SOCIETYOF PETROLEUMENGINEERSOF AIME6200 North CentralExpressway RR SPE 3600Dallas,Texas 75206

THIS IS A PREPRINT’--- SUBJECTTO CORRECTION

A Three-Phase, Experimental and NumericalSimulation Study of the Steam Flood

Process

BYA. Abdalla andJf. H. Coats, Members AIME, INTERCOMP

O Copyright 1971American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc.

This paper was prepared for the 46th Annual Fall Meeting of the Societyof PetroleumEngineersof AIME, to be held in New Orleans,La., Oct. 3-6, 1971. Permissionto copy is restrictedto anabstractof not more than 300 words. Illustrationsmay not be copied. The abstractshould containconspicuousacknowledgmentof where and by whom the paper is presented. Publicationelsewhereafterpublicationin the JOURNALOF PETROLEUMTECHNOLOGYor the SOCIETYOF PETROLEUMENGINEERSJOURNAL isusually grantedupon requestto the Editor of the appropriatejournalprovided agreementto giveproper credit is made. “i

Discussionof this paper is invited. Three copiesof any discussionshouldbe sent to theSocietyof PetroleumEngineersoffice. Such discussionmay be presentedat the above meeting and,with the paper, may be consideredfor publicationin one of the two SPE magazines.

ABSTRACT show earlier steam breakthroughthan those withhigher viscosity.

A numericalmodel of steam-driveoilrecovery was developed and tested. The implicit INTRODUCTIONpressure-explicitsaturation (IMPES)techniquewas used to solve the three-phasefluid flow The first part of the work presented hereequationsfor compressiblefluids. A method is a physical laboratorymodel of steam injec-was developed and applied to determine the tion in a linear system. A constantpressuretemperatureand the rate of steam condensation boundary conditionwas used. Two runs wereimplicitlyfrom the heat-balanceequation. Both performed on the same model using two differenttechniqueswere used in computer simulatorsfor sets of injection and productionpressures. Oillinear and two4iimensionalsystems. recovery and temperaturedistributiondata were

obtained. Each run was repeated to checkA steam-injectionexperimentalstudy was reproducibilityof results.

perfoxmed in a linear model. The results ofthis experimentalwork showed good agreement The second part of this work describes thewith the results obtained from the linear developmentand applicationof numericalnumerical computer simulators. The results from simulationtechniquesto solve equationsthe two-dimensionalnumerical computer simulator describingthe steam-injectionprocess. Thiswas also found to be in good agreementwith simulationmodel was the impli&:9p:essure--..L12 _L_A .L..- 22—.--:---1 -.—--2---A-7 ---..lL-puuALtmeu lJwu-JJIltxls.LulleLL eiiperuilexlu.iu I-eauLIJb.

.5-14 n<+ ..+,,-.+4 *- +-nhrt; “,-c,GA~L&bLU D=UIAL-ULVL4 UGL,lUU~UG .? -

~G ~ofiv.e

difference equationsdescribingthe multiphaseThe numerical simulatorswere also used to flow system. The solution of the heat-balance

study the effect of some parameters on the equationyields the temperatureahead of thesteam-drivenrocess.-.-—. lt was found that n~eri-r------- —-—.— steam front and the rate of steam condensationcal.model results were very sensitiveto behind the front.capillarypressure values. It was also foundthat the relativepermeabilitydata has a minor THE LINEAR PHYSICALMODELeffect on the results obtained. The oilviscosity was found to affect the process to a Few linear physical models have beenlarge etient. Oil recovery from steam-drive reported in the literature. The most recentlyprocess decreases as the oil tiscosity ficreases~ published one is that of Will.manet al.23Also, the recovery curves for low viscosity oils However, they did not publish enough data so the

References and illustrationsat end of paper.numerical simulatordevelopedhere could be

I

A THREE-PHASE.EWEUUWWAL AND NUMERICALVIT17AMli’llMTlPR(Y!T?.S.S----- ----- .-.--—-- SPE 36032. CTMTIT Am-rfm cmmiv m THOJ.L-lUU LAW.,“A“u. “.

properly tested. Therefore,a physical modelwas designed that not only helped in the under-L—..s2.-— -m *L- —------- L..* -1 -- -.-A.- ..4-Ascanang WI me prucesa, Uub CrAau pL-uv J.uGu

sufficientdata to check the simulator.

A schematicdiagram of the apparatusisshown in Rig. 1. It consistedof a condensingsteam trap, filter, adjustablecoil heater,Wet pressure gauge, core holder, thermo-couples, outlet pressure gauge, condenserandbackpressureregulator.

The steam used in the experimentswas asaturated steam from The U. of Texas utilitylines. The injectionpressure was adjustedbya pressure regulatormounted on the steam lines.The steam coming from the pressure regulatorpassed through the condensingsteam trap. Thisknocked out the steam condensate. The steamthen passed through a filter which removedimpuritiesthat could cause clogging of the sanepack. A coil heater was wrapped around theinjection line. The temperatureof the heaterwas adjusted by a variable autotransformerto atemperatureslightlyhigher thsn the saturationtemperatureof the injected steam. TiiiseMmi-nated any possibilityof having condensateinthe injected steam.

The oil used was primol 185 with a viscos-ity of ,43cp at 80°F and at 2600F. Curves ofviscosity and specificgravity vs temperature-—- -1,----. .tiremIuwIk L-I Fig. 2.

The sand used was an unconsolidatedssndof 2.54 darcies permeabilitysnd 35.4 percentporosity.

Two steam injection runs were performed,,.+=ITrli Pfnwnn+ ~~7&ma~&m=r nnndi+.i nne.L4.4.= =+..”. “... The fi?~+.w“..--.--..”. . ..” --- - .

run was performed with an injectionpressureof 40.0 psia and a productionpressure of 28.2psia. The results are plotted in Figs. 3 and 4The second run was performed with injectionpressure of 39.6 psia and productionpressure o:14.7 psia. The results are plotted in Figs. 5and 6. Both runs were repeated and the resultswere in good agreement.

