a water-dodecane-calcite system toward more oil-wetness and stearic acid with a concentration ofapproximately 5 x l0-3 mol/f caused strongly oil-wet surfaces. He found that stearic acidcaused no wettability alteration when quarlz samples were used. Bradley found that aminessuch as octadecylamine CH. (CHr),, NH, alter the wettability of both quartz and calcitetoward oil wetness, especially at concentrations greater than 5 x l0-a mol/{. It should benoted that polar compounds which alter wettability of a given rock type may not alter thewettability of another rock type.


o O . O

a 2 5 , 0 0 0

c 5 0 , O O O

p p m N a C l

W A T E R P H A S E

O I L _ W E T

W A T E R - W E T

6

W A T E R _ P H A S E P H

FIGURE 25. Contact angle as a function of pH.so

VIII. WETTABILITY INFLUENCE ON MULTIPHASE FLOW

The microscopic distribution of fluids in a porous medium is greatly influenced by thedegree of rock preferential wettability. The fluid distribution in virgin reservoirs understrongly water-wet and strongly oil-wet conditions has been described by Pirson.ro2 In astrongly water-wet reservoir, most of the water resides in dead-end pores, in small capillaries,and on the grain surface. In strongly oil-wet reservoirs, water is in the center of the largepores as discontinuous droplets, while oil coats the surfaces of the grains and occupies thesmal ler capi l lar ies.

Under strongly water-wet conditions the effective permeability to the nonwetting phaseat irreducible water saturation is approximately equal to the absolute permeability of therock. On the other hand, in strongly oil-wet systems, the effective permeability to oil atirreducible water saturation is greatly reduced by the water droplets in the larger pores. Razaet aI.66 stated that in some oil-wet reservoirs, water occupies some of the finer pores and istrapped as droplets in the larger ones. Raza et al. analyzed the displacement of oil byadvancing water and the trapping of the residual oil as shown in Figure 26.

In strongly water-wet reservoirs, water traps oil in the larger pores as it advances alongthe walls of the pore, while in strongly oil-wet reservoirs, water moves in large pores andoil is trapped close to the walls of the pores.66

The petroleum industry has long recognized that the wettability of reservoir rock has animportant effect on the multiphase flow of oil, water, and gas through the reservoir. APIProject 27 at the University of Michigan was initiated in l92l to study this problem. The

r 3 0

9 0

7 0

5 0

aC)O

o

uJJ

z

o

Fz

(J

z

z

oI

(ruJF

3

dtsrr m

of rgsc

Thoau\r elurbl

ercnlla

drrea:

S(-hn

trn ilou

icr hetr

*r rm1

Strn

favoral

increas

have n

strongl

73

E 5 0j

O i l - W e t S a n d

FIGURE 26. The trapping process of oil by advancing water'n"

- T E S T 1 w a l e r w e t

- - - T E S T 2 w e t e r w e l

. . . . . . T E S T 3 o i l w e t

\ \\ \\

II

oi1 Brine

f,

nr'eJ hr the

\ r r l f r U n d e f

s . , n t l n a

I r ; p r l l a r i e s ,

o i :nc large

I . . . n r c s t h e

l l t : r I phase

b r i : : r , r i t he

[ l r i , ' o i l a t

p ' r c . . Raza

F' rc . and is

I t , r t o i l by

Bn,Cr a long

F n()rcs and

rrxk has an

F: ' r t r i r . API

Ntt ' , icrn. The

7 55 02 5 1 0 0

B R I N E S A T U R A T I O N

FIGURE 27. Effect of wettabil ity on flow behavior' 'r

dissymmetry of relative permeability curves is attributed largely to the preferential wettability

of reservoir rock.te'es'ro3 As illustrated by Figure 27, Geffen et al.r2 and Donaldson and

Thomas'oa have shown the effect of fluid distributions brought about by rock preferential

wettability on the relative permeability-saturation relationship. As the degree of rock pref-

erential wettability for waier decreases, the oil relative permeability at a given saturation

decreases while the water relative permeability incqeases.

Schneider and Owenssa recognized the fact that rock type appears to have less influence

on flow relationships than does rock wetting preference. However, this may not be the case

for heterogeneous rocks or mixed wettability systems. Owens and Archerrr also confirmed

the importance of preferential wettability on multiphase flow in porous media.

Some investigatorsno have found that relative permeability becomes progressively less

favorable to oil production as a rock becomes less water-wet. The residual oil saturation

increases as a rock becomes less water-wet. Others have shown that weakly water-wet cores

have more favorable relative permeability curves and lower residual oil saturations than

strongly water- or oil-wet rocks. Conceptually, this latter behavior seems reasonable since

74 Relative Permeabilitv of Petroleum Reservoirs

S P L A C E O P H A S E S

4t/

/ l

Ni troqen displacrng Hepti le,Dodecane or bio.tyl eti... up to 49c

llitrogen displacing Water l08o

Dioctyl Ether displacing Nrtrogen I3Io

HepLile displacing Nitroqen above l38c

D i s p l a c e d p h a s e S a t u r a t i o n . p V

il:,yrT.. rt Relative permeability for fluid pairs with various contacr

the capillary forces in strongly water-wet cores are strong. The oil may be bypassed andtrapped in larger pores by the tendency of a water-wet core to imbibe water into the smallercapillaries. The bypassed oil in the large pores is then surrounded by water and is immobileexcept at very high pressure gradients. The saturation interval for two-phase flow under thiscondition is probably short.

As the capillary forces are reduced by reduction in preferential water-wettability of a rock,the tendency toward rapid imbibitional trapping of oil in large pores by movement of waterthrough small pores should also diminish. The zone of two-phase flow should become broaderand oil displacement to a lower residual saturation should be possible. If other factors remainconstant, higher flow rates and lower interfacial tensions are conducive to higher oil recovery;these are changes that diminish the ratio of capillary forces to viscous forces.

Stegemeier and Jensen3T and McCaffery and Bennionr05 reported that wettability alterationsover a relatively wide range produce a negligible effect on the relative permeability curve,as shown by Figure 28. However, other workers did not confirm this finding. Treiber etaI.62 found that relatively small variations in wettability produce considerable effects on therelative permeability curve. Figure 29 shows the effect of contact angles on relative perrne-ability curves for a Torpedo sandstone.

IX. EFFECTS OF SATURATION HISTORY

The relative perrneability-saturation relation is not a unique function of saturation for agiven core, but is subject to hysteresis for porous systems with strong wetting properties.

. 8

Thaton \rI n ar alueincrc-dunni n a rvalueof dra res{u'oul,

Geet a l .relatipermI n a t 'pha-srrelatifrom

Thonlrbit ior

75

1 0 0

\ ."Nt\ A T E R'/

\

I . .

Contact Ancrle^o

nro

. . . . . . 9 0 o

. - ^ o

" ̂ ^o

2 0 4 0 6 0 8 0 1 0 0

sw

FIGURE 29. Imbibition relative permeability with various contact angles."l

That is, the relative permeability of a porous medium to a fluid at a given saturation dependson whether that saturation is obtained by approaching it from a higher value or a lower one.In a displacement process where the wetting-phase saturation is approached from a lowervalue, the resulting relative permeability curve is referred to as an imbibition curve (anincrease in the wetting phase). Examples of imbibition processes are the injection of waterduring waterflooding and coring a water-wet rock with a water-base mud. On the other hand,in a displacement process where the wetting phase saturation is approached from a highervalue, the resulting relative permeability curve is referred to as a drainage curve. Examplesof drainage processes are the displacement of oil by expansion during primary depletion ofa reservoir and the accumulation of hydrocarbons in oil and gas reservoirs; another examplewould be waterflooding an oil-wet reservoir.

Geffen et al. ,r2 Osoba et al. ,r3 Levine,'oo Josendal et al.,r07 Terwil l iger et &1. , 'n' and Coleyet al.8' described the hysteresis phenomenon and verified that both water-oil and gas-oilrelative permeability ratio curves as well as individual wetting and nonwetting phaie relativepermeabil ity of both sandstone and carbonate formations may exhibit hysteresis.rr '22'ro7'roeIn a two-phase system, hysteresis is more prominent in relative permeability to the nonwettingphase than in relative permeability to the wetting phase. ro' I ro The hysteresis in wetting-phaserelative permeability is believed to be very small and thus, sometimes difficult to distinguishfrom normal experimental error, as indicated in Figure 30.

The drainage curve shown in Figure 30 is a primary drainage curve which is applicableonly when drainage occurs before imbibition. When a drainage process occurs after imbi-bition, a secondary drainage curve exists, as shown in Figure 31.

1 0

ol<

. 1

;rp.:..Cd arldr th l . r t ta l le r

i . l : : r rnobi ler u r :Jc r th is

D , ' l . r rock .Rtll , r1 \\ atef

Dci l hroader

ltr r:. rCfflilitl

t t l : l . , r r ery :

; . r l t c ra t i ons! r l r t r curve,

- . I ' rciber

et

Tc, t . t ln the

lr\ c perme-

F u l t t r J l t O f a

;pr , rp,"-r t ies.


S w ( W a t e r - W e t S y s t e m )

FIGURE 30. Primary drainage relative permeability curve

eDW (wa te r -we t sys tem)

:t":rYXt ,t Secondary drainage curve: end-point f low

These curves describe relative permeability when the flow reversal occurs at one of thesaturation end points. The effect of flow reversal at an intermediate saturation value isillustrated by Figure 32.

As shown in Figures 30 and 31, the water (wetting phase) relative permeability curve isessentially the same in strongly water-wet rock for both drainage and imbibition processes.rrHowever, at a given saturation, the nonwetting phase relative permeability of a consolidatedrock is usually less for an imbibition cycle than for a drainage cycle. t2.t3.22.to6 For anunconsolidated rock, the nonwetting phase relative permeability in an imbibtion cycle isusually greater than the coresponding nonwetting phase relative permeability in a drainagecycle. Naar et aL.22 reported that relative permeability relationships for poorly consolidatedformations tend to resemble those for unconsolidated formations.

Figure 33 shows the imbibition and drainage relative permeabilities of a consolidatedrock. It can be seen that the residual nonwetting phase saturation is much greater forimbibition than for drainage. That is, the nonwetting phase loses its mobility at a highersaturation in imbibition than it does in drainage. Figure 34 shows that the imbibition cyclek.o may lie above k.. on the drainage cycle for some systems. This relationship probably isnot typical of petroleum reservoirs.

D r a i n a g e

l m b i b i t i o n

Water

77

o.Y

.=-o$oEL

oo.o

o

.9=ooo-o

be

ov

Sw (wa te r -we t sys tem)

FIGURE 32. Secondary drainase curve: intermediate flowreversal.

1 6 0

1 4 0

120

1 0 0

80

60

40

2 0

00 2 0 4 0 6 0 8 0 1 0 0

Br ine sa tura t io f i , o /o

FIGURE 33. Oil-water flow characteristics of a consolidated rock.12

S e c o n d a r y d r a i n a g e

t t r : l , r i the

; ' . r l u e i s

i r . . r n e i sJ t \ e . r C S . | |

trr1., r lrdated' r ' F t r l 3P

,n . ) c le is

a Jrarnage

111.tr l idated

n . t r l t da ted

Erc.rtr'r for

u .r higher

i t r , 'n c t 'c le

pr,''b.rbll' is

Water

Wate r

l( r o

' a

X* 'n

\--+---r--,^,'


. 5

so

C o n s o l i d a t e d S a n d

. 5 1 . 0

so

U n c o n s o l i d a t e d G l a s s S p h e r e s

FIGURE 34. Relative permeability curves for consolidated sands and unconsolidated glass spheres.rr

The amount of trapped oil in water-wet porous media is given approximately by the areabetween the drainage and imbibition oil relative permeability curves.rr2 It is believed thatthe occurrence of hysteresis is possibly related to the pore size distribution and cementationof a rock. As water is progressively imbibed into oil-filled pores of different sizes, oil isejected from them. The ejection process continues as long as continuous escape paths throughpores still containing oil are available. These escape paths appear to be lost at oil saturationswhich greatly exceed those which occur at the onset of continuity of a nonwetting phase,(e.g.,gas) on the drainage cycle. Thus, the residual oil saturation which results fromwaterflooding a water-wet rock is much greater than the critical gas saturation that char-acterizes the same rock. Apparently oil is trapped on the imbibition cycle. A similar behavioris observed if a preferentially water-wet rock containing free gas is waterflooded.

The imbibition and drainage wetting-phase relative permeabilities of a consolidated orunconsolidated rock are retraced under a succession of imbibition and drainage cycles; in areversal of the saturation change from drainage to imbibition, a distinct path is traced bythe nonwetting phase relative permeability curve (as shown in Figure 32) to a residualnonwetting phase saturation. This path depends on the saturation established in the drainagecycle. Also, the nonwetting phase relative permeability curve in a drainage cycle followingan imbibition cycle retraces the imbibition curve until the previous maximum nonwettingphase saturation is reached. This effect is illustrated by Figure 35.22'13

X. EFFECTS OF OVERBURDEN PRESSURE

Wilsonila reported that a 5000 psi laboratory simulation of overburden pressure at reservoirtemperature reduces the core effective permeabilities to oil and water by about the sameextent as it reduces the single-phase permeability of that core. Consequently, the water andoil relative permeability of a natural core, under 5000 psi overburden pressure, show onlya moderate change from the relative permeability measured under atmospheric conditions,as shown in Figure 36. Wilson also pointed out that an overburden pressure that can produceover 5Vo reduction in porosity of a core can also produce a sufficiently large change in poresize distribution to affect the relative permeability of the core.

In contrast to the work of Wilson, Fatt and Barrettrrs concluded that variation of rockoverburden pressures in the range of 3000 psi does not produce any change on gas relativepermeability in a sandstone gas-oil system. Figure 37 shows the gas relative permeability

1 . O1 . O

o . 5l<

9 . 5j

1 . 0

u'ith an<reporteda l . r r h a rconditiorsimulati<pressure

Wyck,and abs<Dunlap"found n<specificranging Igranulargas saturbil i ty mecurves tovarious p

Botset:

- D r a i n a g e

- - - l m b i b i t i o n

/*-)r o

ta*- *rn.--Y

\ t.',

7

79

1 0 0

*\.

A I R

' - a i r - b r i n e s y s t e m

\\

l\1 0

b a lI

ol.

p ' ' . - '

b r rhc area

e l r c i cd t ha tgCntcntat ion

S l / d : . o i l i s

Ith. thrttugh

I . . r l l ra t ions

I t r r :_ : phase.

R. i , i l . t ' rom

D I l . r t char-

hr ' lhar ior

iJ

F l : . : . r l cd o r

c r . : c . : i n a

b : : . recd by

) .r rcr idual

hc' .lrainage

b l , ' l l r r * i n g

Jlr) l i \ \ c ' t t ing

al rcrc'rt'oif

tl lhc' same

G r . . r tc r and

r .hrr\ \ Ofl lY

c r )nJ i t i ons ,gan prt lduce

nlJ rn pore

ir''n , 'l rock

g: r . rc la t ive

rnncrb i l i ty

. 1

. 0 140 6 0 8 0 1 0 0

Br ine sa tu ra t io r ' , Vo

FIGURE 35. Air flow behavior in two-phase systems, Nellie Bly sand-s tone . r2

with and without the laboratory simulation of overburden pressure. Similar results werereported by Thomas and Ward"6 for a gas-oil system in a low permeability rock. Geffen etal.'2 have shown that the residual gas saturation in a liquid-gas system, under atmosphericconditions, is similar to the resisdual gas saturation measured under a 5000 psi laboratorysimulation of overburden pressure. Merliss et al.r17 concluded that the effect of overburdenpressure on relative permeability was primarily due to changes in interfacial tension.

XI. EFFECTS OF POROSITY AND PERMEABILITY

Wyckoff and Botset3 as well as Leverett and Lewis8 investigated the influences of porosityand absolute permeability on relative permeability and found them to be insignificant.Dunlaprrs used unconsolidated sand packs having permeabil it ies of 3.0,4.5, and 8.0 D andfound no indication that the relative permeability-saturation relationship is a function ofspecific permeability of the sand. Stewart et al.rre found that variations in permeabilitiesranging from 8.5 to 300 mD and porosities from I 5 to 22Vo in limestone cores with inter-granular porosity, caused relative permeability curves to shift up to a maximum of 2Vo ofgas saturation. These investigators employed a solution gas drive, gas-oil relative permea-bility measurement technique in their study. They also reported the relative permeabilitycurves to shift up to a maximum of 47o of gas saturation when fractured limestone cores ofvarious porosities and permeabilities were employed.

Botset2r found that absolute permeabilities ranging from 17 to 260 D had negligible effects


1 0 0

a

o)L

.Y

4 0 6 0

s* ' o/o

1 0 0

FIGURE 36s y s t e m . r r a

Effect of overburden pressure on relative permeability of an oil-brine

o).Y

1 . 0

. 8

. 6

. 4

. 2

o

O B P = 0 p s i g

O B P = 3 O O O p s i g

.t. Ef

rnlsr

r G F

\lv

r d F

rtct Ir G d

.{\rr."{rr\r

lr.crj

rJ;ger

r rlqh

nl gl

I o m

Felrr

.unef

\ TL1

:l�T\ L

FnrJ.( srfi

:rrj r

f{fflr1

elht

0 2 0 4 0 6 0 8 0 1 0 0

s o

FIGURE 37. Effect of overburden pressure on gas relative permeability.rr5

81

1 . 0

oJ

1 0 0

s*

FIGURE 38. Effect of absolute permeability on relative permeability.ro

on the gas-liquid relative permeability-saturation relationship of a consolidated Nichols Buffsandstone. Botset's results were in agreement with the findings of Leverett,a who used sandswith permeabilities ranging from 1.04 to 6.80 D.

