Gassmann's fluid substitution formulas for bulk and shear moduli (Gassmann, 1951) were originally derived for the quasi-static mechanical behavior of fluid saturated rocks. It has been shown recently (Berryman and Wang, 2001) that it is possible to understand deviations from Gassmann's results at higher frequencies when the rock is heterogeneous, and in particular when the rock heterogeneity anywhere is locally anisotropic. On the other hand, a well-known way of generating anisotropy in the earth is through fine layering. Then, Backus' averaging (Backus, 1962) of the mechanical behavior of the layered isotropic media at the microscopic level produces anisotropic mechanical behavior at the macroscopic level. For our present purposes, the Backus averaging concept can also be applied to fluid-saturated porous media, and thereby permits us to study how deviations from Gassmann's predictions could arise in an analytical and rather elementary fashion. We consider both closed-pore and open-pore boundary conditions between layers within this model in order to study in detail how violations of Gassmann's predictions can arise in undrained versus drained conditions, or for high versus low frequency waves.
We review some standard results concerning layered VTI media in the first two sections. Then, we discuss singular value composition of the elastic (or poroelastic) stiffness matrix in order to introduce the interpretation of one shear modulus (out of the five shear moduli present) that has been shown recently (Berryman, 2003) to contain all the important behavior related to pore fluid influence on the shear deformation response. These results are then incorporated into our analysis of the Thomsen parameters for weak anisotropy. For purposes of analysis, expressions are derived for the quasi-P- and quasi-SV-wave speeds and these results are then discussed from this new point of view. Numerical examples show that the approximate analysis presented is completely consistent with the full theory for layered media. Our conclusions are summarized in the final section of the paper.