THE DIFFERENTIALFORM OF THE PROBLEM

Differentialequationsdescribingthe fluifand heat flow for the stesm~rive process arepresented here.

..,-a-–-- .-L.---Fhua !Low Jlql,lazlons

The mathematicalrelationshipsdescribingmultiphase fluid flow a pear in the

fliteratWe05,12717,20t2 The developmentof

such relationshipsis based upon mass balanceand Darcyts law for each phase. When bothrelationshipsare combined,the partial-differentialequationdescribingthe fluid flow

of each phase in the reservoirwill be obtained.

a(pouxo) a(pou ~)

ax - ay + qvo =

a(oposo)

at ;******** ““” (l-A)

for the water phase

a (PWUW)

ax - ay+ ~ + qvc =

and for the steam phase

a(p~ux~) a(p~u ~)

ax - ay ‘%- %.=

where U& and uy_iare given by Darcyts law asfollows:

kxkri apiu= - 6.33 -~i ~“””” (2-A)

r-1

kk. api- 6.33 -&‘yi ‘ ~“”””

(2-B)i

and i = O,w,s.

Substitutionof Eqs. 2 into Eqs. 1 gives

nkxkrOpO apO6.33 &

llo T )L

+a (y 1k kryo ape,

~ ~; + ClvolJo

—.

a(oposo)=

at .“ “ “ ““” “ “ ● “. .(3-A;

a[ k krwpw PPW

)]+ x\ Ilw T + qw + qvc

a(~pWsw)=

at********** “ .(3-B:

SPE 3600 A. AHIALLA end K. H. COATS

a(~psss)=

at””-”””=”””. (3-c)

The saturationsare related as follows:

SO+ SW+ S5 =1..... .00 (4)

The pressures of the differentphases are re-lated by the capillarypressures as follows:

P =PO-PWO. O..co-w

P = Ps -P~* ● . . .co-s

All symbolsused are describedinclature.

Heat Flow Equation

.0.. (5-A)

.,.. (5-B)

the Nomen-

The developmentofrelationshipdescribingmedia is based upon theand Darcy’s equations.combined,the followingobtained.

the mathematicalthe heat flow in porousheat balance, Fourier,When those equationsarcdifferentialequationi:

- ‘,+ ~ (DX%’*(DY%)a

- ~ (Uxphn) - &uyPhn) + qvshinj

= & [$(PSh) + (1 - $) PrCrZ] , (6)

where

uiphn = u. POhn + uiwpwhn10 0 w

+U ispshns “- ● .* ● ● (7)

Sph = Sopoho + Swpwhw + S~p5,S, (8)

snd i = x,y.

Q and ~ are given byEqs 2.

In this study, the functionaldependenciesof the parameters are assumed to be as follows.

1. Densities of water and oil are func-tions of temperatureonly. The density of steanis expressedby the equation

MpsPs = R(T + 460) ‘

i.e., an ideal gas.

2. Viscositiesof the water, oil andsteam depend upon temperatureonly.

3. Water and stesm relativepermeabilitiesare functions of their relative saturations.The oil relativepermeabilityis a functionboth oil and water saturations.

4. The capillarypressure between oilwater is a function of the water saturationCapillarypressure between oil and steam isfunction of both water and oil saturations.

5* The heat loss term is explainedin

of

andonly

a

detail in Appendix A. The differenceform ofthe partialdifferential equationdescribedhereis presented in Appendix B. The applicationofthe IMPES techniqueto solve the difference equation is given in Appendix C. The equationsgiven in both appendicesare for the linear modefor simplicity.

‘DESCRIPTION OF COMPUTER ~~

The Line~SimulatorI

The techniquesdiscussedhere were incor-porated into a Fortran IV computerprogram. Thegrid system used is shown in Fig. 7. Thisprogram computesat each time step the satura-

tions, pressures and temperaturedistributions.Also, it computesthe steam condensationratesin each block and the injectionand productionrates. The check for the convergenceis basedupon the change in pressure, temperature,andsteam condensationrates between two successiveiterations. Between the three checks, the rateof steam condensationis found to be the ccm-trolling one.

The program has a msximum grid-size systemof 100. The executiontimes are dependent onthe weight factor used in the calculationof therate of steam condensationdescribed in AppendixDo A value of 0.85 is found to be most suit-

able. An average executiontime is 0.08 secondsper time step for 10 blocks system on the CDC//--

I bbuu computer.

A generalizedflow chart of the program isgiven in Fig. 8. All the necessarydata otherthan the steam viscosity, specificheat, androck propertiesare read into the program priorto the main computationloop. At the start ofthis loop, the relative permeabilities,thecapillarypressures,the densities,theviscosities,and the transmissibilitiesaredetermined. A table look-up is used for thisprocedure. In the calculationof the trans-missibilities,all the parametersare evaluated100 percent upstream. Calculationof the pres-sure distributionthen follows. The steam

A THREE-PHASE.EXPERIMENT AND NUMERICAL-.——SIMULATIONSTtiY OF !l

saturationtemperaturesare determinedfrom thestesm pressures using a table look-up procedure.Calculationof the saturationsthen follows. ~Computationof the rate of steam condensationorthe temperatureis done using the heat bslanceequation. This is followed by the convergencecheck.

In the program, steam viscosity,rockdensity, and specificheats of oil, water androck are constants. However, stesm viscosityand specificheats of oil and water can be usedin the program as temperaturedependent. Fixingthe former quantitiesis merely due to therelatively small pressure drops used in testingthe model.

The Two-DimensionslSimulator

A computerprogram was written based onthe techniquesdiscussedhere. The grid systemused is shown in Fig. 9. As in the linear simu-lator, the program computespressures, satura-tions and temperaturedistributions. Theprogram slso computes the rate of steam conden~sation and injection and productionrates.Although the controllingparameter in the con-vergence is the rate of steam condensation,the program computesthe change in the threevariables,namely, pressure, temperatureandrate of steam condensation.

