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This final set of examples will use the same model framework as the
preceding example. However, two fluids will be present simultaneously
in this case. If the two fluids (*g*,*l*)
are assumed to saturate only one or the other types of Gassmann
materials, then we have a relatively simple two component model:
(1*g*,2*l*). On the other hand, the setup is now general enough to
permit a variety of other possibilities. For example, porous
material 1 might be saturated with either gas or liquid, and the
same for porous material 2: (1*g*,1*l*,2*g*,2*l*).
We could also suppose that some of the layers have homogeneously (*h*) mixed
saturation of both liquid and gas, *i.e.*, a partially rather than
patchy saturated layer: (1*g*,1*h*,1*l*,2*g*,2*h*,2*l*).
Although this additional complication is not
a problem for the numerical modeling, the large increase in the number
of possible cases needing enumeration becomes a bit too burdensome for the short
presentation envisioned here. (There is an infinite number of ways these
types of materials having homogeneously mixed regions could be incorporated.)
So we will instead limit discussion to the two cases
mentioned before: (a) just two types of patchy saturated layers (1*g*,2*l*), or
(b) four types of patchy saturated layers (1*g*,1*l*,2*g*,2*l*).
Since the case (b) is expected to be more complex but
not expected to contain any new ideas, we will limit the discussion
further to case (a).
In Figure 9 as in Figure 6, the three drained curves for bulk modulus
are so close together that they cannot be distinguished on the scale
of this plot (although they can be distinguished if the plot is
magnified). In contrast to Figure 6, the three undrained bulk modulus
curves can now be distinguished, but they are still quite close
together. The undrained curves start out at the same values as the drained
curves because at zero volume fraction of constituent 2 the only fluid
in the system is air. Then, as the volume fraction of constituent 2
increases, we add liquid up to the point where the final values at
full liquid saturation are the same as in Figure 6. Again
uncorrelated Hashin-Shtrikman bounds are shown for purposes of
comparison, as in Figure 3. The Peselnick-Meister-Watt bounds
on undrained bulk modulus -- making use of the laminated grain/crystal
substructure and the polycrystalline nature of the overall reservoir
model -- clearly are much tighter. Together Figures 3 and 9 also show
that the polycrystalline-based bounding method produces a great
improvement over the uncorrelated Hashin-Shtrikman bounds, whose
microstructural information is limited to volume fraction data. This result
is accomplished without having very detailed knowledge of the spatial
correlations, just by using the fact that the local microstructure is layered.
Knowledge of local layering is therefore a very important piece of
microstructural information that has not been used to greatest advantage
in prior applications of bounding methods for up-scaling purposes.

For Figure 10, the results are not so simple, as
the six curves are all very close to each other. Undrained curves are
always above the corresponding drained curves, but in general there is
little separation to be seen here. Figure 11, like Figure 5, shows that
the shear modulus changes with saturation are not really tightly coupled to
the bulk modulus changes, and especially so at the lowest volume fractions
of liquid, as the changes in shear compliance are again greater in
magnitude there than the changes in the bulk compliance.

**Fig9
**

Figure 9 Illustrating the bulk modulus results for
the random polycrystals of porous laminates model for patchy saturation
when each grain is composed of two
constituents: (1) *K*_{d}^{(1)} = 20.0 GPa, GPa and
(2) *K*_{d}^{(2)} = 50.0 GPa, GPa. Skempton's
coefficient is taken to be *B* = 0.0 for constituent 1 (gas saturated),
and *B* = 1.0 for constituent 2 (liquid saturated). The
effective stress coefficients for the layers are, respectively,
and . Porosity does not
play a direct role in the calculation when we are using *B* as the fluid
substitution parameter. Volume fraction of the layers varies from
0 to 100% of constituent number 2.
To emphasize the accuracy of the polycrystal bounds and
self-consistent estimates, uncorrelated Hashin-Shtrikman bounds
on undrained bulk modulus are also shown.

**Fig10
**

Figure 10 Illustrating the shear modulus results
for the random polycrystals of porous laminates model.
Model parameters are the same as in Figure 9 for patchy saturation.

**Fig11
**

Figure 11 Plot of the ratio *R* from equation (24),
being the ratio of compliance differences due to fluid saturation.
These results are for the same model described in Figure 9
for patchy saturation.
The values of *R* should be compared to those predicted by Mavko
and Jizba (1991) for very low porosity and flat cracks,
when . We find in contrast
that, for partial and patchy saturation, *R* can take any positive
value, or zero. Note that the magnitude of this effect is smaller
than in Figure 5, even though the degree of heterogeneity for bulk
modulus is greater here.

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** Up:** FOUR SCENARIOS
** Previous:** Two Distinct Gassmann Materials,
Stanford Exploration Project

5/3/2005