Introduction

The Transmission Probability Method (TPM) is the main
computational engine inside the LVFPM 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 LVFPM 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 LVFPM 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 LVFPM, since it fits their data extremely well.

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