Relative permeability of petroleum reservoirs

Citation preview

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

''J.: ntJtCnali\ --.:.,'nrhlc cl'lirfl

Ir

- :..F)n\lbilit\

l 5

.\l'lllcn c()n5enl

r

. - .

I

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

Dr. lfcL lhc \ltntrna .{r(arrnl hrr r\rfi.Rr{le

I

tnLlt.rs

t>

nl rstn rrrluhng

drc

h t-;xrlrr

Ti

:

lrrya

I

\lrsr.n.R.i R.{1. [}r }ri (-}rrrrrrr.n r I rcrtr rrltcrj t

.rl

f- lldrr e Fb t)

qrtYln\ll

Erjt

n (tlr.run

DcFtur

r

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.

ctf,

rh t

n

m n l \l fslc CLI

tr I

u I t\ I

rl

ru rltr tt t

u

ll*

tu trl t I I

n I

r|

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.

Tbc I hr crth rrl

r3th\

c{ehlr. \,ilUt-3ll irlurltl thc crr Itrf ft\ thc Ha

ln tt thc tc. drrqlg urcfrr| fa nx

A.h Tht d'er a d ' Frgun nrun alrr P Thc t r alCr Ftrst .r hrs L-Tth

rltc\ rlctcn rnU\\ ktt t rrcrg tlr .i Th than TTE:N

a. flt

Appendix Symbols.

Itr lfi'

....... 137

rnarl ln ci r-all, thYl.

6-i

Chapter I

66 66

MEASUREMENT

OF ROCK RELATIVE

PERMEABILITY

666,\

I. INTRODUCTION

hs h\

6\

-: --l

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

\o \.o

/

> {-i- ^ -o-

o

a

Theoretical saturation gradient

fnf low face

1>

15

20

0 lel -.

. ICsr-

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

a r _ l

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

ns:ii tu t'br

cr';rd crl n .trlf! fl:e rc ;rrbt lrl tw.l. Ilr l{rr 4rS

r

LT TLr r rqSl r&s rl

*&r

bFr hfrl G F r df r f E Hild

rbd

t-q lbr H

FLOWMETER

C:'- .'.

I . B-. li

la_r\'

--i,'.j. . -, ' \v\3

Jl *'

.- ACrC - . -:.rquc

l&' "

'hft

-:l lhc ..--l.rl

l.-.

-

:q;rJ ....ii'n'l

-..

. .h: Cl -,crlxrJ

3a. lqL. : I o: :-i

.jr\e\

:cri br drc ';;.,\i trf

brc:-h:

br

:

r alue

. rimple

tst:n i. lrr h€ lin5 :-.c tcrt :

Ia- .1c...'nhed ^ I an; l-rc'tz. aF:'-::.1tu\ls J ti^..tl{e..ler lct -,-: 'xrtlc't Th. :-i;..urc'

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-

-

- llrr507o

q

O -.1

.

v

g

i rr

;

= :

I

c,

'

a

-

f,u

C

\n o\

o

'=

,c!

H

;

!

/

-aE

E '' u

\n O,

E =

) C

- i '

gl

;,,

C

l

= c ' t(Oo f V

= ; E - !

>

l

F i l a - Y , . -.

E z ' =E P .\./ i Fco

.

;

e # F E

a

)

> e ? = g.qr.>, .\-,/ 1ia3

:

^

{

t

129

(n

U)

2 (n

l

a q;

J

(h

U)

3

R

.o -v a -

,,.J -

-

;

t

R

-i,

E : 7 E P

E E y , E

> . =

> . - €

F - =

OrJ9

0J

0J

=

o!>

E

.

'j ,n u

Q

j ; E?; i 5 e.e 9Eyep .ef€; . ; , v t E . , . iE > : . E

O :

r

-

.

E bO

E . = = = = t E . = - t z

. = ! 2 e. ?- Z ? *

!

