The interacting inclusion model of wave-induced fluid flow

An inclusion-based model for the long-wavelength equivalent medium properties of a rock-like composite should not only be consistent with the Brown-Korringa relations for the (zero-frequency) dependence of the effective compliance tensor on the bulk modulus of the saturating fluid, but also allow for non-dilute concentrations of communicating cavities (pores, cracks, fractures, etc.) and even multiple solid constituents. As discussed by Jakobsen et al. in the proceedings of the 10th International Workshop on Seismic Anisotropy, the standard (stiffness-based) T-matrix approach to wave-induced fluid flow (which represents an extension of the ideas of Hudson and his associates) appears to be consistent with the Brown-Korringa relations if and only if the spatial distributions of inclusions/cavities are the same for all pairs of interacting inclusions/cavities. In the 'revised' (compliance-based) T-matrix approach to wave-induced fluid flow (presented in this paper), however, one can estimate the effective compliances as complex-valued functions of frequency in a manner which satisfies all these criteria (independent of the aspect ratios of the two-point correlation functions of ellipsoidal symmetry) and leads to rather simple analytical results in the limits of extremely low and high frequencies, provided that a certain quantity (depending on the structural correlations among the inclusions) is small. By 'revised' it is meant that a higher-order expression (which takes into account the effects of spatial distribution and strain propagation between two inclusions only) has been used when incorporating the effects of dynamic fluid pressure communication into modelling the effective viscoelastic behaviour of cracked/porous rocks. In other words, the response of a single inclusion/cavity is coupled to the overall properties in the present formalism. Numerical results suggest that the compliance-based formulation of the present paper gives the same results as the stiffness-based T-matrix approach of a previous paper if and only if the frequency is below a critical limit associated with a peak in the attenuation spectrum for a porous medium with embedded cracks that are fully aligned.