Morgan and Gordon28 conducted tests on four sandstone samples from a reservoir rockwith permeabilities ranging from 109 to 213 mD. No clear effect of permeability on oil-water relative permeability curves was observed. Crowell et al.30 studied four different sandswith absolute permeabilities ranging from 3.0 to 8.0 mD and found no correlation betweenabsolute permeability and gas relative permeability in a water-gas system as shown in Figure3 8 .

Keelanr20 observed satisfactory correlations of sandstone air permeability corrected forslippage and the irreducible water saturations as well as end-point relative permeabilityvalues of gas-water systems. Leas et al.r2r noted a correlation between absolute permeabilityand gas relative permeability in particular cases, but believed this relationship not to be truein general.

Felsenthalr22 tested 300 sandstone cores and noted that the gas-oil relative permeabilitycurves became less steep as specific permeability increased. This trend was also reportedby McCord.'23 In Felsenthal's paper an effect of porosity on gas-oil relative permeabilityratio was also noted. This effect was not generally discernible in the study of relativepermeability data for a given reservoir but became apparent when data for sandstone reservoirsof similar lithology but differing average porosity were compared. For example, a definitetrend was observed in a comparison of argillaceous and/or calcareous sandstones from I Ireservoirs ranging in average porosity from l4 to 28Vo, indicating that for a given perme-ability, the gas-oil relative permeability ratio curves became less favorable, (i.e., k1k., increased)

4 0

82 Relative Permeabiliry of Petroleum Reservoirs

as porosity increased. A similar trend was observed for a group of clean sandstones from

five reservoirs ranging in porosity from 15 to2lTa.For a given porosity and permeabil ity,

comparatively clean sandstones gave more favorable gas-oil relative permeability ratio curves

than argillaceous and/or calcareous sandstones or chert reservoirs. The least favorable gas-

oil relative permeability ratio curves were for conglomerates, shaly sandstones, and sand-

stones containing carbonate inclusions. Felsenthal then classified sandstones in three cate-

gories and found a correlation of gas-oil relative permeability ratio for each class. The

parameters used in the correlation were porosity, permeability, and sandstone type, which

are all related to pore geometry. On the other hand, pore geometry may be characterized

by the pore size distribution and Felsenthal found a correlation between gas-oil relative

pl.-.uUitity ratio and pore size distribution. He found that the more favorable gas-oil relative

permeability ratio curves were generally associated with a pore size distribution curve having

a sharp peak among the large pore sizes.

XII. EFFECTS OF TEMPERATURE

Several early studiesr2a-r28 indicated that ineducible water saturation increased with in-

creasing temperature and that residual oil saturation decreased with increasing temperature;

all of these studies employed a dynamic displacement process. Difficulties in evaluating

these results include possible wettability changes due to the core-cleaning procedure,rro

possible changes in absolute permeabil ity, and clay migration.t24't2'7't2'1

Steady-state relative permeabil ity measurements by Lo and Munganr2e indicated that the

relative permeabilities were temperature-dependent when using white oils, but were unaf-

fected by temperature changes when using tetradecane; this finding agrees with the results

of Edmondson.'r. Other variations in results have been attributed to viscosity ratio. Sufi et

al.r30.r3r pointed out that some of the previous results may have significant error due to the

difficulty in measuring relative permeabilities at elevated temperatures and suggested that

t.*p..uture effects possibly result from a combination of measurement difficulties and

laboratory-scaling phenomena, (i.e., end effects in short cores).

Miller and Ramey,32 performed dynamic displacement experiments at elevated tempera-

tures on unconsolidated sand packs and a Berea core. Their results indicated that changes

in temperature do not cause relative permeability changes, but that changes in the flow

capacity at elevated temperatures are due to clay interactions, change in pore structure, etc.

The only change that they observed was an increase in oil relative permeability at irreducible

water saturation and this parameter is relatively unimportant for predicting two-phase flow

behavior. In measuring steam-water relative permeabil it ies, Counsilr33 and Chen et al-r67

also noted the absence of temperature effects.

XIII. EFFECTS OF INTERFACIAL TENSION AND DENSITY

The interfacial forces at fluid-fluid and fluid-solid interfaces are responsible for retention

of residual saturation in porous media. Wyckoff and Botset3 and Leverett4 described a small

but definite effect of interfacial tension within the range of 27 to 72 dyne/cm on relative

permeability. (See Figure 39.) Lefebvre du Preyro3 also identified the interfacial tension of

fluids in a consolidated sample as a factor influencing the relative permeability and residual

saturation values. Crowell et al.30 found that a reduction in interfacial tension of a water-

air system produced an increase in gas recovery and a decrease in residual gas saturation.

IV1uskatr3a discounted the possibility that the interfacial tension within the range of 27 to

72 dynelcm can influence relative permeability. Owens and Archer" concluded that inter-

facial tension has no influence on either the water-oil relative permeability of a water-wet

core or the gas-oil relative permeability of an oil-wet core. They found that water relative

tErrrr..rr txh

\1, r' a f l

. r t

.rlc I IIr

. I : : l s :

'rfftrrrY

. 1 . 1 € ,

rrnr.

L { r ,:E l r r r ,

4x.{r-r'rl r r..

tatesc.rfrso.:.ffC :

b !trLI1

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f:r ;i

:rrh"r.rlJnx

83

nJ-tr)t ' lcs from

| 1 . - r r r rcab i l i ty ,i t r r . r t io curves

fur , ' r l rble gas-

l lc . . . rnd sand-

i rn lhrL'e cate-

r h . l a s s . T h e

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I . h . rnrc ter ized

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g. r r r r l l rc la t ive

n . . r - \ c hav ing

is*.r.r:.1 rr ith in-

It :J: l lP! 'rature;

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g p:, ' . , cdure, l16

I r . . , : . ' .1 that the

t ' . . : .r crc unaf-

t r : : : lhc rc-su l ts

N : . r i i t r . S u f i e t

l r r , ' : r luc t t l the. . . - - : c . t cd t ha t

d r : : . . r l t i c s a n d

i l .r : . ' .1 tcnlpera-

$ : : . ' t changes

lB. . : : thc f low

i . : : . . . t L l f g . e t c .

!r . : : :rrcducible

lu, i . l tase f low

J ( ' : : . 'n c- t a l . r67

S I I \

) l c : , , r rc ten t ion5 r-rl.cd a small,cnr ()n relativeL'r.rl tension of[tr .rnd residualiitrn ..rf a water-

8J. \aturation.

I rrnrr-' of 27 touticJ that inter-trl I \\ ater-wet

I s ltcr relative

1 . 0

ol<

p o i n t s : o - 5 d y n e / c m

\ o

\ l i n e s I o - - 2 4 - g 4 d y n e s / c m /

o \ o o o ' /

\ o /

o r l \ o /

\ . o /\ . /\ o o

\ "

/ w A r E R

\ " /\1." /;eCo

* / , ' ) * *

/

/

00 1 . 0

a" w

FIGURE 39. Effect of interfacial tension on relative permeability.a

permeability of the water-wet core and oil relative permeability of the oil-wet core werecoincident.

Moore and Slobod6T reported a reduction in waterflood residual oil saturation of a water-wet core at lower values of interfacial tension. Pirsonr02 stated that drainage relative perme-

ability is independent of the interfacial tension, but imbibition relative permeability is sen-sitive to interfacial tension. Bardon and Longeronr35 found that a reduction in interfacialtension reduced oil relative permeability at constant gas saturation in an oil-gas drainagecycle of the Fontainebleau formation. (See Figure 40.) The effect of liquid density on relativepermeabil ity has been found to be insignificant.- ' 'r2

XIV. EFFECTS OF VISCOSITY

Leverett et al.a'8 investigated the effect of viscosity variation of an oil-water mixture onrelative permeabil ity of artif icially compacted sands with 417o porosity and 3.2to 6.8 D ofabsolute permeability. He found no systematic variation in relative permeability when theoil viscosity was varied from 0.31 cp (hexane) to 76.5 cp (lubricating oil) and the waterphase viscosity was varied from 0.85 to 32.2 cp. Viscosity ratios employed in the studyranged from 0.051 to 90. The experiments of Leverett et al. were performed under steady-state flow at low pressure gradients. Figures 41 and 42 show the effect of viscosity ratiovariation on water and oil relative permeability curves.

Wyckoff and Botset3 found that moderate variations in viscosities of the fluid phases inunconsolidated sand packs with permeabilities ranging from 3 .2 to 6.0 D failed to produce

any change in the relative permeability values. In their experiment a mixture of water andcarbon dioxide was employed and water viscosity was adjusted between 0.9 and 3.4 cp byaddition of a susar solution to the water.


. 5

ss

FIGURE 40. Effect of low interfacial tensions on gas-oil relative permeability.r15

O M E g O .a 1 . 8 0D 0 . 3 5* 0 . 0 5 7

o 1 . os w

FIGURE 41. Effect of viscosity ratio (M) on water relativepermeability.a

Richardsonr36 found that the water-oil relative permeability ratio is independent of fluidviscosity where the oil viscosity varied from I . 8 to I 5 I cp (see Figure 43) . Johnson et al . r 37

confirmed these results for displaced/displacing viscosity ratios up to 37. Leviner38 found

1 . 0

O . 5

J

1 . 0

t { -

t u : r\ lhgtlri

n {\I

. \G

&

rr irr

rtx!

Fr l'-,\|uri

r{rlt

f.n

.L iri

itrnrlr I

Sr;rtl

l rhr

ra irX

l u

r(:tB

crr0

I rrt

l l

*'a D l

lIllrl

llc dtn r ts

3.Y

o = . 0 O 1 m N / m

85

o M = 9 0

o 1 . 8

o . 3 5v . 0 5 7

r \ '% \ o

s*

FIGURE 42. Effect of viscosity ratio (M) on oil relativepermeability.a

that the relative permeability of a sandstone sample was independent of viscosity ratio inthe range of 1.92 to 22.6. Craigr3e reported that the gas-oil relative permeability ratio of aNellie Bly sandstone sample with 824 mD permeability and 28.l%o porosity showed nosignificant variation with oil viscosities in the range of 1.4 to 125 cp. Results of this studyare illustrated by Figure 44.

Sandberg et al.'ao found that oil and water relative permeabilities of a uniformly saturatedcore are independent of the oil viscosity in the range of 0.398 to 1.683 cp. Donaldson etal.'o' and Geffen et al.ta2 also concluded that relative permeability is independent of viscosityas long as the core wettability is preserved. Wilsonrra found that a 5000 psi fluid pressurewhich caused kerosene viscosity to increase from I .7 to2.l cp and water viscosity to increaseby 17o did not produce any significant effect on water and oil relative permeability values.Muskat et al.27 reported that the effect of viscosity on relative permeability of an uncon-solidated sand was very small and within the limits of experimental accuracy.

Krutter and Day'43 used methane and air as the nonwetting phase in a two-phase systernof oil and gas. The gas was injected into cores saturated with oils with viscosities rangingfrom 2 to 100 cP. They found that the air relative permeability values were slightly lessthan those for methane.

Saraf and Fattro applied Darcy's law to each of the phases of a multi-phase system andconcluded that relative permeability is independent of viscosity. The Saraf and Fatt equationis based on the assumption that different phases flow in different capillaries and do not comein contact with each other.

Yuster,6 however, concluded that relative permeability values for the systems he studiedwere markedly influenced by variation in viscosity ratio, increasing with an increase of theratio. This conclusion was later supported by the work of Morse et al.r44 Odehr45 expandedYuster's work and concluded that the nonwetting phase relative permeability increases withan increase in viscosity ratio. He found that the magnitude of the effect on relative perrne-ability decreases with increase in single-phase perrneability. Odeh found that the deviationin nonwetting phase relative permeability is increased as the nonwetting phase saturation isincreased, with the deviation reaching a maximum at the nonwetting phase residual saturation.He also concluded that the wetting-phase relative permeability is not affected by variationin viscosity ratios. Figure 45 shows the effect of viscosity ratio variation in the range of 0.5

1 . 0

o-Y

1 . 0

\

nnjcnt of fluid

hn . r rn c t a l . r 37

!\ lnc ' ' found


q+6 i . l i ' -q la+6

! ncp,eri-nent

j Waterflood using151 cp. oil

A waterflood usingKerosene

9 0 1 0 0

sw

FIGURE 43 . Compar i son o f s teady -s ta te resu l t s w i th f l ood ingperformance.r36

to 74.5 on water and oil relative permeability curves. Odeh stated that the effect of viscosityratio on relative permeability could be ignored for samples with single-phase permeabilitiesgreater than lD. Yuster's and Odeh's results have been crit icized by other investigators.ra6

Downie and CraneraT reported that oil viscosity could influence the oil effective perme-ability of some rocks. Later, they qualified their statement by saying that once an increasedrelative permeability is obtained by employment of high viscosity oil, it may not be lost byreplacing this oil with one of a lower viscosity. They explained this phenomenon qualitativelyin terms of the movement of colloidal particles at oil-water interfaces.

Hassler et al.r found that lower gas relative permeability values were associated withlower oil viscosity in a Bradford sand. However, they expressed doubt that the variation inrelative permeability could be described by a single factor varying with oil viscosity.

Pirsonro2 stated that the importance of the effect of viscosity ratio on the imbibitionnonwetting phase relative permeability is of second-order magnitude. Ehrlich and Cranera8concluded that the imbibition and drainage relative permeabilities, under a steady conditionof flow, are independent of viscosity ratio. However, they found that the irreducible wetting-phase saturation following a steady-state drainage, when the interfacial effect predominated

6 0

o; 4 1

3j

5 04 03 0 6 0 8 07 0

i \ ( : i

L'' t€1

\L

rvnrsLell

rlr rcPcr

\ l \ - \ !

Px\c.rFri8L'Ct: l !

:lt !j

srfvc

[YF\x. f u h{:-aJL

;wFR

87

1 . 0

0 . 1

o

J

j

0 . 0 1

t r ' i \ r rcos i tyr : : : r c.rhi l i t iesE. : t j . r t t l r s . l 16

f l: . i' [,L*rme-ar. tnarr 'ased

Ft \e l t lst byqu . r J r t a t i ve l y

f \ r . r tcd wi th

I \ . : n J t t o n i nl t - , ' . l l ) .

r r rnh ib i t ionan. l ( - ranera8

dr . , ,nd i t ionil. . ' r i ctt ing-rr ' . i , ' ln inated

0 . 0 0 1o .4

Q"g

FIGURE 44. Relative permeability ratios for Nellie Bly sandstone.rre

over viscous and gravitational effects, decreases with an increase in the ratio of nonwettingto wetting-phase viscosities.

McCafferyse reported that in strongly wetted systems, the imbibition and drainage relativepermeabilities are independent of the viscous forces. He concluded that even though therelative permeability to a phase might be influenced by viscosity variation of that phase,the relative permeability ratio is independent of viscosity.

Perkinsrae concluded that flow in a porous body is governed by relative permeability andviscosity ratio when the ratio of capillary pressure to the applied pressure is negligible.Pickell et al.r-to concluded that only a large variation in viscous forces could have anyappreciable effect on residual oil saturation. Several authors 4'67'rsr rs3 recognized that thewetting and the nonwetting phase relative permeability might be significantly affected bythe ratio of capillary to viscous forces, ocos0/pv, where o represents interfacial tensionexpressed as dynes per centimeter; 0 represents contact angle; p represents viscosity ex-pressed as cp; and v represents fluid velocity expressed as centimeters per second. Lefebvredu Prey'sa made a systematic study of the effect of this ratio on relative permeability bysimultaneously varying the interfacial tension, viscosity, and velocity. He found that relativepermeability decreases as the ratio ocos0/pv increases. He also concluded that the relativepermeability curve is influenced by the viscosity ratio when the wetting phase is displaced


240

o 5 0 1 O O

a-w

FIGURE 45. Ef fect of v iscosi ty rat io (M) on re lat ive permeabi l i ty . r ls

by the nonwetting phase. Bardon and Longelsnr'3-s found that in some gas-oil systems, thedrainage relative permeability and residual oil saturation are strongly affected by the p"vlctratio.

An assumption that the relative permeabil ity values are independent of viscosity impliesthat the system can be represented by a bundle of parallel, noninterconnecting capillarytubes, each of which is f i l led with either the wetting or the nonwetting phase alone. Thus,the nonwetting phase flows through the larger channels while the wetting phase flows throughthe smaller capillaries. However, this model probably does not completely represent theconditions in porous media. An alternative model is the simultaneous flow of two immisciblefluid phases in larger capillaries.

A flow picture more compatible with the present knowledge of fluid behavior is a com-bination of the two models described above, with one dominating over the other dependingprimarily on wettability. OdehT believed that the fluid phases did not flow in separatecapillaries of porous media as Leverett postulated and further stated that the wetting phasemoves microscopically in a sort of sliding motion imparted to it by the shear force causedby motion of the nonwetting phase. From this model he concluded that a decrease in interstitialwetting-phase saturation can be developed as a result of an increase in viscosity, therebyaffecting the relative permeability values.

