SPESPE 200;3

Appraisal of Analytical Steamflood ModeIsH-L. Chen, Texas A&M U., and N,D. Sylvester, U. of Akron

SPE Members

CopW9htWSO,SOCletYof PetroleumErrgirreecaInc.

TIIlspaperweepreparedfor preaantationat the @OthCaliforniaRegionalMeetingheld irrVentura,California,April4-S, 1S90.

Thk paparwasaalectedforpresentationby errSPE Programcommitteefollowlnareviewof inf-tm ~taiti in O@abefrecrWbmiff-f by the authort$).~t~fe ofthe PWftaapreaanted,haverw+beenravkwedbythe societyof PelrofwmEmiri- nd Me wbl~ to OWTOC~Onby thea~~e). ~ mat~al. M Pfe*ntW, * ~t MI=@anYiwainonof theSoaietyofPetroleumEngineers,heoffkem,w mamb- p- Pf-tti at SPEmeetin%eWew@M toP@l~t~ rS~SWbyE~tofial@mIIIMOO’Sof*V*VW&~~m. PmMto~b-dmm~titi- *~-.lwftiNy N@~.~~*M~&n~~of whereand by whomthe paperk preeentad.WritePublicationsManager,$PE, P.O. $0x -, Rkh~*t ~ 7~. Telex,7S0SSSSPEDAL.

$teamflooding in heavy oil reservoirs is one of theprincipal thermal oil recovery methods. This paper evaluatesthe existing analytical steamflood models with respect to theirmechanisms and predk!ive capabilities and compares themwith field data. The three steamflood models selected were: afrontal advance model [Jones (1981)], a modified frontaladvance model [Farouq Ali (1982)], and a vertical gravityoverride model [Miller and Leung (1985)]. Each model wassomewhat modified to improve its ability for the prediction ofproduction rate and/or history match of typical fieldproduction data.

The Jones steamdrive model, with its empiricallydetermined scaling factors, was found to give a reasonablehistory match of oil production for the Kern River field.Fields with different characteristics will require anadjustment of these scaling factors artdlor field property datato achieve an acceptable history match. The modified FarouqAli steamdrive model gives a good history match without needfor empirical factors or adjustable parameters. It is thusrecommended for the prediction of steamdrive oil recoverywhen fisld production data are unavailable. The Miller-Leunggravity override steamflood model, which contains twoadjustable parameters, was found to posses the best W3railhistory matching capabilities and is recommended for thispurpose.

AND I ITFRATLW.BUUW

The injection of steam into heavy or pressure depleted oilreservoirs has been a successful enhanced oii recoveryprocess for more than three dsoedes. A principat applicationof the steam injection is steamflooding which is also termedsteam drive or steam displacement. In this process, steam iscontinuously injected into a number of injection wells, and thedispiaced fluids are produced from the production wells.Ideally, the injected steam forms a steam saturation zonearound the vkinity of the injection welL The temperature inthe steam zone is nearly equal to that of the injected steam.

References and figures at end of paper.

Moving away from the injection well, the steam temperaturedrops graduaily as the steam expands in response to thepressure drop and heat losses to base formations. At a certaindistance, the steam condenses and forms a hot-oil bank. In thesteam zone, oil is displaced by the steam. In the hot oil zoneseveral changes take place which result in oil recovery. Theyinclude heat losses the formation, thermat expansion of the oil,and reduction of oil viscosity. In addition, residual saturationmay decrease and changm in relative permeability may occurdue to the variations of temperature and saturation.

There are three major options available in literature forpredicting the reservoir response to steamflocding. Theseinclude: empirical correlations’ ‘2) , Simple analyticalmodels(l 13-7), and muiticomponent, multiphase numeri~alsimulators(8-11 ). Empirical correlations can be useful forcorrelating data within a field and for predicting performanceof new wells in that or similar fields, However, use of suchcorrelations for situations much different from the ones thatled to their development can result in large discrepancies forhist~. ~ matching. Numerical simulators yield rigoroussolutions to the material and energy balances, However, theirresults are sensitiva to the rock and fluid property input dataand other geological information, some of which may beunattainable. In addition, large computation time is requiredand numerical convergence, and stability problems suggestthat thermai simulators are not appropriate for short-cutdesign and/or preliminary evaluation for steamfloodingprojects. Thus, the incentive to develop simple analyticalmodels which account for the important mechanisms invotvedand for routine or approximate engineering prediction isobvious. The existing analytical steamflood models can bedivided into two categories:

1. Frurrta/advance models: The steam-drive mechanism is

2.

modeled as a horizontal frontal displacement [Figure1(a)]. The steam zone IS assumed to gmw horizontallyand the tendency of the steam to finger beyond the frontis suppressed by condensation.

Verlikal displacemet?t or gravity overz%le models: Theproblem of “gravity override of the steam due to its low

la

. .

