Copyright 2003, Society of Petroleum Engineers Inc.

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AbstractMathematical models to predict oil production accurately bygravity drainage have been few. To this end, an analyticalmodel was developed to determine the ultimate oil recoveryby free-fall gravity drainage. An empirical oil recovery modelwas proposed accordingly to match and predict oil production.The model was tested against experimental, numerical, andfield data of oil production by free-fall gravity drainage. Theresults demonstrated that the oil recovery model could worksatisfactorily in the oil-gas cases studied. Initial oil productionrate, entry capillary pressure, and average residual oilsaturation can be estimated using the oil recovery model. Anapproach was also developed to infer capillary pressure curvesfrom the oil production data by free-fall gravity drainage.

IntroductionThe gravity drainage mechanism is important in thedevelopment of many oil reservoirs with large dip angles.Experimentally it has been found that unexpectedly high oilrecoveries can be obtained by gravity drainage. For example,Dumor and Schols1 reported an extremely low residual oilsaturation of 5% in high permeability sandstone cores aftergravity drainage. Hagoort2 also found experimentally thatgravity drainage could be a very effective oil recovery processin water-wet reservoirs. On the other hand, there have alsobeen oil field production data that showed high oil recoveriesfor reservoirs under gravity drainage. Dykstra3 presented agood example of oil production by strictly free-fall gravitydrainage in the Lakeview Pool, Midway Sunset oilfield. Theoil recovery in the field after 40 years of production was about64%. In a field study, King and Stiles4 demonstrated a veryhigh displacement efficiency of 87% by gravity drainage inthe East Texas Hawkins reservoir. Evidence shows thatgravity drainage is one of the most effective mechanisms ofdeveloping an oil field. Although the gravity drainage

mechanism is important, characterizing and modeling theprocess are still a great challenge.

There have been many reports5-17 on the study of gravitydrainage. Several gravity drainage models have beendeveloped in literature. In these models, capillary pressure isusually either neglected or considered inappropriately.However capillary pressure plays an important role in manycases.

Almost all the gravity drainage models are complicated.Some of the models do not have analytical solutions and haveto be solved numerically. Schechter and Guo15 conducted areview on the papers in the field. There are four main modelsas summarized by Schechter and Guo15. These include theCardwell-Parsons-Dykstra (CPD) model3, 7, Nenniger-Storrow(NS) model9, Pavone-Bruzzi-Verre (PBV) model13, and Luanmodel14. After comparing to experimental data, Schechter andGuo15 concluded that the accuracy of these models to predictthe oil production by gravity drainage is poor. Schechter andGuo15 also developed a gravity drainage model which did notimprove the accuracy significantly.

Because the analytical models do not work well, anempirical model developed to characterize spontaneousimbibition was proposed to model the gravity drainageprocess. The model was suggested originally by Aronofsky etal.18 to match oil production in naturally-fractured reservoirsdeveloped by water flooding. Many applications have beenconducted since then. Schechter and Guo19 used a similarequation to fit the experimental data of spontaneous waterimbibition in oil-saturated rocks by substituting productiontime with the dimensionless time. Baker et al.20 inferred thefracture spacing by matching production data from theSpraberry Trend naturally-fractured reservoir using the modelwith dimensionless time. Li and Horne21 also applied theimbibition model proposed by Aronofsky et al.18 to evaluatewater injection in geothermal reservoirs.

To test the model, both experimental (at core scale) andnumerical simulation (at reservoir scale) data were used. Theresults demonstrated satisfactory consistency between themodel and the experimental data from gravity drainage as wellas the numerical simulation data. Using the model, capillarypressure curves may be inferred from the experimental data ofgas-oil gravity drainage.

We would like to clarify that our study and discussions inthis article are limited to free-fall gravity drainage of gas-oilsystems rather than forced gravity drainage.

