Improved Steamflood Analytical ModelS. Chandra, SPE, and D.D. Mamora, SPE, Texas A&M University

Summary

The Jones (1981) steamflood model incorporates oil displacementby steam as described by Myhill and Stegemeier (1978), and athree-component capture factor based on empirical correlations.The main drawback of the model, however, is the unsatisfactoryprediction of the oil production peak: It is usually significantlylower than the observed value. Our study focuses on improvingthis aspect of the Jones model.

In our study, we simulated the production performance of afive-spot-steamflood-pattern unit and compared the results againstthose based on the Jones model (1981). To obtain a satisfactorymatch between simulation and Jones-analytical-model results, atthe start and height of the production peak, the following refine-ments to the Jones model were necessary. First, the dimensionlesssteam-zone size AcD was modified to account for the decrease inoil viscosity during steamflood and its dependence on the steaminjection rate. Second, the dimensionless volume of displaced oilproduced VoD was modified from its square-root format to anexponential form. The modified model gave very satisfactory re-sults for production performance for up to 20 years of simulatedsteamflood, compared to the original Jones model. Engineers willfind the modified model an improved and useful tool for the pre-diction of steamflood-production performance.

Introduction

Steamflooding is a major enhanced-oil recovery (EOR) processapplied to heavy oil reservoirs. A steamflood typically proceedsthrough four development phases: reservoir screening, pilot tests,fieldwide implementation, and reservoir management (Hong1994). Steamflood-performance prediction is essential to provideinformation for the proper execution of each development phase.Three mathematical models (statistical, numerical, and analyticalmodels) are often used to predict steamflood performance.

Statistical models are based on the historical data of steamfloodperformance from other reservoirs which have similar oil and rockproperties. A statistical model, however, does not include all theflow parameters, and thus may be inaccurate for a particular res-ervoir. Numerical models usually require a large amount of datainput with lengthy calculations using computers; and they are usu-ally CPU-, manpower- and time-consuming and also expensive.They may be extremely comprehensive and better serve as toolsfor research or advanced reservoir analysis. Meanwhile, analyticalmodels are more economical, but at the expense of accuracy andflexibility. They serve as tools for engineering screening of pos-sible reservoir candidates for field testing (Hong 1994).

For many years, attempts have been made to provide analyticalmodels for steamflood-production-performance prediction (Marxand Langenheim 1959; Boberg 1966; Mandl and Volek 1969;Neuman 1975; Myhill and Stegemeier 1978; Gomaa 1980; Jones1981; van Lookeren 1977; Farouq Ali 1970; Miller and Leung1985; Rhee et al. 1978; Aydelotte et al. 1982). None of theseanalytical models gives a comparison with simulation results.Miller and Leung (1985) presented comparison between their ana-lytical model and simulation results for cumulative production vs.time, but the comparison for production rate vs. time is not available.

Jones Model Review. In 1981, Jones presented a model based onwork published by van Lookeren (1977) and Myhill-Stegemeier(1978). In the Jones model (1981), the Myhill-Stagemeier (1978)steamflood-oil-displacement rate is converted to the oil productionrate on the basis of correlation parameters from 14 different steam-flood projects. The Jones model assumes that a steamflood has thefollowing three major stages of production. The first productionstage is dominated by initial oil viscosity, and possibly is affected byreservoir fillup if initial gas saturation exists. The second stage ofproduction is normally dominated by hot-oil mobility and reservoirpermeability. In the second stage, the production rate is essentially thedisplacement rate. The third phase of production is dominated by theremaining mobile fraction of original oil in place (OOIP) (Jones1981).

The oil displacement rate used in the Jones model is based onMyhill-Stegemeiers model (1978). The oil displacement calcula-tion is essentially the same as in the Myhill-Stegemeiers methodbut with some simplifications. Major simplifications are: (1) heatcapacity of base rock and caprock is 1.2 times the heat capacity ofreservoir rock, (2) the simplification of the equation used for over-all reservoir thermal efficiency (Ehs), and (3) use a correlation tocalculate dimensionless critical time (tcD). These simplificationslead to inaccuracies which are removed in the new model by notmaking these simplifications.