2 APPRAISALOF ANALWICAL STEAMFLOODMODELS SPE 20023

density assumes that the principal direction of steamzone propagation is vertically downward [figure2(b)].

An early frontal advance model was that of Marx andLangenheim(l 2) who applied an energy balance of a radiallygrowing steam zone In which one-dimensional conduction heatlosses, uniform steam zone and reservoir temperature wereassumed. Willman el al.(13) presented a model similar tothat of Marx and Langenheim’s but included the Buckley-Leverett equation to estimate oil production from a hot waterzone ahead of the steam zone. Mandle and Volek(l ‘$) extendedthe concepts of Marx-Langenheim by including convective heattransfer from the steam zone into the region ahead of thecondensation front at times greater than a critical time. Themodel was modified by Myhill and Stegemeier(l 5) to calculatethe thermal efficiency after tha critical time to account for thedisparity observed in physical models versus theory.Jones(l) noted that the Myhill and Stegemeier model oftenoverestimates the oil production, especially in the early phaseof a project because of the assumption that the oil displaced bythe steam zone is immediately produced. Thus, there was nolag in oil production due to fill-up of any gas volume, or due tothe development of an oil bank. Jones(l) thus developed amodified predictive model including the results of vanLookeren(l 6) for taking into account the extent of steamoverride, and introduced three empirical factors to account forthe dominant mechanisms during the three stages ofproduction.

Neuman (2S17, and Rhee and Doscher(3) proposed thatthe principal direction of steam zone growth is verticallydownward In the horizontal reservoirs. Neuman’s(17) modelrequires the data of relative permeability to oil and water asfunctions of temperature. Also, oil production from thecondensate zone was determined semi-empirically. Aydelotteand Pope(4) used fractional flow theory and overall energyand material balances to account for changes in oil cut, gasproduction, etc.. Also volumetric sweep efficiency was takeninto account by using van Lookeren’s( 16) vertical sweepefficiency and an empirical correlation given by Farouq

All( 18) for areal sweep efficiency {EA). This model isrestricted to horizontal, homogeneous, isotropic, andincompressible reservoirs and only five spot sweepcorrections were included. Doscher and Ghassemi(f 9)proposed that !he steamflood process consists of the heated oildisplaced by a gas drive mecharrism. Their model showed aninsensitivity of oil recovery to formation thickness, especiallyduring the early stage of production. Their experimentalresults indicated that the oil/steam ratio increases with adecrease of oil viscosity.

Unlike previous models, Vogel(20) proposed that oilproduction was not driven by the growing steam zone, but viceversa, He pointed out the general weakness of predictivemodels based on simple energy balances of a growing steamzone. With a predominantly overriding steam zone, the heatbalance calculations require that the steam produced Inproduction be accounted for as well as the steam that migratedout of pattern, Vogel suggested that the total underground heatrequirement was equal to the heat in the steam chest plus theheat flow upward and downward from the steam chest. Heconcluded ;hat oil production must be determined from someway other than steam zone growth, Miller and Leung(6)utilized the concepts of VogeI(20) and Neuman( 17, todetermine tka oil production rate by conductive heating of theoil below the steam zone.

The purpose of this paper is to evaluate existing analyticalsteamflood models with respect to their mechanisms andpredictive features. Three typical steam flooding models werestudied and modified by Chen(21): Jones’(l) frontal advancedmodel, Farouq Ali’s(5) modified frontal advance model, andMiller and Leung’s(6) vertical gravity ovarride model.History-matching of field data were carried out for each modelto test its applicability.

● ✎

Table 1 summarizes the characteristics and theparameters for three steamflood models. Complete parametersensitivity analyses for each model are available inChen’s(21) dissertation. The major modifications for eachmodel are presented in the Appendix section.

Jones(l) applied van Lookeren’s(l 6, method for theoptimal steam injection rate for a given set of steam andreservoir parameters, and utilized the Myhill andStegemeier(l 5) method to predict oil production. In theMyhill and Stegemeier model, the average thermal efficiencyof the steam zone was calculated by the Marx andLangenheim(l 2) solution at early times while the Mandl andVolek(l 4, method was used to account for heat transferthrough the condensation front after the critical time. Jones’model contains a number of empirical factors (ACD, VODt VPD)

which were obtained through history matching for specificsets of field production data. Thus, the adjustment of field datamay be necessary (TR, ht,hn ,t.toI) to achieve reasonablehistory matching for some projects as shown in Jones’ Table 1.In the original Jones model, the steam injection pressure wascalculated assuming a geometric relationship betweenpressure and injection rate. The optimum steam injection rateis taken to be the steam injection rate which gives themaximum value for the vertical conformance factor (AR D)Unfortunately, steam Injectivity test data is often not availablein the field. Therefore, the computer program written toevaluate the Jones model was modified to allow input of steaminjection rate and pressure, This modification was necessaryto permit comparison of model predictions with actual fielddata.

