Temperature Effects on the Oil Relative Permeability during Tertiary

Oil relative permeability has a significant role in thermal enhanced heavy oil recovery. The present study has investigated the effects of temperature...
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Energy & Fuels 2005, 19, 977-983

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Temperature Effects on the Oil Relative Permeability during Tertiary Gas Oil Gravity Drainage (GOGD) Sh. Ayatollahi,*,† A. Lashanizadegan,‡ and H. Kazemi† School of Chemical and Petroleum Engineering, Shiraz University, P.O. Box 71345-1719, and School of Engineering, Yasuj University, Yasuj, Iran Received February 28, 2004. Revised Manuscript Received January 15, 2005

Oil relative permeability has a significant role in thermal enhanced heavy oil recovery. The present study has investigated the effects of temperature on the heavy oil relative permeability during the tertiary gas oil gravity drainage (GOGD) mechanism. Sand-packed column models were used to examine the temperature effects on the recovery of waterflood residual oil, using the gravity-assisted gas injection process. A new, four-step experimental technique was performed, and an analytical model was used to determine the oil relative permeability from the oil production history at different temperatures. The conclusions from this work indicate that the heavy oil relative permeability is affected by temperature. Aging at the elevated temperature is determined to be the main reason behind the wettability alteration and its effect on heavy oil relative permeability.

Introduction Simultaneous flow of the three phases of oil, water, and gas occurs during different stages of hydrocarbon production from underground reservoirs, especially enhanced oil recovery (EOR) processes. By far, the most important EOR scheme in the United States and all around the world has been thermal EOR. Obviously, the advantage gained by the exponential decrease in oil viscosity, because of the linear increase in temperature, has been regarded as the focal point of all thermal EOR schemes. Because of the very low oil production during the primary oil recovery stage, huge heavy oil reservoirs remain the target of thermal EOR.1 Butler2 indicated that, among different thermal EOR mechanisms, the gravity drainage of heated heavy oil has been given considerable interest by the oil industry. The combined effects of a conventional thermal mechanism and gas oil gravity drainage (GOGD) were observed to be a very effective process: this combined process is called SAGD. Gas injection is also being increasingly applied as a secondary or tertiary oil recovery process in dipping reservoirs, preferably those with high permeability and containing light oil. In such reservoirs, a gravity-stable injection scheme that leads to high sweep efficiencies is possible. Laboratory experiments3-6 and field performance analysis7-9 have proven that very high recovery of residual oil is attainable if gravity drainage is the dominant production * Author to whom correspondence should be addressed. E-mail address: [email protected]. Currently on sabbatical leave to Sultan Qaboos University, Muscat, Oman. † Shiraz University. ‡ Yasuj University. (1) Islam, M. R. Emerging Technologies in Enhanced Oil Recovery. Energy Sources 1999, 21 (1/2), 78-91. (2) Butler, R. M. Thermal Recovery of Oil and Bitumen; PrenticeHall: Englewood Cliffs, NJ, 1991; pp 285-359.

mechanism. The application of thermal GOGD to recover the waterflood residual oil, which is the investigation used in this work, is a new field of interest that needs to be investigated more. In addition to the viscosity reduction, the effects of temperature on the oil relative permeability were the objective of several investigations in the past. Oil relative permeabilities at different temperatures are needed for understanding thermal recovery processes for heavy oil recovery. Over the past three decades, many experimental studies have reported contradictory temperature effects on oil relative permeability using different methods. Edmondson10 found a (3) Chatzis, I.; Kantzas, A.; Dullien, F. A. L. On the Investigation of Gravity Drainage Assisted Inert Gas Injection Using Micromodels, Long Berea Cores and Computer Assisted Tomography. Presented at the 63rd Annual SPE Technology Conference, Houston, TX, October 2-5, 1988, SPE Paper No. 18284. (4) Naylor, P.; Frørup, M. Gravity-Stable Nitrogen Displacement of Oil. Presented at the 64th Annual SPE Technology Conference, San Antonio, TX, October 8-11, 1989, SPE Paper No. 19641. (5) Dullien, F. A. L.; Catalan, L.; Chatzis, I.; Collins, A. Recovery of Waterflood Residual Oil with Low-Pressure Inert Gas Injection, Assisted by Gravity Drainage from Water-Wet and Oil-Wet Cores. Presented at the Petroleum Society of CIM and AOSTRA 1991 Technology Conference, Banff, Alberta, Canada, April 21-24, 1991, Paper No. CIM/AOSTRA 91-1. (6) Oren, P. E.; Pinczewski, W. V. The Effect of Wettability and Spreading Coefficients on the Recovery of Waterflood Residual Oil by Immiscible Gas Flooding. Presented at the 67th Annual SPE Technology Conference, Washington, DC, October 4-7, 1992, SPE Paper No. 24881. (7) King, R. L.; Stile, J. H. A Reservoir Study of the Hawkins Woodbine Field. Presented at the SPE 45th Annual Fall Meeting, Houston, TX, October 4-7, 1970, SPE Paper No. 2972. (8) Carlson, L. O. Performance of Hawkins Field Unit under Gas Drive-Pressure Maintenance Operations and Development of an Enhanced Oil Recovery Project. Presented at the SPE/DOE Enhanced Oil Recovery Symposium, Tulsa, OK, April 17-20, 1988, SPE/DOE Paper No. 17324. (9) Eikmans, H.; Hitchings, V. H. Using 3D Integrated Modeling to Manage the Fractured Natih Field (Oman). Presented at the SPE Middle East Oil Show, Bahrain, February 20-23, 1999, SPE Paper No. 53227.

