The primary question we address in this paper is this: Does the effective shear wave speed of a long-wavelength quasi-SV wave in a finely layered VTI material depend on the fluid in the porous layers, even though Gassmann's results (Gassmann, 1951) say that -- without doubt -- the shear modulus in each individual layer is mechanically independent of the fluid? Perhaps surprisingly, we show that the answer to the question is positive. The quasi-SV wave always does depend on the fluid mechanics, unless the shear modulus of all the layers is exactly a uniform constant. Furthermore, the magnitude of this effect is largest when the layer shear modulus fluctuations are large.
In addition, our analysis leads us to consider some different ways of expanding the formulas for the dispersion relationships for the quasi-P and quasi-SV modes. These secondary results may also have some practical benefits and are illustrated in the Appendix.
Although there are five effective shear moduli for any layered VTI medium, the main result of the paper is that there is just one effective shear modulus for the layered system that contains all the dependence of elastic or poroelastic constants on pore fluids -- all that can be observed in vertically polarized shear waves in VTI media. The relevant modulus Geff is related to uniaxial shear strain and the relevant axis of symmetry is the vertical one, normal to the bedding planes. The pore-fluid effects on this effective shear modulus can be substantial when the medium behaves in an undrained fashion, as might be expected at higher frequencies such as sonic and ultrasonic for well-logging or laboratory experiments, or at seismic frequencies for lower permeability regions of reservoirs. These predictions are clearly illustrated by the example in Figure 2.
The stiffness coefficients a, b, c, and f, all contain contributions from fluid effects for undrained layers. However, only stiffness a and Thomsen parameter contain terms quadratic in layer shear modulus fluctuations, and these contributions are the ones creating the most significant effects on shear waves for strong anisotropy.