A Numerical Example

In Fig. 1, we give an example of Q_p^-1 and v_p as determined using this double-porosity theory. The example models a consolidated sandstone containing pockets (small regions) where the grains are not cemented together. The embedded unconsolidated phase 2 is modeled as 5 cm radius spheres occupying 1.5% of the composite. The frame moduli of this relatively-compressible embedded material are determined using the modified Walton theory given in the Appendix. These moduli are functions of the background effective-stress level P_e. The host phase 1 is modeled as a consolidated sandstone (using $\phi_1=0.15$and c=4 in the model given in the appendix). The permeability of the two phases are taken as k₁=10^-14 m² and k₂ = 10^-12 m². The invariant peak near 10⁵ Hz is that due to the Biot loss (fluid equilibration at the scale of the seismic wavelength) while the principal peak that changes with the effective pressure P_e is that due to mesoscopic-scale equilibration. Figure 1 demonstrates that small amounts of a relatively soft material embedded within a more consolidated rock is capable of producing the level of attenuation measured in field experiments.

The overall magnitude of attenuation in the model is controlled principally by the contrast of compressibilities between the two porous phases; the greater the contrast, the greater the mesoscopic fluid-pressure gradient and the greater the mesoscopic-flow intensity and associated attenuation. The relaxation frequency at which the mesoscopic loss per cycle is maximum is proportional to k₁/L₁². Below this relaxation frequency, Q^-1 increases with frequency as $f \eta /k_1$. Thus, the permeability information in the double-porosity attenuation is principally in the frequency dependence of Q^-1 and not in the overall magnitude of Q^-1 and involves principally the permeability k₁ of the host phase and not the overall permeability of the composite. [See Berryman (1988) for a related discussion.] If phase 2 is well modeled as being small inclusions embedded in phase 1, then k₁ is controlling the overall permeability. If phase 2 corresponds to through going connected joints, then although $Q^{-1}(\omega)$ contains information about k₁, it does not contain information about the overall permeability which is being dominated by k₂ in this case.