Résumé
We consider a two-phase Darcy flow in a fractured and deformable porous medium for which the fractures are described as a network of planar surfaces leading to so-called hybrid-dimensional models. The fractures are assumed open and filled by the fluids and small deformations with a linear elastic constitutive law are considered in the matrix. As opposed to [F. Bonaldi, K. Brenner, J. Droniou and R. Masson, Comput. Math. with Appl. 98 (2021)], the phase pressures are not assumed continuous at matrix fracture interfaces, which raises new challenges in the convergence analysis related to the additional interfacial equations and unknowns for the flow. As shown in [K. Brenner, J. Hennicker, R. Masson and P. Samier, J. Comput. Phys. 357 (2018)], [J. Aghili, K. Brenner, J. Hennicker, R. Masson and L. Trenty, GEM – Int. J. Geomath. 10, (2019)], unlike single-phase flow, discontinuous pressure models for two-phase flows provide a better accuracy than continuous pressure models even for highly permeable fractures. This is due to the fact that fractures fully filled by one phase can act as barriers for the other phase, resulting in a pressure discontinuity at the matrix fracture interface. The model is discretized using the gradient discretization method [J. Droniou, R. Eymard, T. Gallouët, C. Guichard, and R. Herbin, Springer, Mathematics & Applications, 82 (2018)], which covers a large class of conforming and non conforming schemes. This framework allows for a generic convergence analysis of the coupled model using a combination of discrete functional tools. In this work, the gradient discretization of [F. Bonaldi, K. Brenner, J. Droniou and R. Masson, Comput. Math. with Appl. 98 (2021)] is extended to the discontinuous pressure model and the convergence to a weak solution is proved. Numerical solutions provided by the continuous and discontinuous pressure models are compared on gas injection and suction test cases using a Two-Point Flux Approximation (TPFA) finite volume scheme for the flows and ℙ2 finite elements for the mechanics.