Upstream mobility finite volumes for the Richards equation in heterogenous domains

Abstract

This paper is concerned with the Richards equation in a heterogeneous domain, each subdomain of which is homogeneous and represents a rocktype. Our first contribution is to rigorously prove convergence toward a weak solution of cell-centered finite-volume schemes with upstream mobility and without Kirchhoff’s transform. Our second contribution is to numerically demonstrate the relevance of locally refining the grid at the interface between subregions, where discontinuities occur, in order to preserve an acceptable accuracy for the results computed with the schemes under consideration.

On several numerical strategies to solve Richards’ equation in heterogeneous media with finite volumes

“…Further, the rigorous convergence of the scheme as the mesh size and the time steps tend to 0 is also established. However, the numerical results presented in [4] (as well as those presented in what follows) show that the expected first order convergence can be lost in presence of heterogeneities. Methods B, C, and D have been designed as remedies to this loss of accuracy, which takes its origin in the poor approximation of the pressure continuity (1.5) by Method A.…”

“…The scheme (2.9)-(2.11), complemented by the interface fluxes (3.1)-(3.3), has been shown in [4] to be well-posed in the sense that the corresponding nonlinear system admits a unique solution pτ n K q KPT . Further, the rigorous convergence of the scheme as the mesh size and the time steps tend to 0 is also established.…”

“…This method, introduced in [4] consists in adding two thin cells, denoted by I σ,K and I σ,L , of thickness δ B ! d σ on both sides of each face σ " K|L P E Γ located at a rock-type interface, as depicted in Figure 2.…”

“…Let us also mention [44] where the authors solve the transmission condition (1.4)-(1.5) thanks to an iterative procedure stemming from domain decomposition. In the case of cell centered methods, like for instance TPFA finite volumes, convergence can also be assessed without any specific treatment of the interface, as for example done in our recent contribution [4]. However, the pressure continuity is only imposed at convergence w.r.t.…”

“…This specification is the purpose of Section 3. In the case of a homogeneous domain where Γ " H, the resulting system F n pτ n q " 0 which is fully equivalent to (2.2)-(2.6), admits a unique solution τ n (for details see [4,).…”