A Three-Phase, Experimental and Numerical

8/7/2019 A Three-Phase, Experimental and Numerical

1/16

SOCIETYOF PETROLEUMENGINEERSOF AIME6200 North CentralExpressway RR SPE 3 6 0 0Dallas,Texas 75206THIS IS A PREPRINT--- SUBJECTTO CORRECTION

A T h r e e - Ph a s e , E x p e r i me n t a l a n d Nu me r i c a lS i mu l a t i o n S t u d y o f t h e S t e a m F l o o d

P r o c e s sBYA. Abdalla andJf. H. Coats, Members AIME, INTERCOMP

O Copyr igh t 1 97 1Am er ic an In st it ut e o f Min in g, Me t allu rgic a l, a nd Pe t ro leu m En gin ee r s , In c.

This paper was preparedfor the 46thAnnual 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 abstractshouldcontainconspicuousacknowledgmentof where and by whom the paper is presented. Publicationelsewhereafterpublicationin the JOURNALOF PETROLEUMTECHNOLOGYor the SOCIETYOF PETROLEUMENGINEERSJOURNAL isusually grantedupon requestto the Editorof the appropriatejournalprovidedagreementto 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-driveoilrecoverywas developed and tested. The implicit INTRODUCTIONpressure-explicitsaturation (IMPES)techniquewas used to solve the three-phasefluid flow The first part of the work presentedhereequationsfor compressiblefluids. A method is a physicallaboratorymodel 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 computersimulatorsfor sets of injection and productionpressures. Oillinear and two4iimensionalsystems. recovery and temperaturedistributiondata wereobtained. 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 describesthewith the results obtainedfrom the linear developmentand applicationof numericalnumerical computer simulators. The results from simulationtechniquesto solve equationsthe two-dimensionalnumerical computersimulator 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


2/16

A THREE-PHASE.EWEUUWWAL AND NUMERICALVIT17AMlillMTlPR(Y!T?.S.S----- ----- .-.---- SPE 3603CTMTIT Am-rfm cmmiv m THOJ.L-lUU LAW.,Au. .properly tested. Therefore,a physicalmodelwas 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.uGusufficientdata 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 asaturatedsteam from The U. of Texas utilitylines. The injectionpressurewas 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 cloggingof the sanepack. A coil heater was wrapped around theinjection line. The temperatureof the heaterwas adjustedby 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 80F 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 percent

porosity.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 arepresentedhere...,-a--- .-L.---Fhua !Low Jlql,lazlons

The mathematicalrelationshipsdescribingmultiphase fluid flow a pear in thefliteratWe05,12717,20t2 The developmentofsuch relationshipsis based upon mass balanceand Darcyts law for each phase. When bothrelationshipsare combined,the partial-differentialequationdescribing the fluid flow

of each phase in the reservoirwill be obtained.

a(pouxo) a ( p o u ~)ax - ay + qvo =

a(oposo)at ;******** (l-A)

for the water phasea (PWUW)

ax - ay + ~ + qvc =

and for the steam phasea(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-1kk. api- 6.33 -&yi ~ (2-B)i

andi = O,w,s.Substitutionof Eqs. 2 into Eqs. 1 givesnkxkrOpO apO6.33 & llo T )L

+a (y 1k k r y o ape,~ ~; + Cl v ol J o .a(oposo)= at . q . .(3-A;

a[ k krwpw PPW)]+ x\ Ilw T + qw + qvc

a(~pWsw)= at********** .(3-B:


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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-wP = Ps -P~* q . . .co-s

All symbolsused are describedinclature.

Heat Flow Equation

.0.. (5-A)

.,.. (5-B)the Nomen-

The developmentofrelationshipdescribingmedia is based upon theand Darcys equations.combined,the followingobtained.

