A hybridizable discontinuous Galerkin method on unfitted meshes for single-phase Darcy flow in fractured porous media

TL;DR

This paper introduces a new hybridizable discontinuous Galerkin (HDG) method for simulating single-phase Darcy flow in fractured porous media using unfitted meshes, enabling flexible and efficient modeling of complex fracture networks.

Contribution

The paper presents the first truly unfitted finite element scheme for 3D fractured porous media flow with both blocking and conductive fractures, using a simple modification of existing HDG methods.

Findings

Performs well on benchmark tests in 2D and 3D.

Allows for unfitted meshes with fractures modeled by Dirac-$\

Simple to implement and locally mass conservative.

Abstract

We present a novel hybridizable discontinuous Galerkin (HDG) method on unfitted meshes for single-phase Darcy flow in a fractured porous media. In particular we apply the HDG methodology to the recently introduced reinterpreted discrete fracture model (RDFM) [Xu & Yang, 2021 submitted] that use Dirac-$\delta$ functions to model both conductive and blocking fractures. Due to the use of Dirac-$\delta$ function approach for the fractures, our numerical scheme naturally allows for unfitted meshes with respect to the fractures, which is the major novelty of the proposed scheme. Moreover, the scheme is locally mass conservative and is relatively easy to implement comparing with existing work on the subject. In particular, our scheme is a simple modification of an existing regular Darcy flow HDG solver by adding the following two components: (i) locate the co-dimension one fractures in the background mesh and adding the appropriate surface integrals associated with these fractures into the stiffness matrix, (ii) adjust the penalty parameters on cells cut through conductive and blocking fractures (fractured cells). Despite the simplicity of the proposed scheme, it performs extremely well for various benchmark test cases in both two- and three-dimensions. This is the first time that a truly unfitted finite element scheme been applied to complex fractured porous media flow problems in 3D with both blocking and conductive fractures without any restrictions on the meshes.