Wave dispersion and attenuation due to multi-scale wave-induced fluid flow in layered partially saturated pore-crack media

Abstract

The complex heterogeneities of underground rocks will cause wave-induced fluid flows at different scales, which consequently leads to the wave velocity dispersion and energy loss. Mastering and modeling the frequency-dependent elastic and attenuation behaviors are of great significance to characterize underground rocks using multi-scale geophysical data. Following Biot's approach, the constitutive relationship, kinetic energy and dissipation functions in regard to wave induced global fluid flow, interlayer local fluid flow and squirt flow in the annular and penny-shaped cracks are established. From the Lagrange equations, the wave equations considering multi-scale wave induced fluid flow are further derived, which yields three P waves and one S wave. The frequency-dependent velocity and attenuation of fast P wave calculated by the multi-scale wave equations present nice match with that of the single-scale or dual-scale theories in the corresponding frequency bands. Besides, the multi-scale wave theory, under certain circumstance, can be degenerated to the widely known theories including Tang's pore-crack theory, the layered double-porosity theory and Biot's theory, which theoretically illustrates the rationality of the novel wave equation. In order to adjust the low-frequency velocity of the multi-scale wave equations to Gassmann velocity, the dynamic fluid modulus (DFM) is introduced into the multi-scale wave theory. However, the original multi-scale wave theory behaves better fit with the experimental data in comparison with the DFM multi-scale wave theory. The effect of micro-parameters on the dispersion and attenuation calculated by the multi-scale wave theory indicates that the annular crack deforms more with weaker stiffness than the penny-shaped cracks under the same aspect ratio. • We derive the wave equations in layered partially saturated pore-crack media considering multi-scale WIFF. • The frequency-dependent velocity and attenuation are analyzed in wide frequency bands based on the wave equations. • The dispersion and attenuation show fair correspondence with previously known theories and laboratory measurement data.