Two Distinct Gassmann Materials, Patchy Fluid Saturation

This final set of examples will use the same model framework as the preceding example. However, two fluids will be present simultaneously in this case. If the two fluids (g,l) are assumed to saturate only one or the other types of Gassmann materials, then we have a relatively simple two component model: (1g,2l). On the other hand, the setup is now general enough to permit a variety of other possibilities. For example, porous material 1 might be saturated with either gas or liquid, and the same for porous material 2: (1g,1l,2g,2l). We could also suppose that some of the layers have homogeneously (h) mixed saturation of both liquid and gas, i.e., a partially rather than patchy saturated layer: (1g,1h,1l,2g,2h,2l). Although this additional complication is not a problem for the numerical modeling, the large increase in the number of possible cases needing enumeration becomes a bit too burdensome for the short presentation envisioned here. (There is an infinite number of ways these types of materials having homogeneously mixed regions could be incorporated.) So we will instead limit discussion to the two cases mentioned before: (a) just two types of patchy saturated layers (1g,2l), or (b) four types of patchy saturated layers (1g,1l,2g,2l). Since the case (b) is expected to be more complex but not expected to contain any new ideas, we will limit the discussion further to case (a).

In Figure 9 as in Figure 6, the three drained curves for bulk modulus are so close together that they cannot be distinguished on the scale of this plot (although they can be distinguished if the plot is magnified). In contrast to Figure 6, the three undrained bulk modulus curves can now be distinguished, but they are still quite close together. The undrained curves start out at the same values as the drained curves because at zero volume fraction of constituent 2 the only fluid in the system is air. Then, as the volume fraction of constituent 2 increases, we add liquid up to the point where the final values at full liquid saturation are the same as in Figure 6. Again uncorrelated Hashin-Shtrikman bounds are shown for purposes of comparison, as in Figure 3. The Peselnick-Meister-Watt bounds on undrained bulk modulus -- making use of the laminated grain/crystal substructure and the polycrystalline nature of the overall reservoir model -- clearly are much tighter. Together Figures 3 and 9 also show that the polycrystalline-based bounding method produces a great improvement over the uncorrelated Hashin-Shtrikman bounds, whose microstructural information is limited to volume fraction data. This result is accomplished without having very detailed knowledge of the spatial correlations, just by using the fact that the local microstructure is layered. Knowledge of local layering is therefore a very important piece of microstructural information that has not been used to greatest advantage in prior applications of bounding methods for up-scaling purposes.

For Figure 10, the results are not so simple, as the six curves are all very close to each other. Undrained curves are always above the corresponding drained curves, but in general there is little separation to be seen here. Figure 11, like Figure 5, shows that the shear modulus changes with saturation are not really tightly coupled to the bulk modulus changes, and especially so at the lowest volume fractions of liquid, as the changes in shear compliance are again greater in magnitude there than the changes in the bulk compliance.

 

Fig9
Figure 9 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⁾ = 20.0 GPa, $\mu_d^{(1)} = 4.0$ GPa and (2) K_d⁽²⁾ = 50.0 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, respectively, $\alpha^{(1)} = 0.85$ and $\alpha^{(2)} = 0.70$. 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 accuracy of the polycrystal bounds and self-consistent estimates, uncorrelated Hashin-Shtrikman bounds $K_{HS}^\pm$ on undrained bulk modulus are also shown.

 

Fig10
Figure 10 Illustrating the shear modulus results for the random polycrystals of porous laminates model. Model parameters are the same as in Figure 9 for patchy saturation.

 

Fig11
Figure 11 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 9 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. Note that the magnitude of this effect is smaller than in Figure 5, even though the degree of heterogeneity for bulk modulus is greater here.