Simulation of Three-Dimensional, Two-Phase FlowIn Oil and Gas Reservoirs

K. H. COATS 1“THE U. OF TEXASA USTIN, TEX,

R. L. NIELSENMEMBERS AIME

MARY H. TERHUNE

A, G, WEBERMEMBER AIME

ABSTRACT

Two computer-~ rierrted techniques for simulatingthe three-dimensional flow behavior o/ Iwo fluidphases in petroleum reservoirs were developed.Under the first technique the flow equations aresolved to.. model three-dimensional flow in are~ert.uir. ‘The second technique was developed/or modeling ftow in t bree-dimensiorwl media thathave sufficient ly high permeability in the verticaldirection so that vertical f!ow is not seriouslyrestrict ed. Since this latter tecbrrique is a rnodif iedtwo-dimensional areal analysis, suitably structuredthree-dimensional resewoirs can be simulated atconsiderably lower computational expenses thanis required using tbe three.dimensional analysis.A quantitative criterion is provided for deten-nin ingwhen vertical communication is good enough topermit use of the modified two-dimensional arealanaIysis.

The equations solved by botb tecbrriques treatbotb fluids as compressible, and, for gas-oilapplications, provide for the evolution oj dissolvedgas. Accounted for are the ef{ects o~ relativepermeability, capillary pressure and gravity inaddition to reservoir geometry and rock betero-gene it y. Calculations are compared w itb laboratorywaterflood data to indicate the validity of theanalyses. Other resu[ts were calculated with bothtechniques wbicb show tbe equivalence of tbe twosolutions /or reservoirs satisfying the verticalcommunication criterion.

INTRODUCTION

Obtaining the maximum profits from oil and gasreservoirs during all stages of depletion is thefundamental charge to the reservoir engineering

Original manuscript received in Society of P etr.aleumEngineers office Jone 7, 1967. Revised manuscript of SPE1961 received Nov. 7, 1967. @ Copyright 1967 American

Institute of Mining, Metallurgical, and Petroleum Engineers,Inc.

DECEMBER, 1967

t2iS0 PRODUCTION RESEAR&ti CD.HOUSTON, TEX,

AMERICAN AIRLfNESTULSA, OKLA.

ESSO PfiQDUCT/ON RESEA RCII CO.HOUSTON, TEX,

profession. Io recent years much quantitativeassistance in evrdualing field developmentprograms has been provided by computerizedtechniques for predicting reservoir flow behavior.Because of the spatially distributed and dynamicnature of producing operations, automatic optimiza-tion procedures, such as those now in use forplanning refining operations, are not now avaiIablefor planning reservoir development. However,present mathematical simulation techniques dofurnish powe, ful means for making case studiesto help in planning prinrary recovery operations andin selecting and timing supplemental recoveryoperations.

A number of methods have been reported whichsimulate the flow of one, two or three fluid phaseswithin porous media of one or two effective spatialdimensions>-4 However, applying computeranalyses to actual reservoirs have been limitedmostly to two-dimensional areal or cross-sectionalflow studies for two immiscible reservoir fluids.To obtain a three-dimensional picture of reservoirperformance using such two-dimensional tech-niques, it has been necessary to interpret thecalculations by combining somehow the resultsfrom essentially independent areal and cross-sectional studies. To the authors’ knowledge,the only other three-dimensional computationalprocedure, in addition to those presented here,was develcped by Peaceman and Rachford8 tosimulate the behavior of a laboratory waterflood.

Two computational techniques which may beused to simdate three-dimensional flow of twofluid phases are described in this paper. Thefirst method, called the “three-dimensionalanalysis”, employs a fully three-dimensionalmathematical 1 model that treats simultaneouslyboth the areal and cross-sectional aspects ofreservoir flow. The second method, called thevertical equilibrium (VE) analysis”, is applicable

preferences given at end of paper.

977

only to reservoirs having good vertical communi-cation. It adapts a two-dimensional areal calculationby providing special techniques to account forflow and saturation variation in the directionnormal to ret ervoir bedding planes. A quantitativecriterion based on reservoir parameters is providedfor determining when reliable results can beexpected from the second method. The general

character of the analyses and applications arediscussed. Mathematical descriptions and develop-meii’ts are included as appendices.

PROPERTIES OF ANALYSES

The three-dimensional analysis and the VEanalysis are difference equation analogs of thedynamic two-phase flow systems occur!ing in oiland gas reservoirs. The difference equations

employed in the analyses are finite approximationsto the standard differential equations which

govern the simultaneous flow of two fluid phasesin porous media. The following characteristicsare built into each of the analyses: (I) broad

flexibility with respect to reservoir structure,

thickness variation and well locations, (2) capacityfor modeling three-dimensional distributions ofporosity and directional permeability, (3) con-sideration of rock compressibility and pressure-‘dependent fluid densities and viscosities, (4) theability to account fog the evolutiori of dissolvedgas for gas-oil systems which accompanies localIydeclining pressure, (5) treatment of the effects ofrelative permeability and the interaction ofgravitation 1, capillary and viscous forces, (6)use of reflection conditions normal to all boundariesof the calculation grid with influx or efflux accountedfor by assigning source or sink terms to boundaryblocks and (7) simultaneous solution of the linkeddifference equations employing alternating directionimplicit (ADI) procedures in an iterative manner.

