Relative permeability of petroleum reservoirs

RelativePermeability of PetroleumReservoirs Authors Mehdi Honarpour AssociateProfessorof PetroleumEngineering Departme...

Author: Mehdi Honarpour | Leonard Koederitz | A. Herbert Harvey

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RelativePermeability of PetroleumReservoirs

Authors

Mehdi Honarpour AssociateProfessorof PetroleumEngineering Departmentof PetroleumEngineering Montana College of Mineral Scienceand Technology Butte, Montana

A. Herbert Harvey

Leonard Koederitz Professorof PetroleumEngineering Departmentof PetroleumEngineering University of Missouri Rolla. Missouri

Chairman Departmentof PetroleumEngineering University of Missouri Rolla, Missouri

@frc') CRC Press,Inc. Boca Raton, Florida

PREFACE In 1856 Henry P. Darcy determinedthat the rate of flow of water through a sand filter could be describedby the equation q- : K A

h , - h . L

where q representsthe rate at which water flows downward through a vertical sand pack areaA and length L; the terms h, and h, representhydrostaticheadsat with cross-sectional the inlet and outlet, respectively,of the sandfilter, and K is a constant.Darcy's experiments were confined to the flow of water through sand packs which were 1007osaturatedwith water. Later investigatorsdeterminedthat Darcy's law could be modified to describethe flow of fluids other than water, and that the proportionalityconstantK could be replacedby 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 generalform AS

k

dz

dPl

u ' : * Ll-P g o s - d s l where S v Z p g D

dP

Distancein direction of flow, which is taken as positive Volume of flux acrossa unit areaof the porousmedium in unit time along flow path S Vertical coordinate,which is taken as positivedownward Density of the fluid Gravitationalacceleration Pressuregradientalong S at the point to which v. refers

dS

, t

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Ir

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The volumetric flux v. may be further defined as q/A, where q is the volumetric flow rate areaperpendicularto the lines of flow. and A is the averagecross-sectional It can be shown that the permeabilityterm which appearsin Darcy's law has units of length squared.A porousmaterialhas a permeabilityof I D when a single-phasefluid with a viscosityof I cP completelysaturatesthe pore spaceof the medium and will flow through cross-sectionalarea under a pressure it under viscous flow at the rate of I cm3/sec/cm2 gradientof 1 atm/cm. It is important to note the requirementthat the flowing fluid must completelysaturatethe porousmedium. Sincethis conditionis seldommet in a hydrocarbon reservoir,it is evident that further modificationof Darcy's law is neededif the law is to be appliedto the flow of fluids in an oil or gas reservoir. A more useful form of Darcy's law can be obtained if we assurnethat a rock which containsmore than one fluid has an effective permeabilityto each fluid phaseand that the effectivepermeabilityto each fluid is a function of its percentagesaturation.The effective permeabilityof a rock to a fluid with which it is 1007.osaturatedis equal to the absolute permeabilityof the rock. Effective permeabilityto each fluid phase is consideredto be independentof the other fluid phasesand the phasesare consideredto be immiscible. If we define relativepermeabilityas the ratio of effectivepermeabilityto absolutepermeability, Darcy's law may be restatedfor a system which containsthree fluid phasesas tirllows:

Vo.:T(0.,*K-*)

V*.:*(o-'13-t) Vo,:H(o-r#-k) Note that k,,,' where the subscriptso, g, and w representoil, gas' and water, respectively' saturations k.", and k,* arethe relativepermeabilitiesto the threefluid phasesat the respective rock' of the phaseswithin the a hydrocarbon Darcy's law is the basis for almost all calculationsof fluid flow within of permeability relative the determine to necessary is it law, the use reservoir. In order to made throughout the reservoirrock to each of the fluid phases;this determinationmust be in measuring involved problems The encountered. will be that the rangeof fluid saturations A summary investigators. many by studied been have permeability and predictingrelative chapters' following the in presented is research of the major resultsof this

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THE AUTHORS

ltr.' .. \r,tc thlt k..,. re.}.. : r'.. .sturations

Dr. Mehdi "Matt" Honarpour is an associateprofessorof petroleumengineeringat the MontanaCollege of Mineral Scienceand Technology,Butte, Montana. Dr. Honarpour obtainedhis B.S., M.S., and Ph.D. in petroleumengineeringfrom the Universityof Missouri-Rolla.He has authoredmany publicationsin the areaof reservoirengineeringand core analysis.Dr. Honarpourhas worked as reservoirengineer,researchengineer,consultant, and teacherfor the past 15 years. He is a member of severalprofessionalorganizations, including the Societyof PetroleumEngineersof AIME, the honorarysocietyof Sigma Xi, Pi Epsilon Tau and Phi Kappa Phi.

" ., hrJrttarbon Iri:' of tt: . - :.o.':-tlrcahilitl ' .'.ic throughout I h\ !\. . :.: tn lllt'a\uring -: '\ ruilflrof)' [r--::

Leonard F. Koederitz is a Professorof PetroleumEngineeringat the University of fromtheU ni vers it yofM issour iH erecei vedB .S .,M.S ., andP h.D .degrees M i s s o u ri -R o l l a. previously servedas Department and for Atlantic-Richfield has worked Dr. Koederitz Rolla. publicationsand two several technical co-authored or He has authored at Rolla. Chairman reservoir engineering. related to texts

Plc:.

A. Herbert Harvey receivedB.S. and M.S. degreesfrom Colorado School of Mines and a Ph.D. degree from the University of Oklahoma. He has authoredor co-authored numeroustechnicalpublicationson topicsrelatedto the productionof petroleum.Dr. Harvey is Chairman of both the Missouri Oil and Gas Council and the PetroleumEngineering Departmentat the University of Missouri-Rolla.

ACKNOWLEDGMENT The authorswish to acknowledgethe Societyof PetroleumEngineersand the American PetroleumInstitutefor grantingpermissionto usetheir publications.Specialthanksare due J. Josephof Flopetrol Johnstonand A. Manjnath of ReservoirInc. for their contributions and reviews throughoutthe writing of this book.

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TABLE OF CONTENTS n.j thc Anrerican l i : : : . , n k .a r e d u e rr: - 'ntributions

Chapter I Measurement of Rock Relative Permeability . Introduction.. . I. Steady-StateMethods.. . il. Penn-StateMethod A. Single-SampleDynamic Method B. StationaryFluid Methods C. HasslerMethod. D. Hafford Method E. DispersedFeed Method . F. Unsteady-StateMethods III. IV. Capillary PressureMethods V. Centrifuge Methods VI. Calculation from Field Data . R e f e r e n c e.s. . .

I I 1 I 2 4 4 5 5 6 8 9 10 t2

Chapter 2 Two-PhaseRelative Permeability Introduction... I. Rapoportand Leas II. III. Gates,Lietz,andFulcher... F a t t ,D y k s t r a ,a n d B u r d i n e . IV. W y l l i e, S prangl er,and Gardner. V. T i m m e rman,C orey,and Johnson VI. Wahl, Torcaso, and Wyllie VII. VIII. Brooks and Corey XIIX. Wyllie, Gardner,and Torcaso. . . L a n d ,W y l l i e , R o s e ,P i r s o n ,a n d B o a t m a n . . . X. Knopp, Honarpouret al., and Hirasaki XI. References.....

. . . .27 .... . .29 ...... 30 . . . . . .37 ........41

Chapter 3 Factors Affecting Two-Phase Relative Permeability Introduction... I. Two-PhaseRelativePermeabilityCurves il. S tates n. Effe c t sof S aturati on Effectsof Rock Properties IV. V. D e fi n iti onand C ausesof W ettabi l i ty. DeterminationofWettability.... VI. A. ContactAngle Method ImbibitionMethod. B. B u r e a uo f M i n e sM e t h o d C. D. C a p i l l a r i m e t rM i ce t h o d . . . FractionalSurfaceAreaMethod.. E. D y e A d s o r p t i o nM e t h o d F. D r o p T e s tM e t h o d . . G. M e t h o d so f B o b e ke t a l . H. MagneticRelaxationMethod I. ResidualSaturationMethods J.

.... 45 .......45 ....45 . . . . . . 49 .... ... 50 . . . . . . . . 54 .......58 ... 58 .......60 .......63 ......63 ....64 ' ...... .64 .. ...64 ........64 ...64 .. .65

...... 15 .......15 .. ' 15 .....16 ...... 16 . . . . . ' . 19 . . . . . . 20

27

P e r m e a b i l iM t ye t h o d. . . . tye th od Co n n a teW a te r-P e rm e a b i l iM M e th o d.... Re l a ti v ePe rme a b i l i ty Su mma ti o nMethod Re l a ti v eP e rm e a b i l i ty R a ti oMe thod Re l a ti v eP e rm e a b i l i ty W a t e r f l o o dM e t h o d CapillaryPressureMethod a. Re s i s ti v i tyIn d e x M e th o d R. FactorsInfluencing Wettability Evaluation VII. VIII. Wettability Influenceon MultiphaseFlow E f f e c t so f S a t u r a t i oHni s t o r y . . . . IX. Effectsof OverburdenPressure.. X. K ) ( I . E f f ec t sof Po ro s i tya n d P e rm e a b i l i ty ... Effectsof Temperature. XII. XIII. Effects of InterfacialTension and Density .;.... X I V . E f f e c t so f V i s c o s i t y. . . Saturation XV. Effectsof Initial Wetting-Phase XVI. Effects of an Immobile Third Phase XVII. Effects of Other Factors References.....

....... 65 ....... 66 .... 66 ........61 ........67 ....... 68 .... . 68 ... . ... 68 .. . 68 . . .72 ......'74 ... ' .. 78 ......79 . .. .82 . . .82 . .. ' ' 83 ... 89 . '. 90 . . .92 ..-.....97

Chapter 4 Three-PhaseRelative Permeability Introduction... I. DrainageRelativePermeability... il. A. Leverettand Lewis B. Corey, Rathjens,Henderson,and Wyllie Reid. C. Snell. D. Donaldsonand Dean E. Sarem F. S a r a fa n d F a t t G. WyllieandGardner... H. I m bibit io nR e l a ti v eP e rm e a b i l i ty ... m. Caudle,slobod,andBrownscombe A. N a a ra n dW y g a l . . . . . B. Land. C. D. SchneiderandOwens.... Spronsen E. ProbabilityModels IV. V. E x per im e n ta l C o n fi rm a ti o n U\ / I . Labor at o ry Ap p a ra tu s ... PracticalConsiderationsfor LaboratoryTests VII. VIII. ComparisonofModels References""'

... f 03 ......103 ..'.104 ... ' . . 104 .. 105 .. 107 .. l0g .. . . I l0 .......113 ..... I 15 .'ll5 ...117 .......117 ....I 17 .. 120 .....123 .'..123 . .123 .....126 ..127 .... ' 132 ...'133 """'134

K. L. M. N. O. P.

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Appendix Symbols.

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6-i

Chapter I

66 66

MEASUREMENT

OF ROCK RELATIVE

PERMEABILITY

666,\

I. INTRODUCTION

hs h\

6\

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The relative peffneability of a rock to each fluid phasecan be measuredin a core sample "unsteady-state"methods.In the steady-state method, a fixed by either "steady-state" or ratio of fluids is forced through the test sampleuntil saturationand pressureequilibria are established.Numerous techniqueshave been successfullyemployed to obtain a uniform saturation.The primary concern in designingthe experimentis to eliminate or reducethe saturationgradientwhich is causedby capillary pressureeffectsat the outflow boundaryof the core. Steady-state methodsare preferredto unsteady-state methodsby someinvestigators for rocks of intermediatewettability,' althoughsomedifficulty hasbeenreportedin applying the Hasslersteady-state method to this type of rock.2 ln the capillary pressuremethod,only the nonwettingphaseis injectedinto the core during the test. This fluid displacesthe wetting phaseand the saturationsof both fluids change throughout the test. Unsteady-statetechniquesare now employed for most laboratory measurementsof relative permeability.3 Some of the more commonly used laboratory methods for measuringrelative perrneability are describedbelow.

II. STEADY-STATEMETHODS

.le

A. Penn-State Method This steady-statemethod for measuringrelative perrneability was designedby Morse et al.a and later modified by Osobaet aI.,5 Hendersonand Yuster,6Caudleet a1.,7and Geffen et al.8 The version of the apparatuswhich was describedby Geffen et al., is illustrated by Figure l. In order to reduce end effects due to capillary forces, the sample to be tested is mounted between two rock sampleswhich are similar to the test sample. This arrangement also promotes thorough mixing of the two fluid phasesbefore they enter the test sample. The laboratory procedure is begun by saturatingthe sample with one fluid phase (such as water) and adjustingthe flow rate of this phasethrough the sampleuntil a predetermined pressuregradientis obtained.Injection of a secondphase(such as a gas) is then begun at a low rate and flow of the first phaseis reducedslightly so that the pressuredifferential acrossthe systemremainsconstant.After an equilibriumconditionis reached,the two flow rates are recordedand the percentagesaturationof each phasewithin the test sample is determinedby removing the test samplefrom the assernblyand weighing it. This procedure introducesa possible sourceof experimentalerror, since a small amount of fluid may be lost becauseof gas expansionand evaporation.One authorityrecommendsthat the core be wgighedunder oil, eliminating the problem of obtainingthe sameamountof liquid film on the surfaceof the core for each weighing.3 The estimationof water saturationby measuringelectric resistivityis a fasterprocedure than weighing the core. However, the accuracyof saturationsobtained by a resistivity measurementis questionable,sinceresistivitycan be influencedby fluid distributionas well as fluid saturations.The four-electrodeassemblywhich is illustratedby Figure I was used to investigatewater saturationdistributionand to determinewhen flow equilibriumhas been attained.Other methodswhich have been used for in situ determinationof fluid saturation in cores include measurementof electric capacitance,nuclearmagneticresonance,neutron scattering,X-ray absorption,gamma-rayabsorption,volumetric balance,vacuum distillation, and microwavetechniques.

RelativePermeabilin of PetroleumReservoirs El-ectrodes

Outl-et

Differential Taps FIGURE l.

Inlet

Pressure

Inlet

Three-sectioncore assembly.8

After fluid saturationin the core has been determined,the Penn-Stateapparatusis reassembled,a new equilibrium condition is establishedat a higher flow rate for the second phase, and fluid saturationsare determinedas previously described.This procedureis repeated sequentially at higher saturationsof the second phase until the complete relative permeability curve has been established. The Penn-Statemethod can be used to measurerelative permeability at either increasing or decreasingsaturationsof the wetting phaseand it can be applied to both liquid-liquid and gas-liquid systems.The direction of saturationchangeused in the laboratoryshould correspondto field conditions. Good capillary contactbetweenthe test sampleand the adjacent downstream core is essential for accurate measurementsand temperaturemust be held constantduring the test. The time required for a test to reach an equilibrium condition may be I day or more.3

tL*

tl

rEC B. Single-Sample Dynamic Method This technique for steady-statemeasurementof relative permeability was developedby Richardsonet al.,e Josendalet al.,ro and Loomis and Crowell.ttThe apparatusand experimental procedure differ from those used with the Penn-Statetechnique primarily in the handling of end effects. Rather than using a test samplemountedbetweentwo core samples (as illustrated by Figure 1), the two fluid phasesare injectedsimultaneouslythrough a single core. End effects are minimized by using relatively high flow rates, so the region of high wetting-phasesaturationat the outlet faceof the core is small. The theorywhich was presented by Richardson et al. for describing the saturationdistribution within the core may be developed as follows. From Darcy's law, the flow of two phasesthrough a horizontallinear systemcan be describedby the equations -d P* , :

Q*, F*,dL k*, A

(l)

I rr rrl

kir F . rfi

cFr g : f rdt

and Q.i ^Fr" dL - d,nP n : =

tqr er ll

Q)

where the subscriptswt and n denotethe wetting and nonwettingphases,respectively.From the definition of capillary pressure,P", it follows that

G

f,F: 5X

1.0

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/

> {-i- ^ -o-

o

a

Theoretical saturation gradient

fnf low face

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15

20

0 lel -.

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J ii-

*i'trDd ir

CE'.i-:;

5

10

25

Distance from Outflow Face, cffi

[C-

FIGURE 2.

plcir :Jtrtr\r'

Comparison of saturationgradientsat low flow rate.e

I ri,'-..J r-trf' J li. ; .,.: .ric rll

(3)

dP.:dP.-dP*,

3T .:'.:t.t.tIlS id .-;:J end

These three equationsmay be combined to obtain

qP.

nr-' \' hcld tr\. : - mJ\

dL

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

k* ,

(4)

kn //

where dP"/dL is the capillary pressuregradient within the core. Since lc.l. !

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

,i*-J b)

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

D..r:. ' rn thC Cr':;..:::lplCr BJ.-,,.:l'l!ls' " : nrsh f3h Jil. l-: s'ntcrj ! n-:.

(s)

it is evident that dS*, dL

: A |\

/Q*, Fr*, - Q"p.\

k*

I

L" /op.rus*

(6)

re' Jc-

iz.-'.

a(rr

, l t

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Ftt', c.r

From

Richardson et al. concluded from experimentalevidence that the nonwetting phase saturation at the dischargeend 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 saturationthroughout 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 centimetersfrom the outlet. Within this region the wetting phasesaturationincreasesto the equilibriumvalue at the outlet face. Both calculations and experimental evidence show that the region of high wetting-phase saturationat the discharge end of the core is larger at low flow rates than at high rates. Figure 2 illustrates the saturationdistribution for a low flow rate and Figure 3 shows the distribution at a higher rate.

Relative Permeability of Petroleum Reservoirs

1.0

\

't o I -o-o- -o--o-- :- -- : - J

t

o

Theoretical saturation gradient

a

a>l

I n fr o wr a c "

o

5

10

15

20

25

Distance from Outflow Face, ctrl FIGURE 3.

Comparison of saturationgradients at high flow rate.e

Although the flow rate must be high enoughto control capillary pressureeffects at the dischargeend of the core, excessiveratesmust be avoided. Problemswhich can occur at very high rates include nonlaminarflow. C. Stationary Fluid Methods Leas et al.12describeda techniquefor measuringpermeabilityto gaswith the liquid phase held stationarywithin the core by capillary forces. Very low gur flo* ratesmust be used, so the liquid is not displacedduring the test. This techniquewas modified slightly by Osoba et al.,s who held the liquid phasestationarywithin the core by meansof barrierswhich were permeableto gas but not to the liquid. Rapoportand Leasr3employeda similar technique using semipermeablebarrierswhich held the gas phasestationarywhile allowing the liquid phaseto flow. Corey et al.ra extendedthe stationaryfluid methodto a three-phar.ryri.. by using barrierswhich were permeableto water but impermeableto oil and gas. Osobaet al. observed that relative permeability to gas determinedby the stationary liquid method was in good agreementwith values measuredby other techniquesfor some of the cases which were examined. However, they found that relative permeability to gas determinedby the stationary liquid technique was generally lower than by other methodsin the region of equilibrium gas saturation. This situation resulted in an equilibrium gas saturation value which was higher than obtained by the other methods used (Penn-Siate,Single-Sample Dynamic, and Hassler). Saraf and McCaffery consider the stationaryfluid methods to be unrealistic, since all mobile fluids are not permitted to flow simultaneouslyduring the test.2 D. Hassler Method This is a steady-statemethod for relative permeability measurementwhich was described by Hasslerrsin 1944. The technique was later studied and modified by Gates and Lietz,16 Brownscombeet ?1.," Osoba et al.,s and Josendalet al.ro The laboratory apparatusis illustrated by Figure 4. Semipermeablemembranesare installed at each end of the Hassler test assembly.Thesemembraneskeep the two fluid phasesseparatedat the inlet and outlet of the core, but allow both phasesto flow simultaneouslythrough the core. The pressure

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FIGURE 4.

Two-phase relative permeability apparatus.r5

barrier. By adjusting in each fluid phaseis measuredseparatelythrough a semipermeable the flow rate of the nonwettingphase,the pressuregradientsin the two phasescan be made equal, equalizingthe capillary pressuresat the inlet and outlet of the core. This procedure is designedto provide a uniform saturationthroughoutthe length of the core, even at low flow rates, and thus eliminate the capillary end effect. The techniqueworks well under conditionswhere the porousmedium is stronglywet by one of the fluids, but somedifficulty has been reported in using the procedureunder conditionsof intermediatewettability.2'r8 The Hasslermethod is not widely used at this time, since the data can be obtainedmore rapidly with other laboratorytechniques. E. Hafford Method This steady-statetechnique was describedby Richardsonet al.e In this method the nonwetting phase is injected directly into the sample and the wetting phaseis injected through a disc which is impermeableto the nonwetting phase.The central portion of the semipermeable 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 measurethe pressurein the wetting fluid at the inlet of the sample. The nonwetting fluid is injected directly into the sample and its pressureis measuredthrough a standardpressuretap machined into the Lucite@surrounding the sample. The pressuredifference betweenthe wetting and the nonwetting fluid is a measureof the capillary pressurein the sample at the inflow end. The design of the Hafford apparatusfacilitates investigationof boundary effects at the influx end of the core. The outflow boundary effect is minimized by using a high flow rate. F. Dispersed Feed Method This is a steady-statemethod for measuringrelative permeability which was designedby Richardsonet al.e The techniqueis similar to the Hafford and single-sampledynamic meth-

RelativePermeabilin of PetroleumReservoirs GAS

I

G A S P R E S S U R EG A U G E

rtl .r[I

t '.lt PRESSURE

PRESSURE GAS METER

Sn

I

OIL BURETTE FIGURE 5.

. t

Hafford relative permeability apparatus.e

!|t

ods. In the dispersedfeed method, the wetting fluid enters the test sample by first passing through a dispersingsection, which is made of a porous material similar to the test sample. This material does not contain a device for measuringthe input pressureof the wetting phase as does the Hafford apparatus.The dispersingsectiondistributesthe wetting fluid so that it entersthe test samplemore or less uniformly over the inlet face. The nonwettingphaseis introduced into radial grooves which are machined into the outlet face of the dispersing section,at thejunction betweenthe dispersingmaterialand the testsample.Pressuregradients used for the tests are high enough so the boundary effect at the outlet face of the core is not significant.

III. UNSiuoo"-STATEMETHoDS Unsteady-staterelative permeability measurementscan be made more rapidly than steadystate measurements,but the mathematicalanalysisof the unsteady-stateprocedureis more difficult. The theory developed by Buckley and Leverettre and extended by Welge2ois generally used for the measurementof relative permeabilityunder unsteady-stateconditions. The mathematicalbasis for interpretationof the test data may be summarizedas follows: Leverett2rcombined Darcy's law with a definition of capillary pressurein differential form to obtain

'*;h(*-eApsino) f*z

r + In.& k*

(71

Fo

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

7 difference between displacing and displaced fluids. For the case of horizontal flow and negligible capillary pressure,Welge2oshowed that Equation 7 implies llE

S*.u, -

3 ^ - G€

S*z :

f.r, Q*

( 8)

wherethe subscript2 denotesthe outlet end of the core, S*.ouis the averagewater saturation; and Q* is the cumulativewater injected,measuredin pore volumes.SinceQ* and S*.,ucan be measuredexperimentally,f", (fraction oil in the outlet stream)can be determinedfrom the slope of a plot of Q* as a function of S*,ou.By definition l,z:q,,/(q,,*q*)

(e)

By combining this equationwith Darcy's law, it can be shown that I

f,,r:

'

I1.,/K..,

t *

tlOt

tr/.,* Since p" and pw are known, the relative permeability ratio k.o/k.* can be determinedfrom Equation 10. A similar expressioncan be derived for the caseof gas displacingoil. The work of Welge was extendedby Johnsonet a1.22 to obtain a technique (sometimes calledthe JBN method) for calculatingindividual phaserelativepermeabilitiesfrom unsteadystate test data. The equationswhich were derived are lf.'.' ::..rfiS I tc. -:-::iic fc:' bt* lrr-

-

- llrr<' -,-:l :l '' j\'

:\

le .: .:t":.to! Er

k.. :

.(#) /,(a f,,,

(Il)

and

-::,i.cfilr

k.o: ltoo,,,

J :-- -,:c r\

t.z ttr.

(12)

where I,, the ?elative injectivity, is defined as I t--il.1::

-:.'iJr -

I,:

- :a\re

f \\ -"ac,' r\

(l 3 )

(q*,/Ap) at start of injection

:. ltrfln

r

initial injectivity (q*,/Ap)

- i:::()n\ ' . l t r r Ar | .. I .

br::

injectivity

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 by replacingthe subscript"w" with "g" in EquationslI,12, and 13. In designingexperimentsto determinerelative permeabilityby the unsteady-statemethod, it is necessarvthat: l

l. 2. tOt : ::l ilurd b ::.c Jcnsrtr

3. 4.

The pressuregradientbe large enoughto minimize capillary pressureeffects. The pressuredifferential across the core be sufficiently small compared with total operatingpressureso that compressibilityeffects are insignificant. The core be homogeneous. The driving force and fluid propertiesbe held constantduring the test.2

Relative Permeabilin of Petroleum Reservoirs Laboratory equipment is available for making the unsteady-statemeasurementsunder simulated reservoirconditions.2a In addition to the JBN method, several alternative techniquesfor determining relative permeabilityfrom unsteady-state test data have been proposed.Saraf and McCaffery2developed a procedurefor obtainingrelative permeabilitycurves from two parametersdetermined by least squaresfit of oil recovery and pressuredata. The technique is believed to be superior to the JBN method for heterogeneouscarbonatecores. Jones and Roszelle25 developed a graphical technique for evaluation of individual phase relative permeabilities from displacementexperimentaldata which are linearly scalable.Chavent et al. described a method for determining two-phaserelative permeability and capillary pressurefrom two sets of displacementexperiments,one set conductedat a high flow rate and the other at a rate representativeof reservoir conditions. The theory of Welge was extendedby Sarem to describerelative permeabilitiesin a systemcontainingthree fluid phases.Sarememployed a simplifying assumptionthat the relative permeabilityto each phasedependsonly on its own saturation,and the validity of this assumption(particularlywith respectto the oil phase) has beenquestioned.2 Unsteady-staterelative permeability measurementsare frequently used to determine the ratios k*/ko, ks/k", and kr/k*. The ratio k*/k" is usedto predict the performanceof reservoirs which are produced by waterflood or natural water drive; kr/k" is employed to estimatethe production which will be obtained from recovery processeswhere oil is displacedby gas, such as gas injection or solution gas drive. An important use of the ratio k*/k* is in the prediction of performanceof natural gas storagewells, where gas is injectedinto an aquifier. The ratios k*/ko, kg/ko,and kr/k* are usually measuredin a systemwhich containsonly the two fluids for which the relative permeability ratio is to be determined. It is believed that the connatewater in the reservoirmay have an influenceon kg/k.,,expeciallyin sandstones which contain hydratableclay minerals and in low permeabilityrock. For these types of reservoirsit may be advisableto measurek*/k.,in cores which contain an immobile water saturation.2a

tmcl #r},

r;ra;

&-.rsri

t

kr6

rlrn

I

fr* d * b

lr

A't| iltr

IV. CAPILLARYPRESSURE METHODS The techniqueswhich are usedfor calculatingrelative permeabilityfrom capillary pressure data were developedfor drainagesituations,where a nonwettingphase(gas) displacesa wetting phase(oil or water). Thereforeuse of the techniquesis generallylimited to gas-oil or gas-watersystems,where the reservoiris producedby a drainageprocess.Although it is possibleto calculaterelativepermeabilitiesin a water-oil systemfrom capillary pressure data, accuracyof this techniqueis uncertain;the displacementof oil by water in a waterwet rock is an imbibition processrather than a drainageprocess. Although capillary pressuretechniquesarenot usuallythe preferredmethodsfor generating relative permeability data, the methodsare useful for obtaining gas-oil or gas-waterrelative permeabilitieswhen rock samplesare too small for flow testsbut large enough for mercury injection. The techniquesare also useful in rock which has such low permeability that flow testsare impractical and for instanceswhere capillary pressuredata have been measuredbut a sampleof the rock is not availablefor measuringrelative permeability. Another use which has been suggestedfor the capillary pressuretechniquesis in estimating kr/k" ratios for retrogradegas condensatereservoirs, where oil saturationincreasesas pressuredecreases, with an initial oil saturationwhich may be as low as zero. The capillary pressuremethods are recommendedfor this situation becausethe conventionalunsteady-statetest is not designed for very low oil saturations. Data obtainedby mercury injection are customarilyused when relative permeabilityis estimatedby the capillary pressuretechnique.The core is evacuatedand mercury(which is

h h

9 rllTl-

B uLJr'l

Dlnj:i..rllrc' -'. Jcf -:l l-. .:ilCrflc" J i*-..:cicJ

to

the nonwettingphase)is injectedin measuredincrementsat increasingpressures.Approximately 20 datapoints are obtainedin a typical laboratorytestdesignedto yield the complete capillarypressurecurve, which is requiredfor calculatingrelativepermeabilityby the methods describedbelow. Severalinvestigatorshave developedequationsfor estimatingrelative permeabilityfrom capillary pressuredata. Purcell2epresentedthe equations

d R .zcllc':'

fs*i

3fi:.;:^illlts'r

rl --.-:rhcd "

fE

'l'.

k.*, :

l\Atr

"'-': r[ r

lr

:.::;::l

f'.

'

t

::\

dS/Pi

I' ds/p!

JSo i

k.n*,:

-

c ihc lrc'-' Ol '- -J--.,'tr\ f6: '" -r. tilc -.. Xc": !1..

: thc

A.

(l4)

and

.'-.:.i''

h'

fl

dS/pi

ltr

!n " ':' ",cJ J r

l,

Da- -:* - iilc't ;tr.. :..r thc

fl

J,

(l 5 )

dS/pi

where the subscriptswt and nwt denotethe wetting and nonwettingphases,respectively, and n has a value of 2.0. Fatt and Dykstra3odevelopedsimilar equationswith n equal to 3.0. A slightly different result is obtainedby combiningthe equationsdevelopedby Burdine3l with the work of Purcell.2eThe resultsare

br -,--llr.rl ...,:.it. -'r-\,r'

l. r

(l6)

. *- : l J l

D'-

(l7)

b r . : ': ; ..u rc )

;

.,

bJ' .1 "

.:--'\

J

--:.-rrll '- rr

b r - , '- i -,rc

where S, is the total liquid saturation.

f i- .. ^ :iS[-

V. CENTRIFUGEMETHODS

!l :; J:rilng illl'-;.-rllrc' " i':.un lft":

Centrifuge techniquesfor measuringrelative permeability involve monitoring liquids produced from rock sampleswhich were initially saturateduniformly with one or two phases. Liquids are collectedin transparenttubesconnectedto the rock sampleholdersand production is monitored throughout the test. Mathematicaltechniquesfor deriving relative permeability data from these measurementsare describedin References26, 27, and 28. Although the centrifugemethodshave not beenwidely used,they do offer someadvantages over alternativetechniques.The centrifuge methodsare substantiallyfaster than the steadystatetechniquesand they apparentlyare not subjectto the viscousfingering problems which sometimesinterfere with the unsteady-statemeasurements.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 Lease28describean automatedcentrifuge which employs a photodiodearray in conjunction with a microcomputerto image and identify liquids producedduring the test.

lq

:-:l

!ltr$

ts.:. -:'.'J hut tf -r-' u hrch t

t:ltrr: lt)r

E .le. :t'a:€s. Drc ::.cthtrJr !l :. :,'t deI t r K ' - : ^ . . 1 1r\s ''i lr;h is ;:\

t0

Relative Permeabiliy of Petroleum Reservoirs

CAMER CENTRIFUGE

COMPUTER SrmrLrlr o z

o U' IJJ

LIQUID PRODUCTION

LIJ

o o uJ LIJ

oa)

TROBE

tr o o J

rtts:rt r.

:

\&trt or fra g, ;rrrrrrhrr rrrr-l t.-r

CONTROLLER

SPEEDDISK

Thll. tu

SPEED SET POINT

FIGURE 6.

Thc n

Automated centrifuge system.28

:Rr!tn

Stroboscopiclights are located below the rotating tubes and movement of fluid interfaces is monitored by the transmitted light. Fluid collection tubes are square in cross section, since a cylindrical tube would act as a lens and concentratethe light in a narrow band along the major axis of the tube. A schematicdiagram of the apparatusis shown by Figure 6.

It is possibleto calculaterelative permeability ratios directly from field data.23Inmaking the computation it is necessaryto recognize that part of the gas which is produced at the surface was dissolved within the liquid phasein the reservoir. Thus; (free gas) * (solutiongas)

:

kh P.- P -w (r./r*) ln FrB,

?.09-E-e

E

h F'fr' if rttl

t

:u-bil

tr r*l

(18)

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

9g.fr""

nr

.E!

VI. CALCULATION FROM FIELD DATA

(producedgas) :

lr*rj

(l9)

trt I tru: :3rr FFr lr}-rr f$lrI1 hor Fcr -

lst'

I

ll

?

FIGURE 7.

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

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

(20) 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 expressRo, cumulative gas/oil ratio and R,, solution gasioil ratio, in terms of standardcubic foot per stock tank barrel, Equation l8 implies

(2t)

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

Thus, the relative permeability ratio is given by k" _ ( R o- R . ) & - ! ! ko

l|i

'':J:1Jac\

!n-.. \e\ll()n. I ^-:lJ rltr0S I F ..,:l 6

l:

1' :::rltng

Dd--.- :l the

l\r | :', .t::t tx?)

rl9t

(22)

5.615 B. F.

The oil saturationwhich correspondsto this relative permeabilityratio may be determined from a material balance. If we assumethere is no water influx, no water production, no fluid injection, and no gas cap, the materialbalanceequationmay be written

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

s*)

(23)

where minor effects such as changein reservoirpore volume have been assumednegligible. In Equation 23 the symbol N denotesinitial stock tank barrelsof 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 standardconditions. If total liquid saturationin the reservoir is expressedas

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

(*)

(24)

then the relative permeability curve may be obtainedby plotting kr/k" from Equation 22 as a function of S,- from Equation 24. Figure 7 illustrates a convenientformat for tabulating the data. The curve is preparedby plotting column 9 as a flnction of column 6 on semilog paper, with k/k" on the logarithmicscale.The techniqueis useful even if only a few highliquid-saturation data points can be plotted. These kr/k" values can be used to verify the accuracyof relative permeability predicted by empirical or laboratory techniques. Poor agreementbetween relative permeability determined from production data and from laboratory experiments is not uncommon. The causesof these discrepanciesmay include the following:

t2 l. 2.

3.

Relative Permeability of Petroleum Reservoirs The core on which relative permeability is measuredmay not be representativeof the reservoir in regard to such factors as fluid distributions, secondaryporosity, etc. The techniquecustomarily used to compute relative permeability from field data does not allow for the pressureand saturationgradientswhich are presentin the reservoir, nor does it allow for the fact that wells may be producing from several strata which are at various stagesof depletion. The usual techniquefor calculating relative permeability from field data assumesthat Ro at any pressureis constant throughout the oil zone. This assumptioncan lead to computational errors if gravitational effects within the reservoir are significant.

When relative permeability to water is computed from field data, a common source of elror is the production of water from some source other than the hydrocarbon reservoir. These possible sourcesof extraneouswater include casing leaks, fracturesthat extend from the hydrocarbon zone into an aquifer, etc.

tl

Johr

plar'cn l_1 Crid Clrfi.. l l SFr.-t

Jcrl .lr.plr lo Slo5.i rc.hfu, UrS SPL T l r O'llG acotn: Frerr, -\ h tlF*: I r Frt- | :.i

Bra -l_

REFERENCES l. Gorinik, B. and Roebuck, J. F., Formation Evaluation through Extensive Use of Core Analysis, Core L a b o r a t o r i e sI,n c . , D a l l a s ,T e x . , 1 9 7 9 . 2. Saraf, D. N. and McCaffery, F. G., Two- and Three-Phase RelativePermeabilities: a Review, Petroleum Recovery InstituteReport #81-8, Calgary, Alberta, Canada, 1982. 3. Mungan, N., PetroleumConsultantsLtd., personalcommunication,1982. 4. Morse, R. A., Terwilliger, P. L., and Yuster, S. T., Relative permeabilitymeasurementson small s a m p l e s ,O i l G a s J . , 4 6 , 1 0 9 , 1 9 4 7 . 5. Osoba, J. S., Richardson, J. G., Kerver, J. K., Hafford, J. A., and Blair, P. M., Laboratoryrelative permeabilitymeasurements, Trans. AIME, 192, 47, 1951. 6 . H e n d e r s o n ,J . H . a n d Y u s t e r , S . T . , R e l a t i v ep e r m e a b i l i t ys t u d y , W o r l dO i l , 3 , 1 3 9 , 1 9 4 8 . 7. Caudle, B. H., Slobod, R. L., and Brownscombe, E. R. W., Further developmentsin the laboratory determinationof relative permeability,Trans. AIME, 192, 145, 1951. 8. Geffen, T. M., Owens, W. W., Parrish, D. R., and Morse, R. A., Experimentalinvestigationof 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., Laboratorydeterminationof relative permeability,Trans. AIME, 195, 187, 1952. 10. Josendal, V. A., Sandiford, B. B., and Wilson, J. W., Improved multiphaseflow studiesemploying radioactive tracers, Trans. AIME, 195, 65, 1952. I l. Loomis, A. G. and Crowell, D. C., RelativePermeabilityStudies: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., Relativepermeabilityto gas, Trans. AIME, 189,

65,r 9s 0. 13. Rapoport, L. A. and Leas, W. J., Relative permeabilityto liquid in liquid-gassystems,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-phaserelativepermea b i l i t y , J . P e t . T e c h n o l . ,N o v . , 6 3 , 1 9 5 6 . 1 5 . H a s s l e r , G . L . , U . S . P a t e n t2 , 3 4 5 , 9 3 5 , 1 9 4 4 . 16. Gates, J. I. and Leitz, W. T., Relative permeabilitiesof California coresby the capillary-pressuremethod, 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 perrnea b i l i t y ,O i l G a s J . , 4 8 , 9 8 , 1 9 5 0 . 18. Rose, W., Some problemsin applying the Hasslerrelativepermeabilitymethod,J. Pet. Technol.,8, I l6l, 1980. 19. Buckley, S. E. and Leverett, M. C., Mechanismof fluid displacementin sands,Trans. AIME, 146,107, 1942. 20. Welge'H.J.rAsimplifiedmethodforcomputingrecoverybygasorwaterdrive,Trans.A|ME , 5,91, 19 1952. 21. Leverett, M. C., Capillary behaviorin poroussolids, Trans. AIME, 142, 152, 1941.

lv

13 nl.rl:r c ()i thg brt\ . ila ;.'l.r Jtres

bli

dlc' :l.cn .tir. I .i:-:l-: lr [1gI tE..-::tC. thal D .-:i' .c.rJ to lll.. -: l

I|r.

. -:.c

Fn

:-'-<'I\t)lf.

tli

lrr)m

I c\:r-.J

r l

lE-

F.

lrX'r

| :', F I. Er

ls It

lt-

!

JI

";:h,rJ.

F'-'.

[:.' It

:

j'rTnC-

..

rf-

.

tv:

:^

Nt!:

. l f .l . l()7.

-i.91.