Jones Capture Efficiency (1981) converts Myhill-Stegemeiersoil displacement rate (qod) to actual oil production rate (1978).Jones Capture Efficiency, , consists of three elements, AcD, VoD,and VpD. The product of these three elements yields the CaptureEfficiency. That is, = AcD VoD VpD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

Formulas for AcD, VoD, and VpD are given in Eq. 2-4. These for-mulas were determined empirically by Jones using data from nu-merous steamflooded fields.

AcD = AsA0.11 lnoi 100122

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

with limits: 0AcD1.0 and AcD1.0 at oi100 cp.

VoD = 1 NdN SoiSo12

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)with the limit: 0VoD1.0.

VpD = 5.62 Vs,inj43560 AhnSg2

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)with limits: 0VpD1.0 and VpD1.0 at Sg0.Research Objectives and MethodologyThe main objective of this study is to improve the Jones analyticalmodel. The results of the modified model will be tested againstresults on the basis of numerical simulation to verify its accuracyand validity.

A series of simulation runs were conducted to simulate theproduction performance of a five-spot steamflood pattern unit. Thesimulation results were then compared against those based on theJones model. CMG STARS (Computer Modelling Group Ltd.,2005, Calgary) thermal simulator was used for this purpose. Threedifferent reservoir types and fluid properties were simulated using3D-Cartesian noncompositional oil models: Hamaca (9API), SanArdo (12API) and that based on the SPE fourth comparativesolution project (14API) (Aziz et al. 1987). In the first two fieldcases (Hamaca and San Ardo), a 45238 model was used, whichrepresented one-eighth of a 10-acre, five-spot pattern unit, usingtypical rock and reservoir fluid properties. In the SPE project case,three Cartesian models were used: 231212 (2.5 acre), 311612

Copyright 2007 Society of Petroleum Engineers

This paper (SPE 97870) was accepted for presentation at the 2005 SPE-PE/CIM-CHOAInternational Thermal Operations and Heavy Oil Symposium, Calgary, 13 November, andrevised for publication. Original manuscript received for review 13 July 2005. Revisedmanuscript received for review 2 April 2007. Paper peer approved 8 May 2007.

638 December 2007 SPE Reservoir Evaluation & Engineering

(5 acre), and 45238 (10 acre), representing one-eighth of afive-spot pattern unit.

To obtain a satisfactory match between the simulation- andJones-analytical model, results of the start and height of productionpeak, the following refinements to the Jones model were neces-sary. First, the dimensionless steam zone size AcD was modified toaccount for decrease in oil viscosity during steamflood and itsdependence on the steam injection rate. Second, the dimensionlessvolume of displaced oil produced VoD was modified from itssquare-root format to an exponential form.

New Model DevelopmentIn the Jones model (1981), the capture factor consists of threecomponents, AcD, VoD, and VpD. The first two components weremodified to obtain a satisfactory match of oil production rate basedon the new model and simulation. Component VpD in the newmodel is unchanged from that in the Jones model because we haveonly studied the case where initial gas saturation is zero. Modifi-cation of AcD and VoD is described below.

First, as in the Jones model (1981), oil production consists ofthree stages (Stages I, II, and III) as shown in Fig. 1. Stage I isrelated to cold oil production. As described by Jones (1981), thisstage is dominated by initial oil viscosity, and possibly is affectedby reservoir fillup if significant initial gas saturation exists. Duringreservoir fillup, free gas initially in the reservoir is displaced by theinjected steam. This process ends after all of the moveable gas isdisplaced from the reservoir. VpD is the capture-factor componentthat describes this reservoir-fillup phenomenon. This new modeluses the same VpD expression as given by the Jones model (1981):

VpD = 5.62 Vs,inj43560 AhnSg2

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)with limits: 0VpD1.0 and VpD1.0 at Sg0.