In view of the diverse opinions which have been expressed by various investigatorsconcerning the influence of viscosity on relative permeability, it seems best to conduct

t6L

-Y

bhri

ar\lt\

Ttx(-rrllIITUT

!l! rt

thc .uSr

Fffrx?f..rl

Yfi{rL

nrrd

lfct

rrlIrr

G \ |tr.Lrrr

G t f ,

l$ , , \

h - .

r :g,llI

89

W a t e r P r e s e n t a t S t a r t

- - - 5 %

- 1 0 2

- - - - - - 2 0 2

\r o l L

/ W A T E R

-:='

sw

FIGURE 46. Effect of original water saturation on relative permeability.' '

laboratory relative permeability experiments with fluids which do not differ greatly in vis-cosity from the reservoir fluids.

XV. EFFECTS OF INITIAL WETTING-PHASE SATURATION

The amount of initial interstitial water affects the oil-water relative permeability values.Caudle et al.ra investigated this relationship. Figure 46 shows the effect of varying theamount of initial water saturation on water and oil relative permeability. It can be seen thatnot only the starting points, but also the shape of the relative permeability curves vary withthe amount of initial interstitial water.ro'

SaremrT2 found that the presence of initial water saturation tended to shift water-oil relativepermeability ratio curves toward the region of lower oil saturation. The difference in theresidual oil saturation caused by this shift was reported to be about half the difference ininitial water saturation. Thus, a lower residual oil saturation is obtained at higher values ofinitial water saturation.

Henderson et al.3-t'r6s noted that the maximum effect of initial water saturation on therelative permeability curve was a shift of the entire curve laterally approximately 4Tc alongthe saturation axis, in a direction which increased the oil saturation for a given pair ofrelative permeability values. Craig indicated that up to 20Va initial connate water saturationin oil-wet cores had no effect on oil-water relative permeabilities. However, a definite effectwas observed in water-wet cores.

It is suggested that, except for special studies, the amount of water present at the start ofa relative permeability determination should be the irreducible water saturation of the sample.

oJ

1 0 08 06 04 02 0

DO

| . ' . . i : l l l r . t h e

l t . . lhc pr ' , r r

! r ) - ' , . r r n p l i e sI r r * . . r p i l l a r y' . r . ' : . . ' Thus.l l , ' . i . t h r t l ugh

fC : ' : , ' .Cn t the

t t t ; : : : l t t i r c i b l g

/ l r t : l \ J C( )m-

F : . l c p c n d i n g

I . : r . cpa fa te

I c: : r r r phasel , '1 . i ' CaUSed

I t r : : ) l c r s t i t i a l

hr : r . thereby

t n ' . g : [ 1 9 1 1 1 9 1 5

! : . ' J ( )nduc t

I

II*"

^o

";/: l3 l tt /t /

n o c o n n a t e

It/

I: lO l

slil r,!ttl;lI

{t/


1 0 0

6 0 8 0 1 0 0

e" g

FIGURE 47. Effect of connate water on relative permeability ratio.'7*

XVI. EFFECTS OF AN IMMOBILE THIRD PHASE

Many hydrocarbon reservoirs have only two mobile fluid phases. The mobile phases may

be gas and oil in the upper portion of the reservoir and water and oil in the lower portion.

Thus, two-phase relative permeabilities are sufficient to characterize fluid flow behavior in

these reservoirs.Some investigators suggest that the immobile water saturation may be regarded as part

of the rock, and gas and oil saturations may be given in terms of the hydrocarbon pore

space. Owens et al.rss'r73 tested several native-state and cleaned cores, both water-wet and

oil-wet. and found that an immobile connate water saturation had no measurable influence

on the gas-oil relative permeability ratio in the majority of the cases that were studied.

CalhounrTa concluded that low water saturations did not appreciably affect the permeability

ratio, simply because the water occupies space which does not contribute to the flow capacityof the rock. Figure 47 shows the effect of connate water saturation on gas-oil permeability

ratio. Stewart et al.'tt have also shown that in a limestone with intergranular porosity, the

effect of interstitial water on external gas or solution gas drive gas-oil relative permeability

ratio is negligible.Leas et al.'2' reported a close agreement between the gas-oil relative permeability of a

system at various values of interstitial water saturation. This agreement was best in the

oj\. 1

o).:<

. 0 1n+-.:-l.lrFrtr* : :!a.L:

f i . , I

Il|: a,

l r i :

:f"r3-

{ t } \ |

={Trc\r.E rF.&.

ll:,urf :f1.i

ll:uf t ; n6fkra .f$f

2 0 4 0

i l

t

9l

ol<

\\\

\\\

1 5 - 2 5 e a c o n n a t e w a t e r f

""ut" t";' ,r

/ /

\ , / , /\ y / o t f

\ / /G A S

.z'\ ' '--

1 0 08 06 04 0

S o -

.__ sg

FIGURE 48. Efl 'ect of the presence ol' connate water on relativepermeabi l i t ies. l

equilibrium gas saturation region. They concluded that the gas relative permeability isdependent on total liquid saturation. Other investigators have suggested that even thoughthe immobile connate water does not appreciably affect the relative permeability ratio, theamount and distribution of the interstitial water may influence the relative permeabilitycurve. Dunlap,r18 Leverett,a Caudle et al., '" and McCaffery' 'e have indicated a dependencyon connate water saturation. Figure 48 compares the permeability-saturation curves for oiland gas at l5 to 25Vo connate water with the corresponding curves without connate water.

Kyte et al.t7o studied a wide range of core materials and fluid properties that could influenceresidual saturation, to determine the mechanism of oil displacement by water in a partiallygas-saturated porous system. They found that the initial gas saturation is related to thetrapped gas saturation, which plays a beneficial role in reducing residual oil saturation.Mattax and ClotheirtTT concluded that the trapped gas saturation could improve oil-waterrelative permeability values in consolidated water-wet sandstones. (See Figure 49.)

Holmgren and MorserT8 attributed the oil recovery improvement of a sample in the presenceof residual gas to one or more of the following factors:

i [ - : . 1 \C \ n tay

nr i ' : | ' t r f t i t ln .) \ : l . r \ i t l r i n

lrj. ' .: .t: poft

I ; :hrr I ' l P0fea t . : - \ \ c t a n d

b le r r r l l uc 'nce

I c : , . t u d i € d .

1*-. : : : lc . rb i l i tyl l. ' .. . .1pra1,,

J*- : : r rcabi l i tyF ' : , ' . i t r . t h e

J*-: ':rcabil ity

E * ^ r l r t r o f at \ ' . r in the

l. The changes in physical characteristics of oil.2. The selective plugging action of the gas as indicated3. Inclusion of mist in the free gas phase.4. The additional sweeping or driving action of the free

by Kyte.

gas as indicated by Leverett.a.s

Holmgren and Morse concluded that the changes in physical characteristics of oil, withinthe pressure range used for their experimental work, were not sufficient to account for thedifferences in the residual oil saturation which were noted. They further stated that a changein displacement mechanism was the most important cause of the oil recovery improvement.


1 0 0

G A S S A T U R A T I O N , % P VM O B I L E T R A P P E D

o5

9' t 2

so

lfrU.}?"a:.,, water-oil relative permeability ratio improvement due to

Schneider and Owenssa investigated the effect of trapped gas saturation in sandstone andcarbonate rocks and concluded that the trapped gas affected water relative permeability morethan oil relative permeability in oil-wet rocks. These effects are illustrated in Figures 50and 5l . They also concluded that the trapped gas saturation lowered the maximum value ofoil relative permeability. Water relative permeability was also lowered as a result of anincrease in trapped gas saturation. These effects are illustrated by Figure 52.

XVII. EFFECTS OF OTHER FACTORS

The effects of displacement pressure, pressure gradient, and flow rate on the shape ofrelative permeability curves have long been a controversial subject in petroleum-relatedliterature. Some authors believe that the effect of displacment pressure and pressure gradientmay be due to the changes imposed on viscosity, interfacial tension, and other fluid or rockproperties. Others believe that the changes in relative permeability, which appear to resultfrom changes in displacment pressure and pressure gradient, are actually due primarily toan "end effect" developed during laboratory tests.

End effect or boundary effect refers to a discontinuity in the capillary properties of asystem at the time of relative permeability measurement. In a stratum of permeable rock,the capillary forces act uniformly in all directions, and thus negate each other. In a laboratorysample, however, there is a saturation discontinuity at the end of a sample. When the flowingphases are discharged into an open region under atmospheric pressure, a net capillary forcepersists in the sample; this force tends to prevent the wetting phase from leaving the sample.The accumulation of the wetting phase at the outflow face of the sample creates a saturationgradient along the sample which disturbs the relative permeability measurements. For ex-ample, a large difference in saturation at the displacement front causes a large capillarypressure gradient, which in turn causes the water to advance ahead of the flood front andto reduce the capillary pressure gradient in the measured region. The advancing water cannotbe produced when it first reaches the outflow face of a core, because the pressure in the

1 0

3-Y\ 1

oL

. o 11 . O

rll€t 1

:rg f

I C . -

I rcrlrr

t[cr*-tcrE(

tgt

i l r t

cl:r(.tr'sr

t a"r

lilrar1 I

*tr rr& r r|f'c.!*

fl clit&

hAr

I t ' : '

[.c'r€:

ffn

93

1 0 0

FIGURE 50. Ef fectcarbonate) . Ea

1 0

be

olz

. 1

i i l t l " l . 1 , ' nC and

E. r ^ . : l \ mo fe

in I . . r rc: 50l tu" . . r luc ' ofI r ; ' . . . .1 t r t ' an

th. .hapc' ofDk' . . : : r rc latedsu:r - l radientf l - : . 1 t r f fOCk

p . r : l r r r l . su l t

I p : : : : r l r i l r t o

Dft': i tc. t lf a

lltc'.rhlc rock,

l . r l . : h r131g1y

n thc t l ou ' i ng

lpr l l . rn tbrce

! tnc :ampls .

I .r . l luration

inl . FOr ex-luc . . rp i l laryul tr,rnt andtl ulcr e annotlr.-.t 'c in the

s * - 1 ' o

.-so

of t rapped gas saturat ion (o i l wet Grayburg

water just inside the core is lower than the pressure in the oil-filled space around the outflowface. This difference in pressure is equal to the capillary pressure for the existing saturationat the outflow face. Therefore, water accumulates at the outflow end of the core, causinga reduction in the capillary pressure. The water will not be produced until the capillarypressure is overcome and the residual oil saturation (at the outflow face of the core) isreached. The calculation of relative permeability based on the average saturation of thesample produces erroneous results in this case, since the relative permeability varies through-out the core due to the saturation gradient created by the wetting phase accumulation at theoutflow face of the core.

Owens et al.,r-5s Sandberg et al.,to" Kyte and Rappoport,rs6 and Perkinsrae believe thatthe most convenient way of minimizing the boundary effect is the adjustment of capillaryforces to insignificant values, as compared to the viscous forces. This is usually done by aflow rate adjustment. However, the adjusted rate must be low enough so the inertial forcesdo not disturb the laboratory measurement. It is suggested that the higher flow rate alsoincreases the fluid dispersion at the inflow end of the sample, so that fluid mixing is enhanced.An equation has been developedr-57 to predict the extent that a core can be disturbed byboundary effect, at a given rate. Another convenient way of minimizing the boundary effectat the outflow end of a core is to use a more viscous oil in a longer core.rs6

Leverett et al.a'8 reported, then refuted, the influence of flow rate upon relative perme-ability. They eventually attributed the observed deviations in their results to an end effect.

\ * r * v s . S ,

o

k r o v s . S o

T r a p p e d G a s S a t .

o 0 %

. a 1 1 . 8 %

k r o v s . S o

T r a p p e d G a s S a t .

o 0 %

a 1 0 . 3 %o 1 9 . 2 %


1 0 0

s, r -

t_ So

RGURE 51. Effect of trapped gas saturation (oil wet Tensleep sandstone).Ea

such as that previously described by Hassler.r Crowell et al.3o found that a 50-fold variation

of injection rate, within the limits of viscous flow of water and gas, had no effect on residualgas saturation of an Arizona sandstone. Geffen et al.ra2 also concluded that, at reasonable

flow rates, the effect of waterflooding rate on the efficiency of gas displacement was

negligible. Henderson and Yuster3s and Morse et al.r-58 found that relative permeability was

rate-dependent in all gas-liquid systems that were studied. Wyckoff and Botset3 also found

that the gas and liquid relative permeabilities were rate-dependent when the two phases wereallowed to flow through the core under the same pressure gradient.

Caudle et al.'a found that relative permeability decreased with increase in flow rate when

one of the flowing phases was a gas. Labastie et ol.,'-tn however, investigated the effect of

flow rate in a water-wet sandstone and oil-wet carbonate cores and concluded that relative

permeabilities were independent of flow rate except near residual oil saturation. Sandberg

et al . , rao Richardson et a l . , r - t7 Osoba et a l . , r3 and Leas et a l . r2r found that drainage rela-

tive permeability is independent of the flow rate as long as a saturation gradient is not

introduced in the core by the inertial forces. Pirsonr02 concluded that relative permeability

is not rate-sensitive in drainage processes. Ehrlich and Crane'ot examined the effect of flow

rate variation on steady-state relative permeability and concluded that both imbibition and

drainage relative permeability were independent of flow rate.

Handy and Dattar62 found that the imbibition relative permeability values were dependent

on the imbibition procedures; that is, the relative permeability values under free imbibition

be

oY

r€f,C r

:ntrrhri

:srf&

AJ .J

Sr.fr{

tf !l"rr

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were larger than those under a controlled process. The difference between free and controlledimbibition was found to be smaller for more permeable samples. Perkinsrae found that theresidual oil saturation after flooding was independent of the flooding rate and concludedthat capillary forces controlled the microscopic fluid distribution in the core. Moore andSlobod6T reported that waterflood recovery from a water-wet core was practically independentof flooding rate. However, they observed that a significant recovery increase may be obtainedat extremely high rates. Hupplerr6s stated that the waterflood recovery from cores withsignificant heterogeneity was sensitive to flooding rate. Lefebvre du Preyrsa concluded thatthe relative permeability was a function of velocity (v), through the ratio (ocosO/pv), whenthe viscous forces predominate.

Wyckoff and Botset,3 Leverett,a and Henderson et al.3-s'r6s studied the possible effects ofdisplacement pressure and pressure gradient on water-oil relative permeability. They con-cluded that the water and oil relative permeability values were slightly influenced by thesefactors. Muskatr3a and Krutter and Day,'oo however, reported that the gas and oil relativepermeability values of a consolidated sandstone were not affected by changes in differentialpressure. McCafferyrTe indicated that the drainage relative permeability values were notinfluenced by the flow rates which result from apressure gradient in the range of 1.0 to 5.0psi across a 12 in. core. Delclaudr60 also concluded that relative permeability is independentof displacement pressure. Pirson,ro2 however, suggested that the relative permeability in animbibition cycle is sensitive to pressure gradient.

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Krutter and Dayr66 found that ultimate recovery increases with increasing pressure gradient,although the ratio of increased recovery to increased pressure gradient decreases in the regionof high pressure gradients. Brownell and Katzr68 reported that an increase in pressure gradientdecreased the residual saturation toward zero in the systems that were investigated. Geffenet al.ra2 also confirmed that residual gas saturation was a function of pressure gradient.Stegemeier and Jensen3T believed that the residual wetting phase in a drainage process washeld in pendular rings interconnected with only thin wetting-phase layers. They concludedthat this residual wetting phase was trapped by capillary forces and that a higher pressuregradient might overcome the capillary pressure and reduce the residual wetting-phase saturation.

Stewart et al.rre observed that the rate of pressure decline in a nonuniform limestone mightinfluence the gas-oil relative permeability ratio when the solution gas displacement techniquefor relative permeability measurement was employed. Wall and Khuranar6e found the gassaturation developed in a sand pack, at a given rate of pressure decline, was a function ofthe mean particle size and probably a function of permeability. They found that a finer grainsand pack gave rise to higher gas saturations in the solution gas displacement technique.

Crowell et al.30 studied the effect of core dimensions on laboratory measurement of relativepermeability. They found that the residual gas saturation in water-gas systems was almostindependent of length of the core, within limits of the laboratory-scale models used. Theyalso examined cylindrical and rectangular samples, and observed that a 100-fold change inthe ratio of core length to core cross-sectional area of Berea and Boise sandstones did notalter the residual gas saturation of the samples. Moore and SlobodoT also found that fluidrecovery from water-wet cores was not affected by the sample length. Perkinstae andMcCafferyrTe recommended the use of longer cores, to reduce influence of the end effect.

RoserTo studied the effect of gas expansion, created by the pressure gradient along thesample, on gas-liquid flow characteristics. He concluded that a necessary condition forcorrect steady-state measurements of liquid-gas relative permeability was the establishmentof a uniform fluid saturation distribution in the core. Osoba et al. '3 found that gas expansionaffected gas and oil relative permeability values in tests conducted at near-atmosphericpressure. Richardson et al.,r-'7 however, found that the effect of gas expansion on gas andoil relative permeability values was insignificant at the low pressures which were employedin their study.

In the laboratory gas displacement method of relative permeability measurement, a "sta-

bilized zone' ' tends to form when the wetting liquid saturation is sufficiently high to permitits readjustment faster than the imposed displacement by the external drive. The relativepermeability values obtained prior to passage of the stabilized zone are not valid. Therefore,it is advantageous to reduce the range of saturation influenced by the stabilized zone, toobtain valid measurements over as wide a saturation range as possible.