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

- ,+ ~ (DX%*(DY%)a- ~ (Uxphn) - &uyPhn) + qvshinj= & [$(PSh) + (1 - $) PrCrZ] , (6)

whereuiphn = u. POhn + uiwpwhn10 0 w

+U ispshns - q .* q q (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 relativepermeabilities

are 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 saturationCapillarypressurebetween oil and steam isfunctionof both water and oil saturations.

5* The heat loss term is explainedin

of

andonlya

detail in Appendix A. The differenceform ofthe partialdifferential equationdescribedhereis presented in Appendix B. The applicationofthe IMPES techniqueto solve the differenceequation 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-trollingone.

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//--

Ibbuu 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 relativepermeabilities,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


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A THREE-PHASE.EXPERIMENT AND NUMERICAL-.SIMULATIONSTtiY OF !lsaturationtemperaturesare 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 followedby 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 therelativelysmall pressure drops used in testingthe model.

The Two-DimensionslSimulatorA computerprogram was written based on

the techniquesdiscussedhere. The grid systemused is shown in Fig. 9. As in the linear simu-lator, the program computespressures,satura-tions and temperaturedistributions. Theprogram slso computesthe rate of steam conden~sation and injectionand 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 statedinAppendixD. 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 pressuredrop. 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 3600Loss. 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-mentaland 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 computer run was made using the same boundaryconditions. Data used in the program are givenin AppendixE. The value of the surface ove~all.,.--l-.-ems-,--L .._-J:- Al.-....... .....Gnerma cuezzzc~eub usw ML IAe pL-U~L-eJII WU=abooutdouble the value determinedin the labora-tory. However, it was found that the vslue oftheover-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 dependent to some degree. The value ofthis coefficientfor liquid phases was deter-mined expefientally at 1400F using hot waterinjection,while the temperaturein the steaminjection runs reachecvsluesup to 270F 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~atel

ii 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, including the value for thesurface over-allthermal coefficient.Results plotted in Fig. 5 show good agree-

ment between experimentaland calculated


5/16

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

The only published results on two-A:......


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

NOMENCLATUREA=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/lbenthslpyof injected steam, Btu/lbabsolutepermeability,darciesrelativepermeability,dimensionlessheat loss, Btu/Dmolecular weight of steampressure, psiacapillarypressure,psimass injection rate, lb/Dvolumetricinjectionterm, lb/unitbulkreservoirvolume er day

Tgas constant,psia cu ft lb mol Rradiusheat residual,Btu/Dsaturation,dimensionlesstemperature,F if not subscripted,andtransmissibilityif subscripted,lb/Dpsitime, daysDarcys velocity, ft/Dover-allthermal coefficient,Btu/D Sqft Fblock pore volume, cu ftCartesian coordinates,ft

Greekdifference operatorviscosity, cpdimensionlessheightporosity,dimensionlessdensity, lb/cu ftdimensionlesstime

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

REFERENCES1. Buckley, S. E. and Leverett,M. C.: T4ech.anism of Fluid Displacementin Sands,Trans.,AIME (1942)~, 107-116.2. Coats, K. H.: ncomput,ersimulationfThree-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.

21q

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 CalculateTwo-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~Tensionsat Reservoir pres-.sures and Temperatures,Apparatus and thethe Water-MethaneSystem, Trans.,AI14E(1951)~, 57-60. Lauwerier,H. A.: !~~e Trmsport of Heatin an Oil Layer Caused by the Injectionof Hot Fluid;,Applied Sci. Res.-(1955)Sec. A, ~, 145.Leverett,M. C.: ~?capilla~Behavior fiPorous Solids, Trans;,AIME (194.1)l&,152-169.Little: T. w* : f~Recove~.Of ViscOu5 CrudeOil 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 Waterand 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 CrudeOil 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,ChemicalEn 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 GeneralReservoir.:BehaviorSimulation on Digital Computers,Paper SPE 1521-Gpresented at the 35thAnnual SPE Fall Meeting,Denver, Oct. 2-5,1960.Snell, R. W.: t!~ree-phaseRelativePermeabilityin an UnconsolidatedSand,J. Inst. Pet. (March,1962),!@,80.Shutler,N. D.: ~~N~eric~, Three-PhaseSimulationof the Linear SteamfloodProcess, Sot. Pet. EnR. J. (June, 1969)232-246.Shutler,N. D.: t~N~ericd Three-PhaseModel 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


