Homogenizing the acoustic properties of the seabed, part II

We undertake a rigorous derivation of the diphasic Blot's law, describing s mall deformations of a seabed of the characteristic size L-0/epsilon (2) an d containing a pore structure of the characteristic size E. 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 ca se, the contrast of property is of order epsilon (2), i.e., the normal stre ss of the elastic matrix is of the same order as the fluid pressure. We sup pose a periodic matrix and obtain the a priori estimates. Then we let the c haracteristic size of the inhomogeneities tend to zero and pass to the limi t in the sense of the two-scale convergence. The obtained effective equatio ns represent a two-scale system for three velocities and two pressures. We prove uniqueness for the homogenized two-scale system. Then we introduce se veral auxiliary problems and obtain a problem without the fast scale, This new system is diphasic and corresponds to the diphasic effective behavior a lready observed in papers by Blot. In the effective equations, it is possib le to distinguish the velocities of the fluid and the solid part, respectiv ely. The effective stress tenser contains an instantaneous elasticity tense r and there are double porosity terms. We give a detailed study of the effe ctive equations and compare them with the original Blot's poroelasticity eq uations. (C) 2001 Elsevier Science Ltd. All rights reserved.