Homogenizing the acoustic properties of the seabed, part II

We undertake a rigorous derivation of the diphasic Biotʹs law, describing small deformations of a seabed of the characteristic size L0/ε2 and containing a pore structure of the characteristic size ε. The solid part of the seabed (the matrix) is elastic and the pores contain a viscous fluid. The fluid is supposed incompressible or slightly compressible. In this case, 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 suppose 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 velocities and two pressures. We prove uniqueness for the homogenized two-scale system. Then we introduce several auxiliary problems and obtain a problem without the fast scale. This new system is diphasic and corresponds to the diphasic effective behavior already observed in papers by Biot. In the effective equations, it is possible to distinguish the velocities of the fluid and the solid part, respectively. The effective stress tensor contains an instantaneous elasticity tensor and there are double porosity terms. We give a detailed study of the effective equations and compare them with the original Biotʹs poroelasticity equations.