Quantitative Prediction Model for the Water–Oil ... - ACS Publications

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Quantitative Prediction Model for the Water
Oil Relative Permeability Curve and Its Application in Reservoir Numerical Simulation. Part 2: Application Fuquan Luo,†,‡ Jian Hou,*,† Zhenquan Li,§ Shaoxian Bing,§ and Shutao Wang† †

College of Petroleum Engineering, China University of Petroleum, Dongying, Shandong 257061, People’s Republic of China Downhole Operation Company, Jidong Oilfield Company, Tangshan, Hebei 063000, People’s Republic of China § Geoscience Research Institute, Shengli Oilfield Company, Dongying, Shandong 257015, People’s Republic of China ‡

ABSTRACT: On the basis of the quantitative prediction model for the water
oil relative permeability curve established in part 1 (10.1021/ef2008002), the conventional reservoir numerical simulator is modified and a new reservoir numerical simulation program is obtained, which can consider two simulation conditions. One is using different relative permeability curves in different simulation grids, it is called gridding of the relative permeability curve (GRPC) for short. The other is using a constant relative permeability curve in different simulation grids, it is called nongridding of the relative permeability curve (NGRPC) for short. On the basis of establishing the typical reservoir geological models, the reservoir numerical simulations under GRPC and NGRPC are carried out and the results are compared to each other. It is indicated that letting the relative permeability curve vary stochastically throughout the reservoir has little effect on flow characteristic and development performance of the whole reservoir, while it greatly affects the saturation of different permeability positions. Specifically, low-permeability grid cells are beneficial to the flow of the oil phase; therefore, the oil saturation will decrease. High-permeability grid cells are beneficial to the flow of the water phase; therefore, the oil saturation will increase. When the injection well is located in the low-permeability position and the production well is located in the high-permeability position, the distribution width of reservoir oil saturation under GRPC will become larger than that under NGRPC. In other words, reservoir oil saturation distribution will tend to be more dispersed. Meanwhile, the distribution frequency of the high oil saturation value will increase under GRPC, which indicates that the local enrichment of the remaining oil is easier to occur.

1. INTRODUCTION The water
oil relative permeability curve can reflect the flow characteristics of the water
oil two phase in the porous medium, and it is important data in the oilfield development design and reservoir numerical simulation.1
5 In the reservoir numerical simulation, the following methods are usually used to describe the reservoir flow characteristic: (1) The whole reservoir uses a single average relative permeability curve. (2) Different sedimentary facies use different relative permeability curves. However, with regard to the reservoir with a high degree of heterogeneity, the flow characteristic in different positions of the reservoir varies so much that the above methods cannot accurately describe it, which will cause great difficulties for the remaining oil prediction and potential tapping.6
9 To refine the water
oil relative permeability curve to each gird cell, ECLIPSE as a commercial reservoir numerical simulation software provides the end-point scaling technology of the relative permeability curve. The principles of this technology are summarized as follows: First, scale the endpoint values of the relative permeability curve for each grid cell. Then, according to the deviation rules of end-point values between the relative permeability curve of each grid cell and the average relative permeability curve, shift the average relative permeability curve as a whole to the left or right. Consequently, the relative permeability curve for each grid cell is obtained. Unfortunately, this software only presents the relationship between the relative permeability curve and its end-point values, but it r 2011 American Chemical Society

does not explain how to obtain the end-point values of the relative permeability curve for each grid cell, which leads to difficulties for application. Therefore, it is essential to build a prediction model for the water
oil relative permeability curve. On this basis, different relative permeability curves can be assigned to different grid cells in reservoir numerical simulation, which can give a much more precise description of water
oil flow properties and reveal the enrichment law of the remaining oil. At present, there are few studies on the reservoir numerical simulation under gridding of the relative permeability curve (GRPC). Tjølsen et al.10,11 calculated the spatial distribution of the relative permeability curve based on the relationship between the relative permeability curve and the absolute permeability. Then, they compared the production characteristic of the two cases including constant and varying relative permeability curves in each grid cell. They found that considering the spatial distribution of the relative permeability curve had little effect on the production performance. This is the first time studying the reservoir numerical simulation under GRPC. On the basis of this research, Tjølsen et al. summarized the correlation between the water shock front velocity and the absolute permeability for different depositional environments based on the fractional flow theory. Received: June 3, 2011 Revised: August 14, 2011 Published: September 09, 2011 4414

