Effective elastic properties and S-wave anisotropy for rocks containing any oriented penny-shaped cracks in transversely isotropic background

Fractures have a significant influence on rock elastic properties. The existing elastic models for cracked medium mainly consider the crack dip angle. In this work, based on the previous transversely isotropic (TI) background and arbitrary dip crack model, a theoretical model is derived using the Bond transform matrix to quantify the effects of any oriented penny-shaped cracks on the elastic properties of rocks with a TI background. Based on this model, the effects of crack orientation on rock elastic properties are studied. Particularly, a special case with random crack orientation is investigated by using the spherical averaging of the crack compliance matrix with any orientations. Furthermore, the S-wave anisotropy for rocks containing multiple crack sets also is analyzed. The results indicate that the crack orientation has a great influence on rock elastic properties. Especially, the rock elastic properties indicate nonlinear changes with crack azimuth angles when cracks are inclined with respect to the TI background isotropic plane. The randomly oriented cracks decrease the overall rock elastic moduli but have little effect on rock anisotropy. For rocks containing multiple crack sets, parallel crack sets have great effects on the S-wave anisotropy, whereas the effects of orthogonal crack sets are negligible. Using this model, a complete workflow for the inversion of fracture properties using field data measured by acoustic logging is introduced. To explain the effect of crack azimuth angle, the crack density is inverted without considering crack orientation, only considering crack dip angle, and considering both crack dip and azimuth angles. The results indicate that the inversion results considering the crack dip and azimuth angles are in good agreement with the resistivity imaging results. This work provides the theoretical basis for fracture properties inversion for rocks with a TI background.