Effective fluid flow in a porous medium containing a thin fissure

Abstract

We consider the behavior of the solution to Stokes system in porous medium, with characteristic size of the pores ε and containing a thin fissure {0≤x_n≤η}, of width η. The limit when ε and η tend to zero gives the effective (or homogenized) behavior for the flow. Depending on the comparison between ε and η we obtain three possible limits:

– When η<κε^2/3 the fissure is not giving any contribution, i.e., the effective fluid flow is governed by the Darcy law.

– When η~κε^2/3 the coupling effect appears and the effective flow behaves like Darcy flow in the porous medium coupled with the tangential flow on the surface {x_n = 0}. Compared to the previous case η<ε^2/3, the effective velocity has now an additional tangential component concentrated on {x_n = 0}.

– When η>κε^2/3 the fissure is dominant and the tangential flow at {x_n = 0} is dominant. Meanwhile in the other part of the porous medium the effective velocity is equal to zero.