Farouq Ati’s(5) model is a modified fontal advar ]d modelwhich considers the effect of steam gravity override using vanLookeren’s( 16) method. At any instant of time during theproduction, the model predicts both oil and water production-displacement rates, the steam zone volume-thickness, theheated zone average temperature and the water and oilsaturations. An advantage of the model is that It simulates thedominant mechanistic features by material and energybalances and does not employ empirical factors. However, thisproduces the modet’s primary disadvantage in that severalparameters such as Sorst, Sor, Sst and Swir are requiredwhich, unfortunately, are normally unknown and need to beassumed or defaulted by using acceptable values. Also, it hasbeen shown by Chen(21 ) that the water saturation duringproduction affects the relative permeabilities and theproduction rate, and the model predictions are very sensitiveto the accuracy of the Krw and Kro versus SW* which aredifficult to obtain through experiments. Even though theexperimental difficulties can be overcome, the data may notrepresent the actual relative permeability versus saturation

I----

3 H. L. Chen and N. D. Sylvester SPE 20023

relations due to the effect ot temperature and reservoirheterogeneity. ,. I.} relative permeabllities versus saturationsequations presented by Farouq Ali were based on the curvespresented by Gomma(22), These normalized curves wereobtained through history matching of the Kem River field datareported by Chu and Trimble (23), The prediction of theoriginal Farouq Ati model for the Kern River A field productiondata indicates it to be totally inadequate at long times (>1.5years). Several important modifications were able to take intoaccount heat losses [Figure 2(a)] and displacement mechanism[Figure 2(b)] to improve its deficiencies. These are discussedin Appendix (b).

Miller and Leung(6) developed a simple gravity overridemodel which assumed a complete vertical overlaying steamzone with a steam-condensate zone between the steam zone andthe oil zone below. ‘They used one-dimensional, unsteady stateheat conduction to calculate the temperature distributioninside the COIIdWISate and oil zones, and employed thaNeuman(l 7) method to determine condensate zone thickness asa function of fract;on of condensed steam that is produced fromthe reservoir (fcp: 0.7-O.95). They ciaimed that the modeioverrxedicts the oil twoduction rate for fieid cases with iargepatterns (> 10 acres)” because the steam override may not befully developed in those cases. Therefore, another trmPiricalfactor, the areal sweep efficiency (EA: 0.4-1,0) presented by

Aydelotte and Pope(4) was introduced for the field cases withIar9e pattern area. Chen(21 ) has shown that both values offcp and EA have substantial effects on the predicted oilproduction rate. in addition, the heat baiance whichdetermines the optimum steam injection rate was modified byChan(21) to take into account the fact that the steam injectionrate should be based orI cold water fed to a steam generator noton saturated steam.

The five of fieid projects listed in Table 2 were chosen forhistory matching. They represent smail [Kern-A(23), Kern-Canfield(24), and Kern-San Joaquin(24)], medium [Kern-Ten Pattern], and large pattern areas [Tia Juan].The field production history data for each fieid case wasadapted from the Enhanced 011Recovery Fiei6 Report (27). Atime ir?crement of 1.2 month was used for the prediction ofKern River A project, and 1.5 month for Kern-Canfield andKern-San Joaquin matches. The time increment us ‘d inmedals for Kern-Ten Pattern and Tia Juana was chosen ~~beone month because the production history data was reportedmonthly. it is noted that the Kern River A field data was theoniy used to test the performance prediction for the modifiedFarouq Alimodel because of the availability of reiati~epermeability versus saturation relations which are requiredby this model. The other four field production histories wereused to compare the predictive performance of the Jones andMiiler-Leung models.

Figure 3(a) shows, the performance prediction for theKern River A field using the modiflad Farouq Ali model. Alsoshown are the predictions obtained using the Myhill andWegemeier (15) model, the numerical simulation results ofChu and Trimbie(2~), and the actual field data. Figure 3(b)compares the calculated cumulative production versus timeresults to the field data. The agreement Is good with adifference after 5 years of only 5.5% for cumulative oilproduction. It is apparent in Figure 3(a) that the modifiedFarouq All model gives superior predictions to those ofMyhill-Stegemeler and Chu-Trimble.

Figure 4(a) shows that the Jones model predicts a loweroil production rate at the beginning and a highar productionrate for the longer times for Kern-Canfieid project. Figure4(b) shows that aithough the Jones modei underpredicts thecumulative oil production, the prediction improves as timeincreasp. As shown in Table 3, at the eyf of the 7.5 years, theJones modei overestimates the cumulative production by2.1 EYO. it is seen in Figure 4 that the prediction of theMiiier-Leung model is superior to the Jones model for thisfieN case.

