Motion of wetting fronts moving into partially pre-wet soil
Introduction
We consider the problem of vertical infiltration of water into a semi-finite layer of homogeneous soil. Specifically, we study the motion of the wetting front marking the leading edge of the infiltrating flow as it advances into dry or nearly-dry regions in the soil. Our goal is to construct a simple approximate solution that can accurately describe the position and moisture distribution at the wetting front while incorporating the influences of boundary effects and pre-existing initial conditions in the soil.
This problem will be analyzed in terms of Richards’ equation as a model for the one-dimensional vertical infiltration and redistribution of moisture content, θ(z, t), in semi-infinite layers, 0 ⩽ z < ∞, of unsaturated soils,where K(θ) is the conductivity and D(θ) is the diffusivity of the soil. This model for water transport neglects any evaporation and hysteresis that may occur in the soil. As such, Richards’ equation is a simplified and idealized model, but nonetheless has been very useful and widely studied in hydrology.
Influenced and motivated by many of the works of Parlange [11], [18], [19], [20], [22] on problems describing the behavior of wetting fronts, my recent work presented asymptotic analysis of the interaction of wetting fronts with impervious boundaries [32], [34]. This article describes the influences of surface boundary conditions and initial conditions on wetting fronts. We examine (1.1) with a boundary condition describing a constant moisture concentration (normalized to value one) at the surface of the soil,
The soil layer is assumed to be initially dry or nearly dry. That is, at t = 0 initial conditions are moisture distributions on 0 < z < ∞ with much lower concentrations than the infiltrating flow,
We expect that the problem with with constant flux boundary conditions [7], [26], Q = K(θ) − D(θ)∂zθ at z = 0, replacing (1.2) could be studied using the same approach and would yield similar results.
While the redistribution of finite volumes of water [10] and other classes of problems are described by more complicated solutions, we show that for problems described by (1.2), (1.3) the solutions approach traveling waves for long times. Using the cumulative infiltration, we show that the influences of (1.2), (1.3) must be taken into account to accurately predict the position of the wetting front.
Section snippets
The linear case: the shifted wetting front for the advection–diffusion equation
We begin by examining the solution of the initial-boundary value problem for the linear version of Richards’ equation, the advection–diffusion equation. This is a classical model for transport processes [5], [15] and allows for construction of exact solutions to problems via various mathematical techniques [8], [14], [15]. Careful study and interpretation of the solution to this elementary problem will be used to gain insight into properties that extend to the nonlinear problem for Richards’
Shifted wetting fronts for Richards’ equation
Having thoroughly examined the behavior of wetting fronts in the initial-boundary value problem for the linear advection–diffusion equation, we study the corresponding problem for Richards’ equation. While most of the fundamental ideas described above will carry over from the linear to the nonlinear problem, some of the analysis becomes more complicated in the nonlinear case, but surprisingly some of the results will become more clear.
We will focus on Richards’ equation for Brooks–Corey model
Acknowledgment
I thank the reviewers for many helpful suggestions on improving the presentation in the article.
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