22. Johnson, E. F., Bossler, D. P., and Naumann, V. O., Calculationof relative permeabilityfrom displacementexperiments,Trans. AIME, 216,310, 1959. 23. Crichlow, H. B., Ed., Modern ReservoirEngineering- A SimulationApproaclr, Prentice-Hall,Englewood Cliffs, 1977, chap. 7. 24. SpecialCore Analysis, Core Laboratories,Inc., Dallas, 1976. 25. Jones, S. C. and Roszelle, W. O., Graphical techniquesfor determining relative permeability from displacementexperiments,J. Pet. Technol., 5, 807, 1978. 26. Slobod, R. L., Chambers, A., and Prehn, W. L., Use of centrifugefor determiningconnate water, residualoil, and capillary pressurecurvesof small core samples,Trans. AIME, 192, 127, 1952. 27 . Yan Spronsen, E., Three-phaserelative permeabilitymeasurements using the CentrifugeMethod, Paper SPE/DOE 10688presentedat the Third Joint Symposium,Tulsa, Okla., 1982. 28. O'Mera, D. J., Jr. and Lease, W. O., Multiphaserelativepermeabilitymeasurements using an automated centrifuge,PaperSPE 12128presentedat the SPE 58th Annual TechnicalConferenceand Exhibition, San Francisco.1983. - their measurement 29. Purcell, W. R., Capillarypressures usingmercuryand the calculationof permeability therefrom, Trans. AIME, 186, 39. 1949. 30. Fatt, I. and Dyksta, H.,,Relative permeabilitystudies,Trans. AIME, 192,41, 1951. 31. Burdine, N. T., RelativePermeabilityCalculationsfrom Pore Size DistributionData, Trans. AIME, lg8, 7t,1953.

l5 Chapter 2 TWO-PHASE RELATIVE

PERMEABILITY

I. INTRODUCTION Direct experimentalmeasurementto determinerelative permeabilityof porous rock has long been recordedin petroleumrelatedliterature.However, empirical methodsfor determining relative permeabilityare becomingmore widely used, particularlywith the advent of digital reservoirsimulators.The generalshapeof the relative permeabilitycurves may - S*)"'; where A, be approximatedby the following equations:k.* : A(S*)'; k.., : B(l B. n. and m are constants. Most relative permeability mathematicalmodels may be classifiedunder one of four categories: Capillary models - Are basedon the assumptionthat a porous medium consistsof a bundleof capillarytubesof variousdiameterswith a fluid path lengthlongerthan the sample. Capillary models ignore the interconnectednatureof porous media and frequentlydo not provide realisticresults. Statistical models - Are also basedon the modeling of porous media by a bundle of capillary tubes with various diametersdistributedrandomly. The modelsmay be described as being divided into a large number of thin slicesby planesperpendicularto the axes of the tubes. The slices are imagined to be rearrangedand reassembledrandomly. Again, statisticalmodels have the same deficiencyof not being able to model the interconnected natureof porous media. Empirical models - Are basedon proposedempirical relationshipsdescribingexperimentallydeterminedrelativepermeabilitiesand in general,haveprovi{ed the most successful approximations. Netwoik models - Are frequentlybasedon the modelingof fluid flow in porousmedia using a network of electric resistorsas an analogcomputer.Network models are probably the best tools for understandingfluid flow in porousmedia'r'aa The hydrodynamiclaws generallybear little use in the solutionof problemsconcerning single-phasefluid flow through porous media, let alone multiphasefluid flow, due to the complexity of the porous system. One of the early attemptsto relate severallaboratorymeasuredparametersto rock permeabilitywas the Kozeny-Carmenequation.2This equation expressesthe permeabilityof a porousmaterialas a function of the productof the effective path lengthof the flowing fluid and the meanhydraulicradiusof the channelsthroughwhich the fluid flows. Purcell3formulated an equation for the permeability of a porous system in terms of the porosity and capillary pressuredesaturationcurve of that systemby simply consideringthe porousmedium as a bundle of capillary tubesof varying sizes. adaptedthe relationsdevelopedby Kozeny-Carmenand Purcellto the Severalauthorsa-r6 computationof relativepermeability.They all proposedmodelson the basisof the assumption that a porous medium consistsof a bundle of capillariesin order to apply Darcy's and Poiseuille'sequationsin their derivations.They used the tortuosityconceptor texture parametersto take into accountthe tortuouspath of the flow channelsas opposedto the concept of capillary tubes. They tried to determinetortuosityempirically in order to obtain a close approximation of experimentaldata.

II. RAPOPORTAND LEAS Rapoportand Leasepresentedtwo equationsfor relativepermeabilityto the wettingphase.

16

RelativePermeabilin of PetroleumReservoirs

Theseequationswere basedon surfaceenergyrelationshipsand the Kozeny-Carmenequation. The equationswere presentedas defining limits for wetting-phase relativepermeability. The maximum and minimum wetting-phaserelativepermeabilitypresentedby Rapoport and Leas are

k.*,(max) :

(l) fs*

Jr*,t'ot

,['*'

P. dS

(tj)(T#)'

and

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

.['*' fs-

I J

P. dS fS*,

P . d s +|

r

'

J

r

(2)

R.as

whereS- representsthe minimum irreduciblesaturationof the wetting phasefrom a drainage capillary pressurecurve, expressedas a fraction;S*, representsthe saturationof the wetting phasefor which the wetting-phaserelativepermeabilityis evaluated,expressedas a fraction; P. representsthe drainagecapillary pressureexpressedin psi and S representsthe porosity expressedas a fraction.

u hcre r iun -tr.r Tlr c t\

III. GATES. LIETZ. AND FULCHER Gatesand Lietzsdevelopedthe following expressionbasedon Purcell'smodel for wettingphaserelative permeability:

t. K.*r

_ -

ru I$

(3)

Fulcher et al.,ashave investigatedthe influenceof capillary number (ratio of viscousto capillary forces)on two-phaseoil-water relativepermeabilitycurves.

IV. FATT, DYKSTRA,AND BURDINE Fatt and Dykstrarr developedan expressionfor relativepermeabilityfollowing the basic methodof Purcell for calculatingthe permeabilityof a porousmedium. They considereda lithology factor (a correction for deviation of the path length from the length of the porous medium) to be a function of saturation.They assumedthat the radius of the path of the conductingpores was relatedto the lithology factor, tr, by the equation:

\ : -

a ro

Fan E^t,rat.

(4)

Ttrr rt rflfl

Thc 1 ilrrrrd &nJ Dillrd

!,! hr Drfi crlr cr Ffm

L7 Table I CALCULATION OF WETTING.PHASERELATIVE PERMEABILITY BASED ON THE FATT AND DYKSTRA EQUATION

,aI-' -:l cLluaP C : - '. . r . r l l t r . hi i..'j'prp1

S*, Vo

Area from 0 to S*, in.2

l/P"'], (cm Hg)-t

P", cm Hg

k.*,, Vo

, l r 100 90 80 '70 60 s0 40 30 20 ' " _

[l

n.25

0.0156 0.0110 0.0080 0.0060 0.0046 0.0033 0.0024 0 . 0 01 5 0.0005

7.88 5.54 3.80 2.49 t.50 0.75 0.30 0.20

100.0 70.0, 49.2b 33.8 22.1 13.3 6.1 2.7 0.4

7 . 8 8 / 1 1 . 2x5 1 0 0: 7 0 . 0 . 5 . 5 4 1 1 1 . 2x5 l 0 O : 4 9 . 2 .

l

., ,l:r:nJlC

J:-.

4.0 4.5 5.0 5.5 6.0 6.7 7.s 8.7 13.0

where r representsthe radius of a pore, a and b representmaterial constants,and }, is a function of saturation. The equationfor the wetting-phase relativepermeability,k.*,, reportedby Fatt and Dykstra is

.i.'1tlnS

ft*'

F ' - ' : ' : l ( r n i li- r :'. :, r\ll\

t

,

K.*, :

Jn

ds -

P2(l

l.r

+ b)

dS

(5)

* b) Jo P2(|

Fatt and Dykstra found good agreementwith observeddata when b : Equation 5 to

ft*' ds Jo P:

r-rl

ir5 ::.r hasic srr:...-:ercda J i:; Frrttus i f;i:. ,'l the

r-l)

TF

r/r, reducing

(6)

They statedthat their equation fit their own data as well as the data of Gatesand Lietz more accuratelythan other proposedmodels. The procedurefor the calculation of relative permeability from capillary pressuredata is illustrated by Table I and the results are shown in Figures I and 2. Burdine'3 reportedequationsfor computing relative perrneabilityfor both the wetting and nonwettingphases.His equationscan be shown to reduceto a form similar to thosedeveloped by Purcell. Burdine's contribution is principally useful in handling tortuosity. Defining the tortuosity factor for a pore as L when the porous medium is saturatedwith only one fluid and using the symbol tr*, for the wetting-phasetortuosity factor when two phasesare present, a tortuosity ratio can be defined as

T tr.*,: ;

(7)

l8

RelativePermeabilitvof PetroleumReservoirs

9 I | 7 Pol (cm Hg) 6 5 4 3 2 I

oo' lo 20

40 50 60 70 80 Sw+

FIGURE 1.

Capillary pressureas a function of water saturation.

then

/'*' {^,*,)'ds/(\)'(P.)' kr*,

(8)

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

where S The rela phaseto

If tr is a constantfor the porous medium and tr,*t dependsonly on the final saturation,then fS*'

k.*t :

t

ds/(P.)r (9)

(tr.*.)' rl

t ds/(p")l

where S The e the exp

In a similar fashion, the relative permeabilityto the nonwetting phasecan be expressed tortuosity ratio, tr,,*,, utilizing a nonwetting-phase fl

^

k.n*,:

I dst1e.)'

JS*t

(l0)

(trrn*,)'

J"

ds/(P.)2

Burdine has shown that

Arwt

-

S*,- S1 - S -

(lt)

Wylli comput

l9

r60 r50 r40 r30 t20 l l

roo 90

I

I

t

Pc3 | (CmHqi3

70 60 50 40 30 20

to

o5

lo 20 30 40 50 60 70 Sw

-+

pressure)r as a functionof water of (capillary Reciprocal il,;yul}: where S- representsthe minimum wetting-phasesaturationfrom a capillary-pressurecurve. The relative perrneability is assumedto approach zero at this saturation. The nonwetting phasetortuosity can be approximatedby fa::

thcn

Sn*t-- S' \ -r^n. .w , . :t . l-s*-s" r9)

f3 . r lli'rred

(12)

where S. is the equilibrium saturationto the nonwetting phase. The expressionfor the wetting phase(Equation 9) fit the data presentedmuch better than the expressionfor the nonwetting phase (Equation 10).

V. WYLLIE, SPRANGLER,AND GARDNER Wyllie and Spranglertz reported equations similar to those presentedby Burdine for computing oil and gas relative permeability. Their equationscan be expressedas follows:

rl0)

k,,,:(iil' rll)

fs" J os"rp;

/' or",rl

(l 3 )

Relative Permeabilin of Petroleum Reservoirs

Wetrine o WYLLIE ond SPANGLER

A I a

GATESond LIETZ i

l

|

|

t

l

B E R E AN O . 4

I

E

o.

FIGURE 3. data.rT

\\'etring

Reciprocalof (capillary pressure)ras a function of saturationfor normalized

k,*(r-r+" _ S*,/ )' !Yor",r3

(t4l

/'

where S- representsthe lowest oil saturationat which the gas phase is discontinuous:S: (l - S " . ) . The above equations for oil and gas relative permeabilities may be evaluated when a reliable drainage capillary pressurecurve of the porous medium is available, so that a plot of llP"2 as a function of oil saturationcan be constructed.Obviously, reliable valuesof Sand So.are also neededfor the oil and gas relative permeability evaluation. Figure 3 shows some examplesof llP.2 vs. saturationcurves.rT Wyllie and GardnerrTdeveloped equationsfor oil and gas relative permeabilitiesin the presenceof an ineducible water saturation, with the water consideredas part of the rock matrix:

ft'ds*

k,.:(H), +*

.s;

f' ds*

f*

\alunlllo

rt is fau

cquation drarnaec Pressure tri the ci S.trI

hart: tri I \3luralKr

Jr*,Pi

k,, (*)'

Corsl ut ilit r ar

'6)

Jr*, Pi

where Sl representstotal liquid saturation.Note that theseequationsmay be applied only when the water saturationis at the irreduciblelevel.

VI. TIMMERMAN,COREY,AND JOHNSON Timmermanr8suggeststhe following equationsbasedon the water-oil drainagecapillary pressure,for the calculationof low valuesof water-oil relativepermeability.

2l Wetting-PhaseDrainageProcess:

fl'"H.1"

Injection Curve

k.o :

S.

LTFI

(t7)

Injection Curve

f[Hl"

InjectionCurve

k.* :

g

S*

tc

LrFl

(l8)

lnjection Curve

Wetting-PhaseImbibition Process: b.

kro

:

So

* '.1. S,,

lc;

.i hcn a

ilri:e . rn the I 0: ::.i r(Ek

l-5)

So

Lrsl

(20)

Injection Curve

D l:.:: .r plot ta. -.'. ,tl- S.,, t .ht)\\ S

[[H]"

Curve Trap-Hysteresis

k.o :

JUr.

LTFj

(le)

Injection Curve

l-lt

Jt:

[l'"H 1"

Injection Curve

Coreyrecombined the work of Purcell3and Burdiner3into a form that has considerable utility and is widely acceptedfor its simplicity. It requireslimited input data (sinceresidual saturationis the only parameterneededto developa set of relativepermeabilitycurves)and it is fairly accuratefor consolidatedporous media with intergranularporosity. Corey's equationsare often used for calculationof relative permeabilityin reservoirssubjectto a drainageprocessor externalgasdrive. His methodof calculationwas derivedfrom capillary pressureconceptsand the fact that for certaincases,l/P"2is approximatelya linear function of the effective saturationover a considerablerangeof saturations;i.e. , llP"2 : C [(S" S".)/(1 - S",)] where C is a constantand S" is an oil saturationgreaterthan S.,,.On the basisof this observationand the findingsof Burdiner3concerningthe natureof the tortuositysaturationfunction, the following expressionswere derived:

(2r) rl6)

(22) tpp. reJ o n l y

\o: k,o lgc ..rprllary

[S'

- S'*lo

Lr - s * J

(23)

22

Relative Permeability of Petroleum Reservoirs

where S'- is the total liquid saturationand equal to (l - Sr); S- is the lowest oil saturation (fraction) at which the gas phaseis discontinuous;and Sr* is the residualliquid saturation expressed as a fraction. Corey and Rathjens2o studiedthe effect of permeabilityvariationin porous media on the value of the S- factor in Corey's equations.They confirmed that S,,,is essentiallyequal to unity for uniform and isotropic porous media; however, values of S,, were found to be greaterthan unity when there was stratificationperpendicularto the direction of flow and less than unity in the presenceof stratificationparallel to the direction of flow. They also concludedthat oil relative permeabilitieswere less sensitiveto stratificationthan the gas relativepermeabilities. The gas-oil relative permeabilityequationis often used for testing, extrapolation,and smoothingexperimentaldata.It is also a convenientexpressionthat may be usedin computer simulationof reservoirperformance. Corey's gas-oil relative permeabilityratio equationcan be solved if only two points on the k,r/k,.,vs. S* curve are available.However, the algebraicsolutionof the k,g/k..,equation when two points are availableis very tediousand the graphicalsolution that Corey offers in his original paperrequireslengthygraphicalconstructionaswell asnumericalcomputation. Johnson2rhas offered a greatly simplified and useful methodfor determinationof Corey's constant. Johnsonconstructedthree plots by assumingvaluesof Sr*, S,,, and k.s/k..,by calculating the gas saturation,(1 - S,_),using Corey's equations.The calculationwas carriedout for variousSr* and S- combinationsand for k.s/k,ovaluesof l0 to 0.1, 1.0 to 0.01, and 0. I to 0.001. Johnson'sgraphs may be used to plot a more completek.g/k,,,curve basedon limited experimentaldata. The spanof the experimentaldata determineswhich of the three figures should be selected. The suggestedprocedurefor k.g/k.,calculation,basedon Corey's equation,is as follows: l. 2.

3. 4. 5.

6.

Plot the experimentalk.r/k," vs. S, on semilog paper with k,*/k,oon the logarithmic scale. From the experimentaldata determinethe gas saturationat k.r/k,oequal to 10.0 and 0. 1, 1. 0 and 0 .0 1 , o r 0 .1 a n d 0 .0 0 1 .(T h e l i stedpai rsof val uescorrespond to Fi gures 4,5, and 6 of Johnson'sdata, respectively,and the rangeof the experimentaldata dictateswhich figure is to be employed.Note that if the data do not span the entire permeabilityratio intervalof 10.0 to 1.0, Figure 4 may not be employedfirst; instead Figure 5 with the k,*/k.ointerval of 1.0 to 0.01 or Figure 6 with the k.*/k,.,interval of 0. 10 t o 0. 00 1 ma y b e u s e dfi rs t.) Enter the appropriateFigure (4,5, or 6) using the gas saturationscorrespondingto the pair of k.r/k.ovaluesselectedin step 2. Pick a unique S.* and S- at the intersectionof the gas saturationvalues;interpolate if necessary. Using these S.* and S- values and employing the two other figures of Johnson, determinetwo more gas saturationvalues and the k,*/k," ratio indicatedon the axes of each figure. Add thesepoints to the experimentalplot for obtainingthe relativepermeabilityratio over the region of interest.

This procedureprovidesvaluesof gas saturationat k.*/k.oratiosof 10.0, 1.0, 0.10, 0.01, and 0.001, which are sufficient to plot an expandedk.s/k.ocurve. It should be noted that if the data cover a wide range of permeabilityratios, multiple determinationsof Sr* and S- can be made. If the calculatedvaluesdiffer from the experimental data, the discrepancyindicatesthat thereis no singleCorey curve which will fit all

t -

5rq11 rilustnl ('rtTr'

rrrrrahl

FJ

Ehc\

rk S3tuJr C;ttr

23

o tl I o) J

I o) U)

' . . ,

t \ \

\ :

20

.- :trlilc

N : in..' ln':

l::urcs

lsi'

FIGURE 4.

:-:. Jata . intlre ::.tc',itd

fi:-'

;

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

,lnd

lr

'-

:'.ll of

' .::ng tO

the points; an averageof the values for each constantshould yield a better curve fit. Figure 7 illustratesthe graphicaltechniqueof Johnson. Corey's equationsfor drainageoil and gas relativepermeabilitiesand the gas-oil relative permeabilityratio in the simplestform are as follows: k.o :

l: ," ':3rl31s tri

i .'-

Corey equationconstants.2l

(24)

(s".)o

k.r:(l-S".)2x(l-S3")

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

(2s)

and they are related through I

B;^

l .'

:r ratio

r .( ) . 0 1 ,

i'. '::ultiple ln : : J c\p e rEl'. ,,. I iit all

k.. k. : (S * X - (l - S ;y

I

(26)

where So.representsthe lowest oil saturationat which the gas tortuosity is infinite; S". is defined as (S" - S",)/(l - S".). Corey's equationsin the presenceof irreducible water saturationtake the following form: k,o :

(s*)o

(27)

Relative Permeabilitv of Petroleum Reservoirs

q tl

o

-t

o) -g

(U aQ o) U)

Sg, %, at krn /kro=0.01 nrll(rtl. n FIGURE5. Coreyequation constants.2t

: k,n '

f S 1 2 I t ;---""-^ | " )--)*iJ L

fl

-

S*)2

t(rr\trnal oi the p

(28)

where S- is a constantrelated to ( I - S*") and as a first approximationS- can be assumed to be unity. This is a good approximation,sinceS*"is lessthan5Voinrocks with intergranular porosity. In theseequations,S* : S"/(l - S*,) and S" is the oil saturationrepresentedas a fraction of the pore volume of the rock; S*, is the irreduciblewater saturation,alsoexpressed as a fraction of the pore volume. Theseequationsare linked by the relationship

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

(zs)

drrtntrutl $ tl

lrtrn\

scrB pm ehqrlutcl Cael

;.ffstrlXt hrr result .trr-ludc tV Ctlt1R{

-{pflx nE:Ilut3t

\

rnr

lirr{rc5

Corey et al. plotted severalhundredcapillary pressure-saturation curves for consolidated rocks and only a few of them met the linear relationshiprequirement.However, comparison of Corey's predicted relative permeabilitieswith experimentalvalues for a large number of samplesshowed close agreement,indicating that Corey's predictedrelative permeabilities are not very sensitiveto the shapeof the capillary pressurecurves. Equation 24 may be employed to calculatewater relative permeability if the oil saturation and the residual oil saturationare replacedby water saturationand irreducible water satu-

n crl-gr t

h

r rt

)< 0.9

o o o J

o) .:< (U Ae o U)

5

l

o

Sg, %, at krg/kro of O.O01 FIGURE 6.

Corey equationconstants.2l

ft'r ;, 'nsolidated

ration, respectively.The exponentof Corey's water relative permeabilityequation is approximatelyfour for consolidatedrocks, but dependssomewhaton the size and arrangement of the pores. The exponent has a value of three for rocks with perfectly uniform pore size distribution. Severalother authorshave proposedsimilar water relative permeabilityequations with different exponentsfor other types of porous media. Values of 3.022and 3.521 were proposedfor unconsolidatedsands with a single grain structurewhich may not be absolutelyuniform in pore size but should have a nalrow rangeof pore sizes. Corey compared the calculated values of oil and gas relative permeabilities for poorly values and obtainedgood results. However, consolidatedsandswith laboratory-measured his resultsshowedsome deviationat low gas saturationsfor consolidatedsandstone.Corey concludedthat the equationsare not valid when stratification,solutionchannels,fractures, or extensiveconsolidationis present. Application of Corey's equationpermits oil relative permeabilityto be calculatedfrom are easily made while of gas relative permeability.Since k., measurements measurements k.o measurementsare made with difficulty, Corey's equationis quite useful. The procedure involves the measurementof gas relative permeability at severalvalues of gas saturationin an oil-gas systemand then performing the following steps:

!1c:. itrmparison lar-ic number of

1.

(28) crr, lrc assumed dtl: :ntcrgranular n r.lrc\ented as l. ;..,'CrpreSSed

(2e)

E ;\.-rnteabilities tx- ,rtl saturation ihic u ater satu-

P r e p a r e a n a c c u r a t e p l o t tohf e f u n c t i o n k . r : ( l - S " " ) 2x ( l - S . " ' ) b y a s s u m i n g arbitrary values of So., the effective saturation,which is defined as

26

RelativePermeabilin of PetroleumReservoirs

o l<

o) .:. o n
--

o

o.lo

FIGURE 7.

2. 3. 4.

5. 6.

Data Data

of

Vlelge

points

o.20 0.30 0.40 0.50 0.60 0.70 Sg Example of the use of the Corey equations.rl

Preparea tabulation of k., vs. So" for values of k,, ranging from 0.001 to 0.99 in stepwisefashion. Determinevaluesof So"for eachexperimentalvalueof k., by usingthe above-described tabulation. Plot these values of So. againstthe values of S" coffespondingto the k., values on rectangularcoordinatepaper. The plot should be a straightline between50 and 807o oil saturation. Construct a straight line through the points in this range and extrapolateto S.* : 0. The value of S" at this point correspondsto S".. (SeeFigure 8.) Employ Equation 24, k,o : (So")oand the value of S.,.obtainedin the previousstep to calculatek,o valuesfor assumedvaluesof S".

Corey-typeequationsfor drainagegas-oilrelativepermeability(gasdrive) in the presence of connatewater saturationhave been suggestedas follows:

\\ 3l ttfnF;

n trre :r'tr-ll(r Ttrf.

k ." :

(l - S )u

(30)

Cr{UJllt rr{Tl$l

k.,

s3(2- s)

(31)

where S represents(Sr)/(l - S*,). Corey's equationsfor the drainagecycle in water-wet sandstones as well as carbonate formations are as follows:

,K - . - : ll --l l - s * 1 r Ll - S*,1

(32)

'trLrtn

Brtr rr l,ll ttr it{htut

27

60

50

a o o @

ro Sor

)70

ob

t

20

60

40 So,

0t :

'99 in

r:

.:.LIC\ ()n

.,nJ tl07c

\. - 0. ,'Jr \teP

I th.r'lrc:c'nCe

(-10) ( 31 )

roo

basedon effective of residualoil saturation FIGURE8. Determination oil saturation. k.*:

O\ ( .:. .. rtbcd

\

80

o/o

(S**)o

(33)

VII. WAHL. TORCASO. AND WYLLIE Wahl et al.2asuggestedthe use of the following equationfor drainagegas-oil relative of sandstonereservoirs: permeabilityratios basedon field measurements

*

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

(341

where rf represents( I - S*. - S. - Sg.)/(S,,- C); Sr. is the critical gas saturationas a fraction of total pore space;and C is a constantequal to 0.25. Torcasoand Wylliett comparedgas-oilrelativepermeabilityratios calculatedby Corey's equationwith thoseobtainedfrom Wahl et al. for variousirreduciblewater saturations.This comparisonsuggestedthat Corey's work was theoreticallysound,sinceit agreedwith values by Wahl et al. (seeFigure 9).t^ obtainedfrom field measurements VIII. BROOKS AND COREY

a- ..rrhonate

(32)

modified Corey's original drainagecapillary pressure-saturation Brooks and Corey26'27 relationshipand combined the modified equationwith Burdine's equationto develop the following expressionthat predictsdrainagerelativepermeabilityfor any pore sizedistribution:

28

Relative Permeabilin of Petroleum Reservoirs

trt e hrg l,S. Ttbc r alrr grr rrth rx itrrt it{ crFsll

roo 50 30

9 y y ;= o ' 3

\t-r lh .rl rclrlr

to 5 3

o .g

o, J

to o.5 o.3 o.l

o.

o.03 Iltr

o.ol o.oo5

I

rtttD

=

(*,

o.ootL

-

la--

o

20

60

40

80

5

roo

*

s

=

::

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

: (l)^ s**

(3s)

for P. i Po

where tr, and Po are constants characteristicof the media; ), is a measure of pore size distributionof the media, and Po is a measureof maximum pore size (minimum drainage capillary pressureat which a continuousnonwettingphaseexists). Using this relationship, two-phaserelative permeabilitiesare given by

k

"rwt

2 + l A r

- / S * l \vw

(36)

,

and

k . n * ,:

(l -'t**)'

- (S**) [t

,.l J

(37)

where k.*, and k-*, are wetting and nonwettingphaserelative permeabilitiesrespectively. The valuesof tr and Po are obtainedby plotting (S* - S*,)/(l - S*,) vs. capillary pr.rrur.

rl& rrt

k l lf.r lrel rb'rq

29 on a log-log scaleand establishinga straightline with L as the slopeand Poas the intercept a t (S* - S* i ) /(l - S * ,) : 1. Theseequationsreduceto Equations24 and 25 for \ : 2. Theoretically\ may have any value greaterthan zero, being large for media with relative uniformity and small for media with wide pore size variation. The commonly encounteredrangefor L is betweentwo and Talash28obtainedsimilar equationswith somewhatdifferent four for various sandstones.2t exponents.

IX. WYLLIE, GARDNER,AND TORCASO Wyllie and GardnerrThave presentedthe following expressionsfor the drainage wateroil relative permeability:

k,.:(H)'H

(38)

k,.:(5;)'$i11

(3e)

'

ds*/P.' Jr*,

/' or*,1r";'

More general expressionsfor any wetting and nonwetting relative permeability may be written where kr*r k.n*, S*i SL

Relative permeability to wetting phase(k,* and k,"). Nonwetting phaserelative permeability(k,r). Irreduciblewater saturation. Total liquid saturation: (l - Sr).

(40)

r3 5 ) I ,'l ltrC siZe lru::. Jrainage ] ( ' - '. - t (r t r n s h i P ,

(36)

(41)

Wyllie and Gardner have also suggestedthe following equationfor relative permeability to water or oil when one relative permeability is available: k.* :

(3 7 ) 1-.J*-utively. pressure

[an

(S**)' -

k,o (S**/(1 -

S**))'

(42)

where S**, which is defined as (S* - S*,)/(1 - S*,), is the mobile wetting-phasesaturation in a water-wetsystem. Basedon the linear relationbetweenl/P"2and S"/(l - S*,), they obtaineda drainagewater relative permeability equation for water-wet rocks with intergranularporosity as follows:

Relative Permeabilitv of Petroleum Reservoirs

k,* : (s**)o

(43)

Togpaso and Wyllie2s suggestedthe following equation for calculation of gas-oil relativepermeabilityof water-wetsandstone,where l/P.2 is approximatelya linear function of effective saturation.Their derivationwas basedon the relationdevelopedby Corey:

\=: k.,,

( l - s * ) ,( l - s * , ) (s*)o

(44)

where S* representseffectiveoil saturationand is equalto S.,/(l - S*,). Obviously, a reliable value of irreduciblewater saturation,S*r, needsto be known to calculatethe gas-oilrelative permeabilityratio. X. LAND, WYLLIE,

ROSE, PIRSON, AND BOATMAN

Land2ereportedthat an appreciableadjustmentof experimentalparameterswas required to avoid a discrepancybetweenexperimentaland calculatedtwo-phaserelative permeabilities. A large numberof the relativepermeabilitypredictionmethodsare basedon derivation of pore size distribution factors from the saturationand drainagecapillary pressurerelationship. Some authors3o believethat the employmentof capillary pressurerelationshipsfor the prediction of relative permeabilityis not advisable,since capillary pressureis derived from experimentsperformed under static conditions, whereasrelative permeability is a dynamic phenomenon.McCaffery3rin his thesisarguesthat the surfaceor capillary forces are ordersof magnitudelargerthan forcesarisingfrom the fluid flow and thus, predominate in controllingthe microscopicdistributionof the fluid phasesin many oil reservoirsituations. Brown's32results from the measurementof capillary pressureunder static and dynamic conditionssupportMcCaffery's argument. Severalrelative permeabilityprediction methodswhich are basedon drainagecapillary pressurecurves assumethat pore size distributioncan be derived from thesecurves.These proposedmodels can only be applied when a strongwetting preferenceis known to exist. Additionally, relativepermeabilitycalculationsfrom capillarypressuredataare developed for a capillary drainagesituationwhere a nonwettingphase,suchas gas, displacesa wetting phase(oil in a gas-oil system,or water in a gas-watersystem).They are developedprimarily for gas-oil or gas-condensate relative permeabilitycalculations;however, water-oil relative permeabilitycan be calculatedwith a lessercertainty. Wyllie in Frick's PetroleumProduction Handbook33suggestedsimple empirical gas-oil and water-oilrelativepermeabilityequationsfor drainagein consolidatedand unconsolidated sandsas well as oolitic limestonerocks. Theseequationsare presentedin Tables 2 and3. The oil-gasand water-oilrelativepermeabilityrelationsfor varioustypesof rockspresented in Tables 2 and 3 may be usedto producek.g/k.ocurvesat various S*, when k., measurements are unavailable. It should be noted that the k,.,/k.* values obtainedapply only if water is the wetting phase and is decreasingfrom an initial value of unity by increasingthe oil saturation.This is contrary to what happensduring natural water drive or waterflooding processes;however, Figures l0 through l4 also apply to preferentiallyoil-wet systemson the drainagecycle with respectto oil if the curves were simply relabeled. Rose6developeda useful method of calculatinga relative permeabilityrelationshipon the basisof analysisof the physical interrelationshipbetweenthe fluid flow phenomenain porous media and the static and residual saturationvalues. The equationsfor the wetting and nonwetting relative permeabilitesare

rrblc htr:

rE l-n 5:I

rrr G

*:: ;! }ftl, nr L f i

3l Table 2 OIL-GAS RELATIVE PERMEABILITIES (FOR DRAINAGE CYCLE RELATIVE TO OIL)33

r J 3) : :.rr-oil n i n c . , : : . . n et i t l n b \ ( ' : e\ .

Unconsolidatedsand, well sorted Unconsolidatedsand, poorly sorted Cementedsandstone,oolitic limestone,rocks with vugular porosity"

r _l_l)

nl.. .,:cliable r c l a t i re P'

(S*)'

(l -

(Sxlt :

( l - 5 x ; :( l -

(S*)'

0 -

Note: In theserelationsthe quantity Sx :

{

k.e

k"o

Type of formation

5x;r

sx), (l -

5x's) 5x:1

S,,/(l - S*,).

Application to vugular rocks is possibleonly when the size of the vugs is small by comparisonwith 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 3 WATER.OIL RELATIVE PERMEABILITIES (FOR DRAINAGE CYCLE RELATIVE TO WATER)33

Unconsolidatedsand, well sorted U n c o n s o l i d a t esda n d ,p o o r l y sorted C e m e n t e ds a n d s t o n eo, o l i t i c limestone

fai, ..rprllary C u: ', . '. Th e se D \ a ' t : , . 'C f i S t . lrc .:i\.'loped I-i- .: $ctting

k*=

pr:...:l gas-oil

Ei. :':-c.ented Dc'-:..irCil9OtS 'Ec::.nrphase It;,,:: This is It'.

llt ttt eVgf ,

h . t ; : : . , r cc y ' c l e i l . r : : , ' n . h i ps 1 1 p h r . : t , , l t t c n ian f :-. setting

(s**)'

(l - S**)' (l -

S**)' (l -

S**'')

(S**)tt

(l -

S**;z (l -

5"x:;

(S**)o

Note'. In these relationsthe quantity S** : (S* S*, is the ineducible water saturation.

|f*.*.::.rrrnarily tsr . rc'lative

tx-:.,tlidated bl.'. I and 3.

k.*

k"o

Type of formation

k-:

S"i)/(l -

S*,), where

-s* -) l 6si (s* -s* _)t(l t2si(2- 3s*.) + 3S*S*-(3S*. 2) + S**(4 5S*,,)1'

- S"-)'(I -,lr* - S.-) l653*,(5"*, -2rlt*3S.-)+ 3S"*,S"-(3S n -2* 2,lr*)+ S,-(l - r!*X4-,lr* - 5S".)]' [253*,(2

(4s) (46)

where S* and Sn*,representwetting and nonwettingsaturations,respectively,expressedas fractions;S*- and S.- representminimum wetting and nonwettingsaturationvaluesattained under dynamic flow conditions,expressedas fractions;they are the dynamic equivalentsof S*, and S". obtainedfrom statictests.The symbol qr* representsan immobile wetting-phase saturationexpressedas a fraction. It is that part of the wetting-phasesaturationwhich does not interfere with the nonwetting phase mobility and it is the maximum wetting-phase saturation at which the nonwetting relative permeability is unity. Note that Equation 46 reducesto Equation 45 for r.|l* : 0. The minimum wetting saturation,S**, dependson flow conditionsand may be obtainedby the Brownell and Katz3arelationshipof S*- : (1/86.3) [V(g o cos 0) dP/dx]-o264where g is the accelerationdue to gravity, o is the interfacial tension, 0 is the contactangle, k is the permeability,and dP/dx is the pressuregradient. The principal disadvantageof Rose'smethodis that the residualsaturationof both phases must be known fairlv accuratelv.

32

Relative Permeabilin of Petroleum Reservoirs

o j

o) .Y

o

20

40

60

80

too

Q vr

L r?El€ FIGURE 10. Wyllie curves for water-wetcementedsandstones, oolitic limestones,or vugular systems.rl

Pirson3sderived equationsfrom petrophysicalconsiderationsfor the wetting and nonwetting phaserelative permeabilitiesin clean, water-wet,granularrocks for both drainage and imbibition processes.The water relativepermeabilityfor the imbibition cycle was given

F

,.lu'l

5.

rcl

::Ttrr

Ttr ;rclc

AS

k.*, :

(S**)"'

(R.,/R,)3/2

(41)

k,*t :

(S**)t"

(R"/R,)3/2

(48)

((

later modified to rtrt

and k.*, :

(S**)t"

Si

(4e)

T\

Water relative permeability for the drainagecycle was given by k .* , :

(S* * )t"

Si

(s0)

rF: po..e.l€

33

o v o) l<

@

st tt

water-wet unconsolidated Wylliecurvesfor poorlysorted

i*Ylt

h-i .::'lJ nonXlt: . l r l r n a g e rlc . i . 1 .g i v e n

where R.,representselectricalresistivityof the test core at l00%obrine saturationexpressed as ohm-meters;R, representselectricalresistivityof the test core expressedas ohm-meters; S*, representsirreduciblewetting-phasesaturation;and S* representswater saturationas a fraction of pore space. The nonwetting phase relative permeabilityin clean, water-wetrocks for the drainage cycle was found to be k**, :

(-17)

(l

-

S**) [1 -

S**r'4(R"/R,)r'4]2

( s1 )

or k-*, :

(1 -

S**) (l -S**r/4 Sr/2)2

(s2)

which was later modified to

(48) k,n*, :

(l -

S**Xl

-

S**t/4 Su2)tt2

(53)

The nonwetting phase relative permeability in an imbibition cycle given by

(49)

S* - S*,

krn*, : [t

(s0)

l'

l-s-,-s*J

(54)

where S** represents(S* -S*,)/(l - S*,) and S.*, representsthe irreduciblenonwetting phasesaturationas a fraction of pore space.Pirson also derived equationsfor the wetting

Relative Permeabilin of Petroleum Reservoirs

I,OOO Swi

roo

ro

o J

o) j-

o.l

0.ol

o.ool o.0ootoL

60

40

20

80

loo

s,L FIGURE 12. Wyllie curvesfor well-sorted water-wetunconsolidated } cores. and nonwetting phase relative permeabilities in clean, oil-wet rocks for both drainase and imbibition processes: kr., :

(5s1

(S.r")"' S:

where S.* is defined as (S" - S.,.)/(I - S".) and S.. representsirreducibleoil saturationand is the equilvalentof of ( I - S*') for a clean,water-wetrock; S" represents total oil saturation obtainedby differencesfrom (l - S*). The nonwettingphaserelativepermeabilityin clean,oil-wet rocksfor the imbibition cycle was found to be

:

krr*,

So -

[' L

So,

l-S..-S*,

t'

(s6)

:

(

l --

s,.)u , - s:,.-sl,,.l'

(s7)

!