The fundamental dissimilarity between the twoanalyses is that the three-dimensional nature ofcalculations in the former is reduced to two-dimensional in the latter through the VE concept.This concept is developed mathematically inAppendix B.

Owing to the three-dimensional vs two-dimensionalnature of the calculations performed, the twoanalyses employ somewhat different ADI solutionprocedures. ht the three-dimensional analysis,for which difference equations and method ofsolution are detailed in Appendix A, the Douglas-Rachford technique 5 is used. However, thetwo-dimensional, modified areal model embodiedin the VE analysis employs the Peaceman-Rachfordtechnique which is similar to that described forincompressible fluids. 1

VE CONCEPT

Reservoir flow systems are termed in VEwhenever rhe pressure in each fluid phase varieshydrostatically along any line traversing the sand

thickness normal to its underlying and overlyingstructural confinements and/or bedding planes.Experience eained from numerous two-dimensional.cross-sectional and three-dimensional simulationsof reservoir and laboratory model behavior haveshown that VE frequently occurs. Even in caseswhere significant underrunning by water or over-riding by gas were calculated, the observance ofVE has been quite common.

Factors which favor, VE are (1) low resistanceto flow normal to the bedding planes, (2) sandsthin in the direction normal to the bedding planesand (3) low areal rates of fluid movement. Aquantitative criterion, which may be used toassess when the VE assumption is valid, ispresented in terms of the above variables in thenext section.

Physically, the occurrence of VE implies thatthe rate of redistribution toward a capillary-gravitational equilibrium configuration within adip-normal column of fluids is high relative tothe rate at which saturation fronts advance areally.Alternatively, VE implies that the dip-normalcomponents of gravity and capillary force arebalanced and that, for all practical purposes, thedip-norms 1 component of viscous force is zero.

Provided that VE obtains, knowledge of thevalue of capillary pressure at any reference pointwithin the sand Iayer establishes the variation ofcapillary pressure along the dip-normal linethrough that point and, hence, the phase saturationsalong that line. Therefore, to simulate the flowbehavior for any qualifying three-dimensionalporous medium, multiphase calculations must becarried out only over a two-dimensional referencesurface within the medium.

Eqs. 1 and 2 express the VE condition mathe-matically.

(1)

z

rP= (x, y, z) = PCP (x,y) - cos ad (p~w-p~,i)dz

J0

. . . . . . . . . . . . . . . . . . . . (2)

The z coordinate is oriented in the dip-normaldirection (vertical if the dip angle is zero) andmeasures distance below the reference surface.For convenience, the reference surface was takento connect the mid-points of all dip-normal linesegments passing through the sand. Eq. 2 followsfrom Eq. 1 and the definitions of flow potential andcapi]lary pressure, In Eq. 2, pep is the value of

capillary pressure at any areal point (x, y ) on thereference surface. Eqs. 1 and 2 apply at all pointswithin the three-dimensional medium.

Implementation of the VE concept involvesintegration over the third dimension (z, or reservoirthickness) of each term remaining after substitutionof Eq. 1 into the three-dimensional partial differ-ential equations governing two-phase flow. The

378 SOCIETY OF PETROLEUM EXGI~EERS JOURNAL

.

integration process (Appendix B) effectivelymodifies the functional relationships between.relative permeability, capillary pressure andsaturation. The resulting relationships allow thetwo-dimensional calculations to account for thedip-normal variation of saturation, and therebysimulate both the vertical and areal character ofreservoir flow behavior if YE prevails, Eqs. 3and 4 illustrate how the integration process is

a,pplied to obtain a volumetrically averaged satura.tlon and an effective relative permeability forwetting phase i’!ow parallel to the x-z plane at anyareal point on the reference surface.

h/2

f

h/2

ZW = @ (z) Sw (z) ~zs

+ (z) dz

-h/2 -b/2

. . . . . . . . . . . . . . . . . . . . (3)

h/2

lJf

b/2

kprw =s

kxy (z) krw (Z)~Z kxy (z)dz

-b/2 / -b/2

. . . . . . . . . . . . . . . . . . . (4)

k Eqs. 3 and 4 b is the dip-normal sand thickness.Porosity + and absolute permeability for flowparallel to the x - y plane k

’1’vary with z to

acccmnt for stratification within t e sand thickness.If anisotropy exists in the x-y planes, absolutepermeabilities in the x and y directions may behandled separately . Saturation Sw and relativepermeability k,w are indirect functions of zsince each vary functionally with capillary pressure,whi.ch in turn varies directly with z according toEq. 2. Relative permeability for nonwetting phaseflow is modified in a manner similar to that shownin Eq. 4.