SPE 84184

Prediction of Oil Production by Gravity DrainageKewen Li, SPE, and Roland N. Horne, SPE, Stanford University

2 SPE 84184

MathematicsSince free-fall gravity drainage is a gravity-dominated processand the only resistance is the capillary pressure force, the oilproduction depends significantly on the properties of theporous media, fluids, and their interactions. These includepermeability and relative permeability of the porous media,pore structure, matrix sizes, fluid viscosities, initial watersaturation, the wettability of the rock-fluid systems, and theinterfacial tension. It is difficult to include all these importantparameters in an analytical model. This may be why theexisting analytical models do not work well in characterizingand modeling the gravity drainage process, as describedpreviously.

The empirical model suggested by Aronofsky et al.18 wasused in this study to match the oil production by gravitydrainage. The model is expressed as follows:

te = 1 (1)

where is the recovery in terms of recoverable oil, is aconstant governing the rate of convergence, and t is theproduction time.

In Eq. 1, is the recoverable recovery. It is necessary toobtain the value of the residual oil saturation in order tocalculate . But the residual oil saturation may not beavailable in many cases. Aronofsky et al.18 considered that thevariation in residual oil saturation was sufficiently small to beignored. This may bring about great error in many cases. Tosolve the problem, residual oil saturation is included in Eq. 1explicitly and may be inferred from the match to oilproduction data. In this case, Eq. 1 is expressed as follows:

)1(1

1 twi

orwi eS

SSR

= (2)

where R is the oil recovery in the units of oil originally inplace (OOIP). Swi and orS are the initial water saturation andthe average residual oil saturation in the core sample or in thereservoir. and orS can be obtained simultaneously using aregression analysis technique with the experimental data fromgravity drainage.

The values of residual oil saturation in the core sample orin the reservoir are different at different depth and may not beequal to the residual oil saturation determined from thecapillary pressure curves or the relative permeability curves.The average residual oil saturation orS in a core samplepositioned vertically can be calculated theoretically based onthe equilibrium between gravity and capillary pressure forcesafter gravity drainage process is completed. This is discussedas follows.

The Brooks-Corey model22 is used frequently to representdrainage capillary pressure curves and is expressed as follows:

/1*)( = oec SpP (3)

where pe is the entry capillary pressure. *oS is the normalizedoil saturation. The normalized oil saturation is calculated asfollows:

orwi

oroo SS

SSS

=

1* (4)

where So is the oil saturation and Sor is the residual oilsaturation determined from the capillary pressure curve. Sor isless than orS . It was assumed that the initial gas saturation iszero in this study.

When the equilibrium between gravity and capillarypressure forces is reached after gravity drainage, gravity isequal to capillary pressure at any position of z. The ultimateoil produced by free-fall gravity drainage can be calculatedusing the following equation:

)1

11

1)(1(

ccorwippo zzSSVN

+

=

(5)

where Npo is the ultimate oil produced by free-fall gravitydrainage, Vp is the pore volume (=AL). A and L are the cross-section area and the length of the core sample, is theporosity, is the pore size distribution index. zc is expressedas follows:

LzL

z ec

= (6)

where ze is the depth corresponding to the entry capillarypressure, pe.

The mathematical derivation of Eq. 5 is presented inAppendix A.

The average residual oil saturation orS in a core samplecan then be calculated:

)1

11

)(1(

ccorwioror zzSSSS

+= (7)

It is often assumed2-3 that the oil saturation at z=0 is equal toSor, which may not be true in many cases. This assumptionwas removed to derive Eq. 7 (see Appendix A). Instead, theoil saturation at z=0 was calculated according to the Brooks-Corey model22 (see Eq. 3).

According to Eq. 7, orS approaches to Sor when zcapproaches to zero (this implies that capillary pressureapproaches to zero), which is reasonable.

The ultimate oil recovery in the units of OOIP can becalculated easily based on Eq. 7 (see Appendix A).

