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 deviations from Gassmann's results at higher frequencies, especially for shear modes, can be understood when the rock is heterogeneous on the microscale, 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 study layers of isotropic elastic/poroelastic materials because this is a simple, explicitly calculable model that nevertheless produces surprising results on the overall poroelastic shear modulus behavior. [If we considered instead layers of anisotropic poroelastic materials, the effects we want to study here concerning fluid-shear interactions would arrive before we begin, because they are often automatically present in anisotropic poroelastic materials as was shown earlier by Gassmann (1951) and others (Schoenberg and Douma, 1988; Sayers, 2002). So we could not show what we have set out to show here concerning the fluid effects by considering such inherently anisotropic models.] By studying both closed-pore and open-pore boundary conditions between layers within the chosen model, we learn in great detail just 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, 2004) 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 (originally derived for weak anisotropy, but used here for arbitrary levels of 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.