Case: $\mu^* \le \mu$

The presence of liquid in the pores may alter the mechanical behavior of rocks under shear deformations in at least two quite distinct ways: (a) It is often observed that a very small amount of some liquids can cause chemical interactions that tend to soften the binding material present among the grains of such a system. When this happens, the shear modulus is usually observed to decrease. (See for example FIG.3 for Sierra White granite.) So this situation implies that $\mu^* \ge \mu$, contrary to Gassmann's results. Although this situation is well-known in practice, we will ignore it in our modeling efforts. Our justification for this will be that the medium we are calling ``dry'' should in fact be termed ``drained'' in the sense that it has been wetted previously and therefore has these chemical softening effects already factored into the modulus $\mu^*$. In any case, our goal is not to fit data for specific rocks, but rather to understand general trends. (b) The other situation that can also occur in practice - particularly at higher frequencies - is that the liquid saturating the porous material can have a nonnegligible mechanical effect (Berryman and Wang, 2001) that tends to strengthen the medium under shear loading so that $\mu \ge \mu^*$. If this strengthening effect is great enough (and there are experimental results (see FIG.4) that confirm this does happen in practice (Berryman et al., 2002), then it is possible the density effect is more than counterbalanced by the enhanced shear modulus effect with the result that the speed of shear wave propagation in the liquid saturated medium is greater than that in the air saturated case. Depending on details of the liquid distribution in the pores, either of these cases can be included in the analysis that we now pursue in this subsection.

For Massillon sandstone, Murphy (1982,1984) also measured extensional and shear wave velocities at f = 200 kHz over a range of partial saturations produced using the drainage method. Relevant properties of this sandstone were listed before in TABLE 2.

 

swg200vs
Figure 3 Shear wave velocities as a function of water saturation for drainage experiments of Murphy (1982) in Sierra White granite.

 

knight_vs
Figure 4 Shear wave velocities as a function of water saturation for drainage experiments of Knight and Nolen-Hoeksema (1990) in Spirit River sandstone.