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E36c0 A. ABDALIA and K. H. COATS 722 (June, 1968) 627-638.

q Stone, H. L. and Garder,A. D., Jr.:!?An~ysisof Gas-Cap or Dissolved-GasDrive-Reservoirs,Sot. Pet. E%q. J. (June,1961) 92-104.23. -Willmsn, B. T., et al.: LaboratoryStudies of Oil Recoveryby Steam InjectiontJ. Pet. Tech. (July,1961) 681-690.

APPENDIXAHeat Loss Calculation

In the computersimulatordeveloped 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 PhysicaiModeis

Physicalmodels are made with limited in-sulationthickness. 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 = outsidediameter of the insulation,fiU= over-allthermal coefficient,Btu/Dsq ft FZ = differencein temperatureacross theinsulation,Fx = block length, ft

Since the over-allthermal coefficientof=+.eam i Q di ffaven+. +.hnm+.hn+.of~iq~d~, a-.-..-- --...v....... . .....weighted average value was used in-thisstudyand 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 infiniteinsu-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. reservoirI

o ~_._ ._._. .

Kz a2T aT= Prcr=. . . . . . q . (A-3)az2The boundary conditionsare

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

T+T aas z+rn) . . q . . . . . (A-5)The initial conditionis

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

4 Kztc= . . ...* . . . . . (A-7)H20rcr. (A-8)~= e

Z=T-T q q q q q q q q q q * q (A-9)aThen Eqs. A-3 through A-6 will be

a2z az=E q 2 (A-1Oan2=2 iatrl= 1 and -r= Ti . . (A-n

Z+l) asq +m* q q q q q (A-12

z = O at T = O and allne w q 4(A-13

PerformingLaplace transform on Eqs. A-10and A-ii and using Eq. A-13 we geta z - S~=O. . . . . . . . . . . (A-14

The solutionto Eq.Z=cle Q/-is

~~: A-lz uives ~j, =be 0---

=land~=~i. .(A-15A-14 is+c2e -r)v%- .0.. (A-16

~ . c e -T- ds2 . ..*. . . . . . (A-17Using Eq. A-15 in Eq. A-17, we getZ i -/53--=c2e

then z i&_=+C2 s


8/16


9/16

m 3600 A. ABDAILA ancAt (pOSOhOl = (Pos&)n+l - (~osoho) n. . . . . . . . . . . . .

A:zn =

APPENDIX C

Zn -2zn+i+l i

.*.. (B-lo)Zn q (B-II)

i-1

IMPES ApplicationIMPES 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 Inthis anaiysistlnistechniq~ is applied in amanner describedby Coats.Eq. B-4 can be rewrittenas follows.AxToAxpo =P &~ ~ o n + l * t s ( 3 + At On tpO (C-la)AxTwAxpCg + qc =P v~Pw AtSw +%ApAt Wn tw . (C-lb)n+ 1AxTsAxp~ - 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 getAXTOAXPO + alAxTwAxp,d + a3AxTsAxps

+ (a, - a3)qC = ~ p. AtSOL At n+l

1+ alpwn+l tsw + a3psn+1 tss..

