As we learned in the previous section, large fluctuations in the layer shear moduli are required before the predicted deviations from Gassmann's quasi-static constant shear modulus result will become noticeable. To generate a model that demonstrates these results, I made use of a code of V. Grechka [used previously in a joint publication (Berryman et al., 1999)] and then I arbitrarily picked one of the models that seemed to be most interesting for the present purposes. The parameters of this model are displayed in TABLE 1. The results of the calculations are shown in Figures 1 and 2.
The model calculations were simplified in one way, which is that the value of the Biot-Willis parameter was chosen to be a uniform value of in all layers. We could have actually computed a value of from the other layer parameters, but to do so would require another assumption about the porosity values in each layer. Doing this seemed of little use because we are just trying to show in a simple way that the formulas given here really do produce the results predicted. Furthermore, if is a constant, then it is only the product that matters. Since we are using B as the plotting parameter, whatever choice of constant is made, it mainly determines the maximum value of the product for B in the range [0, 1]. So, for a parameter study, it is only important not to choose too a small value of ,which is why the choice was made. This means that the maximum amplification of the bulk modulus due to fluid effects can be as high as a factor of 5 for the present example.
TABLE 1. Layer parameters for the three materials in the simple layered medium used to produce the examples in Figures 1 and 2.
|Constituent||K (GPa)||(GPa)||z (m/m) 1|
Figure 1 Bulk modulus as a function of Skempton's coefficient B. The Biot-Willis parameter was chosen to be , constant in all layers.
Figure 2 Shear modulus as a function of Skempton's coefficient B. The Biot-Willis parameter was chosen to be , constant in all layers.
The results for bulk modulus in Figure 1 show that K=KR and are always quite close in value. The Voigt upper bound does bound both KR and from above as expected. Furthermore, the simple Gassmann estimate based on the value of the drained constant (B=0) gives a remarkably good fit to these estimates with no free fitting parameters. This good agreement with Gassmann may depend in part on our choosing to be a constant. (Without this choice, a direct comparison in fact could not have been made here.)
The results for the effective shear moduli Geff(2), Geff(1), and the quasi-shear mode eigenvalue contribution , show that , although it actually does bound this parameter from above. The third shear modulus estimate Geff(1) is far from the other two. We also display a constant value of shear modulus based on the Gassmann-like prediction of constant shear modulus equaling the drained modulus, but it is clear that this labeling is not fair to Gassmann as his result was for microhomogeneous materials, and therefore does not strictly speaking apply to layered materials at all. Nevertheless, we see that Gassmann's result for bulk modulus did a very creditable job of matching the bulk modulus values, even though that too was not a fair comparison for exactly the same reasons. This shows that Gassmann's results are much more sensitive to heterogeneity in shear modulus than they are to heterogeneity in the bulk modulus for these layered materials.