Mathematics

I will discuss the discretization of two-phase Darcy flows in a fractured and deformable porous medium, including frictional contact at matrix–fracture interfaces. Fractures are described as a network of planar surfaces leading to the so-called mixed-dimensional models. Small displacements and a linear elastic behavior are considered for the matrix.To simulate the coupled model, we employ a Two-Point Flux Approximation (TPFA) finite volume scheme for the flow and second-order (P2) finite elements for the mechanical displacement coupled with face-wise constant (P0) Lagrange multipliers on fractures, representing normal and tangential stresses, to discretize frictional contact conditions. This choice allows to circumvent possible singularities at tips, corners, or intersections between fractures, and provides a local expression for the contact conditions. We present numerical simulations of one benchmark example and one realistic test case based on a drying model in a radioactive waste geological storage structure, for which the dataset was provided by Andra (the French National Agency for the management of radioactive wastes). The talk is based on joint works with Jérôme Droniou (Monash University, Australia) and Roland Masson (Université Côte d’Azur, Nice, France).