Numerical Examples

In Fig. 2, we compare the prediction of Johnson (2001) for K_U to our own for a consolidated sandstone (frame properties as determined in the Appendix with k=100 mD, c=10, $\phi=0.20$) in which phase 1 is saturated with water and phase 2 is taken to be spherical regions saturated with air. The two estimates have identical asymptotic dependence in both the limits of high and low frequencies. In the cross-over range, the physics is not precisely modeled in either approach. However, even in the cross-over range, the differences in the two models is slight.

 

pridejohn
Figure 2 The undrained bulk modulus $K_U(\omega)$ in both the patchy-saturation model presented in this chapter and the model of Johnson (2001). The top graph is ${\rm Re}\{K_U\}$ while the bottom graph is $Q_K^{-1} = - 2 {\rm Im} \{K_U\} /{\rm Re} \{K_U \}$. The physical model is 10 cm spherical air pockets embedded within a water-saturated region. The volume fraction of gas saturated rock is 3% in this example. The properties of the rock correspond to a 100 mD consolidated sandstone.

Figure 3 gives the P-velocity and attenuation for a model in which the frame properties correspond to k=10 mD, c=15, and $\phi =0.15$. Phase 2 is saturated by air and is taken to be isolated spheres of radius a=1 cm. Phase 1 is saturated with water. The volume fraction v₂ occupied by these 1 cm spheres of gas is as shown in the figure. Even tiny amounts of gas saturation yields rather large amounts of attenuation and dispersion.

 

vq
Figure 3 The P-wave velocity and attenuation of a sandstone saturated with water and containing small spherical pockets of gas having radius 1 cm and occuping a fraction of the volume v₂ as shown.