Seismic dispersion and attenuation in saturated porous rocks with aligned fractures of finite thickness: Theory and numerical simulations — Part 2: Frequency-dependent anisotropy

Numerous theoretical models have been proposed for computing seismic wave dispersion and attenuation in rocks with aligned fractures due to wave-induced fluid flow between the fractures and the embedding background. However, all these models rely on certain assumptions, for example, infinitesimal fracture thickness or dilute fracture concentration, which rarely hold in real reservoirs and, thus, limit their applicability. To alleviate this issue, theoretical models for periodically or randomly spaced planar fractures and penny-shaped cracks were recently extended by the authors to the case of finite fracture thickness for P-waves propagating perpendicular to the fracture plane. Theoretical predictions under low and relatively high fracture density were then assessed by comparing with corresponding numerical simulations. However, the case of arbitrary incidence angles as well as the behaviors of S-waves remained unexplored. In this work, we therefore extended the prediction results to the full stiffness matrix through two theoretical approaches. The first approach uses an interpolation between the low- and high-frequency limits using a relaxation function obtained from the normal-incidence solution. The second approach is based on the linear slip theory with a frequency-dependent fracture compliance. Both derivations rely on the fact that all the stiffness coefficients are controlled by the same relaxation function. With the full stiffness matrix, anisotropic seismic properties can then be studied. P- and S-wave velocities and attenuations at different frequencies and incidence angles and also corresponding anisotropy parameters are calculated for one synthetic 2D rock sample. The results indicate that the predictions provided by the two theoretical approaches are in good agreement with each other and also indicate a good agreement with the corresponding numerical simulations. The extended theoretical models presented in this work are easy to apply and computationally much cheaper than numerical simulations and, hence, can be used in the seismic characterization of fractured reservoirs.