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### Low-frequency limit of

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 aij 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 Bi is equivalent to saying that at low frequencies, the fluid bulk modulus is given by 1/Kf = v1 /Kf1 + v2 /Kf2. 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 L1 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