Plane wave solution for elastic wave scattering by a heterogeneous fracture

A plane-wave method for computing the three-dimensional scattering of propagating elastic waves by a planar fracture with heterogeneous fracture compliance distribution is presented. This method is based upon the spatial Fourier transform of the seismic displacement-discontinuity (SDD) boundary conditions (also called linear slip interface conditions), and therefore, called the wave-number-domain SDD method (wd-SDD method). The resulting boundary conditions explicitly show the coupling between plane waves with an incident wave number component (specular component) and scattered waves which do not follow Snell’s law (nonspecular components) if the fracture is viewed as a planar boundary. For a spatially periodic fracture compliance distribution, these boundary conditions can be cast into a linear system of equations that can be solved for the amplitudes of individual wave modes and wave numbers. We demonstrate the developed technique for a simulated fracture with a stochastic (correlated) surface compliance distribution. Low- and high-frequency solutions of the method are also compared to the predictions by low-order Born series in the weak and strong scattering limit.