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 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.

Figure 1

Figure 2

Figure 3

TABLE 3.Computed values of the Biot-Willis parameter 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*_{i}and the resulting porosity is . 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.

v_{i} |
0.00 |

Figure 4

Figure 5

Figure 6

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.

v_{i} |
0.00 |

Figure 7

Figure 8

Figure 9

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.

v_{i} |
0.00 |

Figure 10

Figure 11

Figure 12

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.

v_{i} |
0.00 |

7/5/1998