Abstract
Abstract The estimation of numerical equivalent conductivity remains a crucial issue for the accuracy and stability of the solution of the non-linear 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.
Highlights
Modeling water flow in variably saturated soils is of great interest to many scientific research and engineering applications involved in the management of water resources
When numerical methods are used to solve a physical problem modeled by the Richards’ equation (RE), the differential equation is integrated over the solution domain Ω, which is decomposed into a set of non-overlapping smaller subdomains Ωe
This study focuses on the issue of monotonicity when solving variably saturated flow problems modeled by the non-linear RE
Summary
Modeling water flow in variably saturated soils is of great interest to many scientific research and engineering applications involved in the management of water resources. Many studies dealing with RE focus on the numerical expression of the mass matrix for time-dependent problems and conclude that the diagonalized (or lumped) form is preferred to avoid oscillations (Neuman, 1972; Cooley, 1983; Milly, 1985; Celia et al, 1990; Pan et al, 1996; Ju and Kung, 1997) In this context, the M-matrix property (i.e., a non-singular matrix with positive diagonal and negative off-diagonal coefficients) is often used to establish conditions that ensure consideration of the maximum principle (Windisch, 1989; Wood, 1996; Thomas, 1997; Hoteit, 2002; Belfort and Lehmann, 2005; Younes et al, 2006). The main objectives of this study are as follows: 1) review the different estimations of equivalent conductivity, 2) analyze their ability to yield monotonic results from a mathematical viewpoint, 3) test a new switching algorithm for a multidimensional implementation and 4) study the efficiency of the different averaging techniques by considering several 1D and 2D test cases
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