The progrsm has a maximum grid-size systemof 20 x 20. Execution times are dependentuponthe weight factor used in the calculationofthe rate of steam.condensationas stated inAppendixD. A value of 0.$35was found to besuitable. An average executiontime is 0.25second per time step on the CDC 6600 computerf~~ .aj ~ ~ g~~~ ~y~~~~,~~udyo

A generalizedflow chart of the program isgiven h Fig. 8. The program follows the sameoutltie as the linear simulator. However, thevalues of the parametersin the transmissi-bilities calculationare taken at the blockunder considerationexcept for the relativepermeabilities,which are 100 percentstreamed.

COMPARISONWITH EXPERIMENTALRESULTS

The Linear Model

up-

As mentioned earlier,two experimentalrunswith differentboundary conditionshave beenperformed. The differencebetween the runs wasthe pressure drop. This gave differentinjec-tion rates, which in turn affectedthe cumula-tive heat loss. The pressure level has greatsignificancein the stesm-injectionprocess.Saturationtemperatureand steam enthalpiesarefunctions of pressure level. The higher thepressure level, the higher the temperaturelevel, which, in turn, gives larger rate of heat

; STEAMFLOODPROCESS SPE 3600

Loss. Data used in the computerprogram forboth experimentalruns are given in Appendix F.

The first expertientwas performed with apressuredrop of about 11.8 psi and an injec-tion pressure of about 40.0 psia. The experi-mental and calculatedresults are plotted inFigs. 3 and 4. The experimentwas terminatedapproximately40 hours from the start. Although3 PV had been produced, only about one-half ofthe model had saturatedsteam temperaturelevel(Fig. 4). About 84 percent of the oil in placewas produced by the end of the experiment.

To test the linear numerical simulator,a computerrun was made using the same boundaryconditions. Data used in the program are givenin Appendix E. The value of the surface ove~all.,.–-——-l-.-ems-,--L .._-J:- Al.-....... .....Gnerma cuezzzc~eub usw ML IAe pL-U~L-eJII WU=

aboout double the value determinedin the labora-tory. However, it was found that the vslue ofthe”over-allthermal coefficientused behind thesteam front is the one that is important ingetting a good agreementbetween the calculatedand the experimentalresults. Accordingly,thedifferencein values can be due to two factors:(1) the over-allthermal coefficientis tempera-ture dependentto some degree. The value ofthis coefficientfor liquid phases was deter-mined expefientally at 1400F using hot waterinjection,while the temperaturein the steaminjection runs reachec”vsluesup to 270°F and(2) the over-allthermal coefficientfor steamis small comparedwith that for liquids. Steamcondensatemight have developed a thin layeraround the inside wall of the core holder in theregion behind the steam front. This will in-crease the coefficientfor this region to somedegree.

Results plotted in Fig. 3 show that experi-mental and calculatedresults agree closelywhen the proper value of the over-allthermalcoefficientis used.

The second experimentwas performedwitha pressure drop of 24.9 psi and an injectionpressure of 39.6 psia. Both experimental.mdcalculatd results are plotted in Figs. 5 and 6.The experimentwas terminatedafter appro~atelii ‘hours. ““” ‘- ‘-‘--1--m -,,------——-_l.._._An~cnougn ouy ~ rv were prouuv=u,three-fourthsof the model had reached steamtemperature (Fig. 6). Comparingthis resultwith the one in the former experimentshows theeffect of the pressure drop on the heat loss.About 80 percent of oil in place was producedby the end of the experiment.

The linear simulatorwas run for the bound-ary conditionsof the second experiment. Allthe parametersused were the same as those usedfor the first tun, includingthe value for thesurface over-allthermal coefficient.

Results plotted in Fig. 5 show good agree-ment between experimentaland calculated

results when the proper value of the over-allthermal coefficientis used.

The Two-DimensionalModel

The only published results on two-A:.....”.<-“..1 --A-1 _ --- -4..-,.. L.. ck..41 -- 20 ~fiUL111C11O-LV11U Uluuc.1-a d~-c &vcll Uy CJIJUULCI-.

his publication,he listed the parameters andthe recovery curve for one-eighthof a five-spotmodel. No temperaturedistributionwas reportedThe data are given in Appendix E.

The two-dimensionalsimulatorwas used withthe data reported. Fig. 10 shows the experi-mental and the calculatedresults. A good matchbetween both results is evident.

DISCUSSION

CapillaryPressure

To determine the importanceof the.m.capillarypressure in tinesteam-drivemodei, two

computer runs have been performedusing thetwo-dimensionalexperimentdata given inAppendix E. One run uses the capillarypres-sures as tabulatedin the above mentioned appen-dix, and the second run uses scaled values, suchthat

1? —— PCscaled Ctabulated.

10The recovery curves are shown in Fig. 10.

The recovery curve of the run which uses scaledcapillarypressure values shows a delay in thewater breakthrough,and an early steam break-through when comparedwith the recovery curve ofthe run which uses the tabulatedvalues. Thismight be due to the fact that low capillarypressure values give low steam pressure,which,in turn, give low steam saturationtemperature.This will decrease the heat loss that is afunction of the temperaturelevels and accountsfor an early steam breakthrough. In this case,more heat will be used to heat the producingzone, giving low oil-to-waterviscosity ratiowhich will result in a delay in the water break-throllgh:

The above discussion shows the importanceof the capillarypressure values in the steam-drive model. The recovery curve, resultingfromthe use of scaled capillarypressure values, iscloser to the experimentalresults than the onedeterminedthrough the use of the tabulatedvalues. To explain such a trend in the results,a comparisonwas made between the values tabu-lated and values calculatedfrom Leverett’s9imbibitionJ-curve using values of the inter-racial tensions at atmosphericconditions. It-L-,.,-A +ho+ Lfi+h .WIA nP ~~,e ~&T,e ~~uer ofSLIUWGU ULia U UU IAL alc UL

magnitude. However, Hough et al.7 shows thatthe value of the interracialtension at thetemperatureand pressure used in the experimentdrops to as low as one-third of its value at

atmosphericconditions. This tends to givelower capillarypressure values than the onetabulated.

