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APPENDIX: PRODUCT FORMULAS

This Appendix will clarify the derivation of the product formulas relating the bulk modulus K = KR and Geff(2). The product formula for KV and Geff(1) is just the dual, and is obtained in a very similar fashion. Since the relevant excitation for Geff(2) has been shown to be a shear strain proportional to (1, 1, -2)T, consider
   \begin{eqnarray}
\left(\begin{array}
{ccc}
C_{11} & C_{12} & C_{13} \
C_{12} & C...
 ...11} + C_{12} - 2C_{13} 
-2(C_{33}- C_{13}) \\ end{array}\right).
 \end{eqnarray} (56)
Then, by applying the inverse of the matrix in (62) to the left side of the equation, we get the useful formula:
   \begin{eqnarray}
\left(\begin{array}
{c}
1 1 -2 \\ end{array}\right)
=
\left(\be...
 ...11} + C_{12} - 2C_{13} 
-2(C_{33}- C_{13}) \\ end{array}\right),
 \end{eqnarray} (57)
which supplies two independent identities among the elastic coefficients. These are

 
1 = (<I>S11I>+<I>S12I>)(<I>C11I>+<I>C12I>-2<I>C13I>) - 2<I>S13I>(<I>C33I>-<I>C13I>)      (58)
and
 
-1 = <I>S13I>(<I>C11I>+<I>C12I>-2<I>C13I>) - <I>S33I>(<I>C33I>-<I>C13I>).      (59)
Adding these together and switching to the a,b,c notation for stiffnesses, we find
   \begin{eqnarray}
S_{11} + S_{12} + S_{13} = (S_{33}+2S_{13})
{{c-f}\over{a+b-2f}}.
 \end{eqnarray} (60)
Recalling that
   \begin{eqnarray}
{{1}\over{K}} = 2(S_{11}+S_{12}+S_{13}) + (S_{33}+2S_{13}),
 \end{eqnarray} (61)
and then substituting (66), we find
   \begin{eqnarray}
{{1}\over{K}} = (S_{33}+2S_{13}){{(a-m-f) + (c-f)}\over{a-m-f}}.
 \end{eqnarray} (62)
Then, since
   \begin{eqnarray}
S_{33} + 2S_{13} = {{2(a-m-f)}\over{\omega_+\omega_-}},
 \end{eqnarray} (63)
we find immediately that
   \begin{eqnarray}
6KG_{eff}^{(2)} = \omega_+\omega_-,
 \end{eqnarray} (64)
because Geff(2) = [(a-m-f)+(c-f)]/3.
next up previous print clean
Next: REFERENCES Up: Berryman: Poroelastic shear modulus Previous: ACKNOWLEDGMENTS
Stanford Exploration Project
7/8/2003