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Their further research12 showed that, when there was a positive correlation between the water shock front velocity and the absolute permeability, the breakthrough time became earlier, comparing the case of the varying relative permeability curve in each grid cell to the case of keeping the relative permeability curve constant in each grid cell. When there is a negative correlation, a later breakthrough time will be expected. The above conclusion was inclined to appear only when the reservoir was highly heterogeneous, and it was not obvious in most reservoirs. In other words, it is still at the preliminary stage for the reservoir numerical simulation under GRPC, and no significant recognitions have been obtained thus far. On the basis of the quantitative prediction model for the water
oil relative permeability curve established in part 1 (10.1021/ef2008002), the conventional reservoir numerical simulator is modified and a new reservoir numerical simulation program is obtained, which can consider both GRPC and nongridding of the relative permeability curve (NGRPC) simulation conditions. On the basis of establishing one- and threedimensional reservoir geological models, the modified program is used to carry out reservoir numerical simulations under GRPC and NGRPC. The results of which are compared to each other. Eventually, new recognitions of the reservoir development rule under GRPC are obtained.

2. ESTABLISHMENT AND VALIDATION OF THE MODIFIED RESERVOIR NUMERICAL SIMULATION PROGRAM The modified reservoir numerical program is established, and the calculation procedure is shown in Figure 1. Under GRPC, it needs to calculate the normalized average relative permeability curve and the spatial variation of the relative permeability curve. Taking the one-dimensional reservoir numerical simulation for example, the simulation results of this program under GRPC and NGRPC are compared to those of ECLIPSE. The parameters of the one-dimensional geological model are as follows: the reservoir length, width, and thickness are 210, 10, and 15 m, respectively, and the average grid size is 10  10  15 m, with a total of 21 grid cells. The reservoir and fluid parameters are given in Table 1. The well pattern is one injector and one producer, producing and injecting at a constant surface liquid rate of 0.1 pore volume (PV) per year. Porosity is uniform in the whole reservoir, while permeability gradually increases from the injector to the producer. The ratio of the maximum and minimum permeability is called the permeability ratio for short. Here, the permeability ratio is 3. The larger the permeability ratio, the stronger the reservoir heterogeneity. Input for the reservoir average relative permeability curve, which represents that the absolute permeability, is 1 μm2, and the porosity is 0.3. Then, the quantitative prediction model for the water
oil relative permeability curve is used to calculate the relative permeability curves in different permeability positions of the reservoir, as shown in Figure 2. It can be found that, when the grid permeability is lower than the reservoir average permeability, the grid relative permeability curve shifts to the right of the average relative permeability curve, which indicates that the oil-phase flow capacity will reinforce. When the grid permeability is higher than the reservoir average permeability, the grid relative permeability curve shifts to the left, which indicates that the water-phase flow capacity will reinforce. When the permeability of the grid cell is equal to the reservoir average permeability, the obtained relative permeability curve is the same as the reservoir average relative permeability curve. Figures 3 and 4 compare the simulation results between the modified program and ECLIPSE. It is indicated that the results of this program under GRPC and NGRPC are both fitted well with those of ECLIPSE. Therefore, this program is highly reliable. In comparison to ECLIPSE,

Figure 1. Calculation procedure of the modified reservoir numerical simulation program.

Table 1. Reservoir and Fluid Parameters of the One-Dimensional Simulation Model parameter

value

parameter

value

average porosity (fraction) 0.3 average permeability (μm2) 1

reservoir pressure (MPa) bubble point pressure (MPa)

19.2 8.0

injection production well

200

reservoir oil density (g/cm3)

0.8

2000

reservoir oil viscosity (mPa s) 30

spacing (m) top depth (m)

Figure 2. Input for the reservoir average relative permeability curve and calculated relative permeability curves corresponding to different permeability positions of the reservoir. 4415

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Figure 3. Comparison of reservoir development performance curves.

Figure 4. Comparison of water saturation when the water cut reaches 98%.

Figure 5. Comparison of reservoir development performance curves for the one-dimensional model. this program is not restricted to the region numbers of the relative permeability curve. Even if there are a large number of grid cells, it is still able to ensure that each grid cell has its own relative permeability curve.

