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Sayers and Kachanov (1991) introduced a convenient method of analyzing
fractured (but otherwise) elastic systems. I showed here that their
method can be successfully generalized to fluidsaturated fractures.
Furthermore,
when their method is used in conjunction with Thomsen's anisotropy
parameters (Thomsen, 1986), we find not only analytical results that
aid our intuition about these complex problems, but also a means to
deconstruct velocity data and then to interpret the nature
(approximate crack density) of fractures in the system being studied.
The magnitudes of the parameters and can be
determined in a straightforward way using any effective medium theory
we trust (Kachanov, 1994; Prat and Baant, 1997; Grechka, 2005);
and also this calculation
can be done just for the isotropic (and, therefore, the simplest)
case. For examples, see TABLE 1. These parameter values do not
change. Only the crack density parameter, the crack orientation
distributions, and possibly the crack shapes, etc., change.
For very dilute fracture systems, any of the standard effective medium
theories will actually produce virtually the same values of the
parameters and .The only variable is the crack shape, which I have assumed here
(as is most commonly done) to be pennyshaped cracks with small
aspect ratios. Values of and can vary with changes
in the assumed microstructure (i.e., other choices of crack shapes), but
values could be tabulated once and for all for the low density limit
with any choices of crack shape we might ever want to consider and then
the numbers would be universally available. Users would not need to
be experts in effective medium theory to make use of these results 
although they would, of course, still need to be experts in the
interpretation of seismic data and, in particular, of the Thomsen
parameters themselves.
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Stanford Exploration Project
4/5/2006