A model for determination of effective permeability from acoustic wavespeed and attenuation in a rigid two-phase porous medium

We propose a two-phase permeability spherical inclusion model of mesoscopic heterogeneity for a rigid, porous acoustic medium, and apply the non-self-consistent theory to the scattered wave field from a single spherical inclusion to obtain the dispersion curves of phase velocity and attenuation factor (inverse quality factor) for the composite. We then determine the effective permeability from the wave frequency, complex velocity and other effective material parameters such as bulk modulus, porosity and fluid viscosity. Unlike other rigid double porosity models, our model shows that the host material and its volume fraction determine the acoustic velocity and attenuation, but there are additional attenuation peaks (we call them resonance peaks) in the inverse quality factor responses which are a possible indicator of the size of inclusions. Since the thermal relaxation frequency of the inclusion material actually does not affect the wave behaviour of the composite, the effective dynamic permeability is easily calculated from the velocity and attenuation. This result is compared with other measures of effective dynamic permeability, deduced from the effective hydraulic permeability by replacing the permeability of the components with their dynamic values as determined from the Johnson model. We find that for frequencies lower than the resonance frequency, this dynamic permeability model is very similar to that determined from the scattering model of this paper. However, the model which uses the arithmetic mean of the constituents is not valid for inclusions having higher permeability than the host phase whereas the model which uses the harmonic mean is not valid for inclusions having lower permeability than that of the host material. We have checked the validity of the effective dynamic permeability and its applicability to rocks encountered in both near surface and petroleum seismic exploration.