Some illustrative examples

Figure 3 shows three examples of the results obtained with plots in the (S, $\lambda/\mu$)-plane and in the ($\rho/\mu$, $\lambda/\mu$)-plane (using $\rho/\mu$ as a proxy for S) for two limestones and one andesite from laboratory data of Cadoret (1993) and Cadoret et al. (1995; 1998). In Figure 3, the true saturation data are used in the Figures on the left and the proxy for saturation ($\rho/\mu$) is used on the right. We therefore call the right hand diagrams ``saturation-proxy'' plots. Using the interpretations arising for our analysis of Gassmann-Domenico partial-saturation theory, we see that Figures 3(a) and (b) indicate homogeneous mixing of liquid and gas, while Figures 3(e) and (f) indicate extremely patchy mixing, and Figures 3(c) and (d) show an intermediate state of mixing for the drainage data, but more homogeneous mixing (as expected) for the depressurization data. The Espeil limestone was observed to be the most dispersive of all those rocks considered in the data sets of Cadoret (1993) and Carodet et al. (1995; 1998). So this case is a very stringent test of the method. In fact, if we were to plot the corresponding data for Espeil limestone at 500 kHz, we would not find such simple and easily interpreted behavior on these plots. Our explanation for this difference between the 500 kHz and 1 kHz results for Espeil limestone is that the dispersion introduces effects not accounted for by the simple Gassmann-Domenico theory, and that there is then no reason to think that our method should work for such high frequencies as 500 kHz. We have found other examples where it does work for frequencies higher than one might expect the method to be valid. The point is that, if we restrict the range of frequencies considered to 1 kHz or less, the method appears to work quite well on most (and perhaps all) samples. [But, at higher frequencies, the solid and fluid can move out of phase and other relations developed by Biot (1956a,b; 1962) and others (O'Connell and Budiansky, 1977; Mavko and Nur, 1978; Berryman, 1981; McCann and McCann, 1985; Johnson et al., 1987; Norris, 1993; Best and McCann, 1995) apply.]