Numerical experiments of fracture-induced velocity and attenuation anisotropy

Authors

  • J. M. Carcione,

    1. Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (OGS), Borgo Grotta Gigante 42c, 34010 Sgonico, Trieste, Italy. E-mail: jcarcione@inogs.it
    Search for more papers by this author
  • S. Picotti,

    1. Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (OGS), Borgo Grotta Gigante 42c, 34010 Sgonico, Trieste, Italy. E-mail: jcarcione@inogs.it
    Search for more papers by this author
  • J. E. Santos

    1. CONICET, Instituto del Gas y del Petróleo, Facultad de Ingeniera, Universidad de Buenos Aires, Av. Las Heras 2214 Piso 3 C1127AAR Buenos Aires, Argentina
    2. Universidad Nacional de La Plata, La Plata, Argentina
    3. Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907-2067, USA
    Search for more papers by this author

SUMMARY

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 inline image, where the brackets denote the discontinuity across the surface, inline image is a fracture stiffness and inline image 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.

Ancillary