Relative Permeability

Dal .++,,a no-e.h+l ++.,.r.l,,a. +h.+ .-a -LLI.GLC.”LVGp’G.u,GcLIJL*A”J v-u=a “..-” a G ~

20 percent off the tabulatedvalues ti AppendixE have been used in the two-dimensionalsimulator. The other parametersare the sane asthose used in the experiment. The recoverycurves obtained from both runs show less than1.2 percent difference. Breakthroughvaluesdid not show any change. This indicatesthatthe steam-drivemodel is not very sensitiveto variations on the order of L 20 percent inrelative permeabilityvalues.

Oil Viscosity

The-steam-driveprocess has been intro-duced to the industry as a solutionto theproblem of producing highly viscous oils; thus,the importanceof investigatingthe effect ofviscosity on the process.

Three different oils (I, II, III) with wideranges of viscosity (Fig. 11) have been used inthe two<imensional simulator. The otherparametersare the same as those of the two-dimensionalexperimentgiven in Appendix F. Thethree resultingrecovery curves are shown inFig. 12. The curves show the following.

1. Recovery curves for low viscosity oilsshow earlier steam breakthroughthan those withhigher viscosity. This is due to the fact thatthe driving front moves more slowly in case ofhigh viscosity oils than it moves in case oflow viscosity ones. This will increase theheat loss which, in turn, delays the steambreakthrough.

2. Recovery curves for high viscosity oilsshowed earlier water breakthroughthan thosewith low viscosity. This is due to the factthat for high viscosity oils, the mobility ofwater is much greater than the mobility of oil!which will acceleratethe water production.

3. Although oil recovery from steam-driveprocess decreases as the oil viscosityincreases,it still gives much higher valuesthan those obtained from the waterfloodprocess.The recovery curve for a waterfloodin a five-spot pattern and for oil-to-waterviscosityratio of ’754is shown in Fig. 10. Such oil iscomparableto the one used in the two-dimensionalexperiment. Comparing the two re-covery curves snows the superiorityof the5tewf100d prOcess over the wat,erf~~~dnrncess.=..--__.However, as mentioned before, the recoverycurves of the steamfloodprocess differ con-siderablywith the magnitude of the heat loss.

6A THREE-PHASE,EXPERIMENTALAND NUMERICALSIMULATIONSTUDY OF THE STEAMFLOODPROCESS SPE 3600

NOMENCLATURE

A=c=f=H=h=

h~j =k=kr =&h =M=

P: :q.qv .

R=r.

R=s=T=

t=u.u=

VP .X,y,z =

~.p=q=~=p=

r=

a.c=i.j=

A?=~.

n+l =

cross-sectionalarea, sq ftspecificheat, Btu/lb day OFfractionalflow, dimensionlessreservoir sand thickness, ftenthalpy,Btu/lbenthslpy of injected steam, Btu/lbabsolutepermeability,darciesrelative permeability,dimensionlessheat loss, Btu/Dmolecular weight of steampressure, psiacapillarypressure, psimass injection rate, lb/Dvolumetricinjectionterm, lb/unit bulkreservoirvolume er day

Tgas constant,psia cu ft lb mol ‘Rradiusheat residual,Btu/Dsaturation,dimensionlesstemperature,‘F if not subscripted,andtransmissibilityif subscripted,lb/Dpsi

time, daysDarcy’s velocity, ft/Dover-all thermal coefficient,Btu/D Sqft ‘F

block pore volume, cu ftCartesian coordinates,ft

Greek

difference operatorviscosity, cpdimensionlessheightporosity,dimensionlessdensity, lb/cu ftdimensionlesstime

Subscripts

ambient conditioncondensategrid index in the x-directiongrid index in the y+irectionliquidold-time stepnew-time step

REFERENCES

1. Buckley, S. E. and Leverett,M. C.: ‘T4ech.anism of Fluid Displacementin Sands”,Trans.,AIME (1942)~, 107-116.

2. Coats, K. H.: ncomput,ersimulation‘f

Three-DimensionalThree-PhaseFlow inReservoirs”,UnpublishedReport, The U.of Texas at Austin (Nov., 1968).

3. Coats, K. H.: P. En. 383.21 class notes,PetroleumEngineertigDept., The U. ofTexas at Austin (1967).

4. Coats, K. H.: Private Cormnunication(196815. Coats, K. H., Nielsen, R. L., Terhune,M.

H. and Weber, A. G.: “Simulationof Three-

6.

7.

8.

9*

10,

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21●

Dtiensional.Two-PhaseFlow in Oil and Gas,.Reservoirs”,Sot. Pet. lhvz.J. (Dec., 1967)377-388.Higgins, R. V. and Leighton,A. J.: “AComputerMethod to Calculate Two-PhaseFlow in Any IrregularlyBounded PorousM&um”, J. Pet. Tech. (June, 1962) 679-

.Hough, E. W., Rzasa, M. J. and Wood, B. B.:l!Interfa~i~Tensions at Reservoir pres-.

sures and Temperatures,Apparatus and thethe Water-MethaneSystem”, Trans.,AI14E(1951)~, 57-60. —Lauwerier,H. A.: !~~e Trmsport of Heat

in an Oil Layer Caused by the Injectionof Hot Fluid;’,Applied Sci. Res.-(1955)Sec. A, ~, 145.Leverett,M. C.: ~?capilla~Behavior fi

Porous Solids”, Trans;,AIME (194.1)l&,152-169.Little: T. w*: f~Recove~.Of ViscOu5 Crude

Oil by In Situ Combustion;”,MS thesis,PetroleumEngineeringDept., The U. ofTexas (July, 1958).Marx, J. W. and Langenheim,R. H.: “Reser-voir Heating by Hot Fluid Injection”,Trans., AIMi (i959)~, 312-315.MacDonald,R. C. and Coats, K. H.:l!Methodsfor Numerical Stitiationof Water

and Gas Coning”, Sot. Pet. En.g.J. (Dec.,1970)L25-436.Rame~. H. J.: Discussion on “ReservoirHeat”&g by Hot Fluid Injection”,Trans.,AIM (1Q5Q) 216. 36L+-365=k-t..{_rRamsey, P. E.: !!Recoveryof Viscous Crude

