Elastic wave propagation and attenuation through deformable porous media containing two immiscible fluids taking into account gravity effect

Abstract

The present study develops a rigorous mathematical framework for describing elastic wave propagation and attenuation in a deformable porous medium containing two immiscible fluids, all under the influence of gravitational forces. Starting from the conservation laws of mass and momentum, we derive a comprehensive set of coupled partial differential equations, employing the displacement vectors of the solid matrix and the two fluids as the primary dependent variables. The classic dynamic equations for dilatational motions, as originally formulated by Biot to incorporate gravitational body forces, emerge as a limiting case of our model when reduced to a single-phase system. A sextic polynomial dispersion equation is subsequently derived, characterizing the relationship between the excitation frequency and the complex-valued dilatational wave number. This generalized dispersion equation includes 66 additional terms beyond existing formulations, thereby enabling a more detailed interpretation of variations in material density and volumetric fraction of both the solid phase and two immiscible interstitial fluids, all subject to gravitational acceleration.