Numerical investigation of flow instabilities using fully unstructured discretization for variably saturated flow problems

Abstract

Richards equation for variably saturated flow can exhibit stability problems due to its nonlinearity. The challenge is to resolve sharp wetting fronts without introducing spurious oscillations, especially for simulations with very dry initial conditions. Flow instabilities at sharp wetting fronts may arise particularly in anisotropic and heterogeneous media, leading to oscillations or convergence problems. The focus of this work is to evaluate and minimize instability problems for simulations with unstructured meshes. To this end, numerical experiments were performed to investigate the accuracy, monotonicity and convergence behavior of numerical solutions for variably saturated flow based on different control volume methods, piecewise gradient reconstruction methods, and flux approximation methods. In particular, nonphysical oscillations at the wetting front were investigated. A novel multi-point flux approximation with multi-point upstream weighting based on piecewise gradient reconstruction was developed. Numerical simulations in both homogeneous and heterogeneous domains with isotropic and anisotropic conductivity tensors demonstrate that the proposed multi-point flux approximation with multi-point upstream weighting avoids spurious oscillations and improves the results for challenging sharp wetting front problems using general unstructured meshes as well as meshes with distortion. The revised Voronoi-dual control volume method has also been found to provide more flexibility than commonly used center-dual and median-dual control volume methods.