# Capillary Pressure and Relative Permeability

Capillary pressure and relative permeability vary by

1 the pore surface properties including wettability, end-point saturations and contact angle, and

2 the net overburden stress effecting the tortuosity, porosity and interconnectivity of pores. Marie (1981) points out that capillary pressure and relative permeability are complicated functions of the properties of the fluids and porous media. By dimensional analysis of an oil-water system in porous media, Marie (1981) has shown that these flow functions can be correlated by means of the pertinent dimensionless groups as:

and

where / is a characteristic dimension of pores, such as the mean pore diameter proportional to -Jk/fy , pj and p2, and |Aj and JI2 denote the densities and viscosities of the fluid phases 1 and 2, respectively, g is the gravitational acceleration, pc is capillary pressure, a is interfacial tension between the fluid phases 1 and 2, 0 is the contact angle, S is the saturation of the fluid phase \,krj denotes the relative permeability of phase j, 7 = 1 for fluid 1 and 2 for fluid 2, and M represents all other characteristics of porous media pertaining to the morphology of pores. In lack of a better approach, frequently, the Leverett (1941) J-function analogy is facilitated to estimate the capillary pressure for an oil/water system during formation damage according to:

where J(SW] is the empirical Leverett J-function, which is a dimensionless function of the water saturation, Sw. Marie (1981) points out that using Equation 4-8 is not rigorously correct because grouping a and 6 as o cos 0 is only valid for cylindrical capillary tubes. Gupta and Civan (1994) have shown that the porous media representative value of the cos 0 term depends on the wettability. The surface tension varies by temperature and species concentration. A quick remedy to apply Equation 4-8 for a nonuniformly-wet porous formation is to define a weighted average of the various wetting fractions of pores as, extending the approach by Cassie and Baxter (1944) and Paterson et al. (1998).

where 0. are the contact angles of the different wetting pore surfaces, a. are the surface fractions of different wetting pores, defined by McDougall and Sorbie (1995). As a simplistic approach, assuming that the Leverett J-function remains unchanged during formation alteration, Equation 4-8 can be applied at a reference state and denoted by subscript "0" and at an instantaneous state during formation damages to obtain:

for which Kl§ can be estimated using one of the methods, such as by the Carman-Kozeny equation. Ajufo et al. (1993) have demonstrated that the capillary pressure data is sensitive to overburden pressure. In poorly sorted and cemented formations, the effect of overburden may create an irreversible decay of the formation integrity. Frequently, the capillary pressure and relative permeability data are correlated by Corey type power law empirical expressions of the normalized saturation given, respectively, by (Mohanty et al., 1995):

where Gjo is the interfacial tension of the jth fluid phase with oil, k^ is the permeability at the end-point saturation of the jth phase, bj and «; are some correlation exponents, and ~Sj is the normalized saturation of the jth phase defined as:

Chang et al. (1997) have resorted to Sigmund and McCaffery (1979) type formulae to represent relative permeabilities, which can be generalized as:

where ra; and a,- are some empirical parameters. Chang et al. (1997) have used the following expression to represent the capillary pressure function:

where F is a scaling factor for the capillary pressure and (3; is an empirical parameter.

Donaldson et al. (1987) propose a hyperbolic expression for capillary pressure as:

where A, 5, and C are some correlation parameters. During formation damage the wettability index and the capillary pressure and relative ermeability curves vary continuously. Therefore, it is reasonable to assume that the parameters of Eqs. 4-14, 15 and 16 can be correlated with respect to the wettability index to obtain dynamic correlations.

McDougall and Sorbie (1995) demonstrate the effect of wettability on capillary pressure and relative permeability. Wang (1988) shows the effect of a wettability alteration on imbibition relative permeability. Tielong et al. (1996) have demonstrated that the oil and water relative permeabilities of cores before and after polymer treatment can be correlated by Eq. 4-12. Tielong et al. (1996) determined the values of the exponents of Eq. 4-12 before and after polymer treatment and showed that they varied. However, they did not determine the exponent values at various intervals during polymer treatment. Therefore, a correlation cannot be derived from their data. Neasham (1977) studied the affect of the morphology of dispersed clay on fluid flow properties in sandstone cores. Neasham present the mineralogical, petrographical and petrophysical properties of the sandstones tested. Neasham demonstrates that different sandstones indicate significantly different capillary pressure behavior.

