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

VANISHING OF THE ANELLIPTICITY PARAMETERS

One observation made immediately upon computing sample results for the model specified here is that the quasi-$SV$-wave propagating in the [$x_1$-${x_3}$]-plane apparently has constant (or very nearly constant to numerical accuracy) wave speed at all angles in this plane (see Table 3). This result is startling when first seen, but it has been remarked upon previously in the literature by Gassmann (1964) and Schoenberg and Sayers (1995). One useful interpretation of this fact is obtained by noting that for quasi-$SV$-waves to have constant velocity in the plane, it is necessary for the pertinent effective anellipticity parameter to vanish for this plane of propagation. If it does not vanish identically, then it must at least vanish to first order in the fracture dependent correction factors (shown here in Table 1) in order to explain the numerical results for the relatively small fracture densities considered here. The condition required for exact vanishing of the pertinent anellipticity parameter in terms of stiffness coefficients turns out to be:

$$c_{11}c_{33} - c_{13}^2 = c_{55}\left(c_{11}+c_{33}+2c_{13}\right),$$ (6)

as was already known to Gassmann (1964). But going farther in the analysis, it takes a fair amount of algebra to show that -- to first order in the correction factors -- the result (6) amounts to a condition relating compliance correction factors:

$$\Delta S_{55} = \Delta S_{11} + \Delta S_{33} - 2\Delta S_{13}.$$ (7)

This condition holds when we can ignore the higher order terms $O(\Delta^2)$, as well as all still higher orders (think of the proportionality $\Delta \propto \rho_f$). Then, we find that (7) is in fact satisfied identically when the expressions in Table 1 are substituted. It is important to notice as well that exact satisfaction of the condition in (7) is true for the general form of the definitions in Table 1, and not just for the restricted definitions used in the examples we computed, where the simplified definitions of (5) were employed to reduce our bookkeeping load. So the result in (7) is more general than just the specific examples I have computed. But the result is certainly not expected to be true for arbitrary compliance matrices. Nevertheless, it does seem to be true for a wide range of compliance matrices having vertical fractures in an otherwise isotropic earth.

$\phi$	$V_{sv}$ (km/s)
0$^o$	0.8602
30$^o$	0.8678
45$^o$	0.8771
60$^o$	0.8896
90$^o$	0.9222

Table 3. Constant $V_{sv}$ wave speeds in the [$x_1$-$x_3$]-plane found for various values of the angle $\phi$ between fracture planes, and for the fixed value of fracture density $\rho _f = 0.20$.

This result surely seems quite interesting all by itself, but the curiously symmetric nature of the full fracture model can be highlighted further by noting that the following two expressions analogous to (7):

$$\Delta S_{44} = \Delta S_{22} + \Delta S_{33} - 2\Delta S_{23},$$ (8)

and

$$\Delta S_{66} = \Delta S_{11} + \Delta S_{22} - 2\Delta S_{12},$$ (9)

which are also both satisfied identically by the same set of expressions found in Table 1. These facts indicate that the model also has vanishing anellipticity factors (at least to the precision at which we are working) in the other orthogonal planes of propagation [$x_2$-$x_3$] and [$x_1$-$x_2$], as well.

As is well-known [see Gassmann (1964)], vanishing of the anellipticity factors means that there will be no triplications of wave arrivals for these models from propagation in any of these planes we have considered. Triplications arise because of complications from taking the derivatives required to compute group velocity from the phase velocities quoted here [also see Berryman (1979) for examples]. Group velocity determines the wave speed of signals and/or pulses of seismic energy [see Brillouin (1946) for a discussion].

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