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Upscaling of unsaturated flow in fractured porous media

Florian List, Kundan Kumar, Iuliu Sorin Pop, Florin A. Radu

TL;DR

This work gives a rigorous proof for the convergence towards effective models in a mathematical model for flow in a unsaturated porous medium containing a fracture by analyzing the case of a fracture having a fixed width-length ratio.

Abstract

In this work, we consider a mathematical model for flow in a unsaturated porous medium containing a fracture. In all subdomains (the fracture and the adjacent matrix blocks) the flow is governed by Richards' equation. The submodels are coupled by physical transmission conditions expressing the continuity of the normal fluxes and of the pressures. We start by analyzing the case of a fracture having a fixed width-length ratio, called $\varepsilon > 0$. Then we take the limit $\varepsilon \to 0$ and give a rigorous proof for the convergence towards effective models. This is done in different regimes, depending on how the ratio of porosities and permeabilities in the fracture, respectively matrix scale with respect to $\varepsilon$, and leads to a variety of effective models. Numerical simulations confirm the theoretical upscaling results.

Upscaling of unsaturated flow in fractured porous media

TL;DR

This work gives a rigorous proof for the convergence towards effective models in a mathematical model for flow in a unsaturated porous medium containing a fracture by analyzing the case of a fracture having a fixed width-length ratio.

Abstract

In this work, we consider a mathematical model for flow in a unsaturated porous medium containing a fracture. In all subdomains (the fracture and the adjacent matrix blocks) the flow is governed by Richards' equation. The submodels are coupled by physical transmission conditions expressing the continuity of the normal fluxes and of the pressures. We start by analyzing the case of a fracture having a fixed width-length ratio, called . Then we take the limit and give a rigorous proof for the convergence towards effective models. This is done in different regimes, depending on how the ratio of porosities and permeabilities in the fracture, respectively matrix scale with respect to , and leads to a variety of effective models. Numerical simulations confirm the theoretical upscaling results.

Paper Structure

This paper contains 22 sections, 16 theorems, 115 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Dimensional model
  4. Non-dimensionalisation
  5. Main result
  6. Effective models: strong formulation
  7. Existence
  8. Notation
  9. Assumptions
  10. Weak solution
  11. Time discretisation
  12. Existence of solution for the time-discrete problem
  13. A priori estimates
  14. Interpolation in time
  15. Rigorous upscaling
  16. ...and 7 more sections

Key Result

Proposition 5

\newlabelprop:H10 Given $\psi \in W^{1,2}(\Omega)$, its restriction to $\Omega_\rho$, $\rho \in \lbrace m_1, m_2, f \rbrace$, defines a triple $\left (\psi_{m_1}, \psi_{m_2}, \psi_{f} \right ) \in \mathcal{V}$. Conversely, given $\left (\psi_{m_1}, \psi_{m_2}, \psi_{f} \right ) \in \mathcal{V}$, $

Figures (5)

  • Figure 1: Dimensional (left) and dimensionless (right) geometry of the fracture and the surrounding matrix blocks
  • Figure 2: Geometry with two-dimensional fracture in rescaled variables (left) and upscaled geometry with one-dimensional fracture (right)
  • Figure 3: Simulation parameters for the realistic example: geometry, initial and boundary conditions, and van Genuchten parameters
  • Figure 4: Solution to the effective model at $t = 0.18$, pressure head (left) and saturation (right)
  • Figure 5: $x$-averaged fracture solution along the fracture for different $\varepsilon$ and for the effective model at $t = T$ (left), $L^2$ error in the fracture and the matrix blocks for different $\varepsilon$ at $t = T$ (right)

Theorems & Definitions (37)

  • Remark 1: Scaling parameters
  • Remark 2: Assumptions
  • Definition 3: Weak solution
  • Definition 4
  • Proposition 5
  • Proof 1
  • Remark 6
  • Definition 7: Weak solution of Problem $\mathcal{P}_\Omega$
  • Lemma 8
  • Lemma 9
  • ...and 27 more