Over-all Heat Transfer Coefficients in Steam And Hot Water Injection Wells
Authors: Willhite G.P. 
Published: December 2007

Abstract
Casing temperatures and wellbore heat losses are critical variables in steam and hot water injection wells. Several papers have been written presenting methods of estimating these parameters if the over-all heat transfer coefficient is known. The over-all heat transfer coefficient for a wellbore is developed from its component terms to promote a better understanding of the concept. Specific methods have been selected from the heat transfer literature for estimating the size of each heat transfer component. Simplified calculation procedures are suggested for determining the over-all heat transfer coefficient. Comparison of calculated and measured casing temperatures during steam injection confirms the basic formulation and applicability of the suggested procedure for engineering calculations.

Introduction
The design of steam and hot water injection projects requires estimation of casing temperatures and wellbore heat loses. Several authors have shown that wellbore heat losses and casing temperatures can be calculated if the over-all heat transfer coefficient is known. This article discusses methods of determining the over-all heat transfer coefficient from the process variables.

Development
The steady-state rate of heat flow through a wellbore Q Btu/hour is proportional to the temperature difference between the fluid and the formation, and the cross-sectional area perpendicular to the direction of heat flow. The proportionally factor, called the over-all heat transfer coefficient, represents the net resistance of the flowing fluid, tubing, casing annulus, casing wall and cement sheath to the flow of heat. Thus, we can write
Q = Uj Aj Tj............................ (1)
Eq. 1 defines Uj, the over-all heat transfer coefficient based on the characteristic area A, and a characteristic temperature difference T,. Subscript j in Eq. 1 identifies the surface upon which these quantities are based. In theory, any radial surface could be used to determine the characteristic area. Some choices are more convenient to work with than others. For example, if hot fluid is injected down tubing it is preferred to let A, be the outside surface area of an incremental length of injection tubing, 2 to L, and let T, be the difference between the temperature of the flowing fluid Tj and the temperature at the cement-formation interface (the drill hole) Th. Then Uj = Uo, referring to the outside tubing surface area, and Eq. 1 would be
Q = 2 to Uto (Tj- Th) L................ (2)
If the fluid is injected down the casing or casing annulus. the characteristic area could be the inside surface area of the casing, and Eq. 1 would be written as
Q = 2 ci Uci (Tj - Th) L..................(3)
Subscript ci refers to the inside casing surface. An expression for the over-all heat transfer coefficient for any well completion can be found by considering the heat transfer mechanisms between the flowing fluid and the cement-formation inter-face. A brief derivation of the over-all heat transfer coefficient is presented in the following paragraphs for the case of a hot fluid flowing through tubing insulated with a dry air annulus. Other cases can be derived easily once the basic concepts are understood. Fig. 1 shows the wellbore model which will be used to derive Uto.

Heat Transfer Mechanisms
The rate of heat transfer between the flowing fluid and inside tubing wall is given by Eq. 4.
Q = 2 ti hj (Tj - Tti) L...................(4)
hf, is defined by Eq. 4 and is the film coefficient for heat transfer based on the inside surface area of the tubing (subscript ti) and the temperature difference between the flowing fluid and the inside tubing wall Tj-Tti. Heat flow through the tubing wall, casing wall and the cement sheath occurs by conduction. Fourier discovered that the rate of heat flow through a body is directly proportional to the temperature gradient in the medium.
JPT
P. 607ˆ