Estimation of the Water–Oil Relative Permeability Curve from Radial

Jun 11, 2012 - Connectionist approach estimates gas–oil relative permeability in petroleum reservoirs: Application to reservoir simulation. Mohammad...
0 downloads 0 Views 2MB Size
Download PDF
Article pubs.acs.org/EF

Estimation of the Water−Oil Relative Permeability Curve from Radial Displacement Experiments. Part 2: Reasonable Experimental Parameters Jian Hou,*,†,‡ Fuquan Luo,§ Daigang Wang,‡ Zhenquan Li,∥ and Shaoxian Bing∥ †

State Key Laboratory of Heavy Oil Processing, and ‡College of Petroleum Engineering, China University of Petroleum, Qingdao, Shandong 266580, 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: The capillary pressure is the key parameter to affect the inversion accuracy of the water−oil relative permeability curve. The existing analytical inversion methods have neglected the influence of capillary pressure, which may cause low precision for the estimated relative permeability curve in some cases. On the basis of the numerical inversion method for the water−oil relative permeability curve established in part 1 (10.1021/ef300018w), taking the one-dimensional radial numerical experiment for example, the rules of relative permeability variation and influence of different displacement conditions on relative permeability deviation when neglecting the capillary pressure are investigated. With regard to water-wet cases whose oil−water viscosity ratio is greater than 1.5, it indicates that the estimated water-phase relative permeability curve is higher and the estimated oil-phase relative permeability curve is lower compared to the true relative permeability curve when the capillary pressure is neglected. The main displacement conditions influencing the inversion accuracy of the relative permeability curve include the injection rate, average permeability, and shape factor of the core sample. As the injection rate increases, the degree of relative permeability deviation caused by neglecting the capillary pressure becomes smaller. Moreover, the deviation trends of the water−oil relative permeability curve are the same as those of the increasing injection rate when average permeability decreases or the shape factor of the core sample increases. Finally, the orthogonal experimental design technique is used to establish the experimental conditions considering the combined effect of multiple factors, and then the water−oil relative permeability curve under every experimental condition is estimated implicitly. On this basis, the multivariate analysis is performed to obtain the threshold value charts of radial displacement experimental parameters, such as the injection rate, average permeability, and shape factor of the core sample, and their corresponding rational value domains are also achieved, which can be used to reduce the influence of neglecting capillary pressure data as much as possible and provide a calculation theory for estimation of the water−oil relative permeability curve accurately.

1. INTRODUCTION The water−oil relative permeability curve can describe the flow characteristics of the water−oil two phase in the porous medium, and it has a great effect on the rules of water cut increase and production variation, which is one important data in oilfield development design and reservoir numerical simulation.1−5 When obtaining the relative permeability curve from unsteady-state radial displacement experiments, the capillary pressure is the key parameter to affect its inversion accuracy. The existing analytical inversion methods have ignored the influence of capillary pressure, which may result in low precision for the estimated water−oil relative permeability curve in some cases.6 For this reason, it is essential to build a numerical inversion method for the water− oil relative permeability curve from radial displacement experiments. On this basis, the rules of relative permeability variation when the capillary pressure is neglected can be investigated. The rational value domains of laboratory experimental parameters can be further obtained, which will reduce the influence of neglecting capillary pressure data greatly in the process of carrying out radial displacement experiments © 2012 American Chemical Society

and give a much more precise estimation of the water−oil relative permeability curve. At present, there are few studies on the rules of relative permeability variation when neglecting the capillary pressure. Chen et al.7 proposed a numerical inversion method for the water−oil relative permeability curve. Then, they investigated the effect of the capillary pressure and heterogeneity on the estimated result with regard to a synthetic case under radial seepage. However, owing to the shortage of optimization algorithm properties and identification of the end-point saturation accurately, no significant recognitions have been achieved. On the basis of the numerical inversion method established in part 1 (10.1021/ef300018w), taking the onedimensional radial numerical experiment for example, this paper makes a thorough study on the rules of relative permeability variation when the capillary pressure is neglected, along with the influence of different displacement conditions on relative permeability deviation. Furthermore, the multivariate Received: January 7, 2012 Revised: June 8, 2012 Published: June 11, 2012 4300

dx.doi.org/10.1021/ef3005866 | Energy Fuels 2012, 26, 4300−4309

Energy & Fuels

Article

analysis is performed according to the estimated relative permeability curve under different experimental conditions using the orthogonal experimental design technique. On this basis, the threshold values charts of radial displacement experimental parameters and their corresponding rational value domains will be obtained.

