Upscaling of unsaturated flow in fractured porous media

TL;DR

This paper develops and analyzes mathematical models for unsaturated flow in fractured porous media, focusing on the limit where fracture width shrinks, and confirms the models with numerical simulations.

Contribution

It provides a rigorous derivation of effective models for flow in fractured media as the fracture width approaches zero, considering different scaling regimes.

Findings

Convergence of models as fracture width tends to zero

Different effective models depending on porosity and permeability scaling

Numerical simulations validate the theoretical upscaling results

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.