Successive applications of Eqs. 3 and 4 over arange of values for P= allow construction of pseudoOT equilibrium relative permeability and capillarypressure curves from corresponding rock or labora-tory curves. Fig. 1 presents typical rock capi!larypressure and relative permeability curves and thepseudocurves which result for two particular sandthicknesses. The figures show that the integrationprocess causes the slope of the pseudo relation-ships to become more nearly linear than thecorresponding rock relationships, and that thetendency toward linearity increases with increasingsand thickness.

As presented quantitatively in Appendix B, theshape of the pseudo relative permeability andcapillary pressure curves is affected by (1) densitydifference p sw - p5ti, (2) dip angle ad, (3) sandthickness b and (4) stratification k(z) and + (z),Several important properties and programmingconsiderations associated with the VE analysisare discussed in the section Computational

Properties.A further application of the VE concept is in

association with the Buckley -Leverett techniquefor modeling two-phase flow in one dimension. Use

of pseudo relative permeability curves in placeof rock curves in constructing the fractional flowfunction permits the technique to partially accountfor the effects of vertical saturation variations onflow behavior.

The VE concept can be extended readily toprovide a concept of “block” relative permeabilityand capillary pressure relationships. Such anextension can be of vzhre in cases where the VEassumption is not valid and rigorous attention mustbe given to vertical flow. In these cases, ‘theprocedure allows coarser grid spacing in the dip-normal direction to be used in two-dimensionalcross-sectional or full three-dimensional calcula-tions than otherwise would be required to modelflow behavior satisfactorily. This extension isbased on assuming equilibrium only within anindividual grid block rather than through theentire sand thickness.

%< I\

7 Lp =.264 PS1/FT _— ROCK CURVE

G:5

#

43&22

~,EaWo

, II I,0

- 1 _.. .~.> lJ

-20 ,. .2 .3 .4 .5 .6 .7 .8 .9 1.0

OIL SA’kJRATION

1.0 I I I t

.9 —- -~ Ap=.264 PS1/FT

~“- ROCK clJR~lE--

.8 o LZ=5’ _~ L2=25’

E .7#

~ Y

~ .6 — ;--- --2 ,--2~ .5 <>& b<\, ,/Y ‘4

A

z5 .3 ‘ww

.2 /# u.1 \

/’ /

o.0 .1 .4 .5 ,6 .7 .8 .9 1.0

OIL SATURATION

FIG. 1 — SENSITIVITY OF EQUILIBRIUM RELATIVEPERMEABILITY AND CAPILLARY PRESSURE CURVES

TO DEPTH OF INTEGRATION,

DECEMBER, 1967 579

VE CRITERION

The VIZ criterion was developed to provide ameasure for predicting the quality of VE resultsfor particular reservoir situations, The functionalform of the criterion was derived analyticallyfrom the partial differential equations whichdescribe two-phase flow due to gravitationalsegregation in a closed, one-dimensional sandcolumn (Appendix C). The dimensionless criteriongroup r defined in Eq. 5 corresponds physically tothe ratio of the time constant associated with thedecay of vertical transients to the time required forrhe fluids CO advance areally through a distanceequai to the reservoir thickness.

r= bu/0.00633kz~ . . . . . . . . . . (5)

% k rru*+ (6)

‘rn Pru+ krwpti “ “ . . . .

The absolute permeability that influences r is thatwhich applies for flow in the dip-normal or zdirection. Since relative permeability and capillarypressure relationships (rock basis) enter thedefinition of ~, this parameter varies withsaturation. The recommended procedure for obtain-ing an appropriate value of @ is to average thevalues that result over the mid-range of saturation.It is important to note that the influence of relativepermeability and viscosity associated with the

. ,.wetting and nonwett:ng phases 1s symmetric, I.e. ,subscripts w and n applied to these parameters inEqs. 5 and 6 may be reversed. This characteristicreveals that high viscosity for either phase affec:sr adversely.

A number of three-dimensional and companion VEcalculations provided the correspondence betweenthe value of r and the quality of VE results shownin Table 1.

COMPUTATIONAL PROPERTIES

Computational procedures for each of theanalyses were developed and programmed inFORTRAN IV for running on an IBM 7044 computerwith 32,000 words of core storage. The extendedgrid size version of the three-dimensional programemploys an IBM 1301 disk file to provide additionalstorage.

MAXIMUMGRID SIZE

The basic version of the three-dimensionalprogram operates in-core and handles up to an800-block calculation grid. Through utilization ofdisk storage, the alternate version of this program

TABLE 1 - SENSITIVITY OF VE QUALITY TO r

value of ‘r Quality of VE Results

Under 50 Very good

50 to 200 Good to fairOver 200 Unreliable

extends the maximum grid size to 69,750 blocks forproblems requiting further definition. The programdeveloped for the VE analysis operates in-core andcan accommodate a 600-block calculation grid.