3. ONE-DIMENSIONAL RESERVOIR NUMERICAL SIMULATION 3.1. Establishment of the One-Dimensional Reservoir Geological Model. Model parameters are the same as those of

the previous one-dimensional model. In GRPC and NGRPC simulation conditions, the saturations are both initialized by means of the equilibrium initialization. Therefore, the initial saturation in each grid cell is kept constant under NGRPC, while the initial saturation varies from different grid cells under GRPC, and depends upon the grid porosity and permeability. The reservoir development rules under GRPC and NGRPC are simulated and compared to each other, as shown in Figures 5

and 6. From Figure 5, reservoir development performance curves under GRPC are almost consistent with those under NGRPC, which indicates that letting the relative permeability curve vary stochastically throughout the reservoir has little effect on flow characteristic and development performance of the whole reservoir. However, as can be seen from Figure 6, the saturation in different positions of the reservoir varies obviously. To be specific, the grid permeability near the injector is low; therefore, the oil-phase flow capacity increases and the water saturation increases under GRPC. The grid permeability near the producer is high; therefore, the water-phase flow capacity increases and the water saturation decreases under GRPC. Consequently, the water saturation distribution tends to disperse, and the fluctuation range increases after considering the spatial variation of the relative permeability curve. 3.2. Correlation between the Saturation Field and Permeability Field. According to saturation data in Figure 6, this paper 4416

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Figure 6. Comparison of water saturation for the one-dimensional model.

Figure 7. Correlation between the oil saturation difference field when the water cut reaches 98% and the dimensionless permeability field.

Figure 8. Permeability distributions under different permeability ratios.

establishes the correlation between the oil saturation difference field when the water cut reaches 98% and dimensionless permeability field, as shown in Figure 7. The correlation is fitted by the logarithmic function and takes the form of the following equations:

of water saturation when the water cut reaches 98% is shown in Figure 6b. For the cases that ξk is 9, 21, 39, and 99, the distribution of water saturation when the water cut reaches 98% is shown in Figure 9. The difference of the maximum and minimum saturation is called the saturation distribution width. The water saturation distribution width is quantitatively analyzed, and the result is given in Figure 10. As can be seen from Figures 6b, 9, and 10, when the injection well is located in the low-permeability position and the production well is located in the high-permeability position, the lowpermeability grid is beneficial to the flow of the oil phase and the water saturation increases under GRPC, while the high-permeability grid is beneficial to the flow of the water phase and the water saturation decreases under GRPC. The water saturation is at a minimum near the producer, where the permeability is the highest; therefore, the minimum oil saturation decreases. The water saturation is at a maximum near the injector, where the permeability is the lowest; therefore, the maximum water saturation increases. As a result, at the same permeability ratio, the water saturation distribution width under GRPC becomes larger than that under NGRPC. As shown in Figure 11, the special grids are selected to represent different reservoir positions, including the injection, center, and production positions. On this basis, the variation rules of water saturation with the permeability ratio in different positions are quantitatively studied, and the results are given in Figures 12 and 13. From Figures 10, 12, and 13, it is indicated that, when the injection well is located in the low-permeability position and the production well is located in the high-permeability position, as the permeability ratio increases, the permeability in the lowpermeability position becomes lower and the ascent extent of water saturation becomes larger, while the permeability in the

ΔSo ¼ 0:0275 lnðkÞ
0:0003 k k ¼ k̅

R 2 ¼ 0:9842

ð1Þ ð2Þ

where ΔSo is the oil saturation difference (GRPC minus NGRPC; fracture), k* is the dimensionless permeability (fracture), k is the grid permeability (10
3 μm2), and k is the reservoir average permeability (10
3 μm2). As can be seen from Figure 7 and eq 1, the determination coefficient is high with a 0.9842 value, which demonstrates that the correlation between the oil saturation field when water cut reaches 98% and the permeability field is very strong. With regard to the low-permeability position of the reservoir, the oil saturation difference is inclined to be negative, which indicates that the oil saturation decreases at the end of development after considering the spatial variation of the relative permeability curve. For the high-permeability position, the variation is reverse. 3.3. Influence Rule of the Permeability Ratio. The permeability ratio is expressed by ξk. Permeability distributions under different permeability ratios are shown in Figure 8. It is found that permeability distribution changes with the permeability ratio for the one-dimensional model, which leads to the change in the spatial distribution of the relative permeability curve. When ξk is 3, 9, 21, 39, and 99, respectively, the saturation distribution rules under GRPC and NGRPC are simulated and compared to each other. For the case that ξk is 3, the distribution

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Figure 9. Influence of the permeability ratio on the water saturation when the water cut reaches 98%.

