Scattering of a compressional wave in a poroelastic medium by an ellipsoidal inclusion

Abstract

SUMMARY We study the interaction of a plane elastic wave in a poroelastic medium with an elliptical heterogeneity of another porous material. The behaviour of both the inclusion and the host medium is described by Biot’s equations of poroelasticity with the standard interface conditions of Deresiewicz and Skalak at the inclusion’s surface. The scattering problem is studied in the Born approximation, which is valid for low contrast of the inclusion’s properties with respect to the host medium. The resulting scattered wavefield consists of the scattered normal compressional and shear waves and a Biot slow wave, which attenuates rapidly with distance from the inclusion. The Born approximation also allows us to derive explicit analytical formulae for the amplitudes of these scattered waves and to compute the amount of energy scattered by the inclusion into these waves. The amplitude and scattering cross-section for the Biot slow wave depend on the relationship between the dimensions of the inclusion and the wavelength of the Biot slow wave. The analytical results for a single inclusion are used to estimate the eVective attenuation of a normal compressional wave in a poroelastic medium with randomly distributed ellipsoidal inclusions. The eVective attenuation due to the elastic scattering of energy by the inclusions is compounded by an additional attenuation caused by poroelasticity, i.e. by the scattering of the incident normal compressional wave into the Biot slow wave. The frequency dependence of this so-called mode conversion attenuation has the form of a relaxation peak, with the maximum of the dimensionless attenuation (inverse quality factor) at a frequency at which the wavelength of the Biot slow wave is approximately equal to the characteristic size of the inclusion. The width and the precise shape of this relaxation peak depend on the aspect ratio of the ellipsoidal inclusion. Physically, the mode conversion attenuation is associated with the wave-induced flow of the pore fluid across the interfaces between the host medium and the inclusions. The results of our study demonstrate how the local flow (or squirt) attenuation can be eVectively modelled within the context of the Biot theory of poroelasticity.