.. 1%=

Calculation of Imbibition Relative Permeability forTwo- and Three-Phase Flow From Rock Properties

CARLON S, LAND I USBt4MEMBER AIME LARAMIE, WYO.ABSTRACT

Relative permeability /urrctiovs are developed {orboth two- arrdthrec.phase systems with tfie satr.:ationchanges in the imbibition direction. An empiriculrelation between residual nonwetting-phase safura.tion ajter water imbibition and initial nonwetting -pbase saturations is /ound from pu.blishcd data.Fro7n this empirical relation, expressions areobtained /or trapped and mobile nonwetting-pb(~.sesaturations which are used in conrzection withestablished theory relating re?at ive permeabiiit y topore-size distribution. Tbe resulting equations yieldrelative permeability as a /unction o/ saturationhaving characteristics beiieved to be representativeof real systems, Tbe relative permeability o/water - wet rocks for both two- and three - pbusesystems, with the saturation cba,nge in theirnbibit ion direction, may be obtained by tijismethod after properly selecting two rock properties:the residual nonwe tt ing-phase sat urvrtion aiter tbecomplete imbibition cycle, and the cap illarypressure curve.

INTRODUCTION

Relative permeability is a function of saturationhistory as well as of saturation. This fact was firstpointed out for two-phase flow by Geffen et al. 1and by Osaba et al, z Hysteresis in the relativepermeability-saturation relation also has beenreported for three-phase systems. 3

Since saturations may change simultaneously intwo directions in a three-phase system, four possiblerelationships arise between relative permeabilityand saturation for a water-wet system. The foursaturation histories of this system were given bySne114 as II, ID, DI and DD. I and D refer to thedirection of saturation change (imbibition anddrainage), with the first letter of the symbol

OrigJnal manuscript received In Society of Petroleun- Engineersoffice July 6, 1967. Revised manuscript received Feb. 21, 1968.Pape~ (SPE 1942) was presented at SPE 42nd AnnuaI F=ll M@et-fng held in Houston, Tex., Oct. 1-4, 1967. @ Copyright 1968American Institute of Mining, Metallurgical, and PetroleumEngineers, Inc.

lRaferences given at end of paper.

This paper wiH be printed in Zransacrions Volume 243, whichwiIl cover 1968.

JUNE, 1968

indicating the direction of change of the Wa erphase. As used in this paper, the second letter ofthe symbol -cfers to the direction of saturationchange of the gas phase, i.e., D and i indicate anincrease and decrease, respectively, in gassaturation.

Only a few thre=-phase relative pernleabi Iitycurves have been published. Leverett and Lewis 5published rhree-phase curves for unconsolidatedsand, and Snel14 reported results of several Englishauthors for both drainage and imbibition three-phaserelative permeability of unconsolidated sands.Three-phase relative permeability curves for aconsolidated sand were published by Caudle et al. sfor increasing water and gas saturations (ID). Coreyef a/.6 reported drainage (DD) three-phase relativepermeability for consolidated sands, Recently,Donaldson and Dean7 and Sarem 8 calculated three-phase relative permeability curves from displacementdata on consolidated sands, also for saturationchanges in the drainage direction.

The only published three -phase relativepermeability curves for consolidated sands withsaturation changes in the imbibition direction (11)are those of Naar and Wygal. g These curves arebased on a theoretical study of the model of Wyllieand Gardner 10 as modified by Naar and Henderson. 1:Interest in three-phase relative permeability hasincreased recently due to the introduction of newrecovery methods and refinements in calculationprocedures brought about by the use of large-scaledigital computers. The scarcity of empiricalrelations for three-phase flow, and the experimentaldifficulty encountered in obtaining such data, havemade the theoretical approach to this problemattractive.

