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DISCUSSION AND CONCLUSIONS

Our main conclusion is that the two best constants to study for our present purposes are the ones derived from the Reuss and Voigt bounds, Geff(1) and Geff(2). Furthermore, these estimates are naturally paired with the Voigt and Reuss bounds on bulk modulus through the product formulas (see the Appendix for a derivation)
   \begin{eqnarray}
(3K_V)(2G_{eff}^{(1)}) = (3K_R)(2G_{eff}^{(2)}) = \omega_+ \omega_-.
 \end{eqnarray} (55)
The product formulas are true for both drained and undrained constants, but of course the numerical values of these constants differ in going from drained to undrained constants. Of these two estimates, Geff(2) has the simplest form and further is paired with the Reuss bound on bulk modulus, which is actually the true bulk modulus of the simple (not polycrystalline) layered system. So we believe, based on theoretical considerations - especially using energy estimates, that this choice is worthy of special attention. But the numerical experiments show that $\omega_-/2$ has quantitatively very similar values while Geff(1) has qualitatively similar behavior in the cases studied. So the practical advantages of this one choice over the others may not always be overwhelming.

Eq.(60) shows explicitly that Geff(2) = m together with generally negative corrections whose magnitude depends strongly on fluctuations in layer shear modulus. The contribution of these correction terms decreases as Skempton's coefficient B increases, so Geff(2) is a monotonically increasing function of B. This is exactly the behavior we were trying to explicate in the present paper, so (60) is one example of the types of aid-to-intuition that we were seeking. The leading term in Geff(2) is also easy to understand, as Geff(2) was first obtained here using the Voigt average, which is an upper bound on the overall behavior; so it is natural that the leading term is m, which is just the Voigt average of the $\mu$'s in the layers.


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Next: ACKNOWLEDGMENTS Up: Berryman: Poroelastic shear modulus Previous: NUMERICAL EXAMPLES
Stanford Exploration Project
7/8/2003