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HORIZONTAL FRACTURES AND VTI SYMMETRY

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 $\eta_1$ and $\eta_2$ are given by  
 \begin{displaymath}
\gamma_h = \frac{c_{66}-c_{44}}{2c_{44}} = \rho_c\eta_2 G_0,
 \end{displaymath} (10)
and  
 \begin{displaymath}
\epsilon_h = \frac{c_{11} - c_{33}}{2c_{33}} =
\rho_c[(1+\nu...
 ...ac{E_0}{(1-\nu_0^2)}
\simeq \frac{2\rho_c\eta_2G_0}{1-\nu_0}.
 \end{displaymath} (11)
The background shear modulus is G0, and the corresponding Poisson ratio is $\nu_0$. Young's modulus is $E_0 = 2(1+\nu_0)G_0$.We also find that $\delta = \epsilon$ to the lowest order in the crack density parameter. We have chosen to neglect the term in $\eta_1$ in the final expression of (11), as this is on the order of a $1\%$ correction to the term retained. Values of $\eta_1$ and $\eta_2$ 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
FIG1
Figure 1
For aligned vertical cracks: examples of anisotropic compressional wave speed (vp) for Poisson's ratio of the host medium $\nu_0 = 0.00$.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.


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FIG2
FIG2
Figure 2
Same as Figure 1 for SH shear wave speed (vsh).


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Stanford Exploration Project
5/6/2007