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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
 (56)
Then, by applying the inverse of the matrix in (62) to the left side of the equation, we get the useful formula:
 (57)
which supplies two independent identities among the elastic coefficients. These are

 1 = (S11I>+S12I>)(C11I>+C12I>-2C13I>) - 2S13I>(C33I>-C13I>) (58)
and

 -1 = S13I>(C11I>+C12I>-2C13I>) - S33I>(C33I>-C13I>). (59)
Adding these together and switching to the a,b,c notation for stiffnesses, we find
 (60)
Recalling that
 (61)
and then substituting (66), we find
 (62)
Then, since
 (63)
we find immediately that
 (64)
because Geff(2) = [(a-m-f)+(c-f)]/3.

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