Introduction

The Transmission Probability Method (TPM) is the main
computational engine inside the LVPM program.  It provides a more
detailed and more accurate description of the propagation of
neutrons and gamma rays in porous media.  TPM interrelates pore
size, pore shape, and laminae bed thicknesses for all neutron
macroscopic scattering and absorption cross sections; and the
gamma ray mass attenuation, mass energy, and linear attenuation
coefficients.

Ultimately, using various logging tool proxy models, program LVPM
becomes the generator of the non-linear mixing rules for all
nuclear formation physical parameters including neutron porosity,
capture cross section, bulk density, and Pe.

TPM converts the original experimental nuclear cross section data
taken in various laboratories using (generally) thin, homogeneous
samples with no pores - in good transmission and scattering
geometries - into a form more useful for computing the response
of nuclear logging tools in thick heterogeneous formations having
pores with finite sizes.  TPM also provides a definite, well
-defined method for converting basic tool responses and shop
calibrations recorded in media with/without pores into ones more
suitable for logging real earth formations with finite pore sizes
and/or laminated beds.

"Transmission" here is not meant to imply that the wellbore
geometry is a transmission geometry for the various nuclear
logging tools, that neutrons and gamma rays travel only as plane
waves in the borehole-formation region, or that somehow the
neutrons and gamma rays travel through laminated beds that are
perfectly parallel or perpendicular to the borehole.  See the
PDFs below for details.

More on Historical Perspectives

The original forward models SNUPAR and MSTAR from
Schlumberger and Halliburton used nuclear cross section data
bases to compute macroscopic absorption and scattering cross
sections at various neutron energies and then applied these
cross sections in their proxy models to obtain the response
of neutron logging tools to formations composed of virtually
any minerals and any fluids.  These models also helped to
delineate departure curves within chart books for various
nuclear logging tools.

Linear mixing rules were used for these models' internal
macroscopic absorption and scattering cross sections.
The models themselves were developed mainly to deal with the
non-linear response of neutron logging tools to porosity.
Central to their operation was the computation of neutron
slowing down length and its use in various proxy models for
neutron porosity.  These models also used linear mixing rules
to deal with bulk density and density porosity.

Doctor Neutron has extended the content and scope of such models
in several significant ways.  He demonstrated that linear mixing
rules imply infinitesimal pore sizes.  He then developed
expressions for the macroscopic cross sections with finite pore
sizes in terms of non-linear mixing rules.  He also extended the
original scope of the forward models to include gamma ray mass
and linear attenuation coefficients for a very wide range of
gamma energies.  Finally, Doctor Neutron extended the scope of
TPM to handle laminated porous media, with variable pore sizes.

The original forward models and those recently developed by
Doctor Neutron have never assumed transmission geometries in
their proxy models.  The Transmission Probability Method simply
develops new expressions for the macroscopic neutron scattering
and absorption cross sections and gamma linear attenuation
coefficients in order to describe heterogeneous porous media.

For both the older and newer models, these cross sections and
attenuation coefficients are used to drive the nuclear logging
tools’ proxies.  Several unexpected results from the newer
models have been the introduction of non-linear mixing rules
for most quantities of physical interest, as well as pore size
and pore shape effects and laminated bed thickness effects
beyond what might be expected from classic/conventional bed
thickness weighting.

Except for special circumstances, linear mixing rules no longer
apply - even for bulk density or neutron capture cross section!

In his 1967 seminal paper, Zakharchenko described the effects of
mercury inclusions in cinnabar on measured values of SIGMA.
His equation is shown above: it is the sine qua non of the
propagation of neutrons and gamma rays in vuggy porous media.
It represents the generalization of the mixing rules of
SIGMArock and SIGMAfluid at porosity PHI for heterogeneous
media.

Analogous expressions are used in the LVPM model for all
internal and external neutron macroscopic scattering and
absorption cross sections and all gamma ray linear attenuation
coefficients.  In his equation, L is NOT the pore size, but a
length closely associated with the pore size - it is pore size
divided by the cube root of the porosity.

As L decreases towards zero, L'Hospital's rule shows that it
correctly reduces to the classic linear volumetric mixing rule
for the neutron capture cross section in homogeneous porous
media.  This is shown in the figure just below.  Thus, it
becomes clear that such a rule is based on capture in a
homogeneous medium with infinitesimal pore sizes.

As mentioned previously, Gabanska and Krynicka-Drozdowicz
constructed a Lucite matrix and loaded silver into its pores
and obtained experimental values of SIGMA.  Their work
confirms both the original theory of Zakharchenko and that
of the forward model LVPM, since it fits their data
extremely well.

Accurate SIGMA values in vuggy porous media must account for
pore sizes and their distribution !!