Calculated Temperature Behavior of Hot-Water Injection Wells
Authors: Squier D.P. , Smith D.D. , Dougherty E.L. 
Published: December 2007

Abstract
A system of differential equations describing the temperature behavior of fluid injected at constant surface temperature in a well is derived and solved analytically. A formula for the fluid temperature at any time and depth is given, as well as a special formula valid for very large times. These formulas are used to calculate temperatures for several typical cases. The results indicate that, initially, the temperature of the water entering the formation is considerably lower than the injection temperature. This condition lasts for only a short period--less than three days for most cases of practical interest. Following this highly transient period, during which the temperature of the fluid entering the formation builds up to about 50 to 75 per cent of the injection temperature, the system enters a quasi-steady state in which the temperature changes are very slow. After several years, the bottom-hole temperature will still be 50 degrees to 100 degrees F lower than the injection temperature, but the heat losses may be tolerable.

Introduction
Predicting the behavior of a hot-water flood requires that the temperature of the water entering the injection interval be estimated. This report describes the development and solution of a system of equations which describes the temperature behavior of the injected water in the wellbore with certain simplifying assumptions. The only previous means known to the authors for describing such a process is that of Moss and White. Their results appear to be close to those obtained by our method in the practical cases which were compared; this agreement is largely due to the fact that in our method temperature soon approaches a quasi-steady state, as was assumed in their method throughout. However, our model covers all times, is continuous (whereas the Moss-White model depends on breaking the depth into discrete intervals) and, we feel, more closely describes the physical problem.

FORMULATION OF THE PROBLEM

PHYSICAL SYSTEM AND ASSUMPTIONS
The injection procedure consists of pumping water at a fixed surface temperaturedown an infinitely long cylindrical well or tubing of inner radius . Any material exterior to the water column such as mud, casing, or cement is regarded as part of the formation. The general behavior of the system may be described qualitatively as follows. when the hot water is first introduced into the system, the temperature difference between the formation and the water is large, resulting in a high rate of heat transfer. As a result, the temperature adjacent to the wellbore rises very quickly. Because the segment of the formation adjacent to the wellbore largely controls the heat transfer rate, the heat transfer rate will become relatively constant when this portion has reached a temperature close to that of the water opposite it. The temperature of the water and formation then increase very slowly with time. The length of the initial highly transient period and the temperature of the water at its conclusion will be functions of depth, injection rate, injection-string radius, surface injection temperature and the physical properties associated with the water-formation system.
The following additional assumptions were made.There is no heat transfer by radiation in the system.There is no heat transfer by conduction in the vertical direction in either the injection stream or the formation.The linear volumetric and mass flow rate of the water is constant throughout the injection stream.No horizontal temperature gradient exists in the injection stream.The product of density and heat capacity is constant for both the water and the formation, and the formation thermal conductivity is constant.Initially, both the water in the wellbore and the reservoir are at a temperature given by the (constant) ambient surface temperature plus the product of depth and geothermal gradient (assumed constant). At large distances for the wellbore ( ), the formation will remain at this temperature.The water temperature and the formation temperature at r=r are equal for all depths D.
JPT
P. 436^