SPE 84184 3

Eq. 5 can be reduced in the case of approaching toinfinity as follows:

)1)(1( corwippo zSSVN = (8)

Similarly the calculation of the average residual oil saturation(Eq. 7) can also be reduced:

corwioror zSSSS )1( += (9)

Eq. 2 can be arranged as:

)1)(1( torwippo eSSVN = (10)

Therefore the oil production rate, qo (=dNpo/dt), can becalculated as follows:

toio eqq

= (11a)

here qoi is the initial oil production rate at t=0 and is expressedas follows:

)1( orwipoi SSVq = (11b)

The values of average residual oil saturation orS and can beobtained by a history match technique once the oil productiondata are available. The initial oil production rate, qoi, can thenbe calculated according to Eq. 11b. The entry capillarypressure, pe, can be inferred from the initial oil production rateaccording to the following equation16:

)1(*

gLpgAkkq e

o

rooi

= (12)

where k is the rock permeability and *rok is the oil phaserelative permeability at Swi. is the density differencebetween oil and gas phases, o is the viscosity of the oil phase.Note that it is assumed in Eq. 12 that the oil productionposition is located at the bottom of the core sample.

The value of zc can be calculated once the value of pe isavailable based on Eq. 6. Then the value of the pore sizedistribution index, , can be inferred according to Eq. 7 if thevalues of Swi and Sor are known or can be determined fromother measurements. These include relative permeabilityexperiments and well logging. According to the Brooks-Coreymodel22 (see Eq. 3), capillary pressure curves can be inferredonce the values of and pe are obtained from the oilproduction data.

ResultsThe oil recovery model (Eq. 2) was tested against differenttype of production data by free-fall gravity drainage. The dataused in this study include experimental, numerical, and fielddata from different sources. The results are discussed in thissection.

Pedrera et al.17 conducted gravity drainage experiments inthe gas-oil-water-rock systems with different wettability. The1m long core sample used by Pedrera et al.17 was positionedvertically and had a permeability of 7000 md and a porosity of41%. The water phase was immobile. The case used in thisstudy was the strongly water-wet system with a wettabilityindex of 1.0 and an initial water saturation of 21%. Fig. 1shows the experimental data of oil recovery, in the units ofOOIP, by gravity drainage. Also shown in Fig. 1 are the modelresults (solid line) of the oil recovery calculated using Eq. 2.One can see the excellent consistency between theexperimental data and the model data. Fig. 1 demonstrates thatthe empirical model (see Eq. 2) can match the experimentaldata of oil recovery by free-fall gravity drainage suitably.

The values of orS and obtained from the match to theexperimental data of oil recovery were 0.393 and 0.000811(minute-1). With these values, the initial oil production rate att=0 was calculated using Eq. 11b and was about 0.259ml/minute. The entry capillary pressure, pe, inferred from thevalue of the initial oil production rate using Eq. 12 was about0.022 atm, very close to the experimental value of 0.026 atmmeasured by Pedrera et al.17. The computation shows that theentry capillary pressure, as an important parameter to estimatecapillary pressure curves, may be inferred accurately from theoil production data by gravity drainage using the models andapproaches developed in this work.

Note that usually the initial oil production rate in gravitydrainage tests was estimated by observing the plot of the oilproduction rate versus time visually. There is an abrupt changein oil production rate when the gas-oil surface touches the topof the core sample because of the effect of the entry capillarypressure (see Eq. 12). This approach to estimate initial oilproduction rate is not convenient and not accurate.

The experimental data of capillary pressure measured byPedrera et al.17 were shown in Fig. 2. The value of Sor wasabout 0.108 and the value of obtained from the model matchusing Eq. 3 was about 6.23. The value of orS calculated usingEq. 7 was about 0.319. Note that the value of orS from thematch to the experimental data of oil recovery using Eq. 2 was0.393.

