Wellbore Temperatures And Heat Losses During Production Or Injection Operations
Authors: Eickmeier J.R. , Ersoy D. , Ramey H.J. 
Published: March 2010

Abstract
A finite difference model has been developed to investigate transient wellboretemperatures and heat losses during injection or production operations. Themodel has particular application for investigating the early transienttemperature performance, during which time both rate of heat loss and wellboretemperatures are subject to considerable change. Tubing string, annulus fluid, casing, cement zone and formation temperatures may be determined. Example caseswhich are presented include temperature profiles resulting from oil productionin permafrost areas and temperature profiles for cold-water injection atacidizing rates.

INTRODUCTION
SEVERAL AUTHORS* have previously studied well tubing and casing temperaturechanges during fluid injection and/or during production operations. Because ofthe assumptions involved in the mathematical models, the analytical methodshave not been applicable for short-time investigations. The short-timetemperature performance is, however, of importance during acidizing operations, when considerable tubing thermal movement occurs, and during well start-upoperations, at which time severe hydrate and wax deposition problems may beencountered. A finite difference model has been developed to investigatewellbore temperatures during the early transient period, when both heat lossesto the formation and wellbore temperatures vary appreciably as functions ofdepth and time. The model can be used to investigate the performance ofdifferent annulus-insulating materials or to determine the effects of thecement zone thermal properties on the temperature in the tubing and casingstrings. Although developed for short-time investigations, the model can alsobe used for long-time investigations.
Program results for oil production and cold-Water injection cases arepresented.

DISCUSSION

Analytical Methods
The original paper by Moss and White(2) dealing with the calculationof temperature profiles in wells resulting from the injection of either hot orcold water, was published in 1959. Their analysis as based on the constant-rateline source solution to the diffusivity equation. They assumed that theinjection fluid temperature was the same as the casing temperature at anyparticular depth. To utilize their approach, it was necessary to divide thewell depths into lengths of 100 to 1,000 feet.
Ramey(3) used a similar approach, but he formulated the problem insuch a manner that it was not necessary to divide the well depth intoincremental lengths; he simplified the calculations by introducingdimensionlessquantities which could be readily evaluated by referring to anappropriate time function, f(t). Ramey also introduced into the calculationsthe concept of an over-all heat transfer co-efficient to account for theeffects of stead-state, wellbore conditions. His results me consideredaccurate for time periods greater than one week for most typicalproblems.
Squier, et al(4) presented an exact short-time solution for thetemperature behaviour of hot water injection wells and also an asymptoticsolution valid for large times. One assumption incorporated into his solutionis that fluid injection temperatures and wellbore formation face temperaturesare the same; i.e the presence of the casing or of the cement zone isneglected.

Numerical Methods
Leutwyler(5) constructed a numerical model to examine tubing andcasing temperatures during steam injection.