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Rigorous Derivation of Discrete Fracture Models for Darcy Flow in the Limit of Vanishing Aperture

Maximilian Hörl, Christian Rohde

Abstract

We consider single-phase flow in a fractured porous medium governed by Darcy's law with spatially varying hydraulic conductivity matrices in both bulk and fractures. The width-to-length ratio of a fracture is of the order of a small parameter $\varepsilon$ and the ratio $K_\mathrm{f}^\star / K_\mathrm{b}^\star$ of the characteristic hydraulic conductivities in the fracture and bulk domains is assumed to scale with $\varepsilon^α$ for a parameter $α\in \mathbb{R}$. The fracture geometry is parameterized by aperture functions on a submanifold of codimension one. Given a fracture, we derive the limit models as $\varepsilon \rightarrow 0$. Depending on the value of $α$, we obtain five different limit models as $\varepsilon \rightarrow 0$, for which we present rigorous convergence results.

Rigorous Derivation of Discrete Fracture Models for Darcy Flow in the Limit of Vanishing Aperture

Abstract

We consider single-phase flow in a fractured porous medium governed by Darcy's law with spatially varying hydraulic conductivity matrices in both bulk and fractures. The width-to-length ratio of a fracture is of the order of a small parameter and the ratio of the characteristic hydraulic conductivities in the fracture and bulk domains is assumed to scale with for a parameter . The fracture geometry is parameterized by aperture functions on a submanifold of codimension one. Given a fracture, we derive the limit models as . Depending on the value of , we obtain five different limit models as , for which we present rigorous convergence results.
Paper Structure (22 sections, 30 theorems, 168 equations, 2 figures)

This paper contains 22 sections, 30 theorems, 168 equations, 2 figures.

Table of Contents

  1. Full-Dimensional Model and Geometry
  2. Full-Dimensional Model in Dimensional Form
  3. Full-Dimensional Model in Non-Dimensional Form
  4. Scaling of Domains and Parameters with Respect to Epsilon
  5. Local Parameterization
  6. Full-Dimensional Problem with Epsilon-Independent Domains
  7. A-Priori Estimates and Weak Convergence
  8. Trace Inequalities
  9. Poincaré-Type Inequalities
  10. Results
  11. Limit Models
  12. Case I: alpha < -1
  13. Case II: alpha = -1
  14. Case III: alpha in (-1,1)
  15. Case IV: alpha = 1
  16. ...and 7 more sections

Key Result

Lemma 1.1

Let $\varepsilon \in ( 0 , \hat{\varepsilon} ]$. Then, ${{\bm{T}}_\mathrm{f}^\varepsilon \colon \Omega_{\mathrm{f}}^\varepsilon \rightarrow \Omega_{\mathrm{f}}^{\hat{\varepsilon}}}$ is a ${\mathcal{C}^1}$-diffeomorphism. Besides, ${{\bm{T}}_\pm^\varepsilon \colon \Omega_{\pm ,\mathrm{in}}^\varepsilo

Figures (2)

  • Figure 1: Sketch of the geometry in the full-dimensional model \ref{['eq:darcydim']} in dimensional form.
  • Figure 2: Sketch of the geometry in the full-dimensional model \ref{['eq:strongdarcyeps']} in non-dimensional form (left) and in the limit of vanishing width-to-length ratio $\varepsilon\rightarrow 0$ (right).

Theorems & Definitions (58)

  • Lemma 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • Lemma 1.5
  • proof
  • Lemma 2.1
  • proof
  • ...and 48 more