Interpretation of the results

Now we have derived all the results needed to interpret Eq. (28) and show how it is related to (27). First, we note some of the main terms missing from (28) are those due to approximations made to $\delta$ and the denominators of (27), which have been approximated as $f \simeq c - 2l$ instead of $f \simeq c(1+\delta) - 2l$.Then, from (56), it is easy to see that the final term in (28) vanishes to lowest order, and that the remainder is given exactly by the shear modulus fluctuation terms in brackets in (53) -- in complete agreement with the final terms of (27). Then, from (60), it follows that the leading contribution to the factor $c\delta + 4l\gamma$ is

$$\begin{eqnarray} c\delta + 4l\gamma \simeq 2c\left<{{m - \mu}\over{\lambda + 2\mu}}\right\gt, \end{eqnarray}$$ (62)

in complete agreement with the second term on the right hand side of (27).

In the case of very strong fluctuations in the layer shear moduli, then (53) and (60) both show that pore fluids effects are magnified due to the fluctuations in layer shear moduli and, therefore, contribute more to the anisotropy correction factors $2c^*(\epsilon^*-\delta^*)$ and $2c^*\delta^*$for undrained porous media. So these effects will be more easily observed in seismic, sonic, or ultrasonic data under these circumstances. When these effects are present, the vertically polarized quasi-shear mode will show the highest magnitude effect, the horizontally polarized shear mode will show no effect, and the quasi-compressional mode will show an effect of intermediate magnitude. It is known that these effects, when present, are always strongest at $45^\circ$, and are diminished when the angle of propagation is either $0^\circ$ or $90^\circ$ relative to the layering direction. We will test these analytical predictions with numerical examples in the next section.

To summarize our main result here: The most significant contributions of the liquid dependence to shear waves comes into the wave dispersion formulas through coefficient a (or equivalently $\epsilon$). Equations (53) and (54) show that

 

<I>aI> = 2<I>fI> - <I>cI> + <I>mI> + 3<I>GI>_<I>effI>. (63)

For small fluctuations in $\mu$, coefficients a and c have comparable magnitude dependence on the fluid effects, but of opposite sign. For large fluctuations, the effects on a are much larger (quadratic) than those on c (linear). Propagation at normal incidence will never show much effect due to the liquids, while propagation at angles closer to $45^\circ$ can show large enhancements in both quasi-P and quasi-SV waves (when shear fluctuations are large), but still no effect on SH waves.