Wellbore Heat Transmission
Authors: Ramey H.J. 
Published: December 2007

Abstract
As fluids move through a wellbore, there is transfer of heat between fluids and the earth due to the difference between fluid and geothermal temperatures. This type of heat transmission is involved in drilling and in all producing operations. In certain cases, quantitative knowledge of wellbore heat transmission is very important. This paper presents an approximate solution to the wellbore heat-transmission problem involved in injection of hot or cold fluids. The solution permits estimation of the temperature of fluids, tubing and casing as a function of depth and lime. The result is expressed in simple algebraic form suitable for slide-rule calculation. The solution assumes that heat transfer in the wellbore is steady-state, while heat transfer to the earth will be unsteady radial conduction. Allowance is made for heat resistances in the wellbore. The method used may be applied to derivation of other heat problems such as flow through multiple strings in a wellbore. Comparisons of computed and field results are presented to establish the usefulness of the solution.

Introduction
During the past few years, considerable interest has been generated in hot-fluid-injection oil-recovery methods. These methods depend upon application of heat to a reservoir by means of a heat-transfer medium heated at the surface. Clearly, heat losses between the surface and the injection interval could be extremely important to this process. Not quite so obvious is the fact that every injection and production operation is accompanied by transmission of heat between wellbore fluids and the earth. Previously, the interpretation of temperature logs has been the main purpose of wellbore heat studies. The only papers dealing specifically with long-time injection operations are those of Moss and White and Lesem, et al. The purpose of the present study is to investigate wellbore heat transmission to provide engineering methods useful in both production and injection operations, and basic techniques useful in all wellbore heat- transmission problems. The approach is similar to that of Moss and White.

DEVELOPMENT
The transient heat-transmission problem under consideration is as follows. Let us consider the injection of a fluid down the tubing in a well which is cased to the top of the injection interval. Assuming fluid is injected at known rates and surface temperatures, determine the temperature of the injected fluid as a function of depth and time. Consideration of the heat transferred from the injected fluid to the formation leads to the following equations. For liquid,
........................(1)
and for gas,
....................(1A)
where
....................(2)
Eqs. 1, 1A and 2 are developed in the Appendix. These equations were developed under the assumption that physical and thermal properties of the earth and wellbore fluids do not vary with temperature, that heat will transfer radially in the earth and that heat transmission in the wellbore is rapid compared to heat flow in the formation and, thus, can be represented by steady-state solutions. Special cases of this development have been presented by Nowak and Moss and White. Both references are recommended for excellent background material. Nowak presents very useful information concerning the effect of a shut-in period on subsequent temperatures. Since one purpose of this paper is to present methods which may be used to derive approximate solutions for heat-transmission problems associated to those specifically considered here, a brief discussion of associated heat problems is also presented in the Appendix. Analysis of the derivation presented in the Appendix will indicate that many terms can be re-defined to modify the solution for application to other problems. Before Eqs. 1, 1A and 2 can be used, it is necessary to consider the significance of the over-all heat-transfer coefficient U and the time function f(t).Thorough discussions of the concept of the over-all heat-transfer coefficient may be found in many references on heat transmission. See McAdams or Jakob, for example. Briefly, the over-all coefficient U considers the net resistance to heat flow offered by fluid inside the tubing, the tubing wall, fluids or solids in the annulus, and the casing wall. The effect of radiant heat transfer from the tubing to the casing and resistance to heat flow caused by scale or wax on the tubing or casing may also be included in the over-all coefficient.
JPT
P. 427^