Gradient scheme discretizations of two-phase porous media flows in fractured porous media

This talk presents the gradient scheme framework for the 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. Two type of discretizations matching the gradient scheme framework are discussed including the extensions of the Hybrid Finite Volume (HFV) and of the Vertex Approximate Gradient (VAG) schemes 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 rock types 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 gradient schemes is obtained for two phase flow models under the assumption that the relative permeabilities are bounded from below by a strictly positive constant. This assumption is needed in the convergence proof only in order to take into account discontinuous capillary pressures in particular at the matrix fracture interfaces. The efficiency of our approach is shown on numerical examples of fracture networks in 2D and 3D.