Fig. 3 shows the comparison of the model results to theexperimental data of the recoverable oil recovery calculatedusing the value of Sor instead of orS . One can see that themodel (Eq. 1) could not match the experimental data. This isbecause the actual residual oil saturation orS by free-fallgravity drainage is greater than the residual oil saturationdetermined from capillary pressure curve (Sor).

Li and Firoozabadi23 conducted oil-gas gravity drainagetests in a Berea sandstone core at different wettability. Tofurther test the oil recovery model (Eq. 2), the experimental

4 SPE 84184

data reported by Li and Firoozabadi23 were used and aredepicted in Fig. 4. The Berea core sample used by Li andFiroozabadi23 was positioned vertically and had a porosity of21.3% and a permeability of 975 md. The wettability of thegas-oil-rock system was altered from strong oil-wetness topreferential neutral gas-wetness by chemical treatment. Theoil recovery by gravity drainage after the chemical treatmentwas greater than that before the chemical treatment because ofthe wettability alteration from strong oil-wetness topreferential neutral gas-wetness (see Fig. 4).

The model (Eq. 2) was used to match the experimentaldata of oil recovery by Li and Firoozabadi23 and the results areshown in Fig. 4. The solid lines represent the model results.Fig. 4 shows that the model matches the experimental data ofoil recovery remarkably well in the rock both with and withoutchemical treatment. In the case without chemical treatment,the values of orS and obtained from the model match were0.867 and 0.02053 (minute-1). The high value of orS might bebecause of the short length (18.9 cm) of the core sample. Thepore volume of the core sample used by Li and Firoozabadi23was 22.03 ml. The initial oil production rate calculated usingEq. 11b was 0.06 ml/minute.

In the case with chemical treatment, the values of orS and obtained from the model match were 0.532 and 0.022934(minute-1). The initial oil production rate calculated using Eq.11b was 0.236 ml/minute.

The results demonstrated that orS decreased and the initialoil production rate increased significantly after the wettabilityof the gas-oil-rock system was altered from strong oil-wetnessto preferential neutral gas-wetness. Note that orS may not beestimated accurately if the gravity drainage time is not longenough.

The previous description shows that the oil recovery modelprovides a way to estimate the effect of wettability or otherparameters on oil production quantitatively. More discussionswill be made later.

Using numerical simulations, Li and Horne24 studied theeffect of pore size distribution index on oil production for thesame core used by Pedrera et al.17 The results are depicted inFig. 5. Also shown in Fig. 5 are the results obtained from thematch using the oil recovery model (Eq. 2) for different valuesof pore size distribution index. One can see that the modelcould match all the oil recovery data satisfactorily.

The values of the average residual oil saturationdetermined from the model match to the oil recovery data fordifferent pore size distribution index are shown in Fig. 6. Theresults demonstrated that the average residual oil saturationincreases with the decrease in pore size distribution index,which is reasonable. Pore size distribution index is arepresentation of rock heterogeneity. The greater the pore sizedistribution index, the more homogeneous the rock. Thereforethe oil recovery by gravity drainage may increase with thepore size distribution index.

Fig. 7 shows the oil recovery data for different values ofentry capillary pressure reported by Li and Horne24 usingnumerical simulations. The value of the pore size distribution

index was equal to 7. The model fits to the oil recovery aregood for all the values of pore size distribution index.

The effect of entry capillary pressure on the averageresidual oil saturation determined from the model match isshown in Fig. 8. The average residual oil saturation increaseswith the entry capillary pressure as expected. Fig. 8demonstrates that the relationship between the averageresidual oil saturation and the entry capillary pressure isalmost linear for a pore size distribution index of 7.

All the data of oil recovery discussed previously wereobtained at core scale. The validity of the oil recovery model(Eq. 2) to match oil production at reservoir scale will bediscussed in the next section. We will first discuss thenumerical simulation results and then discuss the oilproduction data from a real reservoir.