RELATIVE PERMEABILITY AS AFUNCTION OF PORE-SIZE DISTRIBUTION

Purcell used pore sizes obtained from mercury-injection capillary pressure data to calculate thepermeability of porous solids.lz Burdine extendedthe theory by developing a relative permeability-poresize distribution relation containing the correcitortuosity term.ls The works of Purcell and Burdinewere combined by Corey14 into a form that has

149

been accepted widely. These relations have beenderived from the capillary tube model,12 ~13 theKozeny equation 15 and the model of Wyllie andGardner. 10

Corey and co-workers 6 extended the concept ofrelative permeability dependence on poreisizedistribution to three-phase flow with saturationchanges in the drainage direction. Using the modelof Wyllie and Gardner that was adapted to includea trapping mechanism for the nonwetting phase,Naar and Henderson 1 developed equations fortwo-phase imbibition relative permeability. Theextensioo to three - phase imbibition relativepermeability then was made by Naar and Wygal.9

Capillary pressure-saturation relations aresubject co hysteresis, due in part to entrapment ofthe nonwetting phase when the wetting - phasesaturation is increased. Fundamental to thedevelopment of imbibition relative permeabilitycurves-is the concept that pore-size distribution ofa porous medium is defined by drainage capi]larypressure-saturation reIation. If the saturation of thenonwetting phase that is mobile after a saturationchange in the imbibition direction can be asaignedt? the correct pore sizes, imbibition relativepermeability curves can be derived from knowntheory.

For the general case of two- and three-phaseflow, a Corey -Burdine equation can be written,

k rn = . . (1)

where S~B is the mobile or rfree saturation of.,.

phase r.r, whose relative permeability krn isbeing sought; and Sp and ~ are the saturationscorresponding to th= minimu-m and maximum poresizes occupied by S~F. Saturations S; and .$~ areparticular values of S* which is the fraction of the~ffective porosity represented by the sum of al Ipores smaller than the pore size being considered.For simplicity, reduced or effective saturations areused throughout this paper, i.e., saturations arefractions of pore volume, excIuding pore volumeoccupied by the critical (immobi Ie) wetting phase.

TRAPPED GAS SATURATION

Several papers 16-20 have been published relatingthe initial gas saturation to the residual gassaturation left trapped after imbibition. Plots ofdata reported in these papers result in curveshaving a characteristic shape. It is found fromthese data that the difference in the reciprocals ofthe initial and residual gas saturation approximatelyare constant for a given sand. Residual saturationincreases with an increase in the initial gassaturation. When the initial gas saturation is unity,the residual gas saturation is the maximum residual

gas saturation (S& )max. Then.

I 1= I.

-1S~r S~i ($$r)mom

* S~ior

gr = P. . . . . . . . . (2)l+CS~i

where C = 1-1.

(Sg, )maxPlots of the data of several authors are shovn in

Fig. 1. The solid lines are least-squares fits ofEq. 2 to the data. In some cases, the criticalwetting-phase saturation was not reported, and areasonable assumption of this value was made toconvert all saturations to the reduced saturations.The initial effective gas saturations greater thanunity shown in Fig. 1 are results of convertinginitial saturations to effective saturations whenthe data reported were for samples containing nowater. Data from all available sources yield~d agood fit to Eq. 2.

TWO-PHASE IMBIBITIONRELATIVE PERMEABILITY

To develop equations for relative permeabilitywith rhe saturation change in the imbibitiondirection, it is first necessary to be abIe to predictthe saturation of the nonwetting phase that remainsmobile at any saturation. In the following discus.sion, the nonwetting phase is gas at,d the wettingphase is water.

Consider that the porous medium initially containsa gas saturation S* i

fwhich was established by

withdrawal of water drainage). If the water satura-tion is now increased from its initial value SJi toits maximum value 1 - S&, the gas saturation S;assumes every value between S*i and S& as rhesaturation S~i - S&is replacefby water. Duringthis change in saturation, the saturation of gas

IHIIIALGASSAWRAIIOIA,Sei*

FIG. lPLOTS OF PUBLISHED DATA SHOWINq THEFIT OF CURVE l/Sgr* - I/Sal, * = C.

,

1so SOCIETY OF PET ROLEUW ENGINEERS JOURNAL

,that has been trapped St increases from zero at$ S~i co a value of S&the initial conditions Sg =

when S~ = S& At any gas saturation S;, the trappedgas saturation S~t does not contribute to gas flowand is distinguished from the gas saturation thathas remained mobile or free S&. Then the trappedand mobile gas saturations at any instant are partsof the total gas saturation (S; = S:t + S:p) as S;varies from S~i to S * .

