A Discrete Fracture Model for Fluid Flow in Porous Media

Cracks inside a porous medium are a source of anisotropy and heterogeneity. Simulation of fluid flow through this type of medium requires a robust numerical treatment to handle the complex flow pattern in an accurate and efficient way. We present a numerical framework for simulating fluid flow in porous media with arbitrarily embedded cracks. The formulation, cast within the framework of the discrete fracture models, does not make any assumptions on the relative hydraulic conductivities of the host bulk material and the cracks. The ingredients of the proposed approach are the following. Firstly, double, mixed-dimensional vertex-centered finite volume meshes are used for the bulk and crack network domains. The bulk domain is discretized with volumetric elements while the crack network domain is discretized with lower-dimensional fracture elements. The primary mesh for the bulk domain is constructed independent of the geometry of the crack network, but the secondary mesh for the bulk domain is constructed by imposing conformity with the crack network geometry. Secondly, a homogenization procedure is performed for the volumetric elements crossed by a crack in order to define an equivalent, homogenized value of the hydraulic conductivity for these elements. The equivalent hydraulic conductivity ensures correct fluxes between the two bulk control volumes on each side of the crack. The homogenization procedure is based on the definition of an equivalent elemental length in the volumetric elements. This elemental length arises from imposing the equivalence of fluxes given by the homogenized approximation and by a discontinuous description of the pressure field (that aims to reproduce the actual pressure field distribution). Thirdly, an analytical solution, provided by the equivalent problem of fluid conduction within a layered medium having different hydraulic conductivities, is used to define the mass exchange terms between the bulk material and the cracks. The method is locally conservative and suited to unstructured grids, and can accurately resolve the single-phase fluid flow on both the coarse (bulk domain) and fine (crack network) scale fields simultaneously. The accuracy of the model is studied with the aid of two examples that reproduce actual problems: the fluid injection into a well and the drain behavior into a domain with a nearby impervious barrier. Results of the simulations with the discrete fracture formulation agree well with those obtained with the standard Galerkin finite element formulation utilizing extremely fine meshes.