The comparison between the Jones model and Kern-SanJoaqukt field is similar to the Kern-Canfield case. That 1s,theoil production rate is underestimated at short times andoverestimated at long time as shown in Figure 5(a)t while theprediction of the Miller-Leung model is just the reverse. TheMiller-Leung model with a iag time (z) of 61 days is capableof predicting the production up to about 1.75 years. Thecomputer run was terminated after two years bacause thethickness of the condensate and steam zones became iarger thatthe net thickness of the reservoir. Table 3 shows that theJones’ model overestimates the cumulative production by8.1770 at the end of the third y~ar, and the Miller-Leungmodel overestimates the cumulative production by 3.39 % atthe end of the second year.

Figure 6(a) shows that both the Jones and Miiler-Leungmodeis underestimate the oii production rate for Kern-TeilPattern field for the first two years. it also can be observedfrom Figure 8(a) that the Milier-Leung prediction issuperior to Jones modei during this time. Aithough thepredicted production rate of the Miller-Leung model decreasessharpiy after 5.5 years, the MiIler-Leung model gives a moreaccurate cumulative oil production up to about 5 years asshown in Figure 6(a) and Tabie 3.

Figure 7(a) shows that neither model does weli inpredicting the measured oii production rates for the largepattern case of the Tia Juana field aithough Figure 7(b) showsthat both models do reasonably well in predicting thecumulative oil production. It should be noted that the TiaJuana case is a poor candidate for triatory matching because theless productive wells were steam stimulated, there were alarge number of unrepaired welis in the pattern, and the twoproductive zones had oils of different viscosity. This mayexplain the observed decline of oil production rate.

The following conclusions can be drawn form theresuits of the steamflood model modification and evaluation:

1, The Jones modei with input of steam injectivity datacan be used to predict oil production for steamflooding projectswith properties similar to the Kern River field. For othercases, the empirical factors or input data may requireadjustment to achieve better history-matching.

2. The modified Faro~q Ali model is the most realisticsteamflood modei because it simulates both 011and water phasedominant mechanisms (such as the combination 6f frontaladvanoa and steam override) by matedal and energy baiances.In addition, this model gives reasonably good prediction andhistory-matching results without requiring any empiricalfactors or adjustable parameters. However, retatlvepermeablilty versus water saturation data is needed for fieldsotker than Kern River A to obtain reasonable history-matching.

101

4 APWWSAL OF A?’lALvrloAl STEAMFLOODMODELS SPE 20023

3. For history-matchingof field data, the modified Millerand Leung model is better than the Jones model. Carefuladjustment of the parameters fcp and EA yields accuratehistory-matching.

4. Use of the modified Farouq All model is recommendedfor predicting steamflood production when field productionhistory is not available. The Miller and Leung model isrecommended for trtstory matching of steamflood performance.

%4) = dimensionless steam zone size

API

c1q,&fc p

fsdhhfs

= specific gravity of oil at 60 ‘F, dimensionless

= specific heat of phase i, Btu/lbm-°F

= areal sweep efficiency

x vertical sweep efficiency

= tondensed steam produced, fraction

= cownhole steam quality, fraction

= enthalpy of saturated steam at steam temperature,

Btu/lbm

hn = net zone tl ~ickness,ff

hs = steam zone thickness, ft

ht - gross zone thickness, ft

ist = steam injection rate, cold water equivalent BWp O

Kh = thermal oonductfvity of cap rock and base rock,

Kro

Krw

Lvdh

b%N

N‘P

%

qoi

qw

Btu/ft-hr-”Fx relative permeability to oil, fraction

= relative permeability to water, fraction

= latent heat of steam, Btu/lb

= heat capacity of cap rock and base rock, Btu/ft3-°F

9 heat capacity of steam zone, Btu/ft3-°F

= oil originally in place, bbl

= cumulated oil displacement, bbl

= cumulative oil production, bbl

= oil productionrate, BOpD

= pre-steamoil productionrate, BOPD

= water production rate, BWPD

6 = heat Injection rate, Btu/hr

QI - heat bsses to cap rock and steam zone, Btu

G> = oil displacement rate, BOpD

Qw = water displacementrate, BWPD

so = oil saturation, fraction

%c = condensate zone oil saturation, fraction

Soi = initial oil saturation, fraction

Sor = residual oil saturation, fraction‘

Scrst = steamflood residual oil saturation, fraction

%s = steam zone oil saturation, fraction

Sq =steam saturation in the steam zone, fraction

SW =water saturation, fraction

s~” - (~-swir)!(i -Swir-Sorw). dimensionless

Sw/r = irreducible water saturation, fraction

t = time, hr

tc = critical time, hr

‘c D = dimensionless critical time

At - time increment, hr

tB T = steam breakthrough time, hr

T1,2 = temperature at conditions 1 and 2, “F

Ts = steam temperature, ‘F

TR = initial formation temperature, ‘F

v~ = bulk volume of the pattern, ft3

‘B’ = VB -“s(rr+l), fti

“oD = dimensionless displaced oil prtiucad

vpD = Initial pore void filled with steam as water,

dimensionless

Vs(t) = steam zone volume at time t, f@

VsBT = steam zone volume at breakthrough, ft3

a = reservoir thermal diffusivity, ft2/day

@ = porosity, dimensionless

z = constant (=3.14159)