Oil by Steam Injection”,MS thesis, Petro-leum EngineeringDept., The U. of Texas(MaY, 1958).Reid, S.: lt~e Flow of Three ImmiscibleFluids in Porous Media”, PhD dissertation,Chemical En ineeringDept., U. of

fBirmingham 1956).Richtmyer,R. D.: DifferenceMetinodsforInitial-ValueProblems,IntersciencePublishers,New York (1957) 101.Sheldon, J. W., Harris, C. D. and Baviy,D 11AMethod for General Reservoir.:Behavior Simulation on Digital Computers”,Paper SPE 1521-G presented at the 35thAnnual SPE Fall Meeting, Denver, Oct. 2-5,1960.Snell, R. W.: t!~ree-phaseRelative

Permeabilityin an UnconsolidatedSand”,J. Inst. Pet. (March,1962),!@,80.Shutler,N. D.: ~~N~eric~, Three-Phase

Simulationof the Linear SteamfloodProcess”, Sot. Pet. EnR. J. (June, 1969)232-246.Shutler, N. D.: t~N~ericd Three-Phase

Model of the Two-DimensionalSteam FloodProcess”, Sot. Pet. EnR. J. (Dec., 1970)405-417.Spillette,A. G. and Nielsen, R. L.: “Two-DimensionalMethod for PredictingHotWaterflood RecoveryBehavior”, J. Pet. Teck

‘E 36c0 A. ABDALIA and K. H. COATS 7

22 (June, 1968) 627-638.● Stone, H. L. and Garder,A. D., Jr.:

!?An~ysisof Gas-Cap or Dissolved-Gas

Drive-Reservoirs”,Sot. Pet. E%q. J. (June,1961) 92-104.

23. -Willmsn, B. T., et al.: “LaboratoryStudies of Oil Recovery by Steam Injectiont’J. Pet. Tech. (July, 1961) 681-690.

APPENDIX A

Heat Loss Calculation

In the computer simulatordeveloped inthis study two proceduresare used to calculatethe heat loss. One is used in the testing ofthe physical models, and the other is used inthe field case studies.

Heat Loss Caicuiationfor the Physicai Modeis

Physicalmodels are made with limited in-sulation thickness. A representationof theheat loss in terms of an average over-allthermal coefficientthat can be determinedinthe laboratorywill best suit such cases. Thefollowingis the equationused for cylindricalinsulationsaround a cylindricalcore holder:

Heat loss = n dU Z x, . . . . (A-1)

where d = outside diameter of the insulation,fiU= over-allthermal coefficient,Btu/D

sq ft ‘FZ = differencein temperatureacross the

insulation,‘Fx = block length, ft

Since the over-all thermal coefficientof=+.eam i Q di ffaven+. +.hnm+.hn+.of’~iq~d~, a-.“-..-- --..“.v....... . .....weighted average value was used in-this”studyand is given by the following equation:

u = u~s~av

+ Ul(l - Ss) . . .

Heat Loss Calculationfor Field Cases

In field cases, the overburdenand

. (A-2)

theunderburdencan be consideredas infinite insu-lations. The equationsunder considerationarethose describingthe heat flow in a semi-infinite slab. Their solutionis made usingT.-1 . . . 4 . . . . . . . ..#---udp~acc u~-ala~uillla m

Consider a reservoir ~sand of thiclmessh. The ‘z-axis runs parallel tothe heat flowing to theoverbtien as in the fig-ure. The partial-

! overburden ,

differentialequationde- H .....scribingthe heat flow is 7, ‘“ ‘“””‘-”

+r,l 1 n,.,=.as A.uA4.uwa. reservoir

Io ~_._ ._._..

Kza2T aT—= Prcr=. . . . . . ● . (A-3)az2

The boundary conditionsare

T=Ti Hatz=7 and t = ti . . (A-4)

T+T aas z+rn) . . ● . . . . . (A-5)

The initial conditionis

T=T a at t = O and all z . . . (A-6)

Let

4 Kzt‘c= . . ...* . . . . . (A-7)

H20rcr

. (A-8)~= e”””’””””” ““”

Z=T-T ● ● ● ● ● ● ● ● ● ● * ●(A-9)

a

Then Eqs. A-3 through A-6 will be

a2z az—=E” “ “ “ “ “ “ ● “ “ “ “ “2

(A-1Oan

2=2 iatrl= 1 and -r= Ti . . (A-n

Z+l) asq +m* ● “ ● “ ● “ “ ● ●(A-12

z = O at T = O and allne w ● 4 (A-13

PerformingLaplace transform on Eqs. A-10and A-ii and using Eq. A-13 we get

az- S~=O. . . . . . . . . . . (A-14

The solutionto Eq.

Z=cleQ/’-is

~~: A-lz uives ~j, =

be 0-”--

=land~=~i. .(A-15

A-14 is

+c2e-r)v%-

.0.. (A-16

~.ce -T- ds”

2 . ..*. . . . . . (A-17

Using Eq. A-15 in Eq. A-17, we get

Zi -/53--=c2e

then zi&_

=+C2 s

A THREE-~SE , EXPERIMENTALAND NUMERICALSIMULATIONSTUDY OF THE STEAMFLOODPROCESS SPE 3600

-(l-l-l)&~=z% (A-18)iS . ..*.*.

Performingthe invertedLaplace transformLo Eq. A-18, we get

n-lz(~, ~)=zierfc —’.” ● *O (A-19;

26“

Using Leipnitz rule to differentiateEq. A-19,we get

-(T1-1)2

az Zi (n-l) e 4~—=—a~ 2G rT3 “ ““ ““ ‘A-20’

Rate of heat loss per unit volume = Prcr lrn ai1 zL

_(n-i)22Kz Zi 4T 4Kz Zi

= fm(n-l) e .—

h2 Td= 1 h2 d=

. . ...* .. *.** ● ☛☛☛☛✝(A-21

Eq. A-21 gives the rate of heat loss per unitvolume.