2. RULES OF RELATIVE PERMEABILITY VARIATION WHEN NEGLECTING THE CAPILLARY PRESSURE Taking the one-dimensional radial numerical experiment for example, the rules of relative permeability variation when neglecting different levels of capillary pressure data are investigated thoroughly using the numerical inversion method proposed in part 1 (10.1021/ef300018w). The parameters of the one-dimensional geological model are as follows: the core outer diameter, inner diameter, and thickness are 0.1, 0.005, and 0.08 m, respectively. A radial-angular cylindrical grid system is adopted, with a total of 12 grid cells. The reservoir and fluid

Figure 2. Input for the water−oil relative permeability curve. curve when considering the capillary pressure. The results are shown as Figure 4. Water cut variation with the recovery percent under different levels of capillary pressure data is shown in Figure 5. From Figures 3 and 4, it can be found that, in comparison to the estimated result when considering the capillary pressure, the predicted water-phase relative permeability curve is higher and the predicted oilphase relative permeability curve is lower when neglecting the capillary pressure. It also indicates that, the greater the capillary pressure, the bigger the degree of deviation. The main reason is explained as follows: in comparison to that of neglecting the capillary pressure, the breakthrough time is brought forward and water cut increases when the capillary pressure is considered. Moreover, the greater the capillary pressure, the more dramatic water cut increases. To fit the dynamic characteristic of breakthrough time beforehand and water cut rise when considering the capillary pressure, the estimated water-phase relative permeability curve is higher and the estimated oil-phase relative permeability curve is lower when neglecting the capillary pressure.

Table 1. Reservoir and Fluid Parameters of the OneDimensional Radial Numerical Experiment parameter

value

parameter

average porosity (fraction) average permeability (10−3 μm2) core initial pressure (MPa) pressure at outlet face (MPa)

0.3 1000

injection rate (m3/day) water viscosity (mPa s)

0.0015 1.0

value

3.0 2.95

oil viscosity (mPa s) rock compressibility (MPa−1)

1.5 5 × 10−4

parameters are listed in Table 1. The well pattern is one injector located in the inner face and a producer located in the outer face. The production is performed at a constant bottomhole pressure, and the injection is preformed at a constant surface liquid rate, which is the same as the controlling conditions of laboratory displacement experiments most commonly used. Porosity is uniform in the whole reservoir, while permeability gradually decreases from the injector to the producer. The ratio of the maximum and minimum permeabilities is 3. Different levels of the capillary pressure data used in this paper are shown in Figure 1, and the input water−oil relative permeability curve is shown as Figure 2.

3. INFLUENCE OF DISPLACEMENT CONDITIONS ON RELATIVE PERMEABILITY DEVIATION When using the unsteady-state radial displacement experiments to obtain the water−oil relative permeability curve, to reduce the influence of neglecting capillary pressure data and make the estimated curve precisely reflect the flow characteristic of the water−oil two phase in the porous medium, the core sample, properties of fluids, and displacement history should be similar to those of a real reservoir.8−10 Moreover, displacement conditions, such as injection rate, average permeability, and oil−water viscosity ratio, should also be limited to their rational value domains. It is known that the displacement conditions belong to human factors and cannot be ignored.11−13 These factors can alter the pressure gradient at both ends of the core sample and then affect the degree of relative permeability deviation when the capillary pressure is neglected. The displacement conditions investigated in this paper mainly include the injection rate, average permeability, oil−water viscosity ratio, and shape factor of the core sample. Among which, the shape factor of the core sample is defined as the ratio of the core outer diameter and its thickness. The influence of different displacement conditions on the relative permeability deviation when the capillary pressure is neglected will be studied thoroughly. The capillary pressure data neglected in this study is shown in Figure 1, whose maximum capillary pressure is 0.0008 MPa. 3.1. Injection Rate. The injection rate Q of the radial numerical experiment is 0.000 15, 0.000 75, 0.0015, 0.0075, and 0.015 m3/day. The remaining parameters are identical to those of the basic scheme. The influence of different injection rates on relative permeability deviation caused by neglecting the