Fig. 9 shows the oil recovery for different values of entrycapillary pressure at reservoir scale reported by Li and Horne24using numerical simulations. The reservoir was created basedon the parameters of the core reported by Pedrera et al.17. Thereservoir had a porosity of 41% and a permeability of 70 md.The reservoir height was 20 m and the radius was 100 m. Theinitial water saturation was 21%. The oil recovery model (Eq.2) was used to match the oil production from the reservoir forthree different values of entry capillary pressure and theresults are also shown in Fig. 9. It can be seen that the oilrecovery model (Eq. 2) can match the oil production atreservoir scale adequately.

Fig. 10 presents the match of the oil recovery model (Eq.2) to the oil production from the Lakeview Pool, MidwaySunset oilfield reported by Dykstra3. The oil was produced bystrictly free-fall gravity drainage. One can see from Fig. 10that the model can match the oil production from the realreservoir very well. The values of orS and obtained from themodel match were 0.285 and 0.11463 (year-1) for the reservoir.Note that the value of orS is greater than the estimated valueof Sor (0.10) from Dykstra3.

The results discussed in this paper showed that the oilrecovery model (Eq. 2) works satisfactorily for all theexamples presented at both core scale and field scale.

DiscussionsThe results presented previously demonstrate that the oilrecovery model (Eq. 2) can be used in both spontaneousimbibition and free-fall gravity drainage. This may bereasonable because the only two forces involved in the twocases are the same: gravity and capillary pressure. Thedifference is that gravity force is a positive force in free-fallgravity drainage but a negative force in spontaneousimbibition while capillary pressure is a negative force in free-fall gravity drainage but a positive force in spontaneousimbibition.

Although similarity exists between spontaneous imbibitionand free-fall gravity drainage, there are many differences inthe development of analytical solutions to oil recovery.Several analytical oil recovery models25-28 function accuratelyto predict the recovery by spontaneous imbibition in differentcases. However few analytical gravity drainage models worksatisfactorily.

SPE 84184 5

The empirical model expressed in Eq. 2 can match the oilproduction remarkably well in the cases studied. But it wouldbe helpful to find an accurate analytical gravity drainagemodel. More research effort is required in the area.

ConclusionsBased on the present study, the following conclusions may bedrawn:1. A modified model was proposed to match and predict the

oil production by free-fall gravity drainage. The initial oilproduction rate and the average residual oil saturation canbe estimated using the model.

2. The model can match the experimental and numericalsimulation data of oil recovery as well as the oilproduction data from the Lakeview Pool, Midway Sunsetfield.

3. An analytical model was developed to determine theaverage residual oil saturation by free-fall gravitydrainage.

4. The average residual oil saturation increases with theentry capillary pressure but decreases with the increase inpore size distribution index as expected. The relationshipbetween the average residual oil saturation and the entrycapillary pressure is almost linear for a pore sizedistribution index of 7.

5. An approach was developed to infer capillary pressurecurves from the oil production data by free-fall gravitydrainage. The entry capillary pressure can be inferredfrom the initial oil production rate and the pore sizedistribution index can be determined from the averageresidual oil saturation.

AcknowledgementsThis research was conducted with financial support from theUS Department of Energy under grant DE-FG07-02ID14418,the contribution of which is gratefully acknowledged.

NomenclatureA = cross-section area of the core or reservoir, L2g = gravity constant, L/t2k = absolute permeability, L2

*

rok = relative permeability of oil phase at initial oilsaturation

L = core length, LNpo = ultimate oil produced by free-fall gravity drainage,

L3

cP = capillary pressure, m/Lt2

pe = entry capillary pressure, m/Lt2qo = oil production rate, L3/tqoi = initial oil production rate, L3/tR = oil recovery in the units of OOIP

R = ultimate oil recovery in the units of OOIP So = oil saturation

*oS = normalized oil saturation

Sor = residual oil saturation determined from a capillarypressure curve

orS = average residual oil saturation

Swi = initial water saturationt = production time, t

Vp = pore volume, L3z = depth, L

zc = dimensionless length defined in Eq. 6ze = depth corresponding to the entry capillary pressure,

Lo = viscosity of oil phase, m/Lt = porosity = recoverable oil recovery = constant giving the rate of convergence in Eq. 1 = pore size distribution index

= density difference between oil and gas phases, m/L3

References1. Dumor, J.M. and Schols, R.S.: Drainage Capillary Pressure

Function and the Influence of Connate Water, SPEJ (October1974), 437.

