Numerical simulation of Richards equation in partially saturated porous media: under-relaxation and mass balance

Numerical simulation of Richards equation in unsaturated soil is known to be difficult because of the highly non-linear material properties involved. An extensive study was conducted to clarify the role of mass balance and under-relaxation on the rate in which the correct solution is approached. Both finite element and finite difference techniques were considered. For the former, the h-based formulation was compared with a mass-conservative mixed form. The conductivity function (K) was under-relaxed in two ways while the capacity function (C) is computed following the standard mass or non-mass conservative schemes recommended in literature. For fairly coarse discretisation, it was found that large errors were produced when K was under-relaxed by evaluating it at the average of heads from current and previous time step (UR1), regardless of the numerical scheme used. Maintaining global mass balance is found to have little impact on the accuracy. All numerical schemes that under-relaxed K by computing it at the average of two most recent iterations in the current time step (UR2) converged quicker to the correct solution with increasing discretisation, although more iterations per time step than UR1 is needed to achieve a stable solution. An important practical ramification is that it appears to be possible to achieve reasonably accurate and oscillation-free results using fairly coarse discretisation by making only minor modifications (namely, using UR2 for conductivity function) to the h-based finite element formulation and applying some minimum time step criteria.