Constant Drained Bulk Modulus, Patchy Fluid Saturation

To add one level of complication, consider next the same porous framework as before, but now suppose that the saturation is patchy (White, 1975; Berryman et al., 1998; Norris, 1993; Dvorkin et al., 1999; Johnson, 2001; Berryman et al., 2002a), rather than homogeneous. The idea is that some of the layers in the grains will have a liquid saturant having K_f = K_l, while others will have a gas saturant having K_f = K_g. In general I assume that $K_g \ll K_l$ so that for most purposes the gas saturated parts of the system satisfy $K_u \simeq K_d$, i.e., undrained moduli are to a very good approximation the same as the drained moduli for these layers. If this were not so, then I could treat the second saturant in exactly the same way as I will treat the liquid saturant; but there would be no new ideas required to do this, so I will not stress this approach here.

For this system, the drained constants are all the same as in the preceding example. In particular, the overall volume effective stress coefficient $\alpha^*$ is the same.

The undrained constants differ for this system however because the undrained bulk modulus is not constant in the layers. Gassmann's formula does not provide an answer for this overall bulk modulus because the system is not homogeneous. But Backus averaging determines all the elastic constants in a straightforward way for this system [see Berryman (2004a)]. The correct results are obtained for all the constants related to Voigt and Reuss averages [Eqs. (4)-(9) for both bulk and shear moduli] as long as the K's shown explicitly in (2) are properly interpreted as the undrained constants K_u from (17) for the fluids having bulk moduli K_l or K_g in the appropriate layers.

One explicit result found useful to quote from some earlier work (Berryman, 2004a) is  

$$\begin{array} {cc} G_{\rm eff}^v = c_{66} - \frac{4c_{33}^u}... ...left<\frac{\Delta\mu}{K_u+ 4\mu/3}\right\gt^2\Big],\end{array}$$ (25)

where $c_{33}^u = \left<1/(K_u + 4\mu/3)\right\gt^{-1}$ and the bracket notation has the same meaning as in the Backus formulas (2). The difference $\Delta\mu \equiv \mu - c_{66}$ is the deviation of the layer shear modulus locally from the overall average across all the layers. The term in square brackets in (25) is always non-negative. As K_u in the layers ranges (parametrically) from zero to infinity, the corrections from the square bracket term times the factor $\frac{4c_{33}^u}{3}$can be shown to decrease from c₆₆-c₄₄ to zero. Thus, $G_{\rm eff}^v$ in the layered model ranges for all possible layered poroelastic systems from c₄₄ to c₆₆.

Figure 3 shows that the drained bulk modulus does not change with volume fraction, since all the layers have the same drained bulk modulus. The undrained bulk modulus can have some small variations, however, due to variations in the shear modulus, as is shown by the small spread in the bulk modulus bounds. Uncorrelated Hashin-Shtrikman bounds [computed by evaluating (3) at $\mu$'s having the lowest and highest shear modulus values among all those in the layers] are also shown here for comparison purposes. Clearly, the Peselnick-Meister-Watt correlated bounds based on the polycrystals of laminates microstructure are much tighter. Figure 4 shows that the overall shear modulus has only relatively weak dependence (though stronger than that in Figure 3) on patchy saturation when the bulk modulus itself is uniform. Figure 5 shows that shear modulus changes with saturation, while small, are present and not very tightly coupled to the bulk modulus changes (drained to undrained). This observation is seen to be especially significant at the lowest volume fractions of liquid, as the changes in shear compliance are greater here (by a factor of about 3) than the corresponding changes in the bulk compliance.

 

Fig3
Figure 3 Illustrating the bulk modulus results for the random polycrystals of porous laminates model for patchy saturation when each grain is composed of two constituents: (1) K_d⁽¹⁾ = 35.0 GPa, $\mu_d^{(1)} = 4.0$ GPa and (2) K_d⁽²⁾ = 35.0 GPa, $\mu_d^{(2)} = 40.0$ GPa. Skempton's coefficient is taken to be B = 0.0 for constituent 1 (gas saturated), and B = 1.0 for constituent 2 (liquid saturated). The effective stress coefficients for the layers are both given by $\alpha = 0.75$, so $\alpha^* = 0.75$ also. Porosity does not play a direct role in the calculation when we are using B as the fluid substitution parameter. Volume fraction of the layers varies from 0 to 100% of constituent number 2. To emphasize the tightness of the polycrystal (correlated) bounds, uncorrelated Hashin-Shtrikman bounds $K_{HS}^\pm$ on the undrained bulk modulus are also shown.

 

Fig4
Figure 4 Illustrating the shear modulus results for the random polycrystals of porous laminates model. Model parameters are the same as in Figure 3 for patchy saturation.

 

Fig5
Figure 5 Plot of the ratio R from equation (24), being the ratio of compliance differences due to fluid saturation. These results are for the same model described in Figure 3 for patchy saturation. The values of R should be compared to those predicted by Mavko and Jizba (1991) for very low porosity and flat cracks, when $R = 4/15 \simeq 0.267$. We find in contrast that, for partial and patchy saturation, R can take any positive value, or zero.