Figure 10. Influence of the permeability ratio on the water saturation distribution range when the water cut reaches 98%.

high-permeability position becomes higher and the descent extent of water saturation becomes larger. Therefore, after the spatial variation of the relative permeability curve is considered, the larger the permeability ratio, the larger the ascent of the water saturation distribution width.

4. THREE-DIMENSIONAL RESERVOIR NUMERICAL SIMULATION 4.1. Establishment of the Three-Dimensional Reservoir Geological Model. The reservoir length, width, and thickness

are 150, 150, and 15 m, respectively, and the average grid size is 10  10  3 m, with a total of 1125 grid cells. The well pattern is one injector and one producer. The plane permeability distribution is shown in Figure 14, which indicates that the permeability around the injection well is low and the permeability around the production well is high. The plane permeability variation coefficient is 0.4, and the vertical permeability variation coefficient is 0.6. Reservoir parameters, reservoir average relative permeability

curve, initialization method, etc. are consistent with those of the one-dimensional model. The reservoir development rules under GRPC and NGRPC are simulated and compared to each other, as shown in Figures 15
17. The third simulation layer is taken for an example when the saturation is discussed in this paper. From Figure 15, reservoir development performance curves under GRPC are almost consistent with those under NGRPC, which indicates that flow characteristic and development performance of the whole reservoir remain unchanged after considering the spatial variation of the relative permeability curve. From Figures 16 and 17, it seems that the saturation variation in different positions of the reservoir is not obvious. Therefore, the frequency distribution of oil saturation when the water cut reaches 98% is presented, and the result is shown in Figure 18. As can be seen from Figures 17 and 18, the grid permeability near the injector is low; therefore, the oil-phase flow capacity increases, and the oil saturation decreases under GRPC. The grid permeability near the producer is high; therefore, the waterphase flow capacity increases, and the oil saturation increases under GRPC. Consequently, the oil saturation distribution tends to disperse, and the fluctuation range increases comparing GRPC to NGRPC. Meanwhile, the distribution frequency of oil saturation greater than 0.45 increases under GRPC, which indicates that the local enrichment within a certain high saturation interval of the remaining oil is easier to occur. 4.2. Correlation between the Saturation Field and Permeability Field. On the basis of saturation data in Figure 17, this paper establishes the correlation between the oil saturation difference field when the water cut reaches 98% and the dimensionless permeability field, as shown in Figure 19. The correlation is fitted by the logarithmic function and takes the form of eq 3. ΔSo ¼ 0:0251 lnðkÞ þ 0:002

R 2 ¼ 0:9652

ð3Þ

As can be seen from Figure 19 and eq 3, the determination coefficient is high with a 0.9652 value, which demonstrates that the correlation between the oil saturation field when the water 4418

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Figure 11. Special grids corresponding to different positions.

Figure 12. Permeability variations with the permeability ratio in different positions. Figure 14. Plane permeability distribution (from the third simulation layer).

Figure 13. Influence of the permeability ratio on the water saturation in different positions when the water cut reaches 98%.

cut reaches 98% and the permeability field is very strong. After the spatial variation of the relative permeability curve is considered, the oil saturation of the low-permeability position decreases and the oil saturation of the high-permeability position increases at the end of development, which is consistent with rules of the one-dimensional model. 4.3. Influence Rule of the Plane Permeability Variation Coefficient. The plane permeability variation coefficient is expressed by Vk. The permeability distribution changes with the plane permeability variation coefficient for the three-dimensional model, which leads to a change in the spatial distribution of the relative permeability curve. When Vk is 0.2, 0.3, 0.4, 0.5, and 0.6, the saturation distribution rules under GRPC and NGRPC are simulated and compared to each other. For the case that Vk is 0.4, the frequency distribution of oil saturation when the water cut reaches 98% is shown in Figure 18. For the cases that Vk is