P = density of phase i, lbm/ft3

‘s = lag time, days

v = viscosity , cp

Voi = oil viscosity at initkd reservoir condition, cp

(n) = at time step n, dimensionless

avg = average temperature condition

sdh = steam at downhole condition

o = oil phase

s = steam phase

R = rock phase

w = water phase

1. Jones, J.: “Steam Drive Model for Hand-Held~~~~~;able Calculators,” J. Pet. Tech. (Sept. 1981)

.,

2. Neurnan, C.H, “A Mathematical Mo ,el of Steam DriveProcess-Application; paper SPE 47.,7, presented at theCalifornia Regional Meeting of the SPE, Ventura,April 2-4, 1975.

3. Rhee, S.W., Doscher, T.M.: “A Method for Predicting 011Recovery hy Steamflooding Including the Effects ofDlstillatkm and Gravity Overrlde~ Sot. Pet. Eng. J (Aug.1980) 249-66.

mm

●

s H. L. Chen and N. D. Syfvester SPE 20023

—.

I4,

5.

6.

7.

8.

9.

10.

lf.

12.

13,

Aydelotte, S.R, and Pope, G.A.: “A Simplified PredictiveModel for Steamdrive Performance: J. Pet. Tech. (May1983) 991-1002.

Farouq All, S.M.: “Steam Injection Theories - A UnifiedApproacht paper SF2 10746, presented at CaliforniaRegional Meeting of the SPE, San Francisco, March 24-26, 1982.

Miller, M.A. and Leung, W.K.: “A Simple Gravity OverrideModel of Steamdrive paper SPE 14241, presented at the60th Annual Technkal Conference and Exhibition of theSociety of Petroleum Engineers held in Las Vagas, Sept.22-25, 1985.

Wingard, J.S. and Orr, F.M. Jr.: “An Analytical Solutionfor Steam/Oil/Water Displacement; paper SPE 19667,presented at the 64th Annual Technical Conference in SanAntonio, TX, Oct. 8-11, 1989.

Coats, K.H., George W.D., Chu, C. and Marcum, B.E.:“Three-Dimensional Simulation of Steamflooding: Sot.Pet. Eng. J. (Dec. 1974), 573-92.

Crookston, R.B., Culham, W.E., anfi Chen, W.H.: “ANumerical Simulation Model For Thermal Recoveryprocesses,- Sot. Pet. Eng. J. (1979) 19, 37s58.

Vinsome, P.K.W., and Westeweld, J.: *A Simple Methodfor Predicting Cap and Base Rock Heat Losses in ThermalReservoir Simulators,” J. Can, ~e?. Tech., 19, No. 3(1980) 87-90.

Barry, R.: “A General Thermal Model: paper SPE 11713,presented at the California Regional Meeting in Ventura,March 23-25, 1983.

Marx, J.W. and Langenheim, R.H.: “ Resewoir Heating byHot Fluid Injections Trans., AlME (1959) 216, 312-15.

Willman, B.T., Vallerory, V.V, Runberg, G.W. Cornelius.A.J., and Powers, L~W.: “Laboratory Studies of OilRecovery by Steam Injection; J. Pet. Tech. (July 1961j681-90.

14. Mandl, G. and Volek. C.W.: “Heat and Mass Transport InSteam-Drive Processes,” Sot. Pet. Eng. J. (March 1969)46, 59-79; Trans., AIME.

15. Myhlll, N.A. and Stegemeier, G.A.: “Steam-DriveCorrelation and Prediction,” J, Pet. Tech. (Feb.1978)173-182.

16. van Lookeren, J.: “Calculation Methods for Linear andRadial Steam Flow in Oil Resewolr; paper SPE 6788presented at the 52th Technical Conference andExhibition, Denver, Colo. Oct. 9-12, 1977.

17. Neurmn, C.H,: “A Gravity Override Model of Steamdrive,”J,. F ‘. Tech. (Jan. 1985) 163-6%

18. Farouq All, S.M.: ‘Graphical determination of 011Recoveryin a Five-Spot Steamflood paper SPE 2900, presented atthe Rocky Mountain Regional Meeting of SPE, Casper, WY.,June 8-9, 1970.

19. Doscher, T.M, and Gh&ssemi, F.: “The Influence of OilViscosity and Thickness on the Steam Drive; J. Pet. Tech.(Feb. 19S3) 291-98.

20, Vogel, J.V.: “Simplified Heat Calculations forSteamflood,” J. Pet, Tech (July 1984) 1127-35.

