The dual-spaced neutron logging tool to measure formation
porosity consists of a neutron source and count rates from two spaced neutron
detectors NEAR and FAR.A NEAR to FAR
RATIO (R) is developed and a ratio-porosity transform, usually a series
expansion in powers of R, is used to compute actual porosity:

Equation 1: RATIO-POROSITY TRNSFORM

Generally n lies in the range 3 to 6; segmentation into
several porosity intervals is common.The coefficients An are determined by combining laboratory, field data,
and Monte Carlo modeling.

The count rates NEAR and FAR have standard deviations that
lead to a RATIO standard deviation and so a standard deviation on the computed
porosity:

Equation 2: Standard Deviation on POROSITY

The logarithmic derivative in (2) is called the porosity measurement sensitivity (S).

The main objective is to maintain DELTA phi as small as
possible.The historical discussion was
completely dominated by consideration of the fast neutron slowing down length, Ls- the mean distance a neutron
travels as it just reaches thermal energy.It will be seen that neutron sources with highest energies reduced
the numerator of (2), while sources with lowest energies increased the denominator.

The slowing down length scales the count rate seen by
neutron (and gamma!) detectors. Look at equation (3) to see this exponential scaling.Consider a fixed detector spacing:the ratio method’s porosity sensitivity
favors a scale of several/many slowing down lengths, whereas the detector’s
count rate and measurement precision/repeatability favors little/few slowing
down lengths.The slowing down length is
a function of neutron source energy and formation porosity.

Figure 1 from above was taken from US Patent 3,906,224.One of the objectives of its supporting work was
to determine if one particular neutron
source energy offered advantage in dual-spaced epithermal neutron porosity
logging tools.

This figure was based on experimental data acquired with a
dual-spaced He3 logging tool with four different neutron sources having average
neutron energies that ranged from 2.5 Mev for Cf252 to 14 Mev for a neutron
generator in limestone at a fixed porosity of 25%.The He3 detectors were wrapped in cadmium and
had NEAR and FAR spacings from the neutron source of 50 and 80 centimeters,
respectively.

According to the teachings of this patent, the neutron
generator system produced the WORST porosity
measurement sensitivity whereas the Cf252 based system yielded the BEST.Crudely speaking, sensitivity determines the
slope of the ratio-porosity transform on a ratio versus porosity
crossplot.The slope for the 14 Mev
generator system was worse than just being nearly flat: beyond a porosity of
about 60%, it changed signs with the result that the transform became
double-valued!This observation held
whether a continuous or pulsed operational mode was used with the neutron
generator.

For a typical dual-spaced neutron porosity measurement system,
the ratio of NEAR to FAR count rates is given in elementary 2-group theory by

Equation 3: Classical NEAR / FAR RATIO

where R0 is the ratio of the NEAR counting
efficiency to the FAR counting efficiency times the ratio of the FAR spacing to
the NEAR spacing; DELTA is the FAR spacing minus the NEAR spacing; and Ls is
the fast neutron slowing down length.[With R chosen sufficiently large, R is
independent of interference from changes in the thermal neutron diffusion
coefficient (D), the thermal neutron diffusion length (L), and the thermal
neutron capture cross section (SIGMA).]Use
of the chain rule shows that the porosity measurement sensitivity is given by

Values of Ls, for any fluids and any minerals mixed in any
proportion to form any formation, are not that easy to come by: historically, they were the raison d’etre for
nuclear micro geophysical models like SNUPAR, MSTAR, and LVPM.Within
the context of these models, neutron sources with higher energies have larger
slowing down lengths than sources with lower energies.Equation
(4) clearly indicates that, for modest changes in the logarithmic derivative in
Ls for different neutron energies, porosity measurement sensitivity is inversely
proportional to the slowing down length.No wonder that 14-Mev neutron generator systems had the worst
sensitivity when measuring porosity by the dual-spaced ratio method!Conversely, since Cf252 has the lowest mean
fast neutron energy in this study, it follows that it would have the best
sensitivity.

The next topic covered by US Patent 3,906,224 was the impact of the actual values for the
NEAR and FAR count rates and their standard deviations on porosity measurement resolution, i.e. the factor DELTA R / R in
equation (2).From “DATA REDUCTION AND
ARROR ANALYSIS FOR THE PHYSICAL SCIENCES” by Bevington and Robinson, page 46,
equation (3.26), and neglecting the cross term, we can write

Equation 5a: Propigation of Errors on RATIO

Since the count rates NEAR (N) and FAR (F) obey Poisson
statistics, we can write for their standard deviations in time interval DELTA t
just

Equation 5b: Poisson Statistics for NEAR and FAR

so that

Equation 6: Fractional Standard Deviation on RATIO

Combining equations (2), (4), and (6) then yields

Equation 7: One Standard Deviation on POROSITY

It is desirable to maintain DELTA phi as small as possible,
meaning high sensitivity (S) and high NEAR and FAR count rates N and F.In view of the individual count rate
expressions in (3), sources with higher energy neutrons will, for the same
neutron output, have higher count rates at fixed NEAR and FAR spacings – this
means that the 14 Mev neutron generator would be BEST and Cf252 the WORST at
producing a low DELTA phi value in the numerator of (7).This was indeed observed in the experiments
supporting US Patent 3,906,224.For a given spacing from the neutron
source, best is that source that has the longest slowing down length, since the
count rate is scaled by distance divided by slowing down length.

In summary, small values for DELTA phi involved a compromise
between the numerator and denominator of equation (7).In the end, this patent chose the AcBe
neutron source as a compromise between the two extremes represented by Cf252 and
a 14-Mev neutron generator.