Comparing two new transformations for the water content form of Richards' Equation

Abstract

This study will present two new transformations that are applied to the water content (θ) form of Richard's Equation: 1) a reference soil transformation and 2) a constant “relative gradient ratio” transformation. The reference soil transformation enables a model to simulate water flow in a given soil in terms of θ for a reference soil. The main advantage of this transformation is that it provides a continuous θ profile across soil layers. However, the choice of a reference soil could also provide some numerical advantages by, for example, minimising the effect of sharp wetting fronts. An analytical expression will be derived that approximates the effect a transformation has on the gradients within a system. This approximation is based on the notion of relative gradients and will assist in analysing, which reference soils are appropriate for the problem. In addition, the analytical approximation can be used to derive general transformations for partial differential equations (PDEs). For this study, we will derive a constant “relative gradient ratio” transformation that will decrease relative gradients within the system by a constant factor (ε). Using the Method of lines (MOL), numerical solutions will be generated for a test case from the literature for both transformed and non-transformed equations. By comparing these solutions, this paper will analyse whether the new transformations improve the efficiency and/or accuracy of the numerical solution for the test case problem.