Modelling of acoustic waves in homogenized fluid-saturated deforming poroelastic periodic structures under permanent flow

Abstract

Acoustic waves in a poroelastic medium with periodic structure are studied with respect to permanent seepage flow which modifies the wave propagation. The effective medium model is obtained using the homogenization of the linearized fluid–structure interaction problem while respecting the advection phenomenon in the Navier–Stokes equations. For linearization of the micromodel, an acoustic approximation is introduced which yields a problem for the acoustic fluctuations of the solid displacements, the fluid velocity and pressure. An extended Darcy law of the macromodel involves the permeability and advection tensors which both depend on an assumed stationary perfusion of the porous structure. The monochromatic plane wave propagation is described in terms of two quasi-compressional and two quasi-shear modes. Two alternative problem formulations in the frequency domain are discussed. The one defined in terms of displacement and velocity fields leads to generalized eigenvalue problems involving non-Hermitean matrices whose entries are constituted by the homogenized coefficients depending on the incident wave frequencies, whereby degenerate permeabilities can be accounted for. The homogenization procedure and the wave dispersion analysis have been implemented to explore the influence of the advection flow and the microstructure geometry on the wave propagation properties, namely the phase velocity and attenuation. Numerical examples are reported.