It can be shown from the Buckley-Leverett equation that the saturation at which thestabilized zone passes out of a system is inversely related to the viscosity of the displacedliquid. This relationship is based on an assumption that a true stabilized zone forms inlaboratory gas drives on short cores. It can also be shown that the length of the stabilizedzone is inversely related to the injection rate or differential pressure. It has been suggestedthat the stabilized zone will be sufficiently small if the pressure differential is of such amagnitude that a volume of gas approximately equal to one half the pore volume of thesample would be produced in less than 60 sec. This flow rate insures that the portion of thecore in which the capillary effects predominate will be a negligibly small fraction of thetotal pore space. Loomis and CrowellrT' showed experimentally that the influence of thestabilized zone fluid flow is much less marked with relatively viscous oil as the displacedphase.

Botset2' investigated the effect of saturation pressure on gas-oil permeability values andconcluded that the saturation pressure had negligible effect on laboratory relative permeability

mcasudisplasupentests cgener:

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97

measurement. Stewart studied the effect of gas supersaturation on laboratory solution gasdisplacement relative permeability measurements. He indicated that even though very liitlesupersaturation exists under most field conditions, the effect may be significant for laboratorytests conducted at high flow rates. He found that the gas-oil relative permeability ratio wasgenerally independent of the degree of supersaturation in rock with intergranular porosity.

The influence of dispersion on relative permeability was studied by Chilingarian et al.eeThey concluded that an increase in degree of dispersion increased the relativs permeabilityof the porous medium to both the continuous and discontinuous phases. They also concludedthat the degree of dispersion increased with decreasing interfacial tension and increasinetime of coalescence of dispersed-phase droplets.

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r- Bflt n r t

5t( \lorJ (

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6 l . \ le lcl l l,

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6-1. \tu6 5 . A m66. Ra:

' l f t "

67 \to\\ JI

6t( BolC \ J

6e KillR,r

t,, nt\ c r

7 l D u

c l l r

7t \ tu7-1 t-nr

'f c,

7-r Kt,pl.r

75 ( ia

. l l t '76 Jot

\ tc'11

Ad7lt. SID

l 9 :79 [.or

ttr It{o Re:

\a cl

8 t c do i l

f{l Koti-1 Pa

I t .Rc,

8.1. Sclchi

ti-5. ScrTtk

86. Anl'ttr

87 . Cdc()11

t l8. Hrpet

l r l . ,rhtrratoty

l P i . : ' . : h \ n n u a l

l t , . - , . t h r o u g h

B : - " , . J P e t .

' . c r . i t r o f

F- : , , . r i c r d r ive

h - - r . : l h u t i o n ,

l r i - . : ' . . r !Cn len t

x , - ' . . | ' r u n s .

n : . ' - : , ' P , t l o g ) ' .

l (} j .

r h , ' , , l a s o f

f - . : - \ n s l e s .

I ; ' - ' , ' r r r c n a i n

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) , l / , r r . 1 0 .

f , \ . . ' . : : . r h r l i t r .

t

I , . , : . , t n a n d

) ' . : l J r o f

- ! j ' , : [ - t t g .

l - . . r n J l l ( ) n s

] : ' . - - . : . 1 . \ C U

S i , i : \ I \ I E .

T . I I . IT E ,

I i \ ' . . ' r l red ia ,

I . . . - ' : r J g n e t i c

S . ; . : l l o n . ,

i : . - 1 9 6 3 .

I t - . . : , o r l s i n

, ( i . . U E T C /

l r - - " . : . ' l - r t t r t s .

r.

_ 9 9

57 . Boneau, D. F. and Clampitt, R. L., A Surfactant System for the Oil-Wet Sandstone of the North Burbank

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t 9 5 . t . t 9 5 2 .

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84. Schneider; F. N. and Owens, W. W., Sandstone and carbonate, two- and three-phase relative permeability

character is t ics, Soc. Pet . Eng. J. , 3,15, 1970.85. Scrom,H.M.,Signi f icanceof Water-Oi lRelat ivePermeabi l i tyDataCalculatedfromDisplacementTests,

Theory of Fluid Flow in Porous Media Conference, University of Oklahoma, 1959, 189.

86. Amyx, J. W., Bass, D. M., and Whiting, R. L., Petroleum Reservoir Eng,ineering, McGraw-Hill, New

York. 1960.

87. Colpits, G. P. and Hunter, D. E., Laboratory displacement of oil by water under simulated reservoir

conditions, J. Can. Pet. Technol., 3(2), 64, 1964.

88. Haddenhorst, H. G. and Koch, R., Effect of temperature and pressure on the separation of solids from

petroleum, Erdoel Kohle, 2, 12, 1959.


89. Luks, K. D. and Kohn, J. P., The Effect of Methane Under Pressure on the Liquid Solubility of HeavyHydrocarbon Components, Liquid-Vapor and Solid-Liquid-Vapor Behavior, progress Report II, Apl Re-search Project 135, Notre Dame, Indiana, July, t971.

90' Rathmell, J. J., Braun, P. H., and Perkins, T. K., Reservoir waterflood residual oil saturation fromlaboratory tests, J. Pet . Technot. ,225, l i �5 . lg j3.

91. Richardson, J. G., Perkins, F. M., Jr., and osoba, J. S., Difference in behaviorof fresh and aged eastTexas Woodbine cores, Truns. AIME,204, 86. 1955.

92. Mungan, N. , Relat ive permeabi l i ty measuremenr using reservoir f lu ids, Soc. pet . Eng. J. , l2(5) ,39g,1972 .

93. Ehrlich, R., Hasiba, H. H., and Raimondi, P., Alkaline waterflooding for wettabil ityevafuat ion of a porent iat f ie ld appl icar ion, , / . pet . Technot. ,26, 1335, lg j4.

94. DeterminQtion of Residual Oil Saturatior?, Interstate Oil Compact Commission, Oklahomat978.

alteration -

C i t y , Ok la . ,

95. Jennings, H. H., Surface properties of natural and synthetic porous media, prod. Mon., 2l(5). 20. 1957.96' Hough, E. W., Rzasa, M. J., and Wood, B. 8., Interfacial tensions at reservoir pressures and temperatures,

apparatus and the water-methane system, Trans. AIME, 192, 5i, lg5l.97. Poston, s. w., Ysrael, s., Hossain, A. K., Montgomery, E. F., and Ramey, H. J., Jr., The Effect

of Temperature on Relative Permeability of Unconsolidated Sands. paper SPE 1897 presented at the SpE42nd Annual Fal l Meet ing, Houston, Texas. , 1967.

98. Cuiec, L. E., Restoration of the Natural State of Core Samples, paper SPE 5634 presented at the SpE 50thAnnual Meet ing, Dal las, Tex. , 1975.

99. Chifingarian, G. V., Mannon, R. W., and Rieke, H. H., Eds., Oil and Gas production From CqrbonctreRocks, Elsevier , Amsterdam, 1972.

100. BradleY, D. J., The Applicability of Wettability Alteration to Naturally Fracrured Reservoirs anrl lmbibitionWaterf looding, Masters thesis, Univers i ty of Tulsa; 'Oklahoma, 19g3.

l0l' McCafferY, F. G. and Mungan, N.' Contact angle and interfacial tension studies of some hydrocarbonwater sol id systems, J. Cut t . pet . Technot. , j , lg5, 1970.

102. o i l Reserv 'o i r Engineer ing, Pirson, s. J . , Ed. . McGraw-Hi l l , New york. 195g.6g.103' Lefebvre du Prey, E., Deplacements non-miscibles dans les mill ieux poreux influence des parameters

interfaciaux sur les permeabilites relatives, c.R. IV Cotoq. ARTFp puu, 196g.104. Donaldson, E. C. and Thomas, R. D., Microscopic Observations of Oil Displacement in Water-Wet and

Oi l -Wet Format ions, SPE 3555 presented at the 46th SPE Annual Fal l Meet ing, New Orleans, Oct . 3-6.1 9 7 t .

105' McCafferY, F. G. and Bennion, D. W., The effect of wettabil ity on two-phase relative permeabilit ies.J. Can. Pet . Techno1., 10. 42. 1974.

106. Levine, J. S. , Displacement exper iments in a consol idated porous system, Trans. AtME,20l , 57, t9-54.107. Josendal, V. A., Sandford, B. 8., and Wilson, J. W., Improved multiphase flow studies employing

radioact ive t racers, Trons. AIME, I95,65. 1952.108. Terwill iger, P. L., wilsey, L. E., Hall, H. N., Bridges, p. M., and Morse, R. A., Experimental and

theoretical investigation of gravity drainage performance . Trans. AIME, l92, 285, 1951.- 109. Johnson, E. F., Bossler, D. P., and Naumann, V. O., Calculation of relative permeability from dis-

placement experiments, Trans. AIME, 216. 370. lg5g.l l0. Land, C. S. , Compar ison of calculated wi th exper i rnental imbibi t ion re lat ive permeabi l i ty ,Truns. AIME,

25 t . 419 . t97 | .I I l ' Gardner, G. H. F., Messmer, J. H., and Woodside, W., Effective Porosity and Gas Relative permeability

on Liquid Imbibition Cycle. Theory of Fluid Flow in Porous Media Conference, University of Oklahoma.Norman. 1959. 173.

ll2. Shelton, J. L. and Schneider, F. M., The effect of water injection on miscible flooding methods usinghydrocarbons and CO,, paper SPE 4580 presented at the SPE 48th Annual Meeting, Las Vegas, 1973.

5 I l3' Land, C., Calculation of imbibition relative permeability for two- and three-phase flow fiom rock properties,Soc' . Pet . Eng. J. , 6, t49. 1968.

I14. Wi lson, J. W., Determinat ion of Relat ive Permeabi l i ty Under Simulated Reservoir Condi t ions.2 ( t ) , 4 . 1 9 5 6 .

AIChEJ,

- f l5' Fatt, I. and Barrett, R. E., Effect of overburden pressure on relative permeability , Truns. AtME, lgE,325. t953.

I l6' Thomas, R. D. and Ward, D. C., Effect of overburden pressure and water saturation on gas permeabilityof tight sandstone cores, -/. Pet. Te<'hnot., 2, 120, lg'/2.

l l7. Merliss, F. E., Doane, J. D., and Rzasa, M. J., Influence of rock and fluid properties and immisciblefluid-flow behavior in porous media, paper 510-G presented at the AIME Annual Meeting. New orleans.I 955 .

l l 8 . Dun lap ,E .N . , I n f l uenceo f conna tewate ronpermeab i l i t yo f sands too r l ,T rans .A IME, l 2 j . 215 . lg3g .

{

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l e ( " . O k l a . .

l , ' l , i . 1 9 5 7 .d ! . - : r . r a t u r e s .

f r . . i r r [ ' - l ' l 'ectr . : . rhc SPE

I " - r t ' [ : - i 0 t h

f . . , ' | . . , . i t r t t t t l l €

a r ' . r r h r t i o n

f - . . : : , ' . . r r h o n

L ' . : ' . , : . : ' r rd lc r \

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r-

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. " . . l l t c \ .

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: ' . , ' \ I n g

' : . r l u n d

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\ t v L .

l l - - . . r h t l r t r

D t . . : l l r t l l l i . l .

D c ' - . : ' u s i n g

[ : . ' / - . j .

f r ; - - : r c r t i C s .

f f . - . \ l ( hEJ.

. i . t / 1 t 9 t J .

i 3 ' ' . c . r h i l i t y

d " ' : r r . c i b l e

5<' . . r l r leans.

l . : : t 9 - 1 8 .

t 3 2 .

1 3 3 .t 34 .1 3 5 .

t0 l

l l9. Stewart, C. R., Craig, F. F., Jr., and Morse, R. A., Determination of l imestone performance charac-

teristics by model flow tests, Trans AIME, 198, 93, 1953.

120. Keelan, D. K., A practical approach to determination of imbibition gas-water relative permeability , J. Pet.

Technol . , 4, 199, 1976.

l2 l . Leas, W. J. , Jenks, L. H. , and Russel l , C. D. , Relat ive permeabi l i ty to gas, Trans. AIME, 189,65,

r950.122. Felsenthal, M., Correlation of k*/k,, data with sandstone core characteristics, Trans. A1ME,216,258,1959.

123. McCord, D. R., Performance predictions incorporating gravity drainage and gas pressure maintenance -

LL-370 Area, Bolivar coastal field,Trans. AIME' 198, 231, 1953.

124. Edmondson, T. A., Effect of temperature on waterflooding, Can. J. Pet. Tec'hnol., 10, 236, 1965.

125. Poston. S. W.. Ysrael, S., Hossain, A. K. M. S., MontgomerY, E. F., IV, and Ramey, H. J., Jr. '

The effect of temperature on irreducible water saturation and relative permeability of unconsolidated sands,

Soc . Pe t . Eng . J . ,6 , l 7 l , 1970 .

126. Davidson, L. B., The effect of temperature on the permeability ratio of different fluid pairs in two-phase

systems, J. Pet . Technol . , 8, 1037, 1969.

1 2 7 . S i n n o k r o t , A . A . , R a m e y , H . J . , J r . , a n d M a r s d e n , S . S . , J r . , E f f e c t o f t e m p e r a t u r e l e v e l u p o n c a p i l l a r y

pressure curves, Soc. Pet . Et tg. J. , 3. 13. 197 l .

128. Weinbrandt, R. M., Ramey, H. J., Jr., and Cass6, F. J., The effect of temperature on relative and

absolute permeabi l i ty of sandstones, Sor ' . Pet . Eng. J. , 10. 376, 1915.

129. Lo, H. Y. and Mungan, N., Effect of Temperature on Water-Oil Relative Permeabilit ies in Oil-Wet and

Water-Wet Systems, SPE #4505, Las Vegas, Nev. , September 30, 1973.

130. Sufi, A. S., Ramey, H. J., Jr., and Brigham, W. E., Temperature Effects on Relative Permeabilit ies

of Oi l -Water Systems, SPE #l1701, New Orleans, La. , September 26, 1982.

13l . Suf i , A. S. , Ramey, H. J. , Jr . , and Br igham, W. E. , Temperature Ef f 'ects on Oi l -Water Relat ive

Permeabilit ies for Unconsolidated Sands, U.S. Department of Energy, Technical Report, 12056-35. De-

cember. 1982.

Miller, M. A., and Ramey, H. J., Jr., Effect of Temperature on Oil/Water Relative Permeabilit ies of

Unconsol idated and Consol idated Sands, SPE #l2 l16, San Francisco, Cal i f . , October 5, 1983.

Counsil, J. R., Steam-Water Relative Permeability, Ph.D. thesis, Stanford Univ., Stanford, Calif., 1979.

Muskat, M.,, Phvsical Principles of oil Production, McGraw-Hill New York. 1949.

Bardon, C. and Longeron, D., Influence of very low interfacral tensions on relative permeability, paper

SPE 7609 presented at the SPE 53rd Annual Meet ing, Houston, Tex. , 1978.

136. Richardson, J. G., Calculation of waterflood recovery from steady-state relative permeability data,Trans.

A IME. 210 .373 . 1951 .

137. Johnson, E. F., Bossler, D. P., and Nauman, V. O., Calculation of relative permeability from displace-

ment experiments, Trans. AIME, 216. 370, 1959.

138. Levine, J. S. , Displacement exper iments in a consol idated porous system, Trsns. AIME,201,57, 1954.

139. Craig, F. F., Jr., Errors in calculation of gas injection performance from laboratory data, J. Pet. Techrutl.,

8 . 2 3 , 1 9 5 2 .

140. Sandberg, C. R., Gourney, L. S., Suppel, R. F., Effect of fluid flow rate and viscosity on laboratory

determinat ion of o i l -water re lat ive permeabi l i t ies, Trsns. AIME,213, 36. 1958.

14l. Donaldson, E. C., Lorenz, P. 8., and Thomas, R. D., The effect of viscosity and wettability on oil-

water relative permeability, paper SPE 1562 presented at the SPE 4lst Annual Meeting, Dallas, Oct. 2-5,

t966.142. Geffen, T. M., Parrish, D. R., Haynes, G. W., and Morse, R. A., Efficiency of gas displacement from

porous media by liquid flooding, Trans. AIME, 195,29. 1952.

143. Krutter, H. and Day, R. J., Air-drive experiments on long horizontal consolidated cores../. Pet. Technol.,

t 2 , t , t 9 4 3 .144. Morse, R. A., Terwill iger, P. K., and Yuster, S. T., Relative permeability measurements on small core

samples, Oi l Gas J. , 46. 109, 1947.

145. Odeh, A. S., Effect of viscosity ratio on relative permeability, Trans. AIME, 216,346, 1959.

146. Baker, P. E. , Discussion of ef fect of v iscosi ty rat io on re lat ive permeabi l i ty , J . Pet . Technol . ,219,65,

I 960.1 4 7 . D o w n i e , J . a n d C r a n e , F . E . , E f f e c t o f v i s c o s i t y o n r e l a t i v e p e r m e a b i l i t y , s o t ' . P e t . E n g . J . ' 6 ' 5 9 ' 1 9 6 1 .

148. Ehrlich, R. and Crane, F. E. , A model for two-phase flow in consolidated materials , Trans . Al M E , 246,

22 t , t 969 .

149. Perkins, F. M., Jr., An investigation of the role of capillary forces in laboratory waterfloods. J. Pet.

Techno l . , l l , 49 , 1957 .

150. Pickell, J. J., Swanson, B. F., Hickman, W. B., Application of air-mercury and oil-air capillary pressure

data in the study of pore structure and f lu id d ist r ibut ion, Soc. Pet . Eng. J. ,4,55. 1966.

l5l. Warren, J. E. and Calhoun, J. C., A study of waterflood efficiency in oil-wet systems, Truns. AIME.

