Capillary pressure in a porous medium with distinct pore surface and pore volume fractal dimensions

The relationship between capillary pressure and saturation in a porous medium often exhibits a power-law dependence. The physical basis for this relation has been substantiated by assuming that capillary pressure is directly related to the pore radius. When the pore space of a medium exhibits fractal structure this approach results in a power-law relation with an exponent of 3-D-v, where D-v is the pore volume fractal dimension. However, larger values of the exponent than are realistically allowed by this result have long been known to occur. Using a thermodynamic formulation for equilibrium capillary pressure we show that the standard result is a special case of the more general exponent (3-D-v)/(3-D-s) where D-s is the surface fractal dimension of the pores. The analysis reduces to the standard result when D-s=2, indicating a Euclidean relationship between a pore's surface area and the volume it encloses, and allows for a larger value for the exponent than the standard result when D-s>2.