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Analysis for layered media

The analysis presented so far is general for all VTI elastic media. But we can say more by assuming now that the anisotropy arises due to layering. Then, for example, we have the following relations
f + 2l = c\left<{{\lambda + 2l}\over{\lambda + 2\mu}}\right\gt,
 \end{eqnarray} (57)
c - f - 2l = 2c \left<{{\mu - l}\over{\lambda + 2\mu}}\right\gt,
 \end{eqnarray} (58)
a - f - 2l = 2c \left\{\left<{{2m-\mu -l}\over{\lambda + 2\mu}}...
- \left<{{\mu}\over{\lambda + 2\mu}}\right\gt^2\right]\right\}.
 \end{eqnarray} (59)
Eq. (57) is an easy consequence of the Backus averaging formulas. Then, (58) shows that c differs from f + 2l only by a term that measures the difference in the weighted average of $\mu$ and l. Eq. (59) shows that a differs from f + 2l in a more complicated fashion that depends on the difference in the weighted average of (2m-l) and $\mu$, as well as a term that is higher order in the fluctuations of the layer $\mu$ values. Combining these results, we have
G_{eff} = m - {{4c}\over{3}}\left[\left<{{\mu^2}\over{\lambda +...
- \left<{{\mu}\over{\lambda + 2\mu}}\right\gt^2\right],
 \end{eqnarray} (60)
showing that all the interesting behavior (including strong $\mu$fluctuations in the layers together with $\lambda$ dependence) is collected in Geff. Since the product of (58) and (59) is clearly of higher order in the fluctuations of the layer shear moduli, it is not hard to see that, to leading order when these fluctuation effects are small,
{\cal A} \simeq (c-l)(3G_{eff} + m - 4l)
 \end{eqnarray} (61)
from which we can conclude that the important coefficient in (46) is given to a good approximation by
2c(\epsilon-\delta) \simeq 3G_{eff} + m - 4l \sim 4(m-l) = 8l\gamma,
 \end{eqnarray} (62)
where the final expression is a statement about the limiting behavior when either the $\mu$ fluctuations are very small, or when strong undrained behavior is present together with large $\mu$ fluctuations.

To study the fluid effects, the drained Lamé parameter $\lambda$ in each layer should be replaced under undrained conditions by
\lambda^* = K^* - 2\mu/3,
 \end{eqnarray} (63)
where K* was defined by (6). Then, for small fluctuations in $\mu$, Eq. (62) shows that the leading order terms due to these shear modulus variations contributing to $\epsilon - \delta$ actually do not depend on the fluids at all (since m-l does not depend on them). There is an enhancement in the shear wave speed for SV in layered media, just due to the changes in the shear moduli, and independent of any fluids that might be present in that case, but the magnitude of this enhancement is small because the difference m-l is also small. When m-l is large, then the magnitude of the enhancement due to liquids in the pores can be very substantial as we will see in the following examples. So the effects of liquid on Geff will generally be weak when the fluctuations in $\mu$ are weak, and strong when they are strong.

To check the corresponding result for P-waves, we need to estimate $\delta$.Making use of (56), we also have
c\delta = -2c\left<{{\mu - l}\over{\lambda + 2\mu}}\right\gt
 \end{eqnarray} (64)
Working to the same order as we did for the final expression in (62), we can neglect the second term in the square brackets of (64). What remains shows that pore fluids would have an effect on this result. The result is
c^*\delta^* \simeq -2c^*\left<{{\mu - l}\over{\lambda^* +
 \end{eqnarray} (65)
and a similar replacement should also be made for Geff in (60). Eq. (65) shows that, since c* and $\delta^*$ both depend on the $\lambda^*$'s (although in opposite ways, since one increases while the other decreases as $\lambda^*$ increases), the product of these factors will have some dependence on fluids. The degree to which fluctuations in $\lambda^*$ and $\mu$ are correlated or anticorrelated as they vary from layer to layer will also affect these results in predictable ways.

Now we have derived all the results needed to interpret (34) and show how it is related to (33). First, we note the some of the main terms missing from (34) are those due to approximations made to $\delta$ and the denominators of (33), which have been approximated as $f \simeq c - 2l$ instead of $f \simeq c(1+\delta) - 2l$.Then, from (62), it is easy to see that the final term in (34) vanishes to lowest order, and that the remainder is given exactly by the shear modulus fluctuation terms in brackets in (59) -- in complete agreement with the final terms of (33). Then, from (64), it follows that the leading contribution to the factor $c\delta + 4l\gamma$ is
c\delta + 4l\gamma \simeq 2c\left<{{m - \mu}\over{\lambda +
 \end{eqnarray} (66)
in complete agreement with the second term in (33).

In the case of very strong fluctuations in the layer shear moduli, then (59) and (64) both show that effects of the pore fluids can be more strongly felt in the anisotropy correction factors $2c^*(\epsilon^*-\delta^*)$ and $2c^*\delta^*$for undrained porous media, and therefore more easily observed in seismic, sonic, or ultrasonic data. 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: All the liquid dependence in the shear moduli comes into the wave dispersion formulas through coefficient a (or equivalently $\epsilon$). Equations (59) and (60) show that

<I>aI> = 2<I>fI> - <I>cI> + <I>mI> + 3<I>GI><I>effI>.      (67)
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.

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