Wellbore Heat Loss in Production and Injection Wells
Authors: Horne R.N. , Shinohara K. 
Published: December 2007

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Introduction
The classic work by Ramey on wellbore heat transmission derived the temperature distribution in a well used for injecting hot fluid. Ramey later expanded on this to give the rate of heat loss from the well to the formation. However, by assuming that the fluid remains at its inflow temperature, Ramey's analysis effectively gave the heat loss at infinite fluid flow rate - in other words, the maximum possible heat-loss rate. This paper reexamines this problem for finite fluid flow rate and determines the heat-loss rate as a function of fluid properties and fluid flow rate. Because this analysis is valuable when considering geothermal wells, results will be presented for producing and injection wells. This paper presented for producing and injection wells. This paper considers only single-phase fluids flowing in the well. Satter suggested a method for estimating wellbore heat loss when considering condensing steam flow and presented a sample procedure for a given set of reservoir presented a sample procedure for a given set of reservoir properties. His analysis also may be extended to properties. His analysis also may be extended to production wells to obtain heat-flow estimates with two-phase production wells to obtain heat-flow estimates with two-phase flow. For single-phase flow, the formulation permits direct calculation of wellbore heat loss with various production and injection conditions. production and injection conditions. Injection
Using Ramey's analysis and notation, consider a heat balance in the radial direction on a section of a well with height dz, losing heat at rate dq from the casing to the formation. Then,
...............(1)
where T1 is the temperature of the fluid in the tubing, Te is the temperature of the formation, k is the earth thermal conductivity, r, is the inside radius of the tubing, U is the over-all heat-transfer coefficient between the inside of the tubing and the outside of the casing (see Willhite), and f(t) is a dimensionless time function described by Ramey. For long times, f(t) can be approximated as
......................(2)
where r2 is the outside radius of the casing in meters, a is the thermal diffusivity of the earth in square meters per second, and t is the production time in seconds.Performing an over-all heat balance on the well and considering the changing temperature of the fluid as it flows in the tubing, T, can be evaluated as
..................(3)
where az + b is Te, the formation temperature (assuming linear geothermal gradient), b is the surface temperature, and z is measured downward. T0 is the injection temperature. A is a group of variables defined as
..................(4)
where w is the fluid flow rate and c is the specific heat of the fluid.Integrating Eq. 1 (with respect to depth z) and substituting T1 = T0, Ramey obtained
.............(5)
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