Constant Drained Bulk Modulus, Uniform Fluid Saturation

For assumed constant isotropic drained bulk modulus, we have $K^*_d = K^{(n)}_d \equiv K_d$ for all N layers as well as the overall medium, and when N=2 we can prove easily using (20) that $\alpha^* = \alpha^{(1)} = \alpha^{(2)}$. When the fluid is uniform throughout the medium, the undrained bulk moduli also satisfy $K^*_u = K^{(n)}_u \equiv K_u$, since Gassmann's equation depends only on constants that are uniform throughout this model material. Now it has been shown previously (Berryman, 2004b) that when the drained bulk modulus is uniform, a general result for $G_{\rm eff}^v = G_{\rm eff}^r$ is  

$$G^v_d = \left[\sum_{n=1}^N \frac{f_n}{\mu_n + 3K_d/4}\right]^{-1} - 3K_d/4,$$ (21)

f_n being the volume fraction of the layers. This result follows easily from the Backus averages presented in (2) and the formula for $G^v_{\rm eff}$ in (6). In the presence of pore fluid and since each layer is a Gassmann material, the shear moduli of the individual porous layers do not change. So, a second result of the same type is available for the undrained uniaxial shear energy per unit volume $G_{\rm eff}^v$ in this medium:  

$$G^v_u = \left[\sum_{n=1}^N \frac{f_n}{\mu_n + 3K_u/4}\right]^{-1} - 3K_u/4,$$ (22)

f_n again being the volume fraction of the layers.

Neither of these two shear contributions is the overall modulus. They are just contributions of the uniaxial shear component (within each laminated grain) as defined earlier. However, they can be substituted for the term $G_{\rm eff}^v$ in the Peselnick-Meister-Watt bounds defined by (15). Note that it is easy to show from the forms of (21) and (22) that $c_{44} \le G^v_d \le c_{66}$,and similarly that $c_{44} \le G^v_u \le c_{66}$. [Furthermore, since $K_d \le K_u$ and the functionals in (21) and (22) vary monotonically with their arguments K_d and K_u, it is easy to see that $G_d^v \le G_u^v$.] Thus, from (12) and (13), the best choices for shear moduli of the comparison materials are always given by G_- = c₄₄ and G₊ = c₆₆ for this particular model material. So $\zeta_\pm = (G_\pm/6)(9K+8G_\pm)/(K+2G_\pm)$ in (15), where K takes the values K = K_d for the drained case and K = K_u for the undrained case. In both cases, K_R = K_V = K since the drained bulk modulus is uniform, so the form of the shear modulus bounds in (15) also simplifies to  

$$\frac{1}{\mu^\pm + \zeta_\pm} = \frac{1}{5}\left[\frac{1} {... ...rac{2}{c_{44}+\zeta_\pm} + \frac{2}{c_{66}+\zeta_\pm}\right].$$ (23)

We now have upper and lower bounds on the shear modulus in both drained and undrained circumstances by using the appropriate values of $G^v_{\rm eff}$ and $\zeta_\pm$ for each case. It is also possible to generate self-consistent estimates (Berryman, 2004b) for these moduli directly from the form of these bounds by instead making the substitutions $\mu^\pm \to \mu^*$ and $\zeta_\pm \to \zeta^* \equiv (\mu^*/6)(9K + 8\mu^*)/(K+2\mu^*)$. The results of all these formulas are illustrated in Figure 1.

Another important concept in these analyses will be the ratio of compliance differences defined by  

$$R \equiv \frac{1/\mu_d^* - 1/\mu_u^*}{1/K_d^* - 1/K_u^*}.$$ (24)

This quantity has been defined and discussed previously by Berryman et al. (2002b). It is most useful for determining the extent to which an identity derived by Mavko and Jizba (1991) for very low porosity media containing randomly oriented fractures is either satisfied or violated by other types of porous media. For the case studied by Mavko and Jizba (1991), $R \equiv 4/15$. However, it has been shown that for penny-shaped cracks at finite porosities R can be either higher or lower than 4/15, and furthermore that the factor R tends to zero when the pores approach spherical shapes (aspect ratio $\simeq 1$)[see Goertz and Knight (1998) and Berryman et al. (2002b)]. So this ratio is a sensitive measure of the dependence of $\mu_u^*$ on the fluid content of a porous medium, and also to some extent on the microgeometry of the pores.

Figure 1 shows that, for most choices of volume fractions, the drained and undrained values of shear modulus bounds do not overlap. Clearly, as the volume fractions approach zero or unity, the system approaches a pure Gassmann system; but, away from these limiting cases, the results are both qualitatively and quantitatively different from Gassmann's predictions. Graphically speaking, it appears that the lower bound of the undrained constant is always greater than the upper bound on the drained constants, i.e., $\mu_u^- \gt \mu_d^+$. But, when this figure is magnified, we find there are small regions of volume fraction where this inequality is violated slightly. So there is still little doubt that shear modulus is affected by pore fluids in these systems, and for some ranges of volume fraction there is no doubt. This result is a clear indication that Gassmann's results for shear are not generally valid for this model - as expected. Figure 2 shows that the maximum value of R for this case occurs around $f_2 \simeq 0.2$. Furthermore, the magnitude of this value is about 0.32, and therefore greater than 4/15. This shows again [as was shown previously by Berryman et al. (2002b)] that R = 4/15 is also not in general an upper bound on R.

 

Fig1
Figure 1 Illustrating the shear modulus results for the random polycrystals of porous laminates model for homogeneous 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 when the system is gas saturated, and B = 1.0 when the system is fully liquid saturated. The effective stress coefficients for the layers are both $\alpha = 0.75$, and $\alpha^* = 0.75$ also. The computed undrained bulk modulus is K_u = 140 GPa. Volume fraction of the layers varies from 0 to 100% of constituent number 2.

 

Fig2
Figure 2 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 1 for homogeneous 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 the random polycrystals of porous laminates model for the case considered always has $R \le 0.32$.