10.1021/ef0499473 CCC: $30.25 © 2005 American Chemical Society Published on Web 02/26/2005

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shift in the relative permeability ratio of oil to water toward higher water saturation as the temperature was increased. Poston et al.11 investigated the effects of temperature on unconsolidated sand using refined oils. They reported that irreducible water saturation increased as the residual oil saturation decreased with increasing temperature. They concluded that the relative permeabilities to both oil and water increased as the temperature increased. Two further studies12,13 on the effects of temperature on oil relative permeability have reported temperature-independent relative permeability data obtained from unsteady-state experiments in unconsolidated and Berea sandstone. Closmann et al.14 used the steady-state technique to measure the relative permeabilities of altered, unaltered, and deasphalted tar and brine at elevated temperatures, using previously frozen Peace River cores. They found that the tar and water relative permeability curves are shifted toward lower water saturations. Maini and Batcyky15 studied the previously frozen cores from a heavy oil reservoir and stock tank oil at temperatures ranging from room temperature to 522 °F and concluded that oil relative permeability decreased at higher temperatures. Polikar et al.16 used the steady-state technique to measure the relative permeability of Athabasca bitumen, using unconsolidated Ottawa sand as the porous medium. Their results showed no significant effect of temperature on relative permeability and saturation data, up to 392 °F. Akin et al.17 used a new three-step experimental technique to test the effect of temperature on relative permeability. It has been observed that a single set of relative permeability curves can represent both the ambient and high-temperature parts of the experiment. This suggests that relative permeability is not a function of temperature, at least for the system being tested. They summarized that the reasons for the divergence of experimental data were as follows: (1) Errors encountered in saturation measurements (2) Errors caused by neglecting the capillary pressure end effects (3) Wettability variations obtained with different oils and brines (4) Assumptions made to relate experimental procedures and/or calculations (5) Inadequacy of mathematical models used to represent multiphase flow conditions (10) Edmondson, T. A. Effect of Temperature on Waterflooding. J. Can. Pet. Technol. 1965, (October-December), 236-242. (11) Poston, S. W.; Israel, S.; Hossain, A. K. M. S.; Montgomery, E. F., III; Ramey, H. J., Jr. The Effect of Temperature on Irreducible Water Saturation and Relative Permeability of Unconsolidated Sands. Soc. Pet. Eng. J. 1970, (June), 171-180. (12) Sufi, A. H.; Ramey, H. J., Jr.; Brigham, W. E. Temperature Effects on Relative Permeabilities of Oil-Water. Presented at the 57th Annual Fall Technical Conference and Exhibition of Society of Petroleum Engineers, New Orleans, LA, September 26-29, 1982, SPE Paper No. 11071. (13) Miller, M. A.; Ramey, H. J., Jr. Effect of Temperature on Oil/ Water Relative Permeabilities of Unconsolidated and Consolidated Sands. JPT, J. Pet. Technol. 1985, 37 (December), 945-953. (14) Closmann, P. J.; Waxman, M. H.; Deeds, C. T. Steady-State Tar/Water Relative Permeabilities in Peace River Cores at Elevated Temperature. Presented at the 60th Annual Technical Conference and Exhibition of Society of Petroleum Engineers, Las Vegas, NV, September 22-25, 1985, SPE Paper No. 14227. (15) Maini, B. B.; Batycky, J. P. Effect of Temperature on HeavyOil/Water Relative Permeabilities and Vertically Drilled Core Plugs. JPT, J. Pet. Technol. 1985, 37 (August), 1500-1510.

