Figure 2.7 S-wave velocity (experimental and estimated) at different effective pressures.
Figure 2.8 Cross plot of estimated P-wave velocities vs. laboratory measurements.
Figure 2.9 Cross plot of the estimated S-wave velocities vs. laboratory measurements.
Figure 2.8 shows cross plot of the compressional wave velocity measured in the laboratory condition vs. the estimated values along with a best-fitted line. The correlation of the values is 0.95. Figure 2.9 also shows the cross plot of the experimental and estimated shear wave velocities along with a best fitted line. The correlation of the values is 0.96.
Figures 2.10 and 2.11 show the cross plots of the experimental and estimated compressional and shear wave velocities. Mathematical relationship between two sets of wave velocities were obtained. As it shows 90% correlation observed in both modes between compressional and shear wave velocity. The important note is that the slope of the curve is approximately 5 times in the experimental measurements more than the estimated measurement and this is due to larger estimated compressional wave velocity.
Figure 2.10 Plot of experimental shear wave velocity against compressional wave velocity.
Figure 2.11 Plot of estimated shear wave velocity against compressional wave velocity.
Since Greenberg-Castagna model assumptions is ideal for a specially designed environment, there is no great match between this model and laboratory values. But as was mentioned earlier, correspondence between the velocity, the pressure and shear are very close together. It seems environment effective pressure is the essential factor that change is not considered in the model. This effect is highlighted on shear wave velocity and shear wave module.
As it can be seen in Figure 2.12, the behavior of Vs-Experimental/estimated are almost independent of variability of effective pressure. While VP-Experimental/estimated increases with the rising effective pressure. The difference between rates of VP’s is because of some assumptions of Gassmann-Greenberg-Castagna equations, which they guess that media is ideal.
Figure 2.12 Rate of variability of experimental/estimated velocities with increasing effective pressure.
Figure 2.13 Plot of Laboratory vs. estimated Bulk modulus (K) of rock sample.
At this stage, we compared the elastic coefficients of rock samples obtained from experimental velocity values, and those estimated from Greenberg-Castagna model. According to Figure 2.13, the correlation between experimental and estimated values for volume or bulk modulus is very low and is about 0.1. This is because the compressibility of the pores media depends on the minerals type and texture of the rock. Also, it is a function of the amount and shape of media porosity. When the pores are filled with fluid, elastic modulus is affected by many parameters of fluid such as the compressibility, the kind of fluid, distribution, viscosity and also fluid incompressibility.
The fluid used in this study has a patchy saturation and has created anisotropic, heterogeneous environments. Besides that, different effective pressures that applied on a frame stone, impressed the results of Gassmann - Greenberg – Castagna equations.
One of the assumptions of Gassmann equations is that the shear modulus in the dry and wet state is constant. It seems that this assumption is not applicable for most environments. The fact is that the changes to the texture of rock due to the reaction between rock and the fluid cause a change in the shear modulus of saturated rocks. Changes in the shear modulus is the main cause for the difference between the experimental velocity and the calculated velocity utilizing the Gassmann equations. Anyway, these differences in shear modulus cause a decrease in the use of Gassmann theory to estimate the velocity. These differences can be observed in Figure 2.14 for both laboratory and estimated values. Although the correlation between experimental and estimated values are very high and close to 0.96, by replacing the common fluid and use of the estimated shear wave velocity, shear modulus values change a little as well. The reason behind this phenomenon is the defect in the Gassmann hypothesis that the shear modulus for rock is equal in both dry and saturated conditions. The reaction between the fluid and the texture of the rock is consequently causing a change in the shear modulus of saturated rocks.
Figure 2.14 Plot of laboratory vs. estimated shear modulus.
Figure 2.15 Plot of laboratory vs. estimated Young’s modulus.