Anisotropy in seismic velocities and amplitudes from multiple parallel fractures

Many rock structures include multiple, near-parallel, planar discontinuities such as bedding planes or joints. The effects of these nonwelded interfaces on seismic wave propagation are often analyzed using effective moduli, in terms of which seismic wave propagation is independent of frequency and without loss, unless the moduli include imaginary terms. An alternative approach is to treat these interfaces as a boundary condition in the seismic wave equation, across which seismic stress is continuous, but seismic particle displacements are discontinuous. The ratio of the stress to displacement is called the specific stiffness of the interface and characterizes the elastic properties of a fracture. For a completely elastic system this results in frequency-dependent reflection and transmission coefficients for each interface as well as a frequency-dependent group time delay. Using multiple, parallel displacement discontinuities and ignoring converted and reflected waves, expressions derived for transmitted wave amplitudes and group velocities show that these depend on frequency, angle of incidence, and polarization in the case of shear waves. Measurements on a laminated steel block show that shear pulses propagating parallel to the laminations and polarized parallel and perpendicular to the plane of the laminations both travel at the velocity for solid steel, although the spectra of these pulses differ considerably. However, the energy of the pulse polarized perpendicular to the laminations may propagate as an interface wave between each pair of laminations. Predictions of the displacement discontinuity model have features quite distinct from many crustal observations to date. We suggest that we are able to model dense populations of coplanar cracks that cannot be treated by effective moduli methods which require a dilute concentration of cracks.