In this paper we undertake a rigorous derivation of the interface conditions between a poroelastic medium (the pay zone) and an elastic body (the non-pay zone). We assume that the poroelastic medium contains a pore structure of the characteristic size ε and that the fluid/structure interaction regime corresponds to diphasic Biot's law. The question is challenging because the Biot's equations for the poroelastic part contain one unknown more than the Navier equations for the non-pay zone. The solid part of the pay zone (the matrix) is elastic and the pores contain a viscous fluid. The fluid is assumed viscous and slightly compressible. We study the case when the contrast of property is of order ε2, i.e. the normal stress of the elastic matrix is of the same order as the fluid pressure. We assume a periodic matrix and obtain the a priori estimates. Then we let the characteristic size of the inhomogeneities tend to zero and pass to the limit in the sense of the two-scale convergence. The obtained effective equations represent a two-scale system for three displacements and two pressures, coupled through the interface conditions with the Navier equations for the elastic displacement in the non-pay zone. We prove that the appropriate interface conditions at the interface between an elastic and a poroelastic medium are: (i) the effective displacement continuity, (ii) the effective normal stress continuity and (iii) the normal Darcy velocity zero from the poroelastic side. In addition we determine the effective boundary conditions for the contact between a poroelastic body and a rigid obstacle, giving us the effective outer boundary conditions.
Some properties of small perturbations against a stationary solution of the nonlinear Schrödinger equation
