Numerical simulation of wave propagation in media with discrete distributions of fractures: effects of fracture sizes and spatial distributions

We model seismic wave propagation in media with discrete distributions of fractures using the pseudospectral method. The implementation of fractures with a vanishing width in the 2-D finite-difference grids is done using an effective medium theory (that is, the Coates and Schoenberg method). Fractures are treated as highly compliant interfaces inside a solid rock mass. For the physical representation of the fractures the concept of linear slip deformation or the displacement discontinuity method is used. According to this model, the effective compliance of a rock mass with one or several fracture sets can be found as the sum of the compliances of the host (background) rock and those of all the fractures. To first order, the background rock and fracture parameters can be related to the effective anisotropic coefficients, which govern the influence of anisotropy on various seismic signatures. We test the validity of the method and examine the accuracy of the synthetic seismograms by a comparison with theoretical ray traveltimes. We present three numerical examples to show the effects of different fracture distributions. The first example shows that different spatial distributions of the same fractures produce different wavefield characteristics. The second example examines the effects of variation of fracture scale length (size) compared with the wavelength. The final example examines the case of fractures with a power-law (fractal) distribution of sizes and shows how that affects the wavefield propagation in fractured rock. We conclude that characterization of fractured rock based on the concept of seismic anisotropy using effective medium theories must be used with caution. Scale length and the spatial distributions of fractures, which are not properly treated in such theories, have a strong influence on the characteristics of wave propagation.