0.2, 0.3, 0.5, and 0.6, the frequency distribution of oil saturation when the water cut reaches 98% is shown in Figure 20. The oil saturation distribution width is quantitatively analyzed, and the result is given in Figure 21. As can be seen from Figures 18, 20, and 21, when the injection well is located in the low-permeability position and the production well is located in the high-permeability position, the lowpermeability grid is beneficial to the flow of the oil phase and the oil saturation decreases under GRPC, while the high-permeability grid is beneficial to the flow of the water phase and the oil saturation increases under GRPC. The oil saturation is at a minimum near the injector, where the permeability is the lowest; therefore, the minimum oil saturation decreases under GRPC and the descent range is large. The oil saturation is at a maximum near the non-mainstream line, where the permeability is close to the reservoir average permeability; therefore, the maximum oil saturation changes little. Thus, at the same variation coefficient, the oil saturation distribution width under GRPC becomes larger than that under NGRPC. In addition, the frequency distributions of oil saturation have the same characteristics under different variation coefficients. The distribution frequency of high oil saturation increases under GRPC, which indicates that the local enrichment of the remaining oil is easier to occur. As shown in Figure 22, the special grids are selected to represent different reservoir positions, including the injection, center, production, and non-mainstream line positions. On this basis, the variation rules of oil saturation with the plane permeability variation coefficient in different positions are quantitatively studied, and the results are summarized in Figures 23 and 24. From Figures 21, 23, and 24, it is indicated that, when the injection well is located in the low-permeability position and the production well is located in the high-permeability position, 4419

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Figure 15. Comparison of reservoir development performance curves for the three-dimensional model.

Figure 16. Comparison of oil saturation for the three-dimensional model at the initial time.

Figure 17. Comparison of oil saturation for the three-dimensional model when the water cut reaches 98%.

Figure 18. Frequency distribution of oil saturation when the water cut reaches 98%.

Figure 19. Correlation between the oil saturation difference field when the water cut reaches 98% and the dimensionless permeability field. 4420

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Figure 20. Influence of the plane permeability variation coefficient on the oil saturation when the water cut reaches 98%.

Figure 23. Permeability variations with the plane permeability variation coefficient in different positions. Figure 21. Influence of the plane permeability variation coefficient on the oil saturation distribution range when the water cut reaches 98%.

Figure 24. Influence of the plane permeability variation coefficient on the oil saturation in different positions when the water cut reaches 98%.

Figure 22. Special grids corresponding to different positions.

as the variation coefficient increases, the permeability in the low-permeability position becomes lower and the descent extent of oil saturation becomes larger, the permeability in the 4421

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5. CONCLUSION (1) On the basis of the quantitative prediction model for the water
oil relative permeability curve established in part 1 (10.1021/ef2008002), the conventional reservoir numerical simulator is modified and a new reservoir numerical simulation program is obtained, which can consider both GRPC and NGRPC simulation conditions. This program provides an effective tool for describing the reservoir water
oil flow characteristics exactly and carrying out reservoir numerical simulation considering the spatial variation of the relative permeability curve. (2) The results of one-dimensional reservoir numerical simulation are consistent with those of three-dimensional simulation. The results indicate that letting the relative permeability curve vary stochastically throughout the reservoir has little effect on flow characteristic and development performance of the whole reservoir but has great effect on the saturation in different permeability positions. Low-permeability grid cells are beneficial to the flow of the oil phase; therefore, the oil saturation will decrease. High-permeability grid cells are beneficial to the flow of the water phase; therefore, the oil saturation will increase. When the injection well is located in the low-permeability position and the production well is located in the high-permeability position, the saturation distribution tends to be more dispersed and the saturation distribution width becomes larger after considering the spatial variation of the relative permeability curve. (3) For the three-dimensional model, the distribution frequency of high oil saturation will increase after considering the spatial variation of the relative permeability curve, which indicates that the local enrichment of the remaining oil is easier to occur.

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’ AUTHOR INFORMATION Corresponding Author

*Telephone: 86-546-8395660. Fax: 86-546-8395660. E-mail: houjian@ upc.edu.cn.

’ ACKNOWLEDGMENT The authors greatly appreciate the financial support of the National Natural Science Foundation of China (Grants 10772200 and 10972237), the Important National Science and Technology Specific Projects of China (Grant 2011ZX05011), the Fundamental Research Funds for the Central Universities (Grant 10CX03002A), and the Graduate Innovation Fund of the China University of Petroleum (Grant S10-07). ’ REFERENCES (1) Behzadi, H.; Alvarado, V. Impact of three-phase relative permeability model on recovery in mixed media: Miscibility, IFT, and hysteresis issues. Energy Fuels 2010, 24 (10), 5765–5772. (2) Ayatollahi, S. H.; Lashanizadegan, A.; Kazemi, H. Temperature effects on the oil relative permeability during tertiary gas oil gravity drainage (GOGD). Energy Fuels 2005, 19 (3), 977–983. 4422

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