:rfn3 rdr rt&r I

?l4l

:

qiilIlr

Fn.: ln rr rrt h'r

fr k ln Snr

and for the drainagecycle was found to be krn-,

ilktr

35

3 .Y

o ra

Well--Sorted

Grarns

e -w FIGURE 13. Wyllie curves.I

dr,

and

t5-51 illcl-:il()n and

Ot...:iuratiOn D r t . i : ' nc t c l e

trapped-watersaturation,which is determinableby Albert and Butault's whereS*, represents curve be obtainedeither method.36These investigatorssuggestedthat a capillary-pressure with a wetting fluid or with a nonwettingfluid such as mercury to obtain irreduciblenonwetting phasesaturation.They alsoestimatedthat the irreduciblenonwettingphasesaturation is two thirds of net pore volume madeup of capillariesof radii smallerthanthe most common capillary size, when the nonwettingphasedisplacesthe wetting phase. Pirson suggesteda method to determine the in situ trapped nonwetting phase saturation by meansof microresistivitylogging devices,which respondto the flushed zone arounda well bore: Sn*r:1-(1/0)

(5 6 )

(5 7 )

(R-,/R*.,)"t

(58)

where $ representsthe porosity of the reservoir rock and R-r/R^. is the ratio of the mudfiltrate resistivityto flushed zone resistivity. suggestedwater and gas relative permeabilityequationsin terms of core peBoatman3T trophysical propertiesobtained from laboratory data: k,* :

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

(se)

36

Relative Permeabilin of Petroleum Reservoirs n{

rhsnc

Th\r

rbrtr

l

R*x( tw!'rJ

3

rhcr

o L

&lqrr t $rr

L

flE

l<

l<

ti-r F'ffirJ & l,g* I

c";s f u

1.uFc

20

40

60

80

roo

C.ro

e -w

Fsc! lfr 1l

,mnl f, .,,

FIGURE 14. Wyllie curvesfor water-wetcementedsandstones, oolitic limestones,or vugular systems.33

and

k,, :

(1 -

S**t/4 Swt/2)t/2

frgr

r

*fut Hcr

(60)

rtnnt 3rF-il

where

*:s. e * - S * - S * i \'w I - S*,,

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

(S**)"t (R"/R,)2

(61)

\ \ lm-* ' \ \-*' \\,

L.

and k^, :

(l - S*-)'

(62)

where S*- represents(S* - S*,-)/(l - S*,., - S.,.);S** represents(S* - S*,*)/(l - S*,,,). Thornton5proposedthe following equationfor wetting-phaserelative permeability: k.*, :

Sl (PD/P.)2

(01)

where P"/P. representsthe ratio of displacementpressureto drainagecapillary pressure. R o s e a n d Wyl l i eT' 3eproposeda petrophysi calequati onfor w etti ng-p haser elat ive permeability: k.*, :

(Ir/2)

(64)

where I representsresistivity index, R,/R". proposedmathematicalrelationshipsfor water-oil and water-gasrelative permeJonesao abilities as function of S* and S*,, where S* may be determinedfrom well logs, S*, may be estimatedfrom an S* - $ crossplot,and d may be determinedfrom well logs:

k.* : (s**)'

(6s)

k,-:[8H]'

(66)

XI. KNOPP, HONARPOUR ET AL., AND HIRASAKI

(60)

t l"lItr* Sl

(61)

Knoppa' developeda correlation from 107 experimentallydeterminedgas-oil relative permeabilityratios of Venezuelancore samples.The core sampleswere from consolidated as well as poorly consolidatedsandstonereservoirsof high porosity and permeability;the Welge gas-floodprocedurewas used for k.r/k.odetermination. A single correlationwas establishedon the basisof the restored-state water saturationas a correlatingparameter.The correlationis shownas a family of most probablek.s/k,.,curves in Figure 15. Comparisonof Knopp's correlationwith experimentalvaluesis more promisingwhen the geometricmeanof the suiteof k,s/k,ocurvesfor a given reservoiror samplegroupis compared with the correspondingmost probablecurves for the correlation.Knopp also suggesteda procedurefor developing similar correlationsfor various other formations. A comparisonof Knopp's correlationswith the correlationof Corey and Wahl et al. on the basisof l5%owater saturationis shown in Figure 16. Honarpouret al.a2developeda setof empiricalpredictionequationsfor water-oilimbibition relative permeability and gas-oil drainagerelative permeability from a large number of experimentaldata. Their resultsare presentedin Tables4 and 5. Symbolsusedin thesetwo tables are defined as follows: : air permeability,md ku : oil permeability,md ko ko(s*i): oil permeability at irreducible water saturation,md : gas relative permeability,oil and gas system,fraction k., k,e(so,):gas relative permeabilityat residualoil saturation,fraction k,o,* : oil relative permeability,water and oil system,fraction : water relative permeability, water and oil system, fraction k* k.o., : oil relative permeability,oil and gas system,fraction

38

Relative Permeability of Petroleum Reservoirs

roo.o Restored State Water Saturation

o

o

L-

f-

.y, - t

.Y

o,

cr

f-

b

.Y

.-

ooorb

24 30 36 42 48 54 60

ss FIGURE 15. Knopp's correlationof most probablerelative permeability ratios.or

: gas saturation,fraction Ss S*. : critical gas saturation,fraction : oil saturation,fraction S. S..* : residualoil saturationto gas, fraction S..* : residual oil saturationto water, fraction : water saturation,fraction S* S*, : irreduciblewater saturation,fraction : porosity, fraction 0 The data which were usedas a basisfor the study by Honarpour et al. were derivedfr.m oil and gas fields locatedin the continentalU.S., Alaska,Canada, Libya, Iran, Argentina, and the United Arab Republic. Alt of the laboratorytestswere made at room temperature and atmosphericpressure'No attemptwas madeby the authorsto group the data according to laboratorytechniquesusedin measuringrelativepermeability,since this informationwas not availablefor many of the data sets.Each set of relativepermeability data was classified

rsG Sxr h r cirri: a5rF-!

x

39

o l-

o.ol

tooo,o

o.ool

roo.o

o.oool

ro.o

.Y

\

o) l.Y

@

o.ooool

r.o

t2 t 8 24 30 36 42 48 54 60

s g 'o/o FIGURE 16. Comparisonof relative permeabilitycorrelations.*'

"carbonate" or "noncarbonate", but the informationwhich was availablewas not as either sufficientfor more detailedlithologic characterization. "carbonate" or "noncarbonate", a further In additionto the classificationof data setsas classificationwas made on the basis of wettability. This rough classificationwas made accordingto the following arbitrarycriteria: dc:: , ci 1'1nrn l . \ : , l en t i n a , l lc:: ftratufe l.r -:..,,fding f ' I I . r l r ( ) nw a s ra. -..r.:ified

l.

2.

The rock was consideredto be strongly water-wetif k,,,at high oil saturationsin an oil-water system greatly exceededk,o in a gas-oil system at the same saturations, providedk.* in a gas-oil systemgreatlyexceededk,* in an oil-water systemat or near residualoil saturationafter water-flooding. The rock was consideredto be oil-wet when k,o in the oil-water systemwas approximately equal to k,., in the gas-oil system, provided k,* in the gas-oil system was approximatelyequal to k.* in the oil-water system.

Relative Permeability of Petroleum Reservoirs Table 4 EQUATIONS FOR THE PREDICTION OF RELATIVE PERMEABILITY SANDSTONE AND CONGLOMERATE

el

(s* - s"') k... : 0.035388 - S * , - S , , , * -) o.olo874x (l 't"r- t"':' * *,) (water-wer) f . , - S " ,. l " + o . s o s 5 6 ( S * ) r n (- S S S,,,")l Ltt k . , .:

(61)

-'s*' (s* - s"'*) r . 5 8 1 4[ s * l ' " ' - 0 . 5 8 6 1 7 - s * , - S , , ,** ) (l ll-s", I (S" - S*,) - 1.24846( I - S*,)(S* - S*,) (intermediately wet)

(68)

r \/l - ss "'/ \ - s' l , r . k,,,*:0.760671 - t-" s,,,*J ': l'" '- s,,,* [t', t-" =s*, L I +2.63180(l -

S , , . * ) ( S -, , S " , * ) ( a n y w e n a b i l i t y )

t,,=- t,,,,: , k n ' ,: 0 ' 9 8 3 7(l2 + l. (anyweuability) - t-, ). L [ | - s -, - s,,r: ] k .*:

l o ? 2(H )' u ,* ,,,,. .,

(69)

(70)

k*: o2eer. (H)

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

(warer-wet)

(72)

1.2624(H:)

(*)'

k*,e: 0.s37s2 (jil'(ff-o'*: 'sossffik'gts,,,*t

,_)'

utrrc S. S.

ss.

- otztot(ffi;)'-

(S*-S*.)*0.4|325g(*).(intermediate|ywet)

k..* :

ttsrTne shrlied Hrra iollos:

( 7 1)

Table 5 EQUATIONSFOR THE PREDICTIONOF RELATIVE PERMEABILITY IN LIMESTONE AND DOLOMITE k* : 0.002oszs \f

Atler linear n rneasun porosit All u equalio rrxks. The tested in clox \aere ul ln u. relatrre

+ 2.i i g4*

(anY k'g's,,,r' wettabilitY)

\+

T

_1. IN

(13)

(anywettablity)

(74)

(any wettablity)

(75)

L. \.. L \ n

+ 8'oo53x A

-

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

.'

(#)'. - t")' ' - t" (' (t)"' _ i::

I

, (anywettab'ity)

(76)

4r The rock was consideredto be of intermediatewettability when it did not clearly meet either the water-wetor the oil-wet classificationcriteria.

3.

LIT\ I\

After the data sets had been classifiedaccordingto lithology and wettability, stepwise linear regressionanalysiswas employedto developequationswhich would approximatethe measuredrelative permeabilitiesfrom such factors as fluid saturations,permeability, and porosity. All water-oilsystemequationsrefer to displacementof oil by water and the oil-gassystem equationsrefer to drainageprocesses.All experimentaldata were measuredin consolidated rocks. The equationsthat were developedby Honarpouret al. have not yet been extensively tested.However, most of the testswhich have been made indicatedthat the equationsare in closeragreementwith laboratorydatathan the predictionsof publisfredcorrelationswhich were used as a basisfor comparison. In usingempiricalrelationshipssuchas thosepresentedby Honarpouret al., any calculated relative permeability which exceeds l 0 should be assumedequal to 1.0. If a relative permeabilityvalue is known at any water saturation,the relativepermeabilitycurve may be shifted to match the known data point. Hirasakia3has suggesteda relative permeability correlation for fractured reservoirs as follows: S * :

Su -

So.

l-s*-So"

(77)

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

(78)

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

(7e)

L - I 1\ I \ where

r73)

r71)

S* Sd So" So. k.a ko.o k,o k".. n

: : : : : : : : :

Normalizedsaturation. Displacingphasesaturation. Immobile displacingphasesaturation. Residualoil saturation. Displacingphaserelative permeability. Displacingphaserelative permeabilityat residualoil saturation. Relativepermeabilityto oil. Relativepermeabilityto oil at immobile displacingphasesaturation. Exponentparameterfor shapeof relativepermeabilitycurves, said to be equal to one in fractured reservoirs.

r75)

t

-..\

'i i" ,tY

l) rr",/J-

REFERENCES

,_ trt

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

t76)

2. Kozeny, J., Uber Kapillare Leitung desWassersimBoden, Sitzungsber.Akad. Wiss. Wien. Math. Naturwiss. KL., Abt. 2A, 136,2'll, 1927. 3. Purcell, W. R., Capillary pressures- their measurementusing mercury and the calculationof permeability therefrom.Trans. AIME, 186.39, 1949.

42

RelativePermeabilin of PetroleumReservoirs

4. Rose, W. D. and Bruce, W. A., Evaluationof capillary characterin petroleumreservoir rock, Trans. .AIME, t86, 127, t949. 5. Thornton, o. F., valuation of relative permeability,Trans. AIME, 1g6,329, lg4g. 6. Rose, W. D., Theoreticalgeneralizationleadingto the evaluationof relativepermeability,Trans.AIME, 186,1il , 1949. 7. Rose, W. and Wyllie, M. R. J., Theoreticaldescriptionof wetting liquid relarivepermeability Trans. , AIME, 186,329, t949. 8. Gates,J.I.andLeitz,W.J.,RelativepermeabilitiesofCaliforniacoresbythecapillarypressuremethod, paper presentedat the API Meering, Los Angeles,california, May ll, 1950, 296. 9' Rapoport, L. A. and Leas, W. J., Relative permeabilityto liquid in liquid-gassystem, Trans. AIME, 1 9 2 ,9 3 , l 9 5 l . 10. Wyllie' M. R. J., Interrelationshipbetweenwetting and non-wettingphaserelativepermeability,Trans. A I M E , 1 9 2 ,8 3 , 1 9 8 1 . ll. Fatt, I. and Dykstra, H., Relativepermeabilitystudies,Trans. AIME, 192,249, lg5l. 12. Wyllie' M. R. J. and Sprangler, M. B., Application of electricalresistivitymeasurements to problems of fluid flow in porous media, Bull. AApG, 36, 359, 1952. 13. Burdine, N. T., Relative permeabilitycalculationsfrom pore size distributiondata, Trans. AIME, lgg, 7t,1953. 14. Naar, J. and Henderson, J. H., An imbibition model- its applicationto flow behaviorand the prediction o f o i l r e c o v e r y ,T r a n s .A I M E , 2 2 2 , 6 1 , 1 9 6 1 . 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 2129, 2 65 l .4 , 16. Land, C. S., Calculation of imbibition relative permeabilityfor two- and three-phaseflow from rock properties,Soc. Pet. Eng. J., 6, 149, 1968. 17. Wyllie' M. R. J. and Gardner, G. H. F., The generalizedKozeny-Carmenequation,its applicationto problemsof multi-phaseflow in porous media, World Oit, 146, l2l, 1958. 1 8 . T i m m e r m a n , E . H . , B d . , P r a t ' t i L ' aRl e s e r t , o iE r n g i n e e r i n gP , e n w e l lp u b r . , r 9 8 2 , l 0 l . 19. Corey, A. T., The interrelation b e t w e e ng a s a n d o i l r e l a t i v ep e r m e a b i l i t i e sP,r o d . M o n . , 1 9 , 3 8 , 1 9 5 4 . 20. Corey, A. T. and Rathjens, C. H., Effect of stratificationon relativepermeability,Trons.AIME,20j, 358,1956. 21. Johnson, C. E., Jr., Graphicaldeterminationof the constantsin the Corey equationfor gas-oil relative p e r m e a b i l i t yr a t i o , J . P e t . T e c h n o l . ,1 0 , l l l l , 1 9 6 8 . 2 2 . l r m a y , S . , O n t h e h y d r a u l i cc o n d u c t i v i t yo f u n s a t u r a t esdo i l s ,T r a n s .A G U , 3 5 ( 3 ) , 4 6 3 , 1 9 5 4 . 23. Averganov, S. F., About Permeabilitl, ofSubsurfuc'e Soils in Case of IncompleteSaturation, Engineenng Colfection, Vol. 7, 1950, cited by Polubarinova-Kochina, P, in The Theory of Ground Water Movement, E n g l i s ht r a n s l a t i o nb y D e w i e s t ,R . J . M . , P r i n c e t o nU n i v . P r e s s ,P r i n c e t o n N . .J.. t962. 24. Wahl, W. L., Mullins, L. D., and Elfrink, E. 8., Estimationof ultimate recoveryfrom solution gas drive reservoirs,Trctns.AIME, 213, 132, 1958. 25. Torcaso, M. A. and Wyllie, M. R. J., A comparisonof calculatedk.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., Propertiesof porous media affecting fluid flow, "/. Irrig. Drain. Div.. 6.6t. 1966. 28. Talash, A. W., Experimentaland calculatedrelative permeabilitydata for systemscontaining tension additives,Paper5810, Societyof PetroleumEngineers,Dallas, Tx., 1976. 29. Land, C. S., Calculation of imbibition relative permeability for two- and three-phaseflow from rock properties,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 Permeabilityand Imbibition in porous Media, Ph.D. thesis,Universiry of Calgary, Alberta, Canada,1973. 3 2 . B r o w n , H . w . , c a p i l l a r y p r e s s u r ei n v e s t i g a t i o n s , T r a n sA.I M E , 1 g 2 , 6 7 , l g 5 l . 33. Frick, T., Ed., PetroleumProductionHandbook,Vol. 2, Societyof PetroleumEngineersof 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. 3 5 . Pirson, S. J., Ed., Oil ReservoirEngineering,McGraw Hill, New york, 195g. 36. Albert, P. and Butault, L., Etude des CharacteristiquesCapillaries du Reservoir du Cap don par La M e t h o d eP u r c e l l ,P e t . A n n . C o m b u s .L i q . , 1 ( 8 ) , 2 5 0 , 1 9 5 2 . 37. Boatman, E. M., An Experimental Investigation of Some Relative Permeability-RelativeConductivity Relationships,M.S. thesis,University of Texas, Austin, 1961. 38. Pirson, S. J., Boatman, E. M., and Nettle, R. L., Predictionof relativepermeabilitycharacteristics of intergranularreservoirrocks from electricalresistivitymeasurements,Trans. AIME, Z3l,564. 1964.

-19 l1'rllic. Phr

'1t-'

ro. Jones. 11. Knopp I l l r .r {1 Honerl

rc'lattrc -l-r. Hinsrl

Ga.. P. liopfif t-nc l {5 tukha pha.c r

"ll

43 I-rrtns.

R.

\ 1 . \E T. fr-,

'., . 'l'runs.

F.

.-; rttcthod.

.v.vE. '.

I runs.

: ' - ,' h l c r n s t'.1I

I9tt.

..'::Jrctitln

l:

:_- _<:

196L " ,:tt rock

f

,rtl()n [o

t-

,

)r

.. le5-l : r/l_ 107. rclJllve

f -.. | -t

l" lu'

-

:

:nccring ',1 . tttt'ttl t , . .i : r t I l S i t S

h-

. . : '. ' .

J Pet.

' P::.. - . \ o 3 . 'tt

. ,: Div., - tcnsion - 'nr rock

I P

. \tedia,

J \ . \ : l . Dallas, t

LT:

:. I1t.603,

'n Par La

u r { :Juctivity h.:,. : c n ' t i c so f S+': l e6J

relatedto quantitativeevaluationof 39. Wyllie, M. R. J. and Rose, W. D., Some theoreticalconsiderations physicalcharacteristics of reservoirrock from electricallog data, Trans. AIME, 189, 105, 1950. 4 0 . J o n e s , M . A . , W a t e r f l o o dm o b i l i t y c o n t r o l :a c a s eh i s t o r y ,J . P e t . T e c ' h n o l .9, , l l 5 l , 1 9 6 6 . 41. Knopp, C. R., Gas-oil relative permeabilityratio correlationfrom laboratorydata,J. Pet. Technol.,9, llt1,1965. 42. Honarpour, M. M., KoederitzrL. F., and Harvey, A. H., Empiricalequationsfor estimatingtwo-phase relativepermeabilityin consolidatedrock, Trans. AIME, 2'73,2905, 1982. 43. Hirasaki, G. J., Estimationof ReservoirParametersby History Matching Oil Displacementby Water or , a l l a s ,T e x . , 1 9 7 5 . G a s , P a p e r4 2 8 3 , S o c i e t yP e t r o l e u mE n g i n e e r sD 44. Kopli.k, J. and Lasseter, T. J., Two-phaseflow in random network modelsof porous media, Sot'. Pet. Eng.J., 25, 89, 1985. 45. Fulcher, R. A., Ertekin, T., and Stahl, C. D., Effect of cappillarynumberand its constituentson twophaserelative permeabilitycurves,J. Pet. Technol., 2,249, 1985.

Chapter 3 FACTORS AFFECTING TWO-PHASE RELATIVE

PERMEABILITY

I. INTRODUCTION The first publishedinformationconcerningthe simultaneousflow of multiple fluid phases "relative permeability" had not yet been coined was probably by Hassleret al.r The term and Hassleret al. studied only the flow characteristicsof the gas phaseas a function of fluid saturationin consolidatedrocks. The relativepermeabilityconceptwas first postulated by Muskat and Meres.2Their work consistedof extendingDarcy's law to two-phasesystems. For oil reservoirs,the relevanttwo-phasefluid combinationsare water-oil and liquid-gas (usually thought of as oil-gas). Gas-waterrelativepermeabilitycurvesare used to describe the performanceof gas reservoirsand gas-liquidcurvesare usedfor condensatereservoirs. II. TWO.PHASE RELATIVE

PERMEABILITY

CURVES

Water-oil relative permeability is usually plotted as a function of water saturation,as shownby Figure l. At the irreduciblewater saturation(S*.), the water relativepermeability is zero and the oil relative permeabilitywith respectto water is some value less than one. At this point only oil can flow and the capabilityof the oil to flow is reducedby the presence of connatewater. The effect of connatewater in reducingoil flow rate is illustratedschematically by Figure 2. Note that data to the left of the irreducible water saturationare not useful for predicting lessthanS*" arenot encountered. hydrocarbonreservoirperformance,sincewatersaturations As water saturationincreases,the water relative permeabilityincreasesand the oil relative permeability(with respectto water) decreases.A maximum water saturationis reachedat the residualoil saturationand the oil relativepermeabilitybecomeszero. Obviously, aquifer conditionsare representedby a relative permeabilityto water of unity, which occurs at a water saturationof l00%o. Unfortunately,there is an alternatedefinition of relative permeabilitycurrently in use. This terminology(illustratedby Figure 3) definesthe oil relativepermeabilityat irreducible water saturationas having a value of one, and definesabsolutepermeabilityas the effective permeabilityat irreduciblewater saturation.The effectivepermeabilitiesare identical with both definitions of relative permeability and one set of values may be readily convertedto the other. This second definition of relative permeability (k,r) applies to both the oil and water phases. Thesealternateor normalizedvaluesof relative permeabilitymay be convertedto standard valuesby k.srn :

k,2 ku./kusrD

(l)

where k.. : k"o at S*"

{ \

Also note that underthis seconddefinition of relativepermeability,the waterrelativepermeability in an aquiferhas a value greaterthan unity. Essentially,with this alternatedefinition, relative permeability is normalized to the value at irreducible water saturation. Gas-oil relative permeabilityand gas-liquidrelative permeabilityare similar in concept to water-oil relativEpermeability. The preferredrelative permeabilityvalues are those taken with connatewater presentat the ineducible saturationvalue.

46

Relative Permeability of PetSoleumReservoirs

I I I

\

\ oil W a te r I

ftret

/

I

/

Sorw

Swc 0

I

I

I

Svrr-+ (-s

FIGURE l.

,'t Rock )

o

o-

W a t e r - o i lr e l a t i v ep e r m e a b i l i t yc u r v e s .

;Water

oif-

L (

Rock \

-.*-t

FIGURE 2.

Oil flow reductiondue to the presenceof water.

As free gas saturationincreases,the oil relativepermeabilitywith respectto gasdecreases; however, until the critical gas saturation(Sr") is reached,the gas relative permeabilityis zero. The critical gas saturationis the point at which the gas bubblesbecomelarge enough to break through the oil and away from the rock surface.As gas saturationincreases,the gas relative permeabilityincreasesand theoreticallyreachesa value of unity at l00%cgas. A gas-oil relative permeabilitycurve is illustratedby Figure 4. An experimentalprocedureto determinerelativepermeabilityin an unconsolidatedsand was first describedby Wyckoff and Botset.3Their work consistedof injectinga combination conditions.Their resultsare shown of liquids and gasesthrougha sampleundersteady-state in Figure 5, where k.. and k,, are relative permeabilityto oil and gas, respectively.The figure is typical of wetting- and nonwetting phase relative permeabilities,regardlessof whetherthe systemis oil- or water-wet. Figure 5 shows differently shapedrelative permeabilitycurvesfor the two phases.The oil relativepermeabilitycurve is concaveupward while the gas relativepermeabilitycurve has an "S" shape.This figure also showsthat the oil relativepermeabilityat the irreducible

(or criti saturat relative curve tt upward The : reducti rapid de

47

k r e l

Sw FIGURE 3.

Normalizedwater-oil relativepermeabilitycurves

\ \

\1

I

I

Gor

I I

oil I , ,

K rcl

o D .:. . :aJ\gs: l]tc'.:^llrtf iS fgr

lttough

lfe.:.Cs. the 1 [ r r r ' ,g a s . Itti.,:rJ :and ill::t'.ltlation S r:i .h()wn itrrclr. The tgurJlcss of

p h r . e. . Th e b r l r t rc u r v e : rr.'c,.lucible

I

SwcSorg O

- S L 4

t

+-SG

FIGURE 4.

Sg. |

Gas-oil relative permeabilitycurves'

at the ineducible oil (or critical) gas saturationis less than the gas relative permeability apply to water-oil observations general same saturation.Leverett,sworka shows that the permeability relative water the oil, of presence relative permeabilitydata. That is, in the is concave or curve permeability relative curve takes on the shape of the wetting-phase upward. 5 indicatesthat, for a small The shapeof the oil relative permeabilitycurve in Figure permeabilityto oil' This relative in decrease reductionin oil saturation,there is a sizeable by the gasphase'Figure paths flow or pores rapid declineis due to the occupationof larger

l

.

Relative P ermeability of t etroleumReservoirs

ftrel

kts

l't:.:':)'/,. A

SL FIGURE 5.

Relative permeabilitycurves for an unconsolidatedsancl.r

5 alsoindicatesa steepincreasein the gasrelativepermeabilityas the gassaturationincreases abovepoint "A", which is the saturationat which relativepermeabilitiesto the oil and gas phasesbecomeequal. For this unconsolidatedsand,the oil relativepermeabilityat 59Vo orl saturationis equal to gas relative permeabilityat 4l%o gas saturation.The gas relative permeabilityreachesnearly l00%oat a gas saturationlessthan l007o, which 1n.un,that part of the interconnectedpore spacedoes not significantlycontributeto the gas permeabilityof the porousmedium. This figure also showsthat the gasrelativepermeabilityremainsat iero until the gas saturationreachesthe critical gas saturation,point "B". The gas phaseis not mobile at a saturationless than the critical value, but this immobile gas impedesthe flow of oil and reducesoil relative permeability.As oil saturationis increasedfrom an initial value of zero, the oil relative permeabilityremainszero until the oil forms a continu.us phaseat the critical oil saturation,which is represented as point C in Figure5. In a solutiongas-drivereservoir, often the water saturationis small and immobile. Therefore,relative permeabilityvaluesare frequentlyplottedagainstthe liquid saturationratherthan the wetting saturation.Under such a condition, point "C" is the summationof the irreduciblewater saturationand the residualoil saturation,as previouslyindicatedin Figure 4. The sum of the relative permeabilitiesfor all phasesis almost always less than unity becauseof interferenceamongphasessharingflow channels.Thereare a numberof reasons for this interference.One of these reasonsis that part of the pore channelsavailablefor flow of a fluid may be reducedin sizeby the other fluids presentin the rock. Another reason is that immobilized dropletsof one fluid may completelyplug someconstrictionsin a pore channelthrough which anotherfluid would otherwiseflow. Also, somepore channelsmay becomeeffectively plugged by adversecapillary forces if the pressuregradientis too low to push an interface through a constriction. A fourth reason is the trapping of a group of globules that are clustered together and cannot be moved, since the grain configuraiion allows fluid to flow around the trappedglobules without developinga pressuregradient sufficient to move them. This is the phenomenonthat has been referredto as the Jamin effect.

Nou inters mation other r of clar of con occur i Rela the var to wet nomen therefc foral imbibi Relati pressu

Satu At lov wettin pendu transm an app of ind Aba path u phase this re phase betwe the n'e of sat phase ration Flui a poro differe for thi the no reserv as the down i phase It hi to the functi< as wel large c indica

Nowak and Kruegerstestedtwo coresin which the permeabilityto oil in the presenceof interstitialwater was considerablygreaterthan single-phasepermeabilityto syntheticformation water. Yuster6and OdehTboth found the samephenomenonbasedon the resultsof other work. A possibleexplanationfor the high permeabilityto oil is that the distribution of clay varies within the rock and variationsin water saturationcausevariationsin the area of contactbetweenwater and clay minerals.Thus, increasingdegreesof clay swelling may occur at higher water saturationdue to the hydrationof larger amountsof clay minerals. Relativepermeabilityis dependentupon both the fluid saturationand the distributionof the variousfluids in the intersticesof the porousnetwork.This distributionis directly related to wettability characteristicsof the rock, which in turn give rise to capillary pressurephecurves; nomena. It is well known that hysteresisexists in capillary pressure-saturation Thus, expected. also be can curves permeability-saturation in relative hysteresis therefore, is rock that in a measured permeability relative the saturation, given wetting-phase for a draining. rock is while the measured as that phase the same is not imbibing the wetting Relativepermeabilityvaluesalsomay be functionsof factorssuchastemperature,overburden pressure,phaseequilibria,ro'etc. III. EFFECTS OF SATURATION

i thrn unity J rrl t-CSsoDS

u a : l . r h l c f' o r lhcr rcason f\ ur J pore Bnncl\ may I t. lrro low 3 -rroup 6f f,tl rr uration fr' gradient I thc Jamin

STATES

Saturationis a term used to describethe relative volume of fluids in a porous medium. At low saturationsof the fluid that preferentiallytends to wet the grains of a rock, the rings aroundthe grain contactpoints. Theseare called wetting phaseforms doughnut-shaped with each other and pressurecannot be not communicate rings do pendular rings. The such a distributionmay occupy Sometimes another. pendular ring to transmittedfrom one upon the nature and shape depends The amount pore space. of the fraction an appreciable cementation. of and type well as degree as grains, distribution, of individqal Above the critical wetting-phasesaturation,the wetting phaseis mobile through a tortuous path under a pressuredifferential and as the wetting-phasesaturationincreases,the wettingphaserelative permeability increasesas well. The wetting-phasesaturationdistribution in this region is called funicular and up to a point, the relative permeabilityto the wetting phaseis lessthan the relativepermeabilityto the nonwettingphasedue to the adhesionforce between the solid surface and wetting fluid, and the greatertortuosity of the flow path for the wetting phase.The nonwettingphasemoves throughthe larger poreswithin this range of saturation, but as the saturationof the wetting phase further increases,the nonwetting phase breaks down and forms a discontinuousphaseat the critical nonwetting phase satusaturation. ration. This is called an insular stateof nonwetting-phase flow simultaneouslythrough fluids immiscible when that Fluid flow studieshave shown path. flow network changesfor This flow own its a porous medium, each fluid follows phase reduces,the network saturation nonwetting the different ranges of saturationand as stationaryislandsof the remaining discontinuous; becomes for this phasebreaksdown and in hydrocarbon gradients encountered pressure at the nonwetting phasecanrnt be displaced phase saturation. Similarly, nonwetting residual as a reservoirs.This conditionis refened to phase breaks flows which this network through the as the wetting phasesaturationdecreases, wettingas an ineducible referred to This is down and becomesdiscontinuousand immobile. phase saturation. It has been showns-rrthat for strongly water-wetunconsolidatedsandsthe permeability to the wetting phaseis dependentsolely upon its own saturation,(i.e., a plot of k.* as a function of S* has the same shaperegardlessof whether or not the pore spacecontains gas as well as oil). However, in the petroleumrelatedliterature,somesmall'2''3and somequite large deviationsare seenfrom thesefindings for consolidatedrocks. Somepublicationsr4'15 indicate that the nonwetting phaserelative permeability dependson the wetting as well as

I

50

Relative Permeability of P,etroleumReservoirs

average

o J

o) -

Y

minimum

1

.9 (u TE

-o

-

l

(u o

E o o. o .=

-01

(u o (r

0.5

1.0

S L, FIGURE 6.

R e l a t i v ep e r m e a b i l i t yr a t i o sf o r s a n d sa n d s a n d s t o n e s . r s

the nonwetting phase saturationfor strongly water-wet systems.In preferentiallyoil-wet systems,the oil phaserelativepermeabilityis found to be strictlya functionof oil saturation,r6 while in water-wet rocks, the oil phaserelative permeabilityis found to dependon both water and oil saturation.Donaldsonand Dean'7havepointedout that undertwo-phaseflow, relative permeabilityto water was increasedwhen oil, ratherthan gas was the nonaqueous phase,indicatingthat water relativepermeabilityis not solelya functionof water saturation.

IV. EFFECTSOF ROCK PROPERTIES Relativepermeability-saturation relationsare not identicalfor all reservoirrocks, but may vary from formation to formation and from one portion to another of a heterogeneous formation. Arps and Roberts'8have presentedplots of gas-oil relative permeabilityratios for 16 consolidatedsandstonesand 25 dolomites, cherts, and limestones,all with l57o connate water saturation.These plots are presentedas Figures 6 and 7. The maximum curve in Figure 6 seemsto be typical of unconsolidated sandstone,while the minimum curve appears to be more representativeof highly cementedsandstones.The averagecurve can be consideredtypical of the averageconsolidatedsandstone.The minimum curve in Figure 7, which seemsto be the steepestand most unfavorable,is from a fracturedchert core; at the other end of the range, no well-definedmaximum case is apparent.Curve #23, adapted from Bulnes and Fitting's workrerepresenting26samplesof west TexasPermiandolomite, appearsto be the bestmaximum curve. The curve selectedas "average" on Figure7 appears to be typical of vugular limestones. Bulnesand Fitting as well as Stone2o have shownthat the fluid flow behaviorin uniformporositycarbonatesamplesis similar to fluid flow behaviorin consolidatedsandstones, but the differencebecomespronouncedas the rock heterogeneityincreases.

Vario sandsto and the saturat abilitl' r wetting qualita uncons and con effect o in degre system is wide Corer permea in aniso saturat pendicu water-o arTange ogeneit Leve

5l 10 ma x im um

o -g

.\

a verage

\

o) .

Y

nrmum

\ \

l

\

\

\

\

o (!

tr .: -o

I

(u o E q,

oo .:

.01

(!

o

tr

. o o1

0.5

s,

1.0

L

i a .. . , ' r l - r r e t S : l . . I . r l 1 1 t J 1.

lt'

Rn.: ,'lt btlth 'ph.:.c llow. lll\:i.:i.l Llt-'ouS f \.:l'.lfJtlOn.

L..

lLtt ffioy

|fr'icneous llr,'. lttr l6

5 ( " . ()n n a te lnr .urre in ln c .rppears can hc conn I r c u r e7 , C r r t C .a t t h e

J r. .rdapted tl .itrl1t11l1a,

rc

rppears

rn unrl b rmdrt, rnC:. bUt

FIGURE 7. cherts.r*

R e l a t i v ep e r m e a b i l i t yr a t i o sf o r l i m e s t o n e sd. o l o m i t e s ,a n d

Various workss''e'2rhave shown that the gas-oil relative permeability of consolidated sandstoneis qualitativelysimilar to the gas-oil relativepermeabilityof unconsolidatedsand of the two relativepermeabilitiesto oil at high oil and there is a very close coffespondence saturation.It has been found that for consolidatedsand, the wetting-phaserelative permeability drops sharply and the nonwetting phase relative permeabilityrises steeply as the wetting-phasesaturationdecreases.However, Naar et aI.22have shown that there are both qualitative and quantitativedifferencesbetweenrelative permeabilityof consolidatedand unconsolidated sands.Owensand Archerrrindicatedthatpackingasmodifiedby cementation and consolidationaffectsthe equilibrium saturationto the wetting phasebut has a negligible effect on the equilibrium saturationof the nonwettingphase.Nind23statedthat an increase in degreeof consolidationincreasesthe nonwettingphaserelativepermeabilityin a gas-oil system.Severalinvestigatorshave noted that the saturationrangefor a mobile fluid phase is wider in unconsolidatedrock than in consolidatedrock. Corey and Rathjens2a studiedthe effect of rock heterogeneityon drainagegas-oil relative permeability.They investigatedthe flow paralleland perpendicularto obvious stratification in anisotropicBereasandstonecoresand concludedthat the relativepermeabilityat a given saturationfor flow parallel to bedding was greaterthan the analogousvalue for flow perpendicularto the bedding plane, as shown in Figures 8 and 9. Huppler2sfound that the water-oil relativepermeabilityof compositecore changesappreciablywhen the sectionsare arrangedin different orders. Johnsonand Sweeney'oalso studiedthe effect of rock heterogeneityon the gas-oil relative permeabilityratio. Leverettafound a small but systematicchangein the positionof the relativepermeability-

52

Relative Permeabilin of Petroleum Reservoirs

o J

c oo

FIGURE8. Relativepermeability measurements from

an anlsotroprc

sandstone.ra

O

- kro - porpondicul!r

O

- kro

- parEllol to baddlng

O

- trg

- p.rpondlculr.

O

- krg - p!..llol

to boddinO

to b.ddinc

to boddlng

o l(

0

1

so FIGURE 9.

Relative permeability measurementsfrom a Berea sandstone.2a

saturatlo perimen distribu saturati (spheric the shap systems [,even water mt is neces it can h unit volr with larg rations a fluids. T saturatio have larl leavelin permeab Gorrin it is enc conclude nonwett uniform 1 the size relative I higher pr efficientl Botset dependsr and Wyl consequ of pore s

53

Time 1

Time 2

Time 3

E

otL

M

WATER

I

SAND

FIGURE 10. The formation of residualoil by the blocking process.

saturationrelationship due to the employment of different sizes of sand grains in his experiments.Botset2rconfirmed Leverett'sfinding and concludedthat the effect of grain size distributionwas not negligible either on the relationshipbetweenrelativepermeabilityand saturationor on the value of the equilibrium saturation.It was found that the shape" (sphericity),roundness"(angularity),and orientation2a of the grainstendedto influenceboth the shapeof the relative permeability curve and the critical gas saturationvalue in gas-oil systems. Leverettapointed out that the relative permeability of an unconsolidatedsand to an oilwater mixture is relatedto the sandpore size distribution.Muskat et a1.27 suggestedthat it is necessaryto know the pore geometry of a reservoir rock before fluid movement through it can be analyzed. Morgan and Gordon2sfound that pore geometry and surface area per unit volume influencedwater-oil relativepermeabilitycurves.They have shown that rocks with large pores and correspondinglysmall surfaceareashave low irreducible water saturations and therefore have a relatively large amount of pore spaceavailable for the flow of fluids. This conditionallows high relativepermeabilityend pointsto exist and allows a large saturationchangeto occur during two-phaseflow. Correspondingly,rocks with small pores have larger surface areasper unit volume and they have irreducible water saturationsthat leave little room for the flow of hydrocarbons.This condition createsa low initial oil relative permeability as well as a limited saturationrange for two-phaseflow. Gorring2edemonstratedthat oil in a larger pore can be surroundedand blocked off when it is encircledby smaller pores which imbibe the displacingwater by capillary forces. He concluded that both pore size distribution and pore orientation have a direct effect on nonwetting residual equilibrium saturation, as shown by Figure l0; therefore, a perfectly uniform packing of spheresshouldgive a residualsaturationnearzero.Gorring also identified the size of channelsoccupied by the nonwetting phase as an important factor influencing relative permeability. Crowell et al.30indicated that higher initial water saturationyields a higher probability for the nonwettingphaseto be in larger channelsso-thatit can b9 recovered efficiently during wetting-phaseimbibition. Botset2' mentioned as early as 1939 that the relative perrneability-saturationrelation dependson the degreeand the type of interconnections of the pores.Fatt,3rDodd and Kiel,32 and Wyllie33 also concluded that the relative permeability of porous media is a direct consequenceof the network structureof the media. Pathaket al.3aconcludedthat the ratio of pore size to pore throat is a factor which controls the snapping-off of droplets of the

54

Relative Permeability of Petroleum Reservoirs

nonwettingphase,with a high ratio leadingto a high trappedoil saturation.Other workers have investigatedthe possibilityof describingporousmedia as a network of interconnected pore bodies and pore throats. Postdepositional alterationscan form more than one type of reservoirrock from a single original rock type. Alteration may reducepore sizes,thus causinghigher irreduciblewater saturationand a natrow rangeof saturationchangeduring two-phaseflow. The presenceof grains such as feldspar, when partially dissolved,improvesthe reservoirrock quality by forming pores larger than the pores betweengrains not containingfeldspar.This alteration causeshigher relative permeabilityvalues and a larger saturationrange during two-ph4sg flow.tt Reference35 describesalterationsin pore geometry which can occur due to the introductionof reactivefluids in the rock. Land and Baptist36indicatedthat when a reservoirsandstonecontainsmontmorilloniteor mixed-layerclay mineralscontainingexpandablelayers,the watersensitivityof the sandstone is not necessarilya result of pore blockagedue to the increasedvolume occupiedby the swollen montmorillonite.Some sandstones containingtrace amountsof clay mineralsmay exhibit sensitivityto water resultingfrom dispersionand subsequenttransportationof clay mineralsto pore constrictions.Thus, permeabilityreductionmay occur in formationsthat do not contain expandableclay minerals;however, all formationscontainingexpandable clays are probably water-sensitivedue to the easeof dispersionand expansionof this type of clay. Permeabilityreductionin sandscontainingsodiumclays is likely to be higher than the reductionin sandscontainingcalcium clays. Somerock propertiesthat influencerelativepermeabilityvariationsare readily observable with a binocular microscopeor even more clearly under a scanningelectronmicroscope. Therefore,microscopiccore examinationcan be highly usefulfor evaluatingrelativepermeability characteristics.Once the significantrock propertyvariationshave been identified, a reservoir can be subdivided into appropriatereservoir rock types. Within each of such reservoirrocks types, relative permeabilitycharacteristics are usually similar, varying only slightly for rather large changesin air permeabilityor mediangrain size.

