Multilayer Asymptotic Solution for Wetting Fronts in Porous Media with Exponential Moisture Diffusivity

Abstract

We study the asymptotic behaviour of sharp front solutions arising from the nonlinear diffusion equation θ t = (D(θ)θ x ) x , where the diffusivity is an exponential function D(θ) = D o exp(βθ). This problem arises for example in the study of unsaturated flow in porous media where θ represents the liquid saturation. For physical parameters corresponding to actual porous media, the diffusivity at the residual saturation is D(0) = D o ≪ 1 so that the diffusion problem is nearly degenerate. Such problems are characterised by wetting fronts that sharply delineate regions of saturated and unsaturated flow, and that propagate with a well-defined speed. Using matched asymptotic expansions in the limit of large β, we derive an analytical description of the solution that is uniformly valid throughout the wetting front. This is in contrast with most other related analyses that instead truncate the solution at some specific wetting front location, which is then calculated as part of the solution, and beyond that location the solution is undefined. Our asymptotic analysis demonstrates that the solution has a four-layer structure, and by matching through the adjacent layers we obtain an estimate of the wetting front location in terms of the material parameters describing the porous medium. Using numerical simulations of the original nonlinear diffusion equation, we demonstrate that the first few terms in our series solution provide approximations of physical quantities such as wetting front location and speed of propagation that are more accurate (over a wide range of admissible β values) than other asymptotic approximations reported in the literature.

Asymptotical computations for a model of flow in saturated porous media

“…The computations which enable to provide the mentioned results are only possible due to this highly flexible, adaptive finite difference method which is able to cope with extremely unsmooth step-size sequences. The numerical method is sufficiently good to not only confirm the asymptotical calculations in [9] but to also give a clear indication of the structure of further terms, which are currently beyond the reach of theoretical analysis.…”

“…The purpose of this paper is to make a numerical study of the solutions of (3)-(5) in the limit of large γ which corresponds to a problem with β ≫ 1 with large diffusion when u is not small. The motivation for this investigation is to study a series of refined asymptotic estimates developed in [9] which significantly improve the earlier estimates. A second motivation is that the extreme nature of the problem and the existence of true asymptotical results gives an important test and validation of the numerical method described in this paper.…”

“…Indeed, if y < y * then θ = O(1) and if y > y * then θ = O(e −γ 2 ). An asymptotic study of this problem for large values of γ is given in [9], the results of which are summarised in Section 1. One purpose of this paper is to use an accurate high order numerical method to examine the asymptotic predictions of the latter paper and to accurately determine both the location of the wetting front and the value of θ ∞ .…”

“…This problem poses a significant numerical challenge. Whilst the error terms in the asymptotic expansions (say of y * ) decay polynomially in γ as γ increases, it is shown in [9] that the curvature and final value are exponential in γ, that is θ yy (y * ) ∼ e γ 2 and θ ∞ ∼ e −γ 2 .…”

“…The numerical challenge is thus to find discretization schemes and adaptive mesh selection methods which are robust with respect to the dramatic variation in the stepsizes (up to 140 orders of magnitude in our computations reported below) which may be expected due to the predicted solution behavior as the curvature of θ rapidly increases. Earlier attempts in this direction have not been fully satisfactory, as the problem could only be solved for values of γ remote from the asymptotical regime [9].…”