Gassmann's equations (Gassmann, 1951) for fluid substitution state that
K = K_dr + ^2(-)/K_m + /K_f and =_dr, where Km is the bulk modulus of the single solid mineral, Kdr and are the bulk and shear moduli of the drained porous frame. The special combination of moduli defined by is the Biot-Willis parameter (Biot and Willis, 1957). The porosity is , while K and are the effective bulk and shear moduli of the undrained porous medium that is saturated with a fluid mixture having bulk modulus Kf. For partial saturation conditions with homogeneous mixing of liquid and gas, so that all pores contain the same relative proportions of liquid and gas, Domenico (1974) among others shows that
1/K_f = S/K_l + (1-S)/K_g. The saturation level of liquid is S lying in the range . The bulk moduli are: Kl for the liquid, and Kg for the gas. When S is small, (Kf) shows that ,since . As , Kf remains close to Kg until S closely approaches unity. Then, Kf changes rapidly (over a small range of saturations) from Kg to Kl. (Note that the value of Kl may be several orders of magnitude larger than Kg, as in the case of water and air -- 2.25 GPa and 1.45 GPa, respectively.)
Since has no mechanical dependence on the fluid saturation, it is clear that all the fluid dependence of in (Gassmann) resides within the Lamé parameter . Other recent work (Berryman et al., 1999) on layered elastic media indicates that should be considered as an important independent variable for analysis of wave velocities and Gassmann's results provide some confirmation of this deduction (and furthermore provided a great deal of the motivation for the present line of research). The parameters K ()and Kdr ()can be replaced in (Gassmann) by and without changing the validity of the equation. Thus, like K, for increasing saturation values, will be almost constant until the porous medium closely approaches full saturation.
Now, the first problem that arises with field data is that we usually do not know the reason why data collected at two different locations in the earth differ. It could be that the differences are all due to the saturation differences we are concentrating on in this paper. Or it could be that they are due entirely or only partly to differences in the porous solids that contain the fluids. In fact, solid differences easily can mask any fluid differences because the range of detectable solid mechanical behavior is so much greater than that of the fluids (especially when fractures are present).
It is essential to remove such differences due to solid heterogeneity. A related issue concerns differences arising due to porosity changes throughout a system of otherwise homogeneous solids. One way of doing this would be to sort our data into sets having similar porous solid matrix. For simplicity and because of the types of laboratory data sets available, we will use porosity here as our material discriminant.
Considering our three main parameters, , , and ,we see that all three depend on porosity, but only and depend on saturation. Using formulas (rho)-(Kf), we can take partial derivatives of each of these expressions first with respect to while holding S constant, and then with respect to S while holding constant. For now, we are only interested in trends rather than the exact values, and these are displayed in Table 1. The trend for requires the additional reminder that, although this term is always positive, its value is often so small that it may be treated as zero except in the small range of values close to S = 1. Also, using Hashin-Shtrikman bounds (Hashin and Shtrikman, 1962) as a guide, it turns out that it is not possible to make a general statement about the sign of , since the result depends on the particular material constants. (Related differences of sign are also observed in the data we show later in this paper; thus, this ambiguity is definitely real and observable.)
Assuming that the primary variables are , , and (further justification of this choice of primary variables is provided later in the paper), then the two pieces of velocity data we have can be used to construct the following three ratios:
= 1v_p^2-2v_s^2, and
= 1v_s^2. We will consider first of all what happens to these ratios for homogeneous mixing of fluids, and then consider the simpler case of ideal patchy saturation, where some pores in the partially saturated medium are completely filled with liquid and others are completely dry (or filled with gas).