V. DEFINITIONAND CAUSESOF WETTABILITY "Wettability" is a term usedto describethe relativeattractionof one fluid for a solid in the presenceof other immisciblefluids. It is the main factor responsiblefor the microscopic fluid distribution in porous media and it determinesto a great extent the amount of residual oil saturationand the ability of a particular phaseto flow. The relative affinity of a rock to a hydrocarbonin the presenceof water is often describedas "water-wet", "intermediate", or "oil-wet". Examplesof formations with strongly water-wet, strongly oil-wet, and intermediatewettability are the Spraberry formation in west Texas, the Black Bradford sand in Pennsylvania,and the Fairbanksand in south Texas, respectively. Wettability may be representedby the contact angle formed among fluids and a flat solid surfaceor the angle formed betweenthe fluids' interfaceand a glasscapillary tube, as shown by Figure I l. The angle is measuredthroughthe denserfluid. The wettability of a porousmedium is determinedby a combinationof all surfaceforces. A sketchis shown in Figure 12, wherein two liquids, oil and water, are in contactwith a solid. The force exerted by water to spreadlaterally and displaceoil (interfacialtension betweenwater and oil) is opposedby the resultantof the solid and liquid forces (solid-oil and solid-waterinterfacialtensions).This differencein opposingforcesis calledthe adhesion tension: 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. tensions measure A pos surfacei: angle is A neg the solid suring o, evaluate to a surf Under the polar of surfac Stege molecul for the n

55

hcr i. ()rkers rtltlc-Ct€d

tt.,

f)nr .r .ingle P. t'.lc \\ ater p r l . e o C €O f I . 1 . r . r l r tbr y i..tltcration g : , r, , - p h a s e f \:..i to thg f r r . .' , r tt t c o r la..:::Jrttlne p:;.: rr the Itll;:.r,

. t'l'lii)'

'l clal' 'lt. that !r-:: cr:'.:itJable lr' :

ttl

I :1.: t) P€

r h .- : . . ' rt h a n y , , : . c r ra b l e lTll. rrrrCOP9. lltr c P.'a*ai d e : r t r t ' i c ' d .a l, i: ,,l rUCh

WATER-WET

e
OIL-WET

INTERMEDIATE

g >goo

g=9Oo

_w;, ='-Kfig== FIGURE I L

Wettability conditionson flat surfacesand in capillarytubes.

,lI\ Ttrp Vian of a Drop of VJater on a Solid Surface in the Presence of Oil

Ia:-,.:.t ()nly

fx ., . , r l i d i n

Ttrree

Dirrensional

Sdrernatic

View

tTl:.:,.'tCOpiC

| ,': :c.tdual o: .: r(Xk tO !r:::l.irate", lc: .tnd insand fa"::,'r-.1 d . : : . . r ts O l i d be. .r. .hown ItJ. e l()rceS. fil.:.t $ith a X i.:, tc'nsion f,r .trlid-oil thc .tJhesion

(2) Jrt' .'Jhesion

FIGURE 12. Forcesat a water-oil-solidinterface.

tension;oso,o,*, and o*o, respectively,are solid-oil, solid-water,and water-oil interfacial tensions(usually measuredin dyne/cm); 0*" is the contact angle betweenwater and oil measuredthrough the denserliquid phase(usually water). A positivevalue of adhesiontensionmeansthe contactangleis lessthan 90' and the solid surfaceis preferentiallywater-wet. A zerovalue of adhesiontensionindicatesthat the contact angle is equal to 90'; this is intermediatewettability. A negativevalue of adhesiontension meansthe contact angle is greaterthan 90' and that the solid surface is preferentially oil wet. There is no practical laboratory method for measuring trsoor o.*. However, o*o and cos 0 are measurablequantitieswhich can be used to evaluate the wettability of a solid surface. A fluid is referred to as wetting or nonwetting to a surfacedependingon whether the contact angle is less than or greaterthan 90". Understandingthe causesof wettability requires a study of the chemistry of the fluids, the polarity and molecular weight of reservoir hydrocarboncompounds,and the occurrence of surfacechemical processesat the solid-fluid interfaces. experimentally found that the contact angles vary directly with Stegemeierand Jensen3T molecularweight for liquids with similar chemicalstructures.Figure l3 showsthis variation for the normal paraffin seriescompounds.

56

Relative Permeabilin of Petroleum Reservoirs 50 n-ct n-cl n-cl 40

o o o

6

4

2

o)

30

r,u J o z

n-C^ t'

F

o < F

z

v

z o n-c6 10

I

I 50

100

200

150

MOLECULAR WEIGHT

F I G U R E 1 3 . C o n t a c ta n g l ea s a f u n c t i o no f m o l e c u l a rw e i g h t . r i

83o -

Silica

fsooctane

Surface

fsooctane +

Isoqrinoline

Naphthenic

5.7E Isoquinoline

Calcite Sr-rrface FIGURE 14.

Interfacialcontactangles.38

Benner and Bartell38examined various multi-liquid systemsin contact with silica and calcite surfaces.Figure l4 illustratessomeof the findings of this study. It was reportedby theseinvestigatorsthat when water and iso-octaneare used, the silica and calcite surfaces are preferentially wet by water; but when water and naphthenicacid are used, water wet the silica but oil wet the calcite surface.The experimentof Benner and Bartell illustrated the effects of chemical as well as fluid compositionof phaseson wettability of a porous medium. Contact angles as low as 30o and as high as l58o were observedwhen various chemicalswere employedin the study. Salathiel3ediscoveredthat the wettability of mineral surfacesmay be altered not only by adsorbedmonolayersof surface-activepolar compounds,but also by much thicker layers of depositedorganic materials. Severalother workers have reportedthe formation of stable films on solid surfaceswhen the surfacesstandin contactwith certaincrude oils. Reisberg and Doshelo described the deposition on glass or quartz surfaces of highly stable and appreciablythick films of strongly oil-wet material from Ventura crude oil. Early experimentersthought that all oil-bearing formations were strongly water-wet be-

cause al thermon studies: of crude readilr r studiesI One c or depos beenelir geochen crude oi deaspha in soluti rock. Ir quartz iu the resu Despi preSent particula and the Woodbir reservoi Mungan formatio wet whe film left film will can then SuggeSt eitheroil wet rese with airlr the aque Autho reservoir fractiona the fracti the wate of capill: while the "spotted

dition in Gimalud that the r to the ror insular n preferent somepon The er chemistr compoun minerals

i7

d t l : . r l r c aa n d |s :s3rr1sd[y il r - rlc . Urfa Ce S hJ. ri tter wet le l l r llu stra te d ' , 'r r po ro u s rr hcn various )d n,'l only by t h r ; I cr l a ye rs llt,'n of stable o r l. Re i sb e rg lr .t abl e a n d S .r l cr-w € b t e-

causean aqueousphasewas always the fluid initially in contactwith reservoirrock; furthermore, silica and carbonatesare normally water-wet in their clean state. Subsequent studies suggestedthat many oil reservoirsare not strongly water-wet and that the presence agents,suchas asphalticor wax type material of crudeoils containingnaturalsurface-active readily adsorbableby solid-liquid interfaces,can render the solid surfaceoil-wet.arOther studiesprovide evidencethat reservoirrock wetting preferencemay cover a broad spectrum. One criticism of the idea of reservoirrock surfacesbecomingmodified by the adsorption or depositionof polar organicmaterialfrom the oil phaseis that suchmaterialsshouldhave been eliminatedduring migration from the sourcerock to the reservoir.On the other hand, geochemistsare now finding substantialevidenceof variousalterationprocesses which affect crude oils subsequentto their accumulationin reservoirs.[n a discussionof natural gas deasphalting,Evanset al.a2suggesteda reasonablehypothesisthat the more gas a crude has in solutionthe more of its heavyendshave come out of solution,plating out on the reservoir rock. It may be noted in this respectthat Salathiel'sstrongly oil-wet film depositionon quartzand porous rocks from a mixture of evacuatedcrude oil and heptanewas also probably the result of a deasphaltingprocess. Despite uncertaintyas to the causesof reservoir wettability, much evidencehas been presentedin recentyearsto suggestthat many oil reservoirsare not stronglywater-wet.In particular,there are the many brine/crudeoil contact-anglemeasurements of Treiber et aI.62 and the conclusionsof Salathielwith regard to the apparentwetting characteristicsof the Woodbine reservoirin the East Texas Field. Nuttinga3as early as 1934indicatedthat some reservoir rocks are oil-wet. Leach et al.aadescribeda reservoir believed to be oil-wet. Munganasstudied fresh carefully preservedcores from a reservoir and concluded that the formationwas oil-wet. Schmida6hasshownthat stronglywater-wetcoresbecamelesswaterwet when equilibratedwith some crudes.Kusakov et al.a1studiedthe thicknessof a water film left on a quartz surfaceunder crude oil drops and found that for two of the crudes, the film will rupture, bringing the crude oil into direct contactwith the quartz surface;the surface can then be describedas water-wet at some spots and oil-wet at others. Also, Craigas suggestedthat most formations are of intermediatewettability with no strong preferencefor eitheroil or water. There is recentevidenceto suggestthat water may not alwayscompletely wet reservoirrock in gas-waterflow following solvent injection. Soil scientistsconcerned with airlwater/soilsystemshave reportedsituationsin which thereis incompletewetting by the aqueousphase.ae Authors such as Holbrook and Bernard,soand Fatt and Klikoffs' assumedthat wetting of reservoir solids was heterogeneousrather than uniform. Holbrook and Bernard measured fractional wettability by dye adsorption. Brown and Fatts2defined fractional wettability as the fraction of surface area in contact with water. This may not be a constantvalue since the water and oil saturationschangeas a reservoiris produced.Schmida6showedby means of capillary pressure-saturation data, that in preservedcores the fine pores were water-wet while 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 conIwankow,s3Brown and Fatt,s2 dition in oil sandshas also been suggestedby Salathiel,3e Gimaludinov,s4and McGhee and Crocker.ssSeveralof theseinvestigatorshave suggested that the wetting phasecompletely occupiesthe smaller pores of a reservoir rock in addition to the rock surface of the larger pores, while the nonwetting phaseprimarily occupies the insular regions of the larger pores. Evidence suggeststhat some oil reservoirs are partly preferentiallywater-wet and partly preferentiallyoil-wet. Such a condition could arise if someporesare lined with one type of mineral and other poresare lined with anothermineral. The existence of different minerals in porous media can create differences in surface chemistry of the grains, so all grain surfacesdo not have the sameaffinity for surfaceactive compounds. For instance, a tertiary sand reservoir in Alaska contains quartz and siderite minerals which are strongly water-wet and calcite which is strongly oil-wet. The overall

58

Relat iv e P ermeabi I i ty of Petro leum Re'rervoirs

necedi-ng l^later di spLaced by oi1

Water Static

conditrlon

displacing

oil

|st

ti.

Advancmg

tubes. FIGURE 15. Advancing and recedingcontactanglesin capillary

siderite surfacesin rock system is water-wet, probably due to the presenceof quartz and channelsof some flow the in gypsum or anhydrite of presence the main flow channels.The a strongly watercreate to found are minerals These carbonaterock may alter its wettability. conditions' reservoir under oil-wet probably are rocks wet system, while many carbonate in the present are they when oil-wet surface a render to Heavy metal sulfides are known media' flow channelsof Porous is covered by a Wagner and Leachs6stated that in some oil reservoirsthe rock surface would be preferentially firmly attachedbituminousor other organiccoating. Such surfaces Boneau oil-wet in the presenceof oil and water, regardlessof oil and water composition. is reservoir due to and Clampitt.tt-reported that the oil-wet characterof the North Burbank surface' a coating of chamositeclay which coversapproximately77o of the quartz

VI. DETERMINATIONOF WETTABILITY qualitatively' The wettabilityof a rock can be eitherevaluatedexperimentallyor estimated However' laboThere is no satisfactorymethod to determinein si/a reservoirwettability. wettabilityhasbeenusedto evaluatein situ wettability.Many of the widely ratory-measured rock or the used experimentalmethodsof wettability evaluationutilize either the reservoir be related reservoirfluids, but not both. Therefore,a laboratorywettabilityevaluationshould to actual reservoirconditionsusing a greatdeal of caution. {d I I I t

A. Contact Angle Method has received The contactangle methodis usedby a numberof laboratories;the technique measurement' considerableattentionin the literatureas a quantitativemethodof wettability restingon The method consistsof measuringthe contactangle 0 that a drop of pure liquid in immersed a smooth, flat, incompressible,nonporous,homogeneoussolid forms when surface solid the anotherfluid. In most iractical situations,the contactangleformedbetween than a single and the water-oil interface is found to exhibit two limiting values rather contactwith into brought equilibrium value. The value of the contactangle when water is "advancingcontactangle"' oil on a solid surfacepreviouslyin contactwith oil is calledthe into contact with water on a solid surface The value of contact angle when oil is brought "receding contactangle". previouslyin contactwith water is called the in a capillary Figure l5 shows a comparisonof advancingand recedingcontactangles to as referred is tube. The fact that advancing and receding contact angles are not equal roughness, and contactangle hysteresisand it is usually attributedto surfaceheterogeniety As the as well as the presenceof surface-activematerialsssand rate of fluid movement. the provided increase, surfaceroughnessof a rock increases,the contactangle will further the if however, contact angle measuredon the smooth surfaceof the rock is above 90o; roughness surface in contactanglemeasuredon a smoothsurfaceis lessthan90o,the increase to increasein would further decreasethe angle. The smoothsurfacecontactangle is found l80o contact the 0 to of advancingand decreasein receding,on the rough surfaceover most angle range.tn solid-fluid Surface-activematerialsin the fluids may causeadsorptionprocessesat the smooth, a with even interfaceswhich give rise to appreciablecontact angle hysteresis hysangle contact homogeneoussolid. Motion of thl three-phaseline of contactincreases teresisas the rate of movementincreases'

rlcf

thc cr rllu.tr rhri L.ICr trtr

F

rnltrel t\SUf rjren ttrnt

Ttx rtxtr

t

rc.r-[r rnsk Rc irr I a(rr\ txl th ctuLJ cn(!u fTBa\r a nxj

7t'

59 plate

It4ineral Flat ter

/or,

Mineral B .urlrces in lk'1. rrf some Dn. lr \\ ater-

/

Flat

plate

#2

F I G U R E 1 6 . S c h e m a t r cm e a s u r e m e notf c o n t a c ta n g l e s . 5 o

r ..,::Jrtitlns. JB.;::l rn the

't20 . "

Oil-wet

Equilibrium tContact Angle

a

o o o

irri t:.'J b' a ;r':. :lntially br'' [JtrneaU f,r:: . duc to

#1

e

3 8 0 !

o o,

lar 8 E 4 0 o

Water-wet

o

o

1u.:.ri.rtirely. $.'. Jr. laboD t r h. r r i d e l y J : , ' .k o r t h e ll.: ^.' rclated

ll .: ..rpillary BI.::Jd tO AS d r '.rrhness, Jrt'::: .\s the pr, ''. iJc'd the lc',.':. if the fc :,'Lrghness D irircase in | \r I CrtntaCt ts .rrlrd-fluid h .r :t'llOOth, g t . 1 n c l eh y s -

0

20

40

60

Time (hours)

F I G U R EI 7

Influenceof aging on laboratory-measured contactangle.18

Advancingand recedingcontactanglescan be shownin a capillarytube for oil displacing water(recedingangle)and waterdisplacingoil (advancingangle).The procedureto determine the contact angle using a contact angle cell is describedby Wagner and Leachs6 and is illustratedschematicallyby Figure 16. Briefly, samplesof polishea,Rat platesof the mineral which is the main constituentof the reservoirrock are immersedin a sampleof formation water. A drop of reservoiroil is held betweenthe two flat samplesof the mineral and the two plates are moved horizontally so that the water advanceson the surfaceof the plate initially coveredby oil. The contactangleformedbetweenthe interfaceand the newly wateroccupied surfaceof the mineral is a measureof the water advancingcontact angle. The advancingcontactangle is the one that is customarilymeasuredand often reportedwithout being identified as advancing. The contact angle measuredin the laboraotryis often influencedby aging. It has been shown that contactangle increaseswith age of the oil-solid interfaceuntil an Lquilibrium is reached.This may require severaldays and it is one of the disadvantages of the contact angle method.a8Figure l7 shows this effect. Reliable wettability measurementrequiresthat both the reservoirrock and the fluids be free from contaminants.Uncontaminatedreservoirrocks can probably be obtainedif the coresare recoveredwith coring fluid containingno surface-active additivesor with reservoir oil that has not been exposedto oxygen. It has been reportedthat exposureof coresto air could result in alterationfrom water-wetto intermediatewettability. Uncontaminatedreservoir water and oil are easierto obtain than unalteredreservoirrock. Since contactangle measurement can be donewithout a sampleof (uncontaminated) reservoirrock, it hasbecome a widely used method for determiningwettability. Zismanmand other investigatorsstudiedcontactanglesunder controlledconditionsand

60

Relative Permeabilin of Petroleum Reservoirs

expressedvarying opinions concerningthe method's usefulness.Melrose and Brandner6r believed that the contact angles provides the only direct and clear specificationof the wettability property characteristicof a given oil-water-rocksystem.Treiber et a|.62found that the water-advancing contactanglescorrelatewell with other wettabilityindicatorswhile water-recedinganglesdo not. Brown and Fatts2questionedthe ability of the contactangle methodto provide a reliable scale for determiningwettability and suggestedthat the conceptof a contact angle representation of wettability of reservoir rock be abandonedand that this method be replaced with a "fractional surfacearea" method. Morrow et al.63also observedthat severalfactors cast doubt on the utility of the contact angle method. Mungans describedsome of the limitations and pitfalls of contactangle measurementas follows: l.

2.

3.

4.

5.

6. 7.

The mineral chosenfor the contactangle measurementis the principal constituentof the reservoirrock. For the purposeof contactangle measurement,silica or quartz is usedto representa sandstone;calciteis usedto representa carbonateor reef reservoir. Laboratorymeasurementof contactangle or mineral surfacesmay not simulatetrue reservoircontactangle. The contact angle at the water/oil displacementfront is "advancing" while at the leading edge of the oil bank it is "receding". Thesevalues sometimesdiffer by as much as 50o. This variationcan be on the sameorder of magnitudeas the laboratorymeasuredcontactangle. Contactangle measurementshouldbe done when the solid surfaceand a fluid remain in contact for an adequatetime before the secondfluid is introducedover the surface. This is referredto as pre-equilibriumtime and it is of different length for each crude oil-water system.Without adequatepre-equilibriumtime, a stablecontactangle is not reached.In somecasesit hasbeenreportedthat a stablecontactangleis neverobtained if the solid surfacecomes into contactwith some types of crude oils. Contact angle measurementis frequentlytime consuming. Contact angle measurementshould be performedwith actual reservoirfluids, since they are in equilibrium and solubility effectsare negligible;otherwise,the fluids must be equilibratedwith one anotherso that the solubility effectsbecomenegligible. Contactanglemeasurement preferablyshouldbe donewith bottom-holefluid samples; however,becauseof the time and expensesinvolved, flow line samplesare often used. Fluid samplestaken from the storageor treating facilities are not reliable, due to the possibleaccumulationof asphaltenes. When producedwater is not available,synthetic brine is commonly used. Contact angle measurements should be made under controlledconditionsso that the oxidation of crude oil can be prevented. Contact angle measurementrequiresextremecare to assurecleanlinessand inertness of the apparatus.

B. Imbibition Method An imbibition test is a reliabletechniqueof wettability determinationprovidedunaltered reservoir fluids are available. The method consistsof the measurementof rate of flow of a wetting fluid spontaneouslyimbibed into a core and replacinga nonwettingfluid by the action of capillary forces alone. Imbibition testsmay be performedat standardconditionsor at reservoirconditions.Figures 18, 19, and 20 illustrateequipmentthat is usedfor conductingthe testsat ambientconditions. The imbibition test at standardconditionsmay be performedas follows:

I l a

t

a

L l.

A cylindrical plug of reservoirrock I to I tlrin. in diameteris cut with water as a coolantin the cuttingprocess.

T I G

6l I Brrndnero' I t t r ) r 1( ) i t h e 'r iound I rl i c u i . ' ! - .s h i l e &' .r rcliable an i lc repreir* rcplaced r c:.rl t'actors g)r:r. of the

Capillary Tube

th:,; at the dr::lr br as i lrnl )rJtofyflurJ rcmain lhc.urtace. f cJLh crude ,anilc is not tcr,,btained

M e t a l l i cS a m p l e H o l d e r

b n : . , .i l n g l e llu,.:.. stIlC€ I il-:J. ntUst

Tef lon

glrirl.le. Urd..lnlples; ! ( ' | : l c nU s e d . i. Juc to the lG. .r nthetic F I G U R E 1 8 . I m b i b i t i o nc e l l .

D -,, ihat the lhj rncrtness

hJ unaltered Ol :l()\\ Of a fl u r J b1 th e bn. Figures I c , ''nditi o n s.

I r,i.rtc'f ilS &

2. 3. 4. 5.

6. 7.

The sample is placed under water in a beaker and evacuatedto remove trapped gas. The sampleis flushed with water to reducethe oil saturationto residuallevel. The core plug is placedin an imbibition cell underoil and oil imbibition is monitored. The drained water is measured;it is equal to the amount of imbibed oil. Sufficient time should be allowed for the systemto reachequilibrium; this may take severaldays dependingon the permeability of the plug. The plug is then saturatedwith oil to reduce the remaining water to the ineducible level. The sampleis placedin an imbibition cell underwater and water imbibition is monitored by the amount of oil being drained. The fluid that imbibes into the sample (oil or water) is the wetting phase.

The imbibition test under reservoir conditions is more complex. Irreducible water saturation is establishedby flushing the core with live oil and the imbibition tests are made at reservoir pressureand temperature.

62

Relative Permeabilin of Petroleum Reservoirs

hr .rn {cF\ l

:

t

6 to Water Reservoir )7

l

()rl rn fTf3\ff

irtrm . Thc

t c . \ rrxlx.l .ufler F I G U R E 1 9 . I m b i b i t i o nc e l l

ll* rr Ct {m \ltlre

Accumulated

the rm

also rn scll a

ctal' lhe r tr rmhrht aofe s

causer \lur a(rfBa

*rth g le\t K

C. Br The refcrrc oi ttr flurds r hr ca,, br Ga made r

Rubber Stopper -----+

FIGURE 20.

Imbibition cell.

Amott6s developed a quantitative techniquefor defining the degreeof water-wetnessof cores. He expressedthe degreeof water wetnessby a water index, which he defined as the ratio of the volume of water spontaneouslyimbibed into a core to the total volume of oil displacedby a water drive (forced displacementof oil by water). Similarly, an oil index was defined at the ratio of the volume of oil spontaneously imbibed to total water displaced

Ftre\!ul

nrtlxr

D-cq

Jc*rr \rettah'

Jorn\ tl trrth a

63 by an oil drive (forceddisplacementof water by oil). Amott's test consistsof the following steps:

l 2. 3. 4. 5. 6.

Flush the reservoirsamplewith water to reducethe oil saturationto its residuallevel. Immersethe samplein water and evacuateto remove gas. Immersethe samplein kerosene(or reservoiroil) and measurethe volume of water displacedby imbibition of oil after 20 hr. Measure the volume of water displacedwhen the sampleis centrifugedunder oil. Immersethe samplein water and measurethe volume of oil displacedby water after 20 hr. Measure the volume of oil displacedwhen the sampleis centrifugedunder water.

Oil index is the ratio of the volume of fluid measuredin step 3 to the volume of fluid measuredin step 4. Water index is the ratio of fluid volume from step 5 to fluid volume from step 6. The preferentialwettabilityof a rock is determinedby the magnitudeof thesetwo indexes, i.e., strong wettability is indicatedby values approachingone and a weak preferencein indicatedby valuesapproachingzero. A water index of one indicatesa strongly water-wet surfacewhile an oil index of one indicatesa stronglyoil-wet surface.Valuesbetweenthese two extremesor a value nearzero for both ratioscover the rangeof intermediatewettability. Amott's testof wettabilityof porousmediareceivedhigh marksfrom Razaet aI.66,although Moore and Slobad,67 Bobek et aI.,68Killens et al.,6eand RichardsonTo have indicatedthat the imbibition rate cannot be entirely attributedto the wettability of the core, but that it is also influencedby rock porosity, permeability,pore structure,and pore size distribution,as well as viscosityand interfacialtensionof the fluids involved in the experiment.Donaldson et al.7' tried to eliminateextraneouseffectsfrom the wettabilitymeasurement by comparing the volumes of fluids imbibed into preservedreservoircores with the volumes of fluids imbibed in the same cores after extractionand resaturation.Although the use of the same core would appearto offer identicalpore size distributions,the changein fluid distributions causedby the cleaningprocessmay have offset the advantagegained. MunganT2reportedthe use of an imbibition test to evaluatethe wettability of native-state cores. Emery et a1.73used an imbibition test after incubationof cores for up to 1,000 hr with gas-saturated oil under pressure;water was the first phaseto contactthe rock in the test. Kyte et al.7adescribedimbibition testsconductedat reservoirtemperatureand pressure. C. Bureau of Mines Method The U.S. Bureauof Mines methodof wettabilitydeterminationof a porousrock, commonly referredto as the "Centrifuge Method", is basedon the assumptionthat an elementalarea of the internal surfaceof the porous medium is either wettableor nonwettableby one of the fluids involved. The problemis one of determiningthe fractionof the internalsurfacewetted by each fluid. A methodof measuringwettabilitybasedon the abovetheory was suggested by Gatenby and MarsdenTsand was later developedby Donaldson.TrThese investigators made use of the areasobtained from the drainageand imbibition cycles of the capillary pressurecurve to producea numericalrepresentation of wettability. The Bureau of Mines method is quite rapid and it can be employedwith reservoirfluids. ' u c t ncsso f hnc,i as the lu r nc of o i l n ,,rlindex !r J r. place d

D. Capillarimetric Method Johansenand DunningT6recognizedthe importanceof the liquid used in determining wettability of a rock-liquid-brinesystemand suggestedthe use of a capillarimeterwhich joins the two liquid phases,oil and water, through a small diameterglasscapillary tube, with a capillary pressureacrossthe interfacejoining the two phases.Adhesion tensionor

64

Relative Permeabilin of Petroleum Reservoirs

displacementenergy, was calculatedfrom the differencein height of the two liquids in the due to gravity. two armsof the capillarimeter,the differencein densities,and the acceleration forces with either an advancing or receding measuring interfacial The instrumentis capableof reservoir rock are the exclusion of as a factor of this method Major limitations interface. influencingwettability and lack of provision to preventoil from oxidizing. E. Fractional Surface Area Method This method, developedby Brown and Fatt,s2usesmixturesof untreatedsandand sand renderedoil-wet by organosilanevaporsto obtainwettingconditionsrangingfrom completely water-wetto completelyoil-wet. Wettability is representedby the fraction of solid surfacemade artificially oil-wet. Although use of the method to evaluatefield behavioris not in evidence,the conceptof a fractionally wet surfacehas been presentedin the work of other writers.3e F. Dye Adsorption Method This method,developedby Holbrook and Bernard,-50 is basedupon the ability of reservoir rock to adsorba dye suchas methyleneblue from aqueoussolution,while rock surfaceareas covered by contaminantsfrom the oil phase remain unaffected.The test is based on a comparisonof the adsorptioncapacity of the test samplewith that of an adjacentsample extractedby chloroform and methanol.This methodmakesassumptionssimilar to thoseof Brown and Fatts2in their "fractional surface area" method. G. Drop Test Method This method is often used to confirm rock wettability. The procedureinvolves placing drops of oil and water on the surfaceof a fresh break in the core. The fluid that imbibes is the wetting phasewhile the fluid that forms a ball and doesnot wet the surfaceis nonwetting. The drop test is a qualitativedeterminationand is sometimesmisleading. H. Methods of Bobek et al. Bobek et aI.68proposeda laboratorytestto ascertainpreferentialwettability in a qualitative fashion. The techniqueconsistsof determiningwhich fluid will displacethe other from a rock sampleby imbibition. The resultsof this imbibition test are comparedwith thoseof a referenceimbibition test on the samecore sampleafter it has been heatedto 400'F for 24 hr to remove any organic materialsand to make it more water-wet. The assignmentof qualitativewettability designationsis basedon the relativeamountsand ratesof imbibition in the two tests. In the same paper a method for estimatingthe wettability of unconsolidatedmaterial is discussed.A thin layer of the unconsolidatedsandis spreadon a microscopeslide. The oil content of the sand is increasedby adding a clear refined oil. Droplets of water are then placedon the surfaceof the sandgrains and the fluid movementis observed.If the sand is water-wet, the added water will displace oil from the surfacesof the sand grains and the oil will form spherical droplets, indicating that oil is the nonwetting phase. A similar procedureis used to test for oil wettability. I. Magnetic Relaxation Method for determiningthe portionsof A nuclearmagneticrelaxationtechniquewas suggesteds2 the rock surfacearea that are preferentiallywater-wetor oil-wet. A rock sample is first exposedto a strong magneticfield, then to a much weakerfield. The magneticrelaxation rate- that is, the rate at which the initially imposedmagnetismis lost - is then measured. In sandpackscontaining known mixtures of oil-wet and water-wet sand grains, a linear relationship was observedbetween the relaxation rate and the fraction of the surface area

that is a test petrolr routin

J. Res Mc( oil sat a nati\ found ones. be con the co scribe (See I prefen to the a rock fluid s curve

K. Pe The compa at con waterbetwe in the wettab be wei design

65 I l r q u r d si n t h e due to eravity. i n s t r Fr e c e d i n g x[ ar a factor

s.rrrJand sand D nl.\ )nlp l e te l y l ) , " 1 - r r e tA. l I . , l : . c p lo f a

il \ ,, 1rcse rvo i r \ . -.:i.rgc&feos i. ..:.cd on a lp. cnt :ample

0.5

.t

Fraction of Dri-Filmed Sano

FIGURE 21.

Interstitialwater saturationfor sand mixtures.sr

ilar i,' thtlse of

5 1 r l rg r p l a c i n g

h . r i r rrrb i b eis i . n , ' n \ cr t t i n g .

in r .iualitative ! ( ) i i. r i ro m a yril-.th,rseof a ) -lr r r l: t-or 24 l. . r snnte n t o f it,: rrrrhibition is hj rrr.rtcrial s lr Je Th e o i l la tcr lre th e n I t r n c : a n di s p u r n. a n d th e t' '\ similar

h' F'rtions of rnrplc is first ;tre rclaxation Fn nrcasured. lrn.. a linear ! \ urlacea re a

that is oil-wet. Though the authorsreportedno studiesusing naturalcores, they proposed a testingprocedure.Their techniquerequiresspecializedequipmentnot normally found in petroleumlaboratoriesand thereare no indicationsin the literaturethat the methodhasfound routine use. J. Residual Saturation Methods McGheeet al.,ssLorenz et al.7eand Rezniket al.80reporteda correlationbetweenresidual oil saturationand wettability. Treiber et al.62reportedthat the connatewater saturationin a native core can sometimesbe usedas an indicationof formationwetting preference.They found that oil-wet formation have much lower connatewater saturationsthan the water-wet ones. In addition, the connatewater saturationin a stronglyoil-wet reservoirwas found to be constantregardlessof the samplepermeability,while in reservoirsof other wettabilities the connatewater saturationdecreasedwith increasein permeability.Iwankow53also describedthe effect of heterogenoussandwettability in terms of a fraction of drifilmed sand. (See Figure 21.) Drifilm is a solution commonly used in the laboratoryto make sands preferentiallyoil-wet. Coley et al.8r were not successsfulin using the ratio of the wetting relationshipsas to the nonwettingresidual saturationfrom relative permeability-saturation preferential however, found a rock wettability indicator; they that the volume of mobile fluid shown by the spreadbetweenthe residualsaturationvaluesof a relativepermeability curve appearsto decreaseas the oil wettability increases. K. Permeability Method The determinationof wettability of a samplefrom permeabilitydata is accomplishedby comparingthe ratio of water permeabilityat residualoil saturationwith the oil permeability at connatewater saturation.If this ratio is less than 0.3, the sample is consideredto be water-wet,while a value near unity indicatesthat the sampleis oil-wet.82The relationship betweenabsolutepermeabilityand connatewater saturationhas been frequentlymentioned in the petroleum literature and the relationshipbetweenconnatewater saturationand rock wettability has been discussed.Rocks with low connatewater saturationare consideredto be weakly water-wetto oil-wet, while rocks with high connatewater saturationare normally designatedas water-wet.

Relative Permeability of P etroleum Reservoirs

(Water-wet)

ft rel h . al

rl

(r{{uF rlirr

Sw

tbr(f\ edh

TL

(Oit-wet)

crn&t trt lh tT er

dtc rn. It

h!

Fcrrx l*e

krel

trl rel r-cr. rltuot sbrrrr pre.lcf

Sw FIGURE 22. Schematicwettability effecrs on relative permeabilitycurves.

L. Connate Water-Permeability Method A correlationof absolutepermeabilityas a function of water saturationin corescut with oil-basemud hasbeenusedfor qualitativeidentificationof corewettability.6s Watersaturation is measuredin freshly cut coresand absolutepermeabilityis determinedafter extractionand drying. A plot of water saturationas a function of absolutepermeabilityto air is prepared. The curve will have a gentle slope over a large saturationinterval for water-wetsystems, while it will exhibit a nearlyverticalslopeover a narrowsaturationrangefor oil-wet systems. This techniqueis applicableprimarily to thick hydrocarbonreservoirswith sufficientvariation in permeability and water saturationso the required plot can be prepared. M. Relative Permeability Method For a given water saturation,the water relativepermeabilityof a water-wetrock is lower than that of a comparableoil-wet rock. For the systemsstudiedby Owens and Archern it was found that an increasein oil wetness(at constantwater saturation)producedan increase in k,* and a decreasein k,.,. Treiber et aI.62concludedthat water-wetconsolidatedporous media normally have a water relative permeabilitylessthan l5Vo at residualoil saturation, while oil-wet porous media show a 50Voor higher relative permeabilityto water at floodout. Craigasoffers the following heuristicguidelines,which are illustratedby Figure 22:

thc ctl {rtYrg th. g TtE tllg8e rehrr Lncr chang mlcrl{ rhrlc

\. R. Thc

dto I thc ru prefen

O. R. lfo of thc of thc hghcr gas-or fronr r prefen r-anqr

67 Water-wet S*i . k,* -- t[r.* k,* at S.,*

rr-. ;ut $ ith Rr..rluration tI.:. : l()n and i. l:cpared. Iei .'..tgms. lCl .\:tCfflS.

ln: ..lriation

I\-k r. lower | - \ : ' eh c ' r r ri t Ja n r n c r e a s e btc.i porous l..rturatioD, ler .rt tlood;ur-cll:

>20 to 25Va @ S*>507o <0.3

Oil-wet < l 1% a,usual l yl 07o @ S*<507o > 0.5, approachi ng 1.0

In a water-wetrock, residualoil globulesin the large flow channelsblock the easy flow of water and causea low water relativepermeability;however,the oil in an oil-wet system occupiessmaller flow channelsand coatsthe walls of the largerones, causinga minimum disturbanceto water flow and a higher water relativepermeability.ttThis is why an oil-wet reservoirwill waterfloodpoorly, with early water breakthrough,rapid increasein water cut, and high residualoil saturation. The water-oil relative permeabilityrelationshipof native-statecores under steady-state conditionsis one of the best indicatorsof the rock wettability preference.Keelan82pointed out that a sharpdrop in oil relativepermeabilityover a small saturationchangeaccompanied by a rapid rise in relative permeabilityto water, to a terminal value in excessof one third the initial oil relativepermeability,often indicatesoil wetness.Careful sampleexamination or cracked samplesyield relative is essentialin using this technique,for heterogeneous permeabilitydata similar to the data obtainedfrom oil-wet cores. Water relative permeabilitycurves in water-oil systemsshow good agreementwith the oil relativepermeabilitycurve obtainedduring gas-oilrelativepermeabilitytestsin a strongly water-wet core.62'63'84 This effect does not exist under any other wetting condition. In a strongly water-wet core, the water relative permeability curve of a water-oil system also shows good agreementwith the water relative permeabilityof a gas-watersystem in the presenceof residualoil saturation.This agreementwill occur, even though the directionof the changein saturationmay not be the samein the two systems.In the same manner, in stronglyoil-wet cores, the gas relativepermeabilityof a gas-watersystemis comparableto the gasrelativepermeabilityof a gas-watersystemin the presenceof residualoil saturation.8a The point of intersection of the water and oil relative permeability curves has been suggestedas an indication of rock wettability. Owens and Archerrr have shown that the relative permeability intersectionpoint moves toward higher values of water saturationand lower values of relative permeability in a water-oil system as the sample wettability is changedfrom oil-wet to water-wet. As illustratedby Figure 22, a relative permeability intersectionpoint on the left of 507o water saturationindicatesthat the system is oil-wet, while an intersectionto the right of this saturationsuggeststhat the system is water-wet. N. Relative Permeability Summation Method The summation of relative permeabilitiesto the water and oil phaseat fixed saturations also gives some insight into the immiscible flow processes.McCafferysenoted a trend in the minimum values of the sum of relative permeabilitiesof samplesaccordingto their preferential wettabilities. O. Relative Permeability Ratio Method If the ratio of displacing to displacedphaserelative permeability is plotted as a function of the displacing-phasesaturation,the shapeof the plot is related to preferentialwettability of the rock.66It has been shown that the water-oil relative permeability ratio shifts to a higher value as the rock becomesmore oil-wet; furthermore, a semilog plot of water-oil and gas-oil relative permeability indicatesthat the gas-oil relative permeabilityratio curve moves from under to over the water-oil relative permeability ratio curve as the rock becomes preferentially water-wet.sr The water-oil relative permeability ratio curves of rock with variousdegreesof intermediatewettability are found to be practicallythe samein the presence

68

Relative Permeabilin of Petroleum Reservoirs

of constantinitial water saturation.85 Imbibition water-oil relativepermeabilityratio curves in the absenceof initial water saturationsshow higher valuesof residualoil saturationas the cores become more oil-wet.8sSteady-staterelative permeabilitymeasurements should be usedfor determinationof wettability. Unsteady-state methodsmay not allow equilibrium to occur during the flow test; therefore,they may indicatemore oil wettnessthan actually exists. P. Waterflood Method Severalattemptsto find a single correlationof wettability with waterfloodoil recovery for different porous media have failed, even though the tests were carried out under a standardset of conditions.6sHowever, the waterflood performanceof a native-statecore under carefully controlled laboratory conditions has been used as an indication of rock preferentialwettability. It is found that in a strongly water-wetsystem,a large fraction of the oil is producedprior to water breakthroughand very little additionaloil is recovered after breakthrough.For the test to be reliable,an equilibriumwetting conditionmust prevail prior to the passageof the flood front through the core.

ni

.t i

;r<\i rItYl.S b

lg.t.

h. .ti It h.r. rcu-rt tcfnrtr

('ru

rult\ C [vc\f,r .rlltlrr tJJntrl"i r. ll\el

trtl tr I

\arr ln-hnt.

Q. Capillary Pressure Method Both displacementpressureand the ratio of drainageto imbibition displacementpressure have beenproposedas qualitativeindicatorsof preferentialwettabilityof porousmedia. An increasein displacementpressureor in the ratio of drainageto imbibition displacement pressuresignifies a tendencyof the core to becomemore oil-wet. The above techniqueis applicablewhen oil-water capillary tests are made on native-statecores. However, most capillary pressuretestsare either of the mercury injection or air-brinetype, which provide little information concerningwettability.8l R. Resistivity Index Method Formation resistivity obtainedfrom electric logs can be used as a qualitativetechnique for wettability identification. Resistivity index is defined as the ratio of true formation resistivityto resistivityof the formation when 1007asaturatedwith formationwater. A high value of resistivity index indicatesa low water saturationor a discontinuouswater phase, which characterizean oil-wet system.A knowledgeof the water saturationin the rock may yield sufficient information to make a judgementabout rock wettability. There is considerableuncertaintyconcerningthe natureof the wettability characteristics of reservoir rocks in situ. Tests of wettability made on cores taken from reservoirsare not necessarilyvalid indicatorsof subsurfaceconditions, since the coring processitself may alter wettability. Cores cut in oil-base mud, for example, are often renderedentirely or partially preferentiallyoil-wet. Thereforespecialprecautionsmust be observedduring both coring and transportingto minimize the danger of altering the true wettability of the rock. In the absenceof convincingevidenceto the contrary(for example,abnormallyhigh resistivity index) the assumptionof preferentialwater wettability has been frequentlyused.86

VII. FACTORSINFLUENCINGWETTABILITYEVALUATION It has been suggestedthat four factors may influence the results of experimental determination of rock wettability.8TOne of thesefactors is core recoveryand preservation.In the processof core recovery from a reservoir, heavy hydrocarboncomponentsof crude oil become less soluble as the oil loses its associatedsolution gas (as a result of pressure reduction).The heavy hydrocarboncomponentscan precipitateon the rock grains, leading to less water-wetor even oil-wet core behavior.8s-m Drilling fluid containingsurface-active materialsmay drasticallychangea core wettability, but it has been shown that bentonite

rnr'tlf r atc ir,t srrnn$ Ctn ctx|\lrtr rlJny

rxx re! xrphrs Fa-t arrl pn

frrft .t the chr that tlx oi hear

[rR\€n Jenn abrlrtr randst< rn nett rrredn-

rr|3'thaf tr etnbr al."'bt Teras

and as: of this

A rh Stainle In sprt nit h t e lidated heharrr grain s \l'el .