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Figure 1. Schematic diagram for the tertiary gas oil gravity drainage (GOGD) process.18

It is clear that the difficulties in measuring the relative permeability, as a function of saturation and temperature, have a major role in this discrepancy. Therefore, the purpose of this study is to determine the effect of temperature on relative permeability curves for a sand-packed column, using a very simple method under the gravity drainage condition, to examine the precise temperature effect on oil relative permeability by minimizing the previously mentioned errors. Relative Permeability Calculation Three major regions in the course of the tertiary GOGD processsthe upper part or gas-invaded zone, the oil bank zone in the middle, and water zone at the bottomsare shown in Figure 1. The scheme shown in Figure 1 occurs if the production end is set at a very low rate, which permits the oil to accumulate on the top of the water zone. Experimental results revealed that no oil bank was created at a relatively high production rate, e.g., free-fall drainage conditions exist.18 The efficiency of the process is dependent on the oil relative permeability in the gas-invaded zone. The behavior of oil relative permeability in the gas-invaded zone controls the contribution of the oil phase to the oil production, either by increasing the oil bank size or by improving the oil recovery after gas breakthrough. There are two approaches to calculate the fluid relative permeabilities, using the information obtained from the unsteady-state immiscible displacement process. The first is to use the saturation and pressure profiles during the nonwetting phase injection to the porous medium. This technique requires the in situ saturation and pressure distribution measurements, which cannot be easily achieved.19 Many authors have addressed this problem and have used a variety of techniques to measure the saturation profiles, including measurements of electrical resistance, dielectric constant, absorption of neutrons, microwaves, gamma rays, (16) Polikar, M.; Ferracuti, F.; Decastro, V.; Puttagunta, V. R.; Ali, F. S. M. Effect of Temperature on Bitumen-Water End Point Relative Permeabilities and Saturations. J. Can. Pet. Technol. 1986, 25 (September-October), 44-50. (17) Akin, S.; Castanier, L. M.; Brigham, W. E. Effect of Temperature on Heavy Oil/Water Relative Permeabilities. Presented at the 1999 SPE International Thermal Operations Symposium, Bakersfield, CA, March 17-19, 1999, SPE Paper No. 54120. (18) Chatzis, I.; Ayatollahi, S. The Effect of Gas Injection Rate on the Recovery of Waterflood Residual Oil under Gravity Assisted Inert Gas Injection. Presented at the 7th European IOR Symposium, Moscow, Russia, October 27-28, 1993. (19) Dullien, F. A. L. Porous Media: Fluid Transport and Pore Structure, 2nd Edition; Academic Press: San Diego, CA, 1992.

Temperature Effects on Oil Relative Permeability

Figure 2. Schematic picture of vertical downward displacement and saturation profile.30

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ities calculated by this technique correspond to the values of average oil saturation throughout the model. Multiphase fluid during the free-fall GOGD process consists of two stages: (i) bulk water flows as the continuous phase in the region ahead of the gas front, with no oil movement, and (ii) water and oil flow downward in the form of thin films, mainly because of the gravitational forces in the gas-invaded zone. The mechanism of the GOGD process under free-fall conditions stipulates that the oil production starts right after the gas-water contact reaches the height of capillary rise. This fact indicates that the oil produced by this method comes from the gas-invaded zone by film flow; therefore, the calculation of oil relative permeability by the method described in the following section is justified. Experimental results showed that, during the freefall GAIGI process for the recovery of waterflood residual oil, neither is an oil bank created nor is oil produced until the water-gas contact attains the capillary height at the bottom of the column.29 Therefore, the equation of continuity for the oil in the gas-invaded part of the medium shown in Figure 2 becomes

∂Vo ∂S/o ) - φ* ∂z ∂t

(1)

where the reduced porosity of the medium (φ*) and the reduced saturation of the oil (S/o) are

φ* ) φ(1 - Swc - Sorg)

(2)

