In Fig. 1, we give an example of Qp-1 and vp 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 Pe. The host phase 1 is modeled as a consolidated sandstone (using 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 Pe 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 . 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 contains information about k1, it does not contain information about the overall permeability which is being dominated by k2 in this case.