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A fluid-solid interaction problem in porous media

Diego Alonso-Orán, Rafael Granero-Belinchón

TL;DR

The paper analyzes a fluid–structure interaction in a porous medium by deriving reduced interface models for an elastic Muskat problem coupling Darcy flow with a Willmore-type elastic membrane on the interface. It develops two asymptotic regimes: a weakly nonlinear small-slope model that remains nonlocal due to the Dirichlet–to–Neumann expansion and elasticity, and a long-wave lubrication model with a variable mobility that preserves a nonlinear elliptic coupling to the time derivative. The authors establish well-posedness in Wiener spaces for the reduced models, including global existence and decay for small data, and they derive a rigorous leading-order lubrication equation that captures the coupled elasticity and diffusion effects. By bridging Dirichlet–to–Neumann map expansions, geometric elasticity, and lubrication theory, the work provides a rigorous framework for analyzing thin elastic interfaces in porous media and their long-time behavior. Key contributions include explicit nonlocal evolution equations, a robust Wiener-space well-posedness theory, and a global, dissipative thin-film model suitable for porous and geophysical applications.

Abstract

In this work, we derive asymptotic interface models for an elastic Muskat free boundary problem describing Darcy flow beneath an elastic membrane. In a weakly nonlinear regime of small interface steepness, we obtain nonlocal evolution equations that capture the free-boundary dynamics up to quadratic order. In the long-wave thin-film regime, we rewrite the kinematic condition in flux form, flatten the moving domain, and derive a lubrication-type equation. Moreover, we establish well-posedness for these models in suitable Wiener spaces.

A fluid-solid interaction problem in porous media

TL;DR

The paper analyzes a fluid–structure interaction in a porous medium by deriving reduced interface models for an elastic Muskat problem coupling Darcy flow with a Willmore-type elastic membrane on the interface. It develops two asymptotic regimes: a weakly nonlinear small-slope model that remains nonlocal due to the Dirichlet–to–Neumann expansion and elasticity, and a long-wave lubrication model with a variable mobility that preserves a nonlinear elliptic coupling to the time derivative. The authors establish well-posedness in Wiener spaces for the reduced models, including global existence and decay for small data, and they derive a rigorous leading-order lubrication equation that captures the coupled elasticity and diffusion effects. By bridging Dirichlet–to–Neumann map expansions, geometric elasticity, and lubrication theory, the work provides a rigorous framework for analyzing thin elastic interfaces in porous media and their long-time behavior. Key contributions include explicit nonlocal evolution equations, a robust Wiener-space well-posedness theory, and a global, dissipative thin-film model suitable for porous and geophysical applications.

Abstract

In this work, we derive asymptotic interface models for an elastic Muskat free boundary problem describing Darcy flow beneath an elastic membrane. In a weakly nonlinear regime of small interface steepness, we obtain nonlocal evolution equations that capture the free-boundary dynamics up to quadratic order. In the long-wave thin-film regime, we rewrite the kinematic condition in flux form, flatten the moving domain, and derive a lubrication-type equation. Moreover, we establish well-posedness for these models in suitable Wiener spaces.
Paper Structure (12 sections, 4 theorems, 215 equations)

This paper contains 12 sections, 4 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. The elastic Muskat problem and its potential formulation
  3. Geometric setting and eulerian formulation
  4. Potential formulation and dimensionless formulation
  5. Notation, functional spaces, and basic inequalities
  6. A weakly nonlinear model in the small-slope regime
  7. Well-posedness of the weakly nonlinear models for the small slope regime
  8. A lubrication approximation in the thin-film regime
  9. Flux--potential formulation
  10. Flattening the moving strip
  11. A leading-order lubrication model
  12. Well-posedness of the weakly nonlinear models for the thin film regime

Key Result

Theorem 5.1

Let $\lambda=0$ and fix $\chi,\Theta,\sigma>0$. Assume that $h_0\in A^1$ has zero mean. Then there exist constants $\mathcal{C}=\mathcal{C}(\chi,\Theta,\sigma)>0$ and $T=T(\chi,\Theta,\sigma,\|h_0\|_{A^1})>0$ such that, if there exists a (mild) solution to eq:model_sigma_LHS_full_multiplier_1D satisfying the integral formulation where and, for a given profile $h$, the operator $\mathcal{L}_h$

Theorems & Definitions (11)

  • Remark 4.1: Infinite depth models
  • Theorem 5.1
  • proof : Proof of Theorem \ref{['thm:1']}
  • Theorem 5.2
  • proof : Proof of Theorem \ref{['thm:wp_A3_lambda']}
  • Remark 5.3: Infinite-depth case
  • Theorem 5.4
  • proof : Proof of Theorem \ref{['thm:wp_A3_model2']}
  • Remark 6.1: One-dimensional analogue
  • Theorem 7.1
  • ...and 1 more