Our goal is to obtain some new insight into the effective shear modulus of a poroelastic system in order to understand how the shear and bulk modes become coupled in such systems and thereby violate Gassmann's (1951) results for quasi-static systems having nonzero fluid permeability. The purpose of such an analysis will ultimately be to understand why some laboratory ultrasonics data show that the effective shear modulus of porous saturated and partially saturated rocks/systems has a substantial dependence on saturation when Gassmann's result would appear to deny the possibility of such behavior. It was demonstrated by Berryman and Wang (2001) that such deviations from Gassmann's predictions are expected, but they are surely not universal. For example, local elastic isotropy and spherical pores are two cases in which the shear modulus should remain independent of pore-fluid saturation.
For the transversely isotropic system arising from finely layered isotropic layers, we know that four out of five of the shear modes of the system are always independent of fluid saturation. They depend only on the Reuss and Voigt averages of the shear moduli present in the layered system. The remaining two modes are both of mixed character, not being pure shear or pure compression. So at some intellectual level it is clear that this coupling imposed through the eigenvalues is the reason for the shear wave dependence on fluid saturation. But this mathematical statement, although surely correct, is not really helpful in achieving our goal of understanding how the mixing of these effects happens down at the microscale. The best possible way to elucidate this behavior would be to make use of a formula for shear modulus (if one were known), containing the desired effects in it explicitly. But, a rigorous formula of this type is probably not going to be found. So, the next best option is to have in hand a formula that, although it is known to be approximate, still has the right structure and thereby permits analysis to proceed.
What is the right structure? The appropriate shear strain that contains all the effects of interest clearly is of the form (1, 1, -2, 0, 0, 0)T. If we apply this shear strain to the stiffness matrix, the two distinct stresses generated are proportional to 2(a-m-f) [twice] and (c-f) [once]. So the effective shear modulus we seek should depend on these two quantities, each of which acts like in the limit of an isotropic system. But in the nonisotropic cases of most interest, these combinations both include coupling between and of the layers through the Backus formulas -- coupling that is good for our present purposes. Furthermore, since there are two nontrivial constants, it is not obvious what combinations to choose for study. But, the preceding analysis shows that one likely candidate for the effective shear modulus is Geff(1), since it appears naturally in two out of five of the main cases considered above. Geff(2) also appears in two similar cases, as well as in the energy estimate.
The shear moduli Geff(1) and Geff(2) are clearly not eigenvalues; but the most likely candidate eigenvalue relevant for our study is even more difficult to analyze and interpret (both because of its eigenvector's mixed character and because of the complicated formula relating it to the elastic constants) than either Geff(1) or Geff(2). There are other choices that could be made, but we will give preference to Geff(1) and Geff(2) in the following discussion -- both for definiteness and because they do seem to be useful constants to quantify and to help focus our attention. Of these two, Geff(2) is especially appealing because it is the one obtained using energy estimates.