BIT Numerical Mathematics

, Volume 51, Issue 4, pp 845–864

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


    • Queensland University of Technology
  • Tim Moroney
    • Queensland University of Technology
  • Ian Turner
    • Queensland University of Technology

DOI: 10.1007/s10543-011-0335-3

Cite this article as:
Cumming, B., Moroney, T. & Turner, I. Bit Numer Math (2011) 51: 845. doi:10.1007/s10543-011-0335-3


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.


Finite volumeRichards’ equationDAE

Mathematics Subject Classification (2000)

Download to read the full article text

Copyright information

© Springer Science + Business Media B.V. 2011