ABSTRACT
We present an effective technique to model seismic wave propagation in media with discrete distributions of fractures based on the combination of the finite difference and equivalent medium theories. The distribution of discrete fractures with vanishing apertures in the 2D finite difference grids is implemented using the equivalent medium theory. Fractures are treated as highly compliant interfaces inside a solid rock mass. For the physical representation of the fractures the concept of linear slip or displacement discontinuity model is used. The effective compliance of a fractured solid with multiple fracture sets can be found as the sum of the compliances of the matrix solid (background) and extra compliance introduced as a result of the presence of fractures. To the first order, the matrix solid and fracture parameters can be related to the effective anisotropic coefficients, which govern the influence of anisotropy (a result from fractures) on seismic wave propagation. We test the validity of the method by comparison with theoretical ray method. Several numerical examples are presented to demonstrate the effectiveness of our method showing the effects of different fracture distributions. We show that different spatial distributions with the same fracture density can produce significant different wave-field characteristics even when fracture density is small. We also examine the effects of variation of fracture scale-length (size) as compared to the wavelength. In the case of fractures having a power-law (fractal) distribution of sizes, we show how the variation of scale-length affects the wave-field. We conclude that characterization of fractured solids 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 equivalent medium theories, have a strong influence on the characteristics of wave propagations.