Figure 1. Different levels of the capillary pressure data. Production performance under different levels of capillary pressure data can be obtained on the basis of a radial reservoir simulator. When the capillary pressure is neglected, the water−oil relative permeability curve is estimated implicitly and the rules of relative permeability variation will be investigated thoroughly. The results are shown as Figure 3. To evaluate the degree of relative permeability deviation quantitatively when the capillary pressure is neglected, eq 15 shown in part 1 (10.1021/ef300018w) is used to calculate the average absolute error between the estimated result when neglecting the capillary pressure and the estimated water−oil relative permeability 4301

dx.doi.org/10.1021/ef3005866 | Energy Fuels 2012, 26, 4300−4309

Energy & Fuels

Article

Figure 4. Influence of neglecting the capillary pressure on the estimated relative permeability curve.

Figure 5. Water cut variation with the recovery percent under different levels of capillary pressure data.

Figure 6. Estimated relative permeability curve under different injection rates when the capillary pressure is neglected.

Figure 3. Estimated relative permeability curve when different capillary pressure data are neglected.

capillary pressure is investigated according to the production performance obtained under different injection rates. The results are shown as Figures 6 and 7. Water cut variation with the recovery percent under different injection rates is shown as Figure 8. From Figures 6 and 7, it demonstrates that, the lower the injection rate, the greater the degree of relative permeability deviation when the capillary pressure is neglected. The main reason is explained as follows: the rule of displacement difference variation is proportional to that of the injection rate, and the influence of neglecting capillary pressure data is aggravated as the injection rate decreases. To fit the dynamic

Figure 7. Influence of the injection rate on relative permeability deviation caused by neglecting the capillary pressure.

characteristic of breakthrough time beforehand and water cut rise shown as Figure 8, both the degree of oil-phase relative permeability deviation and the degree of water-phase relative 4302

dx.doi.org/10.1021/ef3005866 | Energy Fuels 2012, 26, 4300−4309

Energy & Fuels

Article

Figure 9. Estimated relative permeability curve under different average permeabilities when the capillary pressure is neglected.

Figure 10. Influence of the average permeability on relative permeability deviation caused by neglecting the capillary pressure.

neglected. The main reason is explained as follows: the rule of seepage resistance variation when oil is displaced by water is inversely proportional to that of average permeability, and the influence of neglecting capillary pressure data is weakened as the average permeability decreases. To fit the dynamic characteristic of breakthrough time beforehand and water cut rise shown as Figure 11, both the degree of oil-phase relative permeability deviation andthe degree of water-phase relative permeability deviation become smaller as the average permeability decreases. 3.3. Oil−Water Viscosity Ratio. Water viscosity is limited to a constant value, and the oil−water viscosity ratio μo/μw is 1.5, 5, 10, 15, and 30. The remaining parameters are identical to those of the basic scheme. The influence of different oil−water viscosity ratios on relative permeability deviation caused by neglecting the capillary pressure is investigated according to the production performance obtained under different oil−water viscosity ratios. The results are shown as Figures 12 and 13. Water cut variation with the recovery percent under different oil−water viscosity ratios is shown as Figure 14. From Figures 12 and 13, it can be found that, the greater the oil−water viscosity ratio, the smaller the degree of relative permeability deviation when the capillary pressure is neglected. The main reason is explained as follows: the rule of water mobility variation is proportional to that of oil−water viscosity ratio, and the influence of neglecting capillary pressure data is weakened as the oil−water viscosity ratio increases. To fit the dynamic characteristic of breakthrough time beforehand and water cut rise shown as Figure 14, both the degree of oil-phase relative permeability deviation and the degree of water-phase relative permeability deviation become smaller as the oil−water viscosity ratio increases.

