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We next must obtain the coefficient in the mesoscopic
transport law . Again, the approach is to first obtain the limiting
behaviour at low and high frequencies and then to connect the two limits
by a simple function.
The fluid response in phase 1 (the principal porespace) is governed by
the Navier-Stokes equation
and the compressibility law
where is the local fluid velocity in
the pores.
Since for all frequencies of interest
we have that (note that s^{-1} for liquids and
10^{10} s^{-1} for gases), the fluid pressure in phase 1 is governed
by the wave equation
| |
(96) |
and since the acoustic wavelength in the fluid is always much greater than
the grain sizes,
the fluid pressure in the principal porespace satisfies
(a spatial constant) at all frequencies.
The focus, then, is on determining the flow and fluid pressure
within the cracked grains (phase 2) that is governed by the
local porous-continuum laws and
| |
(97) |
where .
This deformation and pressure change is excited by applying a uniform
normal stress to the surface of the averaging volume
with the fluid pressure satisfying the
boundary conditions
on and
on .