SPECIAL CONSIDERATIONS:VE ANALYSIS

The procedures for constructing the pseudocapilIary pressttre and relative permeabilityrelationships which follow from the VE conditionwere illustrated by Eqs. 3 and 4. In the mostgeneral case, such integration operations mustbe carried out for each block of the area 1 grid ateach time step in the numerical solution becauseof spatial and dynamic variations of fluid densitiesand areal heterogeneity of rock properties. However,in many cases density variations are slight enoughand rock property variations are regular enough topermit use of an externally generated set ofpseudo relative permeability and capillary pressurerelationships. To take advantage of cases cf thistype, the VE program was equipped to handle asinput data up to nine sets of externally generatedpseudo relationships. Where applicable, thisapproach may reduce VE computational timerequireme,lts by as much as 70 percent,

Several advantages which translate into savingsof computational expense are associated with

using the VE analysis instead of the three-dimensional analysis for qualifying reservoirs.The most obvious is due to the reduced size ofthe VE grid as compared with the three-dimensionalgrid for equivalent areal definition because oflayer co.]solidation. Two further advantages aredue to the improved convergence of the VIZ analysisduring the iterative ADI solution because thecapillary pressure-saturation relationship is more

nearly linear than in the three-dimensional analysis,and the length-to-width dimension ratios of theareal grid blocks normally are closer to unitythan the height-to-width and height-to-lengthdimension ratios of three-d.imensional grid blocks.

APPLICATIONS

Results calculated under the three-dimensionalanalysis and the VIZ analysis are presented for alaboratory waterflood and a hypothetical water-drive reservoir. For these cases the values for theVE criterion parameter ? were 15 and 210. Validityof the calculations is indicated by comparing resultswith data from the laboratory waterflood. Calculatedresults for each of the two systems are compared toindicate the quality of the VE results relative to thethree-dimensional results, Yhere samration dis-tributions are compared, the three-dimensionalresults were reduced to an areal basis by depthaveraging.

LABORATORY WATERFLOOD:FIVE-SPOT PATTERN (r= 15)

Oil recovery curves from several laboratorywater floods were presented by Gaucher and

SOCIETY OF PETROLEUM ENGINEERS JOURNAL

. .

Lindley.6 Calculations were made using thethree-dimensional and VE programs to simulatethe flood which was scaled to represent a 42 B/Dwater injection rate into a 16-rnd, 20-acre 5-spot20 ft thick. Properties reported by Gaucher and

Lindley for the fluids and unconsoli&ted sandwere employed in the calculations.

The experimental data, the results calculatedin three-dimensional and under VE and thosecalculated in three-dimensional by Peaceman andRachford several years ago are compared in Fig. 2.The agreement between the experiments 1 data andthe three calculations is generally good althoughthe calculations predict slightly higher recoveriesthan were observed experimentally beyond 0,9 PVof water injection.

The close agreement between the Peaceman-Rachford and the present three-dimensional resultsprovide a favorable cross-check between thetechniques that is especially interesting sincedifferent methods of nu&erica 1 solution were used

in the two programs — leap-frog solution of P andR, equations in the former, and simultaneoussolution of @ ~ and ~,, equations in the latter.Result~ calculated by the present three-dimensionaland VE programs were virtually identical (Fig. 2).Under the conditions of this flood, the value of theVE criterion parameter r was 15. The three-dimensional and VE calculations were performedover 10 x 10 x 5 and 10 x 10 grids, and computationalcosts to simulate l.2 PV injection (50 years offield production) were $250 and $37.

NATURAL WATER DRIVE.HET~ROGENEOUS RESERVOIR (7= 210)

This example illustrates the application ofrhree-dimensional methods to a typically heter-ogeneous reservoir. Production is taken from 19wells distributed over the crest of the structure,and water influx due to a natural water driveoccurs along the southern and eastern boundariesof the hypothetical reservoir (Fig. 3). Withineach of the six areal regions, rock propertiesare layered uniquely (Table 2). Rate schedules for

.7 ! I I I 1 I

II,,, i. I

.6 ‘

i,, .:,

E ! — EXPERII’AENTAL’ , :

‘“EEETPE3o .2 .4 .4 .8 1.0 1.2 1.4 1.6 1.8 2.0

WATER INJECTSD - PV

FIG. 2 — EXPERIMENTAL AND CALCULATEDBEHAVIOR OF LABORATORY FIVE-SPOT WATER-

FLtXID.

the wells and influx points are not tabulated, butduring the 10-year study period production andinflux were essentially in balance at a :ate of

1 PV in 65 years. The areal sweep veiocity atthis influx rate is about 1 ft/day,

The fluid distributions after 10 years of produc-tion as calculated using the three-dimensional andVE programs and the initial saturations arepresented in Fig. 3. The somewhat irregularpattern of fluid movement predicted by the three-dimensional analysis can be seen most ckar!y in

PERMEABILITY REGHONS AND STRUCTURALCONTCAJRMAP

[tQoQu’)--

1

CROSS- SECTIONAL BEHAVIOR

LE

AREAL BEHAVIOR

50% WATER PROFILE-- INITIAL— 30 CALCULATION-10 YEARS.. ... 2D VE CALCULATION -10 YEARS

FIG. 3 — CALCULATED BEHAVIOR OF HETERO.GENEOUS RESERVOIR UNDER NATURAL WATER

DRIVE,

DFCSMll F.R, 1967 381

.

the cross-section al projections taken normc 1 tothe structural surfaces. As expected, the advancingfluids flowed preferentially within the high-permeability streaks. Since the permeabilitygoverning “&ainage out of the loose sand layer

was markedly !ower than that for areal advance(lower by a factor of 500 between a 10- and O.1-darcy layer because of the harmonic averaging)vertical redistribution proceeded slowly. Thus,as shown by Sections A-A ‘ and B-B’ in Fig. 3,where the locwe layer was near the sand top,water override developed in some regions of thereservoir. However, in other regions where theloose layer was near the sand bottom (SectionC-C’), pronounced underrunnitig by water occurred.