Figure 8. Water cut variation with the recovery percent under different injection rates.

permeability deviation become bigger as the injection rate decreases. 3.2. Average Permeability. The average permeability k of the radial numerical experiment is 100 × 10−3, 500 × 10−3, 1000 × 10−3, 5000 × 10−3, and 10000 × 10−3 μm2. The remaining parameters are identical to those of the basic scheme. The influence of the different average permeabilities on relative permeability deviation caused by neglecting the capillary pressure is investigated according to the production performance obtained under different average permeabilities. The results are shown as Figures 9 and 10. Water cut variation with the recovery percent under different average permeabilities is shown as Figure 11. From Figures 9 and 10, it indicates that, the smaller the average permeability of the core sample, the smaller the degree of relative permeability deviation when the capillary pressure is 4303

dx.doi.org/10.1021/ef3005866 | Energy Fuels 2012, 26, 4300−4309

Energy & Fuels

Article

Figure 13. Influence of the oil−water viscosity ratio on relative permeability deviation caused by neglecting the capillary pressure.

Figure 11. Water cut variation with the recovery percent under different average permeabilities.

Figure 12. Estimated relative permeability curve under different oil− water viscosity ratios when the capillary pressure is neglected.

Figure 14. Water cut variation with the recovery percent under different oil−water viscosity ratios.

4304

dx.doi.org/10.1021/ef3005866 | Energy Fuels 2012, 26, 4300−4309

Energy & Fuels

Article

3.4. Shape Factor of the Core Sample. With regard to a disk-shape core sample, when the core volume is limited to a constant value, the greater the ratio of the core outer diameter and its thickness, the more flat the core sample; the smaller the ratio, the more convex the core sample. To describe the differences of the core shape, the shape factor of the core sample M is introduced and takes the form of eq 1 D M= (1) h

Table 2. Sizes of Core Samples under Different Shape Factors scheme

shape factor of the core sample

core outer diameter (m)

core thickness (m)

1 2 3 4 5

0.1 0.5 1.25 5 10

0.044 0.074 0.10 0.16 0.20

0.412 0.146 0.08 0.031 0.02

where M is the shape factor of the core sample (fraction), D is the core outer diameter (m), and h is the core thickness (m). The core volume is limited to a constant value. The shape factor of the core sample M is 0.1, 0.5, 1.25, 5, and 10. The remaining parameters are identical to those of the basic scheme. The influence of different shape factors of the core sample on relative permeability deviation caused by neglecting the capillary pressure is investigated according to the production performance obtained under different shape factors of the core sample. The results are shown as Figures 15 and 16. Among

Figure 15. Estimated relative permeability curve under different shape factors when the capillary pressure is neglected.

Figure 16. Influence of the shape factor on relative permeability deviation caused by neglecting the capillary pressure.

which, sizes of core samples under different shape factors have been summarized in Table 2 and the corresponding water cut variation with the recovery percent is shown as Figure 17. From Figures 15 and 16, it demonstrates that, the greater the shape factor of the core sample, the smaller the degree of relative permeability deviation when the capillary pressure is neglected. The main reason is explained as follows: when the core diameter is identical, the core thickness is reduced and the water adsorption capacity is weakened as the shape factor of the core sample increases. With regard to a constant value of water injection, the rule of displacement difference variation is proportional to that of the shape factor, and the influence of

Figure 17. Water cut variation with the recovery percent under different shape factors of the core sample.

