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With all of the double-porosity coefficients now defined, the compressional
phase velocity and attenuation
may be determined by inserting a plane-wave solution into the effective
single-porosity Biot equations [in the form (1)-(4)].
This gives the standard complex longtitudinal slowness *s* of Biot theory

| |
(29) |

where
| |
(30) |

is simply an auxiliary parameter, and where *H*, *C* and *M* are the
Biot (1962) poroelastic moduli defined in terms of the complex
frequency-dependent parameters of equations (11)-(13) as
| |
(31) |

| (32) |

| (33) |

The complex inertia corresponds to rewriting the relative flow resistance
as an effective inertial effect
| |
(34) |

Taking the minus sign in equation (29)
gives an *s* having an imaginary part much
smaller than the real part and that thus corresponds to the normal
P-wave. Taking the positive sign gives an *s* with real and
imaginary parts of roughly the same amplitude and that thus
corresponds to the slow P-wave (a pure fluid-pressure
diffusion across the seismic band of frequencies).
We are only interested here in the properties of the normal P-wave.

**QandVdp
**

Figure 1 The attenuation and
phase velocity of compressional waves in the double-porosity
model of Pride and Berryman (2003a).
The 5 cm embedded spheres of phase 2 have frame moduli (*K*^{d}_{2} and *G*_{2})
modeled using the modified Walton theory given in the appendex in
which both *K*^{d}_{2} and *G*_{2} vary strongly with the background
effective pressure *P*_{e} (or overburden thickness). These spheres of
porous continuum 2 were embedded into a phase 1 continuum modeled
as a consolidated sandstone.

The P-wave
phase velocity *v*_{p} and the attenuation measure *Q*_{p}^{-1}
are related to the complex slowness *s* as

| |
(35) |

| (36) |

** Next:** A Numerical Example
** Up:** REVIEW OF THE DOUBLE-POROSITY
** Previous:** Double-Porosity Transport
Stanford Exploration Project

10/14/2003