204 .22 . t955 .


152. Caro, R. A. , Calhoun, J. C. , Jr . , and Nielsen, R. F. , Surface act ive agents increase oi l recovery. Oi1

Gus J. , 12. 6. 1952.153. Ojeda, E., Preston, F., and Calhoun, J. C., Jr., Correlation of residuals following surfactant floods,

Prod . Mon . , 12 , 20 , 1953 .- 154. Lefebvre du Prey, E. J., Factors afl 'ecting liquid-liquid relative permeabilit ies of a consolidated porous

medium. Soc. Pet . Ene. J. , 2, 39. 19'13.

155. Owens, W. W., Parr ish, D. R. , and Lamoreaux, W. E. , An evaluat ion of a gas dr ive method for

determining relative permeability relationships, Truns. AIME, 201,275, 1956.

156. Kyte, J. R. and Rapoport, L. A., Linear waterflood behavior and end ef'fects in water-wet porous media,

Trans. AIME, 213. 423. 1958.

157. Richardson, J. G., Kerver, J. K., Hafford, J. A., and Osoba, J. S., Laboratory determinations of

re lat ive permeabi l i ty , Trans. AIME, 195, 187, 19,52.

158. Morse, R. A., Terwill iger, P. K., and Yuster, S. T., Relative Permeability Measurements on Small Core

Samples, Oi l Gas J. , 46. 109, 1941.

159. Labastie, A., Guy, M., Delclaud, J. P., and lff ly, R., Effect of flow rate and wettability on water-oil

relative permeabilit ies and capillary pressure, paper SPE 9236 presented at the SPE Annual Meeting, Dallas.

Tex . , Sep . 2 l -24 , 1980 .159a. McCaffery, F. G., The Effect of Wettability on Relative Permeability and Imbibition in Porous Media.

Ph.D. thesis, Univers i ty of Calgary, Alberta, Canada, 1973.

I 60.

l 6 l .

- t 6 2 .

t 6 3 .

Delclaud, J. P., New results on the displacement of a fluid by another in a porous medium, paper SPE

4103 presented at the SPE 47th Annual Meet ing. San Antonio,Tex. ,1912.

Fetkovitch, M. J., The isochronal testing of oil wells, paper SPE 4529 presented at the 48th Annual Fall

Meet ing of the SPE, Las Vegas, Nevada. 1973.

Handy, L. L. and Datta, P., Fluid distributions during immiscible displacements in porous media. Sot'.

Pe t . Eng . / . , , 10 . 261 , 1966 .

Huppler, J. D., Numerical investigation of the effects of core heterogeneities on waterflood relative

Rc'r lCOl

pha:c' rel

than thc

cnginc-c'n

cartxrn rl

ni trogc'n

.\ll f.r.

thrc'c-ph.. incc' rc.

plctc' l r d

a. a lhrc',

uhrch t t t

.mal l \

includc'.

lIl lTltlrl ,

i : b*-rn!

high arxl

In tl-*-

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ct)mpat:l

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There

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change r

\A ett ing-

drctrons

tri r.rtufi

Drarn

r t udt '

l \ ls

-1 h

sa

lmhrt

permeabi l i ty , Soc' . Pet . Eng. J. , 12, 38 | , 1970.

164. Stewart, C. R. and Owens, W. W., A laboratory study of laminar and turbulent l low in heterogeneous

poros i t y l imes tone , T runs . A IME,2 l3 , 12 l , 1958 .

16,5. Henderson, J. H. and Moldrum, H., Progress report on mul t iphase-f low studies, Prod. Mon.,4. 12,

t949 .

166. Krutter, A. and Day, R. J., Air-drive experiments on long horizontal consolidated cores, -/. Pet. TeL'hnol .,

l l . l . r 9 4 3 .

167. Chen, H. K. , Counsi l , J . R. , and Ramey, H. J. , Jr . , Steam-Water Relat ive Permeabi l i ty . 1978 Geothermal

Resources Counci l Annual Meet ing, Hi lo, Hawai i , July 25-27, 1918.

168. Brownell, L. E. and Katz, D. L., Flow of fluids through porous media - single homogeneous fluids,

Chem. Ens. Pros, . , 43(10), 537. 194'7.

169. Wal l , C. G. and Khurana, A. K. , Saturat ion permeabi l i ty re lat ionshipat low gas saturat ion.J. lnst . Pet . ,

5 1 . 2 6 1 . 1 9 7 1 .

170. Rose, W.D., Flu id d ist r ibut ions character iz ing gas- l iquid f low,Trans. AIME, 192,312, 1951.- l7 l . Loomis, A. G. and Crowel l , D. C. , Relat ive permeabi l i ty studies. I I . Water o i l systems. Prod. Mon.,8.

r 8 . 1 9 5 9 .

172. Sarem, A. M., Significance of water-oil relative permeability data calculated from displacement tests,

Pro<' . , Theory of Flu id Flow in Porous Media Conference, Univers i ty of Oklahoma. Norman, 1959, 189.

173. Owens, W. W., Parrish, D. R., and Lamoreaux, W. E., A comparison of field k*/k,,characteristics and

laboratory ku/k,, test results measured by a new simplified method. paper 518-G presented at the AIME

30th Annul Meet ing, New Orleans, 1955.

174. Calhoun, J. C., Jr., Fundamentals of Reservoir Engineering, University of Oklahoma Press, Norman,

t94'7.

| 75. Stewart, C. R., Craig, F. F., and Morse, R. A., Determination of l imestone performance characteristics

by model flow tests. Truns. AIME, 198, 93, 1953.

176. Kyte, J. R., Stanclift, J. R., Stephan, S. C., Jr., and Rapoport, L. A., Mechanism of waterflooding

in the presence of free gas, Trans. AIME, 101, 215, 1956.

177. Mattax, C. C.and Clotheir , A. T. , Core Analysis of Unconsol idated and Fr iable Sands, paper SPE 4986

presented at the SPE 49th Annual Meet ing, Houston, Tex. . 1974.

178. Holmgren, C. R. and Morse, R. A., Effect of free gas saturation on oil recovery by waterflooding. Trans.

A I M E , 1 9 2 , 1 3 5 , 1 9 5 1 .179. McCaffery, F. G., The Effect of Wettability on Relative Permeability and Imbibition in Porous Media.

Ph.D. thesis, University of Calgary, Alberta, Canada, 1973.

180. Gornik, B. and Roebuck, J. F., Formation Evuluetion through Extensive Use of Core Analysi,s, Core

Laborator ies, Inc. , Dal las, Tx. , 1979. R.,

l \ -

h -

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. . . l l t c d i a .

. r i l r r l l \ O f

. r i l ( - o r e

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103

Chapter 4

THREE-PHASE RELATIVE PERMEABILITY

I. INTRODUCTION

Recent innovations in the field of oil recovery have led to a renewed interest in three-phase relative permeability. Three-phase flow occurs when the water saturation is higherthan the irreducible level, and oil and gas are also present as mobile phases. Detailedengineering calculations of the performance of reservoirs under recovery methods such ascarbon dioxide injection, in situ combustion, steam drive, micellar f luid injection, andnitrogen injection frequently require three-phase relative permeability data.

All factors which influence flow in systems containing two mobile phases also apply tothree-phase systems. Virtually all oil reservoirs constitute potential three-phase systems,since reservoir rocks invariably contain interstitial water, and naturally occurring oils com-pletely devoid of gas are rare. In fact, a two-phase system of oil and gas may be regardedas a three-phase system in which the water phase is immobile. The number of reservoirs inwhich oil, gas, and water are simultaneously mobile during primary production is probablysmall. Nevertheless, three-phase mobility is always possible when a producing intervalincludes part of the oil-water transitional zone in a reservoir. It is probable, however, thatin most cases where oil and free gas are produced with an appreciable water cut, the wateris being produced from layers of the reservoir in which relative permeability to water ishigh and not by true three-phase flow.

In the past, the use of three-phase relative permeability data for conventional reservoirengineering calculations has seldom been necessary. In consequence, considerably less isknown about three-phase relative permeability characteristics of rocks than is known forcomparable two-phase cases. The realization that detailed engineering calculations of theperformance of reservoirs produced by in sitrz combustion processes require three-phase datais quite new. Three-phase relative permeability is useful in the calculation of field perform-ance for reservoirs being produced by simultaneous water and gas drive, and also in analyzingsolution gas drive reservoirs which are partially depleted and are being produced by waterdrive. An increasing interest in three-phase flow phenomena is anticipated.

There are two distinct classes of three-phase relative perrneability data: ( I ) that pertainingto drainage; and (2) that pertaining to imbibition. Drainage refers to the direction of saturationchange in which the wetting-phase saturation decreases. Imbibition refers to an increasingwetting-phase saturation. For the relative permeability data to yield correct reservoir pre-dictions, the direction of saturation change in the reservoir must correspond to the directionof saturation change for which the data were derived.

Drainage relative permeability data should be used in the following situations:

Enhanced recovery processes involving the injection of dry gas, flue gas, carbondioxide, and other gases into watered-out reservoirs.Miscible flood processes in which liquified petroleum gas (LPG) is injected intowatered-out reservoirs.

3. Production from reservoirs in which the water saturation is greater than the ineduciblesaturation.

Imbibition relative permeability data should be used under the following conditions:

l. Reservoirs produced by natural water drive.

h

h

P ' . \ l c d i a .

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FIGURE l. Three-phase relative permeability.r

2. Reservoirs developed by water flood, as well as by processes where the injected watercontains surfactants, polymers, or other additives.

3. Reservoirs developed by recovery processes where water is used to push a slug ofchemicals, LPG, etc.

II. DRAINAGE RELATIVE PERMEABILITY

A. Leverett and LewisMuch of the credit for the classical work in three-phase relative permeability is accorded

to Leverett and Lewis' who were the first to measure three-phase relative permeability of awater-oil-gas system in an unconsolidated sand. These investigators used a steady-statesingle-core dynamic method and ignored end effects and hysteresis. Errors from ignoringcapillary end effects were probably significant, since low flow rates were used. Ring elec-trodes were spaced along the length of a sand pack to measure resistivity of the sample andbrine saturation was assumed to be directly related to resistivity. Gas saturation was deter-mined from pressure and volume measurements. Oil sdturation was obtained by a materialbalance technique. Leverett and Lewis obtained three separate triangular graphs showinglines of constant relative permeability ("isoperms") to the three phases; these were plottedagainst the saturations of the three fluids, as shown by Figure l. They also obtained a plotshowing the region of three-phase flow; Figure 2 shows the region where each componentcomprises at least 5Vo of the flow stream. As shown by the figure, three-phase flow occursin a rather confined region.

Relative permeability to water, k,*, was found to be dependent only on water saturation,S*, and was not affected by the introduction of an additional nonaqueous phase. Relativepermeability to gas, k,r, was found to be slightly less than would be expected for the samegas saturation, S* in two-phase flow. The k., isoperms are convex towards the 1007o S,apex of the triangular diagram. As gas becomes one of the two flowing nonwetting phases,when both oil and water are present, the relative permeability to gas decreases as oil saturation

approarof oil a

The ra g a s ssaturatito its ogas satrmode cthe inteoil andan oil Iwhich <Hower'flow ofof the 1move fat cons

AlsoThis efmay flrthe porparts olresult i

[rvefor var

B. CorThe

a calc i r

105

l O O % g a s

1 0 0 % w a t e r l O O % o i l

ieetcd *'ater

fi -r .lu_s of

It

i . .r . . trrded

ia l ' ' r l r t r o f a

Dlc;Jr -\ tate

nr l rnor ing

Rrng c . l ec -

slrplc and

I l r j r glg[91-

7 .r :naterial

h : .h t rwing

rerc plotted

uncJ a p lo t

c()nlrx)nent

Jltrri occurs

s.rturat ion,

E Rcla t ive

lr lhc same

E l ( x )7c ss

ing phases,

| . ; tura t ion

FIGURE 2. Region of three-phase flow.l

approaches the water saturation value, becoming a minimum when roughly equal saturationsof oil and water are associated with the gas.

The relative permeability to oil is seen to vary in a more complex manner. Starting witha gas saturation of zero, oil relative permeability at constant oil saturation increases as gassaturation increases (except at low oil saturations where k,, remains constant) then decreasesto its original value as more gas is introduced, finally falling well below this value whengas saturation is further increased. In a water-wet system, the presence of gas leaves themode of water flow unchanged, but since the gas tends to occupy the central portions ofthe intergrain spaces (where the oil is also driven by capillary forces) interference betweenoil and gas flow is likely. Visual examination under the microscope shows the presence ofan oil film (in some cases containing a very small amount of finely divided water) throughwhich oil flows around each gas bubble. It is not clear whether all gas bubbles are connected.However, the gas bubbles are observed to move jerkily, as opposed to the generally smoothflow of water (and of oil when gas bubbles are absent or are stationary). This uneven motionof the gas implies a similar motion of at least part of the oil, which would be expected tomove faster than in the absence of gas at the same oil saturation. We see a decrease in k,.at constant S" as S, is increased, especially at low S*.

Also, there is an increase in k.o at constant S" as S* is increased at low values of S*.This effect is evidently due to ihe shifting of oil into parts of the intergrain space where itmay flow more freely. The water introduced tends to occupy the sharply curved parts ofthe pores, forcing oil into the central space vacated by gas. Since fluid in the sharply curvedparts of the pores moves only with difficulty and that in the center moves more readily, theresult is an increase in k,o.

Leverett and Lewis pointed out that they found no effect of oil viscosity on the isopermsfor various saturations of the three phases.

B. Corey, Rathjens, Henderson, and WyllieThe results of the work of Corey et al.2 are shown by Figure 3. These investigators used

a calcium chloride brine. Capillary end effects were minimized by using a core with semi-


g a s

FIGURE 3. Three-phase relative permeability.r

permeable membranes mounted at each end. They measured saturations gravimetrically and

avoided hysteresis effects by using separate cores for each measurement rather than resa-

turating the same core. In an initial conclusion, they reported that when the saturations ofthe wetting phases were equal, the nonwetting phase relative permeability, k,n, was un-changed regardless of whether the nonwetting phase was oil or gas. They used the equivalentliquid permeability as the base value. The oil isoperms of Corey et al. are similar to thoseobtained by Leverett and Lewis, except that Corey's oil isoperms have a greater curvature.Relative permeability to water was not measured, but was calculated on the assumption thatit was a function of water saturation alone and that water permeability in a water-wet systemwas the same as the oil permeability in an oil-wet system. It should be noted that the dataof Cltrey et al. are for oil drainage in an oil-gas system. They also observed that the behaviorof the nonwetting phases was more sensitive to changes in pore geometry than was thebehavior of the wetting phase. The increase in k." (at low S*) with the increase in S* (and

a corresponding decrease in S*) is higher in Corey's consolidated sandstone samples thanin the unconsolidated samples. This is because of the dependence of k.., on the ratio

f t 'I dsl/P.r

J S *

l, dsL/P:

which is usually higher in consolidated rocks than in unconsolidated rocks.Corey et al. extended their two-phase relative permeability relationship to three-phase

flow on the basis of the following approximation:

The dby

where S,As in

especialthe systeat highealone. I rwetting

C. ReidUsing

Reidr otwas igntobtaineddifferent

107

N

2 IFIGURE 4. Three-phase relative permeability.r

8 l : . . t I l r and

Br : : l . r l l feSa-

l l . . - . r l l r r I l S O f

I ' \ . 1 \ un -

E ; . . . r r r a lent

l r l r : i ( r t hose

l f . .1r \ a ture.

Unrn l r r rn that

f- \\ .'l :\ Ste [l

lh.: i thc' data

thc hchav ior

h;:: s aS the

r r : : S . ( a n d

n r : . ; . l c r t hanI r . l l , r

thrc. '-phase

I S , _ - ( S * , . . + S , , . ) ]for S, t

St '' (so i , , + s . , , )

Sr_,l /Pl : g

: o f b r S . s ' ^ t ' ' = , ( l )-

(S*,,, * S",)

The drainage oil phase relative permeability in a water-wet system containing gas is givenbv

(2)

where S., is residual liquid saturation.As in Leverett's data, the oil isoperms tend to be parallel to the oil isosaturation lines,

especially at high S*. At increasing S* and constant S.,, the gas which was previously inthe system is no longer present. Thus, the rate of increase of k.., with increasing S* decreasesat higher values of S*. Corey et al. proposed a method to obtain k.,, and k,*, based on k.*alone. Incidentally, k.* was found to be a function of S* and independent of the relativewetting properties of the fluids within the rock.

C. ReidUsing the same method employed by Leverett and Lewis (single-core dynamic technique),

Reid3 obtaine{ the isoperms shown in Figure 4; He eliminated end effects, but hysteresiswas ignored.'fr ine saturation was measured by resistivity, and'oil and gas saturations wereobtained by gamma ray absorption. His saturation measuremenis possibly were affected bydifferential absorption of gamma rays by oil and water. While Leverett and Lewis obtained


straight lines for the water phase behavior (showing k,* to be independent of the distributionof the nonwetting phases) and oil isoperms concave toward the l}OVa So apex, Reid's resultsindicated concave water isoperms, convex oil isoperms, and slightly concave gas isoperms.These results were interpreted as indicating that the relative permeability to each phase isdependent both upon its own saturation and the saturations of the other phases. His resultsshowed a greater oil permeability when three phases were present than with two phases, ata given oil saturation.