Figure 3. Oil viscosity versus temperature.

and

S/o )

So - Sorg 1 - Swc - Sorg

(3)

where Swc is the connate water saturation, φ the porosity

Figure 4. Oil density versus temperature. Table 1. Oil Properties property

value

SAE specific gravity API viscosity at 20 °C density at 20 °C

50 0.91 24 811.90 cP 906 kg/m3

and X-rays.20-26 In addition to the difficulties of adopting and calibrating these apparatus, some of them even failed to give a unique relative permeability function27 and also could not explain the inconsistency found in temperature effects on the relative permeability in GOGD process.10-17 The second approach is developed to calculate the constants of the Corey expression28 of relative permeability, using the production history of a free-fall gravity drainage process. Note that the oil relative permeabil-

(20) Josendal, V. A.; Sandiford, B.B.; Wilson, J. W. Improved Multiphase Flow Studies Employing Radioactive Tracers. Trans. AIME 1952, 195, 65. (21) Whalen, J. W. A Magnetic Susceptibility Method for the Determination of Liquid Saturation in Porous Materials. Trans. AIME 1954, 201, 203. (22) Bennion, D. W.; Durrani, G. H. Experimental Procedure for Following Flood Fronts Visually and Calculating in Situ Saturation in Opaque Porous Media. J. Can. Pet. Technol. 1974, 13 (1), 42. (23) Parsons, R. W. Microwave Attenuation, A New Tool for Monitoring Saturations in Laboratory Flooding Experiments. SPE J. 1975, 15 (4), 302. (24) Davis, L. A. Computer-Controlled Measurement of Laboratory Areal Flood Saturation Distribution. Presented at the 58th Annual Technical Conference and Exhibition of the SPE, San Francisco, CA, October 5-8, 1983, SPE Paper No. 12037. (25) Brost, D. F.; Davis, L. A. Determination of Oil Saturation Distribution in Field Cores by Microwave Spectroscopy. Presented at the 1981 SPE Annual Technical Conference and Exhibition, San Antonio, TX, October 4-6, 1981, SPE Paper No. 10110. (26) Nichols, C. I.; Heaviside, J. Gamma Ray Attenuation Techniques Improve Analysis of Core Displacement Tests. Presented at the 60th Annual Technical Conference and Exhibition of the SPE, Las Vegas, NV, 1985, September 22-25, 1985, SPE Paper No. 14421. (27) Foulser, R.; Naylor, P.; Seale, C. Relative Permeabilities for the Gravity Stable Displacement of Waterflood Residual Oil by Gas. Chem. Eng. Res. Des. 1990, 68, 307. (28) Corey, T. The Interrelation Between Gas and Oil Relative Permeabilities. Prod. Mon. 1954, 19, 38. (29) Chatzis, I.; Ayatollahi, S. The Effect of Production Rate on the Recovery of Waterflood Residual Oil under Gravity Assisted Inert Gas Injection. Presented at the Fifth Petroleum Conference of the South Saskatchewan Section of the Petroleum Society of CIM, Regina, Saskatchewan, Canada, October 18-19, 1983, Paper No. 32. (30) Hagoort, J. Oil Recovery by Gravity Drainage. SPE J. 1980, 20 (3), 139.

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Figure 5. Experimental setup.

of the medium, Vo the oil velocity in the gas invaded zone, Sorg the residual oil saturation after gas flooding, and So the saturation of oil during free-fall gravity drainage operation. The oil velocity in the gas-invaded zone, where fluid flows in the form of a film when the capillary pressure term is omitted, becomes

Kkro(S/o)Fog Vo ) µo

(4)

where Fo is the density of oil, kro the relative permeability of the oil, µo the viscosity of oil, g the gravitational acceleration, and K the absolute permeability. Note that, in this equation, the capillary pressure term is neglected and the saturation is based on pore volume. Equations 1 and 4 can be combined to give

( )( )

gKFo ∂kro ∂S/o ∂S/o )∂t µoφ* ∂S/ ∂z o Corey’s expression for oil relative here:

permeability28

kro ) k0ro(S/o)n

(5) is used

(6)

( )( ) 1 1 n nk0 t ro D

1/n - 1

(7)

The dimensionless time (tD) is defined as

tD )

∆FoggK t µoφ*L

d ln(1 - Np) 1 )dtD n-1

(9)

