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

• The issue of oscillations in the solutions of Richards’ equation is discussed.
• Different formulations of equivalent conductivity are presented and tested.
• Mathematical and numerical analysis of monotonicity are performed.
• The upstream mean and Szymkiewicz’s algorithm are the most efficient solutions.
• A switching algorithm based on monotonicity test is developed.

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.