neglecting capillary pressure data is weakened as the shape factor of the core sample increases. To fit the dynamic characteristic of breakthrough time beforehand and water cut rise shown as Figure 17, both the degree of water-phase relative permeability deviation and the degree of oil-phase relative 4305

dx.doi.org/10.1021/ef3005866 | Energy Fuels 2012, 26, 4300−4309

Energy & Fuels

Article

of relative permeability inversion results are obtained. (2) Use the orthognal experimental design technique to generate the experimental samples, so that the variation of the sensitive parameters within their rational value domains can be reflected reasonablely by fewer samples. (3) Calculate the degree of relative permeability deviation under every experimental sample. The corresponding average absolute error is computed to evaluate quantitatively the effect of multiple sensitive parameters under different experimental conditions. On the basis of the above analysis, it can be found that the main displacement conditions affecting the estimated result of the water−oil relative permeability curve include the injection rate, average permeability, and shape factor of the core sample. Five different levels are assigned to every sensitive parameter, and then the factor and level graph shown as Table 4 is

permeability deviation become smaller as the shape factor of the core sample increases. On the basis of the above analysis, the influence of different displacement conditions on the relative permeability deviation caused by neglecting the capillary pressure is various. To determine the main influential factors of relative permeability inversion results, eqs 2 and 3 are used to calculate the variation coefficient of relative permeability deviation caused by neglecting the capillary pressure when displacement conditions differ from each other. The results are concluded in Table 3 Table 3. Variation Coefficient of Relative Permeability Deviation Caused by Different Displacement Conditions influential factor 3

injection rate (m /day) average permeability (10−3 μm2) oil−water viscosity ratio shape factor of the core sample

CV =

S − |y |

S=

1 n

range

CV (water phase)

CV (oil phase)

0.00015−0.015 100−10000

0.8254 0.7511

0.7549 0.6107

1.5−30 0.1−10

0.3774 0.7093

0.2891 0.5594

Table 4. Orthogonal Design Factor and Level Graph

(2) n i=1

factor B

factor C

factor D

name scheme

injection rate (m3/day)

average permeability (10−3 μm2)

shape factor of the core sample

1 2 3 4 5

0.00016 0.0004 0.0008 0.0016 0.004

0.00015 0.00075 0.0015 0.0075 0.015

100 500 1000 5000 10000

0.1 1.25 5 10 20

developed. Orthogonal design [L25(56)] is used. A total number of 25 experimental samples are generated, allocating the four influential factors into the first four columns. To evaluate the degree of relative permeability deviation quantitatively caused by neglecting the capillary pressure, eq 15 shown in part 1 (10.1021/ef300018w) is used to calculate the average absolute error of every experimental sample. The orthogonal design results are concluded in Table 5. 4.2. Definition of the Dimensionless Parameter. When compressibility of rock and fluid are neglected and the process of radial displacement reaches a steady state, the injection rate Q can be calculated from the Darcy law,15 which is shown as eq 4

− 2

∑ (yi − y )

factor A capillary pressure (MPa)

(3)

where CV is the variation coefficient of samples, S is the standard deviation of samples (fraction), y ̅ is the mean value of samples (fraction), n is the total number of samples, and yi is the value of the ith sample (fraction). From Table 3, it can be seen that the main influential factors of relative permeability inversion results are the injection rate, average permeability, and shape factor of the core sample and the oil−water viscosity ratio has little effect on the estimated result.

4. ESTABLISHMENT OF REASONABLE EXPERIMENTAL PARAMETER CONDITIONS Considering the fact that the estimated relative permeability curve is affected by multiple factors simultaneously, experimental conditions taking into consideration the combined effect of multiple factors are designed using the orthogonal experimental design technique, and the water−oil relative permeability curve under every experimental condition is calculated implicitly. On this basis, the multivariate analysis is performed to establish the threshold value charts of radial experimental parameters and to obtain their rational value domains. In other words, the reasonable parameter conditions of radial displacement experiments are developed. 4.1. Generation of Experimental Samples. The experimental samples considering the combined effect of multiple factors can be generated from the orthogonal experimental design technique.14 The specific procedure is listed as follows: (1) Select the sensitive parameters for estimation of the relative permeability curve. In other words, on the basis of studies on the influence of different displacement conditions to relative permeability deviation caused by neglecting the capillary pressure, the main influential factors