Considering the degree of heterogeneity in thiseservoir and the definitely non-VE character ofi:he three-dimensionai results, lt is surprising thatI,he VE and three-dimensional restdts agreed so~ell. Of course. where water override occurred,Jw effects were obscured by depth-averagingmaturations, The r value of 210 for shis problemindicates that the VE simplification is invalidand that VE results would not be reliable. Com-putational costs for the three-dimensional anti VE(calculations were approximately $400 and $60.

CONCLUSIONS

1, The validity of the three-dimensional analysiswas indicated by experimental data. Good agreementbetween calculated and experimental oil recoverieswas obtained for a laboratory waterflood.

2. Applicability of the VE analysis for simulatingthree-dimensional flow behavior in reservoirs having

TABLE 2 - DATA FOR HETEROGENEOUS RESE2VOIRUNDER NATURAL WATER DRIVE

Water viscosity .40 Cp

Oil viscosity .55 Cp

Rock compressibility 3 x 10-6 psi-l

L 35,400 ft

L; 25,600 ft

h 300 ftThree-dimensional grid 15 X1OX5VE grid 1s x 10

Pressure(psi) (m., b~STB) (p%%) (res. &/STB) (%/ft)-

600 1.005 .4324 1s316 .29234,000 .9956 .4371 1.244 .3093

Areal Pe~r&~ity (kXy)

Y Pam sity (6) LayersRegion 1 2 3 4 5 12345—. ..— —— ——— —

I 1 10 .111.1 .25 .2 .12 .12II 1 210.11 .2 .12 .1 .25 .12111 21110.1 .12 .12 .25 .2 .1Iv 211110 .2 .12 .1 .12 .25v 1110,21 .12 .1 .25 .2 .12v} 18.1 10 *2 .12 .25 .1 .25 .2

Vertical permeability: k= = 0.1 kxy,

good vertical communication was demonstrated.Results calculated using this modified two.dimensional areal analysis compared favorablywith those from the more expensive but morerigorous three-dimensional analysis in .sever+dexample applications.

3. The VII criterion paramerer provides aquantitative test for determining whether a particularreservoir can be modeled reliably using the VEanalysis. Experience from numerous case studiesindicates that reservoirs having parameter valuesgreater than 200 may require three-dimensionalmodeling.

4. Both the three-dimensional and VE analysesprovide the leservoir engineer with reliable,comprehensive techniques for predicting tbree-dimensional reservoir flow behavior. From acomputational cost standpoint, the VE analysis,if applicable, should be used in preference co thethree-dimensional analysis, However, even forreservoirs which require the three-dimensionalanalysis, computational costs normally need notke prohibitive,

NOMENCLATURE

0.0ffi33 AyAz ~ k ,ti,

((AxW)i+%, ~, ~= — — —

5.614 Ax )PSW bW i+l,~,j,k$

x Pwtransmissibility of wetting phasein x direction, bbl/ft-day

b = formation volume factor, STB/reservoir bbl or Mcf/reservoirbbl

B = formation volume factor, reservoirbbl/STB or reservoir bbl/Mcf

b ‘ = dbldp

Cf = rock compressibility, psi-l

D = depth; or vertical position, ft,measured positively downward o

g = acceleration of gravity, ft/sqsecond

g= = gravitational conversion constant,32.2 Ibm ft/lbf sq second

( b’S’

)G1=Vp PSw Swbw Cf+ ~-~

/ w w

/At, STB/ft-day

[

.fw’ Sn bwG2 = Vp P,w Swbu ~

(~ +’/~w + %~ ‘/

Sw’ bw’ R

1/’-~+~+$

At, STB/

ft-dayr

‘1 = ‘p Psn Sw ‘ bw/At, STB/ft-day

rSW’bn’H2=vpp##eL~-ywS ‘b

1/-Rs~—

~ b:dt, STB/ft-day

Interlock permeability: harmonic mean.

382 sOCIETY OF PETROLEUM ENGINEERS JOURNAL

Hk =

b=

‘n =

iw =

iv=

k=

k,=

k,p =

Lx =

Ly =

iteration parameter for A@itera tion

reservoir thickness, ft

nonwetting phase injection rate,STB/D of Mcf/D

wetting phase injection rate, STB/D

injection rate, STB/D/bbl of bulkreservoir volume

absolute permeability, mcl

relative permeability

pseudo. relative permeability

reservoir length, ft

reservoir width, ft

Pc = capillary pressure, #rn - pti psi

P CP= pseudo capillary pressure, i.e.,

P= at reference plane of

P=’=

p=

R.=

R.’=

Sor Sw=

3.

s’=At =

t=~.

