Neutron flux and axial moments in three-region cylindrical geometry applied for neutron log calibration. Part I: Theoretical description
The scalling procedure the calibration curve (for a dual detectorneuron porosity tool as well as for a single detector) was developed by Profesor Jan A. Czubek in 1992-1995. The general calibration curve was obtained for the whole range of formation lithology and its porosity at any kind of borehole diameter, boreholeand formation sanity, tool stand-off, and drilling fluid physical parameters. Presence of an intermediate zone between the borehole and the formation was taken into account. This paper presents a next of the development of theoretical solutions, which gives a possibility to calculate the apparent neutron slowing down and migration lengths in three-region cylindrical systems which represents the borehole, the intermediate zone (e.g., mud cake at the borehole walls), and the geological formation. A solution to the neutron diffusion equation is given for a three-region cylindrical coaxial geometry. The calculations is performed in two neutron-energy groups which distinguish the thermal and epithermal neuron fluxes in the media irradiated by the fast point neutron source. The solutions in the present paper are applied to the method of semi-empirical calibration of neutron well-logging tools. The three-region cylindrical geometry corresponds to the borehole of radius R1 surrounded by the intermediate region (e.g., a mud cake) of thickness *r2-R1) and finally surrounded by the geological formation which spreads from R2 up to infinity. The cylinders of an infinite length are considered. The paper gives detailed solutions for the 0-th, 2-nd, and 4-thneutron moments of the neurton fluxes for each neutron energy group and in each cylinrical region. A comprehensive list of the solutions for integrals containing Bessel functions or their deratives, wich are absent in common tabels of integrals, is also included.
bibliogr. 10 poz., rys.