21. Chen, H.-L.: “Analytical Modeling of Thermal Oil Recovetyby Steam Simulation and Steamflooding,- Ph.D.Dissertation, The University of Tulsa, Tulsa, Oklahoma(1987),

22. Gomma, E.E,: “Correlation for Predicting Oil RecoveV bySteamflood; J. Pet. Tech. (Feb. 1980) 325-32.

23. Chu, C. and Trimble, A.E.: “Numerical Simulation of SteamDisplacement-Field Performance Applications,” J. Pet.Tech. (June 1975) 765-76.

24. Greaser- G.R. and Shore. R.A.: “Steamffocd Performance in

25.

26.

27.

28.

—----—., —... .—the Kern River Field, “ paper SPE 8834, presented at the1s! Joint SPE/DOE Symposium on Enhanced Oil Recovery,Tulsa, OK, April 20-23, 1980.

Oglesby, K.D., Belvins, T.R., Rogers.E.% and Johnson*W.M.: “Status of the Ten-Pattern Steamflood Kern RiverField, California J. Pet. Tech. (Oct.1982)2251-57.

de Harm, H.J. and van Lookeren: “Early Results of theFirst Large-Scale Steam Soak Project in the Tia JuanaField, West Venezuela” J. Pet Tech. (Jan. 1969) 101-10.

Enhanced 011 Recovery Field Report, 11, 2, Society ofPetroleum Engineers (1986).

Somerton, W.H., Keese, J.A., and Chu, S.L.: “ThermalBehavior of Unconsolidated Oil Sandst” Sm. Pet. Eng. J.(oct. 1974) 513-21.

29. Leung, W.K.: “A Simple Gravity Override PredictiveModel,” M.S. Thesis, The University of Texas, Austin(1986)

APpFNW

The changes made to the Jones model permit direct inputof steam injection rate and pressure, and dimensionlessvolume of displaced oil produced as:

NPSoiVO!3= [’-~(~oi.sor)l; (1)

where Np is used insiead of Nd in the original Jones’ paper(l)

[Eq(A-25)] since VODIs a function of the amount of displacedoil which equals the total amount of mobile oil less thecumulative oil production.

Several modifications have been made to the Farouq Alimodel to improve its predictive capability.

I Tim

The critical time calculation recommended by Mand19 andVolek(14) was used :

t.= [ -xqtcD4 Kh MA

(2)

I

● ✎

6 AWRAISAL OF Analytical STEAMFLOODfvU3DELS SPE 20023

where

et~D effc~ = 1, f~dhLvdh

(3)

CW(T~-TR)

with

Lvdh = 94 (705 - TJO’38 (4)

The thermal conductivity of cap rock as given by Somerton etai.(28)was used.

Kh = 1.04 + 1.3 @+ 0.2$KR(1 -S0) (s)

where KR = 2.75 Btu/ft-hr-”F at 120 “F.

The average heat capacity of the steam zone provided byJones(l )was used.

Ms = (1-Ip)pRCR -t $fsdh(l ‘Sor)%cs+

$S*rpoC* + $-(1 ‘Sor)PwCw (6)

where pR = 165 ibm/ft3, CR = 0.20 Btu/lbm-°F, co = 0,45

Btu/lbm-°F, and Cs = C w = 1.0 Btu/Ibm-°F.

The heat capacity of cap or base rock was taken to be

Mb= p#R (7)

The term etcD erfc~ in Eq (3) was obtained from theJones(l) expression

OtcD erf~ = 0.255K - 0.284K2+1 .421 K3

- 1.453K4 + 1.061 K5 (8)

where

.[ 11..12t~D = K

0.3276(9)

Thus, ~ is obtained as follows:

( i ) Calculate the value of etcD erf~ from Eq (3).( i i ) Solve Eq (8) for K value.(ii i) Determine tcD from Eq (9).

(iv ) Obtain ~ from from Eq (2).

The average temperature of the unswept formation wascalculated by subtracting the heat content of the steam zonefrom the total heat injected and dividing by two times the buikheat capacity of unswept formation:

Tavg = Qit-V~(t)(TS-TR)MS + TR (lo)2t&[VB-Vs(t)]

where TR was not included In the original Farouq All model.

Eq (1O) Is an approximation and Is used for Tavg~ Ts. ifTavg > Ts, then Tavg is set equal to Ts for all future times.

When Vs > VsBT, there is the option of either producingsteam over a given Interval, or shutting off the production.Farouq Ali(5) suggested a simplified treatment which givesVs(t) after breakthrough at any time t

Vs(t)= VsBT+[6i(t-tBT) -2 KhA(43560)EA(Ts-TR)

(~-~) Mm] / Iv& (Ts-TavJ (11)

Eq (11) is modified by perfoming an energy balance acoordingto Vogel(20). Assuming that the reservoir and !he adjacentformation have the same thermal properties, the heat lossesupward and downward from the steam chest are

QI = 4KhA(43560)(TS-TR)~~ (12)

An overall energy balance as shown in Figure 2(a) gives

Qin - Qout = Qaccmulation or

‘iQit-4KhA(43560) (Ts-TR) ~-

= V~MS(T~-TR) (13)

Solving Eq (13) for ~s yields

rhi~-4Kd4(43560) (Ts-Td” ~v.

