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To obtain the *a*_{ij} for the patchy-saturation model, we note that
each patch has the same and *K*. The poroelastic
differences between patches is entirely due to *B*_{1} being different
than *B*_{2}. Upon volume averaging equation (3)
and using ,
where an overline again denotes a volume average over
the appropriate phase, and using the fact that the *a*_{ij} are defined
in the extreme high-frequency limit where
the fluids have no time to traverse the internal
interface (i.e., the *a*_{ij} are
defined under the condition that ),
one has

| |
(37) |

| (38) |

| (39) |

The average confining pressures in each phase are not
*a priori* known; however, they are necessarily linear functions
of the three independent applied pressures of the theory
,
, and . It is straightforward
to demonstrate that if and only if the average confining pressures
take the form
| |
(40) |

| (41) |

will equations (37)-(39) produce *a*_{ij} that
satisfy the thermodynamic symmetry requirement of
*a*_{ij} = *a*_{ji} [i.e., these *a*_{ij} constants are all second
derivatives of a strain-energy function as demonstrated by
Pride and Berryman (2003a). Upon placing equations
(40) and (41) into equations (37)-(39), we then have
| |
(42) |

| (43) |

| (44) |

| (45) |

| (46) |

| (47) |

where is a constant to be determined.
To obtain , we note that in the high-frequency limit,
each local patch of phase *i*
is undrained and thus
characterized by an undrained bulk modulus and
a shear modulus *G* that is the same for all patches. In this limit,
the usual laws of elasticity govern the response of this heterogeneous
composite. Under these precise conditions (elasticity of an isotropic
composite having uniform *G* and all heterogeneity confined
to the bulk modulus which in the present case corresponds to *K*_{i}^{u}),
the theorem of Hill (1963)
applies, which states that the overall
undrained-unrelaxed modulus of the composite *K*_{H} is given exactly by

| |
(48) |

In terms of the
*a*_{ij}, this same undrained-unrelaxed Hill modulus is given by
| |
(49) |

where, upon using and
in equation (8) and then using (42)-(47),
the undrained-unrelaxed pressure ratios are
| |
(50) |

| (51) |

Thus, after some algebra, equation (49) yields the exact result
| |
(52) |

with *K*_{H} given by equation (48). All
the *a*_{ij} are now expressed in terms of known information.

** Next:** Patchy-Saturation Transport
** Up:** PATCHY-SATURATION MODEL
** Previous:** PATCHY-SATURATION MODEL
Stanford Exploration Project

10/14/2003