Bulk Density, Density Porosity, Neutron Porosity, Capture Cross Section, Diffusion Length, Diffusion Coefficient, and Slowing Down Length - All versus Laminated Bed Thickness
In every case, the classic mixing rules predict a smooth, gradual (albeit non-linear) transition from oil sand properties to oil sand + montmorillonite properties, whereas the LVPM heterogeneous calculations predict a much larger and stronger non-linear approach from oil sand properties to oil sand + montmorillonite properties. For example, Figure 1 indicates that the heterogeneous bulk density increases from 2.332 g/cc at a montmorillonite bed thickness of 0.01 cm to 2.401 c/cc at a thickness of just 1 cm, an increase of 0.069 g/cc. Compare this with the homogeneous/classic values: 2.330 g/cc to 2.345 g/cc, an increase of just 0.015 g/cc. Figure 2 echoes these features in the density porosity.
The bulk density differential between the homogeneous and heterogeneous methods reaches its maximum near a montmorillonite bed thickness of 2 cm: it amounts to 0.057 g/cc and corresponds to a porosity differential of 3.4 pu! As the montmorillonite bed thickness continues to increase, the heterogeneous density remains larger than the homogeneous density, but their differential continues to decrease.
Figure 3 details the neutron porosity as a function of montmorillonite bed thickness. See Figure 7 for the underlying neutron slowing-down length. As the montmorillonite bed thickness increases, the homogeneous neutron porosity remains greater than the heterogeneous porosity and this difference increases throught the entire bed thickness range.
Surprisingly, Figure 4 indicates no difference between the homogeneous and heterogeneous neutron capture cross section as the montmorillonite bed thickness increases out to 10 cm! Similar results hold for the thermal neutron diffusion length, as seen in Figure 5. Small differences between the homogeneous and heterogeneous thermal neutron diffusion coefficient are shown in Figure 6.