Vp =

reservoir, psi

dPcldSw

pressure, psi

solution gas ratio, Mcf/STB

dR ~ldp

wetting phase saturation

average wetting phase saturationthrough reservoir thickness

dS ~/dPc

time step, days

time, days

volumetric or Darcy velocity,

2.

3.

4.

5.

6.

7.

8.

volume/sq ft-day

block pore volume,5.614, bbl

Ax, Ay, Az = block dimensions, ft

ad = dip angle

+ = porosity i

@ = flow potential,s

&- q ft

PS.(P]

p~ = specific weight~psi/ft

AP = Psw-Psnp = viscosity, cp

k,n k,w dPc*’- ——

‘rn Pw + ‘rw P n ‘Sw

SUBSCRIPTS

w = wetting phase

n = nonwetting phase

ACKNOWLEDGMENT

The authors wish to express their appreciationto Esso Production Research Co. for permissionto publish this paper.

REFERENCES

1. Douglas, Jim, Jr., Peaceman, D. W, and Rachford,H. H., Jr.: I~A Method for chlCuktkrg Multi-

DECEMBER, 1967

Dimensional Immiscible Displacement t‘, Ttun S

AIME (1959) Vol. 216, 297-308.

‘agh~ R. G. and sr~wm C H,, Jr.: “A NeApproach to the Two-Dimensional, MultiplraseReservoir Simulator”, SOc. Pet. Eng. J. (June, 196175-182,

Goddin, C. S., Jr., Craig, F. F., Jr., Wilkes, J. Qand Tek, M. R.: (IA Numerical Study of WaterfloodPerformance in a Stratified System With Crossflow’},j. Pet. Tech (June, 1966) 765-77I.Nielsen, R. L.: “on the Flow of Two frnrniacible,~cOmpressible Fluids in Porous Media’s, PhDDissertation, The U. of Michigan, Ann Arbor (1962)

Douglas, Jim, Jr., and Rachford, H. H., Jr.: ‘(Othe Numerics 1 Solution of Hest ConductionProblems in Two and Three Space VdriabIesJYTrans., Anr. Math. Sot. (1956) Vol. 82, 421.

Gaucher, D. H. end LindIey, D. C.: “WaterfloodPerformance in a Stratified, Five-SDot Reservoir —A Scaled-Model Study”, ~nrrzs., AIME (1960)VO219,20S-215.

Peacemen, D, W. and Rachford, H,Appl. Math, (1955) Vol. 3, 28.Peaceman, D. W. and Rachford, H.Communication, Esso Production(1963),

APPENDIX A

H., jr.: ], lm

H., Jr.: PritwleResearch Co

THREE-DIMENSIONAL FLOW EQUATIONS

DERIVATION OF EQUATIONS

The basic equations governing two-phase flowin porous media are (1) the continuity or materialbalance equation for each phase

.9 “ (bW ;W) + (iV)

. . . . . . . . . .

-V “ (bW RS ~W +

=w : (0 bw Sw

. . . . . . . . (A-la)

bn ~n) + (iv) ~ =

-?.- (0 bnSn + !3bwR~ SW) . ,(A-1at

(2) Darcy’s law relating superficial velocities tflow potential

k~ Xw=- k—

w P V$W . . . (A-2a

IJw ‘w

k

%n=-k~ p~n V@n O . ..(A-2b)

IJn

and (3) the capillary pressure definition

Pc=pn-Pw . . . . . . . . .. (A-3

Eq. A-3, along with the definition of 0,

3

Pw

‘3 J dp -D= —w

o PSw

P*

@f

q -D= . . . .

n(A-4)

o P sn

allows expression of the saturation derivative

dS w/dt in terms of the potential

at

Substituting Eqs. A-2 and A-5 into Eq. A-1yields two equations in the two dependent variablesOw and @n,

kV’(k ~ bW P*W V*W) + (iV) ~ =

IJw

kV.(k~ bw RS p SW VQW) + v ●

Ww

k

(k * bn Psn V@n) + (iV) ~ =

Pn

b ‘w a$-1-—)1+” !$‘Sn (l- S)bn

bw

S1 “n(—-—- R S’ bw aQn— —)—

1-S bn

s 1-S b atn

. . . . . . . . (A-6b)

where dq5/dpw = Cf+, and S E Sw.

Multiplying Eqs. A-6 by the block volume AxAyAzand writing derivatives in difference form yields

AAWAQW+ i = G1 “ At@w+ HI “ Atlnw

. . . . . . . . (A-7a)

‘2“ AtI?w i- Hz ● At*n o . (A-7b)

where difference notation is defined by A A A @ =Ax Ax~@+A AYAY@+Az AzAz@; A%Ax Ax III=A xi+.% f, k ~Q;+~;&k;~::k~+-l ‘xi-%, j, k(@i,}, k-bi-l, j,k ; ., ., , ‘ai, j,k, m~and Ax, Ay A* Gl, G2,=H1 and H2 are defined inthe Nomenclature. The source terms i are in unitsof STB/D or Mcf/D injected for the block. Allunsubscripted term~ such as i, G1, etc., in Eq.A-7 are understood to apply at spatial positioni, j, ~. The potential @ on the Ieft-hand side isunderstood CO apply at the new rime m + l; Eqs.A-7 are, in this sense, implicit. TransmissibilitiesA ~ Aw R~ and An are all evaluated at old time m.

