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  

$$\gamma_h = \frac{c_{66}-c_{44}}{2c_{44}} = \rho_c\eta_2 G_0,$$ (10)

and  

$$\epsilon_h = \frac{c_{11} - c_{33}}{2c_{33}} = \rho_c[(1+\nu... ...ac{E_0}{(1-\nu_0^2)} \simeq \frac{2\rho_c\eta_2G_0}{1-\nu_0}.$$ (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
Figure 1 For aligned vertical cracks: examples of anisotropic compressional wave speed (v_p) 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.

 

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