Many geological structures of interest are known to exhibit fracturing. Fracturing directly affects seismic wave propagation because, depending on its scale, fracturing may give rise to scattering and/or anisotropy. A fracture may be described mathematically as an interface in nonwelded contact (i.e., as a displacement discontinuity). This poses a difficulty for finite-difference modeling of seismic wave propagation in fractured media, because the standard heterogeneous approach assumes welded contact. In the past, this difficulty has been circumvented by incorporating nonwelded contact into the medium parameters using equivalent medium theory. We present an alternate method based on the homogeneous approach to finite differencing, whereby nonwelded contact boundary conditions are imposed explicitly. For simplicity, we develop the method in the SH-wave case.
In the homogeneous approach, nonwelded contact boundary conditions are discretized by introducing auxiliary, so-called fictitious, grid points. Wavefield values at fictitious grid points are then used in the discrete equation of motion, so that the time-evolved wavefield satisfies the correct boundary conditions. Although not as general as the heterogeneous approach, the homogeneous approach has the advantage of being relatively simple and manifestly satisfying nonwelded contact boundary conditions. For fractures aligned with the numerical grid, the homogeneous and heterogeneous approaches yield identical results. In particular, in both approaches nonwelded contact results in a larger maximum stable time step size than in the welded contact case.