Coupling Darcy and Stokes equations for porous media with cracks

Abstract

Abstract. In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive a priori and a posteriori error estimates. We present some numerical experiments that are in good agreement with the analysis.Mathematics Subject Classification. 65N30, 65N50, 76D07, 76S05.

The Coupling Navier-Stokes/Darcy Systems by Vorticity-Velocity-Pressure Formulation

“…In all the paper, we suppose that f 1 and f 2 the density of body forces in Ω 1 and Ω 2 respectively are in L 2 (Ω) 3 .In fact, f 1 has a value in Ω 1 and vanish in the rest of the domain, and f 2 has a value in Ω 2 and vanish in the rest of the domain.…”

Spectral Discretization of the Stokes Problem with Mixed Boundary Conditions

“…Indeed, this type of mixed boundary conditions appears in a large number of physical situations, the simplest one being a tank closed by a membrane on a part of its boundary (the index "m" in Γ m means membrane). They are also needed for the coupling with other equations, for instance with Darcy's equations when the fluid domain is a crack in a porous medium, see [9] and [10].…”

Error studies of the Coupling Darcy-Stokes System with velocity-pressure formulation

“…Indeed, the flow of a viscous incompressible fluid in a porous medium is usually modelled by Darcy equations and, when the thickness of the crack is too large to be neglected, the Stokes system must be considered in the crack and coupled with these equations. In this work, we consider the following system already studied in [4], [5] and [6]: Let Ω and Ω be bounded connected open domains in IR 3 with Lipschitz-continuous boundaries, such that Ω is contained in Ω. For simplicity, we also assume that Ω is simply connected and has a connected boundary.…”

“…In [4], [5] and [6], the basic idea consists in introducing the vorticity w = curl u as a new unknown on the fluid domain Ω . However, we treat in this work the same systems with the unknowns u and without introducing the vorticity w. Since, we can discretize the pressure and the velocity independently without a discrete inf-sup condition to obtain matrix systems with an optimal dimension and optimal time of resolution.…”