Effects of percolation on the effective conductivity of irregular composite porous media

We develop a phenomenological model for the effective conductivity of porous media that consist of two distinct materials characterized by different values of hydraulic conductivity. Our focus is two-fold: first, to relate the terms and parameters of macroscale conductivity to the spatial variability of local flow and conductivity observed in computational experiments, and second to develop a reduced order model for effective conductivity based on those observations. Darcys law is assumed to hold at local (mesoscopic) and large (macroscopic) scales. At the mesoscale a composite medium is a configuration of irregular subdomains consisting of two different materials. The resulting heterogeneities force fluid to flow along irregular paths, and this produces spatial variability in the magnitude and the orientation of the Darcy flux. This variability, along with the macroscopic effective conductivity, depends on the proportion of the total volume allocated to each material, the ratio of the two conductivity values, and the spatial connectivity of material subvolumes. Computational experiments indicate that the effects of heterogeneity are most pronounced when the two conductivity values are very different, and when the volume fraction of the more conductive material is near a percolation threshold. The percolation threshold is the critical volume fraction of the more conductive material at which the medium is no longer traversed by connected paths through that material. The percolation threshold determines the existence of three regimes in effective conductivity, two where the effective conductivity obeys power laws and is dominated by one of the materials, and a third intermediate regime that interpolates between the power laws.