Fractures are common in the Earth’s crust due to different factors, for instance, tectonic stresses and natural or artificial hydraulic fracturing caused by a pressurized fluid. A dense set of fractures behaves as an effective long-wavelength anisotropic medium, leading to azimuthally varying velocity and attenuation of seismic waves. Effective in this case means that the predominant wavelength is much longer than the fracture spacing. Here, fractures are represented by surface discontinuities in the displacement u and particle velocity v as , where the brackets denote the discontinuity across the surface, is a fracture stiffness and is a fracture viscosity.
We consider an isotropic background medium, where a set of fractures are embedded. There exists an analytical solution—with five stiffness components—for equispaced plane fractures and an homogeneous background medium. The theory predicts that the equivalent medium is transversely isotropic and viscoelastic. We then perform harmonic numerical experiments to compute the stiffness components as a function of frequency, by using a Galerkin finite-element procedure, and obtain the complex velocities of the medium as a function of frequency and propagation direction, which provide the phase velocities, energy velocities (wavefronts) and quality factors. The algorithm is tested with the analytical solution and then used to obtain the stiffness components for general heterogeneous cases, where fractal variations of the fracture compliances and background stiffnesses are considered.