From previous work (Berryman, 2003), we know that large fluctuations in the layer shear moduli are required before significant deviations from Gassmann's quasi-static constant result, thereby showing that the shear modulus dependence on fluid properties can become noticeable. To generate a model that demonstrates these results, again I made use of the same code of V. Grechka as described when presenting Figure 1. But this time I arbitrarily picked just 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 for the various elastic coefficients and Thomsen parameters are displayed in TABLE 2. The results of the calculations for V_{p} and V_{sv} are shown in Figures 1 and 2.
The model calculations were simplified in one way: 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 an exercise of little value because we are just trying to show in a simple way that the formulas given here really do produce the types of results predicted analytically, and also to get a feeling for the magnitude of the effects. Furthermore, if is a constant, then it is only the product that matters. 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 examples.
Constituent | K (GPa) | (GPa) | z (m/m) |
1 | 9.4541 | 0.0965 | 0.477 |
2 | 14.7926 | 4.0290 | 0.276 |
3 | 43.5854 | 8.7785 | 0.247 |
Elastic Parameters | Case | Case | Case |
and Density | B = 0 | B = 1 | |
a (GPa) | 33.8345 | 50.3523 | 132.7003 |
c (GPa) | 33.1948 | 50.4715 | 134.2036 |
f (GPa) | 22.2062 | 38.5857 | 120.7006 |
l (GPa) | 4.0138 | 4.0138 | 4.0138 |
m (GPa) | 6.7777 | 6.7777 | 6.7777 |
G_{eff} (GPa) | 5.2797 | 5.8841 | 6.2417 |
-0.0847 | -0.0733 | -0.0399 | |
0.0943 | 0.0745 | 0.0343 | |
0.3443 | 0.3443 | 0.3443 | |
(kg/m^{3}) | 2120.0 | 2310.0 | 2320.0 |
We took the porosity to be , and the overall density to be , where kg/m^{3}, S is liquid saturation (), and kg/m^{3}. Then, three cases were considered: (1) Gas saturation S=0 and B=0, which is also the drained case, assuming that the effect of the saturating gas on the moduli is negligible. (2) Partial liquid saturation S = 0.95 and [which is intended to model a case of partial liquid saturation], intermediate between the other two cases. For smaller values of liquid saturation, the effect of the liquid might not be noticeable, since the gas-liquid mixture when homogeneously mixed will act much like the pure gas in compression, although the density effect will still be present. When the liquid fills most of the pore-space, and the gas occupies less than about of the entire volume of the rock, the gas starts to become disconnected, and we expect the effect of the liquid to start becoming more noticeable, and therefore we choose to be representative of this case. And, finally, (3) full liquid saturation S = 1 and B=1, which is also the fully undrained case. We assume for the purposes of this example that a fully saturating liquid has the maximum possible stiffening effect on the locally microhomogeneous, isotropic, poroelastic medium.
The results shown in Figures 2 and 3 are in complete qualitative and quantitative agreement with the analytical predictions described, as expected.