-{ '

69 | [.rllr) CUfVeS

s J t u r . r t i ( ) na s Ilc'r t r should r cqurlibrium lh.rn .rctually

ttl,

rc'COVefY

r\ut undef a tc-.t.ttd COfe It, ':t tri rock F ::.rjtionof li,t)\'€fed

lr

I rIl ... i prc'vail

llr'!ll

[trt'Ssgfe

l . : : t c t l i a .A n dr-ll.rcement tc. hniqueis nrc.cr. most rh:.:: provide

n c : e; h n i q u e 1g :, 'rtttation rat.r .\ high r * . r t d Fp h a s e , lhc r,rc[ 63y har.retcristics [tr]!'r afe nOt gr ll.cli ffio!

d c n t r r e l yo r I J .r rrn gb o th I t,l thc- rOCk.

I ;lr

nrghre si su.cd.E6

IO\ mc'ntal deter5('r\ ation. In I trt crude oil I trt Plattuaa t?rn:. leading Urtrcr'-active hat hcntonite

and carboxymethylcellulosehave no observableeffect on rock wettability when they are used in the coring fluid.7aWeatheringand contaminationof coresduring preservationand storageare alsofound to influencecore wettabilities.er Stronglywater-wetcoresmay become less water-wetas a result of air exposure,while cores with intermediatewettability show no significantchange.6sOil-wet cores also may becomewater-wetupon exposureto air.72 It has been suggestedthat alterationdue to air exposurecan be minimized and native-state wettabilitycan be restoredby incubationof the core in reservoiroil for two weeksat reservoir temperature.2s Crude oil is probably the best coring fluid for preservingwettability and maintaining native interstitialwater saturation;e2 however,useof the wetting phaseas a coring fluid may preservethe rock propertiesproperly.2sNaCl brine containingCaCO, powder with no other additivesis considereda good fluid for cutting cores.e3Care must be taken to avoid contaminationof the coring fluid with air, sediments,etc. The useof crudeoil as a coring fluid is likely to introducea fire hazardinto the coring operation,especiallyif a high API gravity oil is used. Native state wettability of cores is obviously the most desirablecondition, and the best techniquefor obtainingcores in this condition is by employing a pressurecore barrel. The method allows coresto be cut and retrievedat reservoirpressure.At the surface,the cores are frozen, cut into sections,and sentto the laboratory.ea Although early attemptsat pressure coring met with limited success,recentdevelopments indicatea success ratio of 80 to90Va. Cores that have been cleaned, dried, and restoredto some saturationand wettability condition are known as "restored state" cores.a8This techniquehas been employed for many yearsand it is an establishedprocedure;unfortunately,quite frequently,the coresare not restoredto their native stateand the useof thesecoresinvalidatesresultsobtainedusing sophisticatedmeasurementtechniques.Put very simply, restoredstatecores are not. Factorsthat influencethe core wettability evaluationincludethe laboratorycore cleaning and preparationprocedure.Mungane2statesthat the cleaning procedureneither changesthe pore size distributionnor the quantity of kaolinite and illite in the core. He concludesthat the changein fluid flow behavioris basicallydue to wettabilityalteration.Salathiel3e reasons that the extractionof a core with strongsolventsdissolvesthe stronglyoil-wet surfacecoating of heavy organic moleculesand therebyaltersfluid displacementbehaviorof many fresh or preservedcores, as shown in Figure 23.28 Jenning'se5 resultsshow a small but measurablechangein the water-oil relative permeability ratio curve after toluene extractionof a variety of core samplesfrom oil-bearing sandstones and limestones.The changesare not thoughtto be causedby significantchanges in wettability. The resultsof Richardsonet al.ershow a higherrateof imbibition and a lower ineducible water saturationwhen East Texas Woodbinecoresare extractedby hexaneand methanol.Morgan and Gordon's28resultsshow that the effect of cleaningprocedureon core wettability may be minimized if reservoirfluids are used as testing fluids. Richardsonet al.et believe a changein fluid flow behavioroccursas a result of repeatedflooding of East Texas Woodbine cores. This change appearsas a decreasein irreducible water saturation and as an increasein residualoil saturation.Furtherwork is necessaryfor betterunderstanding of this problem. A third categoryof factorsthat influencecorewettabilityevaluationis the testingcondition. Stainlesssteel wettability can be alteredby pressureincreasein a methane-watersystem.e6 In spite of decreasein interfacialforces, the oil-water-solidsystembecamemore water-wet with temperatureincreasesin a clean unconsolidatedHouston sand and a natural unconsolidated California oil sand.eTOne explanationfor the effect of temperatureon displacement behavior is that polar componentsof the crude oil may not be adsorbedas readily on the grain surfacesof a rock at elevatedtemperature,so the flow behavior becomesmore waterrA'g1.7+'9a

Relative Permeabilitv of Petroleum Reservoirs

air p e r m e a b i f i t y : -2 2 9 m d .

4

6

o Fresh A Extracted

kto

0'5

j

kr*

-:i -'

sw il:|':?"il,,'t:Hiil:iJT11',:'

permeabi'itv data rrom thesame

A fourth category of factors that influence the core wettability evaluation is the type of fluid used in the test. Carbonatesare very sensitiveto nitrogeneoussurfactantcompounds containing sulfur and oxygen.arSandstonescontaininglarge percentagesof silica possess acid type surfaces.38'er Crude oil containingnormal paraffins are inert and inactive with regard to the surfacesof porous media, while naptheneand aromaticsare more active with porous surfaces.Heterocyclicsand asphaltenescontainingoxygen, nitrogen, sulfur, and metallic atomsare active with regardto the acid or basicsites.Reisbergand Doschelo have indicated that different crude oils probably have different proportions of these compounds of surfaces. which are believedto be responsiblefor the wettability characteristics with for oil increases increasing concentragas and that saturation decreases The critical polar the concentration of compounds polar increasing substances.ee Furthermore, tions of in oil causesthe cumulativewater productionto increaseand cumulativeoil productionto decreasein laboratorytests. Oxidation of crude oil frequentlyappearsto modify the wettabilityof porousmedia. The degreeof modification dependson the amount of oxidizablepolar compoundsin contact Morgan and Gordon28and Cuiece8have with air and wettability may even be reversed.e8 on relativepermeability.Mungane2 handling fluids and laboratory investigatedthe effect of let it sit at reservoir temperaturefor 6 fluid and with reservoir saturatedan extracted core valueswere identicalto those permeability relative days. He discoveredthat the measured purified fluids in place of reservoirfluids a preserved he used of freshly cores; but when in Figure 24. indicated was developed, as core more water-wetcondition in the water alkalinityand hardness,ee The initial fluid saturationin a core,s salinityalteration,eo preferential wettability of a core. Wagner as well as the aging processe'can influencethe and Leach-'6have shown that the wettability of an oil- or intermediatelywet sample of sandstoneor carbonatecan be changedto a more water-wetcondition by the addition of chemicalssuch as hydrochloricacid, sodium hydroxide,and sodium chloride. They inves-

llgilc't

lreale pH vr el-fect Bra angle rep|()r and 1nxks amine t C H -l dtxlec; appror cause such a tou ard noled \r'ettab

7l

0.9

s tle trpe of I .,'lrrpttunds plr..r ptrssess i n . : . l r rc w i t h E .r.lrrcwith r...iilur. and Drsle I-:" have

Sw FIGURE 24.

o.7

Effect of fluid and laboratoryhandlingon relativepermeability."l

B .,'lllPt)UfldS lre' DS .,,ncentraJ .,\lllP()UDdS Jr' \l.le tlon to I i:rrJra. The

d. .r',eontact '' C u r ee h a v e l) \lungane2 pr.rturc for 6 It.Jl to those ;rtrlr tluids a hl d h.rrdness,ee ;rrrc \\'agner Et ..ilnple of r .rJ.lition of . I hcr inves-

tigatedthe influenceof water pH on wettability of a quartzsampleand useda n-octylamine treatedsyntheticoil to producean oil-wet quartz surface.Their resultsindicatedthat lower pH solutionstend to producewater-wetsurfacesunder controlledsalinity conditions.This effect is shown in Figure 25. Bradleyroohas shown that a basic 57oNaCl solutionspontaneously decreases the contact angleof oil-wet coresand as a resultincreasesthe amountof imbibition. Theseeffectswere reportedto be most pronouncedon coresof intermediatewettability.Morrow et al. ,63Wagner and Leach,s6and McCaffery and Munganrorhave shownthat wettabilityof typical reservoir rocks can be easily changedto any desireddegreeby adding polar compoundssuch as aminesor carboxylic acids. Bradleyrmfound that carboxylicacidssuch as stearicacid CH., (CHr)16COOH at concentrationsgreaterthan 10-6 moll( alteredthe wettabilityof a waterdodecane-calcite systemtoward more oil-wetnessand stearicacid with a concentrationof approximately5 x l0-3 mol/f causedstronglyoil-wet surfaces.He found that stearicacid causedno wettability alterationwhen quarlzsampleswere used. Bradley found that amines such as octadecylamineCH. (CHr),, NH, alter the wettability of both quartzand calcite toward oil wetness,especiallyat concentrations greaterthan 5 x l0-a mol/{. It should be noted that polar compoundswhich alter wettability of a given rock type may not alter the wettability of anotherrock type.

72

Relative Permeabilin of Petroleum Reservoirs

a C) O

o

O.O

a

25,000

c

50,OOO

ppm NaCl WATER PHASE

r30

o

uJ J

OIL_WET

z o F

z

90

(J

z z

70

o I

(r uJ F

3

50

W A T ER - W E T

6 WATER_PHASE PH

FIGURE 25.

Contactangle as a function of pH.so

VIII. WETTABILITY INFLUENCEON MULTIPHASEFLOW The microscopicdistribution of fluids in a porous medium is greatly influencedby the degree of rock preferentialwettability. The fluid distribution in virgin reservoirsunder strongly water-wet and strongly oil-wet conditionshas been describedby Pirson.ro2In a stronglywater-wetreservoir,most of the waterresidesin dead-endpores,in smallcapillaries, and on the grain surface.In strongly oil-wet reservoirs,water is in the centerof the large poresas discontinuousdroplets,while oil coatsthe surfacesof the grains and occupiesthe sm allerc apillar i e s . Under strongly water-wetconditionsthe effective permeabilityto the nonwettingphase at irreduciblewater saturationis approximatelyequal to the absolutepermeabilityof the rock. On the other hand, in strongly oil-wet systems,the effective permeabilityto oil at irreduciblewater saturationis greatlyreducedby the waterdropletsin the largerpores.Raza et aI.66statedthat in someoil-wet reservoirs,water occupiessomeof the finer poresand is trapped as droplets in the larger ones. Raza et al. analyzedthe displacementof oil by advancingwater and the trappingof the residualoil as shown in Figure 26. In strongly water-wetreservoirs,water traps oil in the larger poresas it advancesalong the walls of the pore, while in strongly oil-wet reservoirs,water moves in large pores and oil is trappedclose to the walls of the pores.66 The petroleumindustry has long recognizedthat the wettability of reservoirrock has an important effect on the multiphaseflow of oil, water, and gas through the reservoir.API Project 27 at the University of Michigan was initiated in l92l to study this problem. The

dtsrr m of rgsc

Thoau \r elurbl

ercnlla drrea: S(-hn trn ilou

icr hetr *r rm1 Strn favoral

increas have n

strongl

73

Oil-Wet Sand

FIGURE 26.

The trappingprocessof oil by advancingwater'n"

-

TEST ---

wet

1 waler

TEST 2

... ... TEST 3

wel

weter

oil

wet

\\ \\ \ E 5 0 j

I

oi1

f, 25

nr'eJ hr the \rrlfr

Undef

llt: r I phase bri::r ,ri the [lr i,' oil at p'rc.. Raza F'rc. and is It ,rt oil by Bn,Cr along F n()rcs and rrxk has an F:'rtrir. API Ntt',icrn.The

Brine

50

75

100

B R I N ES A T U R A T I O N

s.,n tlna Ir;prllaries, oi :nc large I...nrcs the

I

FIGURE 27.

Effect of wettability on flow behavior''r

dissymmetryof relative permeabilitycurvesis attributedlargely to the preferentialwettability As illustratedby Figure 27, Geffen et al.r2 and Donaldsonand of reservoir rock.te'es'ro3 Thomas'oahave shown the effect of fluid distributions brought about by rock preferential relationship.As the degreeof rock prefwettability on the relative permeability-saturation erential wettability for waier decreases,the oil relative permeabilityat a given saturation decreaseswhile the water relative permeabilityincqeases. Schneiderand Owenssarecognizedthe fact that rock type appearsto have less influence on flow relationshipsthan doesrock wetting preference.However, this may not be the case rocks or mixed wettability systems.Owens and Archerrr also confirmed for heterogeneous the importance of preferential wettability on multiphaseflow in porous media. have found that relative permeabilitybecomesprogressivelyless Some investigatorsno favorable to oil production as a rock becomesless water-wet. The residual oil saturation increasesas a rock becomeslesswater-wet.Othershave shownthat weakly water-wetcores have more favorable relative permeability curves and lower residual oil saturationsthan strongly water- or oil-wet rocks. Conceptually,this latter behaviorseemsreasonablesince

74

Relative Permeabilitv of Petroleum Reservoirs

S P L A C E OP H A S E S

4 t/

/l

Ni troqen Dodecane

displacrng or bio.tyl

Heptile, eti...

llitrogen

displacing

Water

Dioctyl

Ether

HepLile

displacing

displacing

up

to

49c

l08o Nrtrogen

Nitroqen

I3Io above

l38c

.8

Displaced

rt

phase

Saturation.

pV

Relative permeability for fluidpairswithvarious contacr

il:,yrT.. the capillary forces in strongly water-wetcores are strong. The oil may be bypassedand trappedin larger poresby the tendencyof a water-wetcore to imbibe water into the smaller capillaries.The bypassedoil in the large poresis then surroundedby water and is immobile exceptat very high pressuregradients.The saturationintervalfor two-phaseflow underthis condition is probably short. As the capillaryforcesare reducedby reductionin preferentialwater-wettabilityof a rock, the tendencytoward rapid imbibitional trappingof oil in large poresby movementof water throughsmallporesshouldalsodiminish. The zoneof two-phaseflow shouldbecomebroader and oil displacementto a lower residualsaturationshouldbe possible.If other factorsremain constant,higherflow ratesand lower interfacialtensionsareconduciveto higheroil recovery; theseare changesthat diminish the ratio of capillary forcesto viscousforces. StegemeierandJensen3T and McCafferyand Bennionr05 reportedthat wettabilityalterations over a relatively wide rangeproducea negligibleeffect on the relativepermeabilitycurve, as shown by Figure 28. However, other workers did not confirm this finding. Treiber et aI.62found that relatively small variationsin wettabilityproduceconsiderableeffectson the relative permeability curve. Figure 29 shows the effect of contact angleson relative perrneability curves for a Torpedo sandstone.

IX. EFFECTSOF SATURATIONHISTORY The relative perrneability-saturation relation is not a unique function of saturationfor a given core, but is subjectto hysteresisfor porous systemswith strong wetting properties.

That on \r Ina r alue incrc dunn ina value of dr a res u'oul Ge et al relat perm Inat pha-s relat from Th onlr bitior

75 100 \ .

"Nt \

'/

ATER

\ 10

I..

o l<

Contact

Ancrle ^o nro

. . . . . . 9 0 o . -^o " ^^o

.1

20

60

40

80

100

sw FIGURE 29.

;rp.:..Cdarld rthl .rttaller i. l :: r rn o b i l e r u r : J c rt h i s D , ' l . rr o c k . Rtll , r1 \\ atef Dcil hroader ltr r:. rCfflilitl ttl :l.,rr ery: ;.rltcrations ! r l r tr c u r v e , .I'rciber et Tc,t. tln the lr\ c perme-

FulttrJl tOf a

;p r ,r p, " -rti e s.

Imbibitionrelativepermeabilitywith variouscontactangles."l

That is, the relativepermeabilityof a porousmediumto a fluid at a given saturationdepends on whetherthat saturationis obtainedby approachingit from a higher value or a lower one. In a displacementprocesswhere the wetting-phasesaturationis approachedfrom a lower value, the resulting relative permeabilitycurve is referred to as an imbibition curve (an increasein the wetting phase).Examplesof imbibition processesare the injection of water during waterfloodingand coring a water-wetrock with a water-basemud. On the otherhand, in a displacementprocesswhere the wetting phasesaturationis approachedfrom a higher value, the resultingrelative permeabilitycurve is referredto as a drainagecurve. Examples of drainageprocessesare the displacementof oil by expansionduring primary depletionof a reservoirand the accumulationof hydrocarbonsin oil and gasreservoirs;anotherexample would be waterfloodingan oil-wet reservoir. et al.,r07Terwilligeret &1.,'n' andColey Geffenet al. ,r2Osobaet al.,r3Levine,'ooJosendal et al.8' describedthe hysteresisphenomenonand verified that both water-oil and gas-oil relativepermeabilityratio curvesas well as individual wetting and nonwettingphaie relative permeabilityof both sandstoneand carbonateformationsmay exhibit hysteresis.rr'22'ro7'roe In a two-phasesystem,hysteresisis moreprominentin relativepermeabilityto the nonwetting phasethan in relativepermeabilityto the wetting phase.ro'I roThe hysteresisin wetting-phase relativepermeabilityis believedto be very small and thus, sometimesdifficult to distinguish from normal experimentalerror, as indicatedin Figure 30. The drainagecurve shown in Figure 30 is a primary drainagecurve which is applicable only when drainageoccurs before imbibition. When a drainageprocessoccurs after imbibition, a secondarydrainagecurve exists, as shown in Figure 31.

Relative Permeabilin of Petroleum Reservoirs

D r a in a g e lmbibition

Sw (Water-Wet System) FIGURE30. Primarydrainage relativepermeability curve

Water

e W (water-wet system) D ,t

Secondary drainage curve:end-point flow

:t":rYXt Thesecurves describerelative permeabilitywhen the flow reversaloccursat one of the saturation end points. The effect of flow reversal at an intermediate saturation value is illustratedby Figure 32. As shown in Figures30 and 31, the water (wetting phase)relativepermeabilitycurve is essentiallythe samein stronglywater-wetrock for both drainageand imbibition processes.rr However, at a given saturation,the nonwettingphaserelativepermeabilityof a consolidated rock is usually less for an imbibition cycle than for a drainagecycle.t2.t3.22.to6 For an unconsolidatedrock, the nonwetting phaserelative permeabilityin an imbibtion cycle is usually greaterthan the corespondingnonwettingphaserelativepermeabilityin a drainage cycle. Naar et aL.22 reportedthat relativepermeabilityrelationshipsfor poorly consolidated formations tend to resemblethose for unconsolidatedformations. Figure 33 shows the imbibition and drainagerelative permeabilitiesof a consolidated rock. It can be seen that the residual nonwetting phase saturationis much greater for imbibition than for drainage.That is, the nonwettingphaseloses its mobility at a higher saturationin imbibition than it does in drainage.Figure 34 showsthat the imbibition cycle k.o may lie abovek.. on the drainagecycle for some systems.This relationshipprobably is not typical of petroleumreservoirs.

77

Secondary drainage

o .Y

Sw (water-wet system) FIGURE 32. reversal.

Secondarydrainase curve: intermediateflow

160

.=

140

-o $

o

E

120

L

o o. o

100 o

.9 =

o o oo

be ttr:l ,ri the

;

' . r l u ei s

80

60

o

Water

v

40

ir ..rne is Jt\ e.rCS.

||

rlrdated trr1., 'r ' Ftrl 3P ,n .) cle is a Jrarnage

20 Water 0

111.trlidated n.trltdated Erc.rtr'r for u .r higher itr,'n ct'cle pr,''b.rbll'is

0

2

0

4

0

6

0

80

100

Brine saturatiofi, o/o FIGURE 33.

Oil-water flow characteristics of a consolidatedrock.12

Relative Permeabilin of Petroleum Reservoirs 1.O

1.O

X*'n

o . 5 l<

j

9.5

ro

'

a

lmbibition

t.',

7

\ --+---r --,^,' .5

so Consolidated Sand

FIGURE 34.

Drainage

---

ta*- *rn .--Y \

l(

-

1.0

/*-) r o

.5

1.0

so Unconsolidated Glass Spheres

Relativepermeabilitycurves for consolidatedsandsand unconsolidated glassspheres.rr

The amountof trappedoil in water-wetporousmedia is given approximatelyby the area betweenthe drainageand imbibition oil relative permeabilitycurves.rr2It is believedthat the occurrenceof hysteresisis possiblyrelatedto the pore size distributionand cementation of a rock. As water is progressivelyimbibed into oil-filled pores of different sizes, oil is ejectedfrom them. The ejectionprocesscontinuesas long ascontinuousescapepathsthrough poresstill containingoil are available.Theseescapepathsappearto be lost at oil saturations which greatly exceedthose which occur at the onset of continuity of a nonwettingphase, (e.g.,gas) on the drainagecycle. Thus, the residualoil saturationwhich resultsfrom waterfloodinga water-wet rock is much greaterthan the critical gas saturationthat characterizesthe samerock. Apparentlyoil is trappedon the imbibition cycle. A similar behavior is observedif a preferentially water-wet rock containing free gas is waterflooded. The imbibition and drainagewetting-phaserelative permeabilitiesof a consolidatedor unconsolidatedrock are retracedunder a successionof imbibition and drainagecycles;in a reversalof the saturationchangefrom drainageto imbibition, a distinct path is traced by the nonwetting phase relative permeability curve (as shown in Figure 32) to a residual nonwettingphasesaturation.This path dependson the saturationestablishedin the drainage cycle. Also, the nonwettingphaserelativepermeabilitycurve in a drainagecycle following an imbibition cycle retracesthe imbibition curve until the previousmaximum nonwetting phasesaturationis reached.This effect is illustratedby Figure 35.22'13

u'ith an< reported a l . r rh a r conditio simulati pressure

X. EFFECTSOF OVERBURDENPRESSURE Wilsonila reportedthat a 5000 psi laboratorysimulationof overburdenpressureat reservoir temperaturereducesthe core effective permeabilitiesto oil and water by about the same extentas it reducesthe single-phasepermeabilityof that core. Consequently,the water and oil relative permeabilityof a naturalcore, under 5000 psi overburdenpressure,show only a moderatechange from the relative permeability measuredunder atmosphericconditions, as shown in Figure 36. Wilson alsopointedout that an overburdenpressurethat can produce over 5Voreduction in porosity of a core can also producea sufficiently large changein pore size distributionto affect the relative permeabilityof the core. In contrastto the work of Wilson, Fatt and Barrettrrsconcludedthat variation of rock overburdenpressuresin the range of 3000 psi does not produce any changeon gas relative permeabilityin a sandstonegas-oil system.Figure 37 shows the gas relative permeability

Wyck, and abs Dunlap" found n< specific ranging I granular gas satu bility me curvesto various p Botset

79

100 *\.

AIR

10

\\ b

a

l

I

o l.

p''.-'

' - air-brine system

l\

.1

br rhc area e l r c ic d t h a t gCntcntation Sl/d:. oil is Ith. thrttugh I . . r ll r a t i o n s Itrr:_p : hase. R.i,il. t'rom D Il.rt charhr 'lhar ior iJ Fl:.:.rlcdor cr.:c.: in a b : : . r e c db y ) .r rcridual hc' .lrainage b l , ' l l r r *i n g Jlr)li\\c'tting

.01 40

60

80

100

B r i n e s a t u r a t i o r ' ,V o FIGURE 35. stone.r2

Air flow behaviorin two-phasesystems,Nellie Bly sand-

with and without the laboratory simulation of overburdenpressure.Similar results were reportedby Thomasand Ward"6 for a gas-oil systemin a low permeabilityrock. Geffen et al.'2 have shown that the residualgas saturationin a liquid-gassystem,under atmospheric conditions,is similar to the resisdualgas saturationmeasuredunder a 5000 psi laboratory simulationof overburdenpressure.Merliss et al.r17concludedthat the effect of overburden pressureon relative permeabilitywas primarily due to changesin interfacialtension.

XI. EFFECTSOF POROSITYAND PERMEABILITY al rcrc'rt'oif tl lhc' same G r . . r t c ra n d r .hrr\\ OfllY cr)nJitions, gan prtlduce nlJ rn pore ir''n , 'l rock g:r. rclative rnncrbility

Wyckoff and Botset3as well as Leverettand Lewis8investigatedthe influencesof porosity and absolute permeability on relative permeability and found them to be insignificant. Dunlaprrsusedunconsolidated sandpackshavingpermeabilities of 3.0,4.5, and 8.0 D and found no indication that the relative permeability-saturation relationshipis a function of specific permeabilityof the sand. Stewart et al.rre found that variationsin permeabilities ranging from 8.5 to 300 mD and porositiesfrom I 5 to 22Voin limestonecores with intergranularporosity, causedrelative permeabilitycurves to shift up to a maximum of 2Voof gas saturation.These investigatorsemployeda solution gas drive, gas-oil relative permeability measurementtechniquein their study. They also reportedthe relative permeability curvesto shift up to a maximum of 47o of gas saturationwhen fracturedlimestonecoresof variousporositiesand permeabilitieswere employed. Botset2rfound that absolutepermeabilitiesrangingfrom 17 to 260D had negligibleeffects

80

Relative Permeabilin of Petroleum Reservoirs

100

a o) L

.Y

40

' o/o

s* FIGURE 36 system.rra

100

60

Effect of overburdenpressureon relative permeabilityof an oil-brine

1.0

.8

.t. Ef rnlsr rGF \lv

OBP = 0 psig OBP = 3OOO psig

rdF rtct

.6

I

rGd .{\rr."{rr

o) .Y

\r

.4

lr.crj rJ;ger r rlqh nl gl

.2

Iom

o

Felrr

0

20

60

40

80

100

so FIGURE 37.

Effectof overburdenpressure on gasrelativepermeability.rr5

.unef \ TL1 :lT\

L

FnrJ .( srfi :rrj f{fflr1

elht

r

81 1.0

o J

100 40

s* FIGURE 38.

Effect of absolutepermeabilityon relativepermeability.ro

on the gas-liquidrelativepermeability-saturation relationshipof a consolidatedNichols Buff sandstone.Botset'sresultswere in agreementwith the findingsof Leverett,awho usedsands with permeabilitiesranging from 1.04 to 6.80 D. Morgan and Gordon28conductedtestson four sandstonesamplesfrom a reservoirrock with permeabilitiesranging from 109 to 213 mD. No clear effect of permeabilityon oilwater relativepermeabilitycurveswas observed.Crowell et al.30studiedfour differentsands with absolutepermeabilitiesrangingfrom 3.0 to 8.0 mD and found no correlationbetween absolutepermeabilityand gas relativepermeabilityin a water-gassystemas shownin Figure 38. Keelanr20observedsatisfactorycorrelationsof sandstoneair permeabilitycorrectedfor slippage and the irreducible water saturationsas well as end-point relative permeability valuesof gas-watersystems.Leas et al.r2rnoteda correlationbetweenabsolutepermeability and gas relativepermeabilityin particularcases,but believedthis relationshipnot to be true in general. Felsenthalr22 tested300 sandstonecores and noted that the gas-oil relative permeability curves becameless steepas specific permeabilityincreased.This trend was also reported by McCord.'23In Felsenthal'spaper an effect of porosity on gas-oil relative permeability ratio was also noted. This effect was not generally discerniblein the study of relative permeabilitydatafor a givenreservoirbut becameapparentwhendatafor sandstone reservoirs of similar lithology but differing averageporosity were compared.For example,a definite trend was observedin a comparisonof argillaceousand/or calcareoussandstonesfrom I I reservoirsranging in averageporosity from l4 to 28Vo,indicatingthat for a given permeability, the gas-oilrelativepermeabilityratio curvesbecamelessfavorable,(i.e., k1k., increased)

82

RelativePermeabiliryof PetroleumReservoirs

as porosity increased.A similar trend was observedfor a group of clean sandstonesfrom five reservoirsrangingin porosityfrom 15 to2lTa.For a given porosityand permeability, gavemorefavorablegas-oilrelativepermeabilityratio curves comparativelycleansandstones or chert reservoirs.The leastfavorablegasthan argillaceousand/or calcareoussandstones oil relative permeabilityratio curves were for conglomerates,shaly sandstones,and sandstonescontainingcarbonateinclusions.Felsenthalthen classifiedsandstonesin three categories and found a correlation of gas-oil relative permeabilityratio for each class. The parametersused in the correlationwere porosity, permeability,and sandstonetype, which are all relatedto pore geometry. On the other hand, pore geometrymay be characterized by the pore size distribution and Felsenthalfound a correlationbetweengas-oil relative pl.-.uUitity ratio and pore sizedistribution.He found that the morefavorablegas-oilrelative with a pore sizedistributioncurve having permeabilityratio curveswere generallyassociated a sharppeak among the large pore sizes.

XII. EFFECTSOF TEMPERATURE indicatedthat ineducible water saturationincreasedwith inSeveralearly studiesr2a-r28 residualoil saturationdecreasedwith increasingtemperature; that creasingtemperatureand all of these studiesemployed a dynamic displacementprocess.Difficulties in evaluating these results include possible wettability changesdue to the core-cleaningprocedure,rro possiblechangesin absolutepermeability,and clay migration.t24't2'7't2'1 indicatedthat the by Lo and Munganr2e relativepermeabilitymeasurements Steady-state but were unafoils, white when using temperature-dependent were permeabilities relative with the results agrees finding this tetradecane; using when changes by temperature fected Sufi et ratio. viscosity to attributed been have results in variations Other Edmondson.'r. of to the due error significant have may results previous of the some that out pointed al.r30.r3r that suggested and temperatures elevated at permeabilities relative measuring in difficulty and difficulties measurement of a combination from result possibly effects t.*p..uture cores). short in (i.e., effects end phenomena, laboratory-scaling Miller and Ramey,32performeddynamic displacementexperimentsat elevatedtemperatures on unconsolidatedsand packs and a Berea core. Their resultsindicatedthat changes in temperaturedo not cause relative permeabilitychanges,but that changesin the flow capacityat elevatedtemperaturesare due to clay interactions,changein pore structure,etc. The only changethat they observedwas an increasein oil relativepermeabilityat irreducible water saturationand this parameteris relatively unimportantfor predictingtwo-phaseflow and Chen et al-r67 Counsilr33 behavior.In measuringsteam-waterrelativepermeabilities, effects. of temperature absence the noted also XIII. EFFECTS OF INTERFACIAL

TENSION AND DENSITY

The interfacialforces at fluid-fluid and fluid-solid interfacesare responsiblefor retention of residualsaturationin porousmedia. Wyckoff and Botset3and Leverett4describeda small but definite effect of interfacialtensionwithin the range of 27 to 72 dyne/cm on relative permeability. (See Figure 39.) Lefebvre du Preyro3also identified the interfacial tension of fluids in a consolidatedsampleas a factor influencing the relative permeability and residual saturationvalues. Crowell et al.30found that a reductionin interfacialtensionof a waterair system produced an increasein gas recovery and a decreasein residual gas saturation. discountedthe possibility that the interfacialtensionwithin the rangeof 27 to IV1uskatr3a influencerelative permeability.Owens and Archer" concludedthat intercan 72 dynelcm no influence on either the water-oil relative permeability of a water-wet has facial tension relative permeability of an oil-wet core. They found that water relative gas-oil the core or

tErrrr. .rrtxh \1, r 'afl

.rt

.rlc I IIr .I::ls

:

'rfftrrrY .1.1€ ,

rrnr.

L{r, :Elrrr,

4x.{r-r 'rl r r.. tatesc .rfrso.: .ffC

:

b !trLI1 \f r. Jt\-\t! f:r

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83 nJ-tr)t'lcsfrom

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| 1.-rrrrcability, i t r r . r t i oc u r v e s fur , 'rlrble gasllc.. .rnd sand-

points: o -5

/ o -- 24-g4 dynes/cm /

l i n e sI

\

o'/

o\oo \

I . h.rnrcterized !J- ,ril relative

o

/ o /

o r l \

o

\ .

l<

\

o /

./

\ o o

"

\

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is*.r.r:.1rr ith inIt :J:llP!'rature; s :r: c\aluating g p:, '.,cdure, l16

il .r:.'.1tcnlpera$ :: .'t changes lB. .:: thc flow i . : : . . . t L l f g .e t c . !r .:: :rrcducible lu, i.ltase flow ' r67 J ( : : . ' n c - ta l.

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d r : : . . r l t i c sa n d

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\ "

I r . . , : . ' . 1t h a t t h e t'..: .rcrc unaft r::: lhc rc-sults N : . r i i t r .S u f i e t lrr,': rluc ttl the . . . - - : c . t c dt h a t

/

\ o

i rn lhrL'ecaterh .lass. The D ci r p g . w h i c h

g.rr rrllrclative n ..r-\ c having

dyne/cm

*/,')**

1.0

0 "a w FIGURE 39.

Effect of interfacialtensionon relativepermeability.a

permeabilityof the water-wet core and oil relative permeabilityof the oil-wet core were coincident. Moore and Slobod6Treporteda reductionin waterfloodresidualoil saturationof a waterstatedthat drainagerelativepermewet core at lower valuesof interfacialtension.Pirsonr02 ability is independentof the interfacialtension,but imbibition relativepermeabilityis sensitive to interfacial tension. Bardon and Longeronr35found that a reduction in interfacial tension reducedoil relative permeabilityat constantgas saturationin an oil-gas drainage formation.(SeeFigure40.) The effectof liquid densityon relative cycle of the Fontainebleau permeabilityhas beenfound to be insignificant.-''r2

XIV. EFFECTSOF VISCOSITY SII\ ) l c : , , rr c t e n t i o n 5 r-rl.cda small ,cnr ()n relative L'r.rl tensionof [tr .rndresidual iitrn ..rfa water8J. \aturation. I rrnrr-'of 27 to uticJ that intertrl I \\ ater-wet I s ltcr relative

Leverettet al.a'8investigatedthe effect of viscosityvariationof an oil-water mixture on relativepermeabilityof artificiallycompactedsandswith 417oporosityand 3.2to 6.8 D of absolutepermeability.He found no systematicvariation in relativepermeabilitywhen the oil viscosity was varied from 0.31 cp (hexane)to 76.5 cp (lubricatingoil) and the water phaseviscosity was varied from 0.85 to 32.2 cp. Viscosity ratios employed in the study rangedfrom 0.051 to 90. The experimentsof Leverettet al. were performedunder steadystate flow at low pressuregradients.Figures41 and 42 show the effect of viscosity ratio variation on water and oil relative permeabilitycurves. Wyckoff and Botset3found that moderatevariationsin viscositiesof the fluid phasesin unconsolidatedsandpacks with permeabilitiesrangingfrom 3 .2 to 6.0 D failed to produce any changein the relative permeabilityvalues.In their experimenta mixture of water and carbondioxide was employedand water viscositywas adjustedbetween0.9 and 3.4 cp by addition of a susar solution to the water.

84

Relative Permeabilin of Petroleum Reservoirs 1.0

o = .0O1 mN/m

O

.

5

J

.5

t { -

tu:r \ lh

ss

gtlri

FIGURE 40.

Effect of low interfacialtensionson gas-oil relativepermeability.r15

n { \I .\G

1.0

&

rr irr rtx!

Fr l'\|uri

O MEgO. 1.80 a D 0.35 * 0.057

r{rlt

f.n .L iri itrn rlr I

3 .Y

Sr ;rtl lrhr ra irX

o

1.o sw

lu r(:tB crr0 I rrt

FIGURE 41. Effect of viscosity ratio (M) on water relative permeability.a

l l

*' aDl

Richardsonr36found that the water-oil relative permeability ratio is independentof fluid viscositywhere the oil viscosityvaried from I .8 to I 5 I cp (seeFigure43). Johnsonet al . r37 confirmed theseresults for displaced/displacing viscosity ratios up to 37. Leviner38found

lIllrl

llc d tn rts

85 1.0 o M = 9 0 o o v

o -Y

nnjcnt of fluid hn.rrnct al.r37 '' !\ lnc found

.35 .057

r\' %\o

s*

\

1.8

1.0

FIGURE 42. Effect of viscosity ratio (M) on oil relative permeability.a

that the relative permeabilityof a sandstonesamplewas independentof viscosity ratio in the rangeof 1.92 to 22.6. Craigr3ereportedthat the gas-oil relative permeabilityratio of a Nellie Bly sandstonesample with 824 mD permeabilityand 28.l%oporosity showed no significantvariation with oil viscositiesin the rangeof 1.4 to 125 cp. Resultsof this study are illustrated by Figure 44. Sandberget al.'aofound that oil and water relativepermeabilitiesof a uniformly saturated core are independentof the oil viscosity in the range of 0.398 to 1.683 cp. Donaldsonet al.'o' and Geffen et al.ta2alsoconcludedthat relativepermeabilityis independentof viscosity as long as the core wettability is preserved.Wilsonrrafound that a 5000 psi fluid pressure which causedkeroseneviscosityto increasefrom I .7 to2.l cp and waterviscosityto increase by 17odid not produceany significanteffect on water and oil relativepermeabilityvalues. Muskat et al.27reported that the effect of viscosity on relative permeability of an unconsolidatedsand was very small and within the limits of experimentalaccuracy. Krutter and Day'43used methaneand air as the nonwettingphasein a two-phasesystern of oil and gas. The gas was injectedinto cores saturatedwith oils with viscositiesranging from 2 to 100 cP. They found that the air relative permeability values were slightly less than those for methane. Saraf and Fattroapplied Darcy's law to each of the phasesof a multi-phasesystemand concludedthat relative permeability is independentof viscosity. The Saraf and Fatt equation is basedon the assumptionthat different phasesflow in different capillariesand do not come in contact with each other. Yuster,6however, concluded that relative permeability values for the systemshe studied were markedly influenced by variation in viscosity ratio, increasingwith an increaseof the ratio. This conclusionwas later supportedby the work of Morse et al.r44Odehr45expanded Yuster's work and concludedthat the nonwetting phaserelative permeability increaseswith an increasein viscosity ratio. He found that the magnitudeof the effect on relative perrneability decreaseswith increasein single-phaseperrneability. Odeh found that the deviation in nonwetting phaserelative permeability is increasedas the nonwetting phasesaturationis increased,with the deviationreachinga maximum at the nonwettingphaseresidualsaturation. He also concluded that the wetting-phaserelative permeability is not affected by variation in viscosityratios. Figure 45 showsthe effect of viscosityratio variationin the rangeof 0.5

86

Relative Permeabilin of Petroleum Reservoirs 60

o ;

4

1

3 j

q+6i.li'-qla+6

!

ncp,eri-nent

j

Waterflood using 151 cp. oil

A waterflood Kerosene

30

40

50

60

70

80

using

90

100

sw F I G U R E 4 3 . C o m p a r i s o n o f s t e a d y - s t a t er e s u l t s w i t h f l o o d i n g performance.r36

to 74.5 on water and oil relativepermeabilitycurves.Odeh statedthat the effect of viscosity ratio on relativepermeabilitycould be ignoredfor sampleswith single-phase permeabilities greaterthan lD. Yuster'sand Odeh'sresultshave beencriticizedby other investigators.ra6 Downie and CraneraT reportedthat oil viscosity could influencethe oil effective permeability of somerocks. Later, they qualified their statementby sayingthat once an increased relativepermeabilityis obtainedby employmentof high viscosityoil, it may not be lost by replacingthis oil with one of a lower viscosity.They explainedthis phenomenonqualitatively in terms of the movementof colloidal particlesat oil-water interfaces. Hassler et al.r found that lower gas relative permeabilityvalues were associatedwith lower oil viscosity in a Bradford sand. However, they expresseddoubt that the variationin relative permeabilitycould be describedby a single factor varying with oil viscosity. Pirsonro2stated that the importanceof the effect of viscosity ratio on the imbibition nonwettingphaserelative permeabilityis of second-ordermagnitude.Ehrlich and Cranera8 concludedthat the imbibition and drainagerelativepermeabilities,under a steadycondition of flow, are independentof viscosityratio. However,they found that the irreduciblewettingphasesaturationfollowing a steady-state drainage,when the interfacialeffect predominated

i

i\(: L'' t€1

\L rvn rsLell

rlr rc Pcr \ l\-\!

