An Energy-Stable, Bound-Preserving and Locally Conservative Numerical Framework for Multicomponent Gas Flow in Poroelastic Media

TL;DR

This work addresses multicomponent gas flow in poroelastic media by formulating a thermodynamically consistent Maxwell–Stefan–Darcy–Biot model with a Peng–Robinson EOS for the free energy. A stabilized, bound-preserving discretization is developed, combining a semi-implicit time scheme with adaptive stepping and a variable transformation that ensures $0< c_i<1/\beta^*$ and $0< c<1/\beta^*$; spatially, a mixed FE/RT0 upwind scheme is used for mass transport, while a DG method handles the poroelastic momentum to avoid locking. The fully discrete scheme preserves the original energy dissipation law and maintains componentwise and total density bounds through a contractive nonlinear iteration, with rigorous convergence guarantees. Numerical experiments in 2D and 3D demonstrate energy decay, mass conservation for each component, and robust performance in heterogeneous media, highlighting potential applications in carbon sequestration, hydrogen storage, and subsurface transport.

Abstract

In this paper, we propose a robust and efficient numerical framework for simulating multicomponent gas flow in poroelastic media, with a focus on preserving fundamental thermodynamic principles and ensuring computational reliability. The model captures the complex nonlinear coupling between multicomponent transport and solid deformation, while addressing critical numerical challenges such as mass conservation, energy stability, and molar density boundedness. To achieve this, we develop a stabilized discretization approach that guarantees the preservation of the original energy dissipation law and ensures the boundedness of each gas component's molar density. Furthermore, the proposed method incorporates an adaptive time-stepping strategy that dynamically adjusts the time step size based on the system's dynamics, significantly enhancing computational efficiency without compromising stability or accuracy. For spatial discretization, a mixed finite element method combined with an upwind scheme is employed for the flow and transport equations to ensure local mass conservation, while a discontinuous Galerkin (DG) method is utilized for discretizing the momentum equation of poroelasticity to effectively overcome numerical locking phenomena. Numerical experiments are presented to demonstrate the performance, robustness, and applicability of the method in simulating multicomponent gas flow under various scenarios.

Figures (25)

• Figure 1: Example 1: Left: Initial distributions of permeability. Middle: Initial distributions of molar density of CO$_2$. Right: Initial distributions of molar density of CH$_4$.
• Figure 2: Example 1: Left: Distributions of energy at different time steps. Middle: Minimum and maximum values of molar density. Right: Adaptive values of the time step size.
• Figure 3: Example 1: Left: Mass conservation of CO$_2$ at different time steps. Right: Mass conservation of CH$_4$ at different time steps.
• Figure 4: Distributions of molar density of CO$_2$ at different times in Example 1. Top-left: $n = 100$. Top-right: $n = 200$. Bottom-left: $n = 300$. Bottom-right: $n = 400$.
• Figure 5: Distributions of chemical potential of CO$_2$ at different times in Example 1. Top-left: $n = 100$. Top-right: $n = 200$. Bottom-left: $n = 300$. Bottom-right: $n = 400$.

Theorems & Definitions (21)

• Theorem 3.1
• proof
• Remark 4.1
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Lemma 4.3
• Theorem 4.4
• proof