EXAMPLES

When modeling rocks using effective medium theory, or really any approach, it is important to minimize the number of choices (size and shape of the inclusions, etc.) available to the modeler. Especially when dealing with cracks and simultaneously with partial and patchy saturation, the number of possible scenarios multiplies rapidly. For example, it would be entirely realistic to assume that there is a distribution of both crack aspect ratios and sizes present in the rocks. But since this distribution is surely not known, we will assume instead that there is only a single aspect ratio of crack present and choose values to lie in the range $\alpha = 0.001$-0.1. The decision to have only one aspect ratio is arbitrary. But it is motivated by the need to minimize the nonuniqueness inherent in the enterprise of fitting these data. Occam's razor applies here: we try to use the simplest possible model (a single aspect ratio), and if we cannot fit the data then we have learned something about the rock. We will find however that the simplest model is always sufficient when fitting the velocity data alone. The range of values that are considered sensible are based in part on the data of Hadley (1976) on Westerly granite, where it was observed that $\alpha$ ranged from 10^-4 all the way up to unity, with a mode around 10^-3.