Modeling wave propagation in cracked porous media with penny-shaped inclusions

Understanding acoustic wave dispersion and attenuation induced by local (squirt) fluid flow between pores and cracks (compliant pores) is fundamental for better characterization of the porous rocks. To describe this phenomenon, some squirt-flow models have been developed based on the conservation of the fluid mass in the fluid mechanics. By assuming that the cracks are represented by isotropically distributed (i.e., randomly oriented) penny-shaped inclusions, this study applies the periodically oscillating squirt flow through inclusions based on the Biot-Rayleigh theory, so that the local squirt flow and global wave oscillation of rock are analyzed in the same theoretical framework of Hamilton's principle. The governing wave-propagation equations are derived by incorporating all of the crack characteristics (such as the crack radius, crack density, and aspect ratio). In comparison with the previous squirt models, our model predicts the similar characteristics of wave velocity dispersion and attenuation, and our results are in agreement with Gassmann equations at the low-frequency limit. In addition, we find that the fluid viscosity and crack radius only affect the relaxation frequency of the squirt-flow attenuation peak, whereas the crack density and aspect ratio also affect the magnitudes of dispersion and attenuation. The application of this study to experimental data demonstrates that when the differential pressure (the difference between confining pressure and pore pressure) increases, the closure of cracks can lead to a decrease of attenuation. The results confirm that our model can be used to analyze and interpret the observed wave dispersion and attenuation of real rocks.