APPENDIX B

Finite-DifferenceExpansion

Before startingany finite-differenceexpansion,the grid system must be specified.The selectionof such a system depends upon theboundary conditionsto be used.

For the linear model to be developed here,predeterminedinjection and productionpressurewill be used as boundary conditions. The gridsystem shown in Fig. 7 is the most suitablefor+.h;~r.a~e. Au finite-differenceexpansions“A.-s-----that follow in this chapter pertain to such agrid system.

Eqs. 3 contain second-orderspatialderiv-atives and first-ordertime derivatives. Thestandard central-differenceapproximationforthe spatialderivativesis as follows.

.

30. IPO -pOm, -a.(P. - P.

1+1/2\ i+l il i-l\2 i i-1.

AxL. . . . . . . ..*** ● ***** (B-1)

where

kxkro P.

a. ‘— . . . . . . .*.** (B-2)!-l.

The p values in these spatial differences?are unders ood to apply at the new time level

‘n+l”

The backward time differenceapproximationis used for the time derivativesas follows:

(+h)%))n+l- (Qp($())% n&@Poso) = At

,.

. . . . . . .*.*** ● ☛☛☛☛☛(B-3)

Using these finite+ifference approximationsinEq. 3 and multiplyingboth sides of the equationbyAAx, we get

AXTOAXPO = ‘~At(poSo) . (B-4a)

AXTWAXPW + qc = ‘&t (pwsw) . (W4b)

AxT~AxP~ - qc = ‘$At(P&) . (B-4c)

where

AxTOAxpo = To (Po -J?o)i+l\2 i+l i

- To (Po -Po)i-l\2 i i-1

. . . . . . . ..*** ● ***** (B-5)

kxkropo

‘o.= 6.33 ‘=

1+1/2 1-’o i+l\2 ‘M)

and A is the cross-sectionslarea perpendicularto the flow.

Applying the same finite-differenceapproximations to the heat-bslanceEq. 6 and multi–plying both sides by A&, we get

DXA~A;zn - in - AAx(uPhn) + qshinj

v. fiAt(~ph + $+rZ), ● (B-i’)

whereAx(uopoho) = A(uoPo)l+l/2hnoi

-A(uoPo)i_l/2hno

‘:l(B-8). . . . . . . ...0. ● .***

‘(uOpO)i+l/2= TO (Po - PO.)

i+l/2 i+l. ..*.* ● ...*. .***** (B-;)

m 3600 A. ABDAILA anc

At (pOSOhOl = (Pos&)n+l - (~osoho)n

. . . . . . . . . . . . .

A:zn =

APPENDIX C

Zn -2zn+i+l i

.*.. (B-lo)

Zn● (B-II)

i-1

IMPES Application

IMPES is a techniquein which the pressurein the flow term, ATAp, is handled implicitly,while the saturationand saturation-dependentparameters are handled explicitly. This tech-nique is described in the literature.17,22 In

this anaiysis tlnistechniq~ is applied in amanner describedby Coats.

Eq. B-4 can be rewrittenas follows.

AxToAxpo =

‘P &~ ~on+l *ts(3+ At On ‘tpO (C-la)

AxTwAxpCg + qc =

‘Pv

~Pw AtSw +%ApAt Wn tw” . (C-lb)

n+ 1

AxTsAxp~ - qc =

‘P ‘P ~EQsn+l ‘ts~ + m Sn ‘tps “ “ (C-lc)

..-,..rm.uupqimg Eq. C-lb by al and Eq. C-it by a3and adding the three equations,we get

AXTOAXPO + alAxTwAxp,d + a3AxTsAxps

+ (a, - a3)qC = ‘~ p. AtSOL At

n+l

1+ alpwn+l ‘tsw + a3psn+1 ‘tss..’

.&son ‘tpo + Wwn ‘tpw

+ a3%n%ps “ ‘ “ “ “ “ “ “ ●

. (C-2)

.-

Using Eq. 4 and choosing al and a3 such that

}, Atso = () ,- a3ps n+lj

then

‘on+l

a3=~ “ “ ● “ “ “ “ “ “ “ “ ● (c-3a:n+l

P‘n+l

al=a3P “ “ “ “ “ “ “ “ “ ● “ (C-sb:““n+1

SubstitutingEq. 5 in Eq. 13, we get

AXTAXPO -a A TA P1 x w x co_,J

+ a3AxTsAxPC + (al - a3)qCs-Cl

‘P= ~(alsw AtPw + So At~o

n n

+ a3%=Atps), . ● . . c . ● . ● (C-4)

where

‘= alTw+To+a3Ts” “*”””(C-5)

In forming AxT~PO, care must be taken tileaving al and a3 outside the spatial differenc(

Since oil and water densitieshave beenconsideredin this study ljQbe flm.ctiQnsoftemperatureonly and steam density is a func-tion of both temperatureand pressure, the term~Ps can furtherbe expanded as follows:

Atp5 = Atp; + p:Atp , . . . . ● (C-6)

where

Atp; ‘ps(zn+l,Ps ) - Ps(znfP~ )(C-7)n n

Ps(zn+~lPn+l) - Ps(zn+lfP5 )

P: =n

P5 -ps

n+l Ii

. . . . . . . . . . . ...* . . . (c-8)

From Eqs. C-4 and C-6 we get the following:

k+l + ~kAXTAXPO = GAtpo , . . . (C-9)

where kB= (al -a3)q~ - alAxT,dAxPc

o-w+aA?AD3 x s xc s-q

v

A ~~.~~~ ; -IMENTAL AND NUMERICAL) SIMULATIONSTUDY OF THE STEWFLOOD PROCESS SPE 3600

A P*] .

+ son%~o + a3%n t s . (c-lo)

and

G=v$a3ssp~ . . . . . . . . (C-II)n

The superscript(k) shows that the value atthe old iterationis to be used. The super-script (k+l) shows that the value at the newiterationis to be used.

