A Computer Model to Determine the Temperature Distributions In a Wellbore
Authors: Marshall David W. , Bentsen Ramon G. 
Published: March 2010

Abstract
There exists a need for an efficient, accurate method for predicting how temperature distributions about the wellbore change with time. This is particularly true in the Arctic, where the proper handling of problems connected with permafrost and gas hydrates can be critically dependent upon such information. Consequently, a computer model has been developed which utllizes a direct solution technique to solve the finite-difference equations describing transient heat transfer in the wellbore. The solution technique used is considerably more efficient than those used in earlier studies. This enables the model to be used at the wellsite, on a real-time basis, to continuously monitor downhole temperatures. The model is sufficiently versatile to allow for the complicated flow histories involved in drilling a well. Parametric sensitivity testing of the model indicates that a number of assumptions underlying previous models are invalid or unnecessary, and that certain previously ignored factors have a significant effect on wellbore temperature profiles.

Introduction
The present trend of drilling deeper and costlier wells requires an increasingly accurate knowledge of the numerous variables involved. One such variable is temperature. In the past, it has been convenient to ignore the temperature distribution in a well and to assume an isothermal system, primarily because no practical means existed for determining the wellbore temperature profile. An accurate means of estimating the temperature distribution, and its variation with time, would have a number of applications, as follows:Cementing program design.Drilling fluid rheology and design.Determination of casing and tubing thermal stresses.Injection and production operations.Well design in permafrost regions.Control of gas hydrates.
Temperature data cannot be obtained by physical measurement other than at isolated points, and thus analytical means must be employed to model the system. Such methods have formed the basis of previous work in this area. The literature contains studies ranging from the early simplified graphical methods to the more recent intricate computer models.
The earliest study of wellbore temperature is that of Farris(1). He developed charts for predicting bottom-hole circulating temperatures based on measurements made in five shallow Gulf Coast wells. The severe shortcomings of these charts prompted further research into a more precise method of estimating circulating temperatures. Edwardson et al(2) and Tragesser et al(3) developed methods based on an assumed formation temperature distribution after circulation and provided no means of calculating the wellbore temperature distribution directly. Holmes and Swift(4) obtained a steady-state solution to the wellbore heat transfer equations, which cannot be applied to transient behaviour. Raymond(5) developed a method of predicting temperature distributions for both transient and pseudo-steady-state conditions. However, he contended that the pseudo-steady-state solution was adequate for all applications.
Most recent research has dealt with means of solving unsteady-state equations formulated by Raymond, or variations on these, using finite-difference techniques(6,7.8.9,l0). Although the earlier methods are simple and easy to apply, they are very inaccurate. The more recent computer models entail solving the finite-difference equations describing the heat transfer by iterative methods.