Study of Discrete Duality Finite Volume Schemes for the Peaceman Model

Abstract

In this paper, we are interested in the finite volume approximation of a system describing miscible displacement in porous media. This system is made of two coupled equations: an anisotropic diffusion equation on the pressure and a convection-diffusion-dispersion equation on the concentration of invading fluid. The anisotropic diffusion operators in both equations require a special care while discretizing by a finite volume method. We focus here on the numerical approximation by some Discrete Duality Finite Volume methods. After the presentation of the scheme, we establish some a priori estimates satisfied by the numerical solution and prove existence and uniqueness of the solution to the scheme. We show the efficiency of the schemes through numerical experiments.

Convergence analysis of a DDFV scheme for a system describing miscible fluid flows in porous media

“…Lemma 3.1 gives a priori estimates on the pressure, the gradient of the pressure and the Darcy's velocity at the discrete level, while Lemma 3.2 gives a priori estimates on the approximate concentration and its approximate gradient. Thanks to these two lemmas, we get the existence and uniqueness of a solution to the scheme, as in [8]. Then, Lemma 3.3 shows that the reconstructions of the concentration on the primal and dual meshes will necessarily converge to the same limit (when convergence occurs).…”

“…Thanks to Lemma 3.1 and 3.2, we have the existence and uniqueness of a solution (p h,δt , U h,δt , c h,δt ) ∈ H T ,δt × H D,δt × H T ,δt to the scheme (18)- (19) as in [8]. …”

“…In Section 6, we provide some numerical experiments. The efficiency of the DDFV scheme has already been shown in [8]. In this last Section, we just show that the penalization operator introduced for the proof of convergence can be set to 0 in practice.…”

“…Existence and uniqueness of a solution to each linear system has been proved in [8] in the case where λ = 0. This result is based on the a priori estimates satisfied by the discrete pressure and the discrete concentration.…”

“…In [8], we have proposed a DDFV scheme for the Peaceman system (1)- (2). The DDFV scheme requires unknowns on both vertices and "centers" of control volumes.…”