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** Up:** Patchy-Saturation Transport
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As , one can represent the local fields as
asymptotic series in the small parameter

| |
(60) |

| (61) |

and equivalently for .
The zeroth-order response corresponds to uniform
fluid pressure in the pores and is therfore given by and
| |
(62) |

where the patchy-saturation *a*_{ij} have been employed.
The fact that the quasi-static Skempton's coefficient in the patchy-saturation
model is exactly the harmonic average of the constituents *B*_{i} is equivalent to
saying that at low frequencies, the fluid bulk modulus is given by
1/*K*_{f} = *v*_{1} /*K*_{f1} + *v*_{2} /*K*_{f2}. The quasi-static response is
thus completely independent of the spatial geometry of the fluid patches; it depends
only on the volume fractions occupied by the patches.
The leading order correction to uniform fluid pressure is then controlled by
the boundary-value problem

| |
(63) |

| (64) |

| (65) |

| (66) |

| (67) |

It is now assumed that for patchy-saturation
cases of interest (air/water or water/oil), the ratio can be considered small.
To leading order in , equations (63), (66), and (67)
require that (a spatial constant).
The fluid pressure in phase 1 is now rewritten as
| |
(68) |

where, from equations (64), (65) and (67) and
to leading order in , the potential is the solution of
the same elliptic boundary-value problem (25)-(27)
given earlier.
Upon averaging (68) over all of , the
leading order in difference in the average fluid pressures can be written

| |
(69) |

where *L*_{1} is again the length defined by equation (24).
To connect this fluid-pressure difference to the increment we use
the divergence theorem and the no-flow boundary condition on
to write equation (53) as

| |
(70) |

Replacing with using equation (69)
then gives the desired law
with
| |
(71) |

being the low-frequency limit of interest.

** Next:** High-frequency limit of
** Up:** Patchy-Saturation Transport
** Previous:** Mesoscopic flow equations
Stanford Exploration Project

10/14/2003