2. Hagoort, J.: Oil Recovery by Gravity Drainage, SPEJ (June1980), 139-150.

3. Dykstra, H.: The Prediction of Oil Recovery by GravityDrainage, JPT (May 1978), 818-830.

4. King, R.L. and Stiles, J.H.: A Reservoir Study of the HawkinsWoodbine Field, SPE 2972, presented at the SPE 45th AnnualFall Meeting, Houston, Texas, October 4-7, 1970.

5. Leverett, M.C.: Capillary Behavior in Porous Solids, Trans.,AIME (1941), 142, 152.

6. Katz, D.L.: Possibilities of Secondary Recovery for theOklahoma City Wilcox Sand, Trans., AIME (1942), 146, 28.

7. Cardwell, W.T. and Parsons, R.L.: Gravity Drainage Theory,Trans., AIME (1949), 179, 199.

8. Terwilliger, P.L., Wilsey, L.E., Hall, H.N., Bridges, P.M., andMorse, R.A.: An Experimental and Theoretical Investigation ofGravity Drainage Performance, Trans., AIME (1951), 192,285-296.

9. Nenniger, E. and Storrow, J A.: Drainage of Packed Beads inGravitational and Centrifugal-force Fields, AIChE (1958), 4(3),305.

10. Matthews, C.S. and Lefkovits, H.C.: Gravity DrainagePerformance of Depletion-Type Reservoirs in the StripperStage, Trans., AIME, 207 (1956), 265-274.

11. Lefkovits, H.C. and Matthews, C.S.: Application of DeclineCurves to Gravity-Drainage Reservoirs in the Stripper Stage,Trans., AIME, 213 (1958), 275-280.

12. Hamon, G.: Oil/Water Gravity Drainage in Oil-Wet FracturedReservoirs, paper SPE 18366, presented at the SPE EuropeanPetroleum Conference, held in London, UK, October 16-19,1988.

13. Pavone, D., Bruzzi, P. and Verre, R.: Gravity Drainage at LowInterfacial Tension, paper presented at the 5th EuropeanSymposium on Enhanced Oil Recovery, held in Budapest,October 27-29, 1989, 165.

14. Luan, Z.: Some Theoretical Aspects of Gravity Drainage inNaturally Fractured Reservoirs, paper SPE 28641, presented atthe 69th Annual Technical Conference and Exhibition of SPE,held in New Orleans, Louisiana, September 25-28, 1994.

15. Schechter, D.S. and Guo, B.: Mathematical Modeling ofGravity Drainage after Gas Injection into Fractured Reservoirs,SPE/DOE 35170, presented at the SPE/DOE Improved OilRecovery Symposium held in Tulsa, Oklahoma, April 2224,1996.

16. Corra, A.C.F. and Firoozabadi, A.: Concept of GravityDrainage in Layered Porous Media, SPEJ (March 1996), 101-111.

6 SPE 84184

17. Pedrera, B., Betin, H., Hamon, G., and Augustin, A.:"Wettability Effect on Oil Relative Permeability during aGravity Drainage," SPE 77542, presented at the SPE AnnualTechnical Conference and Exhibition, San Antonio, TX, USA,September 29 to October 02, 2002.

18. Aronofsky, J.S., Masse, L., and Natanson, S.G.: A Model forthe Mechanism of Oil Recovery from the Porous Matrix Due toWater Invasion in Fractured Reservoirs, Trans., AIME (1958)213, 17-19.