Q=

⎞ 2πkhΔp ⎛ μw 2πDk Δp ⎜ ⎟⎟ = + k k ⎜ ro rw Re R μw ln R ⎝ μo μw M ln R e ⎠ w

w

⎞ ⎛μ ⎜⎜ w k ro + k rw ⎟⎟ ⎠ ⎝ μo (4)

−3

where k is the average permeability (10 μm ), kro and krw are the relative permeability of the oil and water phases, respectively, μo and μw are the oil and water viscosities, respectively (mPa s), Δp is the displacement pressure difference at both ends of the core sample (MPa), and Rw and Re are the inner and outer boundaries of the core sample, respectively (m). To present the correlation between the capillary pressure data and displacement conditions, including the injection rate, average permeability, and shape factor of the core sample, a dimensionless parameter Ψ is introduced and takes the form of eq 5 pc,max Ψ = Qμ M w

kD

2

(5)

where Ψ is the dimensionless parameter and pc,max is the maximum capillary pressure data (MPa). 4306

dx.doi.org/10.1021/ef3005866 | Energy Fuels 2012, 26, 4300−4309

Energy & Fuels

Article

Table 5. Orthogonal Design Results under Different Experimental Samples experimental condition

orthogonal design result

scheme

factor A

factor B

factor C

factor D

degree of water relative permeability deviation

degree of oil relative permeability deviation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.00016 0.00016 0.00016 0.00016 0.00016 0.0004 0.0004 0.0004 0.0004 0.0004 0.0008 0.0008 0.0008 0.0008 0.0008 0.0016 0.0016 0.0016 0.0016 0.0016 0.004 0.004 0.004 0.004 0.004

0.00015 0.00075 0.0015 0.0075 0.015 0.00015 0.00075 0.0015 0.0075 0.015 0.00015 0.00075 0.0015 0.0075 0.015 0.00015 0.00075 0.0015 0.0075 0.015 0.00015 0.00075 0.0015 0.0075 0.015

100 500 1000 5000 10000 500 1000 5000 10000 100 1000 5000 10000 100 500 5000 10000 100 500 1000 10000 100 500 1000 5000

0.1 1.25 5 10 20 5 10 20 0.1 1.25 20 0.1 1.25 5 10 1.25 5 10 20 0.1 10 20 0.1 1.25 5

0.0334 0.0102 0.0054 0.001 0.0008 0.0242 0.0113 0.0126 0.0552 0.0005 0.0388 0.0825 0.0641 0.0006 0.0007 0.0915 0.0752 0.0032 0.0009 0.0321 0.0982 0.0086 0.0688 0.0265 0.0248

0.0492 0.0183 0.0105 0.0089 0.005 0.0416 0.0192 0.0198 0.0651 0.002 0.0525 0.0896 0.0741 0.0025 0.0046 0.0978 0.0843 0.0094 0.0065 0.0483 0.1068 0.0156 0.0795 0.0442 0.0421

Figure 18. Statistical relationship of the relative permeability deviation and the dimensionless parameter.

carrying out radial displacement experiments to obtain the water−oil relative permeability curve. 4.3. Establishment of Threshold Value Charts of Radial Experimental Parameters. As the displacing fluid, the variation of water viscosity is often to a small extent in different radial displacement experiments. This paper limits it to a constant value of 0.5 mPa s. In the condition of knowing the capillary pressure and core outer diameter, the relationship of three displacement conditions, including the injection rate, average permeability, and shape fator of the core sample, is developed on the basis of the threshold value of the dimensionless parameter of 17.024 and eq 5, and then the threshold value charts of radial experimental parameters are established. Figure 19 presents the threshold value charts of radial experimental parameters when the capillary pressure data are different and the core outer diameter is equal to 0.10 m. In addition, as long as two of the three displacement conditions are known, the threshold value of the three displacement conditions can be determined.