The parameter n then can be determined. For tD ) 1 in eq 7, we obtain the parameter k0ro, using eq 10:

k0ro )

[

n-1 1 n n(1 - Np)

]

n -1

(10)

An experimental procedure is used to find the effect of temperature on the relative permeability (measuring n and k0ro), using the oil production history. Experimental Procedure

where n and k0ro are the exponent and constant of oil relative permeability, respectively. The two parameters n and k0ro are obtained from the production history of a free-fall gravity drainage process as follows. Hagoort30 found the following equation for the oil production in gas-invaded zone, film flow, during the gravity drainage process:

Np ) 1 - 1 -

A close examination of eq 7 shows that a plot of log (1 - Np)versus log tD results in an straight line. The two parameters n and k0ro are obtained readily from the slope and the intercept with the ordinate at tD ) 1. The slope of this plot is

(8)

where ∆Fog is the density difference between the oil and the gas, t is the time, and L is the length of the porous medium.

The gravity drainage experiments were performed in unconsolidated sand-packed columns 91 cm in length and 2.6 cm in diameter with a specific sand grain size of 0.85-1.2 mm. The physical properties of the refined oil used in this investigation are shown in Table 1. Temperature effects on the oil viscosity reduction measured in the laboratory are illustrated in Figure 3. The density is not a function of temperature, at least over the temperature range used in this study (see Figure 4). The experimental procedure involved the following steps: (1) CO2 was injected into the model for ∼20 min, to replace the air in the dry model. This allows the model to be saturated completely with water in the next stage. (2) With the medium fully saturated with water, the porosity and absolute permeability were measured. (3) To establish the connate water conditions, oil was injected from the top of the column. After several pore volumes of oil flooding, the connate water saturation (Swc) and the oil saturation in the porous medium were determined volumetrically. (4) To establish the waterflood residual oil conditions, water was injected from the bottom of the column. The volume of the oil produced from the top of the column was measured volumetrically, and the amount of residual oil was determined. (5) Heating strips were used to warm the model to the desired temperature, which ranged from ambient temperature to 70°C.

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Figure 6. Cumulative oil production for experiments 2 and 4.

Figure 8. Cumulative oil production for experiments 1 and 2. Table 2. Properties of the Sand-Packed Models

Figure 7. Cumulative oil production for experiments 3 and 5. (6) Free-fall gravity drainage tests were performed at different temperatures by opening the valves at the top and bottom of the column to the atmosphere. The experimental setup is shown in Figure 5. Multiphase fluid flow in this process consists of two stages: (a) bulk water flows as the continuous phase in the region ahead of the gas front, and (b) water and oil flow downward in the form of thin films, mainly because of the gravitational forces in the gas-invaded zone.

Results and Discussions The properties of the models used in these experiments are shown in Table 2. The residual oil saturations due to waterflooding (Sorw) were in agreement with the results from the other experiments, indicating that the differences are attributed to the different packing scheme and waterflooding procedure.3 Fresh sand was used for experiment 1 at ambient temperature. Two additional experiments, identified as experiments 2 and 3, were made to examine the aging effects during the GOGD process. After washing the model used for experiment 1 with kerosene, the same sand-packed column was used for the second experiment at 50 °C, which resulted in lower residual oil saturation after gasflooding. In experiment 3, the same procedure was adopted and applied for the same sand-packed column at 70 °C. Comparing the residual oil saturation after gasflooding reveals that the Sorg value has a tendency

experiment number

porosity, φ (%)

K (Darcy)

Swc (%)

Sorw (%)

