RESUMO
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 .
RESUMO
Defining a relation for equilibrium pressure in a porous medium has been difficult to do in terms of readily measurable parameters. We present a simplified analysis of this problem using the first law of thermodynamics combined with a fractal description of a porous system. The results show that the variation in fluid interfacial area with fluid volume, and the respective interfacial surface tensions, are dominant factors determining equilibrium capillary pressure. Departures from equilibrium are seen to occur when fluid-solid contact lines are in movement. By describing the pore space as fractal we are able to obtain an expression for the change in fluid interfacial area with respect to its volume, and the resulting model shows a strong fit to pressure data obtained from a capillary rise experiment conducted in a coarse-grained SiO2 sand.
