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Fractured Poroelastic Media in the Limit of Vanishing Aperture

Maximilian Hörl, Kundan Kumar, Christian Rohde

TL;DR

The paper develops a rigorous limit analysis for poroelastic media containing a thin fracture as its aperture tends to zero. By treating the fracture as a full-dimensional subdomain with ε-dependent material parameters and employing a coordinate transformation to ε-independent spaces, the authors derive a priori estimates and prove convergence to a family of limit models. The limit behavior splits according to the fracture elasticity scaling ν_C and the fracture hydraulic scaling ν_K, yielding discrete-fracture or two-scale limits for both mechanics and flow, with detailed cross-interface coupling and potential reductions to interface conditions. These results unify and extend discrete fracture modeling within poroelasticity by systematically linking parameter scalings to limit models, including several regimes where normal-flow or normal-deformation dominance leads to two-scale problems. The framework provides rigorous justification for commonly used reduced models and clarifies when full fracture resolution or reduced interfacial descriptions are appropriate, offering significant implications for numerical modeling of fractured subsurface systems.

Abstract

We consider a poroelastic medium with a thin heterogeneity, also referred to as a fracture. Fluid flow and mechanical deformation inside both bulk and fracture are governed by the quasi-static Biot equations. The fracture's material parameters, such as hydraulic conductivity and elasticity, are assumed to scale with powers of the width-to-length ratio $\varepsilon$ of the fracture. Based on a priori estimates, we rigorously derive limit models as $\varepsilon \rightarrow 0$ and identify different limit regimes. We obtain five regimes for the hydraulic conductivity and two for the elasticity. While many cases yield discrete fracture models, others result in two-scale limit problems dominated by normal flow or deformation.

Fractured Poroelastic Media in the Limit of Vanishing Aperture

TL;DR

The paper develops a rigorous limit analysis for poroelastic media containing a thin fracture as its aperture tends to zero. By treating the fracture as a full-dimensional subdomain with ε-dependent material parameters and employing a coordinate transformation to ε-independent spaces, the authors derive a priori estimates and prove convergence to a family of limit models. The limit behavior splits according to the fracture elasticity scaling ν_C and the fracture hydraulic scaling ν_K, yielding discrete-fracture or two-scale limits for both mechanics and flow, with detailed cross-interface coupling and potential reductions to interface conditions. These results unify and extend discrete fracture modeling within poroelasticity by systematically linking parameter scalings to limit models, including several regimes where normal-flow or normal-deformation dominance leads to two-scale problems. The framework provides rigorous justification for commonly used reduced models and clarifies when full fracture resolution or reduced interfacial descriptions are appropriate, offering significant implications for numerical modeling of fractured subsurface systems.

Abstract

We consider a poroelastic medium with a thin heterogeneity, also referred to as a fracture. Fluid flow and mechanical deformation inside both bulk and fracture are governed by the quasi-static Biot equations. The fracture's material parameters, such as hydraulic conductivity and elasticity, are assumed to scale with powers of the width-to-length ratio of the fracture. Based on a priori estimates, we rigorously derive limit models as and identify different limit regimes. We obtain five regimes for the hydraulic conductivity and two for the elasticity. While many cases yield discrete fracture models, others result in two-scale limit problems dominated by normal flow or deformation.

Paper Structure

This paper contains 25 sections, 15 theorems, 200 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Poroelasticity with Full-Dimensional Fracture
  3. Notation
  4. Geometric Setting
  5. Full-Dimensional Biot System in Dimensional Form
  6. Non-Dimensionalization
  7. Full-Dimensional Biot System in Dimensionless Form
  8. Coordinate Transformation
  9. A Priori Estimates
  10. Limit Models
  11. Bulk Limit Problem
  12. Fracture Limit Problems for the Mechanics Equation
  13. Limit Models with nuC=1
  14. Full-Dimensional Limit Model
  15. Discrete Fracture Limit Model
  16. ...and 10 more sections

Key Result

Proposition 2.5

Given the asm:biot, the Biot problem eq:fulldim-weak admits unique solutions $p_{\pm{\mathrm{f}}}^\varepsilon \in H^1 ( I ; \Phi^\varepsilon ) \cap L^\infty (I ; \Phi^\varepsilon ) \cap W^{1,\infty } (I ; \Lambda^{\!\varepsilon } )$ and ${\bm{u}}_{\pm{\mathrm{f}}}^\varepsilon \in W^{1,\infty} ( I ;

Figures (2)

  • Figure 1: Geometry in ${\tilde{\Omega}}$ from \ref{['Omega']} for the full-dimensional Biot system \ref{['eq:fulldim-dimensional']}.
  • Figure 2: Sketch of the geometry for the dimensionless full-dimensional Biot system \ref{['eq:fulldim-nondim']} (left) and the corresponding limit geometry as $\varepsilon \rightarrow 0$ (right).

Theorems & Definitions (48)

  • Remark 2.1
  • Proposition 2.5
  • proof
  • Corollary 2.7
  • proof
  • proof
  • proof
  • proof
  • Proposition 3.7
  • proof
  • ...and 38 more