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Pride and Berryman (2003b)
obtain the internal transport coefficient of equation (9) as

| |
(21) |

where the parameter that holds in the final stages of
internal fluid-pressure equilibration
is given by
| |
(22) |

Since the more compressible embedded phase 2 typically has a permeability much
greater than the host phase 1, the *O*(*k*_{1}/*k*_{2}) correction can
be neglected.
The transition frequency corresponds to the onset of a high-frequency
regime in which the fluid-pressure-diffusion penetration distance
between the phases becomes small relative to the scale of the
mesoscopic heterogeneity. It
is given by
| |
(23) |

The length *L*_{1} characterizes the average distance in phase 1 over
which the fluid pressure gradient still exists in the final approach
to equilibration and has the formal mathematical definition
| |
(24) |

where is the region of an averaging volume occupied by
phase 1 and having a volume measure *V*_{1}. The potential has units of length squared and is a solution of
an elliptic boundary-value problem that under conditions where
the harmonic mean is a good approximation for the overall drained
modulus and where the permeability ratio *k*_{1}/*k*_{2} can be considered
small, reduces to
| |
(25) |

| (26) |

| (27) |

where is the external surface of the averaging volume coincident
with phase 1,
and where is the internal interface separating phases 1 and 2.
Multiplying equation (25) by and integrating over ,
establishes that second integral of equation (24).
For complicated geometry,
*L*_{1} can only be determined numerically. For idealized geometries it
can be analytically estimated. For example, if phase 2 is taken to be small spheres
of radius *a* embedded within each sphere *R* of the composite,
Pride and Berryman (2003b)
obtain

| |
(28) |

The volume fraction *v*_{2} of small spheres is then *v*_{2} = (*a*/*R*)^{1/3} which can
be used to eliminate *R* since *R*=*a v*_{2}^{-1/3}.
The other length parameter is the volume-to-surface ratio
*V*/*S* where *S* is the area of in each volume *V* of composite. For the simple spherical-inclusion model, it is
given by *V*/*S*=*R*^{3}/(3*a*^{2})= *a v*_{2}/3.
The coefficient governing shear
generally has a non-zero ``viscosity'' associated with
the mesoscopic fluid transport between the compressional lobes
surrounding a sheared phase 2 inclusion. Both of the frequency
functions and are real and are
Hilbert transforms of each other. The frequency dependence of
was not modeled by Pride and Berryman (2003b).
However, if the inclusions of phase 2 are taken to be spheres,
then exactly and is a constant that
can be approximately modeled using a simple harmonic average
1/*G* = *v*_{1}/*G*_{1} + *v*_{2}/*G*_{2} of the underlying shear moduli of each phase.

Finally, the dynamic permeability to be used in the effective
Biot theory can be modeled in several ways. Perhaps the simplest
modeling choice when phase 2 is modeled as small inclusions
embedded in phase 1 is to again take a harmonic average .

** Next:** Phase Velocity and Attenuation
** Up:** REVIEW OF THE DOUBLE-POROSITY
** Previous:** Double-Porosity a_{ij} Coefficients
Stanford Exploration Project

10/14/2003