In order to separate effects of liquids on Lamé's parameter from well-understood effects of liquids on the density , while taking full advantage of the fluid-effect independence of shear modulus ,we will now combine the vp and vs data into a new type of plot. To take advantage of the predictions of the theory described above, we will plot seismic velocity data in a two-dimensional array with one axis being and the other being the ratio .Now, the ratio acts as a proxy for S which we do not know, but both S and are simply linear functions of S in the region of low frequencies being considered. For porous materials that satisfy Gassmann's homogeneous fluid condition the result should be a straight (horizontal) line until the saturation reaches (around 95% or higher), where the data should quickly rise to a value determined by the velocities at full liquid saturation. This behavior is observed in Figure 1b. Note that, although this behavior is qualitatively similar to that of vp in Figure 1a, we are now using only the seismic velocities themselves (no saturation data are required to generate this plot, although in this case saturation can be inferred at least qualitatively). The behavior we observe here is traditional Gassmann-Domenico predictions (Domenico, 1974) for partial saturation.
If all the other assumptions of the Gassmann model are satisfied, but the liquid and gas are not distributed uniformly (so that different pores have different saturation levels), then we have the circumstances that may better fit the ``patchy saturation'' model (Berryman et al., 1988; Endres and Knight, 1989; Mavko and Nolen-Hoeksema, 1994; Dvorkin and Nur, 1998). In that case, for the plot of vs.,instead of data following a horizontal line with a jump up at the high saturation end (e.g., Figure 1b), the ideal patchy saturation model (for completely segregated liquid and gas pockets) would predict that the data should lie on another straight line connecting to the two end points (dry and saturated) on this plot. These straight lines have been superimposed on the plots [obtained using data from Murphy (1982; 1984) and from Knight and Nolen-Hoeksema (1990)] for Figures 1b, 1d, and 1f. The anticipated behavior has been observed in Figure 1b and in other data not shown here, but two distinctly different types of behavior are observed in Figures 1d and 1f.
Plots of velocity versus saturation and of versus for two sandstones that apparently do not behave according to Gassmann's model are shown in Figures 1c-1f. These apparent deviations from the range of expected behaviors (from purely homogeneous mixed fluids to purely segregated patchy saturation) are resolved by including another display for these three sandstones (Murphy, 1982; 1984; Knight and Nolen-Hoeksema, 1990) in Figures 2a,c,e and corresponding plots for three limestones (Cadoret, 1993; Cadoret et al., 1995; Cadoret et al., 1998) in Figures 2b,d,f. Now the ratio is plotted versus saturation measured in the laboratory, and we observe in all these cases that the basic plot structures we had anticipated for Figures 1b, 1d, and 1f are in fact confirmed. What we learn from this observation is that the quantity , which we wanted to use as a proxy for the saturation S, is not always a good proxy at high frequencies. We can safely attribute the discrepancies in Figures 1d and 1f to effects of high frequency dispersion as predicted by Biot's theory (Biot, 1956a; 1956b; 1962). Even the seemingly odd negative slope of the patchy saturation lines in Figure 1f can be understood as a predicted high frequency effect on the shear velocity (Berryman, 1981).
[ht]2mssdvel_big,mssdpatchxy_big,ftunion200kvel_big,ftunion200kxypatch_big,srvel_big,srxypatch_bigwidth=2.4in,height=2.0inCompressional and shear velocities for Massilon and Ft.Union sandstone measured by Murphy (1982; 1984) and for Spirit River sandstone measured by Knight and Nolen-Hoeksema (1990).
This first new plotting method is limited by the implicit assumptions that the shear modulus is independent of the presence of fluids and that frequency dispersion for shear velocity is negligible. The assumption that the materials' shear properties are independent of the fluid is based on theoretical predictions about mechanical behavior only, and any chemical interactions between fluid and rock that might soften grain contacts could easily account for some of these discrepancies. Fluid-induced swelling of either interstitial or intergranular clays is another possible source of discrepancy as are fluid-induced pressure effects if the fluid is overpressured and therefore tending to severely weaken the rock. All of the chemical effects mentioned should become active with even very small amounts of fluid present, but probably do not have very significant frequency dependence (at least within the seismic frequency band). On the other hand, we must also take into account Biot's theory (Biot, 1956a; 1956b; 1962) of acoustics in porous media, which generalizes Gassmann's theory to higher frequencies and has been shown to be a very reliable predictor of behavior in simple porous systems (Berryman, 1995). There are frequency dependent (dispersion) effects predicted by Biot's theory that can lead to complications difficult to resolve with the severely frequency-band limited data that are normally available.
Fortunately, Cadoret and colleagues (Cadoret, 1993; Cadoret et al., 1995; Cadoret et al., 1998) have in recent years performed a very extensive series of tests on limestones, including both ultrasonic and sonic experiments and with different means of achieving various levels of partial saturation. Figure 3b shows results obtained for an Estaillades limestone at 500 kHz. This material behaves very much like the sandstones we have already considered here, and appears to obey the Gassmann predictions very well all the way up to the ultrasonic frequency regime. There were several other limestones that were found to have similar if not quite such good behavior. On the other hand, there were two limestone samples (a granular Lavoux limestone and an Espeil limestone) that were found to have very strong dispersion in the ultrasonic frequency band. These materials do not behave as expected when the data are plotted as in either Figure 1 or Figure 2. However, since extensional wave and shear wave data (Cadoret, 1993; Cadoret et al., 1995; Cadoret et al., 1998) at 1 kHz were also collected for these same samples, we have computed the necessary quantities using standard formulas and plotted them for these two materials in Figures 2d and 2f. We see that even for these two badly behaved materials (in the ultrasonic band) the plots at lower frequency become easy to interpret again. These results provide a very strong indication that plots such as those in Figure 1 will be readily interpreted for all porous materials at seismic frequencies.