A Generalized Mixed Hybrid Mortar Method for Solving Flow in Stochastic Discrete Fracture Networks

The simulation of flow in fractured media requires handling both a large number of fractures and a complex interconnecting network of these fractures. Networks considered in this paper are three-dimensional domains made up of two-dimensional fractures intersecting each other and randomly generated. Due to the stochastic generation of fractures, intersections can be highly intricate. The numerical method must generate a mesh and define a discrete problem for any discrete fracture network (DFN). A first approach [Erhel, de Dreuzy, and Poirriez, SIAM J. Sci. Comput., 31 (2009), pp. 2688–2705] is to generate a conforming mesh and to apply a mixed hybrid finite element method. However, the resulting linear system becomes very large when the network contains many fractures. Hence a second approach [Pichot, Erhel, and de Dreuzy, Appl. Anal., 89 (2010), pp. 1629–1643] is to generate a nonconforming mesh, using an independent mesh generation for each fracture. Then a Mortar technique applied to the mixed hybrid finite element method deals with the nonmatching grids. When intersections do not cross or overlap, pairwise Mortar relations for each intersection are efficient [Pichot, Erhel, and de Dreuzy, 2010]. But for most random networks, discretized intersections involve more than two fractures. In this paper, we design a new method generalizing the previous one and that is applicable for stochastic networks. The main idea is to combine pairwise Mortar relations with additional relations for the overlapping part. This method still ensures the continuity of fluxes and heads and still yields a symmetric positive definite linear system. Numerical experiments show the efficiency of the method applied to complex stochastic fracture networks. We also study numerical convergence when reducing the mesh step. This method makes it easy to perform mesh optimization and appears to be a very promising tool to simulate flow in multiscale fracture networks.