Reid made no attempt to correlate the three-phase results with those from two-phaseexperiments. He placed emphasis on his conclusions for the oil isoperms and noted thatLeverett's oil phase data showed a substantial amount of scatter. For this reason, he believedthat his oil isoperms were more valid than Leverett's. The work of Rose seems to confirmReid's findings.

D. SnellThree-phase behavior in a water-wet unconsolidated sand was investigated by Snell,a-6

who used radio frequency detection for the determination of S* and a neutron countingmethod for measurement of Sr. Oil saturation was obtained by material balance calculation.His experiments had a repeatability within I lr%o for relative permeability values, with a betterrepeatability for the saturation values. He found that when the wetting phase saturation wasuniform over a length of the test sample, the saturations of the other two phases were alsouniform over the same length.

Although Caudle et al. " did mention hysteresis in their work, the first significant studyon the effect of saturation history on three-phase relative permeability was done by Snell.In describing Snell's work, it is convenient to define four types of liquid saturation histories:

l. Imbibition of water with oil saturation increasing (II).2. Imbibition of water with oil saturation decreasing (ID).3. Drainage of water with oil saturation increasing (DI).4. Drainage of water with oil saturation decreasing (DD).

As seen from his results in Figure 5, k.o values were lower for DD than for the othersaturation histories. Since, in two-phase flow, drainage caused the wetting phape to lose itsmobility at higher saturations, it has been suggested that there is a partial change in wettabilityfrom water-wet to oil-wet during DD. When the system was oil-wet, a larger S,, was requiredfor the same k,., because some of the oil was trapped in the smaller pores. This oil increasesS.,, but it is immobile. He further suggested that this change in wettability may be causedby polar compounds in the oil. Snell's results do not show good agreement with those ofLeverett and Lewis except in the case of k.*.

Oil and water isoperms reported by Snell are similar to those determined by Reid, butSnell's k.. values are higher than Reid's, especially at low water saturation.

In a later work, Snell reinterpreted the results of four earlier studies done on unconsolidatedsands. In these investigations, no hysteresis was found for water isoperms. Oil isopermsshowed hysteresis only when kerosene or a kerosene/lubricating oil mixture was used as theoil phase. Nonpolar oil gave no hysteresis. Reinterpretation of the earlier results was possiblebecause Leverett and Lewis indicated possible enors in their saturation measurements. Reid'ssaturation data might also have been inaccurate because of differential absorption of gammarays by oil and water. Relative permeability to oil was found to be dependent only on thehistories of the liquid phase saturations, although Snell did not rule out dependence on gasphase saturation history. Snell reinterpreted Leverett's data to obtain oil isoperms convextoward the l00%o S,, apex. Oil isoperms then followed the same pattern in all four investi-gations. These results are shown in Figure 6. The curvature of the isoperms of both nonwetting

I1 0 0 %

1 O 0 % w a t r

dr . r r ibu t ionRrJ' . r ! 'sul ts5 i l r t rp t3 t tT lS .

rh phase is

, [ { r . rcsu l ts

o pha res . a t

I I \ \ ( )_phase

I n , ' ted that

hc hc' l ieved

i l r , a( )n f i fm

b r S n L - l l . r 6

tn . \ )unt ing

ca le u l a t i on .

t r th .r hrctter

Uf . : i i ( ln \ . \ ' aS

s i i c rc a lso

l- ls.rnt rtud)

r t r S n e l l .

x r h r . t t l r ies :

r thc other! l \ \ l ( ) \e i t s1u e r r . rb i l i t yra. rcquiredi l rn i rcases

I n- eausedith lhore of

y Re r . l . but

on., ' ' l idatedll r.oP€ITnSu'cd as they3. possibleint . Reid 'sI ( ) l gamma

Dn l r on the

ll luc ttn 935lTIlr r-t)DVeX

Dur rnr est i -Jk,n rr ctt ing

I O O * w a t e t

109

I O O i o i l

1 O 0 % o i l

1 0 0 % o i l

1 0 O i o i l

w a t e r

1 0 O S w a t e r r o O S o i l

1 O O % w a t e r

1 O O % w a t e r

FIGURE 5. Three-phase relative permeability.5

l m b i b i t i o n _

D r a i n a g e - - - -

R € s u l t s o f S n e l l

i l ! t o , D t _

o D - - - -

1 0 0 % o i l 1 0 0 % w a t e r

n o n - p o l a r

1 O 0 % o i l ' l O O % w a t e r

FIGURE 6. Reinterpretation of results by Snell.6

g a s

1 0 0 %

M R)

' l 0 O % g a s

/I ,a'

1 0 0 % g a s

l l 0 Relative Permeabilin of Petroleum Reservoirs

--- ;,--=--_ 4 o : . _ )

\ G a s i n i e c t s d , p o r e v o l u m e s

1 0 0

1 0 0 { - rNr r rAL orL sATURATToN 0

FIGURE 7. Fluid flow experimental aata for Berea sandstone.T

phases (oil and gas) are convex toward the corresponding phase-apex, whereas wetting phaseisoperms are straight lines or are concave toward the l00%o apex of the wetting phase.

E. Donaldson and DeanAn extension of Welge's two-phase unsteady-state technique was used by Donaldson and

DeanT to determine three-phase relative permeabilities of Berea sandstone and Arbucklelimestone. Oil and water in the core were displaced by gas and the flow rates of all threephases were measured simultaneously. Their results for the displacement tests on the twocores starting with various S*, and S.,, are shown in Figures 7 and 8. They minimized endeffects by using a high pressure differential and high flow rates, and they did not accountfor hysteresis effects. The volumes of oil and water displaced were less in the limestonethan in the sandstone for the same S,, (or S*) and the same pore volumes of gas injected.This effect is presumably caused by the larger flow channels in the limestone. The efficiencyof a gas displacement process is greater for a matrix with smaller pores. There is a narrowerrange of saturations for three-phase flow in the limestone because the large vugs may allowgas to flow without transfering energy to oil or water.

The isoperms are presented as functions of terminal rather than average saturations, becausethe former govern the flow of fluids through the core. The results of Donaldson and Dean,shown in Figures 9 through 14, indicate that, at low and constant S' k., for Berea sandstoneinitially decreased with increasing S" until S" reached a value of about 50Vo. Further increasesin So caused an increase in k,r. At S* greater than I 3Vo, k," increased so the isoperms becameconcave toward the gas apex. No explanation of this phenomenon was suggested by theauthors. At a given S* the k., was lower in the presence of water than in the presence ofoil, probably because water adhered more strongly to the rock surface than did the oil. Theflow path of gas is more restricted in the presence of water, since gas can displace oil moreeasily than it can displace water. For the limestone, k,, was always concave toward the gasapex.

i---- to -..\..

[ - - ' " - \ . .i -'--\:\:1..,, /,F---,:-:N*,

O t N t l A L w A T E R s A T U R A T t o N

z j +1

(L

EoooO_ .zob

o. 1

0

The uwas genin the pr

< - L*-' -lI

^ - . 5 - A

J . 1L-.25 ---A

l l l

D a t a P o i n t s :

. 4

€A)

e o

IU,oA'

2 oo-o_!

o- .3Eq)O(g

O - AA . Z

i5

o

t . r , / - o a s

I n l e c t e o . p o r e v o t u m e s"oo,

1 0 0 I N I T I A L O I L S A T U R A T I O N

Rt : : . : phase

l : " . : \c

t L . . l . r r f l O D d

d \ r huck le

t ' l . r l l th ree

r ,n l hc two

I r : : : r zc t l end

n ' : . lLCOUnt

f . ln tc \ tone

F . : r t t c c t e d .E J : : l j l C n c y

I .r :t.rITt)Wer

I r : : . : r a l l ow

n. . hccause

I . rnJ Dean,

a . . rndstone

Er lnar r -ases

ml . hccame

stcJ br the

prCrc. l lCe Of

l hc , ' t l . The

Fc , \ r l mo re

lar. l thc' gas

FIGURE 8. Fluid flow experimental data for Arbuckle limestone.T

uw so

FIGURE 9. Gas relative permeability for Berea sandstone.T

The water isoperms are concave toward the water apex. Relative permeability to waterwas generally higher in the presence of oil than in the presence of gas, but k.* was higherin the presence of gas than in its absence at a constant high S*. Both k." and k.* increased

tt2 Relative Permeability of Petroleum Reservoirs

o^ao/

sw

FIGURE IO

so

Gas relative permeability for Arbuckle limestone.i

5A6ieo

1 o

sw so

FIGURE I l. Oil relative permeability for Berea sandstone.T

at constant S,, and S*, respectively, when S, was increased from 0 to 8Va possibly becausegas was trapped in pores which would otherwise be occupied by immobile wetting phases.Also, k.* increased in the presence of oil because there may be partial oil wetting, so thatwater was displaced into larger pores; this was not the case when gas was present.

F. SarenrUsing ;

did not ccSarem's r

113

sw

FIGURE I2 .

so

Oil relative permeability for Arbuckle limestone.T

n. l l r l ) because

Ic t : r nS phases .

t e t i r ns . so t ha t

f L ' . c n l .

- q Q-w oo

FIGURE 13. Water re lat ive permeabi l i ty for Berea sandstone'-

F. SaremUsing an unsteady-state method, Sarems obtained three-phase data for a Berea core. He

did not consider end effects or saturation history, but his method did account for wettability.Sarem's method, which is an extension of Welge's two-phase technique, is relatively fast.

ss

tt{ Relative Permeabilin of Petroleum Reservoirs

sw so

FIGURE 14. Water relative permeability for Arbuckle limestone.T

The core is saturated first with one liquid and then flooded with an immiscible unreactive

liquid, at least until breakthrough. Then, both liquids are displaced by gas. In the derivation

of his equations, Sarem assumed each relative permeability to be a function of the colre-

sponding saturation alone. Isoperms were therefore parallel to the isosaturation lines. The

relative permeability to gas was assumed to be dependent on total liquid saturation and

independent of the relative wetting properties.The saturation equations are

Soz : S..o,, * f", Q

S*z : S*.ou* * f* , Q

S r r : 1 - S * r - S . , 2

where

Q : cumulative volume of injected fluid (per pore volume)

e : 9 r tLAO

and q, : total volumetric flow rate (cclsec), t : time (seconds), and f : fractional flow.

Subscripts o, w, g, and 2 stand for oil, water, gas, and outlet, respectively. The relative

permeabilities are computed from the following relationships:

(3)

(4)

(5)

Sarr--n

o n k . .

the s:r

and D

G. Sa

A r

Sarai

uscd t

efl-ect

o i l t l r

a\\ulT

salur:r

\ \a\ a

on l r t

O i l r .

rcsu l t ,

shapc

n I'ere

a ca_\e

the as

H. n'Thr

\r aterhare I

follouthis ta

Thrdiagrarelatirare ot

t .2 .

3

d ( l / Q )k.* : I*z -l-4pp4

1t l l I

\Lp"q'Q/

(6)

k.* Or,,H

d ( l / Q )K"' : t"'(-4r!4-r*,,

o,J

115

(7)

( 8 )

Jc . rn r c ' ac t i ve

ir.* .lcrir ation

ol : i rc corre-

r t : l rc . . The

l lu:.r: l rrr l i l f ld

X t i , ' n r l t l ow .

. . l 'hc

rclat ive

( l )

( 4 )

( 5 )

Sarem also concluded that initial saturation conditions affect k.., and k.*, but have little effecton k,r. He found that k."/k.* was influenced by initial saturations in three-phase studies inthe same manner as in two-phase studies. Sarem's results differed from those of Donaldsonand Dean even though both used the same type of sandstone.

G. Saraf and FattA dynamic method using nuclear magnetic resonance (NMR) techniques was used by

Saraf and Fatte to determine liquid saturations in Boise sandstone. A volumetric method wasused to obtain gas saturations. The experimental technique was designed to minimize endeffects. To maintain a constant pressure differential, the gas flow rate was increased as theoil flow rate was decreased. Saraf and Fatt found no theoretical justification for Sarem'sassumption that three-phase relative permeability to each phase was a function only of thesaturation of that phase. In the water-wet Boise sandstone, however, they did find that k,*was a function of S* alone. Using water permeability as the base, they found that k,* dependedonly on the total liquid saturation and was independent of the relative wetting properties.Oil isoperms determined by these investigators were convex toward the oil apex. Theirresults are shown in Figure 15. The explanation given by the authors for this unexpectedshape of the isoperms seems less than convincing. They did state, however, that in studieswhere k,* was a function of both S* and S", the system was not 1007a water-wet. In sucha case, it seems likely that S* did not remain constant when So or S* was increased and thatthe assumption of constant S* could be a source of experimental error.

H. Wyllie and GardnerThree-phase relative permeability equations for preferentially water-wet systems where

water and oil saturations were determined by the drainage cycle rather than by imbibitionhave been given by Wyllie and Gardnerro and are presented in Chapter 2, Table 3. Thefollowing factors should be taken in consideration when using the equations presented intothis table.

l. The k,* values are normalized to absolute permeability.2. The values of k.o and k., calculated from these relationships are both normalized to

the effective hydrocarbon permeability at irreducible water saturation. Inasmuch asthey are normalized to the same base, k.s/k.. values may be calculated directly byusing these equations. This is not true, of course, for water-hydrocarbon relativepermeability ratios.

3. The gas and oil relative permeability equations do not include provision for residualoil saturation. When S* equals S*i.,, k,o is equal to [S"/(l - S*,.,)]o for cementedsandstone, oolitic limestones, and vugular rocks. To handle residual oil saturation,this relationship should be altered to [(S" - S.,,)/(l - S*,'..)].0

The correlations developed by Wyllie and Gardner can be used to construct a ternarydiagram showing the relative permeabilities to oil, gas, and water. In general, the values ofrelative permeability (10, 20,30Vo, etc.) are chosen first and then the values of saturationare obtained from the correlations. As can be seen from Chapter 2, Table 3, some of the

(6)

1 0 0 % g a s

r16 Relative Permeabilin of Petroleum Reservoirs

o r L

1 0 0 % w a t e r 1 0 0 % o i l

W A T E R

w a t e r

G A S

w a t e r

FIGURE 15. Three-phase relative permeability.'

equations are nonlinear. Hence, numerical methods (such as Newton-Raphson) are requiredto solve these equations. Manual interpolation is also possible for plotting relative perme-ability isoperms.

I

o i l

En(xlcs

pfe\ k

the hr\ ()lr\

l) F*- t

Ftr

lx)otA,

the \\

l() ca

A. CrL'sr

c*xarrttre rrt'tf !:aacrtx(

k - stri rru

effc,,-t

S raa

B. \r\iu

g a s

g a s

tl7

W A T E R

o l L

w u t 6 l

G A s

o

FIGURE 16. Three-phase relative permeability data of Caudle et al.rr

Empirical relationships provide reasonable results in some cases and very disappointingones in other situations; consequently, they must be used carefully. Note that most of theprevious relationships were developed for media with intergranular porosity. This points outthe huge problem of determining relative permeability curves for naturally fractured reser-voirs. The difficulty arises primarily from the difficulty (or impossibility) of making thistype of measurement on a fractured core sample.

For totally oil-wet three-phase systems in which oil is the wetting phase, water thenonwetting phase, and gas nonwetting with respect to both, the substitution of S" for S* inthe Wyllie and Gardner equations can be made for estimation of the relative permeabilityto each phase.

III. IMBIBITION RELATIVE PERMEABILITY

A. Caudle, Slobod, and BrownscombeUsing a dynamic displacement method on a consolidated core sample, Caudle et al.rl

obtained isoperms for k.o, k.*, and k,*, as shown in Figure 16. They used distillation to findthe water and oil saturations at each data point, and used material balance for determinationof gas saturation. Caudle et al. employed a pressure differential of 5 to 50 in. of wateracross the core and used water permeability as the base value. Relative permeability to waterk.* was found to be dependent on S*, Sr, and S". These workers recognized the presenceof some form of hysteresis in the three-phase studies, but they ignored the capillary endeffect. They found all relative permeabilities to be approximately at minimum values whenSo was maintained at the value of S*..

B. Naar and WygalNaar and Wygal12 developed a set of equations that was discussed in Chapter 2. Based

) arc rc'quiredhtrr g perme-

l18 Relative Permeabilin of Petroleum Reservoirs

w a t e r - w e t

S 'w

FIGURE 17. Three-phase imbibi t ion.rr

on these equations they plotted isoperms with 1007a reduced saturations at the apexes, as

shown in Figures 17 and 18. The displacement mechanism indicated that at the beginning

of the imbibition process, S** (reduced water saturation) increased at the expense of S, at

constant S", until no more gas was trapped. Thereafter, S* increased at the expense of S.,

at constant S*. This path is traced in Figure 17. The locus of all such paths is also shown.

Unlike the findings of other workers, Naar and Wygal concluded that k,.,/k,* is not a

function of S*, for equal values of oil recovery in three-phase flow. On the other hand, the

ratio was found to be a function of Sr, and wettability. This dependence is shown in Figure

19. The higher the initial gas saturation, the less the influence of wettability on k.,,/k.*. Also,

the water saturation at a given recovery was a function of initial water saturation and initial

gas saturation. The ratio of S*, for a water-wet system to S*, for an oil-wet system increasedwith Sri, and the rate of increase was an incresing function of S*,. For a given S* ratio and

a given recovery, S*, decreased with increasing Sri. With higher S*,, there is less pore space

available and the oil is already pushed out into larger channels because of the higher S*i;

therefore, less water is required for the same recovery.The imbibition water-oil relative permeability equations developed by Naar and Wygal,

based on the assumption that l/P.2 equals CS*, are

S * * . , n , b - s * d s *

kr* . i r rb

S;

o i l - r

and

anJ S

\a;

FCrm(

n herc

IS;

So, i , , , . imb /s*

" " in t t '

P: (e)f ' t - s * d s *J,, P:

I . : : \ , \ ! ' \ . a \

E ^ . , - r n n i n g

l f l . ; , ' l S . a t

l f \ ' : : .C t l f S, ,

a l . , ' . h t t \ \ ' n .

