Because a set of aligned fractures may be associated with an excess fracture compliance matrix (Schoenberg and Sayers, 1995, Geophysics), compliance is more suitable than stiffness for modeling the elastic behavior of fractured media. The effective compliance of fractured rock is conveniently expressed as the sum of the compliance matrix of the (unfractured) host rock and the excess compliance matrices of each of the sets of aligned fractures. Similarly, a set of aligned fractures may be associated with an excess fracture permeability matrix (Schoenberg, 1991, Geophysical Prospecting), and the effective permeability of fractured rock is expressible as the sum of the permeability matrix of the host rock and the excess permeability matrices of each of the sets of aligned fractures. Unfortunately, it is not yet known how excess permeability is related, quantitatively, to excess compliance, although other factors being equal, more compliance generally implies higher permeability.
When the rock mass contains multiple sets of aligned vertical fractures, each of which with its own different excess compliance matrix, the situation becomes more complicated. The presence of two (non-orthogonal) or more sets of fractures yields an effective medium which, in its elastic response, is up-down symmetric, i.e., monoclinic with a horizontal plane of mirror symmetry. However, a vertically fractured medium is not a general monoclinic medium since there are constraints on any excess fracture compliance matrix. In particular, for vertical fractures with the 3-axis assumed vertical, the (1,3), (2,3), (3,3) and (3,6) elements (in 6x6 condensed Voigt notation) of the excess fracture compliance matrix must vanish, yielding two constraints on the elements of the effective compliance matrix of a vertically fractured medium: the vanishing of the (3,6) element and of the difference between the (1,3) and (2,3) elements. Note that these constraints do not carry over to the inverse of the compliance matrix, the stiffness matrix, but in either formulation, such a monoclinic medium has but eleven independent elastic parameters.
Once sufficient measurements of wave speeds in different directions are found, it is sometimes possible to deconstruct the properties of the fractured rock into properties of the host rock and properties of up to two sets of vertical fractures. When this can be done, it is possible to say much more about the fractured rock than just fracture orientation. The overall magnitude of the fracture compliances, relative to the compliance of the host rock, is indicative of the degree of fracturing and the ratio of normal compliance to tangential compliance for any given fracture set is indicative of the fluid content of the fractures. Yet, it is generally not possible to extract sufficient azimuthal wave speed information to carry out such deconstruction. However, 3D seismic data, binned in such a way as to yield amplitude vs. azimuth information at the reservoir, can give not only fracture orientation, but also some estimate of the amount of fracture induced azimuthal anisotropy, and hence, an estimate of the degree of fracturing present in the reservoir.