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bound-preserving Adaptive Time-Stepping Method with Energy Stability for Simulating Compressible Gas Flow in Poroelastic Media

Huangxin Chen, Yuxiang Chen, Jisheng Kou, Shuyu Sun

TL;DR

The paper tackles thermodynamically consistent, compressible gas flow in poroelastic media by developing a stabilized linear energy-stable scheme that exactly preserves the original energy while enforcing molar-density bounds. It introduces a stabilized semi-discrete and a fully discrete finite-element framework with upwind and DG discretizations, proven to converge via a contraction mapping and to dissipate energy, $D_{\tau}E^{n+1}_h\le 0$. An adaptive time-stepping strategy coupled with an explicit stabilization parameter update ensures stability and efficiency, even in highly dynamic regimes. Numerical experiments in 2D and 3D demonstrate convergence, mass conservation, energy dissipation, and robustness under heterogeneous permeability, validating the method’s accuracy and practicality for complex geomechanical applications. The approach lays groundwork for extensions to multi-component and multi-phase flow scenarios in geomechanics and subsurface energy systems.

Abstract

In this paper, we present an efficient numerical method to address a thermodynamically consistent gas flow model in porous media involving compressible gas and deformable rock. The accurate modeling of gas flow in porous media often poses significant challenges due to their inherent nonlinearity, the coupling between gas and rock dynamics, and the need to preserve physical principles such as mass conservation, energy dissipation and molar density boundedness. The system is further complicated by the need to balance computational efficiency with the accuracy and stability of the numerical scheme. To tackle these challenges, we adopt a stabilization approach that is able to preserve the original energy dissipation while achieving linear energy-stable numerical schemes. We also prove the convergence of the adopted linear iterative method. At each time step, the stabilization parameter is adaptively updated using a simple and explicit formula to ensure compliance with the original energy dissipation law. The proposed method uses adaptive time stepping to improve computational efficiency while maintaining solution accuracy and boundedness. The adaptive time step size is calculated explicitly at each iteration, ensuring stability and allowing for efficient handling of highly dynamic scenarios. A mixed finite element method combined with an upwind scheme is employed as spatial discretization to ensure mass conservation and stability. Finally, we conduct a series of numerical experiments to validate the performance and robustness of the proposed numerical method.

bound-preserving Adaptive Time-Stepping Method with Energy Stability for Simulating Compressible Gas Flow in Poroelastic Media

TL;DR

The paper tackles thermodynamically consistent, compressible gas flow in poroelastic media by developing a stabilized linear energy-stable scheme that exactly preserves the original energy while enforcing molar-density bounds. It introduces a stabilized semi-discrete and a fully discrete finite-element framework with upwind and DG discretizations, proven to converge via a contraction mapping and to dissipate energy, . An adaptive time-stepping strategy coupled with an explicit stabilization parameter update ensures stability and efficiency, even in highly dynamic regimes. Numerical experiments in 2D and 3D demonstrate convergence, mass conservation, energy dissipation, and robustness under heterogeneous permeability, validating the method’s accuracy and practicality for complex geomechanical applications. The approach lays groundwork for extensions to multi-component and multi-phase flow scenarios in geomechanics and subsurface energy systems.

Abstract

In this paper, we present an efficient numerical method to address a thermodynamically consistent gas flow model in porous media involving compressible gas and deformable rock. The accurate modeling of gas flow in porous media often poses significant challenges due to their inherent nonlinearity, the coupling between gas and rock dynamics, and the need to preserve physical principles such as mass conservation, energy dissipation and molar density boundedness. The system is further complicated by the need to balance computational efficiency with the accuracy and stability of the numerical scheme. To tackle these challenges, we adopt a stabilization approach that is able to preserve the original energy dissipation while achieving linear energy-stable numerical schemes. We also prove the convergence of the adopted linear iterative method. At each time step, the stabilization parameter is adaptively updated using a simple and explicit formula to ensure compliance with the original energy dissipation law. The proposed method uses adaptive time stepping to improve computational efficiency while maintaining solution accuracy and boundedness. The adaptive time step size is calculated explicitly at each iteration, ensuring stability and allowing for efficient handling of highly dynamic scenarios. A mixed finite element method combined with an upwind scheme is employed as spatial discretization to ensure mass conservation and stability. Finally, we conduct a series of numerical experiments to validate the performance and robustness of the proposed numerical method.
Paper Structure (13 sections, 6 theorems, 110 equations, 16 figures, 4 tables)

This paper contains 13 sections, 6 theorems, 110 equations, 16 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Energy stable and bounds-preserving numerical method
  4. Stable time semi-discrete scheme
  5. Full discretization
  6. Bound-preserving Adaptive Time-Stepping strategies
  7. Convergence of the iterative method and energy stability
  8. Numerical examples
  9. Example 1
  10. Example 2
  11. Example 3
  12. Example 4
  13. Conclusion

Key Result

Lemma 4.1

For two given real constants $a, b$ and $a < b$, for any $c_h \in \mathcal{Q}_{h}$, we define $c_{h,-} = \min(c_h+a, 0)$, $c_{h,+} = \max(c_h-b, 0)$. We have the following inequalities hold

Figures (16)

  • Figure 1: Mesh: Left: Two-dimensional mesh. Right: Three-dimensional mesh.
  • Figure 2: Example 1: Convergence plots of the molar density $c$ in space and time. Left: Spatial, Right: Temporal.
  • Figure 3: Example 2: Distributions of initial molar density and permeability. Left: Initial molar density. Right: Initial permeability.
  • Figure 4: Example 2: Left: Distributions of energy at different time steps. Middle: Mass conservation at different time steps. Right: Minimum and maximum values of molar density.
  • Figure 5: Example 2: Left: Adaptive values of the stabilization parameter at different time steps. Right: Adaptive values of the time step size.
  • ...and 11 more figures

Theorems & Definitions (14)

  • Remark 2.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Remark 4.1
  • Theorem 4.3
  • proof
  • Lemma 5.1
  • proof
  • ...and 4 more