L l . n ( ) t a

r : : t . 1 r 1 J . t h e

l r : r [ i r g u r e

; . ' \ l s o .

J l . : ' . 1 l n l t i a l

!n: : . . rcased

S : . r i t t r d f l d

S l " : J :paC€

I : . : h c r S , , , ;

Naar and Wygal suggestedpermeability:

FIGURE 18. Three-phase drainage.rl

: s;:i (s..,, + 3 Si")- S"o)t (S,, + 2S.,n + 3S*, - 3S*')

( I - S* ' )"

the following approximation for

119

w a t e r

( l0)

imbibition gas relative

and

where

kr,r. i rrb

(s,,

k,*0 .5 - S* * i . i n , r ,

0 . 5

: S * * . d r u i n -

( I - S*i'",n)

l /2 S::, drain;

( l l )

S**, . i - t ,g l t

a i . l \ \ . r ga l ,S _ S .

S * : * 'l - S * ,

s . l , : S " - S " t 'l - S * ,

S l * - S * l S * 'l -

* S.t,

S * i

and S"o is the trapped oil saturation.

o2

t20 Relative Permeability of Petroleum Reservoirs

1 A

S * 1 r , = o ' 3

s w i r r - o . ' l 5

' t 2

3 r o=;L 8

I

o

J

6

o3I

(!

3

;-l

J

^ t ^(Dl u)

{ l {o l == l o :{ l qo l t- l r

l o

. 6. 5. 4. 3. 2. 1

I N I T I A L G A S S A T U R A T I O N

FIGURE 19. Influence of wettabil ity at 40o/c recovery.' l

These models were derived by assuming the random interconnection of straight capillaries,with a provision for blocking of the nonwetting phase by the invading wetting fluid.

The imbibition water, and drainage oil and gas, relative permeabilitt equations developedby Naar and wygal were also presented in the following form:

W a t e

kr* , i , r ,b : (S ; )o

k, . . i *u : ( l - 2S*x; t ' t {2 - ( l - 2S*x; , ,2 t

stncenaturc

Hi :s im i la

[-rn

and

(t2)

( l 3 )

(r4)k,* s_i r: - 2srr)

where

S - , : s F - s * 'F ' l - S * ,

In these equations the subscript "t" stands for "trapped" and,,f" for.,free,,

C. LandIn Land's13 work, equations for imbibition two- and three-phase relative permeabilities

were obtained from rock properties. Land considered residual gas saturation after imbibitionto be directly related to the initial gas saturation. The gas and water imbibition relativepermeabilities were reported to be the same in three-phase systems as in two-phase systems, For S.

12l

G a sI

IId

cI

It

))It

o r L

W a t e r

FIGURE 20. Imbibition k.,, for a mobile gas saturation.rl

since the totally nonwetting and wetting phases occupied the same pores regardless of thenature of the other phases present.

His plots fork.o in the II and ID cases are shown in Figures 20 and 21. The ID plots aresimilar to the plots obtained by Naar and Wygal,12 their system being an II case.

Land's final equations are

X . . r t ' r I la r ies .t ? - I

D r . l - ' r c l t l ped

( l 2 )

( l 3 )

( 1 4 )

m: re . rh i l i t i esi r r ; : rh rb i t ionIr , ' : : rc lat ivel : i .) r tc ' ITlS.

k,*

^ * r I t d s *5 i It s ' J r * s g r p :

t . _Kr* -

k.. :

I tII''EI t

r,'fl.,": Y

( l s )

( l 6 )

( t7)f ' d s *J,, P:

For S* increasing and S* constant:

k.o : S:i [2(S** * S",*) * S"r*] ( 1 8 )

G a s


W a t e r

G A S

Imbibition k,,, for a trapped gas saturation.rl

one obtained by Corey et al.2 for the drainage condition.

o i l

( l e )

Land'of Cc

D . &Scl

cartxrimb ibgas \:on oip€rTnrfoundperTnrUrSlitr

u'ettirthen tbetucrelat i rvalueCorer

E. SrThr

to draare c(co. ishou r

S i na matlrequirmode lNolenconditextrerthe ea

\togas v

This equation isWhen all the gas

k.. :

where

FIGURE 2I.

similar to theis trapped:

s:i(2s** + S"r*) - S,,r*[S*i. + 2/C(S;, + l lC{lnSr,/Sr,})J

S* : ( l _ S* r * )

S. - S.-S.* :

Sr,* :

S.u* :

S.r* :

( - -

l - S * , - S , , , .

S",-l - S * ,

Sot

l - s * ,

S. - Sru

l - S * ,

II

(sr, * ),.u*

G A S

p . , ' n d r t i o n .

( 1 9 )

123

s** :

s :v ( ) m

Sr, :

S* - S*.

l - S * .

minimum residual oil saturation

trapped gas saturation

S.r, : trapped oil saturation

Land's correlations did not consider hysteresis since his derivation was based on the workof Corey et al., which did not include hysteresis effects.

D. Schneider and OwensSchneider and Owensra performed steady-state and unsteady-state tests on a variety of

carbonate and sandstone samples, and found the relative permeability to oil during animbibition process in a water-wet system to be insensitive to the flowing gas phase whengas saturation was increasing. Oil relative permeability was found to be primarily dependenton oil saturation. It was reported that residual oil significantly reduced the gas relativepermeability in a water-wet system. The gas relative permeability in an oil-wet system wasfound to be insensitive to the presence of a residual oil saturation. The nonwetting relativepermeability-saturation relationship in three-phase flow was reported to depend on the sat-uration history of both nonwetting phases and on the ratio of the saturations of the twowetting phases. In some cases the nonwetting relative permeability was found to be lowerthen the two-phase value due either to trapping of a nonwetting phase or to flow interferencebetween the nonwetting phases when both were mobile. For some tests the nonwettingrelative permeability value for three-phase flow was found to be higher than the two-phasevalue. The authors discussed the reasons why their results did not fullv asree with those ofCorey et al.

B. SpronsenThe centrifuge method, already proven for two-phase flow, was extended by Spronsenrs

to drainage three-phase flow in a water-wet system. Oil isoperms determined by Spronsenare concave toward the l00Vo oll apex. He discussed the adverse influence of immiscibleCO, injection on the shape of three-phase oil isoperms. The results of his investigation areshown in Figure 22.

IV. PROBABILITY MODELS

Since the experimental problems associated with three-phase flow are difficult to surmount,a mathematical model appears to be an alternate approach. The correlations discussed earlierrequired some type of experimental three-phase flow data. On the other hand, probabilitymodels as formulated by Stoner6'r7 and modified by Dietrich and Bondor'8 and later byNolen as cited by Molina,'e assume that two-phase flow behaviorcan be used as a l imitingcondition for three-phase flow. Water-oil-gas flow can be bounded by water-oil flow at oneextreme and oil-gas flow at the other. While some of these models can consider hysteresis,the earlier correlations, such as Land's,r3 cannot do so.

Most probability models assume that gas relative permeability is dependent only on thegas saturation:

kry k., (Sr) (20)

g a s


w a t e r

O I L I S O P E R M S

w a t e r o i lW A T E R I S O P E R M S

FIGURE 22. Data of Spronsen for Berea sandstone.r5

Similarly, it is assumed that the relative permeability to water is dependent only on thewater saturation:

k,* : k.*(S*) (2t )

Oil relative permeability, however, varies in a more complex manner. These assumptionshave been confirmed in laboratory investigations for a water-wet system.

In a water-wet system, gas behaves as a completely nonwetting phase, but oil has anintermediate ability to wet the rock. The relative permeability to oil in a water-oil-gas system

wil l t lsaturaStoneresultWatercorTesexprel

where

and

Fayen

where

Storresiduoil rel

whererelatirphase.kr. val

Alttrelativin whiof themost (

g a s

. 0 o 0 0 2

- o o o 1

- o o o 5- o o 1

12s

will therefore be bounded by relative permeability to oil in a water-oil system at low gassaturations and by relative permeability to oil in a gas-oil system at low water saturations.Stone attempted to combine these two terminal relative permeabilities to obtain a three-phaseresult by using the channel flow theory in porous media and simple probability models.Water and gas three-phase relative permeabilities, according to Stone, are the same as theircorresponding two-phase relative permeabilities. In his first model, Stone developed theexpression:

k.. : s;P",F*

where

S.,* :S. - S.,,

l - S * , - S . .

(22)

k_.P * : r = ;

( 2 - P h a s e )

S*.*s * - S* i

l - S * , - S . ,

9, : +T (2-phase)

t rn l r t ln the

( 2 1 )

a..untpt ions

I , r r l has an

l - t . r . r t s tem

and

S , :s . -

[ - s * , - s . . - S r .

Fayers and Matthews26 suggested that

S.. : c{. S,,,* + (l - o,) S,,.o

where

: l - S '- r I - S * . - {

Stone's earlier model did not agree well with data involving the dependence of waterfloodresidual oil saturation on trapped gas saturations. Stone's second model gave three-phaseoil relative permeability as

k,,, : (k..,* + k,*)(k",* + k,s) - (k.* + k,s) (23)

where k,o* and k.* represent oil and water relative permeabilities from two-phase, oil-waterrelative permeability data; k,o, and k,* represent oil and gas relative permeabilities from two-phase, oil-gas relative permeability data. Equation 23 may yield unrealistic results at lowk.o values.

Although it seems reasonable that one should be able to combine the two two-phaserelative permeabilities to arrive at three-phase data at least for water-wet systems, the mannerin which they have been combined in these models may not account for the total physicsof the process. These probability models strongly depend on the assumption that there is atmost one mobile fluid in any channel. That is, Stone's assumption implies that water-oil


capillary pressure and water relative permeability are functions of water saturation alone inthe three-phase system, regardless of the relative saturations of oil and gas. Moreover, theyare the same function in the three-phase system as in the two-phase gas-oil system. Stone'ssecond model generally predicts the correct oil relative permeability in the three-phase systemif the relative permeability at the end points is equal to one. Stone points out that when hissecond model yields a negative k,,,, this implies a complete blockage of oil and as a resultk." equals zero. The Stone models account for hysteresis when water and gas saturationsare changing in the same direction.

Dietrich and Bondorr8 applied Stone's models to published three-phase data and foundthem to be only partially successful. They found that it was necessary to modify Stone'ssecond model for the case where gas/oil relative permeability is measured in the presenceof connate water. They pointed out that, at irreducible water saturation and zero gas satu-ration, Equation 23 reduced to

k,. : (k..,*)(k.,,*)

This expression can be valid only if both k..,* and k.,,, equal unity. Since k,,, at S*. isfrequently less than one, Stone's second model has some limitations.

Dietrich and Bondor adjusted Stone's model by normalizing it with k..,.* to obtain:

k.. : fr t,0,"* * k,*) (k.", * k.r)l - (k.* + k.g)

where k.o.* is the oil relative permeability at connate water saturation. At irreducible watersaturation and zero gas saturation this equation reduces to:

k - . : ( k " ' o ) ( k " ' t )

k..r.*

This model tends to predict incorrect oil relative permeability values (magnitude largerthan unity) for values of k,,,.* < 0.3.

Nolen, as referenced by Molina'e has taken into account this problem and suggested thefollowing model which remains bounded as k.,,.* approaches zero:

k_ t .k.,, : k..,.* )* + k,*:: * k.s - (k.*, + k.s) (25)

Na,, .o ' *

k . . r . *

V. EXPERIMENTAL CONFIRMATION

Three-phase relative permeability studies are still in an early stage of development. Littlehas been done on the experimental confirmation of imbibition correlations and most of thecorrelations available are for imbibition.

Early work was done primarily on unconsolidated sands and the effects of wettability andhysteresis were not recognized until recently. Donaldson and Kayser2o have categorized thereasons for divergence of experimental three-phase relative permeability data as follows:

l. Errors introduced in saturation measurements in various experimental methods.2. Errors introduced by neglect of capillary end effects and saturation hysteresis phenomena.3. Variations caused by use of different oils, brines, and cores which could exhibit

different wettability characteristics.4. Assumptions made to facilitate experimental procedures or calculations.5. Inadequacy of mathematical formulations to represent three-phase flow conditions.

The empirical methods, though seemingly simpler, suffer from simplifying assumptionsthat have limited the range of saturation histories that can be simulated.

(24)

TabperTneiauthorsysterrrock sair . ca

ThrrdieselBerea.stone iis ofter

Perrtical mia1 'abYh. gtProbleu'hen rremo\p€rTne;itored

Brirmeasuand th,l iquidor b1' r

Thebe sturto inflconcer

Bouand pema) 'p lPenn Igatorsstate n

I n arate ofmEasu,may bshouldwith tlthe ganeedlebe meior '* ' i t lis desirSO OS lr

pressur

I t , ' : : . r l t lne in

Drc, , r c r . they

l c : : : S tone ' s

ph. : .e \ \ Stem

hu: ' ,r hcn his

rJ . : . u result

F . . r lu ra t ions

Lt .:n.l tirund

XJ ; : , . S tone ' s

th . l . rcscnce

3 f ' - . r : \ i l t U -

}. .r t S,.. iS

r , x : . r l n :

(24)

lu . . ^ l c *a te r

nr : . . .1c la rger

Ui iJ. t r 'd the

( 1 5 )

pc:Jnt . L i t t le

I r : r , ' . I o f t he

B t l : i . r l i t r and

le g' ' r12a6 ,1ta

I r l , r l l t tws :

3 t h , r J r .

i^*n, ' t t lena.

t ru .J c ' rh ib i t

9( )n ! i l l lons.

a . .unrpt ions

i'il!

i

t

r27

Table I is a chronological listing of the experimentally determined three-phase relativepermeabilities that have been reported.2r In all of the studies included in the tabulation theauthors used refined oils in order to minimize oil-wetting; they assumed a totally water-wetsystem. In cases where a single core was used, the influence of the saturation history of therock sample was frequently ignored. The gases used in the studies listed in Table 1 wereair, carbon dioxide, and nitrogen.

VI. LABORATORY APPARATUS

Three-phase relative permeability studies have been conducted using refined nonpolar oil,diesel oil, Soltrol, kerosene, hydrocarbon fractions, brine, nitrogen, air, and carbon dioxide.Berea, Boise, Torpedo, Tensleep, and Weeks Island sandstones, as well as Arbuckle lime-stone and unconsolidated sand samples have been used for the flow media. Berea sandstoneis often preferred because of its uniformity and general acceptability as an industry standard.

Personal communications with researchers in this field have indicated that the most prac-tical means of saturation measurement is gravimetric. Other modern methods, such as gammaray absorption, X-ray absorption, NMR, etc., are unnecessarily expensive and elaborate.The gravimetric saturation measurements are sufficiently accurate and relatively inexpensive.Problems may be encountered with gravimetric saturation measurements, however, especiallywhen gas is used in the presence of volatile oil. Therefore, core holders which permit rapidremoval of cores (without the removal of rubber sleeves) should be used when relativepermeability is determined by steady-state methods. Wettability of the core should be mon-itored either by the centrifugal technique23 or an alternative method.

Brine saturation may be determined with satisfactory accuracy by',electrical resistivitymeasurement when nonpolar oil is employed. Oil saturation may be obtained gravimetricallyand the gas volume may be computed as the difference between total pore volume and totalliquid volume. The oil and water flow rates may be obtained by a simple burette arrangementor by flowmeters. The gas flow rate may be obtained by use of a gas flowmeter.

The effect of wettability on the relative permeabilities is an important factor that shouldbe studied. The change of wettability in a core from oil-wet to a water-wet has been knownto influence relative permeabilities, but no definite conclusions are found in the literatureconcerning the influence of wettability on three-phase relative permeabilities.

Boundary effects should be eliminated by using core plugs at either end of the test coreand performing the experiment at reasonably high flow rates. A semipermeable membranemay precede the core plug at the inlet end for proper distribution of the phases. A modifiedPenn State method of relative permeability measurement may be used, since most investi-gators believe that the Penn State method gives better results than any of the other steady-state methods.

In addition to saturation measurements for each phase, one needs to measure the flowrate of each fluid and the pressure drop while making the steady-state relative permeabilitymeasurements. A gas dome may inject fluids into the core and a back-pressure regulatormay be used to maintain a constant pressure at the outlet end. Also, the gaseous phaseshould be bubbled through the oil supply tanks. This procedure ensures that the oil is saturatedwith the gas before it enters the core. As a result, there should be no mass transfer betweenthe gaseous phase and the oil inside the core. The gas flow rate may be regulated with aneedle valve, with a large pressure differential across the valve. The rate of gas flow maybe measured either by collecting gas by displacement of water for a known length of time,or with a soap bubble meter or a wet-test meter. For pressure differential measurements, itis desirable that the displacement of fluid into the measuring device be as small as possibleso as to minimize error. Hence, use of a regular manometer is not possible, but differentialpressure transducers may be used. The connections for measuring pressure differential can


U)

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J =

- =x L

E .t"

r!

c..l

.=;

n 2X F. ! E

z

, l

= E i " S r I ar ! . E ' E i j =

9 E J i l H : :i ; h . d i . . 5 =: E - -

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" c 3

< t 5 Er 9

Q - , ^ : ?: = ! . . ;t r t s E ; '

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a)

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i _ : . :(n

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E E F E e = i = E s € i ! r e ! t g s r H s ; :E ! E : t H i H E E E t 3 : ; E Z ; ; : € f i ; ! ri EE i * €E IE€EE; IgEgE i l i ; :E :EEi t g * E E d ' = O o # = o - : + = 3 - i E j € 5 : g ;

EE € E i. E : 8 i r E s i i e =g E i F ; 2 E t € f r EF - F e E E E # y t E

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E - !'= c,

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F i l ( ) c )a - Y , . - C C

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O: . � . o

' j , n r i s . )- u r - . . v - ,

: . E r g ' r t i l ," 3 € . = 1 3 E €3 : 3 = i i E E zE : ; _ E = U I e j' E - a o 9 i' - ' = F ^ . = 2 . ? Z *> !