In this study, we found that the steam injection rate has asignificant effect on the oil production rate in Stage I, and thus onAcD. From Eq. 2, it can be seen that the constant in the denomi-nator, 011, appears to be the main parameter that would vary withthe steam injection rate. In the new model, this constant is replacedby as follows.

AcD = AsA lnoi 100122

, . . . . . . . . . . . . . . . . . . . . . . . . . (6)

with limits: 0AcD1.0 and AcD1.0 at oi100 cp.Correlation between and the steam injection rate was devel-

oped by making several simulation runs of the SPE comparativemodel, with each run having a different steam injection rate. Byprocess of trial and error, we determined that for each steaminjection rate is that which gave the best match of oil productionrate on the basis of the new model and simulation. A graph of vs.steam injection rate is shown in Fig. 2, indicating the followinglinear relationship: = 0.00015 is + 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

Stage II is related to the breakthrough of hot oil (if Sgi0) orthe oil bank (if Sgi>0). In any case, viscosity of the oil in the hot-oilregion is lower than the original oil viscosity. At the beginning ofa steamflood process, a hot-oil region is formed near the injector.The producer still produces cold oil near by, which is not affectedby the steam temperature. As the volume of steam injected in-creases, the hot oil region moves towards the producer. At thesame time, the oil viscosity in the reservoir continues to decrease,increasing oil production rate.

When the oil-bank region arrives at the producer, a largeamount of the hot oil breaks through, and the result is a sharpincrease in oil production. Oil production rate at this time will besignificantly higher than the Myhill-Stegemeier oil displacementrate (1978). This happens because the oil displaced previously,which is not produced but stored in the oil-bank region, is pro-duced at this time.

To account for the viscosity change because of heating, theviscosity value for AcD in Stage II is described as an averageviscosity between the hot-oil region and the cold-oil region asgiven in Eq. 8.

o =oiA As + oTsteam As

A . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)With this average viscosity value, the value of AcD will increasefaster to properly describe the oil bank breakthrough. The AcDexpression for Stage II of the new model is:

AcD =AsA

4

lno1002. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

with limits: 1.0

= 17.93 Nc + 1.3401. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)Nc is the ratio of the volume of moveable oil to that of steaminjection up to the critical time, tc:

Nc =7758 Ahn1 Sor Swc

365 istc, . . . . . . . . . . . . . . . . . . . . . . . (12)

where

tc =tcDht2M12

35040 khM2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)

Table 1 gives comparisons of AcD, VoD, and VpD between newmodel and Jones model.

Results and DiscussionThirteen different cases were conducted to verify the validity ofthe new model, as given in Table 2. Tables 3 through 5 give the

reservoir properties of the SPE comparative model, San Ardomodel, and Hamaca model.

Results on the basis of the improved model agree well withsimulation results for 13 different cases conducted. Figs. 4through 8 give oil-rate-production performances from simulation,the Jones model (1981) and the new model for three cases of SPEcomparative model, San Ardo model, and Hamaca model. Figs. 9through 13 give cumulative-oil/steam-ratio plots for three cases ofthe SPE comparative model, San Ardo model, and Hamaca model.Cumulative oil steam ratio (OSR) is defined as cumulative oilproduced and then divided by the cumulative steam injected.

The oil production rate and cumulative OSR based on the newmodel are in good agreement with simulation results.

In this study, the two components of capture factor, AcD andVoD, were developed using different simulation cases of SPE com-parative model. The new model using these new components wasthen tested for the San Ardo and Hamaca models. Production rateson the basis of the new model and simulation are in good agree-ment, verifying the validity of the new steamflood model.

Conclusions1. The proposed steamflood model improved history-match of

simulation results compared to the Jones model.2. Simulation results (and field data) indicate exponential decline

trends for oil rate in the third production stage. This is differentfrom the square-root trend assumed in the Jones model.

