HTI RESERVOIR SYMMETRY FROM ALIGNED VERTICAL FRACTURES

Now the trick to get from horizontal fractures and VTI to aligned vertical fractures and HTI symmetry is relatively simple. We will not need to make any effort to relabel the c_ij's. Rather we just change the meaning of the labels. As long as we stay mentally oriented in the reference frame of the fractures themselves, we can continue to view the z-direction as the symmetry axis and the xy-plane, as the plane of the fractures. The only change we need to make arises from the fact that the surface, where we shoot our seismic survey, is now at 90^o from the fracture plane, whereas for horizontal fractures the surface was at 0^o from the fracture plane. This observation implies that, wherever the angle $\theta$ (measured in radians) appeared in our previous formulas, now we must replace it by $\frac{\pi}{2} - \theta$ radians. Thus, $\sin^2\theta \to \cos^2\theta$ and vice versa in the formulas. This algorithm is exactly right only for those planes that are vertical and also perpendicular to the fracture plane, i.e., at azimuthal angles $\phi = \pm\frac{\pi}{2}$. For all angles, we actually need to replace $\sin^2\theta$ by $\cos^2\theta\sin^2\phi$.Then, when $\phi = 0$ or $\pi$, there is no angular dependence since we are in the plane of the fracture.

For the $\theta$ dependence, taking $\sin^2\theta \to 1 - \sin^2\theta$,is actually a handier way to proceed, because then we can reduce all the formulas to the same equivalent form as the one Thomsen had originally chosen -- if we choose to do so. It is also helpful to backup one step in the Thomsen derivation and restore squares, thereby ``unexpanding'' the square root. Certain approximations are then undone, and the final formulas we obtain will be more accurate.

If $\epsilon$, $\delta$, and $\gamma$ are the Thomsen parameters for the VTI symmetry (horizontal fracture), then, for example,  

$$v^2_{sh}(\frac{\pi}{2}-\theta) = v_s^2(0)\left[1+ 2\gamma\si... ...gamma)\left[1 - \frac{2\gamma}{1+2\gamma}\sin^2\theta\right].$$ (12)

From this result, we deduce that $\gamma \to -\gamma/(1+2\gamma)$. This is a rigorous statement for the form of the equation considered. Then, the weak anisotropy limit will be $\gamma \to -\gamma$, but this final step is not necessary or recommended for some of the higher crack densities considered here.

Similar calculations for v_p² and v_sv² give  

$$v^2_p(\frac{\pi}{2}-\theta) = v_p^2(0)\left[1 + 2\delta\sin^... ...eta\cos^2\theta +2\epsilon\sin^2(\frac{\pi}{2}-\theta)\right]$$ (13)

and  

$$v_{sv}^2(\frac{\pi}{2}-\theta) = v_s^2(0)\left[1 + 2[v_p^2(0)/v_s^2(0)](\epsilon-\delta)\sin^2\theta\cos^2\theta\right],$$ (14)

which lead to the results $\epsilon \to \frac{-\epsilon}{1+2\epsilon} \simeq -\epsilon$, and $\delta \to \frac{\delta - 2\epsilon}{1+2\epsilon} \simeq \delta - 2\epsilon$. As a consistency check, note that $\epsilon - \delta \to (\epsilon-\delta)/(1+2\epsilon) \simeq (\epsilon - \delta)$. Similarly, the pertinent wave speeds are: $v_p(0) \to \sqrt{c_{33}(1+2\epsilon)/\rho} = \sqrt{c_{11}/\rho}$and $v_s(0) \to \sqrt{c_{44}(1+2\gamma)/\rho} = \sqrt{c_{66}/\rho}$in (12), but the remaining velocity [v_sv(0)] does not change since $v_{sv}(\theta)$(within Thomsen's weak anisotropy approximation) is completely symmetric in $\theta$ and therefore has to remain so, also with the same end points, after the switch from $\theta$ to $\frac{\pi}{2} - \theta$.These results were all known previously and can be found in Rüger (2002), p. 75.

Examples of these results for small ($\rho_c = 0.05$) and higher ($\rho_c = 0.1, 0.2$) crack densities [see Berryman and Grechka (2006) for details of the methods used to obtain all the Sayers and Kachanov crack-influence parameters from simulation data and Berryman (2007) for a full discussion of the reservoir application] are presented in Figures 1-6.

 

FIG3
Figure 3 Same as Figure 1 for SV shear wave speed (v_sv).

 

FIG4
Figure 4 Same as Figure 1, but the value of $\nu_0 = 0.4375$.