On the basis of using the orthogonal experimental design technique to carry out multivariate analysis for the water−oil relative permeability curve, the value of dimensionless parameter Ψ under every experimental sample is calculated. The statistical relationship of the relative permeability deviation and the dimensionless parameter can be further obtained, which is shown as Figure 18. As seen from Figure 18, the degree of both the water- and oil-phase relative permeability deviations will become smaller as the dimensionless parameter decreases. This paper selects 17.024 as the threshold value of the dimensionless parameter, and the corresponding degree of the water- and oil-phase relative permeability deviations is 0.0126 and 0.0198, respectively, both of which are less than 0.02. When the dimensionless parameter is less than or equal to the threshold value of 17.024, the estimated water−oil relative permeability curve is reliable even if the capillary pressure is neglected. For this reason, the requirement Ψ ≤ 17.024 should be met when 4307

dx.doi.org/10.1021/ef3005866 | Energy Fuels 2012, 26, 4300−4309

Energy & Fuels

Article

Figure 19. Threshold value charts of radial experimental parameters.

deviation caused by neglecting the capillary pressure becomes smaller. Moreover, the deviation trends of the water−oil relative permeability curve are the same as those of the increasing injection rate when average permeability decreases or the shape factor of the core sample increases. (2) The orthogonal experimental design technique is used to carry out multivariate analysis for the estimated water−oil relative permeability curve under different displacement experimental conditions. On this basis, the threshold value charts of radial experimental parameters are established, and then their corresponding rational value domains are further obtained. Real values of experimental parameters and their corresponding threshold values should satisfy the following relationship: the real value of the injection rate is greater than its threshold value; the real value of the average permeability is less than its threshold value; and the real value of the core shape factor is greater than its threshold value.

From eq 5 and Figure 19, it can be seen that, to guarantee the inversion accuracy of the estimated water−oil relative permeability curve, the real value of experimental parameters and their corresponding threshold values should satisfy the following relationship: the real value of the injection rate is greater than its threshold value; the real value of the average permeability is less than its threshold value; and the real value of the core shape factor is greater than its threshold value.

5. CONCLUSION (1) On the basis of the numerical inversion method established in part 1 (10.1021/ef300018w), taking the one-dimensional radial numerical experiment for example, the rules of relative permeability variation and influence of different displacement conditions on relative permeability deviation when neglecting the capillary pressure are investigated. With regard to water-wet cases whose oil−water viscosity ratio is greater than 1.5, it indicates that the estimated water-phase relative permeability curve is higher and the estimated oil-phase relative permeability curve is lower compared to the true curve when the capillary pressure is neglected. The main displacement conditions affecting the inversion accuracy of the water−oil relative permeability curve include the injection rate, average permeability, and shape factor of the core sample. As the injection rate increases, the degree of relative permeability

■

AUTHOR INFORMATION

Corresponding Author

*Telephone: 86-546-8395660. Fax: 86-546-8395660. E-mail: [email protected]. Notes

