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Nonisothermal Richards flow in porous media with cross diffusion

Esther S. Daus, Josipa Pina Milišić, Nicola Zamponi

TL;DR

The paper develops a thermodynamically consistent, nonisothermal, multicomponent Richards-type model with cross-diffusion for porous media and proves the global existence of weak variational entropy solutions. It combines entropy methods with Div-Curl compactness to handle strong coupling and degeneracy, deriving uniform a priori estimates and enabling a rigorous passage to the limit in a fully nonlinear system. The results establish the well-posedness of a complex coupled framework featuring dynamic capillary pressure, cross-diffusion fluxes, and temperature effects, providing a solid analytical basis for numerical schemes in groundwater remediation and thermal-energy storage. Overall, the work advances mathematical understanding of multicomponent, nonisothermal flows in porous media and offers a robust foundation for simulations that include cross-diffusion phenomena.

Abstract

The existence of large-data weak entropy solutions to a nonisothermal immiscible compressible two-phase unsaturated flow model in porous media is proved. The model is thermodynamically consistent and includes temperature gradients and cross-diffusion effects. Due to the fact that some terms from the total energy balance are non-integrable in the classical weak sense, we consider so-called variational entropy solutions. A priori estimates are derived from the entropy balance and the total energy balance. The compactness is achieved by using the Div-Curl lemma.

Nonisothermal Richards flow in porous media with cross diffusion

TL;DR

The paper develops a thermodynamically consistent, nonisothermal, multicomponent Richards-type model with cross-diffusion for porous media and proves the global existence of weak variational entropy solutions. It combines entropy methods with Div-Curl compactness to handle strong coupling and degeneracy, deriving uniform a priori estimates and enabling a rigorous passage to the limit in a fully nonlinear system. The results establish the well-posedness of a complex coupled framework featuring dynamic capillary pressure, cross-diffusion fluxes, and temperature effects, providing a solid analytical basis for numerical schemes in groundwater remediation and thermal-energy storage. Overall, the work advances mathematical understanding of multicomponent, nonisothermal flows in porous media and offers a robust foundation for simulations that include cross-diffusion phenomena.

Abstract

The existence of large-data weak entropy solutions to a nonisothermal immiscible compressible two-phase unsaturated flow model in porous media is proved. The model is thermodynamically consistent and includes temperature gradients and cross-diffusion effects. Due to the fact that some terms from the total energy balance are non-integrable in the classical weak sense, we consider so-called variational entropy solutions. A priori estimates are derived from the entropy balance and the total energy balance. The compactness is achieved by using the Div-Curl lemma.

Paper Structure

This paper contains 12 sections, 11 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. The physical quantities and main assumptions
  3. Model equations
  4. Constitutive relations
  5. Hypotheses
  6. Entropy balance equation
  7. Weak formulation and main result
  8. A priori estimates
  9. Proof of Theorem \ref{['thm.ws']}
  10. Convergence results
  11. Passing to the limit in \ref{['weak.1']}--\ref{['weak.EBE']} when $n \to \infty$.
  12. Appendix

Key Result

Theorem 6

Let $(\vec{\rho}^{(n)},T^{(n)},S^{(n)})$ be a sequence of smooth solutions to cons_w--PC.Sw, bc with initial data Assume further that $\rho^{(n)}_i, T^{(n)}, S^{(n)} > 0$ a.e. in $Q_\mathcal{T}$ for $i=1,\ldots,N$ and $n\in{\mathbb N}$. Suppose that the following convergences of the initial data hold: weakly in $L^1(\Omega)$ and Then, up to a subsequence, $(\vec{\rho}^{(n)},T^{(n)},S^{(n)})$ co

Theorems & Definitions (23)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Definition 1
  • Theorem 6: Weak sequential stability
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 13 more