Читать книгу Reservoir Characterization - Группа авторов - Страница 38

To see how closely estimated wave velocity correlates with the experimental velocity values; we replaced the primary CO₂ saturated water with brine (using apparent module of 2.25) using Gassmann equation. Then using the Greenberg-Castagna formula we estimated the compressional wave velocity (Figure 2.6).

In Figure 2.6, the compressional wave velocity obtained from the laboratory experiment and the Gassmann equation is depicted versus effective pressure. It can be noticed that at all effective pressures the estimated velocities have higher values than experimental ones. The difference gradually enlarges as the effective pressure is increased. The rate of velocity increment is faster for estimated velocity than the experimental counterpart. This is because in Gassmann theory environment (the core) is considered a homogeneous and isotropic elastic environment. This theory is often true for Monomineralic rocks such as pure quartz sandstone or clean limestone. But since the sample used in this study developed naturally, possible impurities exist such as clay or fine cracks; so wave scattering is not considered in the Gassmann equations. It is also effective on bulk modulus and often in this hypothesis, it is assumed to be average. Another issue to the assumptions of this model is the fluid replacement instead of several fluids, regardless of their distribution in the environment and is often considered to be the average of the density of the fluids. This is effective on the accuracy of the model especially while the saturation is not uniform (Patchy saturation). Also with the pressure increasing, joints and cracks closed more and reduced permeability. Therefore, fluid flow and the opportunity to achieve balance are much reduced and the operating result is that Gassmann theory pushes farther than expected.

Shear wave velocity of the rock sample was determined at the different pressure from the estimated compressional wave velocity, using Greenberg-Castagna formulas.

Figure 2.7 shows the experimental and estimated shear wave velocity values at different effective pressures. As it can be seen in contrary to the compressional wave, the estimated shear wave velocity is very much in accordance with the corresponding experimental values. Particularly this compatibility is more obvious at low effective pressures. The difference between experimental and estimated velocities and the rate of changes increase as the applied pressure increases. This is because the shear waves are only sensitive to changes solid in the environment. With the increase in effective pressure and reduced porosity and micro cracks, stone resistance increases against deformation (while volume or angle); as a result, it will face increasing shear wave velocity. Since in Greenberg-Castagna equation compressional and shear wave communication waves can be described with porosity and modules, effective stress plays an important role in the trend of estimation. But because of unconsidered pore shape, form, size, and density of fractures, anisotropy, texture and especially effective pressure on the environment in this relationship, differences although small can be seen in the results.

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.

But based on the values in Figure 2.15, the experimental and the estimated values of Young’s modulus are very close to one another and very comparable (%97).