Biot's Poroelastic Equations Research Articles

Homogenizing the acoustic properties of the seabed, part II

We undertake a rigorous derivation of the diphasic Biot's law, describing small deformations of a seabed of the characteristic size L 0/ε 2 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 ε 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 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 Biot's poroelasticity equations.

Full frequency‐range transient solution for compressional waves in a fluid‐saturated viscoacoustic porous medium1

AbstractAn analytical transient solution is obtained for propagation of compressional waves in a homogeneous porous dissipative medium. The solution, based on a generalization of Biot's poroelastic equations, holds for the low‐ and high‐frequency ranges, and includes viscoelastic phenomena of a very general nature, besides the Biot relaxation mechanism. The viscodynamic operator is used to model the dynamic behaviour associated with the relative motion of the fluid in the pores at all frequency ranges. Viscoelasticity is introduced through the standard linear solid which allows the modelling of a general relaxation spectrum. The solution is used to study the influence of the material properties, such as bulk moduli, porosity, viscosity, permeability and intrinsic attenuation, on the kinematic and dynamic characteristics of the two compressional waves supported by the medium. We also obtain snapshots of the static mode arising from the diffusive behaviour of the slow wave at low frequencies.