HISTORICAL PERSPECTIVES

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:

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:

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

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

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

so that

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.