PORE SIZE EFFECTS ON NEUTRON CAPTURE CROSS SECTION MEASUREMENTS
As a final example in the series with the vuggy oil saturated dolomite, the barite was replaced with borax and its saturation was dropped to 10%. The borax has an extremely high thermal neutron capture cross section (SIGMA) and heterogeneous effects are expected to be stronger than in all the other examples. This is crudely visualized as follows. The characteristic length (1/SIGMA) is associated with the thermal neutron absorption process: small SIGMA values have large characteristic lengths and vice versa. The neutron absorption process can "resolve" features of the order of the characteristic length and larger so that in this example in which SIGMA is over 8000 cu = 8 inverse cm, features of the order of 1/8 cm or larger can be resolved.
The Transmission Probability Method (TPM) used by the Laminated Vuggy Porous Media (LVFPM) program deals with these concepts in a more formal manner and accounts for both absorption and scattering during neutron slowing down, diffusion, and thermalization.Recall the equation for absorption:
This formula was first derived by V. F. Zakharchenko in 1967. It represents the non-linear mixing rule for SIGMA in heterogeneous media with finite pore sizes. In Figure 1 below, the red curve represents the classic linear mixing rule for SIGMA: this is the LVFPM homogeneous output for the current example, corresponding to infinitesimal pore sizes. Notice that the dashed gold curve, the LVFPM heterogeneous SIGMA output for a pore size of 0.0001 cm, is indistinguishable from its homogeneous output. Both curves are strictly linear in porosity.
The other two curves in Figure 1 are the LVFPM heterogeneous SIGMA outputs for pore sizes of 0.5 cm and 1.0 cm. Both curves are quadratic in LVFPM model formation porosity. At a porosity of 30%, the droop in SIGMA corresponding to these pore sizes is 11% and 21%. Thus, pore size both alters the relationship between SIGMA and porosity and also causes the relationship to become non-linear.
Recall that SIGMA, the thermal neutron diffusion length (L), and the thermal neutron diffusion coefficient (D) are interrelated by the equation
D = (L)**2 * SIGMA.
Figures 2 and 3 above detail the very strong non-linear behavior of both D and L on porosity and also the effects of pore size on both.
Figures 4 - 7 detail the LVFPM homogeneous and heterogeneous LVFPM outputs for bulk density, density porosity, and neutron porosity. Pore size effects on neutron porosity and its related slowing-down length are very large - in part because of the very high hydrogen content of borax. Notice that even the homogeneous neutron porosity and slowing-down length are non-linear functions of porosity - this was a major impetus for the original development of programs like SNUPAR and MSTAR, the precursors to LVFPM. Figure 8 shows the behavior of the photoelectric factor (Pe).