Applications of percolation theory to porous media with distributed local conductances

Abstract

Critical path analysis and percolation theory are known to predict accurately dc and low frequency ac electrical conductivity in strongly heterogeneous solids, and have some applications in statistics. Much of this work is dedicated to review the current state of these theories. Application to heterogeneous porous media has been slower, though the concept of percolation was invented in that context. The definition of the critical path is that path which traverses an infinitely large system, with no breaks, which has the lowest possible value of the largest resistance on the path. This resistance is called the rate-limiting, or critical, element, R c. Mathematical schemes are known for calculating R c in many cases, but this application is not the focus here. The condition under which critical path analysis and percolation theory are superior to other theories is when heterogeneities are so strong, that transport is largely controlled by a few rate-limiting transitions, and the entire potential field governing the transport is influenced by these individual processes. This is the limit of heterogeneous, deterministic transport, characterized by reproducibility (repeatability). This work goes on to show the issues in which progress with this theoretical approach has been slow (in particular, the relationship between a critical rate, or conductance, and the characteristic conductivity), and what progress is being made towards solving them. It describes applications to saturated and unsaturated flows, some of which are new. The state of knowledge regarding application of cluster statistics of percolation theory to find spatial variability and correlations in the hydraulic conductivity is summarized. Relationships between electrical and hydraulic conductivities are explored. Here, as for the relationship between saturated and unsaturated flows, the approach described includes new applications of existing concepts. The specific case of power-law distributions of pore sizes, a kind of “random” fractal soil is discussed (critical path analysis would not be preferred for calculating the hydraulic conductivity of a regular fractal).