Homogenizing the Darcy/Stokes coupling

Isabelle Gruais (1) , Dan Polisevski (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 IMAR - "Simion Stoilow" Institute of Mathematics

We study a fluid flow traversing a porous medium and obeying the Darcy's law in the case when this medium is fractured by a periodical distribution of fissures filled with a Stokes fluid. These two flows are coupled by a Beavers-Joseph type interface condition. As the small period of the distribution shrinks to zero, the resulting asymptotic behaviour is implicitely described by two underlying macroscopic quantities: the limit of the Stokes velocity and the limit of the Darcy pressure, solutions of a new coupled system obtained by homogenization. The behaviour of the macroscopic two-scale limit of the filtration velocity is given by an explicit two-scale Darcy type law, presenting coupling terms with the gradient of the limit of the Darcy pressure and with the limit of the Stokes velocity.

Mots clés: Fractured porous media; Stokes flow; Beavers-Joseph interface; Homogenization; Two-scale convergence

Domaines: Equations aux dérivées partielles [math.AP]

Isabelle Gruais, Dan Polisevski. Homogenizing the Darcy/Stokes coupling. 2011. ⟨hal-00597970⟩