Asymptotic Behaviour of Wetting Fronts in Porous Media with Exponential Moisture Diffusivity

Abstract. We investigate the asymptotic behaviour of the wetting front arising from unsaturated flow in porous media wherein the liquid saturation θ obeys a nonlinear diffusion equation having moisture diffusivity that is an exponential function D(θ) = Do exp(βθ). For values of the physical parameters corresponding to actual porous media, the diffusivity at the residual value of saturation is D(0) ≪ 1, so that the problem is nearly degenerate. The novelty of our approach stems from our aim to derive a solution that is valid throughout the wetting front, in contrast with most other work that cuts off the solution at the wetting front location. A matched asymptotic analysis indicates that the solution has a three-layer structure, and also provides an estimate of the wetting front location in terms of the material parameters describing the porous medium. Using numerical simulations of the original problem, we demonstrate that the first few terms in our series solution give an accurate approximation of physically relevant quantities such as the wetting front location and speed of propagation, which improves on other asymptotic solutions reported in the literature.