An important paper by Gassmann (1951) concerns the effects of fluids on the mechanical properties of porous rock. His main result is the well-known fluid-substitution formula (that now bears his name) for the bulk modulus in undrained, isotropic poroelastic media. He also postulated that the effective undrained shear modulus would (in contrast to the bulk modulus) be independent of the mechanical properties of the fluid when the medium is isotropic. That the independence of shear modulus from fluid effects is guaranteed for isotropic media at very low or quasistatic frequencies was shown recently by Berryman (1999) to be tightly coupled to the original bulk modulus result of Gassmann; each result implies the other in isotropic media. It has gone mostly without discussion in the literature that Gassmann (1951) also derived general results for anisotropic porous rocks in the same 1951 paper. It is not hard to see that these results imply that, contrary to the isotropic case, some of the overall undrained shear moduli in fact may depend on fluid properties in anisotropic media, thus mimicking the bulk modulus behavior. However, Gassmann's paper does not remark at all on this difference in behavior between isotropic and anisotropic porous rocks. Brown and Korringa (1975) also address the same class of problems, including both isotropic and anisotropic cases, but again they do not remark on the shear modulus results in either case. Norris (1993) studies partial saturation in isotropic layered materials in the low-frequency regime ( 100 Hz) and takes as a fundamental postulate that Gassmann's results hold for the low frequency shear modulus, but it seems that some justification should be provided for such an assumption, and furthermore some indication of its range of validity established.
On the other hand, Hudson (1981), in his early work on cracked solids, explicitly demonstrates differences between fluid-saturated and dry cracks and relates his work to that of Walsh (1969) and O'Connell and Budiansky (1974), but does not make any connection to the work of either Gassmann (1951), or Brown and Korringa (1975). Mukerji and Mavko (1994) show numerical results based on work of Gassmann (1951), Brown and Korringa (1975) and Hudson (1981) demonstrating the fluid dependence of shear in anisotropic rock, but again they do not remark on these results at all. Mavko and Jizba (1991) use a simple reciprocity argument to establish a direct, but approximate, connection between undrained shear response and undrained compressional response in rocks containing cracks. Berryman and Wang (2001) show that deviations from Gassmann's results sufficient to produce shear modulus dependence on fluid mechanical properties require the presence of some anisotropy on the microscale, thereby explicitly violating the microhomogeneous and microisotropy conditions implicit in Gassmann's original derivation. Berryman et al. (2002a) go further and make use of differential effective medium analysis to show explicitly how the undrained, overall isotropic shear modulus can depend on fluid trapped in penny-shaped cracks. Meanwhile, laboratory results for wave propagation [see Berryman et al. (2002b)] show conclusively that the shear modulus does indeed depend on fluid mechanical properties for low-porosity, low-permeability rocks, and high-frequency laboratory experiments (f > 500 kHz).
One thing lacking from all the preceding work is a simple example showing how the presence of anisotropy influences the shear modulus, and specifically when and how the shear modulus becomes fluid dependent. Our main purpose in the present work is therefore to demonstrate, in a set of rather simple examples, how the overall shear behavior becomes coupled to fluid compressional properties at high frequencies in anisotropic media -- even though overall shear modulus is always independent of the fluid properties in microhomogeneous isotropic media at sufficiently low frequencies, whether drained or undrained. Two other distinct but related analyses addressing this topic have been presented recently by the author (Berryman, 2004b,c). Both of these prior papers have made explicit use of layered media, composed of isotropic poroelastic materials, together with exact results for such media based on Backus averaging (Backus, 1962). In contrast, the present analysis does not make use of such a specific model, and is therefore believed to be about as simple as possible, while still achieving the level of understanding desired for this rather subtle technical issue. One important simplification we make here in order to separate the part that is due to poroelastic effects, from the part that would be present in any elastic (i.e., possibly zero permeability porous medium) is to model each material as if the elastic part is entirely isotropic, while the poroelastic effects [i.e., the Biot-Willis coefficients (Biot and Willis, 1957) for the anisotropic overall material] provide the only sources of anisotropy in the system. Thus, we specifically distinguish two possible sources of anisotropy, the elastic or ``hard'' anisotropy that is assumed not to be present here, and the poroelastic or ``soft'' anisotropy that is the source of the effects we want to study in this paper.
Our analysis for general transversely isotropic media is presented in the next three sections. In particular, the section on eigenvectors also introduces the effective undrained shear modulus relevant to our general discussion. Examples are then presented for glass, granite, and sandstone. The paper's results and conclusions are summarized in the final section. Some mathematical details are collected in the Appendix.