Hydraulic Conductivity of Porous Media at Low Water Content

Matric potential ψ and hydraulic conductivity K at low water content θ often obey power laws in θ, but the exponents of these are largely empirical. Theories of fractal geometry and of thin‐film physics provide a basis for the observed power‐law behavior of ψ and K . Specifically, they lead to ψ ∝ θ −1/(3‐ D ) and K ∝ θ 3/ m (3 − D ) , where D is the Hausdorff dimension of the surface between the pore space and grains or matrix, and m is the exponent in the relation of disjoining pressure II and film thickness h , i.e., II ∝ h −m . These power laws may increase the reliability of extrapolating measurements of ψ and K at low θ. Using the data of Nimmo and Akstin (1988) to test our ideas, we found that, in the case of water in soils, m < 1 and, across length scales between 5 µm and 20 µm, 2.1 < D < 2.7. In the limit of smooth pore walls, D = 2. The measured hydraulic conductivities lie between upper and lower bounds of K (θ) that we computed using three trial distributions of pore radius.