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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 and *c*=4 in the model given in the appendix).
The permeability of the two phases are taken as *k*_{1}=10^{-14} m^{2}
and *k*_{2} = 10^{-12} m^{2}. The invariant peak near 10^{5} 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 *k*_{1}/*L*_{1}^{2}.
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 *k*_{1} 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 *k*_{1} is controlling the overall
permeability. If phase 2 corresponds to through going connected joints, then
although contains information about *k*_{1}, it does not contain
information about the overall permeability which is being dominated by *k*_{2} in
this case.

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** Previous:** Phase Velocity and Attenuation
Stanford Exploration Project

10/14/2003