If the water satu!~tion now is increased from itsinitial value so that gas saturatiar S; has a valuebetween S~i and .$;,, the saturation S;i - .$* has

~ andbeen replaced by u%er, S& has been trappe.$F is considered to be unaffected by tbe increasein water saturation. The mobile gas saturation -$;Fis that part of S; that is susceptible to displace-ment and entrapment upon further reduction of S;.If .$; is reduced to S;,, the contribution of S*F to

7the residual gas saturation can be found by app ymgS&F as an initial saturation in Eq. 2.. The gassaturation that is trapped S*t at aay gas saturation

%s:, plus rhe additional gas t at will be trapped (thecontribution of S~F) during a reduction in gassaturation from S* to S* must equal the residualvalue S& associated $;~h S* .. Then the trappedgas saturation can be express% as

Making rhe substitution sit = -$; - S~F, and solvingEq. 3 for S~F,

Eq. 4 is an expression for the gas saturationthat remains mobile after the gas saturation hasbeen reduced from S~i to S;, This gas saturationmay be used in the equation relating gas relativepermeability to pore-size distribution.

f-n

Eq. 5 yields relative permeability - saturationrelations for the nonwetting phase with thesaturation change in the imbibition direction. Forporous media havi~g a pore-size distribution such

hat (w) 2 where A is a pore-size distri-

bution factor, 21 integration of Eq. 5 gives

where c = Z/h + 3.In an oil-water system in rhe absence of gas, the

above equations are equally valid with subscript grepIaced by o if it is assumed that the porousmedium is completely water wet.

In the two-phase system, the entire wetting phaseremains mobiIe. As water saturation increases, thewater invades the next larger size pores, trappingsome gas in pores of the size invaded. Since someof the pores of the size occupied by water containtrapped gas, for a particular water saturationsome water must occupy pores of a larger size thanit would occupy if the trapped gas were not present.

Water relative permeability under these conditionscan be expressed by the equation

k =rwJ&*2Sw () If/$$-

I*

Oc

. . . (7)

P= is a function of S*, the fraction of the effectiveporosity that is the sum of all pores smaller thanthe largest containing water. Since the integrand inthe nu;nerator of E;. 7 is to be integrated withrespect to S*W it is necessary to express either Pcas a function of S; or S: as a function of .S. Pccan be expressed as a function of S$ since S* = 1- -$F and S~F is reiated to S& through Eq. 4. Theresulting capillary pressure as a function of S; isthe imbibition capillary pressure which would beobtained if the contact angIe is the same inimbibition as in drainage.

For the purpose of developing equations ofreIative permeability, S; is treated as a funcrionof S*, Since S* is the sum of the water saturationand the trapped gas saturation, Eq. 7 can bewritten

.

*2Swkrw =

JoEq. 3 can be wrirten in terms of S*, since S~F

=1 - S*. This .sJbstitution results in an equationwhich, when differentiated, yields

Substituting Eq. 9 into Eq. 8 gives

JUNE, 1968 1s1

krw=H S*W+S:ts*-%y

*2 oSw

r

I4S

~

oL

S:+s; r &

/[ II[*C 1- 1- (s;r)mox}s* Q

- (s~:ax w

Jo. . . . . . . . . . . . . . . . . . . (lo)

For the speciaI case corrst./Pc2 = S, integrationof Eq. 10 results in

rk - S:2 (s; * S;1)2rw-

L-

I+

(s;+ S:t)1- !1- (S;r)mox >

I.- ... .

)]

. . (11)1- (S~r)ma~l-, :s;[

Fig. 2 is a plot of imbibition relative permeabilitycalculated from Eqs. 6 and II for c-= 4 and amaximum residual gas saturation (s:, )max = O.A.The parameter in Fig. 2 is the initial gas saturation.Also shown (dashed lines) are drainage ,curvescalculated from Coreys equations for c = 4. Thecurves for gas permeabilit-y shown in Fig, 2 aresimilar in appearance to experimental curves ofGeffen et all A greater permeability to water inthe imbibition direction than the drainage direction(Fig. 2) also has been observed.1 The water curvefor .$~j = 1 is of particular interest since it is thecurve of the complete imbibition cycle. Theequation of this curve in Fig. 2 is

,

r

This equation can be generalized as k,wi = k,wd +f(s~) where /(S~), the last term of Eq. 12, is thedifference in imbibition and drainage relativepermeability brought about by the redistribution ofthe water phase. The function /(s~) is differentfor each pore-size distribution, and the numericalvalues given by the function increase with increas-ing range of pore sizes. When all pores are of thesame size (f = 3), /(S;) = 0.

