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An extension of the box method discrete fracture model (Box-DFM) to include low-permeable barriers with minimal additional degrees of freedom

Ziyao Xu, Dennis Gläser

Abstract

The box method discrete fracture model (Box-DFM) is an important finite volume-based discrete fracture model (DFM) to simulate flows in fractured porous media. In this paper, we investigate a simple but effective extension of the box method discrete fracture model to include low-permeable barriers. The method remains identical to the traditional Box-DFM [41, 48] in the absence of barriers. The inclusion of barriers requires only minimal additional degrees of freedom to accommodate pressure discontinuities and necessitates minor modifications to the original coding framework of the Box-DFM. We use extensive numerical tests on published benchmark problems and comparison with existing finite volume DFMs to demonstrate the validity and performance of the method.

An extension of the box method discrete fracture model (Box-DFM) to include low-permeable barriers with minimal additional degrees of freedom

Abstract

The box method discrete fracture model (Box-DFM) is an important finite volume-based discrete fracture model (DFM) to simulate flows in fractured porous media. In this paper, we investigate a simple but effective extension of the box method discrete fracture model to include low-permeable barriers. The method remains identical to the traditional Box-DFM [41, 48] in the absence of barriers. The inclusion of barriers requires only minimal additional degrees of freedom to accommodate pressure discontinuities and necessitates minor modifications to the original coding framework of the Box-DFM. We use extensive numerical tests on published benchmark problems and comparison with existing finite volume DFMs to demonstrate the validity and performance of the method.
Paper Structure (8 sections, 2 theorems, 19 equations, 16 figures, 1 table)

This paper contains 8 sections, 2 theorems, 19 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem description
  3. Algorithm
  4. A review of the traditional Box-DFM
  5. The inclusion of low-permeable barriers
  6. Numerical experiments
  7. Summary
  8. The symmetry and positive definiteness of the stiffness matrix of \ref{['eq:BoxB']}

Key Result

Lemma A.1

Let $\phi_i\in\overline{V}_h$ the Lagrange basis and $B_{i}\in \overline{\mathcal{B}}$ the box associated with the node $i$, for $i=1,\ldots, \#\overline{\mathcal{B}}$. Then we have where $\mathbf{n}$ denotes the unit outer normal of $\partial B_{i}$.

Figures (16)

  • Figure 2.1: The geometry of an interface model in $\mathbb{R}^2$. $\gamma$ represents the interface of a fracture or barrier. $\Omega^{\pm}$ and $\mathbf{n}^{\pm}$ are the bulk matrix regions and unit outer normal from each region, respectively.
  • Figure 3.1: The meshes used in box methods for unfractured porous media, porous media with high-permeable fractures and low-permeable barriers.
  • Figure 4.1: Example \ref{['ex:convergence']}: convergence test. The grid used for the lowest refinement, consisting of $256$ triangles. The red line represents the barrier interface.
  • Figure 4.2: Example \ref{['ex:single']}: single barrier. Visualization of the solution $p$ (row $1$) obtained with the proposed method, and the difference $\Delta p$ (row $2$) to the solution obtained with the ebox-dfm in glaser2022comparison. In the first scenario (vertical barrier), the grid contains $253$ vertices and $450$ triangles; in the second scenario (slanted barrier), it contains $229$ vertices and $404$ triangles.
  • Figure 4.3: Example \ref{['ex:single']}: single barrier. Pressure profiles along the slice (0, 0.75) -- (1, 0.75) for the vertical barrier and along the slice (0, 0.5) -- (1, 0.5) for the slanted barrier.
  • ...and 11 more figures

Theorems & Definitions (15)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • ...and 5 more