Rigorous Upscaling of Unsaturated Flow in Fractured Porous Media

Abstract

In this work, we consider a mathematical model for flow in an 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 ε > 0. Then we take the limit ε → 0 and give a rigorous proof for the convergence toward effective models. This is done in different regimes, depending on how the ratio of porosities and permeabilities in the fracture, respectively, in the matrix, scale in terms of ε, and leads to a variety of effective models. Numerical simulations confirm the theoretical upscaling results.