19. Schechter, D.S. and Guo, B.: An Integrated Investigation forDesign of a CO2 Pilot in the Naturally Fractured SpraberryTrend Area, West Texas, paper SPE 39881, presented at the1998 SPE International Petroleum Conference and Exhibitionheld in Villahermosa, Mexico, March 3-5, 1998.

20. Baker, R.O., Spenceley, N.K., Guo, B., and Schechter, D.S.:Using an Analytical Decline Model to Characterize NaturallyFractured Reservoirs, SPE 39623, presented at the 1998SPE/DOE Improved Oil Recovery Symposium, Tulsa,Oklahoma, April 19-22, 1998.

21. Li, K. and Horne, R.N.: An Experimental Method forEvaluating Water Injection into Geothermal Reservoirs,presented at the GRC 2000 Annual Meeting, September 24-27,2000, San Francisco, USA; GRC Trans. 24 (2000).

22. Brooks, R. H. and Corey, A. T.: "Properties of Porous MediaAffecting Fluid Flow," J. Irrig. Drain. Div., (1966), 6, 61.

23. Li, K. and Firoozabadi, A.: Experimental Study of WettabilityAlteration to Preferential Gas-Wetness in Porous Media and itsEffect, SPEREE (April 2000), 139-149.

24. Li, K. and Horne, R.N.: Numerical Simulation withoutSpecifying Relative Permeability Functions, SPE 79716,Proceedings of the 2003 SPE Reservoir Simulation Symposiumheld in Houston, Texas, U.S.A., 35 February 2003.

25. Li, K. and Horne, R.N.: Characterization of Spontaneous WaterImbibition into Gas-Saturated Rocks, SPEJ (December 2001),6(4), 375-384.

26. Li, K. and Horne, R.N.: A General Scaling Method forSpontaneous Imbibition, SPE 77544, presented at the 2002 SPEAnnual Technical Conference and Exhibition, San Antonio, TX,USA, September 29 to October 02, 2002.

27. Handy, L.L.: Determination of Effective Capillary Pressuresfor Porous Media from Imbibition Data, Trans., AIME, 219,1960, 75-80.

28. Rangel-Germn, E.R.: Water Infiltration in Fractured PorousMedia: In-Situ Imaging, Analytical Model, And NumericalStudy, PhD thesis, Stanford University, Stanford, CA, USA,2002.

Appendix A: Derivation of Ultimate Oil Recovery byFree-fall Gravity DrainageAn analytical ultimate oil recovery model for free-fall gravitydrainage is derived in this section. It is not assumed in thederivation that the oil saturation at the top of the core sample,So(z=0), is equal to Sor. The main mechanism is the balancebetween the gravity and the capillary pressure forces.

The gravity force is equal to the capillary pressure force atany position in the core after the free-fall gravity drainage iscompleted. Assuming that the capillary pressure curve couldbe represented using the Brooks-Corey model (Eq. 3), thefollowing equation applies in this case for a cylinder-shapecore sample positioned vertically:

1

* ))((

= oe SzLzL (A-1)where z is the distance from the top of the core sample and zeis the position corresponding to the entry capillary pressure.

The ultimate cumulative oil production after theequilibrium between the gravity and capillary pressure forcesis reached can be calculated as follows:

= ==

ezzo

SzoS opo

zdSAN ,0,

(A-2)The following equation can be obtained based on Eq. 4:

*)1( oorwio dSSSdS = (A-3)Substituting Eq. A-3 into Eq. A-2:

= ==

*,

* 0,

*)1( ezzoSzoS

oorwipo zdSSSAN (A-4)

According to Eq. A-1, z can be expressed as:

1

*))((

= oe SzLLz (A-5)and

)(* 0, LzLS ezo

==

(A-6a)

1*,

== ezzo

S (A-6b)Eq. 5 can be obtained by substituting Eq. A-5 into Eq. A-4

and rearranging with Eqs. A-6a and A-6b.The average residual oil saturation after gravity drainage

can be computed as follows:

p

powipor V

NSVS

=

)1( (A-7)

Eq. 7 can be obtained by substituting Eq. 5 into Eq. A-7and rearranging.