3. This model is developed by modifying two of the three com-ponents of the capture factor (AcD and VoD) in the Jones model.

4. Modifications to AcD and VoD were based on simulation resultsusing SPE comparative project case model (14 API). The factthat these modifications improved the history-matches for SanArdo (12 API) and Hamaca (9 API) simulation cases is veri-fication of the robustness of the improved model.

5. Results on the basis of the improved model agree well withsimulation results for thirteen different cases that include three

Fig. 3 vs. Nc relationship.

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different sets of reservoir and fluid properties: that of the SPEfourth comparative project, San Ardo field, and Hamaca field.

6. Engineers will find the new model even more useful because ofits improved accuracy in the prediction of steamflood produc-tion performance.

NomenclatureA effective pattern area, acres

AcD dimensionless steam zone size, dimensionlessAs steam zone size, acres

Ehs average thermal efficiency of steam zone, defined byEq. A-8 in Jones (1981)

hn net zone thickness, ftht gross formation thickness, ftis steam injection rate, cold water equivalent, B/D.

kh bulk thermal conductivity of cap rock and base rock,BTU/ft-hr-oF.

M1 average heat capacity of steam zone, BTU/cu ft-oFM2 average heat capacity of cap rock and base rock,

BTU/cu ft-oFN oil originally-in-place, STB

Nc ratio of the volume of moveable oil to that of steaminjection up to the critical time, tc

Nd cumulative oil displacement, RBqod oil displacement rate, RB/D, defined by Eq. A-22 in

Jones (1981)Sg gas saturation, fractionSo oil saturation, fraction

Soi initial oil saturation, fractionSor residual oil saturation, fractionSwc connate water saturation, fraction

tc critical time, yrtcD time of steam injection at onset of convective heat

transport through condensation front, dimensionlessTsteam steam temperature, F

VoD volume of displaced oil produced, fractionVpD initial pore void filled with steam as water, fraction

Vs,inj cumulative steam injection, RBSo change in oil saturation before/after steam front

passage, fraction correlation parameter defined by Eq. 7 correlation parameter defined by Eq. 11 capture efficiencyo average viscosity, cpoi initial oil viscosity, cp

oTsteam oil viscosity at steam temperature,cp porosity, fraction

Subscriptsmax maximum

ReferencesAydelotte, S.R. and Pope, G.A. 1983. A Simplified Predictive Model for

Steamdrive Performance. JPT 35 (5): 9911002. SPE-10748-PA. DOI:10.2118/10748-PA.

Aziz, K., Ramesh, A.B., and Woo, P.T. 1987. Fourth SPE ComparativeSolution Project: Comparison of Steam Injection Simulators. JPT 39(12): 15761584. SPE-13510-PA. DOI: 10.2118/13510-PA.

Boberg, T.C. 1966. Calculation of the Production Rate of a ThermallyStimulated Well. JPT 18 (12): 16131623. SPE-1578-PA. DOI:10.2118/1578-PA.

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Effinger, A.W. and Wasson, J.A. 1969. Applying Marx and LangenheimCalculations to the Prediction of Oil Recovery by Steamflooding inVenango Sands. USBM Information Circular 8432,US Dept. of theInterior, Washington,DC.

Ejiogu, G.C. and Fiori, M. 1987. High-Pressure Saturated-Steam Correla-tions. JPT 39 (12): 15851590. SPE-15405-PA. DOI: 10.2118/15405-PA.

Farouq Ali, S.M. 1970. Graphical Determination of Oil Recovery in aFive-Spot Steamflood. Paper SPE 2900 presented at the SPE RockyMountain Regional Meeting, Casper, Wyoming, 89 June. DOI:10.2118/2900-MS.

Gomaa, E.E. 1980. Correlations for Predicting Oil Recovery by Steam-flood. JPT 32 (2): 325332. SPE-6169-PA. DOI: 10.2118/6169-PA.