1 2 3 4 5

49.6 49.6 49.6 49.6 49.6

377 400 400 405 410

12 12 12 10.43 12.5

58 53.3 53.54 43.3 54

to decrease as the temperature increases. To investigate the aging effect, fresh sands were used in the sandpacked columns to repeat the GOGD process at 50 and 70 °C. Comparing the results of experiment 4 to those of experiment 2 in one hand and the results of experiment 5 to experiment 3 on the other hand demonstrates that reduced Sorg values, by more than 65% and 30%, respectively, were accomplished. The differences found here are attributed to the wettability alteration that is due to the aging. The increase in absolute permeability of the porous media in experiments 2 and 3 are also another indication of wettability alteration to partially oil-wet in the course of experimentation. The differences in the absolute permeabilities for the sand-packed models in experiments 4 and 5 are thought to be due to differences in the packing procedure. To check the validity of Hagoort’s method for the tertiary GOGD process, the oil production history for all of the experiments are shown in Figures 6-8 as plots of (1 - Np) vs tD, which show the linear relation represented by eq 7. The relative permeability parameters (n and k0ro) were determined easily, using the plotted data and eqs 8 and 9, respectively. The measured parameters (n and k0ro) are shown in Table 3 for different experiments. Comparison between n values for different experiments shows that increasing the temperature results in decreasing the value of n. To magnify the effects of temperature on relative permeability, the calculated relative permeability curves are shown in Figures 9-12. The results of experiments 2 and 3 shown in Figure 9 illustrate the heating effects on oil relative permeability values for the nonfresh, partially oil-wet media. These results clearly indicate that the oil relative permeability

Table 3. Values of the Exponent (n) and Constant (k0ro) of the Corey Expression Value parameter

expt 1

expt 2

expt 3

expt 4

expt 5

temp, T (°C) sand type n k0ro Sorg (%)

room temperature fresh sands 6.474 0.950 16.70

50 nonfresh sands 5.300 0.196 13.79

70 nonfresh sands 3.873 0.444 10.33

50 fresh sands 3.073 0.409 4.58

70 fresh sands 2.833 0.287 7.23

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Figure 9. Relative permeability versus oil saturation for experiments 2 and 3. Figure 12. Relative permeability versus oil saturation for experiments 1 and 2.

shown in Table 2, although this is not supported by comparing the results of experiments 4 and 5. This discrepancy is thought to be due to the difference in pore size structure of the model used in experiment 5, because it was a new model, compared to models used in other experiments. Conclusions Figure 10. Relative permeability versus oil saturation for experiments 3 and 4.

Figure 11. Relative permeability versus oil saturation for experiments 3 and 5.

increases in the tertiary GOGD process as the temperature increases. Additional results are shown to further investigate the effects of both temperature and aging on oil relative permeability. Figures 10 and 11, which contain the results of experiments 3, 4, and 5, indicate that the aging has a greater effect than the temperature on the oil relative permeability in these experiments. Figure 12 shows the results of experiments 1 and 2. As this figure clearly indicates, the relative permeabilities in experiments 1 and 2 are almost the same, because of the more oil-wet condition of the sands used for experiment 2, compared to experiment 1. These two opposite effectsslower relative permeability in oil-wet media (see Figures 10 and 11) and higher relative permeability in heated media (see Figure 9)sare compensated for experiments 1 and 2, as shown in Figure 12. The same scenario is repeated for experiments 3 and 4, and the results are shown in Figure 10. Using the same media, experiments 1-3 yielded lower Sorg values by increasing the temperature of the experiments. These data are

(1) The oil relative permeability during tertiary gas oil gravity drainage (GOGD) process, to recover waterflood residual oil, can be measured using the two-phase relative permeability analytical model proposed by Hagoort.30 (2) As the temperature increases, the oil relative permeability in tertiary GOGD process increases if the wettability does not change during the experiments, because of the aging effects. (3) The aging process during the thermal enhanced oil recovery (EOR) experiments has a more important role than temperature in regard to the oil relative permeability. (4) If the wettability of the media is changed, the increase in temperature does not affect the relative permeability, unless it compensates the wettability alteration. (5) Some of the diversities reported in the literature, reviewed here, are now thought to be due to the wettability alteration occurs during the course of the experiments. (6) With an increase in temperature, the residual oil saturation due to the gasflooding is decreased if the wettability does not change. This amount is dependent directly on the extent of wettability alteration of the media from water-wet, clean sand to partially oil-wet, used sands. Nomenclature g ) gravity acceleration (m/s2) K ) absolute permeability (Darcy) kro ) relative permeability of oil k0ro ) constant of oil relative permeability L ) length of porous medium (cm) Np ) oil production history (cm3) n ) exponent of oil relative permeability So ) saturation of oil during free fall gravity drainage operation

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Scw ) connate water saturation

Subscripts

Sorg ) residual oil saturation after gas flooding

Fo ) density of oil (kg/m3)

o ) oil r ) residual g ) gas c ) connate w ) water p ) production

µo ) viscosity of oil (cP)

EF0499473

t ) time (s) Vo ) flow velocity of oil in under saturated region (cm/s) φ ) porosity of medium