.&s on t p o + Wwn tpw+ a3%n%ps q . (C-2).-

Using Eq. 4 and choosing al and a3 such that

}, At so = () ,- a3ps n+ljthen

on+la3=~ q q (c-3a:n+lP n + l

al=a3P q (C-sb:n+1SubstitutingEq. 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~on n+ a3%= Atps), . q . . c . q . q (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.ctiQnsof

temperatureonly and steam density is a func-tion of both temperatureand pressure, the term~Ps can furtherbe expanded as follows:Atp5 = Atp; + p:Atp , . . . . q (C-6)

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

Ps(zn+~lPn+l) - Ps(zn+lfP5 )P: = nP5 -p s

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,dAxPco-w+aA?AD3 x s xc s-qv


10/16

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 CondensationThe 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 have

free 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 -PSIY.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-UAfter solving the fluid flow equationsforthe pressuredistribution,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 residualof 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 CalculationsThis appendixcontainsdata used in theoperationalmodels. The relativeperme-abilities,capillarypressuresand dispersioncoefficientsfor the linear model study areobtainedfrom Shutler.20

E.1E.2E.3

E.12.54.354

3=42

Linear ExperimentILinear ExperimentIITwo-dimensionalexperiment

E.22.54

E.3132

.3723.9.83

UL(Btu/day.ft. F)Us(Btu\day.ft. F)

Cr(Btu\lb. F)Ta ( F)Sw

i

6.2.204

801


11/16

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

Tinj ( F)Pinj ( Psi)prod ( Psi)

E.1 and E.2Sw.2287.30.40.50.60.70.90

SW.2287.3.4.5.6.7.9

so.2.4.5.6.7.8

E. 1

267.2525.313.5

Pco-w2.21.0.7.52.37.23.1

krwo.002.009.012.019.022.042

kros-o.0008.01.04.125.38.7

E.2 .E.3266.63 40024.9 2600 190

so.3.4.5.6,7.8

kroo-w1.0.922.8.58.26.06

0.0

krs.175.105.05.01.001.0

Pc s -o..3 8

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12/16

A THREE-FHASEEXPERIMENTALAND NUMERICALSIMULATIONSTUDY OF THE STEAMFLOODPROCESS SPE 3600

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13/16

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

Temperature Viscosity ( Cp )80 800

100 330140 1101$?n... ~~240 18280 11360 5.26

450 2.9

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

Pressure TemperatureF(Psi) FluidsTime Produced(cc)(min.) DistancefromInletIn- out- 011 Total 1.2 8.9 15.6 23.3 31!1let let 38.7

3:6090120150180210240..,sClu~%3603904204504s0510570600640680720760000850n9409701000103010601090112011s0~2~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.S12.912.813.21313.213.113.113.113.313.113.113.213.213.213.213.213.113.213.213.013.212.912.813.3131312.8~z.g12.912.812.812.312.213.013.013.0

06.613.420.327.935.842.95057.868.277.587.698.5109.2121134.8149.8171191.8212221.3226.5232.5237241.5247.6250.9254.9259.1263.6268.4272.5277.3282.3287.4295.8298,2301.8305.3308.9312.9317.4321324326.2

06.613.420.327.935.842.95057.8~~ ~77;587.698.5109.2121134.8149.8171191.8219.5260.5309.3358.5408454503564599.2636.7676.6715.5753.9790.6830.6874.8917.41008,11050,61095.41140.11186.71251.71310.21360.51406.41451.6

808496107116124132137138~~~147154161168174183192200205258279279279279279279279279279279279279279279279279279279279279279279279279279279

808080808080848588w92939698100101105108112117125144180223247275275275275275275275275275275275275275275275275275275275275275

808080008080@8080

80818283050687889098106115128143152162172182192199211228244266266266266266266266266266266

80808080808080f%

80808080:;%828384%9599102106109117119124129135149157164170175184198205210212

%00808080808080w80808000t%8080808080,z8080808282838486899092941::104106108110114123127130134

