Temperature distribution in flowing liquid wells

Abstract

The temperature distribution in flowing liquid wells is usually calculated from a simple equation, advanced by Boldizsar (1958) and Ramey (1962), under the assumption that the vertical temperature gradient is negligibly disturbed by the fluid flow. The crucial parameter in this equation is a time function derived by considering the heat conduction in a region bounded internally by a circular cylinder. The present results, obtained by the alternating direction, implicit, finite‐difference method, confirm the validity of this equation for elapsed time of intermediate range from days to months. Caution must be exercised, however, when the equation is used for very small or very large times. For the impractical case of very small times (< 1 day), the time function should be calculated from the correct model of heat conduction which takes into account the nonzero heat capacity of the borehole fluid. On the other hand, for very large times (> 1 year), the assumption of negligible net heat transfer in the vertical direction, upon which the equation is based, ceases to hold. This leads to the introduction of systematic errors which reach 10 percent after about 11 years, when the equation is used to estimate the formation temperature gradient and/or thermal conductivity.