HIGH FREQUENCIES

Figure 1 displays data from one granite and two sandstones in the frequency range 200-1000 kHz (also see Table 1). These examples were chosen for common display to emphasize the fact that there can be very clear deviations from the Gassmann-Domenico predictions at high frequencies. In particular, we see the startling difference in the right-hand subplots (Figure 1b,d,f) that the slopes of the patchy saturation lines (i.e., lines connecting data points for fully dry and fully saturated samples) in all three cases are negative, instead of positive as predicted for low-frequency behavior [see Berryman et al. (2002a)]. Nevertheless, all three plots on the left (Figure 1a,c,e) seem to behave in a manner consistent with the Gassmann-Domenico ideas. Sierra White granite and Schuler-Cotton Valley sandstone (both measured at about 200 kHz) show behavior consistent with our interpretation of nearly ideal patchy saturation, again consistent with the drainage method of producing the changes in saturation.

TABLE 1.Some physical parameters of the samples considered in Figure 1. (Note: 1 mD $\simeq 1\times 10^{-15}$ m².)

^aMurphy (1982) ^bKnight and Nolen-Hoeksema (1990); Murphy (1982); Walls (1982)

On the other hand, the Spirit River sandstone (Figures 1e,f) was measured in the 600 kHz to 1 MHz frequency range, and two distinct methods of saturation were employed. The drainage method in this case again seems to show patchy saturation content, although it is not very close to the ideal patchy saturation line. The imbibition data are expected to produce a more uniform distribution of gas and liquid in the pores than that obtainable in most cases with a drainage method. Thus, imbibition data should behave much as predicted by Gassmann-Domenico, at least at low frequencies. Here we observe in Figure 1e that the imbibition data do indeed mimic the predicted behavior of Gassmann-Domenico, even though we are at high frequencies. Taken together, these results seem to suggest that something fairly simple is happening to produce these data, and that the main issue in Figure 1b,d,f is probably the actual violation of the Gassmann's very low frequency result that the shear modulus is not influenced by the presence of the fluid, and/or how such behavior can be modelled.

Seifert et al.(1999), working at about 1 MHz, chose to use the symmetric effective medium theory of Berryman (1980a) to model their data. The frequencies used are low enough so that a typical wavelength is 2 mm, while the grain sizes for the sands studied range from 210 to 250 $\mu$m, so the wavelength is an order of magnitude larger than the grain sizes and effective medium theory can safely be used. For an unconsolidated sand fully saturated with liquid, such a system is fairly closely approximated by a fluid suspension and therefore the self-consistent scheme (Berryman, 1980a) is appropriate for their problem. However, it would not be appropriate for our partial saturation problem where the pore fluid is sometimes all gas, and the solid frame always plays the major role in supporting both compressional and shear stresses. The better choice for such problems is a differential effective medium (DEM) theory [see Berge et al. (1995) for a more complete discussion of the advantages and disadvantages of these methods]. Then the solid can be treated correctly as the host medium and the gas and liquid constituents are treated strictly as inclusion phases -- a requirement for this problem.

Our calculation for patchy saturation first uses DEM to compute the bulk and shear moduli for a porous solid saturated with gas only, and then repeats the calculation for bulk and shear moduli for a porous solid saturated with liquid only. In both cases, the shear modulus starts out at the shear modulus of the solid host medium and this is gradually replaced by (zero) inclusion shear modulus as the final desired porosity is attained. Nevertheless, the results in the gas- and liquid-saturated cases differ in these calculations because even though they have the same value for inclusion shear modulus, they do not have the same value for inclusion bulk modulus. This difference is important to the computed results. The physical reason for the difference is that in a random medium when a shear stress is applied macroscopically, it is resolved microscopically into both shear and compressional component stresses [see Berryman and Wang (2001) for an analysis of this aspect of the problem]. Trapped liquid can support some of those resolved compressional stresses resulting from an applied shear stress and therefore makes the saturated porous medium stronger in both shear and compression than when the same medium is saturated instead with a gas. Thus, the theory shows that $\mu_{dr} \ne \mu_{sat}$ when the saturating fluid is a liquid. This result disagrees with Gassmann, but does not contradict Gassmann. The point is that Gassmann's result is quasi-static and therefore pertinent for much lower frequencies, wherein the fluid can respond to the applied shear field by simply moving out of the way. But for trapped fluids or relatively rapid wave propagation through the medium, the result just described must hold.

At high enough frequencies, adding liquid to a partially saturated system will in fact increase the effective shear modulus of the system. Thus, when we plot $\lambda/\mu$ versus $\rho/\mu$, it is no longer the case that $\rho/\mu$ is a monotonic function of saturation. The density $\rho$is still a monotonic increasing function of saturation S as before, but now $\mu$ is also a monotonic increasing function of saturation. Therefore, the ratio $\rho/\mu$ is not necessarily monotonic and its behavior depends on which of the competing changes in the numerator and denominator dominate. The results for Sierra White granite (very low porosity) in Figure 1b clearly show that the main effect of addition of liquid to the system is to produce changes in $\mu$ at low porosities, with the result that the patchy saturation line has the opposite sign of slope (seen in Figure 1b) as that predicted by Gassmann-Domenico (seen in Figure 1a) and the data tend to fall along this line. The results are similar but not quite so well behaved for the Schuler-Cotton Valley sandstone in Figure 1d.

The most interesting behavior is observed for the Spirit River sandstone in Figure 1f. Here we see very clearly that as the liquid saturation increases, at first we have an increase in $\rho/\mu$ and then, when some special value of saturation (near 40% for the drainage data) is achieved, the influence of liquid on the shear modulus becomes more important and dominates the remainder of the curves up to full saturation.

Of the examples shown here, all three deviate dramatically from the predicted Gassmann-Domenico behavior. All these cases have the lowest porosities and permeabilities of the examples considered by Berryman et al. (2002a). This effect is presumably related therefore to the influence of permeability on the inability of the pore fluid pressure to equilibrate during the passage time of the wave, i.e., having a higher likelihood of acoustically disconnected porosity.