%00808080808080On%8080::8080::808080808080::80808080808080838485858686:;92939498


14/16

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

?mfi FluId* TemPeP.ture FTime P,oiueeai,.)(min.) DistancefrcaInletIn- out- 011 Total 1.2 8, 9 16.6 23.? 31 30.7let let

o40so1201542U02402203203604s04404903205606SCI640

24,825,324.024.724,724.524.324,624, e24.024,924.924,624.924,024,924,9

00000000000000000

0 020.4 20.441,2 41,265 6591.5 91.5122.7 122.7160.3 160.3lm.2 222207.5 305.5222.1 381,0237.3 448247.2 514,6262,6 394.e275.5 677.7226.4 76E.22J7 .3 849.63043,1 929.1

W608060Wll o 00 W 00 00133 23 00 00 60lmzzboeom176 96 W 00 80210104 s4m2a279 11629WW279 170 92 E4 20279 234 102 06 00279 275 172 90 20279 275 264 106 6~279 275 265 144 90279 275 26S 170 98279 2?5 265 254 120279 275 265 2S 186279 275 265 234 214279 275 265 254 236

N3WeoNo

240084606.380enw208 096102124

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

1.a* SO -

&q R..u= too -~ \: A tSipERlUEt4TAL qp ,s0- -o- CALCULATED \

.5-:g .4-

Z .~-~ .e-a-1 J -Zio

o .4 .6 l.z 1,6 Zo 6.4 MCUMULATIVE TOTAL FLUIDS PRODUCED-PORE VOLUMESFig. 5 - Experimental md calculated results of linear !kPeriMent 2,

AP =24.9 Psi.


15/16

A

A EXPERIMENTAL-o- CALCULATE

50 I I I I 1 1 I ~0 5 10 Is 20 25 30 35 L~141inchc$OISTANCE FROM INJECTION POINTS - INCHES

Fig.6 - Temperaturedistributionat theendof Experiment2.

1 2 3 N-2 N-1 NFig. 7 - Grid system used in the linear numerical simulator.

Read DataInitialize the Model

tPrtnt Dataand Initialized Conditions

I+Compute:

- Pore volume;- Inltfal oil andwater Iln place3 - Flow ratecoefficient;1

+~-Compute f rom tabl( e look-up:

1 - Re la ti ve p (e rmeabi ll ty2 - Capillary [pressure- Viscosities: - DensitiesCompute t ransm issibi l it iesI

t~1Establish F1OW Rate r

Compute enthlalples andsaturat ion temperaturesII Compute saturations I

t

I Compute temperature andralte of steam.condensationfrom the bleat balance-lc L 1Check Convergence

t

I Compute material andheat ba[lances It

I I nc rememt t imeand print resultslL#Updatedensities

L ~Fig. 8 - Nlurnerica .1 simulatorsflowchart.


16/16

r I I I I !

B

q **.*

q **OS i,.i+lq q *

i - ; , j i ~ j i+l ,jwFig. ; - Gr i d w.t,n used in thetho-airqensicmal numericals i r . u i a t o r .

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~ 2s$ 5 /! ,4 /

STEAM S.T. A

[1 .x Y> .-a:8

: /;-r

a .2d en- Celcu loted using tabulated5

-c- Calculated using Pc t.ab.lo?edltOH, @g i n s , . ! 01 wa t e r f l e d ( 11 )

.2 .4 .6 .8 1.0 1.2 1.4 1.6PoRE VOLUMES PROOUCEO

I0,000, ! 1 I 1 d

N iI JMozI: 100zov~>

10

I I I I I I I50 100 150 200 250 300 350 400~EMpzR~~,~:~ .~

Fig. 11 - Viscosity vs t e mp e r a t u r e

Fig. 10 - Experimental and calculated results ofthe t w o - d i me ns i o n al e x pe r i me nt .

0.6

~ 0.5S~o>~ 0.4agIq 0.3-g: 0.2JG

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

.RtMwL gil Waco sitv fat RoomTomverotwe, c K II

PORE VOLUMES PROOUCEO

m, J Q . rmn->,+.d Gil r e cQv er y c u r v es for different oil viscosities.. .=. . . .. ...=....-

Documents

A Three-Phase, Experimental and Numerical