Scaling Invariant Effects on the Permeability of Fractal Porous Media

Porous media are interconnected systems, in which the distribution of pore sizes might follow scaling invariant property and will affect the fluid flow through it significantly. Thus, except for a detailed understanding of the fundamental mechanism at pore scale, hard is it to determine the appropriate relationship between the permeability and the basic properties. In this study, in terms of the size distribution and spatial arrangement of the pores, we analytically derived a permeability model using series-parallel flow resistance mode firstly. And then, together with the scaling invariant characteristics of the porosity, specific area and hydraulic tortuosity, the analytical permeability model is reformulated into a fractal permeability-pore structure relationship. The results indicate that: (1) the square of the porosity () is proportional to the permeability in a fractal porous media, not the cubic law described in Kozeny-Carman (KC) equation; (2) the hydraulic tortuosity is a power law model of the minimum particle size with the exponent , where and d are the pore size fractal dimension and Euclidean space dimension, respectively, while is a parameter characterizing the spatial arrangement of pores; (3) the KC numerical prefactor is not a constant in fractal porous media. Its value, however, increases linearly with the size ratio of the minimum to the maximum pores but decreases exponentially with . More importantly, it is found to be a parameter characterizing the difference of fluid flow in porous media from that in a straight tube described by the Poiseuille law. The performance of the new fractal permeability-pore model is verified by lattice Boltzmann simulations, and the numerical prefactor universality is examined as well.