M, (Ts-TR)(14)

Since

Qi tBT = VSBT & (Ts-TR) (15)

We can write

~ Qi (t-tBT) + VSBTMS(TS-TR) (16)

Substituting Eq (16) into Eq (14) gives

Vs (t) = vsEw+lhi(t-tf3d-4 KtrA(43560)

(Ts-TR) H/=] / Ms (Ts-TR) (17)

From a time step At [t(n) to t(n+l )], the steam volumefrom which the oil and water “are displaced due to expansionand displacement of fluids is

1

..-

-1

..

7 H. L. Chen and N. D. Syfvester SPE 20023

In Figure 2(b), the solid line indicates the extent ofdisplacement by steam, The displaced volume Is the volume ~between the dashed and solid lines. The material baiance forthe displacement element is given below.

The only modification made for the Miller-Leung modelis m the calculation of optimum steam injection rate whkh

The oil displacement rate, ~o, is given by:was originally presented by Leung (29) as:

QiQ.= Av~$(S~)-SOr$t) (19) ist= (27)

5.6146 PWLvdhAt

The water displacement rate, Qw, is given by: To account for the fact that the sleam injection rate should bebased on cold water fed to a steam generator, Eq (27) becomes

C)w = AVS@[St)-(l -Sst -S.rst)]is t- Qi

(28)= AVS$(S$)- 1+Sst -+Sor$t) (20) 5.6146 p~[hfs+fsdhLvdh-& (TR-32)]

Then, the overail material balance on oil-water zone between where the amount of heat injected, Q i is calculated byt(n) and t(n+f ) is as follows: Vogel(20) as:

For oil: Qj=4K~A(Ts-TR)@+ AhsMs(Ts-TR) (29 )

Qo - qoAt = [VB-VY’)]I$[S$+’)- S$”)] (21 )

Assume that VB - @“+’)= v;, then for wate~

Qw - qwAt = V:@[s$+’ )- s!?] .

= V:o[(l -s!’’+’)- Sg)-(1-swsg)]

= v@@lw’’+’)] (22)

From Eqs (21) and (22) we have

*= W&[sy+’w’q (23)

‘w Qw-v@o(n)--s$+l)l

From the fractional fiow eqution, we can write

fw=~=~ (24)qo+q~ 1+Kro~w

Krwpo

Let,

qo Kro~w c—=— . . (25)qw K~oVo

(n+l)Substituting Eq (25) into (23) and solving for So gives

Sy (1+C)+ Qo-CQw

Sy+l) = t$v; (26)1+C

-..111

Wb ~qp~g~

Me 1

Summary of Steamflooding Models

Jones (1981) Farouq Ali (1982) Miller and Leung (1985)

rype of the Modal FrontalAdvanoe Modified Frontal Advance Vertbal Advanoe Gravity Override

Cftaracteristios 1. Predkts ~,~, Ehs, and Fos. 1, Prediits ~,~,qw,So,~, and 1. Pradiote ~ and ist.

and TaW 2. Adjustment of fW SW, %s. ~, ad

2. Empirkal coefficientssuch as 2. Requires defaulted values for and EA values maybe nacaasaryfor

AcDtVOD, Vpo areused. Data Sorst,Sor, %irl ad %t. for reasonablehistory-rnafohiftg.

suchas TR,hn,~i mayneed 3. Km, Kmvs. &data needed 3. Tuning of field data for history=

to be adjusted to obtain good when a ffefd 0sss other than matching is not necessary.

history matching for some Kern Riier-A field is evaluated.

field cases. Tuning of hn may needed for

reasonable history-matching.

Comparison d Underpradots ~ at short times Was notevaluated for field cases Setter pradktbn than Jones’ model

Predictive Ability and over-shootsthe measured other than Kern River-A project. especially for large ~atterrt area fieftf

values at bnger time for large cases (see Tabte 3).

.,. pattern area oases.

[see Figures 6(a) and 7(a)]

Sensitive isto Soit ‘sdh ist~ ‘sdh$orst fcP EA, ~i, hi, S~, ~c

Parameters

lBbles Z

Data Used for History Matching

Field T~ TR kaI(TI ) WOI(T2) ~01 qoi f~dh API Soi ht hn

(:F) (“~) [cp(”F)j [GP(”F)] (CP) (BOPD) (ft) (ft)

Kern River A 380 95 1380(100) 47(200) 1380 25 0.7 15 0.5 75 9rJ

iChu.Trlmble(23)]

Kern-Canfield 300 100 1700(100) 10[230) f700 15 0.7 13.5 0.51 125 80

iQreaaar-Shoro(24)]

.

