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

Abstract

Abstract. 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.

Stochastic modeling of flow and conservative transport in threedimensional discrete fracture networks

“…Successful characterizations of flow and contaminant transport in fractured geologic formations depend on adequate descriptions of complex geometrical structures, which comprise a wide variety of fractures and their connections (Ahmed et al, 2015;Pichot et al, 2012;Weng et al, 2014). The fracture characteristics can be quantified by using various statistical parameters, including the fracture orientation, length, shape, and permeability alongside the fracture intensity and 25 connectivity (Bonnet et al, 2001;Botros et al, 2008;Bour et al, 2002;Koike et al, 2015;Stephens et al, 2015).…”

“…CC BY 4.0 License. numerous models that are based on the DFN approach to model the flow or transport in fractured formations (Cacas et al, 1990a;Cacas et al, 1990b;de Dreuzy et al, 2013;Hyman et al, 2015a;Liu and Neretnieks, 2006;Long et al, 1985;Pichot et al, 2012;Xu and Dowd, 2010). Advanced 3D DFN approaches typically include procedures of fracture generation, DFN meshing, and flow and transport or particle tracking (de Dreuzy et al, 2013;Erhel et al, 2009;Hyman et al, 2014;Pichot et al, 2012;Xu and Dowd, 2010;Zhang, 2015).…”

“…numerous models that are based on the DFN approach to model the flow or transport in fractured formations (Cacas et al, 1990a;Cacas et al, 1990b;de Dreuzy et al, 2013;Hyman et al, 2015a;Liu and Neretnieks, 2006;Long et al, 1985;Pichot et al, 2012;Xu and Dowd, 2010). Advanced 3D DFN approaches typically include procedures of fracture generation, DFN meshing, and flow and transport or particle tracking (de Dreuzy et al, 2013;Erhel et al, 2009;Hyman et al, 2014;Pichot et al, 2012;Xu and Dowd, 2010;Zhang, 2015). Particle tracking algorithms are usually preferred to simulate DFN transport 5 and have recently been widely implemented to evaluate the time resistance of contaminants for fractured formations (Hyman et al, 2015a;Hyman et al, 2015b;Makedonska et al, 2015;Painter et al, 2008;Stalgorova and Babadagli, 2015;Wang and Cardenas, 2015).…”

The Virtual Element Method for Discrete Fracture Network Flow and Transport Simulations

“…In this work we recall some results concerning the application of the Virtual Element Method [4,3,2] to the steady state simulation of the flow in DFNs [1,22,27,30,24,32,23,15,16,17,18,19,9,10,8,13]. In this approach we can exploit the flexibility of VEM in order to tackle the geometrical complexity.…”

“…In this approach we can exploit the flexibility of VEM in order to tackle the geometrical complexity. Indeed, a crucial issue in DFN flow simulations is the need to provide on each fracture a good quality mesh [24,20,29,30] on any randomly generated configuration. Namely, if classical triangular or quadrilateral meshes on the fractures are required to be conforming to the traces (fracture intersections), and also conforming each other, the meshing process for each fracture is not independent of the others, thus yielding in practice a quite demanding computational effort for the mesh generation process.…”