E

-a'*

Ere

O

E 9

> = cJ .

brJ= F

.

t r

i

l F

+

R

l

-

=

i

u

qJ =

i=E f

*=E*€

r v

s t

V a

x t z =

4 a

-

-

C)

-

v)

t t Z

'E*,H:

biH

",

u o

B g

. . : = - i

c!i; sE€lEeEr=:;EEEg ' 5 E : gi ; s : : * E ! ; s ; : * E s s + * ; t E! e :; : ! E$ * s; ; E sE€ E * $E F r 3 E E F E= > 3 i : € B ! 3 ; i ' P E

-

9 r y >. g)

L J

P

E g U E E : E "€ f = E H E I E = 3

g

I-lJ

I

'

: 1

Z a i t ' Z i . ' .

E ; E - €E. EE E

.=

s.)

,

' - ' =L F* ^

>

9 . = F i i o o l = E - 4 r i ; . l b P

oo

r -i

v

= 13E€ i i E E jz = UI e

i-e-+E=32 q) L

.

: . Er

t > 9 ! =

. "

o

g'rtil,

€. "3 3: 3 = 'E E - :a ;o 9 _ i E

E

=

.

- l t -

e.,

F . - 4 _

F . - E 4 l

gEe? lri=-=

Z y?EE i=E ' .& P r -I ,; ' i E ! +=.E Zi-==.7=;='=

iF si E" _! a i

1 a

l

F . g *

S

b

HE f i - . i E H *

: : i ; E T q ?* U !

s t I i ; = € n [ g t * [ i t n l i .€ , ; =€ g j=;=3-;;l:-$t € l x e €El : g i $ $ i g l n

(.)

c.l o\

|r)

t-A

q..)

I

U)

(h

6

d

trc)

o -

(n

130

Relative Permeability of Petroleum Reservoirs

U)

ra d

(na

l o l t o l

(n ._i -

(t)

A L (n

>.

:itr . o E . !

!

r;l t s

L1

FI

a

z 3 rh

3 a r-'! -

tI

j , ; o .r, u Y v t

II

-

3

T ; E

= g E ) z J

J

)

1

tr

z -l

? -

3 Fl

tr .c)

j

-

^)

O F F € J

O F

= < = n x Y r-r -t U) o ( .o* Enr F 'Jr -

& 3 FF H

rfr

z-

E-r (t)

>';

9 O

; J E t r ( D

Fr

z

& -

E F

()

rrt

r -

F 8 E : L t ' , >,

9

S

U

=

.

u - - - 2 F 9'6 '

i

v

LIJ

'

d

a 0

)

E -

A

€s -5?5 :; $ !1

i.= t

U)

3

3 q) r Q ) o =

>.-'

.r, I .l

8.s

€ . Y Z

> i i : /

E E E :

iL l

X c t r

. = e

. o ; :

-

F U 3 g ,;,()

L

ri

E F,9

9 q r S olJ

E X

.E;

ii ' U)

x

tl J

a E =

rco

(.)

L q )

@

g " A -

v

.

' rt

E

l-

-

" d

s

I

t^

O

#

a

E 9 E : v (h

v

i fi: _

5 s = 3 ;, g , €5geTU;e 2 =

a.l co

: F A o ,

E 4

c

t = r " 3F

;

9 i x : Y! i l ' ,: ,; , a ; s I

I q,,

,

E I, -: ; : x . E , . ,E _y;X

V ^ N

=

tl rF FI

o.r O

t 't , ,=' gFg E* Egl; lE: i- ,Y( Ei E

o Z x 9 r "

2

.

H J?o

q)

lr( rh

z

d

i E "

; ; g 9 J - t r H > ' 13 o) 92i-= = d - t E t r i F

E#* :':

c : i

X

.

-

Fl

-

q)

L

U)

be ma< ducerl Uns vantag used ir data re pnase deicnt perrne labora obtairx threein the Asr ureme measu ti-onal elimin

Pt &11-

valves are pr( valves shou'n Aur t em . lt air an