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The Existence, uniqueness, and regularity of weak solutions for a thermodynamically consistent two-phase flow model in porous media

Huangxin Chen, Jisheng Kou, Haitao Leng, Shuyu Sun, Hai Zhao

TL;DR

The paper addresses the well-posedness of a thermodynamically consistent two-phase flow model in porous media by constructing fully implicit time semi-discrete and fully discrete schemes, then proving existence of weak solutions for the discrete and continuous problems via the zeros-of-a-vector-field theorem and energy stability. It establishes uniqueness under Lipschitz conditions on the chemical potential and related coefficients, and derives regularity results through artificial/complementary pressures and elliptic PDE theory, culminating in $S_w\in L^\infty(0,T;H^1(\Omega))$ and $\mu_w,p\in L^2(0,T;H^2(\Omega))$. The analysis hinges on energy estimates, compactness (Aubin–Lions), and careful handling of nonlinearities, providing the first rigorous well-posedness and regularity results for Kou et al.'s thermodynamically consistent model. These results support reliable numerical approximation and have potential impact on simulation fidelity in geo-energy applications and groundwater management.

Abstract

Thermodynamically consistent models for two-phase flow in porous media have attracted significant attention in recent years. In this paper, we prove the existence, uniqueness and regularity of the weak solution to such a recent model proposed in [25,35]. To this end, firstly, we introduce a fully implicit time semi-discrete approximation and a fully discrete approximation for an appropriate weak formulation of the thermodynamically consistent model. Next, by using the zeros of a vector field theorem, we prove the existence of the weak solution for the fully discrete approximation. Then the existence of weak solutions for the fully implicit time semi-discrete approximation and the weak formulation of the model are derived by the weak convergence technique and the energy stability estimate. Subsequently, by the Gr{\" o}nwall inequality, we prove the uniqueness result under the smoothness assumption on the chemical potential. Finally, combined with the regularity theory of elliptic partial differential equations (PDE), the regularity of the weak solution for the model with complete Neumann boundary conditions is established.

The Existence, uniqueness, and regularity of weak solutions for a thermodynamically consistent two-phase flow model in porous media

TL;DR

The paper addresses the well-posedness of a thermodynamically consistent two-phase flow model in porous media by constructing fully implicit time semi-discrete and fully discrete schemes, then proving existence of weak solutions for the discrete and continuous problems via the zeros-of-a-vector-field theorem and energy stability. It establishes uniqueness under Lipschitz conditions on the chemical potential and related coefficients, and derives regularity results through artificial/complementary pressures and elliptic PDE theory, culminating in and . The analysis hinges on energy estimates, compactness (Aubin–Lions), and careful handling of nonlinearities, providing the first rigorous well-posedness and regularity results for Kou et al.'s thermodynamically consistent model. These results support reliable numerical approximation and have potential impact on simulation fidelity in geo-energy applications and groundwater management.

Abstract

Thermodynamically consistent models for two-phase flow in porous media have attracted significant attention in recent years. In this paper, we prove the existence, uniqueness and regularity of the weak solution to such a recent model proposed in [25,35]. To this end, firstly, we introduce a fully implicit time semi-discrete approximation and a fully discrete approximation for an appropriate weak formulation of the thermodynamically consistent model. Next, by using the zeros of a vector field theorem, we prove the existence of the weak solution for the fully discrete approximation. Then the existence of weak solutions for the fully implicit time semi-discrete approximation and the weak formulation of the model are derived by the weak convergence technique and the energy stability estimate. Subsequently, by the Gr{\" o}nwall inequality, we prove the uniqueness result under the smoothness assumption on the chemical potential. Finally, combined with the regularity theory of elliptic partial differential equations (PDE), the regularity of the weak solution for the model with complete Neumann boundary conditions is established.
Paper Structure (12 sections, 13 theorems, 132 equations)

This paper contains 12 sections, 13 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Mathematical model and preliminary
  3. Approximation of the continuous model
  4. Fully implicit time semi-discrete approximation
  5. Fully discrete approximation
  6. The existence of weak solutions
  7. The proof of Theorem \ref{['thm 3.1']}
  8. The proof of Theorem \ref{['thm semi3.2']}
  9. The proof of the existence for Theorem \ref{['thm 2.1']}
  10. Uniqueness of the weak solution
  11. Regularity of the weak solution
  12. Conclusion

Key Result

Theorem 2.1

Under Assumption assum 4, let a triplet $(S_w,\mu_w,p)$ be a weak solution of equF20, then the saturation $S_w$ satisfies $S_\epsilon\leq S_w\leq 1-S_\epsilon$.

Theorems & Definitions (28)

  • Remark 1
  • Definition 1
  • Remark 2
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Definition 2
  • Theorem 3.1
  • Definition 3
  • Theorem 3.2
  • ...and 18 more