An Interior Penalty Discontinuous Galerkin Method for an Interface Model of Flow in Fractured Porous Media

TL;DR

This paper introduces an interior penalty discontinuous Galerkin method for interface modeling in fractured porous media, offering stability, error bounds, and validation through numerical experiments, including two-phase flows.

Contribution

It presents a novel IPDG approach that models fractures without extra degrees of freedom, with proven stability, error analysis, and extension to two-phase flow simulations.

Findings

Optimal error bounds in energy and L2 norms.

Robust performance validated by benchmark tests.

Method successfully extended to two-phase flow modeling.

Abstract

Discrete fracture models with reduced-dimensional treatment of conductive and blocking fractures are widely used to simulate fluid flow in fractured porous media. Among these, numerical methods based on interface models are intensively studied, where the fractures are treated as co-dimension one manifolds in a bulk matrix with low-dimensional governing equations. In this paper, we propose a simple yet effective treatment for modeling the fractures on fitted grids in the interior penalty discontinuous Galerkin (IPDG) methods without introducing any additional degrees of freedom or equations on the interfaces. We conduct stability and {\em hp}-analysis for the proposed IPDG method, deriving optimal a priori error bounds concerning mesh size ($h$) and sub-optimal bounds for polynomial degree ($k$) in both the energy norm and the $L^2$ norm. Numerical experiments involving published benchmarks validate our theoretical analysis and demonstrate the method's robust performance. Furthermore, we extend our method to two-phase flows and use numerical tests to confirm the algorithm's validity.