Ramey's Wellbore Heat Transmission Revisited
- Jacques Hagoort (Hagoort & Assocs. B.V.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2004
- Document Type
- Journal Paper
- 465 - 474
- 2004. Society of Petroleum Engineers
- 1.14 Casing and Cementing, 5.6.4 Drillstem/Well Testing, 5.9.2 Geothermal Resources, 4.1.5 Processing Equipment, 6.5.2 Water use, produced water discharge and disposal, 4.1.2 Separation and Treating, 5.3.2 Multiphase Flow, 4.3.4 Scale
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In this work we assess Ramey's classic method for the calculation of temperatures in injection and production wells. We show that Ramey's method is an excellent approximation, except for an early transient period in which the calculated temperatures are significantly overestimated. We present a simple graphical correlation to estimate the length of this early transient period. We further demonstrate that Ramey's procedure for the estimation of overall heat-transfer coefficients holds good only for large values of the Fourier dimensionless time. Finally, we illustrate the results of this work by an example calculation of wellhead temperature in a flowing oil well.
Estimation of temperatures in a wellbore during injection or production is a recurring problem in petroleum engineering. Examples are the prediction of bottomhole temperatures of injection fluids (water, gas, and steam) and of wellhead temperatures in oil and gas wells. Almost all practical methods for the calculation of temperatures in wellbores go back to the classic paper by Ramey1 on wellbore heat transmission published in the early 1960s. In that paper, Ramey presented a simple analytical equation for wellbore temperatures based on a grossly simplified heat balance. Apart from this analytical temperature equation, Ramey also proposed a simple procedure to estimate an overall heat-transfer coefficient for wellbore heat losses comprising both transient heat resistance in the formation and near-wellbore heat resistance.2 Despite its simplifications, in practice Ramey's approach seems to work remarkably well.
In 1990, Wu and Pruess3 presented an analytical solution for wellbore heat transmission in a layered formation with different thermal properties without introducing the simplifying assumptions of Ramey. From their example calculations, they observed that the Ramey method is valid at long times but can generate large errors at early times. However, quantification of the conditions under which Ramey's method could be applied was beyond the scope of their paper.
The objective of this work is to assess the Ramey method and to establish criteria for its applicability. To this end we have first performed an inspection analysis of the basic wellbore heat transmission equations without the simplifying Ramey assumptions. This entails formulation of the equations in dimensionless form and identification of the governing dimensionless numbers. Subsequently, we have developed a rigorous solution of these dimensionless equations and have compared this solution with the Ramey solution for various ranges of the dimensionless numbers. We have also checked Ramey's procedure for estimating overall heat-transfer coefficients. To illustrate the results of the paper, we conclude with an example calculation of the estimation of wellhead temperature in a flowing oil well.
Our focus is primarily on heat transmission between the wellbore tubing and the formation. We have therefore kept the heat transmission within the tubing as simple as possible by assuming a constant fluid density and heat capacity, no heat conduction in the flow direction, and no frictional heating. In addition, we have restricted ourselves to the temperature distribution in production wells. Extension of the analysis to injection wells is straightforward.
The physical model that underlies the equations describing wellbore heat transmission consists of a straight, cased well that is cemented to the formation and equipped with tubing for the transfer of fluids to the surface. Both casing and tubing have a constant diameter. The diameter of the tubing is small with respect to the length of the tubing. Initially, the tubing is filled with a fluid in thermal equilibrium with the formation. At time zero, fluid starts flowing from the bottom of the tubing to the top at a constant flow rate. The fluid has a constant density and a constant heat capacity. Heat conduction in the flow direction and frictional heating in the tubing are negligibly small. The fluid that is initially present in the tubing is the same as the fluid that enters the bottom of the tubing. The temperature of the fluid at the bottom is equal to the formation temperature at the bottom. Flow in the tubing is 1D (i.e., temperature and fluid velocity depend only on the distance along the tubing). As the fluid moves up the tubing, it loses heat to the colder formation. Heat losses to the formation take place through heat conduction in the radial direction only. The effect of the tubing wall, the annular space between casing and tubing, the casing wall, and the cement zone on the heat transmission is included through a single, steady-state heat-transfer coefficient. The formation surrounding the wellbore is of infinite extent. The initial temperature of the formation increases linearly with depth, reflecting a constant geothermal gradient.
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