Amplitude of Biot's slow wave scattered by a spherical inclusion in a fluid-saturated poroelastic medium

Abstract

Spatial heterogeneity of hydrocarbon reservoirs causes significant attenuation and dispersion of seismic waves due to wave-induced flow of the pore fluid between more compliant to less compliant areas. This paper investigates the interaction between a plane elastic wave in a poroelastic medium with a spherical inhomogeneity of another porous material. The behaviour of both the inclusion and the background medium is described by the low frequency variant of Biot's equations of poroelasticity with the standard boundary conditions at the inclusion surface and for the inclusion size much smaller than the wavelength of the fast compressional wave. The scattering problem is formulated as a series expansion of displacements expressed in the spherical symmetry. The resulting scattered wavefield consists of the scattered normal compressional and shear waves and Biot's slow wave, which attenuates rapidly with distance from the inclusion and represents the main difference with the elastic case. This study concentrates on the attenuation effects caused by the mode conversion into Biot's slow wave. The obtained solution for Biot's slow wave is well described by two terms of the order n=0 and n=2 of the scattering series. The scattering amplitude for the order n=0 is given by a simple expression. The full expression for the order n=2 is very complicated, but can be simplified assuming that the amplitude of the scattered fast (normal) compressional and shear waves are well approximated by the solution of the equivalent elastic problem. This assumption yields a simple approximation for the amplitude of the scattered slow wave, which is quite accurate for a wide range of material properties and is sufficient for the analysis of the scattering amplitude as a function of frequency.In the low frequency limit the scattering amplitude of the slow wave scales with , and reduces to the asymptotic long-wavelength solution of Berryman (1985), which is valid for inclusions much smaller than the wavelength of Biot' slow wave. For inclusions larger than the wavelength of Biot's slow wave, the scattering amplitude is proportional to , which is consistent with the results of Gurevich et al. (1988), which were derived by Born approximation and therefore were limited to weak contrast between the inclusion and the background medium. However, our general solution does not require these assumptions on frequency and material properties. The obtained results can be used in the analysis of the effective properties, attenuation and dispersion of elastic waves in randomly inhomogeneous porous materials.