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 (LVPM) 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 LVPM homogeneous output for the current
example, corresponding to infinitesimal pore sizes.  Notice
that the dashed gold curve, the LVPM 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 LVPM heterogeneous
SIGMA outputs for pore sizes of 0.5 cm and 1.0 cm.  Both curves
are quadratic in LVPM 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 LVPM homogeneous and heterogeneous
LVPM 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 LVPM.  Figure 8 shows the behavior
of the photoelectric factor (Pe).