Fluids in fractal porous media: scaling of transport properties

This work describes recent progress in modeling transport properties of natural porous media at low saturations of a wetting phase, i.e. when total wetting phase saturation Sw is the sum of thin-films and pendular structures inventories. Capillary pressure Pc, hydraulic conductivity Kw, electrical conductivity σw, and the capillary dispersion coefficient Dc have been observed to obey power laws in the wetting phase saturation. We relate power-law behavior at low wetting phase saturations, i.e. at high capillary pressures, to the thin-film physics of the wetting phase and the fractal character of the pore space of natural porous media. If wetting phase inventory is primarily pendular structures, and if thin films control the hydraulic resistance of wetting phase we deduce the power laws X = SbXw, with X = Pc, Kw, σw and Dc, where for capillary pressure bPc = -1/(3 − D), for hydraulic conductivity bKw = 3/m(3 − D), for electrical conductivity bσw = 1/m(3 − D), and for capillary dispersion coefficient bDc = [3 − m(4 − D)]/m(3 − D), where m is the exponent in the relation of disjoining pressure and film thickness and D is the fractal dimension of the surface between the pore space and solid matrix. Recent experimental work lends support to these scaling laws in the cases of natural sandstones and clayey soils. Recent displacement experiments show anomalously rapid spreading of wetting liquid during imbibition into a prewet porous medium. We explain this phenomenon, called hyperdispersion, as viscous flow along fractal pore walls in thin films of thickness h governed by disjoining forces and capillarity. Asymptotic analysis of the “capillary diffusion” equation indicates hyperdispersive behavior for -2<bDc <0.