Hyperdispersive flow of liquid thin films in fractal porous media

Abstract

Recent displacement experiments show anomalously rapid spreading of water 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. At high capillary pressure P c , the total wetting phase saturation S w is the sum of wetting phases in thin films S tf and in pendular structures S ps . In many cases, the disjoining pressure Π is inversely proportional to a power m of the film thickness h, i.e. Πh -m , so that S tf P c -1/m . The contribution of fractal pendular structures to wetting phase saturation often obeys a power law S ps P c (3-D) , where D is the Hausdorff or fractal dimension of the pore wall roughness. Hence, if wetting phase inventory is primarily pendular structures, and if thin films control the hydraulic resistance of wetting phase, then the capillary dispersion coefficient obeys D c S v w , where v=[3-m(4-D)]/m(3-D). The spreading is hyperdispersive, i.e. D c (S w ) rises as wetting phase saturation approaches zero, if m>3/(4-D), hypodispersive, i.e. D c (S w ) falls as wetting phase saturation tends to zero, if m<3/(4-D), and diffusion-like if m=3/(4-D). Asymptotic analysis of the capillary diffusion equation indicates hyperdispersive behavior for -2 <v<0. In addition, we also predict the values of D c by Monte Carlo simulation in porous media which are idealized as networks of pore segments.