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.''