The ultimate oil recovery in the units of OOIP is expressedas follows:

)1( wippo

SVN

R

=

(A-8)

where R is the ultimate oil recovery.Substituting Eq. 5 into Eq. A-8, the ultimate oil recovery

by free-fall gravity drainage can be calculated as follows:

)1

11

1(1

1

cc

wi

orwi zzS

SSR

+

=

(A-9)

According to Eq. A-9, the ultimate oil recovery by free-fallgravity drainage depends on the residual oil saturation (Sor),the pore size distribution index, and the entry capillarypressure.

SPE 84184 7

0.0

0.2

0.4

0.6

0.8

1.0

0 5000 10000 15000 20000 25000Time, minute

Oil

Reco

ver

y, O

OIP Experimental

Model

Fig. 1: Comparison of the oil recovery calculated by the modifiedmodel to experimental data17.

0.00

0.02

0.04

0.06

0.08

0.10

0 20 40 60 80 100Oil Saturation (%)

Capi

llary

Pre

ssu

re (a

t) Brooks-Corey modelExperimental

Fig. 2: Experimental data of capillary pressure and the fitting bythe Brooks-Corey model24.

0.0

0.2

0.4

0.6

0.8

1.0

0 5000 10000 15000 20000 25000Time, minute

Rec

ov

erab

le O

il Re

cover

y

ExperimentalModel

Fig. 3: Comparison of the oil recovery calculated by the existingmodel to experimental data.

0.0

0.2

0.4

0.6

0.8

1.0

0 200 400 600 800Time, minute

Oil

Rec

over

y, O

OIP

With Chemical, ExperimentalWith Chemical, Model Without Chemical, ExperimentalWithout Chemical, Model

Fig. 4: Comparison of the calculated oil recovery at differentwettability to experimental data23.

0.0

0.2

0.4

0.6

0.8

1.0

0 5000 10000 15000 20000Time, minute

Oil

Rec

over

y, O

OIP

=1, Simulation =2, Simulation =3, Simulation =7, Simulation Model

Fig. 5: Model fit to the numerical simulation results of oil recoveryin core samples at different values of 24.

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8Pore Size Distribution Index

Res

idua

l Oil

Satu

ratio

n, fra

ctio

n

Fig. 6: Effect of the pore size distribution index on the averageresidual oil saturation in core samples.

8 SPE 84184

0.0

0.2

0.4

0.6

0.8

1.0

0 5000 10000 15000 20000Time, minute

Oil

Rec

ov

ery,

O

OIP

pe = 0.1*pempe = 0.5*pem

pe = 1.0*pempe = 2.0*pem

Fig. 7: Model fit to the numerical results of oil recovery in coresamples at different pe (pem=0.0259 atm).

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.01 0.02 0.03 0.04 0.05 0.06Entry Capillary Pressure, atm

Res

idua

l Oil

Satu

ratio

n,

fract

ion

Fig. 8: Effect of the entry capillary pressure on the averageresidual oil saturation in core samples.

0.0

0.2

0.4

0.6

0.8

1.0

0 5000 10000 15000Time, day

Oil

Reco

ver

y, O

OIP

pe = 0.1*pecpe = 1.0*pecpe = 2.0*pec

Fig. 9: Model fit to the numerical results of oil recovery atreservoir scale for different pe (pec=0.259 atm, k=70 md).

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50Production Time, year

Oil

Reco

ver

y, O

OIP Field

Model

Fig. 10: Model fit to the oil production data from the LakeviewPool, Midway Sunset field.