A Hybrid High-Order Method for Darcy Flows in Fractured Porous Media

Abstract

We develop a novel Hybrid High-Order method for the simulation of Darcy flows in fractured porous media. The discretization hinges on a mixed formulation in the bulk region and a primal formulation inside the fracture. Salient features of the method include a seamless treatment of nonconforming discretizations of the fracture, as well as the support of arbitrary approximation orders on fairly general meshes. For the version of the method corresponding to a polynomial degree k ě 0, we prove convergence in h k`1 of the discretization error measured in an energy-like norm. In the error estimate, we explicitly track the dependence of the constants on the problem data, showing that the method is fully robust with respect to the heterogeneity of the permeability coefficients, and it exhibits only a mild dependence on the square root of the local anisotropy of the bulk permeability. The numerical validation on a comprehensive set of test cases confirms the theoretical results.

Multipoint flux mixed finite element methods for slightly compressible flow in porous media

“…Figure 4(a) shows the numerical pressure (left) and the logarithm of the norm of the numerical velocity (right), which are symmetric about the diagonal y = x, as expected for this homogeneous permeability tensor. In fact, the pressure plot is qualitatively similar to that shown in [54], where an incompressible model is considered. In order to check the behaviour of the method when considering a discontinuous full permeability tensor, we considerK to be defined aŝ…”

A Hybrid High-Order method for passive transport in fractured porous media

“…T , and that it is equivalent to [15 15,Eq. (19)] with discrete divergence operator expanded according to its definition.…”

Mixed-Dimensional Geometric Multigrid Methods for Single-Phase Flow in Fractured Porous Media

“…The so-called extended finite element methods (XFEM), which permit to mesh the entire domain independently of the fractures, are described in [17,21] and references therein. In addition, further discretization schemes have been proposed for handling general elements and distorted grids, namely: mimetic finite difference methods [6,23], discontinuous Galerkin methods [5], virtual element methods [10,26], hybrid high-order methods [16], or multipoint flux approximation methods [1,45].…”