Fluid flows through fractured porous media along Beavers-Joseph interfaces

We study a fluid flow traversing a porous medium and obeying the Darcy's law in the case when this medium is fractured in blocks by an $\veps$-periodic ($\veps>0$) distribution of fissures filled with a Stokes fluid. These two flows are coupled by a Beavers-Joseph type interface condition. The existence and uniqueness of this flow in our $\veps$-periodic structure are proved. As the small period of the distribution shrinks to zero, we study the asymptotic behaviour of the flow when the permeability and the entire contribution on the interface of the Beavers-Joseph transfer coefficients are of unity order. We find the homogenized problem verified by the two-scale limits of the coupled velocities and pressures. It is well-posed and provides the corresponding classical homogenized problem