Trace analysis for fracture networks with anisotropic orientations and heterogeneous distributions

Since only intersections with lines or planes are usually available to quantify the properties of real fracture networks, a stereological analysis of these intersections is a crucial issue. This article-the second of a series-is devoted to the derivation of the direct relations between the properties and the observable quantities. First, this derivation is achieved for anisotropic networks whose orientations obey a Fisher probability distribution function; second, it is extended to networks which are heterogeneous in space, i.e., whose density decays according to an exponential law. Five major quantities are determined: the excluded volume, the average number of intersections with a line and with a plane, the average trace length and the surface density of trace intersections. Some of these relations are valid for any convex fracture shape and some only for circular disks; however, numerical simulations show that excellent approximations are obtained by considering disks with the same area as the noncircular fractures.