An Efficient XFEM Approximation of Darcy Flows in Arbitrarily Fractured Porous Media

Abstract

Re´sume´-Une approximation efficace par XFEM pour e´coulements de Darcy dans les milieux poreux arbitrairement fracture´s -Les e´coulements souterrains sont fortement influence´s par la pre´sence de failles et de grandes fractures qui modifient la perme´abilite´du milieu agissant comme des barrie`res ou des conduits pour l'e´coulement. Une description pre´cise des proprie´te´s hydrauliques des fractures est donc essentielle pour la mode´lisation de la migration du pe´trole ou de l'exploitation de ressources non conventionnelles. Toutefois, la largeur de fractures est souvent petite par rapport a`la taille typique des e´le´ments du maillage. Pour re´soudre le proble`me par une approximation nume´rique obtenue sans raffinement du maillage, on remplace les fractures par des surfaces immerge´es dans la matrice poreuse. Par ailleurs, on veut permettre aux surfaces d'eˆtre inde´pendantes par rapport a`la structure du maillage, en manipulant les discontinuite´s à l'inte´rieur des e´le´ments par la me´thode des e´le´ments finis e´tendus (XFEM Extended Finite Element Method). On utilise une technique similaire pour obtenir un outil plus flexible, apte a`la re´solution du proble`me dans les re´gions d'intersection potentielles entre les fractures.Abstract -An Efficient XFEM Approximation of Darcy Flows in Arbitrarily Fractured Porous Media -Subsurface flows are influenced by the presence of faults and large fractures which act as preferential paths or barriers for the flow. In literature models were proposed to handle fractures in a porous medium as objects of codimension 1. In this work we consider the case of a network of intersecting fractures, with the aim of deriving physically consistent and effective interface conditions to impose at the intersection between fractures. This new model accounts for the angle between fractures at the intersections and allows for jumps of pressure across the intersection. This latter property permits to describe more accurately the flow when fractures are characterised by different properties, than other models that impose pressure continuity. The main mathematical properties of the model, derived in the two-dimensional setting, are analysed. As concerns the numerical discretization we allow the grids of the fractures to be independent, thus in general non-matching at the intersection, by means of the Extended Finite Element Method (XFEM), to increase the flexibility of the method in the case of complex geometries characterized by a high number of fractures.

Robust Discretization of Flow in Fractured Porous Media

“…Modeling intersections is non-trivial for a variety of fracture flow schemes and typically calls for special considerations (see e.g. [15,17,31,32]). In contrast, for the method we present, the implementation of an intersection follows naturally due to the dimensional decomposition.…”

Dimensionally reduced flow models in fractured porous media: crossings and boundaries

“…To be able to represent the discontinuities in the solution, i.e., the jump and the average term, we use the extended finite element method as it is described for example by Dolbow et al [13], Hansbo and Hansbo [17], and Mohammadi [22]. XFEM is used in the context of fractured porous media for example in D'Angelo and Scotti [9], Fumagalli and Scotti [16] for lower dimensional fractures introducing a discontinuous solution in the matrix, for lower dimensional fracture networks having different permeabilities in the network and therefore also discontinuities [8], as well as for thin heterogeneities (equi-dimensional) which are not resolved directly with the grid but rather with the XFEM, [19]. The system is completely assembled and solved with a direct solver.…”

Dual Virtual Element Method for Discrete Fractures Networks

“…where ·| ξ l is the trace operator from a one co-dimensional intersection to ξ l and T m n l is a unit vector, pointing outward from ξ l , and lying on γ m . While it is possible to extend the model to allow for jumps in pressure and velocity over ξ l , see [20,22], this is not considered herein.…”