Effective medium modeling of velocity dispersion and attenuation in isotropic rocks

Abstract

Calculation of the velocity of elastic waves that propagate in saturated rocks is complicated by many dispersion and attenuation mechanisms. Wave-induced fluid flow at pore and mesoscopic scales is responsible for substantial dispersion and attenuation effects. In addition, dispersion and attenuation phenomena in rocks are due to spatial heterogeneities in density and elastic properties. Whenever the wavelength of a propagating wave is of the same scale as the size of the pores, dispersion takes place in the form of multiple scattering. We have developed an effective medium model that reproduced the combined effects on acoustic velocities of four wave-attenuation mechanisms by replicating results predicted by existing models in saturated isotropic rocks, Biot’s and squirt flow, and acoustic scattering from spherical pores and randomly oriented penny-shaped cracks. First, the strengthening of rocks and wave velocity increase with frequency associated with fluid-related dispersion was modeled by introducing the concept of complex-valued and frequency-dependent equivalent moduli for the solid background. Effective properties were calculated by invoking the newly defined dynamic solid moduli and the classic self-consistent approximation theory. Second, we replicated the geometric effect of idealized spherical and spheroidal porous inclusions on propagating waves with two existing dynamic self-consistent embedding schemes. In doing so, we explicitly derived the equations of the self-consistent model previously developed for the analysis of waves that propagated in a rock containing randomly oriented cracks. Finally, we established an original procedure to combine all dispersion effects into one simple workflow. We examined the performance of the model for a variety of cases, and consideration was given to the possibility of quantifying petrophysical and rock-physics parameters, such as the presence and shape of porous inclusions, permeability, and fluid viscosity, from seismic, sonic, and ultrasonic measurements.