Application of the Douglas-R achford 5 altemating-direction procedure to Eqs. A-7 will now bedescribed. Eqs. A-7 can be written more simply inmatrix form as

AAAQ+.i, =GA~@....8)-8)——

where

Aw o

A=[ AWRS A 1

n

ii=

[1win

[

‘1G=

‘2

‘1

‘21

@@ = [1‘3W . . . . . .(A-9)

n

984 sOCIETY OF PEI’ROLEL’M ENCJNEERS JOURNAL

.

The x-direction calculation is

+ AzAzAz:k +~

- @) + H (:* - :k) .(A-H3a)= G (1* _m

The y- and z-direction calculations are

AXAXAX @* + AYAYAY 4**

+ AZAZAZ Qk + i--

** - !k)**-~m)+H(~=G(~

. . . . .(A-10b)

and

AXAXAX I* + AYAYAY 4** I-

AAA @k+l+izzz -

= G (@k+l k+ 1.4m) -t- H (: - !k)

. . . . . . .(A-1OC)

The matrix H is given by

‘k ~ ‘W o

H=[

1HkXAwR~ Hk~An A

. . . . . . . .(A-II)

where the Z term denotes summation of the sixtransmissibilities on the six faces of the block. Theterm Hk is an iteration parameter discussed below.Index k, is iteration number; Qk is the kth iterate ofQ. The solution of equations identical in form toEqs. A-10a, A-10b and A-1OC forQ*, Q** and @k+l,respectively, is described in detail in the appendixto Ref. 1.

Iteration parameters are obtained from a relationthat results from an analysis of convergence ofthe two-dimensional (Peacemsrt-Rach ford) iterativeADI.1 The convergence analysis is beyond thescope of this paper. 7 The relation employed here

for ~inimum iteration pararn crer is

i-r’ 1H

(A-n)

min ‘2N2 kx Ax’

3+=—

where N% is number of blocks in the x direction,kx and k z are rock permeabilities in the x andz directions, respectively, and Ax and Az areblock dimensions. Maximum Hh is unity andintermediate values are spaced geometrically:One cycle ia defined as, say, six iterations whereeach iteration involves solving Eqs. A-10 witha single Hk value. Cycles are then repestedusing the same set of six iteration parametersuntil convergence is strained.

APPENDIX B

VE EQUATIONS ANDSATURATION FUNCTIONS

ADJUSTMENT OF RELATIVE PERMEABILITYAND CAPILLARY PRESSURE CURVES TOREFLECT EQUILIBRATED SATURATIONDISTRIBUTION

Mathematical expression of the VE assumptionsimply involves integration over z (reservoirthickness direction) of each term in the three-dimensional partial differential equations govern-ing two-phase flow. This htegration process

yields integrals of the type1

k krwdz andb/2 2=-?3/2

J1$S dz, where b is reservoir-thickness. The

- b/2first integral relates to e//ective permeability forareal wetting phase flow, taking into account thedistribution of absolute and relative permeabilitythrough reservoir thickness. The second integralrelates ro the average saturation of the areal

block corresponding to a capillary pressure valueof Pcp at block center. Eq. 2 gives dz = - dpc /Ap cos ad where Ap is p~w - psn so that these

integraLs can be written as

1

AP cos ad

Ph+ AP Cos ad —

J

Cp2

k (Z) k=W d P=

P - Ap cos ad +-Cp

1

Ap COS Ud

Ph

+ Ap cos ad —Cp

J

2

@(z) SdPc

Ph- Ap COS Ud ~

Cp

k Az&x

DECEMBER. 1967

Equilibrium or pseudo relative pexrneability andcapillary pressure curves are defined as

k1

prw = Ap COS t%d

AP

P +75

J

Cp

PAP

ccp-—

2

/

-+

k (Z) krwd Pc f k(z) dz

-h—

2

. . . . . . . . . . . . . . . . . . . .(M)

APCP +—Cp 2P

1E= J APC

Ap COS U~ Pcp ->

h

/

2-

@(z) S d P= J h $(Z) dz.—

2

,, . . . . . . . . . . . . . . . . . .(B-2)

where AP= e Ap cos ad b/2.These equations define kprw - ~ and Pcp - ~

curves which reflect reservoir stratification and~ertica 1 saturation distribution, The integrationsare easily performed since k ,W and S ate single-valued functions of P= (from rock curves), andz is a function of P= from Eq. 2.

These k ~,w - ~ and Pcp - Y relationships aredependent upon (1) density difference Ap, (2) dipangle ad, (3) reservoir thickness b and (4) strati-fication k(z), +(z). Thus, the general case ofcompressible fluids and areal variation ofsttatificati~n :equires separate pseudo relative

permeability and capillary pressure curves for eachareal grid point.

