A mass-conservative control volume-finite element method for solving Richards’ equation in heterogeneous porous media

Abstract

We present a mass-conservative vertex-centred finite volume method for efficiently solving the mixed form of Richards' equation in heterogeneous porous media. The spatial discretisation is particularly well-suited to heterogeneous media because it produces consistent flux approximations at quadrature points where material properties are continuous. Combined with the method of lines, the spatial discretisation gives a set of differential algebraic equations amenable to solution using higher-order implicit solvers. We investigate the solution of the mixed form using a Jacobian-free inexact Newton solver, which requires the solution of an extra variable for each node in the mesh compared to the pressure-head form. By exploiting the structure of the Jacobian for the mixed form, the size of the preconditioner is reduced to that for the pressure-head form, and there is minimal computational overhead for solving the mixed form.The proposed formulation is tested on two challenging test problems. The solutions from the new formulation offer conservation of mass at least one order of magnitude more accurate than a pressure head formulation, and the higher-order temporal integration significantly improves both the mass balance and computational efficiency of the solution.

A force‐balanced control volume finite element method for multi‐phase porous media flow modelling

“…Cumming et al . demonstrated that a CVFEM‐based discretisation could be used to solve the Richards equation (coupled mass conservation and Darcy equations) in heterogeneous porous media with relatively small computational overhead, compared with traditional, coupled velocity pressure‐based formulations. Mass balance was enforced as described by Kirkland et al .…”

Implicit and semi-implicit second-order time stepping methods for the Richards equation

“…We propose implicit and semi-implicit time stepping discretizations for each of the formulations (3.4), (3.6) and (3.9). Few second order accurate time-stepping schemes were proposed for Richards equation based on the Crank-Nicolson method [19,79,80] and BDF methods [7,26] in their fully implicit form requering Newton or Picard iterations. The Crank-Nicolson method is A-stable but lacks L-stability, and may lead to non-monotone solutions for larger time steps.…”

An adjoint-based a posteriori analysis of numerical approximation of Richards equation

“…Space-time finite element methods have been previously used for parabolic equations (see for example [20]) and for reaction-diffusion system (see for example [15]). A recent work on application of control volume finite element in combination with method of lines for solving Richards Equation is recorded in [10]. To the best of the author's knowledge, there has not been any attempt to apply space-time finite element methods technique to Richards Equation.…”