Summary: We undertake a rigorous derivation of the two-phase Biot’s law describing small deformations of a seabed of the characteristic size \(L_0/\epsilon^2\) and containing a pore structure of the characteristic size rounded \(\epsilon\). 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 rounded \(\epsilon^2\), 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 two-phase system corresponds to the two-phase 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 Biot’s poroelasticity equations.
For Part I, cf. R. P. Gilbert and A. Mikelic, Nonlinear Anal., Theory Methods Appl. 40A, No. 1-8, 185–212 (2000; Zbl 0958.35108).