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 theo-retically 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.
NSTL