Hong, K.C. 1994. Steamflood Reservoir Management. Tulsa: PennWellBooks.

Jones, J. 1981. Steam Drive Model for Hand-Held Programmable Calcu-lators. JPT 33 (9): 15831598. SPE-8882-PA. DOI: 10.2118/8882-PA.

Mandl, G. and Volek, C.W. 1967. Heat and Mass Transport in Steam-DriveProcesses. Paper SPE 1896 presented at the Fall Meeting of the Societyof Petroleum Engineers of AIME, New Orleans, 14 October. DOI:10.2118/1896-MS.

Marx, J.W and Langenheim, R.H. 1959. Reservoir Heating by Hot FluidInjection, Trans, AIME 216: 312315.

Miller, M.A. and Leung, W.K. 1985. A Simple Gravity Override Model ofSteamdrive. Paper SPE 14241 presented at the SPE Annual TechnicalConference and Exhibition, Las Vegas, Nevada, 2225 September.DOI: 10.2118/14241-MS.

Myhill, N.A. and Stegemeier, G.L. 1978. Steam-Drive Correlation andPrediction. JPT 30 (2): 173182. SPE-5572-PA. DOI: 10.2118/5572-PA.

Neuman, C.H. 1974. A Mathematical Model of the Steam Drive ProcessApplications. Paper SPE 4757 presented at the SPE Improved OilRecovery Symposium, Tulsa, 2224 April. DOI: 10.2118/4757-MS.

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ery by Steamflooding Including the Effects of Distillation and GravityOverride. SPEJ 20 (4): 249-266. SPE 7547. DOI: 10.2118/7547-PA

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SI Metric Conversion Factorsacre 4.046 873 E+03 m2

API 141.5/(131.5+API) g/cm3Btu 1.055 056 E+00 kJ

ft 3.048* E01 mF (F32)/1.8 C

* Conversion factor is exact.

Suandy Chandra is a reservoir engineer with P.T. Caltex,basedin Duri,Indonesia. He holds a BS degree from Institute Tech-nology Bandung and an MS degree from Texas A&M U.,bothin petroleum engineering. Email: csua@chevron.com. DaulatD. Mamora is the Rob L. Adams professor of petroleumengineering at Texas A&M U.,College Station,Texas. Beforejoining academia,Mamora worked internationally for RoyalDutch/Shell for 15 years, where he became reservoir engineer-ing manager. He holds a BS Honors degree in applied physicsf r o m M a l a y a U . , a n d M S a n d P h D d e g r e e s f r o mStanford U. in petroleum engineering. Email: daulat.mamora@pe.tamu.edu.

Fig. 6Oil-production rate (SPE comparative model, area=10acres, injection rate=1,400 B/D).

Fig. 7Oil-production rate (San Ardo model, area=10 acres, in-jection rate=1,600 B/D).

Fig. 4Oil-production rate (SPE comparative model, area=2.5acres, injection rate=400 B/D). Fig. 5Oil-production rate (SPE comparative model, area=5.0acres, injection rate=1,000 B/D).

642 December 2007 SPE Reservoir Evaluation & Engineering

Fig. 8Oil-production rate (Hamaca model, area=10 acres, in-jection rate=1,600 B/D). Fig. 9Cumulative oil/steam ratio (SPE comparative model,

area=2.5 acres, injection rate=400 B/D).

Fig. 10Cumulative oil/steam ratio (SPE comparative model,area=5.0 acres, injection rate=1,000 B/D). Fig. 11Cumulative oil/steam ratio (SPE comparative model,area=10 acres, injection rate=1,400 B/D).

Fig. 12Cumulative oil/steam ratio (San Ardo model, area=10acres, injection rate=1,600 B/D).

Fig. 13Cumulative oil/steam ratio (Hamaca model, area=10acres, injection rate=1,600 B/D).

643December 2007 SPE Reservoir Evaluation & Engineering