THREE-PHASE IMBIBITIONRELATIVE PERMEABILITY

When three phases saturate the porous medium,the equat?ons for imbibition gas relative permeabilityare the same as for the two-phase system. This isbecause the gas, being nonwetring, occupies thesame pores regardless of the nature of the liquidspresent. For similar reasons, if the porous solid iswater wet, the imbibition relative permeability towater is the same function of water saturation forboth the two- and three-phase systems.

In a three-phase water-wet system of water, oiland gas, both oil and gas are trapped as watersaturation is increased. To develop the followingequations for the relative permeability to oil, theassumption is made that rhe maximum residualhydrocarbon saturation is the same, regardless ofwhether the initial hydrocarbon saturation is gas,oil or both. This usually is considered to be trqe;

,

FIG. 2TWO-PHASE IMBIBITION RELATIVE PERME-ABILITY SHOWING THE INFLUENCE OF INITIAL

SATURATION.

SO CISTY OF PErROLkll M EXGIXEER> ]oIIR!iAL1s2

l.

however, other assumptions could be made whichwould lead to different sets of equations. Theabove assumption, when applied to Eq. 2, results in

I I I. . - 1

S;r + S:r * - (%)maxS;i + sol

s;,S~i~. .

or s:. . . . . . . (13)I t C(S~, + S~i)

Making use of the procedure previously used toarrive at the expression for the free gas saturation,the free oil saturation is fcund to be

4 (s:- S:r)~? i)

. . . . . . . .

C S:,(14)

Imbibition relative permeability to oil is found from

[

s:t s;d S:

~

ks:; s:+ s:+

ro =... . . (15)

nl

Substituting the relations S~ + .f:t = S* aod as:t(Sfir)max ~-f*

~-{1-(sg,) ~ax}~.] ~ Yieds

kto=s;;

JUNE, 1968

. . . . . . . . . . . . . . . . . . . (16)

When 1/P=2 is a linear function of .S, Eq. 16results in

k ro

[

s:: ( I - s;F)2- [s; t s:t)2

(W:$-_2/c In. hr)mox

... ., . . . . . . . . . . . . . . . (17)If water saturation increases at the expense Of

oil saturation while the gas saturation remainsconstant (or is increasing), Eq, 17 simplifies to

rO=s;t(

las a parameter. In Fig. 3, which is a plot of Eq.19, the gas saturation is trapped; in Fig. 4,obtained from Eq. 18, the gas saturation is mobile.Eqs, 18 and 19 are presented in the conventionaltriangular diagram as Figs. 5 and 6, respectively.AH the curves were constructed for the imbibitioncycle starting with an initial water saturationof zero. Eq. 17 may be used to plot oil relativepermeability curves for any history of gas saturationbetween these two extremes, and for initial watersaturations other than zero.

DISCUSSION OF EQUATIONS[Unfortunately, experimental results are not

available with which to check the accuracy of the

0.9

8 [\ 1

.2

I

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0WATER SATURATION,S:

FIG, 3 RELATIVE PERMEABILITY TO OIL AS AFUNCTION OF WATER SATURATION, SHOWING THE

EFFECT OF A TRAPPED GAS SATURATION.

o 0.1 0.2 0.3 Q.4 0.5 0.6 0.7 0.8 0.9 1,0WATER SATURATION, S:

FIG. 4 RELATIVE PERMEABILITY TO OIL AS AFUNCTION OF WATER SATURATION, SHOWING THE

EFFECT OF A MOBILE GAS SATURATION.