The authors declare no competing financial interest. 4308

dx.doi.org/10.1021/ef3005866 | Energy Fuels 2012, 26, 4300−4309

Energy & Fuels

■

Article

(9) Eqermann, P.; Lombard, J. M.; Fichen, C.; Rosenberq, E.; Tachet, E.; Lenormand, R. A new experimental method to determine interval of confidence for capillary pressure and relative-permeability curves. Proceedings of the Society of Petroleum Engineering (SPE) Annual Technical Conference and Exhibition; Dallas, TX, Oct 9−12, 2005; SPE Paper 96896. (10) Lafhaj, Z.; Skoczylas, F.; Dubais, T.; Dana, E. Experimental determination of relative permeability of sand. J. ASTM Int. 2005, 2 (7), 35−47. (11) Chukwudeme, E. A.; Ingebret, F.; Kumuduni, A.; Arild, L. Effect of interfacial tension on water/oil relative permeability and remaining saturation with consideration of capillary pressure. Proceedings of the Society of Petroleum Engineering (SPE) EUROPEC/EAGE Annual Conference and Exhibition; Vienna, Austria, May 23−26, 2011; SPE 143028. (12) Li, K.; Horne, R. N. Experimental verification of methods to calculate relative permeability using capillary pressure data. Proceedings of the Society of Petroleum Engineering (SPE) Western Regional/AAPG Pacific Section Joint Meeting; Anchorage, AK, May 20−22, 2002; SPE 76757. (13) Nquyen, V. H.; Sheppard, A. P.; Knackstedt, M. A.; Val, P. W. The effect of displacement rate on imbition relative permeability and residual saturation. J. Pet. Sci. Eng. 2006, 52 (1−4), 54−70. (14) Zhang, J.; Zhu, H.; Li, Y.; Yang, C. Optimization design of multiphase pump impeller based on orthogonal design method. J. China Univ. Pet. 2009, 33 (6), 105−110 (In Chinese). (15) Jiang, H. Q.; Yao, J.; Jiang, R. Z. Principles and Methods of Reservoir Engineering; University of Petroleum Press: Dongying, China, 2006 (In Chinese).

ACKNOWLEDGMENTS The authors greatly appreciate the financial support of the National Natural Science Foundation of China (Grant 10972237), the Important National Science and Technology Specific Projects of China (Grant 2011ZX05011), the Natural Science Foundation for Distinguished Young Scholars of Shandong Province, China (Grant JQ201115), the Program for New Century Excellent Talents in University (Grant NCET-11-0734), and the Programme of Introducing Talents of Discipline to Universities (Grant B08028).

■

NOMENCLATURE M = shape factor of the core sample D = core outer diameter h = core thickness CV = variation coefficient of samples S = standard deviation of samples y ̅ = mean value of samples n = total number of samples yi = value of the ith sample Q = injection rate k = average permeability kro = oil relative permeability krw = water relative permeability μo = oil viscosity μw = water viscosity Δp = displacement pressure difference at both ends of the core sample Rw = inner boundary of the core sample Re = outer boundary of the core sample Ψ = dimensionless parameter pc,max = maximum capillary pressure data

■

REFERENCES

(1) Masihi, M.; Javanbakht, L.; Bahaloo, H. F.; Rasaei, M. R. Experimental investigation and evaluation of three phase relative permeability models. J. Pet. Sci. Eng. 2011, 79 (1−2), 45−53. (2) Hussain, F.; Cinar, Y.; Bedrikovetsky, P. Comparison of methods for drainage relative permeability estimation from displacement tests. Proceedings of the Society of Petroleum Engineers (SPE) Symposium on Improved Oil Recovery; Tulsa, OK, April 24−28, 2010; SPE Paper 129678. (3) 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), 5765−5722. (4) 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. (5) Arhuoma, M.; Yang, D. Y.; Dong, M. Z.; Li, H.; Idem, R. Numerical simulation of displacement mechanisms for enhancing heavy oil recovery during alkaline flooding. Energy Fuels 2009, 23 (12), 5995−6002. (6) Toth, J.; Bodi, T.; Szucs, P.; Civan, F. Determining relative permeability from unsteady-state radial fluid displacements. Proceedings of the Society of Petroleum Engineering (SPE) Annual Technical Conference and Exhibition; Dallas, TX, Oct 9−12, 2005; SPE Paper 94994. (7) Chen, S.; Li, G.; Peres, A.; Reynolds, A. C. A well test for in situ determination of relative permeability curves. SPE Reservoir Eval. Eng. 2008, 11 (1), 95−107. (8) Wang, J. X.; Yang, P. H.; Liu, Q. J.; Guo, H. S. Determination of relative permeability curves using data measured with rate-controlled mercury penetration. J. Univ. Pet. China 2003, 27 (4), 66−69 (In Chinese). 4309

dx.doi.org/10.1021/ef3005866 | Energy Fuels 2012, 26, 4300−4309