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Deformable porous media with degenerate hysteresis in gravity field

Chiara Gavioli, Pavel Krejčí

TL;DR

This work analyzes flow in a deformable unsaturated porous medium where the pressure-saturation relation shows strong hysteresis that degenerates at turning points and where gravity drives moisture flux. The authors formulate a coupled PDE-ODE system with a hysteresis operator $G[u]$ acting on the saturation, and a viscoelastic solid skeleton coupled to fluid pressure via a gravity term. Existence of a weak solution is established using a time-discretization scheme, a hysteresis-variant Moser iteration to obtain uniform bounds, and a convexity/compactness framework for passing to the limit in Orlicz spaces, under a convexifiable Preisach structure for $G$. The results extend prior degenerate diffusion analyses to gravity-driven, deformable porous media and provide a robust mathematical foundation with potential relevance to hydrogeology and geomechanics, along with avenues for numerical implementation and model refinements.

Abstract

Hysteresis in the pressure-saturation relation in unsaturated porous media, which is due to surface tension on the liquid-gas interface, exhibits strong degeneracy in the resulting mass balance equation. Solutions to such degenerate equations have been recently constructed by the method of convexification even if the permeability coefficient depends on the hysteretic saturation. The model is extended here to the case that the solid matrix material is viscoelastic and that the system is coupled with a gravity driven moisture flux. The existence of a solution is proved by compact anisotropic embedding involving Orlicz spaces with respect to the time variable.

Deformable porous media with degenerate hysteresis in gravity field

TL;DR

This work analyzes flow in a deformable unsaturated porous medium where the pressure-saturation relation shows strong hysteresis that degenerates at turning points and where gravity drives moisture flux. The authors formulate a coupled PDE-ODE system with a hysteresis operator acting on the saturation, and a viscoelastic solid skeleton coupled to fluid pressure via a gravity term. Existence of a weak solution is established using a time-discretization scheme, a hysteresis-variant Moser iteration to obtain uniform bounds, and a convexity/compactness framework for passing to the limit in Orlicz spaces, under a convexifiable Preisach structure for . The results extend prior degenerate diffusion analyses to gravity-driven, deformable porous media and provide a robust mathematical foundation with potential relevance to hydrogeology and geomechanics, along with avenues for numerical implementation and model refinements.

Abstract

Hysteresis in the pressure-saturation relation in unsaturated porous media, which is due to surface tension on the liquid-gas interface, exhibits strong degeneracy in the resulting mass balance equation. Solutions to such degenerate equations have been recently constructed by the method of convexification even if the permeability coefficient depends on the hysteretic saturation. The model is extended here to the case that the solid matrix material is viscoelastic and that the system is coupled with a gravity driven moisture flux. The existence of a solution is proved by compact anisotropic embedding involving Orlicz spaces with respect to the time variable.
Paper Structure (8 sections, 4 theorems, 148 equations, 1 figure, 1 table)

This paper contains 8 sections, 4 theorems, 148 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Derivation of the model
  3. Statement of the problem
  4. Time discretization
  5. Uniform upper bounds
  6. Convexity estimate
  7. Proof of Theorem \ref{['t1']}
  8. Conclusions

Key Result

Proposition 3.2

Let $G$ be a regular Preisach operator in the sense of Definition dpr. Then it can be extended to a Lipschitz continuous mapping $G: L^p(\Omega; C[0,T]) \to L^p(\Omega; C[0,T])$ for every $p \in [1,\infty)$.

Figures (1)

  • Figure 1: Typical experimental hysteresis dependence in porous media between the logarithm soil suction $\psi$ and the volumetric water content $\theta$.

Theorems & Definitions (7)

  • Definition 3.1
  • Proposition 3.2
  • Definition 3.3
  • Theorem 3.7
  • Proposition 4.1
  • proof
  • Proposition 6.1