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To use the patchy-saturation model, appropriate values for
the two geometric terms *L*_{1} and *V*/*S* must be specified.
Immiscible fluid distributions in the earth have very complicated
geometries since they arise from slow flow that often produces
fractal patch distributions. In particular,
analytical solutions of the
boundary-value problem (25)-(27) that defines
*L*_{1} for such real-earth situations are impossible.
Recall that *L*_{1} is a characteristic length of phase 1 (the phase having
the smaller fluid mobility ) that defines the distance overwhich the
fluid-pressure gradient is defined during the final stages of
equilibration. For complicated geometries it may either be numerically
determined, guessed at, or treated as a target parameter for a full-waveform
inversion of seismic data. In the numerical examples that follow, we
simply assume that the individual patches correspond to
disconnected spheres for which simple analytical results are available
for *L*_{1} and *V*/*S*.

If we consider phase 2 (porous continuum saturated by the
less viscous fluid) to be in the form of spheres of radius *a*
embedded
within each radius *R* sphere of the two-phase composite, then
*v*_{2} = (*a*/*R*)^{3}, *V*/*S* = *a v*_{2}/3, and *L*_{1}^{2} = 9 *v*_{2}^{-2/3} *a*^{2}/14
[1 - 7 *v*_{2}^{1/3}/6]. This model is
particularly appropriate when
.
Since the
fluid 2 patches are disconnected,
the definitions (11)-(13)
of the effective poroelastic
moduli again hold. Further, fluid 2 may be taken to
be immobile relative to the framework of grains
in the wavelength-scale Biot equilibration so
that the inertial properties of equations
(29) and (30) are identified as
, and
.

In situations where it is more appropriate to treat
fluid 1 (the more viscous fluid) as occuping disconnected patches
(e.g., when ),
the effective poroelastic moduli are defined by
replacing 2 with 3 (and 3 with 2)
in the subscripts of equations (11)-(13).
Again assuming the phase-1 patches to be spheres of radius *a*
embedded within radius *R* sphere of the two-phase composite,
we have that *v*_{1} = (*a*/*R*)^{3} and *V*/*S* = *a v*_{1}/3. The
elliptic boundary-value problem (25)-(27)
can be solved in this case to give *L*_{1}^{2} = *a*^{2}/15. Furthermore, the
effective inertial coefficients in the Biot theory are defined
, , and .

In situations where both phases form continuous paths across each
averaging volume, it is best to determine the attenuation and phase velocity
by seeking the plane longtitudinal-wave solution of
non-reduced ``double-porosity'' governing equations of the form
(6)-(10). However, this approach is not pursued here.
We conclude by noting that if the embedded fluid is fractally distributed,
the lengths *L*_{1} will remain finite while
as the fractal surface area *S* becomes large
(however, *V*/*S* never reaches zero because the fractality
has a small-scale cutoff fixed by the grain size of the material).

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Stanford Exploration Project

10/14/2003