Effective conductivity of conduit networks with random conductivities

Abstract: The effective conductivity ($T^{eff}$) of 2D and 3D Random Resistor Networks (RRNs) with random edge conductivity are studied. The combined influence of geometrical disorder, which controls the overall connectivity of the medium, and leads to percolation effects, and conductivity randomness is investigated. A formula incorporating connectivity aspects and second-order averaging methods, widely used in the stochastic hydrology community, is derived and extrapolated to higher orders using a power averaging formula based on a mean-field argument. This approach highlights the role of the so-called resistance distance, introduced by graph theorists. Simulations are performed on various RRN geometries constructed from 2D and 3D bond-percolation lattices. The results confirm the robustness of the power averaging technique and the relevance of the mean-field assumption.