Numerical Examples

 

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Figure 4 The squirt-flow model of P-wave attenuation when the grains are modeled as being spherical of radius R and containing microcracks having effective aperatures h. The overall drained modulus of the rock corresponds to a consolidated sandstone.

In Fig. 4, we plot the P-wave attenuation predicted using the above model when the overall grain packing corresponds to a consolidated sandstone (v₁=0.2 and c=5) having a permeability of 10 mD. For the grain properties we take $\sigma=0.8/(5\times 10^{-3})$, 3N_c /(4N_R²) = 1, and K_s = 38 GPa (quartz) as fixed constants. This $\sigma$ value was chosen so that there would be an important peak in attenuation at ultrasonic frequencies and is taken to be the same for all values of h/R. The various curves can be thought of as being due to the application of effective stress. The peak in Q^-1 near 1 MHz that is invariant to h/R is that due to the macroscopic Biot loss (fluid pressure equilibration at the scale of the wavelength). The peak that shifts with h/R is that due to the squirt flow.

This figure indicates that although the squirt mechanism is probably operative and perhaps even dominant at ultrasonic frequencies, it does not seem to be involved in explaining the observed levels of intrinsic attenuation in exploration work. For real cracks inside of real grains, the $\sigma$ value will diminish with effective stress (i.e., with h/R), so that the effects of squirt in the seismic band are likely to be even less than shown in Fig. 4.

 

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Figure 5 The undrained bulk modulus $K_U(\omega)$ in the squirt model of the present study and in the model of Dvorkin et al. (1995). 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 frame properties are the same as in Fig. 4. The curves having a smaller relaxation frequency ($\simeq 10^3$ Hz) and almost no dispersion correspond to $h/R = 2\times 10^{-4}$while the curves having the larger critical frequency ($\simeq 10^5$Hz) and more dispersion correspond to $h/R = 5 \times10^{-3}$.

We next introduce the grain parameters k₂, $\phi_2$, and K^d₂ as modeled here along with the same overall drained modulus K into the model of Dvorkin et al. (1995) and compare the results to our own model for two different values of h/R (Fig. 5). Although both models have similar dependencies on the various material properties involved, there are nonetheless significant differences. These are principally due to the fact that the Dvorkin et al. (1995) model requires the grains to be in the form of effective cylinders of radius R, while in Fig. 5 we use a geometric parameter L₂ and volume-to-surface ratio V/S that are appropriate for spherically-shaped grains. However, in various limits as the frequency and/or fluid bulk modulus become either large or small, we have verified that both models yield qualitatively similar results.