EXAMPLE: WEBER SANDSTONE

 

weberK Figure 3 Bulk modulus bounds and self-consistent estimates for the random polycrystal of porous laminates model of a Weber sandstone reservoir.

 

weberG Figure 4 Shear modulus bounds and self-consistent estimates for the random polycrystal of porous laminates model of a Weber sandstone reservoir.

 

weberaij Figure 5 Values of double-porosity coefficients aij for a system similar to Weber sandstone. Values used for the input parameters are listed in TABLE 1. For each coefficient, three curves are shown, depending on which estimate of the overall bulk modulus is used: lower bound (dot-dash line), self-consistent (solid line), or upper bound (dashed line).

Weber sandstone is one possible host rock for which the required elastic constants have been measured by Coyner (1984). TABLE 1 displays the values needed in the double-porosity theory presented here. These values follow from an analysis of Coyner's data if I assume the stiffer phase occupies about 92% of the volume and the more compliant phase the remaining 8% of total volume. The drained bulk moduli of the storage and fracture phases are used in the effective medium theory of the previous section to determine the overall drained and undrained bulk moduli of the random polycrystal of laminates system. Results for the self-consistent estimates (Berryman, 2004b), and the upper and lower bounds for the bulk moduli are all displayed in Figure 3. I see the undrained moduli are nearly indistinguishable, but the drained constants show some dispersion. Similarly, I show bounds and self-consistent estimates for the overall shear modulus of this model reservoir in Figure 4. Both undrained and drained shear moduli show some dispersion. Note that a correction must be applied to (31) before computing the self-consistent effective constants. The self-consistent estimates for bulk modulus are found correctly from the bounds (26) by taking $K_\pm \to K^*$, $G_\pm \to G^*$, and therefore $\zeta_\pm \to \zeta^*$. The resulting formula is  

$$K^* = K_V\frac{(G_{\rm eff}^r + \zeta^*)}{(G_{\rm eff}^v + \zeta^*)}.$$ (33)

The self-consistent formula for shear modulus requires more effort. The difficulty is that the formula given in (31) has already made use of a constraint that is only true along the bounding curves defining the upper and lower bounds on shear modulus. Since the self-consistent estimate always falls at points away from this curve, a more general result must be employed. When the inappropriate constraint is replaced by the general formula and then (33) is substituted, I find instead that the self-consistent formula for shear modulus is given by  

$$\frac{1}{G^* + \zeta^*} = \frac{1}{5}\left(\frac{1+\gamma^*(... ...rac{2}{c_{44} + \zeta^*} + \frac{2}{c_{66} + \zeta^*}\right),$$ (34)

where $\gamma^* = 1/(K^*+4G^*/3)$. The main difference is that the denominator of the first term on the right hand side is simpler than it is in the formulas for the shear modulus bounds. Observed dispersion is small over the range of volume fractions considered. Then these drained values K_d^*, $K_d^\pm$ are used in the formulas of the second section to determine both estimates and bounds on the double-porosity coefficients. These results are then displayed in Figure 5, which is also the main result of this paper. Note that the curves for a₁₁ essentially repeat results shown in Figure 3, but for the compliance 1/K_d^*, instead of the stiffness K_d^*. The coefficients a₁₂, a₂₂, and a₂₃ show little dispersion. This is natural for a₁₂ and a₂₂ because the storage material contains no fractures, and therefore is not sensitive to fracture compliance, whereas those mechanical effects on the overall reservoir response can be very large. The behavior of a₂₃ also shows little dispersion as this value is always very close to zero (Berryman and Wang, 1995; Berryman and Pride, 2002). The two remaining coefficients show a significant level of dispersion are a₁₃ and a₃₃, where the third stress is the pore pressure p_f⁽²⁾ of the fracture or joint phase. I generally expect that the joint phase is most tightly coupled to, and therefore most sensitive to, the fluctuations in overall drained bulk modulus K_d^*. So all these results are qualitatively consistent with our intuition. Since I have analytical formulas for all the a_ij's, it is straightforward to check that the observed dispersion in a₁₃ and a₃₃ is directly proportional to the dispersion in 1/K_d^* (or, equivalently, a₁₁).