Px\c .rFri8 L'Ct:l! :lt

!j

srfvc [YF\x

.fuh {:-aJL

;w FR

87 1.0

0.1

o J

j

0 . 01

0 . 0 01

o.4 Q

"g FIGURE 44.

t r'i \ rrcosity r:: : r c. rh i l i ti e s E . : t j . r t t l r s l.1 6 f l: . i' [,L*rmear. tnarr'ased Ft \e ltlst by qu.rJrtatively f \ r . r t c dw i t h I \ . : n J t t o ni n lt-, '.ll) .

r rrnhibition a n . l (-ra n e ra 8 dr .,,ndition il. .' ri cttingrr '. i, 'lni n a te d

Relativepermeabilityratios for Nellie Bly sandstone.rre

over viscousand gravitationaleffects,decreases with an increasein the ratio of nonwetting to wetting-phaseviscosities. McCafferysereportedthat in stronglywettedsystems,the imbibition and drainagerelative permeabilitiesare independentof the viscous forces. He concludedthat even though the relative permeabilityto a phasemight be influencedby viscosity variation of that phase, the relative permeabilityratio is independentof viscosity. Perkinsrae concludedthat flow in a porousbody is governedby relativepermeabilityand viscosity ratio when the ratio of capillary pressureto the applied pressureis negligible. Pickell et al.r-toconcluded that only a large variation in viscous forces could have any rs3recognizedthat the appreciableeffect on residualoil saturation.Severalauthors4'67'rsr wetting and the nonwetting phaserelative permeabilitymight be significantly affectedby the ratio of capillary to viscous forces, ocos0/pv, where o representsinterfacialtension expressedas dynes per centimeter;0 representscontact angle; p representsviscosity expressedas cp; and v representsfluid velocity expressedas centimetersper second.Lefebvre du Prey'samade a systematicstudy of the effect of this ratio on relative permeabilityby simultaneouslyvarying the interfacialtension,viscosity,and velocity. He found that relative permeabilitydecreasesas the ratio ocos0/pv increases.He also concludedthat the relative permeabilitycurve is influencedby the viscosityratio when the wetting phaseis displaced

88

Relative Permeabilin of Petroleum Reservoirs

240

t 6 L

-Y

o

50

1OO

-aw bhri FIGURE 45.

E f f e c t o f v i s c o s i t yr a t i o ( M ) o n r e l a t i v ep e r m e a b i l i t y . r l s

by the nonwettingphase.Bardon and Longelsnr'3-s found that in some gas-oil systems,the drainagerelative permeabilityand residualoil saturationare stronglyaffectedby the p"vlct ratio. An assumptionthat the relativepermeabilityvaluesare independent of viscosityimplies that the system can be representedby a bundle of parallel, noninterconnecting capillary tubes,eachof which is filled with eitherthe wettingor the nonwettingphasealone.Thus, the nonwettingphaseflows throughthe largerchannelswhile the wettingphaseflows through the smaller capillaries. However, this model probably does not completely representthe conditionsin porousmedia.An alternativemodel is the simultaneous flow of two immiscible fluid phasesin larger capillaries. A flow picture more compatiblewith the presentknowledgeof fluid behavioris a combinationof the two modelsdescribedabove,with one dominatingover the other depending primarily on wettability. OdehTbelieved that the fluid phasesdid not flow in separate capillariesof porous media as Leverettpostulatedand further statedthat the wetting phase moves microscopicallyin a sort of sliding motion impartedto it by the shearforce caused by motion of the nonwettingphase.From this modelhe concludedthata decrease in interstitial wetting-phasesaturationcan be developedas a result of an increasein viscosity, thereby affectingthe relative permeabilityvalues. In view of the diverse opinions which have been expressedby various investigators concerningthe influence of viscosity on relative permeability, it seemsbest to conduct

ar\lt\

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rt

thc .u Sr Fffrx ?f..rl Yfi{rL

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89

Water

Present

---

at

Start

5%

-102 ------202

\r

olL

o J

/

WATER

-:=' 20

60

40

80

100

sw

DO FIGURE 46.

Effect of original water saturationon relativepermeability.''

laboratoryrelative permeabilityexperimentswith fluids which do not differ greatly in viscosity from the reservoirfluids. | .'..i:lllr.

the

XV. EFFECTS OF INITIAL WETTING-PHASE SATURATION

l t . . lhc p r',rr ! r ) - ' , .r r n p l i e s Irr*..rpillary ' . r . ' : . . ' Th u s. ll,' .i . thrtlugh fC:':,'.Cnt the ttt;:::lttirciblg /lrt: l\ J C()mF: .lcpcnding I

.:r .cpafate

I c : :r r r p h a se l , ' 1 .i ' C a U S e d I tr ::)lcrstitial hr:r . thereby tn'. g:[19111915

! :.' J()nduct

The amount of initial interstitialwater affectsthe oil-water relativepermeabilityvalues. Caudle et al.ra investigatedthis relationship.Figure 46 shows the effect of varying the amountof initial water saturationon water and oil relativepermeability.It can be seenthat not only the startingpoints, but also the shapeof the relativepermeabilitycurvesvary with the amountof initial interstitialwater.ro' found that the presenceof initial water saturationtendedto shift water-oilrelative SaremrT2 permeabilityratio curves toward the region of lower oil saturation.The differencein the residualoil saturationcausedby this shift was reportedto be about half the differencein initial water saturation.Thus, a lower residualoil saturationis obtainedat higher valuesof initial water saturation. noted that the maximum effect of initial water saturationon the Hendersonet al.3-t'r6s relativepermeabilitycurve was a shift of the entire curve laterallyapproximately4Tc along the saturationaxis, in a direction which increasedthe oil saturationfor a given pair of relativepermeabilityvalues.Craig indicatedthat up to 20Vainitial connatewater saturation in oil-wet coreshad no effect on oil-waterrelativepermeabilities.However, a definiteeffect was observedin water-wetcores. It is suggestedthat, exceptfor specialstudies,the amountof water presentat the startof a relativepermeabilitydeterminationshouldbe the irreduciblewatersaturationof the sample.

90

RelativePermeabilin of PetroleumReservoirs 100

I

*"I

I

^o

";/ :l

no

connate

3 l t

t/

t/

I

t/

o

I

j

\.

o) .:<

1

:l

O l

sl

ilr, !t

tl

;l I

{t /

.01 20

60

40

80

100

"eg

n+-.:-l .lrFrtr *:: !a.L:

fi .,I

FIGURE 47.

Effect of connatewater on relativepermeabilityratio.'7*

XVI. EFFECTS OF AN IMMOBILE

THIRD PHASE

Many hydrocarbonreservoirshave only two mobile fluid phases.The mobile phasesmay be gas and oil in the upper portion of the reservoirand water and oil in the lower portion. Thus, two-phaserelative permeabilitiesare sufficientto characterizefluid flow behaviorin thesereservoirs. Some investigatorssuggestthat the immobile water saturationmay be regardedas part of the rock, and gas and oil saturationsmay be given in terms of the hydrocarbonpore testedseveralnative-stateand cleanedcores,both water-wetand space.Owens et al.rss'r73 oil-wet. and found that an immobile connatewater saturationhad no measurableinfluence on the gas-oil relative permeability ratio in the majority of the casesthat were studied. concludedthat low water saturationsdid not appreciablyaffect the permeability CalhounrTa ratio, simply becausethe wateroccupiesspacewhich doesnot contributeto the flow capacity of the rock. Figure 47 showsthe effect of connatewater saturationon gas-oil permeability ratio. Stewartet al.'tt have also shown that in a limestonewith intergranularporosity, the effect of interstitialwater on externalgas or solutiongas drive gas-oilrelativepermeability ratio is negligible. Leas et al.'2' reporteda close agreementbetweenthe gas-oil relative permeabilityof a system at various values of interstitial water saturation.This agreementwas best in the

Il|:

a, lr i:

:f"r3{t}\|

={Trc

\r.E r F.&. ll:,u rf

:f1.i

i t

ll:u ft;n 6fkr a .f$f

l

9l

\\ \

15-25eaconnate

""ut"t";'

,r

\ \

/ \ \

o

water f

,

/

, / y/ot f

\

l< GAS

/

\

/

/

.z'\''-60

40

80

100

So-

.__ sg FIGURE 48. Efl'ect of the presence ol' connate water on relative permeabilities.l

i [-: .1\C\ntay nr i': |'trftitln. ) \:l.r\itlr

lrj.'.:

in

.t: poft

I;:hrrI'l at.:-\\ct

P0fe and

ble rrrlluc'nce Ic:,

.tudi€d.

1 * - .: : : lc.rb i l i ty ll.'.. . .1pra1,, J *- :: rrca b i l i ty F ' : , ' . i t r. t h e J*-:':rcability E * ^ r l r t ro f a t \ '. r in th e

equilibrium gas saturationregion. They concluded that the gas relative permeability is dependenton total liquid saturation.Other investigatorshave suggestedthat even though the immobile connatewater does not appreciablyaffect the relative permeabilityratio, the amount and distribution of the interstitial water may influence the relative permeability curve. Dunlap,r18 Leverett,aCaudleet al.,'" and McCaffery''ehave indicateda dependency on connatewater saturation.Figure 48 comparesthe permeability-saturation curves for oil and gas at l5 to 25Voconnatewater with the correspondingcurveswithout connatewater. Kyte et al.t7ostudieda wide rangeof corematerialsand fluid propertiesthatcould influence residualsaturation,to determinethe mechanismof oil displacementby water in a partially gas-saturatedporous system. They found that the initial gas saturationis related to the trappedgas saturation,which plays a beneficial role in reducing residual oil saturation. Mattax and ClotheirtTTconcludedthat the trappedgas saturationcould improve oil-water relative permeabilityvalues in consolidatedwater-wetsandstones.(SeeFigure 49.) Holmgrenand MorserT8 attributedthe oil recoveryimprovementof a samplein the presence of residualgas to one or more of the following factors: l. 2. 3. 4.

The changesin physicalcharacteristics of oil. The selectiveplugging action of the gas as indicatedby Kyte. Inclusionof mist in the free gas phase. The additionalsweepingor driving action of the free gas as indicatedby Leverett.a.s

Holmgrenand Morse concludedthat the changesin physicalcharacteristics of oil, within the pressurerange used for their experimentalwork, were not sufficient to account for the differencesin the residualoil saturationwhich were noted.They further statedthat a change in displacementmechanismwas the most importantcauseof the oil recoveryimprovement.

92

Relative Permeability of Petroleum Reservoirs 100 G A S S A T U R A T I O N ,% P V MOBILE

TRAPPED

o 5

10

9 't2

3 -Y \

1

o L

.o1 1.O

so lfrU.}?"a:.,,

water-oilrelative permeability ratioimprovement dueto

Schneiderand Owenssainvestigatedthe effect of trappedgas saturationin sandstoneand carbonaterocks and concludedthat the trappedgasaffectedwater relativepermeabilitymore than oil relative permeabilityin oil-wet rocks. These effects are illustratedin Figures 50 and 5l . They also concludedthat the trappedgas saturationloweredthe maximum value of oil relative permeability. Water relative permeabilitywas also lowered as a result of an increasein trappedgas saturation.Theseeffectsare illustratedby Figure 52. XVII. EFFECTS OF OTHER FACTORS The effects of displacementpressure,pressuregradient,and flow rate on the shapeof relative permeability curves have long been a controversialsubject in petroleum-related literature.Someauthorsbelievethat the effect of displacmentpressureand pressuregradient may be due to the changesimposedon viscosity,interfacialtension,and other fluid or rock properties.Others believe that the changesin relativepermeability,which appearto result from changesin displacmentpressureand pressuregradient,are actually due primarily to an "end effect" developedduring laboratorytests. End effect or boundary effect refers to a discontinuityin the capillary propertiesof a systemat the time of relative permeabilitymeasurement.In a stratumof permeablerock, the capillaryforcesact uniformly in all directions,and thusnegateeachother. In a laboratory sample,however,thereis a saturationdiscontinuityat the end of a sample.When the flowing phasesare dischargedinto an open region under atmosphericpressure,a net capillary force persistsin the sample;this force tendsto preventthe wetting phasefrom leavingthe sample. The accumulationof the wetting phaseat the outflow face of the samplecreatesa saturation gradientalong the sample which disturbsthe relative permeabilitymeasurements. For example, a large difference in saturationat the displacementfront causesa large capillary pressuregradient, which in turn causesthe water to advanceaheadof the flood front and to reducethe capillary pressuregradientin the measuredregion.The advancingwatercannot be producedwhen it first reachesthe outflow face of a core, becausethe pressurein the

rll€t

1

:rg

f

I C.I rcrlrr t[cr*tcrE( tgt ilrt

cl:r (.tr'sr a"r

t lilrar1

I

*tr rr &rr |f'c.!* fl

clit&

hAr It':' [.c'r€: ffn

93

100 \

*r* vs. S,

o

10 be o lz kro vs. So

Trapped Gas Sat. o 0 % .a 1 1.8%

.1

1'o s*-

i i l t l " l. 1 , ' n C a n d E.r^ .:l\ mofe

in I . . r rc: 5 0 lt u" . . rl u c'o f I r ; ' .. . . 1t r t 'a n

th. .hapc'of D k ' .. : :rcl r a te d s u : r - lr a d i e n t fl-:.1 trf fOCk p.r:

lrr rl.sult

I p::::rlrilr to Dft':i tc. tlf a lltc'.rhlc rock, l.r l.:hr131g1y n thc tlou'ing

lp r ll. r n tb rce ! t n c: a m p l s . I .r .lluration in l . F Ore xlu c . . r p i l l a ry ul tr,rnt and tl ulcr eannot lr.-.t'c in the

.-so FIGURE 50. carbonate).Ea

Effect of trapped gas saturation (oil wet Grayburg

waterjust insidethe core is lower than the pressurein the oil-filled spacearoundthe outflow face. This differencein pressureis equal to the capillary pressurefor the existingsaturation at the outflow face. Therefore,water accumulatesat the outflow end of the core, causing a reduction in the capillary pressure.The water will not be produceduntil the capillary pressureis overcome and the residual oil saturation(at the outflow face of the core) is reached.The calculation of relative permeability basedon the averagesaturationof the sampleproduceserroneousresultsin this case,sincethe relativepermeabilityvariesthroughout the core due to the saturationgradientcreatedby the wetting phaseaccumulationat the outflow face of the core. Owens et al.,r-5s Sandberget al.,to"Kyte and Rappoport,rs6 and Perkinsrae believethat the most convenientway of minimizing the boundaryeffect is the adjustmentof capillary forcesto insignificantvalues, as comparedto the viscousforces. This is usually done by a flow rate adjustment.However, the adjustedrate must be low enoughso the inertial forces do not disturb the laboratory measurement.It is suggestedthat the higher flow rate also increasesthe fluid dispersionat 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 disturbedby boundaryeffect, at a given rate. Another convenientway of minimizing the boundaryeffect at the outflow end of a core is to use a more viscousoil in a longercore.rs6 Leverett et al.a'8reported,then refuted, the influenceof flow rate upon relative permeability. They eventuallyattributedthe observeddeviationsin their resultsto an end effect.

Relative Permeability of Petroleum Reservoirs

100

be o kro vs. So

Y

T r a p p e dG a s S a t . o 0 % a 10.3% o 19.2%

s,rt_

So

(oilwetTensleep sandstone).Ea gassaturation oftrapped RGURE 51. Effect suchas that previouslydescribedby Hassler.rCrowell et al.3ofound that a 50-fold variation of injectionrate, within the limits of viscousflow of waterand gas,had no effect on residual gas saturationof an Arizona sandstone.Geffen et al.ra2also concludedthat, at reasonable flow rates, the effect of waterflooding rate on the efficiency of gas displacementwas found that relativepermeabilitywas negligible. Hendersonand Yuster3sand Morse et al.r-58 in all gas-liquidsystemsthat were studied.Wyckoff and Botset3also found rate-dependent when the two phaseswere that the gasand liquid relativepermeabilitieswere rate-dependent allowed to flow through the core under the samepressuregradient. Caudleet al.'a found that relativepermeabilitydecreasedwith increasein flow rate when one of the flowing phaseswas a gas. Labastieet ol.,'-tnhowever, investigatedthe effect of flow rate in a water-wetsandstoneand oil-wet carbonatecoresand concludedthat relative permeabilitieswere independentof flow rate exceptnear residualoil saturation.Sandberg O s o b ae t a l .,r3and Leas et al .r2rfoundthat drai nagerel aet al. , r aoRic ha rd s o ne t a l .,r-t7 tive permeability is independentof the flow rate as long as a saturationgradient is not introducedin the core by the inertial forces. Pirsonr02concludedthat relativepermeability is not rate-sensitivein drainageprocesses.Ehrlich and Crane'otexaminedthe effect of flow rate variation on steady-staterelative permeability and concluded that both imbibition and drainagerelative permeabilitywere independentof flow rate. Handy and Dattar62found that the imbibition relativepermeabilityvalueswere dependent on the imbibition procedures;that is, the relativepermeabilityvaluesunder free imbibition

r€f,C r :ntrrhri :srf& AJ

.J

Sr.fr{ tf !l"rr {

crtt

\ ili.rtr

trci rir

k

\f, r. .!{*r ; f.rJnl :.rtr.r ffi. trFrlul c:},rt tr* I: -t Jr*{ -::ftiq

95

100

kro vs

so

o\s o

Kr* us. S*

-Y

s*---_ -so Effectof trappedgas saturation (warerwet Tensleep il*"r,::.,i,]. It'

.: r.rriation

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were largerthan thoseundera controlledprocess.The differencebetweenfree and controlled imbibition was found to be smaller for more permeablesamples.Perkinsrae found that the residualoil saturationafter flooding was independentof the flooding rate and concluded that capillary forces controlled the microscopicfluid distribution in the core. Moore and Slobod6Treportedthat waterflood recoveryfrom a water-wetcore was practically independent of flooding rate. However,they observedthat a significantrecoveryincreasemay be obtained at extremely high rates. Hupplerr6sstated that the waterflood recovery from cores with significantheterogeneitywas sensitiveto flooding rate. Lefebvredu Preyrsaconcludedthat the relativepermeabilitywas a function of velocity (v), throughthe ratio (ocosO/pv),when the viscousforces predominate. Wyckoff and Botset,3Leverett,aand Hendersonet al.3-s'r6s studiedthe possibleeffectsof displacementpressureand pressuregradienton water-oil relative permeability.They concluded that the water and oil relative permeabilityvalueswere slightly influencedby these factors. Muskatr3aand Krutter and Day,'oohowever, reportedthat the gas and oil relative permeabilityvaluesof a consolidatedsandstonewere not affectedby changesin differential pressure.McCafferyrTeindicated that the drainagerelative permeabilityvalues were not influencedby the flow rateswhich result from apressuregradientin the rangeof 1.0 to 5.0 psi acrossa 12 in. core. Delclaudr60 also concludedthat relativepermeabilityis independent of displacementpressure.Pirson,ro2however, suggestedthat the relativepermeabilityin an imbibition cycle is sensitiveto pressuregradient.

96

Relative Permeabilin of Petroleum Reservoirs

Krutter andDayr66found that ultimaterecoveryincreases with increasingpressuregradient, althoughthe ratio of increasedrecoveryto increasedpressuregradientdecreases in the region of high pressuregradients.Brownell and Katzr68reportedthat an increasein pressuregradient decreasedthe residualsaturationtoward zero in the systemsthat were investigated.Geffen et al.ra2also confirmed that residual gas saturationwas a function of pressuregradient. Stegemeierand Jensen3T believedthat the residualwetting phasein a drainageprocesswas held in pendularrings interconnectedwith only thin wetting-phaselayers. They concluded that this residualwetting phasewas trappedby capillary forces and that a higher pressure gradientmight overcomethecapillarypressureandreducetheresidualwetting-phase saturation. Stewartet al.rreobservedthat the rateof pressure declinein a nonuniformlimestonemight influencethe gas-oilrelativepermeabilityratio when the solutiongasdisplacementtechnique for relative permeabilitymeasurementwas employed.Wall and Khuranar6e found the gas saturationdevelopedin a sand pack, at a given rate of pressuredecline, was a function of the meanparticlesize and probablya functionof permeability.They found that a finer grain sandpack gave rise to higher gas saturationsin the solutiongas displacementtechnique. Crowell et al.30studiedthe effect of coredimensionson laboratorymeasurement of relative permeability.They found that the residualgas saturationin water-gassystemswas almost independentof length of the core, within limits of the laboratory-scale models used. They also examinedcylindrical and rectangularsamples,and observedthat a 100-foldchangein the ratio of core length to core cross-sectional area of Berea and Boise sandstones did not alter the residualgas saturationof the samples.Moore and SlobodoTalso found that fluid recovery from water-wet cores was not affected by the sample length. Perkinstaeand McCafferyrTerecommendedthe use of longer cores, to reduceinfluenceof the end effect. RoserTostudied the effect of gas expansion,createdby the pressuregradientalong the sample, on gas-liquid flow characteristics.He concludedthat a necessarycondition for correct steady-state measurements of liquid-gasrelativepermeabilitywas the establishment of a uniform fluid saturationdistributionin the core. Osobaet al. '3 found that gasexpansion affected gas and oil relative permeability values in tests conductedat near-atmospheric pressure.Richardsonet al.,r-'7however, found that the effect of gas expansionon gas and oil relativepermeabilityvalueswas insignificantat the low pressureswhich were employed in their study. In the laboratorygas displacementmethodof relativepermeabilitymeasurement,a "stabilized zone'' tendsto form when the wetting liquid saturationis sufficientlyhigh to permit its readjustmentfaster than the imposed displacementby the externaldrive. The relative permeabilityvaluesobtainedprior to passageof the stabilizedzone arenot valid. Therefore, it is advantageousto reduce the range of saturationinfluencedby the stabilizedzone, to obtain valid measurements over as wide a saturationrangeas possible. It can be shown from the Buckley-Leverettequation that the saturationat which the stabilizedzone passesout of a systemis inverselyrelatedto the viscosityof the displaced liquid. This relationshipis based on an assumptionthat a true stabilizedzone forms in laboratorygas drives on short cores. It can also be shown that the length of the stabilized zone is inverselyrelatedto the injection rate or differentialpressure.It has been suggested that the stabilizedzone will be sufficiently small if the pressuredifferential is of such a magnitudethat a volume of gas approximatelyequal to one half the pore volume of the samplewould be producedin lessthan 60 sec. This flow rate insuresthat the portion of the core in which the capillary effects predominatewill be a negligibly small fraction of the total pore space. Loomis and CrowellrT' showedexperimentallythat the influence of the stabilizedzone fluid flow is much less marked with relatively viscousoil as the displaced phase. Botset2' investigatedthe effect of saturationpressureon gas-oil permeability values and concludedthat the saturationpressurehadnegligibleeffecton laboratoryrelativepermeability

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measurement.Stewart studiedthe effect of gas supersaturation on laboratorysolution gas displacementrelative permeabilitymeasurements. He indicatedthat even though very liitle supersaturation existsundermost field conditions,the effect may be significantfor laboratory testsconductedat high flow rates.He found that the gas-oil relativepermeabilityratio was generallyindependentof the degreeof supersaturation in rock with intergranularporosity. The influenceof dispersionon relative permeabilitywas studiedby Chilingarianet al.ee They concludedthat an increasein degreeof dispersionincreasedthe relativs permeability of the porousmediumto both the continuousanddiscontinuous phases.They alsoconcluded that the degreeof dispersionincreasedwith decreasinginterfacialtension and increasine time of coalescenceof dispersed-phase droplets.

REFERENCES l' Hassler, G. L., Rice, R. R., and Leeman, E. H., Investigationsof recovery on the oil from sandstones b y g a s - d r i v e ,T r a n s .A I M E , I 1 8 , l 1 6 , 1 9 3 6 . 2. Muskat, M. and Meres, M. W., phv-sics, j, 346. 1936. 3 ' W y c k o f f , R . D . a n d B o t s e t , H . G . , F l o w o f g a s l i q u i d m i x t u r e st h r o u g hs a n d s , p h y s i c s j,, 3 2 5 , 1936. 4 ' L e v e r e t t , M . C . , F l o w o f o i l - w a t e r m i x t u r e s t h r o u g h u n c o n s o l i d a t e d s a nTdr as n , s.AIME, l32, l49,lg3g. 5' Nowak, T. J. and Krueger, R. P., The effect of mud filtratesand mud particles upon the permeability of cores, Proceedingsof the Spring Apr Meeting, Los Angeres, 1955. 6' Yuster, S. T., TheoreticalConsiderationof MultiphaseFlow in ldealizedCapillary System, proceedings of the Third world Petroleumcongress, Hague, Netherlands,lgsl, (z\ 43i. 7. Odeh, A. S., Relative PermeabilityStudies,Mastersthesis,Universityof California, Los Angeles, 1953. 8' Leverett, M. C. and Lewis, W. 8., Steadyflow of gas-oil-watermixturesthrough unconsolidated sands, Trans. AIME, 142. 107. t94t. 9. Sarem, A. M., Three-phaserelativepermeabilitymeasurements by unsteady-state methods, Soc.pet. Eng. J.,9. t99. t966. l0' Saraf, D. N. and Fatt, I., Three-phaserelativepermeabilitymeasurementusing a N.M.R. techniquefor estimatingfluid sarurarion,Soc. pet. Eng. J., 9,235, lg6j. ll' Owens, W. W. and Archer, D. L., The effect of rock wettability on oil-water relative permeability relationships , Trans. AIME, 251, 8j3, lgjl. l2' Geffen, T. M., owens, W. W., Parrish, D. R., and Morse, R. A., Experimental investigationof factors affectinglaboratoryrelative permeabilitymeasurements, Trans. AIME, lg2, gg, lg5l. 13. Osoba, J. S., Richardson, J. G., Kerver, J. K., Hafford, J. A., and Blair, p. M., Laboratorymeasurementsof relative permeability,Trans. AIME, 192, 47, lg5l. 14' Caudle, B. H., Slobod, R. L., and Brownscombe, E. R., Further developments in the laboratory determinationof relative permeability,Trans. AIME, 192, 145, 1951. 15. Snell, R. W., Measurementof gas-phasesaturationin porousmedia, J. Inst.pet.45, (4Zg), lg5g. l6' Emmett, W. R., Beaver, K. W., and McCaleb, J. A., Little Buffalo basin Tensleep heterogeneityand i t s i n f l u e n c eo n d r i l l i n g a n d s e c o n d a r yr e c o v e r y J, . p e r . T e c h n o l . , 2 , 1 6 l . l g 7 l . l7' Donaldson, E. C. and Dean, G. W., Two- and Three-PhaseRelative Permeability Studies, report# 6g26, u.s. Departmentof the Interior, Bureauof Mines, Bartlesville,okla., 1966. 18. Arps, J. J. and Roberts, T. G., The effect of the relative permeabilityratio, the oil gravity, and the solutiongas-oil ratio on the primary recoveryfrom depletiontype reservoir,Trans.AIME,204,120, lg5l. 19' Bulnes, A. C. and Fittings, R. U., An introductory discussion of reservoir performance of limestone formations, Trans. AIME, 160, 179, 1945. 20' Stone, H.L., Probabilitymodel forestimating three-phase relativepermeability,Trans.AIME,24g,Zl4, t970. 2 1 . B o t s e t , H . G . , F l o w o f g a s l i q u i d m i x t u r e st h r o u g hc o n s o l i d a t e ds a n d , T r a n s . AIME,136,91 ,1940. 22' Naar, J., Wygal, R. J., and Henderson, J. H., Imbibition relativepermeability in unconsolidated porous media, Trans. AIME, 225. t3. t962. 23. Nind, T. E. w., Ed., Principles of oil production, McGraw Hill, New york. 1964. 24' Corey, A. T. and Rathjens, C.H., Effect of stratificationon relativepermeability,Trans. AIME,207, ( 3 s 8 ) ,6 9 , 1 9 5 6 . 25' Huppler, J. D., Numerical investigation of the effects of core heterogeneities on waterflood relative permeability,,Soc.Pet. Eng. J., 10, 381, 1970. ,

98

Relative Permeabilin of Petroleum Reservoirs

26. Johnson, C. E., Jr. and Sweeney,S. A., Quantitativemeasurementof flow heterogeneityin laboratory core samplesand its effect on fluid flow characteristics, papersPE 3610 presentedat the SpE 46th Annual M e e t i n g ,N e w O r l e a n s ,O c t o b e r3 , l g l l . 27. Muskat, M., Wyckoff, R. D., Botset, H. G., and Meres, M. W., Flow of gas-liquidmixturesthrough sands,Trans. AIME, 123, 69, 193i. 28. Morgan, T. J. and Gordon, D. T., Influenceof pore geometryon water-oilrelativepermeability J . pet. , T e c h n o l . ,l 1 9 9 , 4 0 7 . 1 9 7 0 . 29. Gorring, R. L., Multiphase Flow of Immiscible Fluids in porous M e d i a , P h . D . t h e s i s ,U n i v e r s i t yo f Michigan, Ann Arbor, 1962. 30. Crowell, D. C., Dean, G. W., and Loomis, A. G., Efficiency of gas
r-

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57 . Boneau, D. F. and Clampitt, R. L., A SurfactantSystemfor the Oil-Wet Sandstoneof the North Burbank U n i t , S y m p o s i u mo n I m p r o v e dO i l R e c o v e r y ,T u l s a , A r i z o n a , M a r c h , 1 9 7 6 . 58. Morrow, N., The effectsof surfaceroughnesson contactanglewith specialreferenceto petroleumrecovery, J . C a n . P e t . T e c h n o l . .1 0 . 4 2 , 1 9 7 5 . 59. McCaffery, F. G., The Effect of Wettability on Relative Permeabilityand Imbibition in Porous Media, P h . D . t h e s i s ,U n i v e r s i t yo f C a l g a r y ,A l b e r t a ,C a n a d a ,1 9 7 3 . , m . C h e m . S o c ' . ,4 3 , 6 0 . Z i s m a n , W . A . , C o n t a c tA n g l e W e t t a b i l i t ya n d A d h e s i o nA d v a n c e si n C h e m i s t r y A t. 1964. 61. Melrose, J. C. and Brandner, C. F., Role of capillaryforcesin determinationof microscopicdisplacement efTiciencyfbr oil recoveryby water flooding, J. Can. Pet. Technol., 10, 54, 1914. 62. Treiber, L. E., Archer, D. L., and Owens, W. W., A laboratoryevaluationof the wettability of fifty o i l p r o d u c i n gr e s e r v o i r sS, o t ' . P e t . E n g . J . , 1 2 ( 6 ) ,5 3 1 , 1 9 7 2 6 3 . M o r r o w , N . R . , C r a m , P . J . , a n d M c C a f f e r y , F . G . , D i s p l a c e m e nstt u d i e si n d o l o m i t ew i t h w e t t a b i l i t y c o n t r o lb y o c t a n o i ca c i d , S o r ' .P e t . E n g . / . , l 3 ( 4 ) , 2 2 1 , 1 9 1 3 . 6 4 . M u n g a n , N . , E n h a n c e do i l r e c o v e r yu s i n g w a t e r a s a d r i v i n g f l u i d . W o r l d O i l , 3 , 1 7 . 1 9 8 1 . 65. Amott, E., Observationsrelating to the wettability of porousrock. frrut.r. AIME,216, 156. I959. 66. Raza, S. H., Treiber, L. E., and Archer, D. L., Wettability of reservoirrocks and its evaluation,Prod. Mon.. 32. 156.1968. 67. Moore, T. F. and Slobod, R. L., The effect of viscosity and capillarity on the displacementoi oil by w a t e r ,P r o d . M o n . , 8 , 2 0 , 1 9 5 6 . 68. Bobek, J. E., Mattax, C. C., and Denekas, M. O., Reservoirrock wettability - its significanceand e v a l u a t i o nT, r a n s .A I M E , 2 1 3 , 1 5 5 , 1 9 5 8 . 69. Killens, C. R., Nielsen, R. F., and Calhoun, J. C., Capillary Desaturationand Imbibition in Porous R o c k M i n e r a l I n d u s t r i e sE, x p e r i m e n t aSl t a t i o nB u l l e t i n# 6 2 , P e n n S t a t eU n i v e r s i t y ,U n i v e r s i t yP a r k , l 9 - 5 3 , 55. 70. Richardson, S. G., Flow Through PorousMedia, Hundbookof Ftuid Dt'namic'sSectiort/6, McGraw-Hill. New York. 1961. 7l . Donaldsol, E. C., Thomas, R. D., and Lorenz, P. B., Wettabilitydeterminationand its effecton recovery e f f i c i e n c y ,S o c . P e t . E n g . J . , 3 , 1 3 , 1 9 6 9 . . et. Eng.J.,6, | 15, 1964. 7 2 . M u n g a n , N . , R o l e o f w e t t a b i l i t ya n d i n t e r f a c i atle n s i o ni n w a t e r f l o o d i n gS, o c ' P 73. Emery, L. W., Mungan, N., and Nicholson, R. W., Causticslug injectionin the Singletonfield, J. Pet. T e c h n o l . ,1 2 , 1 5 6 9 , 1 9 7 0 . 74. Kyte, J. R., Nuamann, V. O., and Mattax, C. C., Effect of reservoirenvirclnmenton water-oil disp l a c e m e n tJ,. P e t . T e c h n o l . , 6 , 5 ' 7 9 ,1 9 6 1 . of syntheticporous media, Prod. 75. Gatenby, W. A. and Marsden, S. S., Some wettability characteristics Mon.. 22. 5. 1957. 76. Johansen, R. T. and Dunning, H. N., Relative Wetting Tendenciesof Crude Oils by Capillarimetric M e t h o d . U . S . B u r e a uo f M i n e s . 1 9 6 1 . 5 ' 7 5 2 . 3rd ed., Oxford Univ. Press,London, 1959, 192. 77. Adams, N. K., The Physicsand Chemistryof Surfat'e.s, 78. Slobod, R. L. and Blum, H. A., Method for determiningwettability of reservoirrocks, Truns. AIME, t95. t. t952. of Wettability 79. Lorenz, P. 8., Donaldson, E. C., and Thomas, R. D., Use of CentrifugeMeasurements t o P r e d i c tO i l R e c o v e r y ,U . S . B u r e a uo f M i n e s , 1 9 1 4 , ' 7 8 7 3 . 80. Reznik, A. A., Fulton, P. F., and Colbeck, S. C., Jr., A mathematicalimbibition model with fractionalw e t t a b i l i t yc h a r a c t e r i s t i c P s ,r o d . M o n . , 3 l ( 9 ) , 2 2 , 1 9 6 7 . 8l . Coley, F. H., Marsden, S. S., and Calhoun, J. C., Jr., Study of the eff'ectof wettabilityon the behavior of fluids in syntheticporous media, Prod. Mon., 20(8), 29, 1956. 8 2 . K e e l a n , D . K . , A c r i t i c a lr e v i e w o f c o r e a n a l y s i st e c h n i q u e sJ,. C o n . P e t . T e c h n o l . , 6 , 4 2 , 1 9 1 2 . 83. Poettmann, F. H., Caudle, B. H., Craig, F. F., Jr., Crawford, P. 8., Bond, D. C., Farouq Ali, S. M., Holott, C. R., Johansen, R. T., Mungan, N., and Dowd, W. T., Secondaryand Tertiary Oil RecoveryProcesses,lnterstateOil CompactCommission,OklahomaCity, Okla., September1974. relativepermeability 84. Schneider; F. N. and Owens, W. W., Sandstoneand carbonate,two- and three-phase c h a r a c t e r i s t i cS s ,o c . P e t . E n g . J . , 3 , 1 5 , 1 9 7 0 . 85. Scrom,H.M.,Significanceof Water-OilRelativePermeabilityDataCalculatedfromDisplacementTests, Theory of Fluid Flow in PorousMedia Conference,University of Oklahoma, 1959, 189. 86. Amyx, J. W., Bass, D. M., and Whiting, R. L., PetroleumReservoirEng,ineering,McGraw-Hill, New York. 1960. 87. Colpits, G. P. and Hunter, D. E., Laboratorydisplacementof oil by water under simulatedreservoir conditions,J. Can. Pet. Technol., 3(2), 64, 1964. 88. Haddenhorst, H. G. and Koch, R., Effect of temperatureand pressureon the separationof solids from petroleum,Erdoel Kohle, 2, 12, 1959.