APPENDIX D

Rate of Steam Condensation

The calculationof the rate of steam con-densation is made by the use of two sets ofequations. The first set is for blocks thathave no free steam, i.e., their temperaturesarebelow the saturationtemperaturesof steam. Thesecond set of equationsis for blocks that havefree stesm, i.e., their temperaturesare equalto the saturationtemperaturesof steam.

Blocks with No Free Steam

In these blocks, all the steam coming infrom adjacentblocks is condensing,i.e.,

q= = Ts (Ps -PSIx. “ i-l,j i,j1-1/2 ,j

+T (PS -PSI‘Y.l,j-l\2 i,j-1 i,j

. . . . . . . ..*.. . ..*.. (D-1)

k (darcys)

H (ft.)

D-1 --1.- ..G+L V--* C+a.mD-LuGfi3 W.1.L/LL XLGG UVV-U

After solving the fluid flow equationsforthe pressure distribution,the steam saturationtemperaturesfor blocks with free steam aredetermined. The use of these temperaturesinthe heat-balanceequationwill result in resid-uals. These residuslsare due to the use of therate of steam condensationat the old iterationin solvin~ the fluid flow equations. Correctionof such v~ues will reduce ~he residualstowithin limits of tolerance.

Denoting the residual of the heat-balanceequationat any grid point by R, we then have

where (wf) is a weight factor to be chosen ina way that will acceleratethe convergence.

APPENDIX E

Data Used for Calculations

This appendix containsdata used in theoperationalmodels. The relativeperme-abilities,capillarypressures and dispersioncoefficientsfor the linear model study areobtained from Shutler.20

E.1

E.2

E.3

E.1

2.54

.354

3=42

Linear ExperimentI

Linear ExperimentII

Two-dimensionalexperiment

E.2

‘2.54

E.3

132

.372

3.9

.83

UL(Btu/day.ft. F)

Us(Btu\day.ft. F)

Cr(Btu\lb. F)

Ta ( F)

Swi

6.2

.204

80

1<7A“#

.1855

80

.229

6.2

.204

80 24

167 167

.1855 .2156

80 80

.229 .1

m 3600 A. AEDALIA and K. H. COATS 11

Tinj ( F)

Pinj ( Psi)

‘prod( Psi)

E.1 and E.2

Sw—

.2287

.30

.40

.50

.60

.70

.90

SW—

.2287

.3

.4

.5

.6

.7

.9

so—

.2

.4

.5

.6

.7

.8

E.1

267.25

‘25.3

13.5

Pco-w

2.2

1.0

.7

.52

.37

.23

.1

krw

o

.002

.009

.012

.019

.022

.042

kro

s-o

.0008

.01

.04

.125

.38

.7

E.2 .E.3

266.63 400

24.9 260

0 190

so—.3

.4

.5

.6

,7

.8

kro

o-w

1.0

.922

.8

.58

.26

.06

0.0

krs

.175

.105

.05

.01

.001

.0

Pc

s -o.—

.38

.29

.21

. 16

12___

.11

A THREE-FHASEEXPERIMENTALAND NUMERICAL

12 SIMULATIONSTUDY OF THE STEAMFLOODPROCESSSPE 3600

E.3

Sw—

.1

.2

. 3

.4

.5

.6

.7

. 8

. 86

.1

.2

.3

.4

.5

.6

.7

. 8

. 86”

—

.1

.2

. 3

.4

.5

.6

.7

. 8

. 89

4.1

.095

.072

.061

A,-,.U3L

.041

.031

.021

.011

krw

o

.0016

.0081

.0259

. 0672

. 1

. 14

.20

.25

kro

s -o

0

.009

. 031

.062

.11

.19

. 335

.570

1.0

50—

.1

.2

.3

.4

.5

.6

.7

. 8

. 89

kroo-w

1.0

.875

.735

.590

.42

.21

.07

.016

G

krs

.52

.41

.31

.22

.14

.08

.03

.005

0

Pcs-o

4.517

.067

.042

.02

-.022

-.043

-.064

-.085

SPE 3600 A. ABDALLA and K. H. COATS 1:

Temperature Viscosity (Cp)

80 800

100 330

140 110

1$?n... ~~

240 18

280 11

360 5.26

‘450 2.9

TABLE 1 - RESULTS OF EXPERIMENT1, LINEAR MODEL, PORE VOLUME = 494.14cc

Pressure TemperatureF(Psi) Fluids

Time Produced(cc)(min.) DistancefromInlet

In- out- 011 Total 1.2” 8.9” 15.6” 23.3” 31!1let let

38.7”

3:6090120150180210240..,sClu

~%

3603904204504s0510

570600640680720760000850

n9409701000103010601090112011s0~2~Q12401270130513451385143014601500

25.225.12525.125.225.225.125.225.3~~ ~

25:325.225.225.225.225.225.224,824.924.924.924.924.924.824.824.924.924.924.924.424.924.924.824.424.824.924.824.824.824.024.824.824.024.824.924.8

13.413.313.213,212.913,213.313.113.212.S

12.912.813.21313.213.113.113.113.313.113.113.213.213.213.213.213.113.213.213.013.212.912.813.3131312.8~z.g12.912.812.812.312.213.013.013.0

06.613.420.327.935.842.95057.868.277.587.698.5

109.2121134.8149.8171191.8

212221.3226.5232.5237241.5247.6250.9254.9259.1263.6268.4272.5277.3282.3287.4295.8298,2301.8305.3308.9312.9317.4321324326.2

06.613.420.327.935.842.95057.8~~ ~

77;587.698.5109.2121134.8149.8171191.8219.5260.5309.3358.5408454503564599.2636.7676.6715.5753.9790.6830.6874.8917.4

1008,11050,61095.41140.11186.71251.71310.21360.51406.41451.6

808496107116124132137138~~~

147154161168174183192200205258279279279279279279279279279279279279279279279279279279279279279279279279279279

