Permeability model for fractal porous media with rough surfaces

Abstract

In this paper, a permeability model is derived for porous media, which are assumed to be comprised of a bundle of tortuous capillaries whose size distribution and roughness of surfaces follow the fractal scaling laws. The proposed model includes the effects of the fractal dimensions for size distributions of capillaries, for tortuosity of tortuous capillaries and for roughness of surfaces on the permeability. The analytical expression for permeability is a function of the relative roughness, the fractal dimensions for tortuosity and sizes of capillaries and for roughness of surfaces, as well as the microstructural parameters (such as the characteristic length, the maximum and minimum pore diameters and the fractal dimensions). The proposed model can properly reveal some mechanisms that affect the permeability. Every parameter in the proposed model has specific physical meaning. The ratio of the permeability ( $$ K_{\text{R}} $$ ) for rough capillaries to that (K) for smooth capillaries is found to be a function of the relative roughness and follows the quadruplicate power law, i.e., $$ K_{\text{R}} /K = (1 - \varepsilon )^{4} $$ , where $$ \varepsilon $$ is the relative roughness of surfaces of capillaries.