DISCUSSION AND CONCLUSIONS

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 fluid-saturated 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 $\eta_1$ and $\eta_2$ can be determined in a straightforward way using any effective medium theory we trust (Kachanov, 1994; Prat and Ba$\tilde{z}$ant, 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 $\eta_1$ and $\eta_2$.The only variable is the crack shape, which I have assumed here (as is most commonly done) to be penny-shaped cracks with small aspect ratios. Values of $\eta_1$ and $\eta_2$ 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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