154

concepts and equations developed here. The onlythree-rdlase imbibition re!ative Dermeabilitv curves

. . ,

that were found in the literature are the theoreticalcurves of Naar and Wygal. 10

Although the two - phase curves of Naar andHenderson were derived from a model study, theequations for these curves may be derived by theprocedure given here if the relation of residual gassaturation to initial gas saturation is taken as

The mobile gas saturation from this relation is

When this expression-for S~F is used in Eq. 5, theresulting relative permeability satisfies the rela-tionship of Naar and Henderson between imbibitionand drainage saturations at equal gas permeability.However, in this development, the maximumresidual gas saturation is restricted to a value of0.5.

l$Q

FIG. 5 OIL ISOPERMS AS FUNCTIONS OF WATER,OIL AND MOBILE GAS SATURATIONS FOR THE

IMBIBITION CYCLE STARTING WITH .SWi = 0,

FIG. 6 OJL ISOPERMS AS FUNCTIONS OF WATER,oIL AND TRAPPED GAS SATURATIONS FOR THE

IMBIBITION CYCLE STARTING WITH Swi = O.

SOCIETY OF PETROLEUM ENGINEERS JOURNAL

A similar procedure for three-phase oil permea-bility does not yield the results of Naar and Wygal,although S~F cart be expressed similarly. Toobtain their equation for three-phase oil relativepermeability, Naar and Wygal integrated an equationthat differs from Eq, 15. h-t comparing the oilcurves of Naar and Wygal with the curves reportedhere, it is noticed that the curves of Naar andWygal for the 11 case are more nearly representedby the ID case of this paper. The upper limit ofintegration of the equation of Naar and Wygalplaces the trapped gas in the largest pores, whichis also the po-s>tion occupied by a gas saturationfrom the drainage cycle. For this reason ordy oneset of curves gives oil relative permeability that isindependent of the gas-saturation history.

If the oil relative permeability is influenced bythe behavior of the gas saturation, the direction ofchange of the gas saturation, as well as that of thewater saturation, must be considered in selectingthe correct curves to be appIied to a particularprocess. The ID case is beIieved to apply to oilrecovery by in-.=.itu combustion, while the 11 caseapplies to waterflooding a reservoir having aninitial gas saturation.

In this paper, equations are developed andplotted using S* = const./Pc2 and (S~r)max = 0.4.These equations are used as examples becausethey are considered as fairly representative ofslightly consolidated porous media. However,similar curves may be constructed using any desiredvalue of (sir )max and any pore-size distribution.Eqs. 5, 7 and 15 are integrated readily by numericalmethods when experimental capillary pressurecurves are used.

Geffen 1 reported that although hysteresis isobserved in the relative ~ermeabilitv function when

. ,

reversing the direction of saturation from drainageto imbibition, the imbibition curve is reversible.Drainage relative permeability after imbibitionfollows the imbibition relative permeability curveuntil the maximum nonwetting saturation of aprevious drainage cycIe is reached. In the derivedrelative permeability functions, the trapped satura-tions are distributed within a range of pore sizes insuch a way that if the direction of saturation changeis reversed from imbibition to drainage, the trappedsaturation is reIeased in the reverse order ofentrapment. With this saturation distribution, atrapped saturation becomes mobile when pores ofthe size containing the trapped saturation areinvaded by a mobi Ie saturation of the same phase.The imbibition relative permeability functions canbe considered reversible without the necessity ofreassigning pore sizes to mapped saturations, oraccounting for a trapped saturation and a mobilesaturation of the same phase occupying the samepore sizes, as would be necessary- ior othersaturation distributions.

CONCLUSIONS

The relations developed here for two- and

three-phase imbibition relative permeability werederived from established theory of the dependenceof relative permeability on pore-size distribution.In addition, the equations saeiafy an empiricalrelation of residual nonwetting-phase saturation toinitial nonwetting-phase saturation for the imbibitioncycle. As a consequence of the combination of thesetwo fundamental requirements ,zhe derived relationshave properties which agree with reported laboratoryobservations. FolIowing are characteristic of th~derived functions for two-phase flow which agreewith laboratory observations.

1. The wetting-phase relative permeability in theimbibition direction is greater than ia the ,drainagedirection.

2. wtre~ the direction of saturation change isreversed from draiuage to imbibition, a distinctpath is traced by the nonwettirtg-phase relativepermeability as a function of saturation to a residualnonwetting-phase saturation that depends on thesaturation established in the drainage direction.

3. The path followed by relative permeability asa function of saturation when the saturation changeis in rhe imbibition direction is reversible.

