EXAMPLES

A series of Tables and Figures will now be presented to illustrate the results obtained using the four methods discussed above. Two pf these methods (CPA and DEM) are known to be realizable (Milton, 1985; Avellaneda, 1987). The other two are not. The particular examples were computed assuming four different shapes for the inclusion, but the method is not restricted to the shapes chosen.

Input parameters are from Table 1 for a clay and Kayenta sandstone mixture. Solid sandstone grains occupy from 100% to 60% of the volume, while porous clay (with fixed porosity of 40%) occupies the remaining 0% to 40%. The overall porosity $\phi$ therefore ranges from 0% to 16%. CPA treats both components symmetrically, neither being singled out as a host material. The other three models have been computed assuming the strong component (the sand grains) is the host and the weak component (porous clay) is the inclusion. The tables then present results for various choices of inclusion shape: spheres, needles, disks, and penny-shaped cracks.

TABLE 2. Material constants for a clay and a Kayenta sandstone.

$$0.15in] \par \begin{tabular} {\vert c\vert c\vert c\vert c\v... ...d & 0.0 & 37.88 & 37.88 & 29.0 & 29.0 \hline\hline\end{tabular}$$

 

ssk
Figure 1 Bulk modulus as a function of clay volume fraction for four models: CPA (solid line), DEM (dot-dash line), Kuster-Toksöz (dotted line), and Mori-Tanaka (dashed line). Both host and inclusion shape factors are assumed to be those for spheres. Note that Kuster-Toksöz and Mori-Tanaka give the same results for this particular case.

 

ssm
Figure 2 Shear modulus as a function of clay colume fraction for the same four models as in Figure 1.

 

ssa
Figure 3 Biot-Willis parameter as a fucntion of clay volume fraction for the same four models as in Figure 1.

TABLE 3.Computed values of the Biot-Willis parameter $\alpha^*$using CPA, DEM, Kuster-Toksöz, and Mori-Tanaka, assuming both host and inclusion materials are spherical in shape. The volume fraction of the inclusion is v_iand the resulting porosity is $\phi$. The three models that distinguish host and inclusion have used sand as host and clay as inclusion for this computation. Note that Kuster-Toksöz and Mori-Tanaka give identical results for this case as expected.

$$\phi \alpha_{CPA}^* \alpha_{DEM}^* \alpha_{KT}^* \alpha_{MT}^*$$ v_i

 

snk
Figure 4 Bulk modulus as a function of clay volume fraction for four models: CPA (solid line), DEM (dot-dash line), Kuster-Toksöz (dotted line), and Mori-Tanaka (dashed line). Host and inclusion shape factors are assumed to be those for spheres and needles, respectively. Note that Kuster-Toksöz and Mori-Tanaka do not give the same results for this case.

 

snm
Figure 5 Shear modulus as a function of clay colume fraction for the same four models as in Figure 4.

 

sna
Figure 6 Biot-Willis parameter as a fucntion of clay volume fraction for the same four models as in Figure 4.

TABLE 4.Same as Table 2 except that the host (sand) is assumed to be spherical in shape, while the clay is assumed to be needle-shaped inclusions. Note that Kuster-Toksöz and Mori-Tanaka do not give identical results for this case.

$$\phi \alpha_{CPA}^* \alpha_{DEM}^* \alpha_{KT}^* \alpha_{MT}^*$$ v_i

 

sdk
Figure 7 Bulk modulus as a function of clay volume fraction for four models: CPA (solid line), Kuster-Toksöz (dotted line), and Mori-Tanaka (dashed line). Host and inclusion shape factors are assumed to be those for spheres and disks, respectively. DEM is not shown because it predicts that the moduli and Biot-Willis parameter all shift to the values of the clay at very low volume fractions of clay. Note that Kuster-Toksöz and Mori-Tanaka do not give the same results for this case.

 

sdm
Figure 8 Shear modulus as a function of clay colume fraction for the same three models as in Figure 7.

 

sda
Figure 9 Biot-Willis parameter as a fucntion of clay volume fraction for the same three models as in Figure 7.

TABLE 5.Same as Table 2 except that the host (sand) is assumed to be spherical in shape, while the clay is assumed to be disk-shaped inclusions. Note that Kuster-Toksöz does not give sensible results for this case except at extremely low volume fractions.

$$\phi \alpha_{CPA}^* \alpha_{DEM}^* \alpha_{KT}^* \alpha_{MT}^*$$ v_i

 

spk
Figure 10 Bulk modulus as a function of clay volume fraction for four models: CPA (solid line), DEM (dot-dash line), Kuster-Toksöz (dotted line), and Mori-Tanaka (dashed line). Host and inclusion shape factors are assumed to be those for spheres and penny cracks, respectively. Note that Kuster-Toksöz and Mori-Tanaka do not give the same results for this case.

 

spm
Figure 11 Shear modulus as a function of clay colume fraction for the same four models as in Figure 10.

 

spa
Figure 12 Biot-Willis parameter as a fucntion of clay volume fraction for the same four models as in Figure 10.

TABLE 6.Same as Table 2 except that the host (sand) is assumed to be spherical in shape, while the clay inclusions are assumed to fill penny-shaped cracks. Note that Kuster-Toksöz does not give sensible results for this case, except at very low volume fractions.

$$\phi \alpha_{CPA}^* \alpha_{DEM}^* \alpha_{KT}^* \alpha_{MT}^*$$ v_i