Schoenberg's angle on fractures and anisotropy: A study in orthotropy

FRACTURE ANALYSIS

For waves propagating in the [$x_1$-$x_3$]-plane with wavenumbers $k_1 = k\sin\theta$ and $k_3 = k\cos\theta$ where $k^2 = k^2_1 + k^2_3$, Tsvankin (1997) shows that we have the following equations [patterned here after the notation of Berryman (1979)]:

$$\rho\omega^2_{\pm} = \frac{1}{2}\left[(c_{11}+c_{55})k^2_{1} + (c_{33}+c_{55})k^2_{3} \pm R\right],$$ (1)

where

$$R \equiv \sqrt{\left[(c_{11}-c_{55})k^2_{1} - (c_{33}-c_{55})k^2_{3}\right]^2 + 4(c_{13}+c_{55})^2k^2_{1}k^2_{3}}$$ (2)

and where $\rho$ (with no subscript) is the inertial density. Equation (1) determines the two wave speeds

$$V^2_{\pm} = \frac{\omega^2_{\pm}}{k^2}.$$ (3)

The quantities $\omega_\pm$ have dimensions of angular frequency, but they are introduced mostly to simplify the form of the equations. The pertinent phase speeds $V_+$ for quasi-$P$-waves and $V_-$ for quasi-$SV$-waves are given respectively by values corresponding to the $+$ and $-$ subscripts in the velocity equation (3). Group velocities [Brillouin (1946); Tsvankin (2005); Rüger (2002)] can be computed relatively easily as well when using the methods outlined earlier by Berryman (1979).

Sayers and Kachanov (1991) consider a model with two sets of possibly nonorthogonal fractures, also possibly having two different fracture density values $\rho_a$ and $\rho_b$. Total fracture density is therefore $\rho_f = \rho_a + \rho_b$. These authors found that the pertinent fracture influence parameters were multiplied in this situation, when the angle between the fracture sets is $\phi$, either solely by $\rho_f$ itself or by one of the two factors:

$$\begin{array}{c} A = \rho_f + \left[\rho_f^2 - 4\rho_a\rho_b... ...t[\rho_f^2 - 4\rho_a\rho_b\sin^2\phi\right]^{1/2}. \end{array}$$ (4)

Table 1 shows the Sayers and Kachanov (1991) results for corrections to the isotropic background values of compliance (in Voigt $6\times6$ matrix notation -- the original paper had results expressed in terms of tensor notation). Those background values are specfically for one model considered having Poisson's ratio $\nu = 0.4375$ (dimensionless), effective bulk modulus $K = 16.87$, shear modulus $\mu = 2.20$, and Young's modulus $E = 6.325$, with all moduli measured in units of GPa. For the assumed inertial density $\rho = 2200.0$ kg/m$^3$, the resulting isotropic background compressional wave speed is $V_p = 3.0$ km/s and shear wave speed is $V_s = 1.0$ km/s. For our computations, we also need the isotropic background compliance values, which are $S_{11} = S_{22} = S_{33} = 6.325$, $S_{12} = S_{13} = S_{23} = - 2.767$, and $S_{44} = S_{55} = S_{66} = 0.4545$. The fracture influence factors $\eta_1$ and $\eta_2$, found for this specific model by Berryman and Grechka (2006), are displayed in Table 2. Some higher order fracture-influence factors were also obtained in the earlier work, but I will not be considering such factors in this short paper.

$\Delta S_{11}$	$=$	$(\eta_1 + \eta_2)A$
$\Delta S_{22}$	$=$	$(\eta_1 + \eta_2)B$
$\Delta S_{33}$	$=$	$0$
$\Delta S_{12}$	$=$	$\eta_1\rho_f$
$\Delta S_{13}$	$=$	$\eta_1 A/2$
$\Delta S_{23}$	$=$	$\eta_1 B/2$
$\Delta S_{44}$	$=$	$\eta_2 B$
$\Delta S_{55}$	$=$	$\eta_2 A$
$\Delta S_{66}$	$=$	$2\eta_2\rho_f$

Table 1. Compliance matrix correction values for the vertical fracture model considered in Eq. 4, which are also true for the specific limit of Eq. 5, as a special case of the general result.

In the examples that follow, I will consider only the case of equal fracture densities $\rho_a = \rho_b = \rho_f/2$. For this somewhat simpler situation, I also have

$$\begin{array}{c} A = \rho_f(1+\cos\phi),\cr B = \rho_f(1-\cos\phi). \end{array}$$ (5)

Fracture parameter	GPa$^{-1}$
$\eta_1$	$-0.0192$
$\eta_2$	$0.3944$

Table 2. Fracture-influence parameters (see Table 1 for usage) in a model reservoir having isotropic background with Poisson's ratio $\nu = 0.4375$, $V_p = 3.0$ km/s, and $V_s = 1.0$ km/s.

There may also be some uncertainty about exactly which of these factors is which in this degenerate case, because of the sign ambiguity in taking the square root of $\cos^2\phi$. But I will not concern myself with this detail here.

Schoenberg's angle on fractures and anisotropy: A study in orthotropy