100

Relative Permeability of Petroleum Reservoirs

89. Luks, K. D. and Kohn, J. P., The Effect of MethaneUnder Pressureon the Liquid Solubility of Heavy HydrocarbonComponents,Liquid-Vapor and Solid-Liquid-VaporBehavior, progressReport II, Apl Res e a r c hP r o j e c t1 3 5 , N o t r e D a m e , I n d i a n a ,J u l y , t 9 7 1 . 90' Rathmell, J. J., Braun, P. H., and Perkins, T. K., Reservoirwaterfloodresidualoil saturation from l a b o r a t o r yt e s t s ,J . P e t . T e c h n o t . , 2 2 5 , l i 5 . l g j 3 . 91. Richardson, J. G., Perkins, F. M., Jr., and osoba, J. S., Differencein behaviorof fresh and agedeast Texas Woodbine cores, Truns. AIME,204, 86. 1955. 9 2 . M u n g a n , N . , R e l a t i v ep e r m e a b i l i t ym e a s u r e m e nurs i n g r e s e r v o i rf l u i d s , S o c .p e t . E n g . J . , l 2 ( 5 ) , 3 9 g , 1972. 93. Ehrlich, R., Hasiba, H. H., and Raimondi, P., Alkaline waterfloodingfor wettability alteratione v a f u a t i o no f a p o r e n t i a ft i e l d a p p l i c a r i o n , , /p. e t . T e c h n o t . , 2 6 , 1 3 3 5 ,l g j 4 . 94. DeterminQtion of Residual Oil Saturatior?,Interstate Oil Compact Commission, Oklahoma City, Okla., t978. 95. Jennings, H. H., Surfacepropertiesof naturaland syntheticporousmedia, prod. Mon., 2l(5). 20. 1957. 96' Hough, E. W., Rzasa,M. J., and Wood, B. 8., Interfacialtensionsat reservoirpressures andtemperatures, apparatusand the water-methanesystem,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 Temperatureon Relative Permeabilityof UnconsolidatedSands.paper SPE 1897 presentedat the SpE 4 2 n d A n n u a l F a l l M e e t i n g ,H o u s t o n ,T e x a s,. 1 9 6 7 . 98. Cuiec, L. E., Restorationof the Natural Stateof Core Samples,paperSPE 5634 presentedat the SpE 50th A n n u a l M e e t i n g ,D a l l a s ,T e x . , 1 9 7 5 . 99. Chifingarian, G. V., Mannon, R. W., and Rieke, H. H., Eds., Oil and Gas productionFrom Cqrbonctre R o c k s ,E l s e v i e r ,A m s t e r d a m ,1 9 7 2 . 100. BradleY, D. J., The Applicability of WettabilityAlterationto NaturallyFracruredReservoirs anrllmbibition W a t e r f l o o d i n gM , a s t e r st h e s i s ,U n i v e r s i t yo f T u l s a ; ' O k l a h o m a1, 9 g 3 . l0l' McCafferY, F. G. and Mungan, N.' Contactangle and interfacialtensionstudiesof some hydrocarbon w a t e r s o l i d s y s t e m sJ, . C u t t . p e t . T e c h n o t . ,j , l g 5 , 1 9 7 0 . 1 0 2 . o i l R e s e r v ' o iEr n g i n e e r i n g P , i r s o n ,s . J . , E d . . M c G r a w - H i l l , N e w y o r k . 1 9 5 g . 6 g . 103' Lefebvre du Prey, E., Deplacementsnon-misciblesdans les millieux poreux influence des parameters interfaciauxsur les permeabilitesrelatives,c.R. IV Cotoq. ARTFp puu, 196g. 104. Donaldson, E. C. and Thomas, R. D., MicroscopicObservationsof Oil Displacementin Water-Wet and O i l - W e t F o r m a t i o n sS , P E 3 5 5 5 p r e s e n t e da t t h e 4 6 t h S P E A n n u a l F a l l M e e t i n g ,N e w O r l e a n s ,O c t . 3 - 6 . 197t. 105' McCafferY, F. G. and Bennion, D. W., The effect of wettability on two-phaserelative permeabilities. J. Can. Pet. Techno1.1 , 0.42. 1974. 1 0 6 . L e v i n e , J . S . , D i s p l a c e m e net x p e r i m e n t si n a c o n s o l i d a t e pd o r o u ss y s t e m ,T r a n s .A t M E , 2 0 l , 57, t9-54. 107. Josendal, V. A., Sandford, B. 8., and Wilson, J. W., Improved multiphaseflow studies employing r a d i o a c t i v et r a c e r s ,T r o n s .A I M E , I 9 5 , 6 5 . 1 9 5 2 . 108. Terwilliger, P. L., wilsey, L. E., Hall, H. N., Bridges, p. M., and Morse, R. A., Experimental and theoreticalinvestigationof gravity drainageperformance . Trans. AIME, l92, 285, 1951. - 109. Johnson, E. F., Bossler, D. P., and Naumann, V. O., Calculationof relative permeability from displacementexperiments,Trans. AIME, 216. 370. lg5g. l l 0 . L a n d , C . S . , C o m p a r i s o no f c a l c u l a t e dw i t h e x p e r i r n e n t ai m l b i b i t i o nr e l a t i v ep e r m e a b i l i t y , T r u n sA. I M E , 2 5 t . 4 1 9 . t 9 7| . I I l ' Gardner, G. H. F., Messmer, J. H., and Woodside, W., EffectivePorosityandGas Relativepermeability on Liquid Imbibition Cycle. Theory of Fluid Flow in PorousMedia Conference,University of Oklahoma. Norman. 1959. 173. ll2. Shelton, J. L. and Schneider, F. M., The effect of water injectionon miscible flooding methods using hydrocarbonsand CO,, paper SPE 4580 presentedat the SPE 48th Annual Meeting, Las Vegas, 1973. 5 I l3' Land, C., Calculationof imbibitionrelativepermeabilityfor two- and three-phase flow fiom rock properties, S o c ' .P e t . E n g . J . , 6 , t 4 9 . 1 9 6 8 . I 1 4 . W i l s o n , J . W . , D e t e r m i n a t i o no f R e l a t i v eP e r m e a b i l i t yU n d e r S i m u l a t e dR e s e r v o i r C o n d i t i o n s .AIChEJ, 2(t), 4. 1956. - f l5' Fatt, I. and Barrett, R. E., Effect of overburdenpressure on relative permeability, Truns. AtME, lgE, 325. t953. I l6' Thomas, R. D. and Ward, D. C., Effect of overburdenpressureand water saturationon gas permeability of tight sandstonecores,-/. Pet. Te<'hnot.,2, 120, lg'/2. ll7. Merliss, F. E., Doane, J. D., and Rzasa, M. J., Influenceof rock and fluid propertiesand immiscible fluid-flow behaviorin porous media, paper 510-G presentedat the AIME Annual Meeting. New orleans. I955. l l 8 . D u n l a p , E . N . , I n f l u e n c e o cf o n n a t e w a t e r o n p e r m e a b i l i t ysoafn d st o o r l , T r a n s . A I M E , l 2 j . 2 1 5 . l g 3 g .

h! tea.

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l,' l,i. 1957. d!.-:r.ratures. fr.. irr ['-l'l'ect r.: . rhcSPE rt'[:-i0th

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ll9. Stewart, C. R., Craig, F. F., Jr., and Morse, R. A., Determinationof limestoneperformancecharacteristicsby model flow tests,Trans AIME, 198, 93, 1953. 120. Keelan, D. K., A practicalapproachto determinationof imbibition gas-waterrelativepermeability, J. Pet. T e c h n o l . ,4 , 1 9 9 , 1 9 7 6 . l 2 l . L e a s , W . J . , J e n k s , L . H . , a n d R u s s e l l ,C . D . , R e l a t i v ep e r m e a b i l i t yt o g a s , T r a n s . A I M E , 1 8 9 , 6 5 ,

r9 5 0 . Trans.A1ME,216,258,1959. core characteristics, 122. Felsenthal,M., Correlationof k*/k,,datawith sandstone 123. McCord, D. R., Performancepredictionsincorporatinggravity drainageand gas pressuremaintenance LL-370 Area, Bolivar coastal field,Trans. AIME' 198, 231, 1953. 124. Edmondson, T. A., Effect of temperatureon 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 temperatureon irreducible water saturationand relative permeability of unconsolidatedsands, S o c .P e t . E n g . J . , 6 , l 7 l , 1 9 7 0 . 126. Davidson, L. B., The effect of temperatureon the permeabilityratio of different fluid pairs in two-phase s y s t e m sJ, . P e t . T e c h n o l . ,8 , 1 0 3 7 , 1 9 6 9 . 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 tf 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 p r e s s u r ec u r v e s ,S o c . P e t . E t t g . J . , 3 . 1 3 . 1 9 7l . 128. Weinbrandt, R. M., Ramey, H. J., Jr., and Cass6, F. J., The effect of temperatureon relative and a b s o l u t ep e r m e a b i l i t yo f s a n d s t o n e sS,o r ' .P e t . E n g . J . , 1 0 . 3 7 6 , 1 9 1 5 . 129. Lo, H. Y. and Mungan, N., Effect of Temperatureon Water-Oil RelativePermeabilitiesin Oil-Wet and W a t e r - W e tS y s t e m s ,S P E # 4 5 0 5 , L a s V e g a s ,N e v . , S e p t e m b e3r 0 , 1 9 7 3 . 130. Sufi, A. S., Ramey, H. J., Jr., and Brigham, W. E., TemperatureEffects on Relative Permeabilities o f O i l - W a t e rS y s t e m s ,S P E # l 1 7 0 1 , N e w O r l e a n s ,L a . , S e p t e m b e2r 6 , 1 9 8 2 . 1 3 l . S u f i , A . S . , R a m e y , H . J . , J r . , a n d B r i g h a m , W . E . , T e m p e r a t u r eE f f ' e c t so n O i l - W a t e r R e l a t i v e Permeabilitiesfor UnconsolidatedSands, U.S. Departmentof Energy, Technical Report, 12056-35.Dec e m b e r .1 9 8 2 .

t 3 2 .Miller, M. A., and Ramey, H. J., Jr., Effect of Temperatureon Oil/Water Relative Permeabilitiesof , a l i f . , O c t o b e r5 , 1 9 8 3 . S a n d s ,S P E # l 2 l 1 6 , S a n F r a n c i s c oC U n c o n s o l i d a t eadn d C o n s o l i d a t e d L ' . : ' . , :. : ' r r d l c r \

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1 3 3 Counsil, . J. R., Steam-WaterRelativePermeability,Ph.D. thesis,StanfordUniv., Stanford,Calif., 1979. t 3 4 .Muskat, M.,, PhvsicalPrinciples of oil Production, McGraw-Hill New York. 1949. C. and Longeron, D., Influenceof very low interfacraltensionson relativepermeability,paper 1 3 5 Bardon, . S P E 7 6 0 9 p r e s e n t e da t t h e S P E 5 3 r d A n n u a l M e e t i n g ,H o u s t o n ,T e x . , 1 9 7 8 . relativepermeabilitydata,Trans. 136. Richardson, J. G., Calculationof waterfloodrecoveryfrom steady-state AIME. 210.373. 1951. 137. Johnson, E. F., Bossler, D. P., and Nauman, V. O., Calculationof relativepermeabilityfrom displacement experiments,Trans. AIME, 216. 370, 1959. 1 3 8 . L e v i n e , J . S . , D i s p l a c e m e net x p e r i m e n t isn a c o n s o l i d a t e pd o r o u ss y s t e m ,T r s n s .A I M E , 2 0 1 , 5 7 , 1 9 5 4 . 139. Craig, F. F., Jr., Errors in calculationof gas injectionperformancefrom laboratorydata,J. Pet. Techrutl., 8.23, 1952. 140. Sandberg, C. R., Gourney, L. S., Suppel, R. F., Effect of fluid flow rate and viscosityon laboratory d e t e r m i n a t i oo n f o i l - w a t e rr e l a t i v ep e r m e a b i l i t i e sT,r s n s .A I M E , 2 1 3 , 3 6 . 1 9 5 8 . 14l. Donaldson, E. C., Lorenz, P. 8., and Thomas, R. D., The effect of viscosity and wettability on oilwater relativepermeability,paper SPE 1562 presentedat 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 displacementfrom porous media by liquid flooding, Trans. AIME, 195,29. 1952. 143. Krutter, H. and Day, R. J., Air-drive experimentson long horizontalconsolidatedcores../.Pet. Technol., t2, t, t943. on small core 144. Morse, R. A., Terwilliger, P. K., and Yuster, S. T., Relativepermeabilitymeasurements samplesO , il GasJ., 46. 109, 1947. 145. Odeh, A. S., Effect of viscosityratio on relativepermeability,Trans. AIME, 216,346, 1959. 1 4 6 . B a k e r , P . E . , D i s c u s s i o no f e f f e c t o f v i s c o s i t yr a t i o o n r e l a t i v ep e r m e a b i l i t yJ, . P e t . T e c h n o l . , 2 1 9 , 6 5 , I 960. 1 '95691' . 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 fv 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 M E 246, Trans . Al materials flow in consolidated , for two-phase E. A model , 148. Ehrlich, R. and Crane, F. , 22t, t969. 149. Perkins, F. M., Jr., An investigationof the role of capillary forces in laboratorywaterfloods.J. Pet. T e c h n o l . ,l l , 4 9 , 1 9 5 7 . 150. Pickell, J. J., Swanson, B. F., Hickman, W. B., Applicationof air-mercuryand oil-air capillarypressure d a t a i n t h e s t u d y o f p o r e s t r u c t u r ea n d f l u i d d i s t r i b u t i o n ,S o c .P e t . E n g . J . , 4 , 5 5 . 1 9 6 6 . l5l. Warren, J. E. and Calhoun, J. C., A study of waterfloodefficiency in oil-wet systems,Truns. AIME. 204.22. t955.

102

Relative Permeabilin of Petroleum Reservoirs

1 5 2 . C a r o , R . A . , C a l h o u n , J . C . , J r . , a n d N i e l s e n ,R . F . , S u r f a c ea c t i v ea g e n t si n c r e a s eo i l r e c o v e r y .O i 1 GusJ., 12. 6. 1952. 153. Ojeda, E., Preston, F., and Calhoun, J. C., Jr., Correlationof residualsfollowing surfactantfloods, Prod.Mon., 12,20, 1953. 154. Lefebvre du Prey, E. J., Factorsafl'ectingliquid-liquid relative permeabilitiesof a consolidatedporous m e d i u m .S o c . P e t . E n e . J . , 2 , 3 9 . 1 9 ' 1 3 . 1 5 5 . O w e n s , W . W . , P a r r i s h , D . R . , a n d L a m o r e a u x , W . E . , A n e v a l u a t i o no f a g a s d r i v e m e t h o d f o r determiningrelative permeabilityrelationships,Truns. AIME, 201,275, 1956. 156. Kyte, J. R. and Rapoport, L. A., Linear waterfloodbehaviorand end ef'fectsin water-wetporousmedia, Trans. AIME, 213. 423. 1958. 157. Richardson, J. G., Kerver, J. K., Hafford, J. A., and Osoba, J. S., Laboratorydeterminationsof , r a n s .A I M E , 1 9 5 , 1 8 7 , 1 9 , 5 2 . r e l a t i v ep e r m e a b i l i t y T on Small Core 158. Morse, R. A., Terwilliger, P. K., and Yuster, S. T., RelativePermeabilityMeasurements SamplesO , il GasJ., 46. 109, 1941. 159. Labastie, A., Guy, M., Delclaud, J. P., and lffly, R., Effect of flow rate and wettability on water-oil relativepermeabilitiesand capillarypressure,paperSPE 9236 presentedat the SPE Annual Meeting, Dallas.

-

T e x . , S e p .2 l - 2 4 , 1 9 8 0 . 159a. McCaffery, F. G., The Effect of Wettability on Relative Permeabilityand Imbibition in PorousMedia. P h . D . t h e s i s ,U n i v e r s i t yo f C a l g a r y ,A l b e r t a ,C a n a d a ,1 9 7 3 . I 6 0 . Delclaud, J. P., New resultson the displacementof a fluid by anotherin a porous medium, paper SPE 4 1 0 3 p r e s e n t e da t t h e S P E 4 7 t h A n n u a l M e e t i n g .S a n A n t o n i o , T e x . , 1 9 1 2 . l 6 l . Fetkovitch, M. J., The isochronaltestingof oil wells, paper SPE 4529 presentedat the 48th Annual Fall -

M e e t i n go f t h e S P E , L a s V e g a s ,N e v a d a . 1 9 7 3 . t 6 2 . Handy, L. L. and Datta, P., Fluid distributionsduring immiscibledisplacementsin porous media. Sot'. Pet.Eng./.,, 10.261, 1966. on waterflood relative t 6 3 . Huppler, J. D., Numerical investigationof the effects of core heterogeneities p e r m e a b i l i t,yS o c ' .P e t . E n g . J . , 1 2 , 3 8 | , 1 9 7 0 . 164. Stewart, C. R. and Owens, W. W., A laboratorystudy of laminar and turbulentllow in heterogeneous p o r o s i t yl i m e s t o n eT, r u n s .A I M E , 2 l 3 , 1 2 l , 1 9 5 8 . s t u d i e s ,P r o d . M o n . , 4 . 1 2 , 1 6 , 5 .H e n d e r s o n ,J . H . a n d M o l d r u m , H . , P r o g r e s sr e p o r to n m u l t i p h a s e - f l o w t949. ., 166. Krutter, A. and Day, R. J., Air-drive experimentson long horizontalconsolidatedcores,-/. Pet. TeL'hnol l l. l. r943. 1 6 7 . C h e n , H . K . , C o u n s i l ,J . R . , a n d R a m e y , H . J . , J r . , S t e a m - W a t eRr e l a t i v eP e r m e a b i l i t y1. 9 7 8G e o t h e r m a l R e s o u r c eC s o u n c i l A n n u a l M e e t i n g ,H i l o , H a w a i i , J u l y 2 5 - 2 7 , 1 9 1 8 . 168. Brownell, L. E. and Katz, D. L., Flow of fluids through porous media - single homogeneousfluids, C h e m . E n s . P r o s , . ,4 3 ( 1 0 ) , 5 3 7 . 1 9 4 ' 7 . p e r m e a b i l i t yr e l a t i o n s h i p alto w g a ss a t u r a t i o n . J . l n s t P . et., 169. Wall, C. G. and Khurana, A. K., Saturation

5 1 .2 6 1 .1 9 7 1 . 1 7 0 . R o s e , W . D . , F l u i d d i s t r i b u t i o n cs h a r a c t e r i z i ngga s - l i q u i df l o w , T r a n s . A I M E , 1 9 2 , 3 1 2 , 1 9 5 1 . - l 7 l . L o o m i s , A . G . a n d C r o w e l l , D . C . , R e l a t i v ep e r m e a b i l i t ys t u d i e s I. I . W a t e r o i l s y s t e m sP . rod. Mon.,8. r8. 1959. 172. Sarem, A. M., Significanceof water-oil relative permeabilitydata calculatedfrom displacementtests, , n i v e r s i t yo f O k l a h o m a .N o r m a n , 1 9 5 9 , 1 8 9 . P r o < ' . ,T h e o r y o f F l u i d F l o w i n P o r o u sM e d i a C o n f e r e n c eU and 173. Owens, W. W., Parrish, D. R., and Lamoreaux, W. E., A comparisonof field k*/k,,characteristics laboratoryku/k,,test results measuredby a new simplified method. paper 518-G presentedat the AIME 3 0 t h A n n u l M e e t i n g , N e w O r l e a n s ,1 9 5 5 . 174. Calhoun, J. C., Jr., Fundamentalsof ReservoirEngineering,University of Oklahoma Press,Norman, t94'7. | 75. Stewart, C. R., Craig, F. F., and Morse, R. A., Determinationof limestoneperformancecharacteristics by model flow tests. Truns. AIME, 198, 93, 1953. 176. Kyte, J. R., Stanclift, J. R., Stephan, S. C., Jr., and Rapoport, L. A., Mechanismof waterflooding in the presenceof free gas,Trans. AIME, 101, 215, 1956. 1 7 7 . M a t t a x , C . C . a n d C l o t h e i r , A . T . , C o r e A n a l y s i so f U n c o n s o l i d a t eadn d F r i a b l eS a n d s ,p a p e rS P E 4 9 8 6 presented a t t h e S P E 4 9 t h A n n u a l M e e t i n g ,H o u s t o n ,T e x . . 1 9 7 4 . 178. Holmgren, C. R. and Morse, R. A., Effect of free gas saturationon oil recoveryby 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 Permeabilityand Imbibition in PorousMedia. 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 L a b o r a t o r i e sI,n c . , D a l l a s ,T x . , 1 9 7 9 .

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I. INTRODUCTION

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Recent innovationsin the field of oil recovery have led to a renewedinterestin threephaserelative permeability. Three-phaseflow occurs when the water saturationis higher than the irreducible level, and oil and gas are also presentas mobile phases.Detailed engineeringcalculationsof the performanceof reservoirsunder recoverymethodssuch as carbon dioxide injection, in situ combustion,steamdrive, micellar fluid injection, and nitrogen injection frequentlyrequirethree-phaserelativepermeabilitydata. All factors which influenceflow in systemscontainingtwo mobile phasesalso apply to systems, systems.Virtually all oil reservoirsconstitutepotentialthree-phase three-phase naturally oils cominterstitial water, and occurring sincereservoirrocks invariably contain pletely devoid of gas are rare. In fact, a two-phasesystemof oil and gas may be regarded as a three-phasesystemin which the water phaseis immobile. The numberof reservoirsin which oil, gas, and water are simultaneouslymobile during primary productionis probably small. Nevertheless,three-phasemobility is always possible when a producing interval includespart of the oil-water transitionalzone in a reservoir.It is probable,however, that in most caseswhere oil and free gas are producedwith an appreciablewater cut, the water is being producedfrom layers of the reservoir in which relative permeabilityto water is high and not by true three-phaseflow. In the past, the use of three-phaserelative permeabilitydata for conventionalreservoir engineeringcalculationshas seldom been necessary.In consequence,considerablyless is known about three-phaserelative permeability characteristicsof rocks than is known for comparabletwo-phasecases.The realizationthat detailedengineeringcalculationsof the performanceof reservoirsproducedby in sitrzcombustionprocesses requirethree-phase data is quite new. Three-phaserelativepermeabilityis useful in the calculationof field performwaterandgasdrive, and alsoin analyzing ancefor reservoirsbeingproducedby simultaneous solution gas drive reservoirs which are partially depletedand are being produced by water drive. An increasinginterestin three-phaseflow phenomenais anticipated. There are two distinct classesof three-phaserelative perrneabilitydata: ( I ) that pertaining to drainage;and (2) that pertainingto imbibition. Drainagerefersto the direction of saturation changein which the wetting-phasesaturationdecreases.Imbibition refers to an increasing wetting-phasesaturation.For the relative permeabilitydata to yield correct reservoirpredictions, the directionof saturationchangein the reservoirmust correspondto the direction of saturationchange for which the data were derived. Drainagerelative permeabilitydata should be used in the following situations: l.

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Enhancedrecovery processesinvolving the injection of dry gas, flue gas, carbon dioxide, and other gasesinto watered-outreservoirs. Miscible flood processesin which liquified petroleum gas (LPG) is injected into watered-outreservoirs. Productionfrom reservoirsin which the water saturationis greaterthan the ineducible saturation. Imbibition relative permeabilitydata should be used under the following conditions:

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104

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Relative Permeabilin of Petroleum Reservoirs

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2. 3.

Three-phaserelative permeability.r

Reservoirsdevelopedby water flood, as well as by processeswherethe injectedwater containssurfactants,polymers, or other additives. Reservoirsdevelopedby recovery processeswhere water is used to push a slug of chemicals,LPG, etc.

II. DRAINAGERELATIVEPERMEABILITY A. Leverett and Lewis Much of the credit for the classicalwork in three-phase relativepermeabilityis accorded to Leverettand Lewis' who were the first to measurethree-phase relativepermeabilityof a water-oil-gassystem in an unconsolidatedsand. These investigatorsused a steady-state single-core dynamic method and ignored end effects and hysteresis.Errors from ignoring capillary end effects were probably significant,since low flow rateswere used. Ring electrodes were spacedalong the length of a sand pack to measureresistivity of the sample and brine saturationwas assumedto be directly relatedto resistivity. Gas saturationwas determined from pressureand volume measurements. Oil sdturationwas obtainedby a material balancetechnique.Leverett and Lewis obtainedthree separatetriangulargraphs showing lines of constantrelative permeability("isoperms") to the three phases;thesewere plotted againstthe saturationsof the three fluids, as shown by Figure l. They also obtaineda plot showing the region of three-phaseflow; Figure 2 shows the region where each component comprisesat least5Voof the flow stream.As shown by the figure, three-phaseflow occurs in a rather confined region. Relativepermeabilityto water, k,*, was found to be dependentonly on water saturation, S*, and was not affected by the introduction of an additional nonaqueousphase. Relative permeability to gas, k,r, was found to be slightly less than would be expectedfor the same gas saturation, S* in two-phase flow. The k., isoperms are convex towards the 1007oS, apex of the triangulardiagram. As gas becomesone of the two flowing nonwettingphases, whenboth oil and waterarepresent,the relativepermeabilityto gasdecreases asoil saturation

approa of oil a The r agass saturat to its o gas sat mode c the inte oil and an oil I which < Hower flow of of the 1 move f at cons Also This ef may flr the por parts ol result i [rve for var

B. Cor The a calci

105 lOO%gas

100% water

lOO%oil

FIGURE 2.

ieetcd*'ater fi -r .lu_sof I t

i. .r..trrded i a l ' ' r l r t ro f a Dlc;Jr -\tate nr lrnoring Rrng c.lecslrplc I lr jr

and glg[91-

7 .r :naterial h: .htrwing rerc plotted uncJ a plot c()nlrx)nent Jltrri occurs s.rturation, E Rclative lr lhc same E l ( x ) 7 cs s ing phases, | .;turation

Region of three-phaseflow.l

approaches the water saturationvalue, becominga minimum when roughly equal saturations of oil and water are associatedwith the gas. The relative permeability to oil is seento vary in a more complex manner. Starting with a gas saturationof zero, oil relative permeabilityat constantoil saturationincreasesas gas saturationincreases(except at low oil saturationswhere k,, remainsconstant)then decreases to its original value as more gas is introduced,finally falling well below this value when gas saturationis further increased.In a water-wet system,the presenceof gas leavesthe mode of water flow unchanged,but since the gas tends to occupy the central portions of the intergrainspaces(where the oil is also driven by capillary forces)interferencebetween oil and gas flow is likely. Visual examinationunder the microscopeshowsthe presenceof an oil film (in some casescontaininga very small amountof finely divided water) through which oil flows aroundeachgasbubble.It is not clearwhetherall gasbubblesare connected. However, the gas bubblesare observedto move jerkily, as opposedto the generallysmooth flow of water (and of oil when gasbubblesare absentor are stationary).This unevenmotion of the gas implies a similar motion of at leastpart of the oil, which would be expectedto move faster than in the absenceof gas at the same oil saturation.We see a decreasein k,. at constantS" as S, is increased,especiallyat low S*. Also, there is an increasein k.o at constantS" as S* is increasedat low values of S*. This effect is evidently due to ihe shifting of oil into parts of the intergrain spacewhere it may flow more freely. The water introduced tends to occupy the sharply curved parts of the pores,forcing oil into the centralspacevacatedby gas. Sincefluid in the sharplycurved parts of the poresmoves only with difficulty and that in the centermoves more readily, the result is an increasein k,o. Leverettand Lewis pointed out that they found no effect of oil viscosityon the isoperms for various saturationsof the three phases. B. Corey, Rathjens, Henderson, and Wyllie The resultsof the work of Corey et al.2are shown by Figure 3. Theseinvestigatorsused a calcium chloride brine. Capillary end effects were minimized by using a core with semi-

106

Relative Permeabilitv of Petroleum Reservoirs

gas FIGURE 3.

Three-phaserelative permeability.r

permeablemembranesmountedat eachend. They measuredsaturationsgravimetricallyand avoided hysteresiseffects by using separatecores for each measurementrather than resaturating the samecore. In an initial conclusion,they reportedthat when the saturationsof the wetting phaseswere equal, the nonwettingphase relative permeability,k,n, was unchangedregardlessof whetherthe nonwettingphasewas oil or gas.They usedthe equivalent liquid permeabilityas the basevalue. The oil isopermsof Corey et al. are similar to those obtainedby Leverettand Lewis, exceptthat Corey's oil isopermshave a greatercurvature. Relativepermeabilityto water was not measured,but was calculatedon the assumptionthat it was a function of water saturationaloneand that waterpermeabilityin a water-wetsystem was the sameas the oil permeabilityin an oil-wet system.It shouldbe noted that the data of Cltrey et al. are for oil drainagein an oil-gassystem.They alsoobservedthat the behavior of the nonwetting phaseswas more sensitiveto changesin pore geometry than was the behaviorof the wetting phase.The increasein k." (at low S*) with the increasein S* (and a correspondingdecreasein S*) is higher in Corey's consolidatedsandstonesamplesthan in the unconsolidatedsamples.This is becauseof the dependenceof k..,on the ratio ft' I dsl/P.r

JS*

l,

dsL/P:

which is usually higher in consolidatedrocks than in unconsolidatedrocks. Corey et al. extendedtheir two-phaserelative permeabilityrelationshipto three-phase flow on the basisof the following approximation:

The d by

where S As in especia the syst at highe alone.I wetting

C. Reid Using Reidr ot was ign obtaine differen

107

N 2

I FIGURE 4.

8 l : . . tI l r a n d B r : : l . r l lf e S a -

l/Pl : g

Three-phaserelative permeability.r

Sr_,

for S,' t

IS,_-(S*,..+S,,.)]

St' (soi,, + s.,,)

l l . . - . r l l r r I l SO f

'\.1\unI E ; . . . rr r a l e n t lrlr: i(r those l f . . 1 r \a t u r e . U n r n l r r r nt h a t f- \\ .'l :\ Ste[l lh.:i thc' data thc hchavior h;:: s aS the r r : :S . ( a n d n r : . ; . l c rt h a n I

r . l l , r

thrc.'-phase

t''=, : o f b r S . s- ' ^ (S*,,, * S",)

(l)

The drainageoil phaserelativepermeabilityin a water-wetsystemcontaininggas is given bv

(2)

where S., is residualliquid saturation. As in Leverett's data, the oil isopermstend to be parallel to the oil isosaturationlines, especiallyat high S*. At increasingS* and constantS.,, the gas which was previously in the systemis no longerpresent.Thus, the rate of increaseof k..,with increasingS* decreases at higher valuesof S*. Corey et al. proposeda methodto obtain k.,,and k,*, basedon k.* alone. Incidentally, k.* was found to be a function of S* and independentof the relative wetting propertiesof the fluids within the rock. C. Reid Using the samemethodemployedby Leverettand Lewis (single-coredynamictechnique), Reid3obtaine{ the isopermsshown in Figure 4; He eliminatedend effects, but hysteresis was ignored.'frine saturationwas measuredby resistivity,and'oil and gas saturations were obtainedby gamma ray absorption.His saturationmeasuremenis possiblywere affectedby differentialabsorptionof gamma rays by oil and water. While Leverettand Lewis obtained

108

Relative Permeability of Petroleum Reservoirs

straightlines for the water phasebehavior(showingk,* to be independentof the distribution of the nonwettingphases)and oil isopermsconcavetoward the l}OVaSoapex, Reid's results indicatedconcavewater isoperms,convex oil isoperms,and slightly concavegas isoperms. These results were interpreted as indicating that the relative permeability to each phase is dependentboth upon its own saturationand the saturationsof the other phases.His results showeda greateroil permeabilitywhen threephaseswere presentthan with two phases,at a given oil saturation. Reid made no attempt to correlate the three-phaseresults with those from two-phase experiments.He placed emphasison his conclusionsfor the oil isopermsand noted that Leverett'soil phasedata showeda substantialamountof scatter.For this reason,he believed that his oil isopermswere more valid than Leverett's.The work of Rose seemsto confirm Reid's findings. D. Snell Three-phasebehavior in a water-wet unconsolidatedsand was investigatedby Snell,a-6 who used radio frequency detection for the determination of S* and a neutron counting methodfor measurementof Sr. Oil saturationwas obtainedby materialbalancecalculation. His experimentshad a repeatabilitywithin Ilr%ofor relativepermeabilityvalues,with a better repeatabilityfor the saturationvalues.He found that when the wetting phasesaturationwas uniform over a length of the test sample,the saturationsof the other two phaseswere also uniform over the samelength. Although Caudle et al. " did mention hysteresisin their work, the first significantstudy on the effect of saturationhistory on three-phaserelative permeabilitywas done by Snell. In describingSnell's work, it is convenientto definefour typesof liquid saturationhistories: l. 2. 3. 4.

Imbibition of water with oil saturationincreasing(II). Imbibition of water with oil saturationdecreasing(ID). Drainageof water with oil saturationincreasing(DI). Drainageof water with oil saturationdecreasing(DD).

As seen from his results in Figure 5, k.o values were lower for DD than for the other saturationhistories.Since, in two-phaseflow, drainagecausedthe wetting phapeto lose its mobility at highersaturations,it hasbeensuggested that thereis a partialchangein wettability from water-wetto oil-wet during DD. When the systemwas oil-wet, a largerS,,was required for the samek,., becausesomeof the oil was trappedin the smallerpores.This oil increases S.,,but it is immobile. He further suggestedthat this changein wettability may be caused by polar compoundsin the oil. Snell's resultsdo not show good agreementwith those of Leverettand Lewis except in the caseof k.*. Oil and water isoperms reported by Snell are similar to those determinedby Reid, but Snell's k.. valuesare higher than Reid's, especiallyat low water saturation. In a later work, Snell reinterpretedthe resultsof four earlier studiesdoneon unconsolidated sands.In these investigations,no hysteresiswas found for water isoperms.Oil isoperms showed hysteresisonly when keroseneor a kerosene/lubricatingoil mixture was used as the oil phase.Nonpolaroil gaveno hysteresis.Reinterpretation of the earlierresultswas possible becauseLeverettand Lewis indicatedpossibleenors in their saturationmeasurements. Reid's saturationdata might also have been inaccuratebecauseof differential absorptionof gamma rays by oil and water. Relativepermeabilityto oil was found to be dependentonly on the historiesof the liquid phasesaturations,althoughSnell did not rule out dependenceon gas phasesaturationhistory. Snell reinterpretedLeverett'sdata to obtain oil isopermsconvex toward the l00%oS,,apex. Oil isopermsthen followed the samepatternin all four investigations.Theseresultsareshownin Figure6. The curvatureof the isopermsof both nonwetting

I

10 0 %

1O0% watr

109 dr.rribution R r J '. r! 'su l ts 5i lrtrpt3ttTlS.

rh phaseis , [{r. rcsults o p h a r e s .a t I I\\ ()_phase I n,'ted that

I OO*

watet

10Oi

hc hc'lieved i lr, a()nfifm

br

oil

water

S n L - l l . r6

tn . \)unting c a l eu l a t i o n . t rth .r hrctter Uf .:i i(ln \.\'aS

s ii crc also 10OS water

M

roOS

R) oil

I OOi

oil

gas

l-ls.rntrtud) r tr Snell. xr hr.ttlries:

FIGURE 5. 'l0O%

Three-phaserelative permeability.5

gas

100% gas

lmbibition _ - - - -

D ra ina ge

r thc o th e r ! l \ \ l ( ) \ ei t s 1 uer r . r b i l i t y ra. rcquired il r nir ca se s I n- eaused ith lhore of

R€sults of Snell

1OO%water

il!to,Dt

/I,a'

oD

_ ----

100% oil

100% water

10 0 %

1O0% oil

y Rer . l. b u t o n ., ''lid a te d ll r.oP€ITnS u'cd as the y3.possible in t. Re i d 's

non-polar

I ()l gamma Dnlr on the llluc ttn 935 lTIlr r-t)DVeX

D urr nre sti Jk,nrrctting

1OO%water

1O0% oil

FIGURE 6.

'l

OO% water

Reinterpretationof resultsby Snell.6

100% oil

ll0

Relative Permeabilin of Petroleum Reservoirs

---

_) _ 4 o :;,--=-. Gas iniectsd, pore

\

zj+

volumes

i---- to-..\..

E

o o o .z

F---,:-:N*, *-'

b

o

-'--\:\:1..,, i /, [--'"-\..

(L

O_ o

1

^-.5

.1

L-.25

0 O

100 FIGURE 7.

tNtlAL wATERsATURATtoN

{-

< - L

-l I

-A

J . 1

---A

100

rNrrrAL orL sATURATToN

0

Fluid flow experimentalaata for Berea sandstone.T

phases(oil and gas)are convextoward the correspondingphase-apex, whereaswetting phase isopermsare straightlines or are concavetoward the l00%oapex of the wetting phase. E. Donaldson and Dean An extensionof Welge's two-phaseunsteady-state techniquewas usedby Donaldsonand DeanTto determinethree-phaserelative permeabilitiesof Berea sandstoneand Arbuckle limestone.Oil and water in the core were displacedby gas and the flow ratesof all three phaseswere measuredsimultaneously.Their resultsfor the displacementtestson the two cores startingwith various S*, and S.,,are shown in Figures7 and 8. They minimized end effects by using a high pressuredifferential and high flow rates, and they did not account for hysteresiseffects. The volumes of oil and water displacedwere less in the limestone than in the sandstonefor the same S,,(or S*) and the samepore volumes of gas injected. This effect is presumablycausedby the largerflow channelsin the limestone.The efficiency of a gas displacementprocessis greaterfor a matrix with smallerpores.There is a narrower rangeof saturationsfor three-phaseflow in the limestonebecausethe large vugs may allow gas to flow without transferingenergy to oil or water. The isopermsarepresentedasfunctionsof terminalratherthanaveragesaturations,because the former govern the flow of fluids throughthe core. The resultsof Donaldsonand Dean, shown in Figures9 through 14, indicatethat, at low and constantS' k., for Bereasandstone initially decreased with increasingS" until S" reacheda valueof about50Vo.Furtherincreases in Socausedan increasein k,r. At S* greaterthan I 3Vo,k," increasedso the isopermsbecame concavetoward the gas apex. No explanationof this phenomenonwas suggestedby the authors.At a given S* the k., was lower in the presenceof water than in the presenceof oil, probably becausewater adheredmore stronglyto the rock surfacethan did the oil. The flow path of gas is more restrictedin the presenceof water, sincegas can displaceoil more easily than it can displacewater. For the limestone,k,, was alwaysconcavetoward the gas apex.

The u was gen in the pr

lll

DataPoints:

.4 €

A)

o-

.3

e

E q) O (g O A

t

-

I

Inlecteo. pore votumes

"oo,

.r, /-oas

U, o

A .Z

2

i5 o

100 FIGURE 8.

o

A'

o o -o_ !

INITIAL OIL SATURATION

Fluid flow experimentaldata for Arbuckle limestone.T

Rt::.: phase l :" .:\c

tL..l .rrfl ODd

d \rhuckle t'l .rll three r,n lhc two I r : : : r z c t el n d n ': .lLCOUnt f .l n t c \ t o n e F.:rttccted. E J: : lj lCncy I .r :t.rITt)Wer I r::.:r allow

n.. hccause I .rnJ Dean, a ..rndstone Er lnarr-ases ml. hccame stcJ br the prCrc.llCeOf lhc ,'tl. The Fc ,\rlmore lar.l thc' gas

uw FIGURE 9.

so Gas relative permeabilityfor Berea sandstone.T

The water isoperms are concave toward the water apex. Relative permeability to water was generallyhigher in the presenceof oil than in the presenceof gas, but k.* was higher in the presenceof gas than in its absenceat a constanthigh S*. Both k." and k.* increased

tt2

Relative Permeability of Petroleum Reservoirs

o^ ao/

sw FIGURE IO

so Gas relative permeabilityfor Arbuckle limestone.i

5

A6

ieo

1o

sw FIGURE I l.

so Oil relative permeabilityfor Berea sandstone.T

at constantS,,and S*, respectively,when S, was increasedfrom 0 to 8Vapossiblybecause gas was trappedin poreswhich would otherwisebe occupiedby immobile wetting phases. Also, k.* increasedin the presenceof oil becausethere may be partial oil wetting, so that water was displacedinto larger pores;this was not the casewhen gas was present.

F. Saren Using ; did not cc Sarem'sr

113

so

sw

F I G U R EI 2 . Oil relative permeabilityfor Arbuckle limestone.T

ss

--w

q

Q

oo

F I G U R E 1 3 . W a t e r r e l a t i v ep e r m e a b i l i t yf o r B e r e as a n d s t o n e ' -

n.llrl) because I c t : r n Sp h a s e s . tetirns.so that fL'.cnl.

F. Sarem data for a Bereacore. He method, Saremsobtainedthree-phase Using an unsteady-state did not considerend effectsor saturationhistory, but his methoddid accountfor wettability. Sarem'smethod, which is an extensionof Welge's two-phasetechnique,is relatively fast.

tt{

Relative Permeabilin of Petroleum Reservoirs

Sarr-on k.. the s:r

and D

G. Sa A r Sarai

uscd t

efl-ect oil tlr

so

sw

a\\ulT salur:r \\a\ a

FIGURE 14. Water relative permeabilityfor Arbuckle limestone.T

onlr t

The core is saturatedfirst with one liquid and then flooded with an immiscible unreactive liquid, at leastuntil breakthrough.Then, both liquids are displacedby gas. In the derivation of his equations,Sarem assumedeach relative permeabilityto be a function of the colrespondingsaturationalone. Isopermswere thereforeparallel to the isosaturationlines. The relative permeability to gas was assumedto be dependenton total liquid saturationand independentof the relative wetting properties. The saturationequationsare

Oil r. r c s ul t , shapc n I'ere

a ca_\e the as

H. n' Soz :

S..o,, *

S*z :

S*.ou* *

f", Q

f*, Q

S r r : 1 - S * r - S . , 2

(3) (4)

(5)

Thr \r ater hare I follou t his t a

t. 2.

where Q :

cumulativevolume of injectedfluid (per pore volume)

e : 9 LAO rt

3

and q, : total volumetric flow rate (cclsec),t : time (seconds),and f : fractionalflow. Subscriptso, w, g, and 2 standfor oil, water, gas, and outlet, respectively.The relative permeabilitiesare computed from the following relationships: d (l/Q) k.* : I*z -l-4pp4 1 tl l I \L p " q' Q/

(6)

Thr diagra relati are ot

115

k.* Or,,H d ( l/Q) K"': t"'(-4r!4-r*,, o,J

(7)

(8)

Saremalso concludedthat initial saturationconditionsaffect k..,and k.*, but havelittle effect on k,r. He found that k."/k.* was influencedby initial saturationsin three-phasestudiesin the samemanneras in two-phasestudies.Sarem'sresultsdiffered from thoseof Donaldson and Dean even though both used the sametype of sandstone.

