By making two seemingly small changes in the method of display, we now
arrive at one of the main points of this paper. Since the expected behavior
for Gassmann materials as observed in Figure 1b is a horizontal
straight line for most values of saturation (*i.e.,* is expected to be almost constant until high saturation levels are
reached), it is natural to consider
dividing by , and then plotting the points
again in the (, )-plane.
In the straight-line portion of the curve from Figure 1b, the only
effect will be a change
of scale, but large changes will result in the points
representing full saturation or nearly full saturation.
The results of this new plotting method
are displayed in Figure 3.
We observe that in all
cases the result is apparently a straight line. This linear behavior
is expected for a Gassmann material, since is just a scaling
factor, is unaffected by saturation, and is linearly
dependent on saturation. It would also be expected for a non-Gassmann
material in which the effect of fluids on was negligible
compared to the effect on .It may also be expected for the case of patchy saturation if chemical
interactions cause to change with saturation, because then
for the porous medium would be some weighted average of for the dry case and for the fully saturated, chemically
altered portions of the rock.

[ht]2mssdpatchsy_big,estddppatchsy_big,ftunion200kdpatchsy_big,elvxddppatchsy_big,srsypatch_big,eespddppatchsy_bigwidth=2.4in,height=2.0in
Ratio versus saturation for the three
sandstones (Murphy, 1982; 1984; Knight and Nolen-Hoeksema, 1990)
of Figure 1 and for three limestones (Cadoret, 1993;
Cadoret *et al.*, 1995; Cadoret *et al.*, 1998).

Figure 3a shows the same sandstone data as Figure 1.
Similar data for five limestone samples (Cadoret, 1993;
Cadoret *et al.*, 1995; Cadoret *et al.*, 1998)
are plotted in Figure 3b.
The straight line correlation of the data in this display
is clearly confirmed by the limestone data. Numerous other examples
of the correlation have been observed. No examples of appropriate
data for partially saturated samples with major deviations
from this behavior have been observed, although an extensive survey
of available data sets has been performed for materials including
limestones, sandstones, granites, unconsolidated sands, and some
artificial materials such as ceramics and glass beads.
This straight line correlation is a very robust
feature of partial saturation data. The mathematical trick
that brings about this behavior will be explained in simple terms below
following a brief discussion of the usefulness of this display.

We now have two new methods of data display, both using the Lamé parameter in a critical role. We therefore call this pair of plotting methods the ``-diagrams.''

10/25/1999