Convergence of a multi-point flux approximation finite volume scheme for a sharp–diffuse interfaces model for seawater intrusion

Abstract

This paper deals with development and analysis of a finite volume (FV) method for the coupled system describing seawater intrusion in coastal aquifers. The problem is modeled by the recently sharp–diffuse interfaces approach. This process is formulated by a coupled system of two nonlinear parabolic of cross-diffusion type equations describing two immiscible phase seawater/freshwater flow tacking into account the width of transition zones. A fully coupled, fully implicit approach is developed to discretize the coupled system. The method combines advantages of the MPFA method to accurately solve fluxes and diffusive terms and upstream for advective terms with implicit Euler’s time discretization. The non-negativity of the discrete solution is proved and an existence result is shown using a fixed point theorem. Based on a priori estimates and compactness arguments, we prove the convergence of the numerical approximation to the weak solution. We have developed and implemented this scheme in a new module in the context of the open source platform DuMuX. Two numerical experiments are presented to demonstrate the efficiency of this scheme, one of which is related to flows in a fractured porous aquifer.