Error Estimates for a Time Discretization Method for the Richards’ Equation

We present a numerical analysis of an implicit time discretization method applied to Richards’ equation. Written in its saturation-based form, this nonlinear parabolic equation models water flow into unsaturated porous media. Depending on the soil parameters, the diffusion coefficient may vanish or explode, leading to degeneracy in the original parabolic equation. The numerical approach is based on an implicit Euler time discretization scheme and includes a regularization step, combined with the Kirchhoff transform. Convergence is shown by obtaining error estimates in terms of the time step and of the regularization parameter. 1. Physical motivation Richards’ equation (see, for example, [4]) is a popular model for saturated-unsaturated flow in porous media. Assuming a constant air pressure in the medium, two physical quantities are unknown in this equation, the saturation and the pressure in the fluid phase. Depending on the regime of flow (unsaturated, or completely saturated), one has to decide which of the two is the primary unknown. This leads to three main forms of the Richards’ equation, saturation based, pressure based, or mixed. Here we are interested in the unsaturated case. If the porous medium is wetted by a liquid (water) of density ρ, Darcy law relates the flow velocity q to the permeability of the medium K and the pressure inside the fluid Ψ, (1.1) q = −K∇(Ψ + z), with z denoting the vertical coordinate in the medium. The continuity condition ∂t(ρΦ) +∇ · (ρq) = 0 combined with Darcy law (1.1) leads to Richards’ equation (1.2) ∂t(ρΦ)−∇ · (ρK∇(Ψ + z)) = 0. Assuming the wetting phase has a constant density ρ, the equation above contains two unknowns, namely the saturation Φ and the pressure Ψ. Moreover, in case the medium is not completely saturated, the permeability K depends on the saturation of the medium. Based on experimental results, different retention curves relating the permeability and the pressure to the saturation have been proposed in the literature. Taking into account these relationships we can re-formulate equation (1.2) in terms of the dimensionless fluid