Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media

This paper presents a finite volume discretization of two-phase Darcy flows in discrete fracture networks taking into account the mass exchange between the matrix and the fracture. We consider the asymptotic model for which the fractures are represented as interfaces of codimension one immersed in the matrix domain, leading to the so called hybrid dimensional Darcy flow model. The pressures at the interfaces between the matrix and the fracture network are continuous corresponding to a ratio between the normal permeability of the fracture and the width of the fracture assumed to be large compared with the ratio between the permeability of the matrix and the size of the domain. The discretization is an extension of the Vertex Approximate Gradient (VAG) scheme to the case of hybrid dimensional Darcy flow models. Compared with Control Volume Finite Element (CVFE) approaches, the VAG scheme has the advantage to avoid the mixing of the fracture and matrix rocktypes at the interfaces between the matrix and the fractures, while keeping the low cost of a nodal discretization on unstructured meshes. The convergence of the scheme is proved under the assumption that the relative permeabilities are bounded from below by a strictly positive constant. This assumption is needed in the convergence proof in order to take into account discontinuous capillary pressures in particular at the matrix fracture interfaces. The efficiency of our approach compared with CVFE discretizations is shown on two numerical examples of fracture networks in 2D and 3D. Introduction This article deals with the discretization of two-phase Darcy flows in fractured porous media for which the fractures are modelized as interfaces of codimension one. In this framework, the d − 1 dimensional flow in the fractures is coupled with the d dimensional flow in the matrix leading to the so called, hybrid dimensional Darcy flow model. We focus on the particular case where the pressure is continuous at the interfaces between the fractures and the matrix domain. This type of hybrid dimensional Darcy flow model has been introduced in [1] for single phase Darcy flows and in [29], [28], [21] for two-phase Darcy flows. It corresponds physically to pervious fractures for which the ratio of the transversal permeability of the fracture to the width of the fracture is large compared with the ratio of the permeability of the matrix to the size of the domain. Note that it does not cover the case of fractures acting as barriers for which the pressure is discontinuous at the matrix fracture interfaces (see [18], [26], [3] for discontinuous pressure models). It is also assumed in our model that the pressure is continuous at the fracture intersections. It corresponds to a high ratio assumption between the permeability at the fracture intersections and the Laboratoire de Mathématiques J.A. Dieudonné, UMR 7351 CNRS, University Nice Sophia Antipolis, and team COFFEE, INRIA Sophia Antipolis Méditerranée, Parc Valrose 06108 Nice Cedex 02, France, [email protected] Laboratoire de Mathématiques J.A. Dieudonné, UMR 7351 CNRS, University Nice Sophia Antipolis, and team COFFEE, INRIA Sophia Antipolis Méditerranée, Parc Valrose 06108 Nice Cedex 02, France, [email protected] Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, CNRS, Laboratoire Jacques-Louis Lions, F-75005, Paris, and INRIA, ANGE project-team, Rocquencourt B.P. 105, F78153 Le Chesnay cedex, and CEREMA, ANGE projectteam, 134 rue de Beauvais, F-60280 Margny-Lès-Compiègne, France, [email protected] Laboratoire de Mathématiques J.A. Dieudonné, UMR 7351 CNRS, University Nice Sophia Antipolis, and team COFFEE, INRIA Sophia Antipolis Méditerranée, Parc Valrose 06108 Nice Cedex 02, France, [email protected]