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Next consider horizontal fractures, as illustrated by the correction matrix
(9). The axis of
fracture symmetry is uniformly vertical, and so such a reservoir would
exhibit VTI symmetry. The resulting expressions for the Thomsen
parameters in terms of the Sayers and Kachanov (1991) parameters
and are given by

| |
(10) |

and
| |
(11) |

The background shear modulus is *G*_{0}, and the corresponding Poisson ratio
is . Young's modulus is .We also find that to the lowest
order in the crack density parameter. We have chosen to neglect the
term in in the final expression of (11),
as this is on the order of a correction to the term retained.
Values of and can be determined from simulations
and/or effective medium theories (Zimmerman, 1991; Berryman and Grechka, 2006).
They depend on the elastic constants of the background medium, and
on the shape of the cracks (assumed to be penny-shaped in these examples).

**FIG1
**

Figure 1 For aligned vertical cracks:
examples of anisotropic compressional wave speed (*v*_{p}) for
Poisson's ratio of the host medium .Velocity curves in black are exact for the fracture model discussed
in the text. The Thomsen weak anisotropy velocity curves for the same
fracture model were then overlain in blue.

**FIG2
**

Figure 2 Same as Figure 1 for SH shear wave speed (*v*_{sh}).

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Stanford Exploration Project

5/6/2007