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- a ' * P B g: 1 . . : = - iZ a i t ' Z i . ' .E g U E E : E € "f = E H E I E = 3

g

t >. " 9 ! =i - e - + E = 3 29 . = F i i o o l =E - 4 r i ; . l b P

E ; E - E € . E E EI-lJ

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E r e * = E * € i = E fc ! i ; s E € l E e E r = : ; E E E g' 5 E : g i ; s : : * E ! ; s ; : * E s s +E s E € E * ; t E ! e : ; : ! E $ * s ; ;* $ E F r 3 E E F E = > 3 i : € B ! 3 ; i ' P E

b i H ' E * , H :

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c J " ,L V

J . i + R l au t l - = xo r F i u t zb r J = q J = C )F - r s 4 -

' . = v t a - -

v )

t t Z - l t - 1 a l b S

: : i ; E T q ? * U ! H E f i - . i E H *t * [ i t n l i . € , ;=g €s t I i ;=€n [g t€ l x e E € l : g i $ $ i g l j = ; = 3 - ; ; l : - $ t

t--

(n

A

6 dt r c )

o -

|r)

I

(h

c.lo\

q..)

U)

n

(.)

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(n

JU)

q)L

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.=

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9 r y

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(t)

>.

: i t r. oE . !

r-adr;ll o lt o lt s lL1

t-III1tr

(n

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€ - ? : $ !s 5 5 ; 1 3U)

3

J

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()

O

z

X c : i- . 9 S U .

u - - - 2 F; ; g 9 J - t r H > '1 - 3 o ) 9 2 i - == d - t E t r i Fi 9 ' 6 ' i . = t

LIJ

o .r,u Y v t

T ; E= g Ez -

FI

az-3r h-3ar-'!-

z-l

3Fl

j---

&---

^ )E FO F

= <

= nx -Y r-r-t U)o (. o *E n rF 'Jr-

-&FFH

3-

r f r

z-E-r(t)-t lJ

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Fl

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tr .c)O FF € J

; J Et r ( DFr

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> i i li : / L

X c t r

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9 q rS olJ

ii o.r' OU)

a . . , cE = ; t =

H J ? o ' r t r " 3 Ft ' , = F * E l l : i - , ( E EE t , ' g g E I g ; E - : Y i i; : x . E , . , , E _ y ; X f i : _9 i x : Y ! i l ' , , : ; , a ; s I2 = " d s I5 s = 3 , , ; g € 5 g e T U ; e

r-co

q)

oZ x

2 9 r "L @ Vq ) ^ N

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l-

F 8 E: L t ' ,

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3q)

r Q )o =

8 . s

E E. = e

E :

3 g

EF , 9E X. E ;

x

a.lco

q)

L

U)

q,, -: FA o ,

t^

O # a

E E 94 E :

v v(h

131

C O R E H O L O E R

FIGURE 23. Schematic diagram of three-phase relative permeability apparatus.

be made through semipermeable membrane ports. The capillary tubes connecting the trans-ducer may be inserted and cemented in place about I in. from each end of the test apparatus.

Unsteady-state methods of three-phase relative permeability measurements have the ad-vantage of being rapid. Oil and water may be displaced by gas to duplicate gas drive processesused in enhanced recovery methods. However, the calculation of isoperms from laboratorydata requires analytical solutions of the partial differential equations describing the three-phase fluid flow. Some early studies have made erroneous simplifying assumptions indescribing the dynamic condition of the unsteady-state process. Reliable values of relativepermeability as a function of saturations may be obtained by mathematical simulation oflaboratory data using finite difference calculations.20 Capillary pressure data should beobtained for gas-oil, water-oil, and water-gas systems to provide basic data necessary forthree-phase relative permeability calculations. Solubility of the gas in the liquids employedin the study should be determined before these calculations are performed.

A schematic diagram of the apparatus used for three-phase relative permeability meas-urement is shown in Figure 23. The core holder, which has ports for differential pressuremeasurements, allows rapid retrieval of the core. Temperature is controlled with a Propor-tional Controller connected to a heating tape wrapped around the core holder. In order toeliminate pulsation of flow normally associated with pumps, fluids are injected by applyinggas pressure on top of the fluid in a tank equipped with appropriate relief valves. Solenoidvalves and level controllers maintain a constant head of fluid in the supply tanks. Filtersare provided in the supply lines of each phase being injected into the core holder. Checkvalves prevent backflow of each of the three phases. A cross section of the core holder isshown in Figure 24.

Auxiliary equipment includes an accurate balance, electrical resistivity measurement sys-tem, level controller, chart recorder, differential pressure transducer, cylinders, compressedair and regulators, and a humidity oven.


D I F F E R E N T I A L P R E S S U R E P O R T S

2 "<-._

A N N U L A R

P R E S S U R EP O R T

rON

cv)

-- - 1 . 1 2 5 >

+1JJ\1 9 "

l .

2 .

3 .

4 .

5 .

6 .

7 .

FIGURE 24. Diagram of a core holder.

VII. PRACTICAL CONSIDERATIONS FOR LABORATORY TESTS

The literature cited contains a large amount of information on factors affecting the. lab-oratory investigation of relative permeability. The following listing, however, cites iomepractical considerations that have not been widely discussed in the literature:

If a pump is used to inject fluids into the core, the packing material should preferablybe Teflon@. Most other packing materials contain silicon and carbon which maydissolve in injected fluids and affect the wettability of the core.When brine is used as one of the fluids, all metal parts of the system should be ofstainless steel. One-eighth-in. tubing offers excellent handling characteristics. Tygontubing is recommended if the pressure is not too high.Most electronic differential pressure transducers have good linearity and hysteresischaracteristics; however, if possible, the transducer should be recalibrated at least onceper month.While changing pressures on the liquid storage tanks, it is important not to exceed thebackpressure rating of the solenoid valves.Every effort should be made to ensure l00%o saturation of the wetting phase beforestarting injection of the nonwetting phase.In a steady-state experiment, input flow rate should equal the output flow rate for eachphase. In many cases, this condition is tedious to achieve.Some extraneous material may be noticed in the output lines. It must be determinedwhether the particles are fines from the test sample or bacterial matter. A bactericidemay be used with caution not to alter either the wettability or the resistivity of thecore.Often the resistivity meter utilizes chamois leather contacts at either end of the coreholder. The contacts should be kept immersed in brine to prevent changes in thereadings.It has been noticed that the position of the outlet tubes going into the measuringcylinders affects the pressure differential readings. It is recommended that the tubingoutlet be kept at the same level as the core holder to eliminate gravitational effects.

10. cd

l l . E12. l r

dp

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9 .

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r t jctcrmined

t hu i tc r ic ide

Irr rtr t l f the

I t r i thc ' COf€

nsc . in the

: rnca:uringI t r rc tubingurl ct 't 'ects.

r33

FIGURE 25. Comparison of three-phase oil relative permeability determinations.

10. Gas in the transducer lines seriously affects pressure differential readings. The trans-ducer should be bled of gas at frequent intervals.

I l. Every effort should be made to eliminate end effects as described by Batycky et a1.2212. If possible, the wetting characteristics of the core should be frequently monitored

during the relative permeability experiments. The centrifuge method23 may be em-ployed for monitoring wettability.

VIII. COMPARISON OF MODELS

The following section presents a comparison of some of the models discussed earlier.The equation of Corey et al.2 for three-phase k,,, values is valid for a system in which oil

is displaced by a gas. Donaldson and DeanT obtained three-phase k.., values following thesame displacement mechanism. Thus, we have an opportunity to observe how well theequation of Corey et al.2 fits data provided by other workers. Three-phase oil relativepermeability values calculated by the equation of Corey et al.2 were compared with Donaldsonand Dean's data. The isoperms obtained are shown in Figure 25 along with Donaldson andDean's data as a basis for comparison. The Corey et al.2 equation gives higher k,. valuesthan those obtained by Donaldson and Dean. Isoperms by Corey et al3 are less concavetowards l00%o oil saturation. Both methods are in agreement in predicting that the isopermsbecome concave toward l00Vo S" and decreasing S*. The Donaldson et a1.23'2a data showk.., increasing up to an optimum S, value and then decreasing. This is evident for values ofS" between 30 and 6OVo on this Berea core. The Corey et al. correlations give isopermswhich show k,o to increase as S., increases at the expense of S*. The discrepancy betweenthe two methods is larger at low S., values.

In the second comparison, data of Schneider and Owens2s have been used to obtain


N o l e n , s M o d e l

-o--<r- Meihod of Naar & Wygal

FIGURE 26. Comparison of three-phase oil relative permeability determinations.

isoperms by Nolen's modelre and by Naar and Wygal's correlation.r2 Few data are availablein the literature that show how the latter method compares with experimental values or othercorrelations. Figure 26, however, provides such a comparison. Schneider and Owens ob-tained gas-oil drainage data in the absence of connate water; their oil-water imbibition datais for a water-wet system. Theoretically, the Dietrich and Bondorr8 or the Nolen modelshould give the same results as Stone's second model, since gas-oil data used in thiscomparison have been obtained in the absence of connate water, i.e., k.o"* equals unity. Asin the earlier comparison, the discrepancy between the two methods is evident at low S"values. Another point to note is the evidence that k," depends only on So values, especiallyat low S" in Naar and Wygal's correlations. There is a slight indication in both methodsthat k," isoperms become convex towards the l}OVo So apex at high S".

REFERENCES

l . Leverett, M. S. and Lewis, W.8., Steady flow of gas-oil-watermixtures through unconsolidated sands,Trans. AIME, 142. 107.1941.

2. Corey, A. T., Rathjens, C. H., Henderson, J. H., and Wyllie, M. R. J., Three-phase relative perrne-ability, Trans. AIME, 201,349. 1956.

3. Reid, S. , The Flow of Three Immiscib le Flu ids in Porous Media, Ph.D. thesis, Univers i tv of Birminsham.England 1956.

4. Snell, R. W., Measurements of gas-phase saturation in a porous medium, "/.l 959 .

5. Snell, R. W., Three-phase relative permeability in an unconsolidated sand, "/.t 962 .

6. Snel171 .

7. Dontt.l ' .l l

8 . SanJ . . t

9. Sarzre\()

10. \ ' l ' r l

Pnrhl | � C a u

dete12 . Naa

19613. L.an

PrttP14. Schr

char15 . \ ' an

o f P16. Stor17. Stol

t : .18. IXcr

at th19. l ltol

SPE20. Dorl

rcF\

21 . l l a r

Prcv22. Bat-r

and rl9t r I

23. Dooefllc

24. Donoi l '

25. Schrchar,

26. Falrrelat

Inst. Pet., 45(428), 259,

Inst. Pet., 48(459), 80,

b : .

' \ : . . 4 W y g a l

arc r\ ai lable

i luc. trr other

| ( ) r icns ob-

br i . r t ron data

f r r r lc I model

u\ ' ! l in th is

t l . . r n i t v . As

nt . r t low S"

s . c .pec ia l l y

rrth rncthods

Fl . : . , : . ' J sands ,

It r ' ' i Perme-

I i r " ' r n g h a m .

. 3 i j ' r r r 5 Q

, . : . - : i 9 ) . 8 0 .

135

6. Snell, R. W., The saturation history dependence of three-phase oil relative permeability, J. Inst. pet.,59,

4 '71. 1963.7. Donaldson, E. C. and Dean, G. W., Two- and Three-Phase Relative Permeability Studies, U.S. Bureou

of Mines, Washingron, D.C. , #6826, 1966.8. Sarem, A. M., Three-phase relative permeability measurements by unsteady-state methods, Soc.. pet. Eng.

J . . 9 . 1 9 9 . 1 9 6 6 .9. Saraf, D. N. and Fatt, I., Three-phase relative permeability measurement using a nuclear magnetic

resonance technique for est imat ing f lu id saturat ion, sor ' . Pet . Eng. J. ,9,235. 1967.10. Wyllie' M. R. J. and Gardner, G. H. F., The generalized Kozeny-Carman equation, its application to

problems of mul t i -phase f low in porous media, World Oi l , 146, l2 l . 1958.ll. Caudle, B. H., Slobod, R. L., and Brownscombe, E. R., Further developments rn the laboratory

determination of relative permeability , Trans. AIME, 192. 145, l95l .12. Naar, J. and Wygal , R. J. , Three-phase imbibi t ion re lat ive permeabi l i ty , So<' . Pet . Eng. J. , 12,254.

r 9 6 r .13. Land, C. S., Calculation of imbibition relative permeability for two- and three-phase flow fiom rock

properties, Soc. Pet. Eng. J., 6, 149, 1968.14. Schneider, F. N. and Owens, W. W., Sandstone and carbonate two- and three-phase relative permeabrlity

character is t ics, Soc. Pet . Eng. J. , 3, 75, 1910.15. Van Spronsen, E., Three-Phase Relative Permeability Measurements Using the Centrifuge Method. Society

of Petro leum Engineers/Department of Energy, Tulsa, Okla. , #10688, 1982.16. Stone, H. L. , Est imat ion of three-phase re lat ive permeabi l i ty , J . Pet . Tech. , 2,214, 1970.17. Stone, H. L. , Est imat ionof three-phaserelat ivepermeabi l i tyandresidualoi ldata, J. of Can. Pet .Technol . ,

t 2 , 5 3 , t 9 1 3 .18. Dietrich' J. K. and Bondor, P. 8., Three-phase oilrelative permeability models, paper SPE 6044 presenred

at the 5lst Annual Fall Technrcal Conference and Exhibition of the SPE, New Orleans. 1976.19. Molina, N. N., A systematic approach to the relative permeability problems in reservoir simulation, paper

SPE 9234 presented at the 55th Annual Fall Technical Conference and Exhibition of the SPE, Dallas, 1980.20. Donaldson, E. C. and Kayser, M.8., Three-Phase Fluid Flow in Porous Media. DOE/BETCilC-8)t4.

report publ ished by the U.S. Department of Energy. Bart lesvi l le , Okla. , Apr i l . 1981.21. Manjnath, A. and Honarpour, M. M., Investigation of three-phase relative permeability, SPE 12915

presented at the Rocky Mountain Regional Meeting of the SPE, Casper, May 20-23, 1984.22. Batycky, J. P., McCaffery, F. G., Hodgous, P. K., and Fisher, D. 8., Interpreting capillary pressures

androckwet t i ngcharac te r i s t i cs f romuns teady -s ta ted isp lacementmeasurements ,so r ' . Pe t .Eng .J . ,6 ,296 ,t 9 8 l .

23. Donaldson, E. C., Thomas, R. D., and Lorenz, P. 8., Wettability determination and its ef-fect on recoveryef f ic iency, So<' . Pet . Eng. J. , 3, 13, 1969.

24. Donaldson, E. C. and Dean, G. W., Two- and Three-Phase Relative Permeability Studies, U.S. Bureauof Mines, Washington, D.C. , #6826. 1966.

25. Schneider, F. N. and Owens, W. W., Sandstone and carbonate two- and three-phase relative permeabilitycharacter is t ics, Sor ' . Pet . Ens. J. , 3,75, 1910.

26. Fayers, F. J. and Matthews, J. D., Evaluation of normalized Stone's methods for estimating three-phaserelat ive permeabi l i t ies. Sor ' . Pet . Eng. J. , 4. 224, 1984.

r37

APPENDIX

SYMBOLS

A : area: constant

A, : adhesion tensiona : material constantB : formation volume factor

: constantb : material constantC : constantF : fractiong : gravitational accelerationh : thicknessI - injectivity

: resistivity index: permeability: length: exponent: number of barrels of oil: exponent: pressure: volume: volumetric rate: radius: resistivity: radius: saturation: distance in direction of

flow: reduced saturation: total liquid saturation: t ime: velocity: vertical coordinate: constant: constant: angle: lithology factor: viscosity: surface or interfacial

tension: porosity: immobile saturation

Subscriptsa : absoluteav : averagec - critical

: capil larycw : connate waterD : displacementd : displacing phasede : immobile displacing phasee - equilibrium

: external (radius): effective

f : freeg : g a s

i - init ial: index number: irreducible

imb : imbibitionirr : irreducibleL : l iquidLR : residual liquidm : minimummf : mud filtraten : nonwettingo : o i l

: measured at 1007o S*(resistivity)

ob : trapped oilp : producedr - relative

: residuals - solutionSL : total l iquidSTD : standard conditionT : totalt - trappedw : water

: wellwt : wettingxo : flushed zone

kLmNnP

aqR

rS

s*S L

T

Zct

p0\f.ro

0.1,

honarpour - relative permeability of petroleum reservoir

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