## References

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2 Amott, E., "Observations Relating to the Wettability of Porous Rock," Trans. A/ME, Vol. 216, 1959, pp. 156-162.

3 Cassie, A. B. D., & Baxter, S., "Wettability of Porous Surfaces," Trans. Faraday Soc., Vol. 40, 1944, pp. 546-551.

4 Chang, Y. C., Mohanty, K. K., Huang, D. D., & Honarpour, M. M., "The Impact of Wettability and Core-Scale Heterogeneities on Relative Permeability," /. of Petroleum Science and Engineering, Vol. 18, Nos. 1/2, 1997, pp. 1-19.

5 Civan, R, & Donaldson, E. C., "Relative Permeability from Unsteady- State Displacements: An Analytical Interpretation," SPE Paper 16200, Proceedings of the SPE Production Operations Symposium held in Oklahoma City, Oklahoma, March 8-10, 1987, pp. 139-155.

6 Collins, E. R., Flow of Fluids Through Porous Materials, Penn Well Publishing Co., Tulsa, Oklahoma, 1961, 270 p.

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8 Donaldson, E. C., "Use of Capillary Pressure Curves for Analysis of Production Well Formation Damage," SPE 13809 paper, Proceedings of the SPE Production Operations Symposium held in Oklahoma City, Oklahoma, March 10-12, 1985, pp. 157-163.

9 Donaldson, E. C., & Crocker, M. E., "Characterization of the Crude Oil Polar Compound Extract," DOE/BETC/RI-80/5, NTIS, Springfield, Virginia 22161, 1980, 27 p.

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13 Ertekin, T., & Watson, D., EOR, DOE/BC-90/4 Progress Review, September 30, 1990.

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16 Hirasaki, G. J., "Wettability: Fundamentals and Surface Forces," SPE Formation Evaluation, June 1991, pp. 217-226.

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23 McDougall, S. R., & Sorbie, K. S., "The Impact of Wettability on Waterflooding: Pore Scale Simulation," SPE Reservoir Engineering, August 1995, pp. 208-213.

24 Mohanty, K. K., Masino Jr., W. H., Ma, T. D., & Nash, L. J., "Role of Three-Hydrocarbon-Phase Flow in a Gas-Displacement Process," SPE Reservoir Engineering, August 1995, pp. 214-221.

25 Neasham, J. W., "The Morphology of Dispersed Clay in Sandstone Reservoirs and Its Effect on Sandstone Shaliness, Pore Space and Fluid Flow Properties," SPE 6858, Proceedings of the 52nd Annual Fall Technical Conference and Exhibition of the SPE of AIME held in Denver, Colorado, October 9-12, 1977, pp. 184-191.

26 Paterson, A., Robin, M., Fermigier, M., Jenffer, P., and Hulin, J. P., "Effect of Density and Spatial Distribution of Wettability Heterogeneities on Contact Angle," Journal of Petroleum Science and Engineering, Vol. 20, No. 3/4, pp. 127-132, 1998.

27 Robin, M., Rosenberg, E., & Fassi-Fihri, O., "Wettability Studies at the Pore Level: A New Approach by use of Cryo-SEM," SPE Formation Evaluation, March 1995, pp. 11-19.

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29 Sigmund, P. M., & McCaffery, F. G., "An Improved Unsteady-State Procedure for Determining the Relative Permeability Characteristics of eterogeneous Porous Media," Soc. Pet. Eng. J., Vol. 19, 1979, pp. 15-28.

30 Tielong, C., Yong, Z., Kezong, P., & Wanfeng, P., "Experimental Studies and Field Trials of Relative Permeability Modifier for Water Control of Gas Wells in Low Permeability Reservoir," SPE Paper 35617, Proceedings of the SPE Gas Technology Conference held in Calgary, Alberta, Canada, April 28-May 1, 1996, pp. 385-392.

31 Wang, F. H. L., "Effect of Wettability Alteration on Water/Oil Relative Permeability, Dispersion, and Flowable Saturation in Porous Media," SPE Reservoir Engineering, May 1988, pp. 617-628.

32 Yan, J., Plancher, H., & Morrow, N. R., "Wettability Changes Induced by Adsorption of Asphaltenes," SPE paper 37232, Proceedings of the 1997 SPE International Symposium on Oilfield Chemistry held in Houston, TX, February 18-21, 1997, pp. 213-227.