808080808080848588w92939698100101105108112117125144180223247275275275275275275275275275275275275275275275275275275275275275

808080008080@8080

80818283050687889098106115128143152162172182192199211228244266266266266266266266266266266

80808080808080

f%

80808080

:;

%828384

%9599102106109117119124129135149157164170175184198205210212

%00808080808080w80808000

t%8080808080,

z808080828283848689909294

1::104106108110114123127130134

%00808080808080On

%

8080

::8080

::808080808080

::80808080808080838485858686

:;92939498

TMERMOCWPLESTABLE 2 - RESULTS OF EXPERIMENT 2, LINEAR MODEL, POKE VOLUMi = 494.14CC

?mfi FluId*TemPeP.ture F

Time P,oiueeai,.)(min.) Distancefrca Inlet

In- out- 011 Total 1.2” 8, 9“ 16.6” 23.?” 31” 30.7”let let

o

40

so

120

154

2U0

240

220

320

360

4s0

440

490

320

560

6SCI

640

24,8

25,3

24.0

24.7

24,7

24.5

24.3

24,6

24, e

24.0

24,9

24.9

24,6

24.9

24,0

24,9

24,9

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

20.4 20.4

41,2 41,2

65 65

91.5 91.5

122.7 122.7

160.3 160.3

lm.2 222

207.5 305.5

222.1 381,0

237.3 448

247.2 514,6

262,6 394.e

275.5 677.7

226.4 76E.2

2’J7 .3 849.6

3043,1 929.1

W608060W

llo 00 W 00 00

133 23 00 00 60

lmzzboeom

176 96 W 00 80

210104 s4m2a

279 11629WW

279 170 92 E4 20

279 234 102 06 00

279 275 172 90 20

279 275 264 106 6~

279 275 265 144 90

279 275 26S 170 98

279 2?5 265 254 120

279 275 265 2S 186

279 275 265 234 214

279 275 265 254 236

N3

W

eo

No

24

00

84

60

6.3

80

en

w

20

80

96

102

124

Fig. 2 - Viscosity emd specific gravity vs temperature for Primol 185.

1.a

*SO -

&● ‘R..u= too -~

\: A tSipERlUEt4TAL ●

p ,s0 --o- CALCULATED

\

<._100 -

L———————0 s to 1s 20 !?6 so 66 L. a Inctms

OISTANCE FROM tN.3ECT10N POINTS - INCHES

Fig. h - Temperature distribution at the end of Experiment 1.

INLET

CONOENSEO PRESSURE

sTEAMTRAP

.....

!

MC K- PRESSUFIE

TO COIL HEATII13 REWLATOR

u

Fig. 1 - Schematic diagram of the epparatus.

E/L77

[: //

f!

.1

s

o .4 .a hp. 1.6 do M hCUMPiJLATIV”-TOTALFLuIOS PRODUCED-PORE VOLUMES

Fig. 3 - Experimental and calculated results of linear Experiment 1,

AP =11.8 Psi.

SW

r

A AEI. TEMPERATUREAT 8,9’● $!50 FROMlNJECTIONPOINT

&’J AAm 100#/

“~ A =

m .?T-.

b .6b

o>

.5-:

g .4-

Z .~-

~ .e-

a

-1J -

Zio

o .4 .6 l.z 1,6 Zo 6.4 M

CUMULATIVE TOTAL FLUIDS PRODUCED- PORE VOLUMES

Fig. 5 - Experimental md calculated results of linear !kPeriMent 2,

AP =24.9 Psi.

A

A EXPERIMENTAL

-o- CALCULATE

50 I I I I 1 1 I ~—

0 5 10 Is 20 25 30 35 L~141 inchc$

OIST’ANCE FROM INJECTION POINTS - INCHES

Fig.6 - Temperaturedistributionat theendof Experiment2.

1 2 3 N-2 N-1 N

Fig. 7 - Grid system used in the linear numerical simulator.

Read DataInitialize the Model

t

Prtnt Dataand Initialized Conditions

I+

Compute:

- Pore volume

;- Inltfal oil andwater Iln place

3 - Flow ratecoefficient;1

+~-

Compute from tabl(e look-up:

1 - Relative p(ermeabillty2 - Capillary [pressure

- Viscosities: - Densities

Compute transmissibil ities

I

t

~1Establish F1OW Rate

r—

Compute enthlalples andsaturation temperatures

II Compute saturations I

t

I Compute temperature andralte of steam.condensation

from the bleat balance

-———————l——————

c L—1Check Convergence

t

I Compute material andheat ba[lances I

t

I Incrememt timeand print results

l’———

L#Update

densities

L —~

Fig. 8 - Nlurnerica.1 simulatorsflowchart.

r I I I I !

B

● **.*

● **OSi,.i+l

●● *

i-; ,j i~j i+l ,j

wFig. ; - Grid w.t,n used in the

tho-airqensicmal numericalsir.uiator.

.6,

\

~ 2–s’’’”’””$ ‘5

/

! ,4 /

STEAM S.T. A

[1 .x Y> .-

a

:8

: /;-’r

a .2d en- Celculoted using tabulated

5

-c- Calculated using Pc t.ab.lo?edltO

H,@gins, .!01 water fled (11)

.2 .4 .6 .8 1.0 1.2 1.4 1.6

PoRE VOLUMES PROOUCEO

I0,000, ! 1 I 1 d

N iI JMo

zI

: 100

zov~>

10

I I I I I I I

50 100 150 200 250 300 350 400

~EMpzR~~,~:~ .~

Fig. 11 - Viscosity vs temperature

Fig. 10 - Experimental and calculated results ofthe two-dimensional experiment.

0.6

~ 0.5S~o>~ 0.4ag

I● 0.3-

g

: 0.2

JG

0.1

00 0,2 0.4 0.6 0.s 1.0 1,2 1.4 106

.RtMwL gil Waco sitv fat Room

Tomverotwe, cK II

PORE VOLUMES PROOUCEO

m, “ J Q . rmn->,+.d Gil recQvery curves for different oil viscosities...=. .. .....=....-