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A coupled HDG/DG method for porous media with conducting/sealing faults

Aycil Cesmelioglu, Miroslav Kuchta, Jeonghun J. Lee, Sander Rhebergen

TL;DR

This work addresses coupled subsurface flow in porous media with faults by formulating a $(\dim)$-dimensional Darcy system discretized with a dual mixed $HDG$ method and a $(\dim-1)$-dimensional IPDG discretization on faults. The key contributions are a consistent, well-posed coupled HDG/DG discretization and a rigorous a priori error analysis that yields optimal convergence rates; numerical experiments across conducting/sealing faults, immersion scenarios, and intersecting fault networks validate the theory and demonstrate robustness. The results show that the method achieves $\mathcal{O}(h^{k+1})$ convergence for velocity and fault pressures under suitable assumptions, with guidance on polynomial orders $k$ and $k_f$ to attain optimal rates. Overall, the coupled HDG/DG approach provides a flexible, accurate framework for inter-dimensional subsurface flow modeling with faults, with potential implications for reservoir simulation, groundwater management, and geothermal applications.

Abstract

We introduce and analyze a coupled hybridizable discontinuous Galerkin/discontinuous Galerkin (HDG/DG) method for porous media in which we allow fully and partly immersed faults, and faults that separate the domain into two disjoint subdomains. We prove well-posedness and present an a priori error analysis of the discretization. Numerical examples verify our analysis.

A coupled HDG/DG method for porous media with conducting/sealing faults

TL;DR

This work addresses coupled subsurface flow in porous media with faults by formulating a -dimensional Darcy system discretized with a dual mixed method and a -dimensional IPDG discretization on faults. The key contributions are a consistent, well-posed coupled HDG/DG discretization and a rigorous a priori error analysis that yields optimal convergence rates; numerical experiments across conducting/sealing faults, immersion scenarios, and intersecting fault networks validate the theory and demonstrate robustness. The results show that the method achieves convergence for velocity and fault pressures under suitable assumptions, with guidance on polynomial orders and to attain optimal rates. Overall, the coupled HDG/DG approach provides a flexible, accurate framework for inter-dimensional subsurface flow modeling with faults, with potential implications for reservoir simulation, groundwater management, and geothermal applications.

Abstract

We introduce and analyze a coupled hybridizable discontinuous Galerkin/discontinuous Galerkin (HDG/DG) method for porous media in which we allow fully and partly immersed faults, and faults that separate the domain into two disjoint subdomains. We prove well-posedness and present an a priori error analysis of the discretization. Numerical examples verify our analysis.

Paper Structure

This paper contains 12 sections, 7 theorems, 67 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Porous media with faults
  3. The discretization
  4. Well-posedness of the discrete problem
  5. Error Analysis
  6. Numerical results
  7. Error convergence
  8. Conducting faults
  9. Sealing faults
  10. Intersecting faults
  11. Conclusions
  12. Two alternative hybridizable discretizations

Key Result

Lemma 1

Let $u \in [H^1(\mathring{\Omega})]^{\dim}$, $p\in H^1(\mathring{\Omega})$, and $p_f \in H^{s}(\Gamma_c)$ with $s > 3/2$ satisfy eq:fault_problemeq:fault_problem_bcs. Denote the average of $p$ on $\mathcal{F}_0 \cup \mathcal{F}_s^f$ by $\{\!\!\{p\}\!\!\}$. Then $u$ and $\boldsymbol{p}:=(p,\{\!\!\{p\

Figures (13)

  • Figure 1: Illustration of the domain notation in $\mathbb{R}^2$.
  • Figure 2: Setup of the error convergence study in \ref{['ex:mms']}. (Left) Geometry with a conducting fault in red, sealing fault in blue, and a sample mesh. (Right) Discrete pressures on the finest mesh ($h=1/64$) with the HDG/DG discretization choosing $k=k_f=1$.
  • Figure 3: Setup of the error convergence study in \ref{['rmrk:nonconvex']}. (Left) Geometry with a non-convex conducting fault in red, and a sample mesh. (Right) Discrete pressures on the finest mesh ($h=1/64$) with the HDG/DG discretization choosing $k=k_f=1$.
  • Figure 4: Fully immersed conducting fault problem described in \ref{['ex:conducting']}. (Left) Problem setup together with the initial mesh. The conducting fault is depicted by a red line. To compare solutions of different discretizations we sample the pressure along the brown dashed line. (Center) The pressure distribution obtained by the HDG/DG method with $\xi=0.75$. (Right) Comparison of HDG/DG solutions for three different values of $\xi$ to a reference $\textit{P}_1$ solution of the fracture model alboin1999domain in terms of pressure values sampled along the brown dashed line in the left panel. The conforming scheme reflects the modeling assumption $p_{+}=p_{-}=p_f$ on $\Gamma_c$alboin1999domain. Note that no value of $\xi$ is needed for the reference solution. The HDG/DG solution was computed on a coarse mesh with 2176 cells while a fine mesh consisting of 30056 cells is used to compute the $\textit{P}_1$ reference solution.
  • Figure 5: Conducting fault with discontinuous permeability $\overline{\kappa}_{f, n}=\overline{\kappa}_{f, \tau}$ as described in \ref{['ex:conducting']}. (Left) Problem setup together with the initial mesh. The conducting fault is depicted by a red line. To compare solutions of different discretizations we sample the pressure along the brown dashed line. (Center) The pressure distribution obtained by the HDG/DG method with $\xi=0.75$. (Right) Comparison of HDG/DG solutions for three different values of $\xi$ to a reference $\textit{RT}_0$-$\textit{P}_0$ solution Martin:2005 where $\xi=1$ in terms of pressure values sampled along the brown dashed line in the left panel. The HDG/DG solution was computed on a coarse mesh with 2286 cells while a fine mesh with 32354 cells was used for the reference solution. For clarity, the reference solution is shown as a thin red line as well as using markers.
  • ...and 8 more figures

Theorems & Definitions (16)

  • Lemma 1: Consistency
  • proof
  • Lemma 2: Coercivity and boundedness of $a_h$
  • proof
  • Lemma 3: $c_h$ is positive semi-definite
  • proof
  • Lemma 4
  • Theorem 1: Well-posedness
  • proof
  • Remark 1
  • ...and 6 more