i

Kern.SanJoaquin 300 90 1000(100) 10(250) 1000 10 0.75 14.5 0.52 33 29

iGraaaar-Shoro(24)]

I Kern-1O Pattern 400 SO 2710(85) 4(350) 2710 230 0.7 14 0.50 97 97

iO@aabyet 4J2S)]

(acres) (BWPD) {tt2/D) (BTU/tt30F)

2.5

2.7

2.7

60.7

137

0.345 225

0.31 300

0.2s 300

0.33 6000

0.33 5s000

0.96

1.097

1.097

0.870

0.9s

35.0

3s.4

38.4

35.7

35,0TlaJuana 400 113 27S0(1 13) S{350) 2780 1S40 0.6 15 0.71 250 200

[da Haan6

m Lookardq

-112

II

—

Fmb$aNp

mKem.Canfield 132677

(7.5)

Kern.San 28928(2.0)

Joaquln 37507(3.0)

Kern-Ten 334346a

Pauorn (6.0)

TiaJuana 10414373

(5.5)

Comparisonof \hs History Match Roaulta forUlllmato Cumulative 011 Production

%% NP %

udU@@Qcs Lkl?lsl~124053 -6.50 136531 2.t5

(7.5) (7. s)

42912 1s.21 40571 8.17

(3.0) (3.0)

3131158 -9.35 3572606 6.84

(6.0) (6.0)

110800s4 6.39---- ----(5.5)

Np

136198(7.5)

30943(2.0)

3313459

(6.0)

10073073

(5.5)

%

2.65

3.39

-0.90

.3.28

1. ThenumberInaldatheparentfwalsirdkatestheuitimateoilpmdwtbnyearbythefielddataor predkfivemdal.2. % difference- [(Np,model. NpMd)1fJP.f~~) x 1~

Heat condidlon to cap rock

4

Stec.rn ZO= 011zone+

+

Heatcanductlon ta base rack

(a) Frontal AdVCWICedDisplacement

Heat ccmductkrn to caD rack

Steam zone

4Haofnowto

undeftyfngzone

Condenwte & 011zone

= 20023

----- —-- --‘1-T*EASEROCK

(a) Heai Losses

xV,an --

,/R,H

0“ 0:’W*71W,+’

INITIAL (1”)%.$:

SW*8W”

fkal (1”+’1so● swat$W”I-seidor.t

(b) Dlsplaoement Meohanism

Figure 2. Control Volumo for Energy end Material Eralencee(b) Vertical or Gravity Overrtde Dtsptacement (Modified Farouq Ali Model]

Note: ~ k the dkectlon of heat transfer

~ k the dkecflonof steam@owth

Ffguro1. lhe Mechonf$rnof St-m Displocemonf

1- 1I I I I I t !‘Otn

IS*VTk[IYCA’RM

(a) oil Production Ratevs. Time (At = 0.1 year)

(b) Cumulative Oil Production ve. Time (At.= 0.1 year)

Figure 3, History Match of Kern River - A Data

I .a

TIK IWRIr

(a) Oil Production Rate vs. Time (At -0.125 year)

dm I .m m.m . . . S.n .

Tna w

(b) Cumulative Oil Production vs. Time (At = 0.125 year)

B

FigUre 4. HietorY Matoh ot Kern - Cenf’e’d ‘at a

m

● *E 20023

d 1

i

‘1 —f:euklo!mr-w~● rmlmalw’s -

d,,4aKe”rlaa

I I.* 1.40 *mm .40a ●aw 1.40

TM! -

(a) 011Production Rate vs. Time (At -0.125 year)

(b) Cumulative 011 Production vs. Time (At -0.125 year)

Figure 5. History Matoh of Karn - San Joaquin Data

am

- .b

# -s

4rn ./

#,$

r“ : ●*Ira

●

:Mr4 . 9

~om .

I “/ —iia. sF:a.m W*O PA-

#● mllm+nM”s -,maa”mmm.

l.a 9.41 4.* s.o 4.U

Tin? m

(a) Oil Production Rate vs. Time (At -1 Month)

+

i d .a

Ii : —:Xsl,m Wm4-lo Mm

i! .~*-

i●4QEa’ -

*4n.m S.U 4.n s.n - .m

● .a 1.48

11= -

(b) Cumulative 011 Production vs. Time (At -1 Month)

Figure 6. History Match of Kern . Ten Pattern Data

sPE 20029 ● ‘ “

‘3 h=8”””8-S417-t

=~

8- -L88●88 — F!- TIA _

m87‘ /“ , Mw.utuws”s-.SOW , J@ES”-

@

TIME (YSAM)

(a) Oil Production Rate vs. Time (At = 1 Month)

L

_ FIEIG 71A JWJU

. “WSF1-mM’s -

8 , J-’ MOOSL

TX= -

(b) Cumulative Oil Production vs. Time (At = 1 Month)

Figure 7. History Match of Tia Juana Data

116