Wave propagation in 3-D poroelastic media including gradient effects

Abstract

In the framework of the theory of mixtures, the governing equations of motion of a fluid-saturated poroelastic medium including microstructural (for both the solid and the fluid) and micro-inertia (for the solid) effects are derived. This is accomplished by appropriately combining the conservation of mass and linear momentum equations with the constitutive equations for both the solid and the fluid constituents. The solid is assumed to be gradient elastic, that is, its stress tensor depends on the strain and the second gradient of strain tensor. The fluid is assumed to have an analogous behavior, that is, its stress tensor depends on the pressure and the second gradient of pressure. A micro-inertia term in the form of the second gradient of the acceleration of the solid is also included in the equations of motion. The equations of motion in three dimensions are seven equations with seven unknowns, the six displacement components for the solid and the fluid and the pore-fluid pressure. Because of the microstructural effects, the order of these equations is two degrees higher than in the classical case. Application of the divergence and the rot operations on these equations enable one to study the propagation of plane harmonic waves in the infinitely extended medium separately in the form of dilatational and rotational dispersive waves. The effects of the microstructure and the micro-inertia on the dispersion curves are determined and discussed.