Equivalent medium representation of fractured rock

A similarity may be found between various:approaches for determining the effects of parallel fractures or aligned cracks on seismic wave propagation at wavelengths that are long compared with the scale length of the cracks. Fractures can be modeled using an empirical linear slip condition; however, natural fracture surfaces can also be simulated directly as planar distributions of small isolated areas of slip (cracks) (model 1) or, conversely, as planar distributions of imperfect interfacial contacts (model 2). An alternative is plane surfaces separated by thin continuous layers of viscous fluid or a soft material (model 3). We present analytic expressions for the fracture compliances for these three models and, using these-analytic results, compute the effective compliances and stiffnesses of the fractured material. As a result, it is possible to relate: the measured compliances or stiffnesses directly to the statistics of the microstructural details of a fracture, given appropriate a priori information on the fracture surfaces. The results for model 1 are equivalent to a volume distribution of cracks as studied by Hudson [1980, 1981] for small crack density; the results for model 2 are basically the same as those given by White [1983] for a packing of spheres; and finally, the results for model 3 are in agreement with those given by Backus [1962] for combinations of two constituent layers. These results can be extended to the case Of nonaligned fractures and to allow for fluid flow between cracks and into a porous matrix rock. Finally, it is shown that the ratio of the normal to shear fracture compliance is a good indicator of the properties of the fracture infill.