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The constants *a*_{ij} are all real and
correspond to the high-frequency response for which no internal fluid-pressure
relaxation can take place. They are given exactly as (Pride and
Berryman, 2003a)

| |
(14) |

| (15) |

| (16) |

| (17) |

| (18) |

| (19) |

where the *Q*_{i} are auxiliary constants given by
| |
(20) |

Here, *v*_{1} and *v*_{2} are the volume fractions of each phase within
an averaging volume of the composite. The one constant that has not
yet been defined is the overall drained modulus *K*=1/*a*_{11} of the
two-phase composite (the modulus defined in the quasi-static limit
where the local fluid pressure throughout the composite
is everywhere unchanged).
It is through *K* that the *a*_{ij} potentially
depend on the mesoscopic geometry of the two porous phases. However,
a reasonable modeling choice when phase 2 is embedded within
phase 1 is to simply take the geometry-independent
harmonic mean 1/*K* = *v*_{1}/*K*^{d}_{1} + *v*_{2}/*K*^{d}_{2}. Although this choice
actually violates the Hashin-Shitrikman bounds
(Hashin and Shtrikman, 1961) for truly isotropic
media, it is nevertheless a reasonable choice for earth systems
where the assumed isotropy is itself an approximation.
This choice is also a particularly convenient one
because it results in *Q*_{1}=*Q*_{2}=1 as well as *a*_{23}=0.
All dependence on the fluid's bulk modulus
is contained within the two
Skempton's coefficients *B*_{1} and *B*_{2} and is thus restricted to
*a*_{22} and *a*_{33}. In the quasi-static limit
(fluid pressure everywhere uniform throughout the composite),
equations (12) and (13)
reduce to the known exact results of Berryman and Milton (1991)
once equations (14)-(19) are employed.

** Next:** Double-Porosity Transport
** Up:** REVIEW OF THE DOUBLE-POROSITY
** Previous:** Reduction to an Effective
Stanford Exploration Project

10/14/2003