Jc.rnrc'active ir.* .lcrir ation ol :irc corrert :lrc.. The llu:.r:lrrrl ilfld

(l) (4)

(5)

G. Saraf and Fatt A dynamic method using nuclearmagneticresonance(NMR) techniqueswas used by Sarafand Fatteto determineliquid saturationsin Boise sandstone.A volumetricmethodwas used to obtain gas saturations.The experimentaltechniquewas designedto minimize end effects.To maintaina constantpressuredifferential,the gas flow rate was increasedas the oil flow rate was decreased.Saraf and Fatt found no theoreticaljustification for Sarem's assumptionthat three-phaserelative permeabilityto each phasewas a function only of the saturationof that phase.In the water-wetBoise sandstone,however, they did find that k,* was a functionof S* alone.Using waterpermeabilityasthe base,they found thatk,* depended only on the total liquid saturationand was independentof the relative wetting properties. Oil isopermsdeterminedby these investigatorswere convex toward the oil apex. Their resultsare shown in Figure 15. The explanationgiven by the authorsfor this unexpected shapeof the isopermsseemsless than convincing.They did state,however,that in studies where k,* was a function of both S* and S", the systemwas not 1007awater-wet.In such a case,it seemslikely that S* did not remain constantwhen Soor S* was increasedand that the assumptionof constantS* could be a sourceof experimentalerror. H. Wyllie and Gardner Three-phaserelative permeabilityequationsfor preferentiallywater-wet systemswhere water and oil saturationswere determinedby the drainagecycle rather than by imbibition have been given by Wyllie and Gardnerroand are presentedin Chapter 2, Table 3. The following factors should be taken in considerationwhen using the equationspresentedinto this table. l. 2.

3. Xti,'nrl tlow. .l'hc rclative .

(6)

The k,* valuesare normalizedto absolutepermeability. The values of k.o and k., calculatedfrom theserelationshipsare both normalizedto the effective hydrocarbon permeability at irreducible water saturation. Inasmuch as they are normalized to the same base, k.s/k.. values may be calculateddirectly by using these equations.This is not true, of course, for water-hydrocarbonrelative permeabilityratios. The gas and oil relative permeabilityequationsdo not include provision for residual oil saturation.When S* equals S*i.,, k,o is equal to [S"/(l - S*,.,)]ofor cemented sandstone,oolitic limestones,and vugular rocks. To handle residualoil saturation, this relationshipshould be alteredto [(S" - S.,,)/(l - S*,'..)].0

The correlations developed by Wyllie and Gardner can be used to construct a ternary diagramshowing the relative permeabilitiesto oil, gas, and water. In general,the valuesof relative permeability(10, 20,30Vo, etc.) are chosenfirst and then the valuesof saturation are obtained from the correlations. As can be seen from Chapter 2, Table 3, some of the

r16

Relative Permeabilin of Petroleum Reservoirs 10 0 % g a s

orL

100% water

100% oil

gas

I WATER En (xlcs

pfe\ k

the hr \ ()lr\

water

l) F*- t Ftr lx)otA the \\

gas

l() ca

GAS

A. Cr L's c*xarr ttre rr t'tf !:a acrtx(

water

oil

FIGURE 15. Three-phaserelative permeability.'

equationsare nonlinear.Hence, numericalmethods(suchas Newton-Raphson)are required to solve theseequations.Manual interpolationis also possiblefor plotting relative permeability isoperms.

k-s tri rru

effc,,-t S raa

B. \r \iu

tl7

WATER

wut6l olL

o GAs

FIGURE 16. Three-phaserelative permeabilitydata of Caudleet al.rr

Empirical relationshipsprovide reasonableresultsin some casesand very disappointing ones in other situations;consequently,they must be used carefully. Note that most of the previousrelationshipswere developedfor media with intergranularporosity.This points out the huge problem of determining relative permeability curves for naturally fractured reservoirs. The difficulty arisesprimarily from the difficulty (or impossibility)of making this type of measurementon a fracturedcore sample. For totally oil-wet three-phasesystemsin which oil is the wetting phase, water the nonwettingphase,and gas nonwettingwith respectto both, the substitutionof S" for S* in the Wyllie and Gardner equationscan be made for estimationof the relative permeability to each phase.

III. IMBIBITION RELATIVEPERMEABILITY A. Caudle, Slobod, and Brownscombe Using a dynamic displacementmethod on a consolidatedcore sample, Caudle et al.rl obtainedisopermsfor k.o, k.*, and k,*, as shown in Figure 16. They useddistillationto find the water and oil saturationsat eachdata point, and usedmaterialbalancefor determination of gas saturation.Caudle et al. employed a pressuredifferential of 5 to 50 in. of water acrossthe core and usedwater permeabilityas the basevalue. Relativepermeabilityto water k.* was found to be dependenton S*, Sr, and S". These workers recognizedthe presence of some form of hysteresisin the three-phasestudies,but they ignored the capillary end effect. They found all relative permeabilitiesto be approximatelyat minimum valueswhen So was maintainedat the value of S*.. ) arc rc'quired htrr g perme-

B. Naar and Wygal Naar and Wygal12developeda set of equationsthat was discussedin Chapter2. Based

l18

RelativePermeabilin of PetroleumReservoirs

oil-r water-wet

S;

S' w

I

S;

F I G U R E 1 7 . T h r e e - p h a siem b i b i t i o n . r r

on theseequationsthey plotted isopermswith 1007areducedsaturationsat the apexes,as mechanismindicatedthat at the beginning shown in Figures17 and 18. The displacement of the imbibition process,S** (reducedwater saturation)increasedat the expenseof S, at constantS", until no more gas was trapped.Thereafter,S* increasedat the expenseof S., at constantS*. This path is tracedin Figure 17. The locus of all such paths is also shown. Unlike the findings of other workers, Naar and Wygal concludedthat k,.,/k,* is not a flow. On the other hand, the function of S*, for equal valuesof oil recoveryin three-phase is shown in Figure ratio was found to be a function of Sr, and wettability. This dependence 19. The higher the initial gas saturation,the lessthe influenceof wettabilityon k.,,/k.*.Also, the water saturationat a given recoverywas a function of initial water saturationand initial gas saturation.The ratio of S*, for a water-wetsystemto S*, for an oil-wet systemincreased with Sri, and the rate of increasewas an incresingfunction of S*,. For a given S* ratio and with increasingSri. With higherS*,,thereis lesspore space a given recovery,S*, decreased availableand the oil is alreadypushedout into larger channelsbecauseof the higher S*i; therefore,less water is requiredfor the samerecovery. The imbibition water-oil relative permeabilityequationsdevelopedby Naar and Wygal, basedon the assumptionthat l/P.2 equalsCS*, are ""intt' So,i,,,.imb /s*

kr*.irrb

and

\a;

FCrm(

n herc

S * * . , n -, b s * d s * P:

f' t - s* ds* J,, P:

(e)

anJ S

119

o2

water

I .::\, \ !'\.

a\

FIGURE 18. Three-phasedrainage.rl

E ^.,-rnning lfl.; ,'l S. at

and

lf\'::.C tlf S,,

kr,r.irrb

al.,' .htt\\'n. L

: s;:i(s..,, + 3 Si")

(s,,-

l. n()t a

r : : t . 1 r 1 Jt.h e :r [irgure '\lso. ;. Jl .:' .1 lnltial lr

:paC€

I : . : h c r S g, l,t , ;

ai.l \\.rgal,

-

0.5

k,*

:.rittr dfld

S l":J

(l0)

Naar and Wygal suggestedthe following approximationfor imbibitiongas relative permeability:

!n: :.. rcased S

S"o)t(S,, + 2S.,n+ 3S*, - 3S*') ( I - S * ' )"

S**i.in,r,

( I - S*i'",n)

0.5

where S**,.i-t,

:

S* :

S**.druin

l/2 S::, drain;

-

S _ S* '. l -S * ,

s.l,:S"-S"t' l - S * ,

Sl*-S*lS*' l -

and S"ois the trappedoil saturation.

*

S*i

S.t,

(ll)

t20

Relative Permeability of Petroleum Reservoirs 1 A

S*1r,= o'3

't2 o 3

3 ro = 3 ; ; I

(!

L

-l

I

J

J

^(Dlt ^

s w i r r - o . ' l5

{ o =

8

o

u) l { l= lo:

{ lq - l t lr

o

6

l o

.1

.2

.3

.4

.5

.6

I N I T I A LG A S S A T U R A T I O N FIGURE 19.

Influenceof wettability at 40o/crecovery.'l

Wate

Thesemodelswere derivedby assumingthe randominterconnection of straightcapillaries, with a provision for blocking of the nonwettingphaseby the invading wetting fluid. The imbibition water, and drainageoil and gas, relativepermeabilittequationsdeveloped by Naar and wygal were also presentedin the following form: kr*,i,r,b :

k,..i*u :

(l -

(t2)

(S;)o

2S*x;t't {2 -

(l -

2S*x;,,2t

stnce natur Hi: simil [-rn

(l 3 )

and

k,* s_ir: - 2srr)

(r4)

where

SF -' , : s F - s * ' l - S * ,

In theseequationsthe subscript"t" standsfor "trapped" and,,f" for.,free,, C. Land In Land's13work, equationsfor imbibition two- and three-phaserelative permeabilities were obtainedfrom rock properties.Land consideredresidualgas saturationafter imbibition to be directly related to the initial gas saturation.The gas and water imbibition relative permeabilitieswere reportedto be the samein three-phase systemsas in two-phasesystems,

For S.

12l Gas I

I

I d

c I I t

) )

orL

I t

Water X . . rt ' rI l a r i e s . t?-

FIGURE 20.

Imbibition k.,,for a mobile gas saturation.rl

I

Dr .l-'rcltlped

(l2)

since the totally nonwetting and wetting phasesoccupiedthe samepores regardlessof the natureof the other phasespresent. His plots fork.o in the II and ID casesare shown in Figures20 and21. The ID plots are similar to the plots obtainedby Naar and Wygal,12their systembeing an II case. Land's final equationsare

(l3) ^*r It

ds*

5 i I ts'Jr*sgr p:

k,*

It

(14)

t. Kr*

_ -

k.. : m : r e. r h i l i t i e s ir r;:rhrbition Ir ,': : r cla ti ve l:i

.) rtc'ITlS.

II''E

It

r,'fl.,": Y f'ds*

(ls)

(l6)

(t7)

J,, P: For S* increasingand S* constant: k.o :

S:i [2(S** * S",*) * S"r*]

(18)

122

Relative Permeabilitv of Petroleum Reservoirs Gas

GAS

Land of Cc

D.&

Sc cartxr imbib gas\: on oi p€rTn found perTn

UrSlitr

Water

oil

F IGU R2EI.

Imbibition k,,,for a trappedgas saturation.rl

This equationis similar to the one obtainedby Corey et al.2 for the drainagecondition. When all the gas is trapped: k.. :

s:i(2s**+

S"r*) - S,,r*[S*i.+ 2/C(S;, + llC{lnSr,/Sr,})J

where (l _ S*r*)

S .* :

S. - S.l-S*, -S,,,.

(--

Sin a mat requi model Nolen condi extre the ea \to gas v

Sot

l-s*, S. - Sru l-S*, I (sr,* ),.u*

Thr to dra are c(

shour

S", Sr,* : l -S *,

S.r* :

E. Sr

co. i

S* :

S.u* :

(l e )

u'etti t hent bet uc r elat value Corer

I

123

s**: s

v()m

GAS

p .,'ndrtion.

(19)

S* - S*. l-S*.

: minimum residualoil saturation

Sr, :

trappedgas saturation

S.r, :

trappedoil saturation

Land's correlationsdid not considerhysteresissincehis derivationwas basedon the work of Corey et al., which did not include hysteresiseffects. D. Schneider and Owens Schneiderand Owensraperformed steady-stateand unsteady-state tests on a variety of carbonateand sandstonesamples, and found the relative permeability to oil during an imbibition processin a water-wet systemto be insensitiveto the flowing gas phasewhen gas saturationwas increasing.Oil relativepermeabilitywas found to be primarily dependent on oil saturation.It was reported that residual oil significantly reducedthe gas relative permeabilityin a water-wetsystem.The gas relativepermeabilityin an oil-wet systemwas found to be insensitiveto the presenceof a residualoil saturation.The nonwettingrelative permeability-saturation relationshipin three-phaseflow was reportedto dependon the saturation history of both nonwetting phasesand on the ratio of the saturationsof the two wetting phases.In some casesthe nonwettingrelative permeabilitywas found to be lower then the two-phasevalue due eitherto trappingof a nonwettingphaseor to flow interference between the nonwetting phaseswhen both were mobile. For some tests the nonwetting relative permeabilityvalue for three-phaseflow was found to be higher than the two-phase value. The authorsdiscussedthe reasonswhy their resultsdid not fullv asreewith thoseof Corey et al. B. Spronsen The centrifugemethod, alreadyproven for two-phaseflow, was extendedby Spronsenrs to drainagethree-phaseflow in a water-wetsystem.Oil isopermsdeterminedby Spronsen are concavetoward the l00Vo oll apex. He discussedthe adverseinfluenceof immiscible CO, injection on the shapeof three-phaseoil isoperms.The resultsof his investigationare shown in Figure 22.

IV. PROBABILITYMODELS Sincethe experimentalproblemsassociated with three-phase flow aredifficult to surmount, a mathematicalmodel appearsto be an alternateapproach.The correlationsdiscussedearlier required some type of experimental three-phaseflow data. On the other hand, probability models as formulated by Stoner6'r7and modified by Dietrich and Bondor'8 and later by Nolen as cited by Molina,'eassumethat two-phaseflow behaviorcanbe usedas a limiting condition for three-phaseflow. Water-oil-gasflow can be boundedby water-oil flow at one extremeand oil-gas flow at the other. While someof thesemodelscan considerhysteresis, the earlier correlations,such as Land's,r3cannotdo so. Most probability models assumethat gas relative permeabilityis dependentonly on the gas saturation: kry

k., (Sr)

(20)

124

Relative Permeabilin of Petroleum Reservoirs gas

will t satur Stone result Wate corTe expre

where

water OIL ISOPERMS

gas and

.0o002

Fayen

- o o o1 -ooo5 - o o1

where

water

Sto resid oil rel

oil WATER ISOPERMS

FIGURE 22.

Data of Spronsenfor Berea sandstone.r5

Similarly, it is assumedthat the relative permeabilityto water is dependentonly on the water saturation: k,* : k.*(S*)

(2t)

Oil relative permeability,however, varies in a more complex manner.Theseassumptions have been confirmed in laboratoryinvestigationsfor a water-wetsystem. In a water-wet system, gas behavesas a completelynonwettingphase, but oil has an intermediateability to wet the rock. The relativepermeabilityto oil in a water-oil-gassystem

where relatir phase kr. val Altt relativ in whi of the most (

12s will thereforebe boundedby relative permeabilityto oil in a water-oil systemat low gas saturationsand by relative permeabilityto oil in a gas-oil systemat low water saturations. Stoneattemptedto combinethesetwo terminalrelativepermeabilitiesto obtaina three-phase result by using the channel flow theory in porous media and simple probability models. Water and gas three-phaserelativepermeabilities,accordingto Stone,are the sameas their correspondingtwo-phaserelative permeabilities.In his first model, Stone developedthe expression:

k.. : s;P",F*

(22)

where S. - S.,,

S.,* :

l-S*,-S.. k_.

P*:

r=;

(2-Phase)

s * - S*i

S*.*

l-S*, -S.,

9, : +T

(2-phase)

and

S,:

s .[-s*, -s..-Sr.

Fayersand Matthews26suggestedthat S.. :

c{.S,,,* + (l - o,) S,,.o

where -: lr -

S ' I -S*.-{

Stone'searliermodel did not agreewell with datainvolving the dependence of waterflood residualoil saturationon trappedgas saturations.Stone's secondmodel gave three-phase oil relative permeabilityas k,,, :

trnlr tln the

( 2 1) a..untptions I ,rrl has an l- t.r. rt stem

(k..,* + k,*)(k",* + k,s) - (k.* + k,s)

(23)

where k,o* and k.* representoil and water relativepermeabilitiesfrom two-phase,oil-water relativepermeabilitydata;k,o,and k,* representoil and gasrelativepermeabilitiesfrom twophase,oil-gas relative permeabilitydata. Equation 23 may yield unrealisticresultsat low k.o values. Although it seemsreasonablethat one should be able to combine the two two-phase relativepermeabilitiesto arrive at three-phase dataat leastfor water-wetsystems,the manner in which they have been combined in thesemodels may not accountfor the total physics of the process.Theseprobability modelsstronglydependon the assumptionthat there is at most one mobile fluid in any channel.That is, Stone's assumptionimplies that water-oil

126

Relative Permeabilin of Petroleum Reservoirs

capillary pressureand water relative permeabilityare functionsof water saturationalone in the three-phasesystem,regardlessof the relativesaturationsof oil and gas. Moreover, they are the samefunction in the three-phasesystemas in the two-phasegas-oil system.Stone's secondmodel generallypredictsthe correctoil relativepermeabilityin the three-phase system if the relative permeability at the end points is equal to one. Stone points out that when his secondmodel yields a negativek,,,, this implies a completeblockageof oil and as a result k." equals zero. The Stone models accountfor hysteresiswhen water and gas saturations are changingin the samedirection. Dietrich and Bondorr8applied Stone's models to publishedthree-phasedata and found them to be only partially successful.They found that it was necessaryto modify Stone's secondmodel for the casewhere gas/oil relative permeabilityis measuredin the presence of connatewater. They pointed out that, at irreduciblewater saturationand zero gas saturation, Equation 23 reducedto k,. :

(k..,*)(k.,,*)

This expressioncan be valid only if both k..,* and k.,,, equal unity. Since k,,, at S*. is frequentlyless than one, Stone's secondmodel has some limitations. Dietrich and Bondor adjustedStone'smodel by normalizingit with k..,.* to obtain:

k..:

fr

t,0,"** k,*)(k.",* k.r)l- (k.*+ k.g)

(24)

where k.o.* is the oil relative permeabilityat connatewater saturation.At irreduciblewater saturationand zero gas saturationthis equationreducesto: k-. :

(k"'o )(k"'t ) k..r.*

This model tends to predict incorrectoil relative permeabilityvalues (magnitudelarger than unity) for valuesof k,,,.* < 0.3. Nolen, as referencedby Molina'e has taken into accountthis problem and suggestedthe following model which remainsboundedas k.,,.* approacheszero: k.,, :

k_

k..,.*)*

Na,,.o

t.

+ k,*:: '*

* k.s - (k.*, + k.s)

(25)

k..r.*

V. EXPERIMENTALCONFIRMATION Three-phaserelativepermeabilitystudiesare still in an early stageof development.Little has been done on the experimentalconfirmationof imbibition correlationsand most of the correlationsavailableare for imbibition. Early work was done primarily on unconsolidated sandsand the effectsof wettability and hysteresiswere not recognizeduntil recently. Donaldsonand Kayser2ohave categorizedthe reasonsfor divergenceof experimentalthree-phaserelative permeabilitydata as follows: l. 2. 3. 4. 5.

Errors introducedin saturationmeasurements in variousexperimentalmethods. Errors introducedby neglectof capillary end effectsand saturationhysteresisphenomena. Variations caused by use of different oils, brines, and cores which could exhibit different wettability characteristics. Assumptionsmade to facilitate experimentalproceduresor calculations. Inadequacyof mathematicalformulationsto representthree-phaseflow conditions.

Tab perTne autho syster rock s air . ca

Thrr diesel Berea stonei is ofte Per tical m ia1'ab Yh. gt Proble u'hen r remo\ p€rTne itored Brir measu and th liquid or b1' r The be stu to infl conce Bou and pe ma)'p Penn I gators staten Ina rate of mEasu may b should with tl the ga needle be mei or '*'it l is des

SO OS lr

The empirical methods,though seeminglysimpler, suffer from simplifying assumptions that have limited the rangeof saturationhistoriesthat can be simulated.

press

r27 Table I is a chronologicallisting of the experimentallydeterminedthree-phaserelative permeabilitiesthat have been reported.2rIn all of the studiesincludedin the tabulationthe authorsusedrefined oils in order to minimize oil-wetting;they assumeda totally water-wet system.In caseswhere a single core was used,the influenceof the saturationhistory of the rock samplewas frequently ignored. The gasesused in the studieslisted in Table 1 were air, carbon dioxide, and nitrogen.

I t , ' : : . r l t l n ei n D r c , , cr r . t h e y lc::: Stone's p h . : . e\ \ S t e m hu: ',rhcn his rJ .:. u result F ..rlurations

VI. LABORATORYAPPARATUS Lt .:n.l tirund X J ; : , .S t o n e ' s th. l.rcscnce 3f'

}.

-.r:

\iltU-

.rt S,.. iS

r , x : .lrn :

(24) l u . . ^ l c* a t e r

n r : . . . 1lca r g e r U i i J . t r 'd th e

(15)

pc:Jnt. Little I r : r , ' . Io f t h e

i' il

!

B t l : i . r l i t ra n d le g' 'r12a6,1ta Ir l,rllttws: 3th,rJr. i^*n,'ttlena. tru.J c'rhibit

9()n!illlons. a..unrptions

i

t

relativepermeabilitystudieshavebeenconductedusing refinednonpolaroil, Three-phase dieseloil, Soltrol, kerosene,hydrocarbonfractions,brine, nitrogen,air, and carbondioxide. as well as Arbuckle limeBerea, Boise, Torpedo, Tensleep,and Weeks Island sandstones, stoneand unconsolidatedsandsampleshave beenusedfor the flow media. Bereasandstone is often preferredbecauseof its uniformity and generalacceptabilityas an industrystandard. in this field have indicatedthat the most pracPersonalcommunicationswith researchers is gravimetric.Othermodernmethods,suchas gamma tical meansof saturationmeasurement ray absorption,X-ray absorption,NMR, etc., are unnecessarilyexpensiveand elaborate. aresufficientlyaccurateand relativelyinexpensive. The gravimetricsaturationmeasurements Problemsmay be encounteredwith gravimetricsaturationmeasurements, however,especially when gas is used in the presenceof volatile oil. Therefore,core holderswhich permit rapid removal of cores (without the removal of rubber sleeves)should be used when relative permeabilityis determinedby steady-state methods.Wettability of the core shouldbe monitored either by the centrifugaltechnique23 or an alternativemethod. Brine saturationmay be determinedwith satisfactoryaccuracyby',electricalresistivity when nonpolaroil is employed.Oil saturationmay be obtainedgravimetrically measurement and the gas volume may be computedas the differencebetweentotal pore volume and total liquid volume. The oil and water flow ratesmay be obtainedby a simpleburettearrangement or by flowmeters.The gas flow rate may be obtainedby use of a gas flowmeter. The effect of wettability on the relative permeabilitiesis an importantfactor that should be studied.The changeof wettability in a core from oil-wet to a water-wethas beenknown to influencerelative permeabilities,but no definite conclusionsare found in the literature relativepermeabilities. concerningthe influenceof wettability on three-phase Boundary effects should be eliminatedby using core plugs at either end of the test core and performing the experimentat reasonablyhigh flow rates.A semipermeablemembrane may precedethe core plug at the inlet end for properdistributionof the phases.A modified Penn Statemethod of relative permeabilitymeasurementmay be used, since most investigatorsbelieve that the Penn Statemethodgives betterresultsthan any of the other steadystatemethods. for each phase,one needsto measurethe flow In addition to saturationmeasurements rate of each fluid and the pressuredrop while making the steady-state relativepermeability measurements.A gas dome may inject fluids into the core and a back-pressureregulator may be used to maintain a constantpressureat the outlet end. Also, the gaseousphase shouldbe bubbledthrough the oil supply tanks. This procedureensuresthat the oil is saturated with the gas before it enters the core. As a result, there should be no masstransfer between the gaseousphase and the oil inside the core. The gas flow rate may be regulated with a needle valve, with a large pressuredifferential acrossthe valve. The rate of gas flow may be measuredeither by collecting gas by displacementof water for a known length of time, it or with a soapbubble meter or a wet-testmeter. For pressuredifferentialmeasurements, possible into measuring device be small of fluid the as as is desirablethat the displacement so as to minimize error. Hence, use of a regularmanometeris not possible,but differential pressuretransducersmay be used. The connectionsfor measuringpressuredifferential can

r28

Relative Permeabilin of Petroleum Reservoirs a

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Relative Permeability of Petroleum Reservoirs

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131

CORE HOLOER

FIGURE 23.

Schematicdiagram of three-phaserelative permeabilityapparatus.

be made through semipermeablemembraneports. The capillary tubesconnectingthe transducermay be insertedand cementedin placeabout I in. from eachend of the testapparatus. have the admethodsof three-phaserelative permeabilitymeasurements Unsteady-state vantageof beingrapid. Oil and watermay be displacedby gasto duplicategasdrive processes used in enhancedrecoverymethods.However, the calculationof isopermsfrom laboratory data requiresanalytical solutionsof the partial differentialequationsdescribingthe threephase fluid flow. Some early studies have made erroneoussimplifying assumptionsin process.Reliablevaluesof relative describingthe dynamic condition of the unsteady-state permeabilityas a function of saturationsmay be obtainedby mathematicalsimulation of laboratory data using finite difference calculations.20 Capillary pressuredata should be obtainedfor gas-oil, water-oil, and water-gassystemsto provide basic data necessaryfor three-phaserelative permeabilitycalculations.Solubility of the gas in the liquids employed in the study should be determinedbefore thesecalculationsare performed. A schematicdiagram of the apparatusused for three-phaserelative permeabilitymeasurementis shown in Figure 23. The core holder, which has ports for differentialpressure allows rapid retrievalof the core. Temperatureis controlledwith a Propormeasurements, tional Controller connectedto a heatingtape wrappedaroundthe core holder. In order to eliminatepulsationof flow normally associatedwith pumps, fluids are injectedby applying gas pressureon top of the fluid in a tank equippedwith appropriaterelief valves. Solenoid valves and level controllersmaintain a constanthead of fluid in the supply tanks. Filters are provided in the supply lines of each phasebeing injectedinto the core holder. Check valves preventbackflow of each of the three phases.A cross sectionof the core holder is shown in Figure 24. Auxiliary equipmentincludesan accuratebalance,electricalresistivitymeasurementsystem, level controller,chart recorder,differentialpressuretransducer,cylinders,compressed air and regulators,and a humidity oven.

r32

Relative Permeabilin of Petroleum Reservoirs

D I F F E R E N T I AP L R E S S U R EP O R T S ANNULAR P R ES S U R E PORT

2"

<-._

rO N cv)

-1.125>

-19" FIGURE 24.

+1JJ\

Diagram of a core holder.

VII. PRACTICALCONSIDERATIONS FOR LABORATORYTESTS The literature cited contains a large amount of information on factors affecting the.laboratory investigationof relative permeability.The following listing, however, cites iome practicalconsiderationsthat have not been widely discussedin the literature: l.

2.

3.

4. 5. 6.

7.

8.

9.

If a pump is used to inject fluids into the core, the packingmaterialshouldpreferably be [email protected] other packing materialscontain silicon and carbon which may dissolvein injected fluids and affect the wettability of the core. When brine is used as one of the fluids, all metal parts of the systemshould be of stainlesssteel. One-eighth-in.tubing offers excellenthandlingcharacteristics. Tygon tubing is recommendedif the pressureis not too high. Most electronic differential pressuretransducershave good linearity and hysteresis characteristics; however,if possible,the transducershouldbe recalibratedat leastonce per month. While changingpressureson the liquid storagetanks, it is importantnot to exceedthe backpressurerating of the solenoidvalves. Every effort should be made to ensure l00%osaturationof the wetting phase before startinginjection of the nonwettingphase. In a steady-stateexperiment, input flow rate should equal the output flow rate for each phase.In many cases,this condition is tediousto achieve. Some extraneousmaterial may be noticed in the output lines. It must be determined whether the particles are fines from the test sampleor bacterial matter. A bactericide may be used with caution not to alter either the wettability or the resistivity of the core. Often the resistivity meter utilizes chamoisleathercontactsat either end of the core holder. The contacts should be kept immersed in brine to prevent changesin the readings. It has been noticed that the position of the outlet tubes going into the measuring cylinders affects the pressuredifferential readings.It is recommendedthat the tubing outlet be kept at the same level as the core holder to eliminate gravitational effects.

10. c

d ll. E 12. lr d p

The The is disp same c equatio perTne and De Dean's than th toward becom k,,, inct S" betr which the tw< In tl

r33

+---:5 - 5 ' .

-

ESfS trnr thelab'. . ila\ SOme

FIGURE 25.

Comparisonof three-phaseoil relative permeabilitydeterminations.

10. kl prclcrably rr:rih may f1,,ultl be of str. . Tr gon ld nr.teresis a l i er \ t o n c e o cr;ccd the lhr.c betbre l'3tc ltr[ gOCh r tjctcrmined t huitcricide Irr rtr tlf the I tri thc' COf€

nsc. in the : rnca:uring I tr rct u b i n g url ct't'ects.

Gas in the transducerlines seriouslyaffectspressuredifferentialreadings.The transducer should be bled of gas at frequentintervals. I l. Every effort should be made to eliminateend effectsas describedby Batycky et a1.22 12. If possible, the wetting characteristicsof the core should be frequently monitored during the relative permeability experiments.The centrifugemethod23may be employed for monitoring wettability.

VIII. COMPARISON OF MODELS The following sectionpresentsa comparisonof some of the modelsdiscussedearlier. The equationof Corey et al.2 for three-phasek,,,valuesis valid for a systemin which oil is displacedby a gas. Donaldsonand DeanTobtainedthree-phasek..,valuesfollowing the same displacementmechanism.Thus, we have an opportunity to observehow well the equation of Corey et al.2 fits data provided by other workers. Three-phaseoil relative permeabilityvaluescalculatedby theequationof Coreyet al.2werecomparedwith Donaldson and Dean's data. The isopermsobtainedare shown in Figure 25 along with Donaldsonand Dean's data as a basis for comparison.The Corey et al.2equationgives higher k,. values than those obtainedby Donaldsonand Dean. Isopermsby Corey et al3 are less concave towards l00%ooil saturation.Both methodsare in agreementin predictingthat the isoperms becomeconcavetoward l00Vo S" and decreasingS*. The Donaldsonet a1.23'2a data show k..,increasingup to an optimum S, value and then decreasing.This is evidentfor valuesof S" between30 and 6OVoon this Berea core. The Corey et al. correlationsgive isoperms which show k,o to increaseas S.,increasesat the expenseof S*. The discrepancybetween the two methodsis larger at low S.,values. In the second comparison, data of Schneiderand Owens2shave been used to obtain

134

Relative Permeabilin of Petroleum Reservoirs

6. Snel 1 7 1. 7. Don tt.l' .ll

8. San J..t 9. Sarz re\() 10. \'l'rl

Nolen,s Model -o--
Meihod

of Naar & Wygal

Pnrh Cau dete 12. Naa 196 13. L.an l|

PrttP 14. Sch char 15. \'an ofP 16. Stor 17. Stol t:. 18. IXcr at th 19. lltol SPE 20. Dorl rcF\ 21. llar FIGURE 26.

Comparisonof three-phaseoil relativepermeabilitydeterminations.

isopermsby Nolen's modelreand by Naar and Wygal's correlation.r2Few dataare available in the literature that show how the latter methodcompareswith experimentalvaluesor other correlations.Figure 26, however, provides such a comparison.Schneiderand Owens obtainedgas-oil drainagedata in the absenceof connatewater; their oil-water imbibition data is for a water-wet system. Theoretically, the Dietrich and Bondorr8or the Nolen model should give the same results as Stone's second model, since gas-oil data used in this comparisonhave beenobtainedin the absenceof connatewater, i.e., k.o"*equalsunity. As in the earlier comparison,the discrepancybetweenthe two methodsis evident at low S" values.Another point to note is the evidencethat k," dependsonly on Sovalues,especially at low S" in Naar and Wygal's correlations.There is a slight indication in both methods that k," isopermsbecomeconvex towards the l}OVo So apex at high S".

REFERENCES l. Leverett, M. S. and Lewis, W.8., Steadyflow of gas-oil-watermixturesthroughunconsolidatedsands, T r a n s .A I M E , 1 4 2 . 1 0 7 . 1 9 4 1 . 2. Corey, A. T., Rathjens, C. H., Henderson, J. H., and Wyllie, M. R. J., Three-phaserelativeperrneability, Trans. AIME, 201,349. 1956. 3 . R e i d , S . , T h e F l o w o f T h r e eI m m i s c i b l eF l u i d si n P o r o u sM e d i a , P h . D . t h e s i s ,U n i v e r s i t vo f B i r m i n s h a m . E n g l a n d1 9 5 6 . 4. Snell, R. W., Measurementsof gas-phasesaturationin a porous medium, "/. Inst. Pet., 45(428), 259, l 959. 5. Snell, R. W., Three-phaserelative permeabilityin an unconsolidatedsand, "/. Inst. Pet., 48(459), 80, t962.

Prcv 22. Batand r l 9 t rI 23. Doo efllc 24. Don oi l' 25. Schr char 26. Falr relat

135

b:. ' \:

.. 4 Wygal

6. Snell, R. W., The saturationhistory dependenceof three-phase oil relativepermeability,J. Inst.pet.,59, 4 ' 7 1 .1 9 6 3 . 7. Donaldson, E. C. and Dean, G. W., Two- and Three-Phase RelativePermeabilityStudies,U.S. Bureou of Mines, Washingron,D.C. , #6826, 1966. 8. Sarem, A. M., Three-phaserelativepermeabilitymeasurements by unsteady-state methods,Soc..pet. Eng. J . . 9 . 1 9 9 .1 9 6 6 . 9. Saraf, D. N. and Fatt, I., Three-phaserelative permeability measurementusing a nuclear magnetic resonance t e c h n i q u ef o r e s t i m a t i n gf l u i d s a t u r a t i o ns, o r ' . P e t . E n g . J . , 9 , 2 3 5 . 1 9 6 7 . 10. Wyllie' M. R. J. and Gardner, G. H. F., The generalizedKozeny-Carmanequation,its applicationto p r o b l e m so f m u l t i - p h a s ef l o w i n p o r o u sm e d i a , W o r l d O i l , 1 4 6 , l 2 l . 1 9 5 8 . ll. Caudle, B. H., Slobod, R. L., and Brownscombe, E. R., Further developmentsrn the laboratory determinationof relative permeability, Trans. AIME, 192. 145, l95l . 1 2 . N a a r , J . a n d W y g a l , R . J . , T h r e e - p h a s iem b i b i t i o nr e l a t i v ep e r m e a b i l i t y ,S o < 'P . et. Eng. J., 12,254.

r96r.

arc r\ ailable iluc. trr other | ( )ricns obb r i . r t r o nd a t a frrrlcI model u\'!l in this t l . . r n i t v .A s nt .rt low S" s. c.pecially rrth rncthods

F l . : . , : . ' Js a n d s , It r' 'i Perme"'rngham. I ir .3i j'rr ,. :.

r5Q

-:i9). 80.

13. Land, C. S., Calculation of imbibition relative permeability for two- and three-phaseflow fiom rock properties,Soc. Pet. Eng. J., 6, 149, 1968. 14. Schneider, F. N. and Owens, W. W., Sandstoneand carbonatetwo- and three-phase relativepermeabrlity c h a r a c t e r i s t i cS s ,o c . P e t . E n g . J . , 3 , 7 5 , 1 9 1 0 . 15. Van Spronsen, E., Three-Phase RelativePermeabilityMeasurements Using the CentrifugeMethod. Society o f P e t r o l e u mE n g i n e e r s / D e p a r t m eonf tE n e r g y ,T u l s a ,O k l a . , # 1 0 6 8 8 , 1 9 8 2 . 1 6 . S t o n e , H . L . , E s t i m a t i o no f t h r e e - p h a sree l a t i v ep e r m e a b i l i t yJ, . P e t . T e c h . , 2 , 2 1 4 , 1 9 7 0 . 1 7 . S t o n e ,H . L . , E s t i m a t i o n o ft h r e e - p h a s e r e l a t i v e p e r m e a b i l i t y a n d r e s i d u a l o i l dJ a. toaf ,C a n . P e t . T e c h n o l . , t2, 53, t913. 18. Dietrich' J. K. and Bondor, P. 8., Three-phase oilrelative permeabilitymodels,paperSPE 6044 presenred at the 5lst Annual Fall TechnrcalConferenceand Exhibition of the SPE, New Orleans. 1976. 19. Molina, N. N., A systematicapproachto the relativepermeabilityproblemsin reservoirsimulation,paper SPE 9234 presentedat the 55th Annual Fall TechnicalConferenceand Exhibition of the SPE, Dallas, 1980. 20. Donaldson, E. C. and Kayser, M.8., Three-PhaseFluid Flow in PorousMedia. DOE/BETCilC-8)t4. r e p o r tp u b l i s h e db y t h e U . S . D e p a r t m e not f E n e r g y .B a r t l e s v i l l eO , kla., April. 1981. 21. Manjnath, A. and Honarpour, M. M., Investigationof three-phaserelative permeability, SPE 12915 presentedat the Rocky Mountain RegionalMeeting of the SPE, Casper,May 20-23, 1984. 22. Batycky, J. P., McCaffery, F. G., Hodgous, P. K., and Fisher, D. 8., Interpretingcapillary pressures androckwettingcharacteristicsfromunsteady-statedisplacementmeasureP meetn. E t sn, g so . Jr .' ., 6 , 2 9 6 , t 9 8 l. 23. Donaldson, E. C., Thomas, R. D., and Lorenz, P. 8., Wettabilitydeterminationand its ef-fecton recovery e f f i c i e n c y ,S o < 'P . et. Eng. J., 3, 13, 1969. 24. Donaldson, E. C. and Dean, G. W., Two- and Three-PhaseRelativePermeabilityStudies,U.S. Bureau of Mines, WashingtonD , .C., #6826. 1966. 25. Schneider, F. N. and Owens, W. W., Sandstoneand carbonatetwo- and three-phase relativepermeability c h a r a c t e r i s t i cS s ,o r ' .P e t . E n s . J . , 3 , 7 5 , 1 9 1 0 . 26. Fayers, F. J. and Matthews, J. D., Evaluationof normalizedStone'smethodsfor estimatingthree-phase r e l a t i v ep e r m e a b i l i t i e sS. o r ' .P e t . E n g . J . , 4 . 2 2 4 , 1 9 8 4 .

r37 APPENDIX SYMBOLS

A A, a B b C F g h I k L m N n P

a q R r S

s*

: : : : : : : : : : : : : : : : : : : : : : : : :

o

: : : : : : : : : : :

0 .1,

: :

SL

T Z ct

p 0 \ f.r

area constant adhesiontension materialconstant formation volume factor constant materialconstant constant fraction gravitationalacceleration thickness injectivity resistivityindex permeability length exponent number of barrelsof oil exponent pressure volume volumetric rate radius resistivity radius saturation distancein directionof flow reducedsaturation total liquid saturation time velocity vertical coordinate constant constant angle lithology factor viscosity surfaceor interfacial tension porosity immobile saturation

Subscripts : absolute a av : average - critical c : capillary cw : connatewater : displacement D : displacingphase d de : immobile displacingphase - equilibrium e : external(radius) : effective : free f :gas g - initial i : index number : irreducible imb : imbibition irr : irreducible : liquid L LR : residualliquid : minimum m mf : mud filtrate : nonwetting n :oil o : measuredat 1007oS* (resistivity) ob : trappedoil : produced p - relative r : residual - solution s SL : total liquid STD : standardcondition : total T - trapped t : water w : well wt : wetting xo : flushed zone