Equations and fundamental characteristics of compressional waves propagating in fluid-saturated porous materials

Abstract

The main purpose of the paper is to present an extended analysis of the coupled system of linear equations describing compressional wave propagation in fluid-saturated porous materials, and to determine their fundamental characteristics. The present paper is a continuation of the analysis of equations for transverse waves that is presented in the manuscript (Cieszko and Kubik, 2020). The starting point of considerations in both works are nonlinear equations describing the dynamic behavior of the two-phase medium formulated in the spirit of the theory of interacting continua in which the parameters of an isotropic pore space structure are explicitly present in the model, and constitutive equations are formulated for each physical component (porous solid and fluid) separately. This system of equations contains all of the fundamental mechanical couplings between the components of such a medium in the general nonlinear form and of clear physical meaning. The linear coupled equations of this theory for a viscous barotropic fluid and a hyperelastic skeleton are equivalent to Biot's model, widely used in the description and analysis of wave propagation in saturated porous materials and wave interactions with the boundary of such media, and they are the subject of the ongoing analysis. A new method of analysis of the system of differential equations written in matrix form is proposed, based on the spectral decomposition of this system in the vector space of dilatations of the solid and fluid particles, and on the method of differential operator decomposition. Such an approach enabled to obtain two separated differential equations describing the propagation of two attenuated compressional waves called the fast and slow wave. This also allowed expressions for parameters characterizing both elastic compressional waves to be derived: velocities, effective mass densities and elastic modules, and also wave impedances which have been determined for the first time for saturated porous materials. In turn, the separated equations for attenuated waves enabled direct derivation of the dispersion equations for the plane harmonic waves of both types and determination of their frequency characteristics, i.e. phase velocities and attenuation coefficients. The obtained frequency characteristics of the fast and slow waves are fully defined by their velocities in the elastic and in the compact medium, and the coefficient characterizing the viscous interaction of the fluid with the skeleton. The results show that the phase velocity of the long slow wave can be greater than the phase velocity of the long fast wave, and the attenuation of the fast wave can be greater than attenuation of the slow wave. It was also proven that both waves can be anomalously dispersed which means that some porous materials saturated with fluid are metamaterials.