On equivalent hydraulic conductivity for oscillation–free solutions of Richard’s equation

Abstract

The estimation of numerical equivalent conductivity remains a crucial issue for the accuracy and stability of the solution of the nonlinear Richards' equation (RE) when modeling variably saturated flow. In the literature, it appears that this topic has been typically considered for one-dimensional discretization despite the growing interest in multidimensional problems. After reviewing different possibilities of equivalent hydraulic conductivity estimation, we evaluate their ability to yield monotonic results. Hence, the monotonicity analysis provided by Forsyth and Kropinski (1997) has been generalized for the different equivalent conductivity formulations. On one hand, the upstream mean is unconditionally stable but is also known to overestimate the conductivity. On the other hand, other formulations, including Darcian mean approximations, can be accurate and straightforward to adapt in multidimensional codes but do not always provide monotonic solutions of the RE. An adaptive algorithm is presented, which adapts the conductivity in function of the monotonicity condition, i.e., a variable criterion based on the conductivity at nodal points, the conductivity averaging technique and the piezometric head variation. The proposed numerical method can be implemented in existing multidimensional codes. Numerical investigations in steady state and time-varying conditions, 1D and 2D cases, and homogeneous and heterogeneous media confirm the interest in the proposed algorithm.

Modeling Variably Saturated Flow in Stratified Soils With Explicit Tracking of Wetting Front and Water Table Locations

“…In a homogeneous soil the water flux between z u and z l is approximated by where Δ z = z l − z u ; h u = h ( z u ), h l = h ( z l ). Evaluation of q hm is crucial for the stability of the numerical solution of Richards' equation (Belfort et al, ). Equation is usually written in the form of the product of equivalent hydraulic conductivity k hm and potential gradient as …”

Second-order accurate finite volume schemes with the discrete maximum principle for solving Richards’ equation on unstructured meshes

“…We consider the rectangular computational domain Ω = [0, 2] × [0, 10] and set g = 2, ρ = µ = 1, and p atm = 0. The computational domain is divided into equal subdomains Ω i , i = 1, 2, as follows: [10,5] and Ω 2 = [0, 2] × [5,0]. The permeability tensor is a constant function in each subdomain, K(x, z) = 0.5 in Ω 1 and K(x, z) = 2 in Ω 2 .…”

Discontinuous Galerkin Method for Steady-State Richards Equation

“…The latter takes into account the actions of gravity (advection) and capillarity (diffusion) but neglecting the flow of the nonwetting phase, namely the air [7]. Richards equation has been widely used to simulate water flow in unsaturated porous media [8][9][10]. Nevertheless, its derivation is generally rather roughly introduced, which leads us to first set out the main steps to underline some issues in regards to our model problem.…”