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

TL;DR

The paper develops an interior penalty discontinuous Galerkin method for a hybrid-dimensional interface model of flow in fractured porous media, using fitted meshes that avoid introducing interface degrees of freedom. It proves stability and hp-error estimates, achieving optimal convergence in mesh size $h$ and suboptimal convergence in polynomial degree $k$ for single-phase flow, with extensions to two-phase Buckley–Leverett dynamics validated on benchmarks. Numerical experiments across conductive fractures, blocking barriers, and complex fracture networks confirm the theoretical rates and demonstrate robustness and accuracy, competitive with established discrete fracture models. The approach offers a computationally efficient high-order framework suitable for simulating flow in complex fracture networks, with potential enhancements via enriched Galerkin spaces to further reduce degrees of freedom.

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.

Figures (17)

• Figure 2.1: The geometry of an interface model involving fractures.
• Figure 3.1: Local area of a body-fitted mesh near the fracture $\gamma_2$.
• Figure 5.1: Example \ref{['ex:convergence']}: convergence test. The distribution of the fracture (indicated by the black thick line segment) and the grid with $h=1/8$ used in the computation.
• Figure 5.2: Example \ref{['ex:single']}: single fracture. The distributions of the fracture (indicated by the black thick line segments) and the grids used in the computation. The grid for the vertical fracture contains $450$ cells, and the grid for the slanted fracture contains $404$ cells.
• Figure 5.3: Example \ref{['ex:single']}: single fracture. Results of the $P^1$-SIPG method for the single conductive fracture computed on the grids are shown in Figure \ref{['fig:single_grid_medium']}. The slices of pressure in (b) and (d) are taken along $y=0.75$ and $y=0.5$, respectively. The reference solutions are obtained from the Box-DFM xu2024extension with $23, 306$ and $23, 455$ cells for the vertical and slanted fractures, respectively.

Theorems & Definitions (17)

• Remark 3.1
• Remark 3.2
• Remark 3.3
• Lemma 4.1
• Theorem 4.1
• proof
• Lemma 4.2
• Theorem 4.2
• proof
• Remark 4.1