Misalignment of the orientation of fractures and the principal axes forPandSwaves in rocks containing multiple non-orthogonal fracture sets

Azimuthal anisotropy in rocks can result from the presence of one or more sets of partially aligned fractures with orientations determined by the stress history of the rock. A shear wave propagating in an azimuthally anisotropic medium splits into two components with different polarizations if the source polarization is not aligned with the principal axes of the medium. In the presence of two or more non-orthogonal sets of vertical fractures, the symmetry of the medium may be approximated as monoclinic with a horizontal plane of mirror symmetry if, in the absence of fractures, the rock is transversely isotropic (TI) with the symmetry axis perpendicular to the bedding plane. For such a medium, the fast and slow polarization directions for vertically propagating shear waves are not parallel or perpendicular to any of the fracture planes but lie in directions given by the principal axes of a second-rank fracture compliance tensor. This tensor is independent of the normal compliance of the fractures. It follows that for vertical propagation of shear waves in a vertically fractured TI medium, any number of arbitrarily oriented vertical fracture sets is equivalent to two mutually perpendicular fracture sets, provided that the seismic wavelength is large compared to the size of the fractures. For offsets typical of surface seismic acquisition, the P-wave velocity at fixed offset varies with azimuth as cos2(φ − φ0 ), where φ is the azimuth measured with respect to the fast polarization direction for a vertically polarized shear wave; φ0 depends on both the normal and the shear compliance of the fractures and may differ from zero if the ratio of the normal to shear compliance of the fractures varies significantly between fracture sets. If this ratio is similar for all fractures, φ0 ≈ 0 and the principal axes of the variation in P-wave velocity with azimuth for fixed offset are determined by the principal axes of the second-rank fracture compliance tensor.