Finite volume method for coupled subsurface flow problems, I: Darcy problem

The article introduces a finite-volume method for the Darcy problem in heterogeneous anisotropic media. The method is based on the mixed formulation for the pressure and its gradient. The method is stable despite collocation of both pressure and its gradient at cell centers and demonstrates the first order convergence on numerous benchmarks as well as good monotonicity property. The method produces quasi-definite matrix, which is numerically shown to have good asymptotics of the condition number. Our flux discretization method is a realization of our more general concept of stable flux discretization for saddle-point systems with vector of several unknowns. In this paper this vector is composed of pressure and its gradient and the saddle-point system is the mixed formulation of the Darcy problem. •A heterogeneous Darcy problem in saddle-point formulation is considered.•A linear finite-volume method with two-point flux approximation is proposed for saddle-point system.•The method is consistent and first order accurate on general meshes with heterogeneous full tensor permeability.•The method is found to satisfy the discrete maximum principle.