A new theorem is stated on the construction and use of orthogonal sets for solving boundary value problems in a composite medium where the boundary function traverses more than one region. The concept of quasi-orthogonality is introduced and employed in a rigorous expansion of the boundary function into a series of the nonorthogonal eigenset that arises from separation of variables in such instances. Orthogonal sets are constructed from the nonorthogonal (quasi-orthogonal) eigenset by the use of orthogonality factors for each region derived from the orthogonality condition. Orthogonality factors for representative problems in neutron diffusion, heat conduction, scalar wave propagation, and nonrelativistic quantum mechanics are computed for rectangular, spherical, and cylindrical coordinates. The theorem allows a solution of multi-region problems by a direct expansion without resort to integral transformation or Green's functions. It is an extension of the method of separation of variables and generalized Fourier analysis to cases heretofore beyond its scope.

Two‐group neutron diffusion theory has been applied to two concentric cylindrical regions representing borehole and formation to show that the ratio of thermal neutron flux at two appropriately large distances along the axis is virtually identical to the epithermal flux ratio, and is a measure of a single epithermal parameter of the formation, viz., the slowing down length. The slowing down length is most strongly dependent on hydrogen content. It is shown that variations in salinity of the liquid in the formation or the borehole and uncertainties in borehole size have relatively little effect on the flux ratio. Experiments in a borehole model showed that the thermal flux ratio does not depend on eccentricity of location of the logging tool in the borehole. Fits of theory to experiment allowed determination of slowing down lengths for Pu‐Be neutrons in salt water filled sandstone at two porosities for which experimental values have not previously been available. These investigations suggest that an improved porosity log can be obtained by use of 70‐ and 90‐cm source‐detector spacings with a neutron source of [Formula: see text] neutrons/sec or greater. Proportional counters containing [Formula: see text] are recommended as detectors. Such a system employing flux ratios is expected to be insensitive to casing and cement in cased holes.

An analytical theory of epithermal neutron logging is presented. One‐group diffusion theory is applied to the slowing down of neutrons from a point fast neutron source in infinite continuous media, in a single cylinder, and in concentric cylinders representing a fluid‐filled borehole and the surrounding formation. Numerical results are given for the epithermal neutron flux in a water‐filled hole six inches in diameter, passing through limestone of 10 percent or 30 percent porosity. Preliminary semiquantitative agreement is obtained with the relative response of a commercial logging instrument in the range of 10 to 100 percent porosity.

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Experiments were performed to determine the distribution of thermal neutrons and of indium resonance neutrons in continuous hydrogenous media and in pipes passing through hydrogenous media. Included in the study were water, brine, mixtures of sand and water, and mixtures of sand and brine. Experiments in a continuous typical barite drilling mud showed that the neutron distributions were essentially the same as in water. Also, from the point of view of these experiments, oil and fresh water are nearly identical. These experiments show that well fluid (and, by inference, cement) imposes serious limitations on the sensitivity and accuracy of the neutron‐neutron logging method. The indium resonance neutron response (or, in general, the intensity of epithermal neutrons) is a more reliable indicator of hydrogen content of the formation than is the thermal neutron response. The neutron‐neutron method of chlorine determination was found to be not sensitive enough to be useful with brines of the concentrations ordinarily found in reservoirs.