If fluids are incompressible, or nearly so, andthe reservoir is homogeneous with constantthickness and dip angle, then the pseudo relation-ships are unique over the entire reservoir. In thiscase, letting a = h Ap cos ad,

P +~Cp z

k1

f

,(B-3)

prw = ~a

k ~W d PcPcp-~

P ++Cp

J

2

S=Aa

SdPa P

c. . —Cp

2

. . . . . . . . . . .(B-4)

andfhese‘@rro - z Pcp - ~ relationships may befed mto the VE program and used directly, i.e.,the internal integration can be bypassed.-

The pseudo or equilibrium twlative permeabilityand capillary pressure curves (Eqs. B-2 throughB-4) are compared with rock curves in Fig. 1. Thepseudo relationships are always more nearlylinear than the rock curves,

APPffNDIx C

DERIVATION OF A CRITERIONFOR VALIDITY OF VE ASSUMPTION

Consider a sand column, closed at both ends,nearly vertical and saturated with a mixtu:e oftwo incompressible” immiscible fluids. Such acolumn may be approximately identified with acolumn of fluid moving areally through a reservoi~,As this column mo~es, the fluid distribution tendstoward one of capillary-gravitational equilibriumsince the ends are reprtiaented by the top andbottom of the reservoir and are therefore closed. Theexistence of equilibrated fluid distributions throughreservoir thickness will depend on the timenecessary for transients in such a closed columnto decay,

If z denotes distance down the column, thenDarcy’s law for two-phase flow gives phasevelocities as

k ap

u=w -= k(=- PSWQ)

IJw az 32

. . . . . . ,., .(C-l)

krn

apu = -—-

n VJ-P~n+)P az

n

,. .,.. . .(C-2)

where k is permeability in the direction of thecolumn.

Since fluids are incompressible and the columnis closed at both ends, the sum of these velocitiesmust be zero,

Uw+un

=0”” ”””’ ”(C-3)

Substituting Eqa. C-1 and C-2 into Eq, C-3 andusing the definition Pc = pti - pw yields

SS6 SOCIETY OF PETROLEUM ENGINEERS JOURNAL

. .

apw

[

k krn= (— P,n + -& P*W)

az ~ w

1

aD ~:rn apC .—-— ~bz ~

Tazn

[

k krw rn 1i——..(C-4)

IJw Mn

The continuity or material balance equation forwater flow is

auw as =0 . . . ..(C-5)-1- @—

r at

Combining Eqs. C-1, C-4 and C-5 yields

where

krw krn——

IJw M$1=-

n-

k krw rn

—+—IJw IJn

SI = dS/dPc

P= PC+ APD . . . ..(C-7)

Ap = p~w - P~n

Taking mean values of ~ ~ and S‘ for the purposeof analysis, Eq. C-6 can be written

32P ap ,O. .,. ..S)-S)

q ‘<

where z D = zlh, tD = k#l t/~ h 2 .S; and @l and S‘. .

now denote mean values. h IS cohunn length, I.e.,reservoir thickness. The boundarj; conditions for

Eq. C-8 are expressions of the no-flow conditionsat the closed column ends ZD = O, 1. Theseconditions are

ap—=0 EltzD = o,1 . . .(CazD

The solution of Eq. C-8 subject to boundaryconditions in Eq. C-9 and a nonequilibriuminitialcondition

P(ZD, o) = f(~) . . . . . (C-IO).

is

P(ZD, tD) = AO + n~l A.n

-n2n2tCOS nllz e

D D-’ .(c-I 1

where

1

AO =J

f(zD) dzD

o

1

J ‘(zD) Cos nl’rzD

oA=n ,

J

fCos

2 nlTzD dzD

o

. . . . . . . . . . . . ., . (C-12

At large time (equilibrium), P is equal to AO aall z ~ The distance from equilibrium at any ZD, tDis therefore

P(zD, tD) - A. =;Ain

-n2n2tcos nTTz e

D D .s. (c-13

The time necessary for I/e decay toward equilibriuof the primary harmonic is then

l-r2t= loo.....D

. . (C-14

DECEMBER, 1967 S

or, using the definition of dimensionless time t~, is a measure of the ratio of time necessary forvertical direction transient decay to time necessaryfor a given areal advance. A convenient measure

t . Ohz_ . . . . . . . ..(c-l5”) 1?~ is simply h, meaning that we compare time

k$rr2necessary for vertical transient decay to timenecessary to advance an areal distance equal to

where reservoir thickness. Thus, the validity of thevertical equilibriumassumption should be inversely

d’P proportional IO the value of-.

‘+ = -= *l””””””””@)dS

Eq. C-15 gives an estimate of the time necessaryfor a I/e decay of a nonequilibrium fluid distribu-tion. If u is an average areal superficial velocity,then f%#u is a meaaureof the time to flow i!% Thisfeet areally. Thus, laws

‘r. hu— . . . . . . . . .. (C-la)

k$

group scales compatibly with the scalingfor reservoirs. That is, if two reservoirs

are scaled, they will have the same value of r.

Th2u . (C-17)

***=— . . . . . . . . . .

k! i‘x

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