In extending the concepts to three-phase flow,equations arise that predict behavior that, aIthou8hnot experimentally observed, seems reasonable.This behavior results in the difference in the IDand H cases. In the water-wet, three-phase system,oil relative permeability is influenced by thedirection of the change of gas saturation. Thiseffect appears to be a necessary result of thedistribution of gas saturation, and is analogous tothe difference in the relative permeability to waterobserved in the imbibition and drainage directions.

Due to the lack of experimental data, accuracyof the equations has not been evaluated. However,since the relative permeability relations weredeveloped to meet required conditions of nonwetting-phase entrapment, they are thought to be sufficientlyaccurate for many reservoir calculations. With theproper choice of residual hydrocarbon saturationand pore-size distribution, relative permeability asa function of saturation may be calculated torepresent the reIative permeability characteristicsof any water-wet sand.

NOMENCLATURE

Standard SPE Ietter symbols are used, with thefollowing exceptions.

S* = effective saturation, fraction of effective

pore space, S* = w - w=, S* =w- Swc g

L, S;= SO11 - SW= 1 - SW=

(S;,)m.x = residual effective gas saturation left inthe sand after completion of animbibition cycle starting with aneffective gas saturation of unity

c= 1-1, a trapping characteristic,

(S;,)max

JUNE, 196a 1ss

constant for each saridA = pore-size distribution factor, slope of a

plot of log l/Pc-vs-Iog Sfic = 2/A + 3, exponent of saturation in the

equation for drainage wetting-phaserelative permeability, krw = S~c

SUBSCRIPTSF = free or mobilet = trappedd = drainage (used with relative permeabil-

ity)i = imbibition (used with reIative pertrfea -

bility)

n = phase n

ACKNOWLEDGMENT

The research from which rhia report was derivedwas conducted under a cooperative agreementbetween the U.S. Dept. nf the Interior, U.S. Bureauof Mines and the U. of Wyoming.

1.

2.

3.

4.

s.

6.

7.

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Osoba, J. S., Richardson, J. G., Kerver, J, K.,Ha fford, J, A. and Blair, P. M.: gLaboratory Measure.ments of Relative Permeability ~, Trarzs., AIME (1951)Vol. 192,47.56.Caudle, B. H,, Slobod, R. L. and Brownscombe, E. R.:tIFurther Developments in the Laboratory Determin-ation of Relative Permeability!, Trans., AIME (1951)vol. 192, 145-150.Snell, R. W.: ~~The Satwation History Dependence ofThree-Phase Oil Relative Permeability, ]. Jnst,Pet, (March, 1963) Vol, 49, 81,

Leverett, M, C. and Lewis, W. B.; Steady F1OW ofGas-Oil-Rater Mixtures Through UnconsolidatedSands, Trarrs., AIME (1941) Vol. 142, 107-116.Corey, A. T., Rathjens, C, H,, Henderson, J, H, andWyllie, M, R. J,: Three-Phase Relative Permea-bility, Trrztis., AIME (1956)VOL 207, 349-351.Donaldaon, E. C. and Dean, G. W.: {Two- end

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1954) vol. 19, 38.Wyllie, M, R. J. and Spangler, M. B,: Applicationof Electrical Resiativity Measurement to Problemsof Fluid Flow in Porous Media)], Bu1l,, AAPG (Feb.,1952) Vol. 36, 359.Holmgren, C. R. and Morse, R, A.: Effect of FreeGas Saturation on Oil Recovery by Water Flooding,~fr?nS., AIME (1951) Vol. 192, 135-140.Dyea, A, B,: (rfaroduction of Water-Driven ReservoirsBelow Their Bubble Point, Trans., AIME (1954)Vol. 201, 240-244.Kyte, J. R., Stanclift, R, J., Jr,, Stephan, S, C., Jr.and Rapoport, L, A.: Mechanism of Watcrfloodingin the Presence of Free Gss, Trans., AIME (1956)Vol. 207, 21.5-221,Dardaganian, S, G.: ~~The Displacement of Gas byOil in the Presence of Cormste Water, MS Thesis,Texas A&M U., College Station (1957).Crowell, D. C., Desn, G. W, and Loomis, A. G,:t