Continuum percolation theory and Archie's law

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Abstract
Results for the unsaturated hydraulic conductivity, K, are adapted to generate the electrical conductivity, σ, as a function of moisture content, θ. The result for σ, evaluated θ = ϕ, (ϕ the porosity), yields σ ∝ ϕm, known as Archie's Law. This result is based on the fact that σ must vanish as the critical moisture content for percolation, θt, is approached, and on the apparent proportionality of θt to ϕ, at least in coarser porous media (with insignificant clay content). In three dimensions the power is expected to be 1.88, but in two dimensions, approximately 1.35. The former result is compatible with data summarized in Thompson et al., giving m = 1.86 ± 0.19, while the latter appears compatible with two‐dimensional simulations by Kuentz et al., who found m = 1.28 ± 0.07. The same analysis demonstrates that K cannot be written as a power of the porosity.