Heat loss from the wellbore fluid depends on the temperature distribution in the formation. Formation temperature distribution around a well was modeled by Ramey (1962) by assuming a vanishingly small wellbore radius. The assumption, while robust for some cases, could lead to unrealistic predictions of early-time behavior. This paper uses a rigorous model of heat transfer paper uses a rigorous model of heat transfer developed with consideration for the appropriate boundary conditions; that is, the heat transfer at the formation/wellbore interface is represented by the Fourier law of heat conduction. The superposition principle is used to account for the gradual change in heat transfer rate between the wellbore and the formation.
The results of the new analysis are in agreement with the classical work of Ramey for large times (dimensionless time, tD greater than 10) However, significant differences are noted between the proposed solution and that of Ramey's log-linear approximation at small (tD less than 1.0) times. These differences may have considerable effect on certain applications, such as static earth temperature estimation from temperature logs, and bottomhole temperature estimation for cyclic steam injection.
We also present an approximate algebraic expression for the rigorous integral solution of dimensionless formation temperature, TD. This simplified expression is accurate for most engineering calculations. Use of this solution is shown in the second part of the paper where the wellbore fluid temperature distribution is computed during two-phase flow.
The importance of various aspects of heat transfer between wellbore fluid and the earth has generated a rich literature on the subject. Carslaw and Jaeger have covered such fundamental concepts of thermal diffusion as linear flow of heat in a rod with constant temperature and flow of heat in regions bounded by a cylinder. These concepts were later modified and extended to the cases of heat loss in both oil and geothermal wells.