The preceding analysis centered on homogeneous saturation of porous media.
On the other hand, consider a porous medium containing gas and liquid
mixed in a heterogeneous manner, so that patches of the medium
hold only gas while other patches hold only liquid in the pores.
Then, the theory
predicts that, depending to some extent on the spatial distribution
of the patches, the results will deviate overall from Gassmann's results
(although Gassmann's results will hold locally in each individual patch).
If we consider the most extreme cases of spatial distribution possible,
which are laminated regions of alternating liquid saturation and
gas saturation, then the effective bulk modulus
will be determined by an average of the two extreme values
of (Gassmann): *K*|_{S=0} = *K*_{dr} and *K*|_{S=1}.
Using saturation as the weighting factor, the harmonic mean and the mean are
the two well-known extremes of behavior
(Hill, 1952). Of these two, the one
that differs most from (Gassmann) for 0 < *S* < 1 is the mean.
And, because of *K*'s linear dependence on both and ,and 's independence of *S*, we therefore have

_patchy(S) = (1-S)_dr + S|_S=1.
So, on our plot in the (, )-plane,
the results for the mean will again lie along
a straight line,
but now the line goes directly from the unsaturated value (*S* = 0)
to the fully saturated value (*S* = 1) [*e.g.*, Figure 3(e)].
The two straight lines described [the one given by (patchy)
and the horizontal one discussed in the preceding paragraph for
saturations up to about 95%]
are rigorous results of the theory, and form two sides of
a triangle that will contain all data for partially saturated systems,
regardless of the type of saturation present. The third side of this
triangle provides a rigorous bound on the behavior as full saturation
is approached (it just corresponds to the physical requirement that
, so values with *S* > 1 have no physical significance).
In general, heterogeneous
fluid distribution can produce points anywhere within the resulting
triangle, but not outside the triangle (within normal experimental
error).

A brief presentation of some examples (Figure 3) will now follow a reminder of an important and well-known caveat.

4/28/2000