Analytical Definition Of The Overall Heat Transfer Coefficient
Authors: Zolotukhin A.B. 
Published: December 2007

Abstract
It is a well-known fact that the solution of many problems connected with heat transfer in porous media problems connected with heat transfer in porous media can be simplified efficiently by using the overall heat transfer coefficient concept to represent the heat loss from a reservoir into the adjacent strata. This idea is not a new one, and is efficiently used by many authors. The value of the overall heat transfer coefficient, however, has been assumed to be constant.
A new approach to define the value of the overall coefficient is proposed in this study. It is shown that the value of the overall heat transfer coefficient is not constant, but changes with respect to time. The use of the overall heat transfer coefficient has shown the applicability of this coefficient for solving some problems connected with thermal recovery and that high problems connected with thermal recovery and that high accuracy solutions are obtained.

Introduction
In many oil reservoirs exploited by thermal methods, heat loss plays a significant role. In some cases thermal recovery methods cannot be used at all because of heat loss. Sometimes, numerical evaluations of heat loss have to be done to forecast reservoir behavior during thermal recovery processes and to estimate the efficiency of such processes.
To calculate heat loss during the thermal recovery process accurately, one has to solve a two dimension (for one-dimension flow geometry) or three-dimension (for two-dimension flow geometry) energy equation, which is a very difficult problem.
Solutions of many problems connected with heat transfer can be simplified efficiently by supposing the heat transfer between strata can be described by Newton's Law:
(1)
where U is the overall heat transfer coefficient.
It is obvious that use of the overall heat transfer coefficient reduces the number of dimensions. Some authors have used this expression successfully to solve complicated mathematical problems.
The value of the overall heat transfer coefficient has been assumed to be constant. Nevertheless, some experiments to measure the value of this coefficient have shown that its value decreases with respect to time during the experiments.
The purpose of this study is to obtain, if possible, a simple analytical equation for the overall heat possible, a simple analytical equation for the overall heat transfer coefficient dependent on time for processes such as hot water or steam injection, or in-situ combustion. The value of the overall heat transfer coefficient was obtained by using an accurate expression for the total value of heat losses from the reservoir to the adjacent strata. The values were compared with some experimental results, showing a good agreement between experimental and theoretical results.
As an example of the use of the overall heat transfer coefficient dependent on time, the problem of hot water injection into a homogeneous water-saturated reservoir was solved analytically. Comparing results obtained through the accurate analytical solution with use of the time-dependent overall heat transfer coefficient indicated high accuracy in the latter solution. As a second example, the problem of hot water injection into a multilayered reservoir was solved analytically. Comparison with numerical solution of the same problem obtained by Martuzan in 1964 has shown the applicability of the time-dependent overall heat transfer coefficient for solution of certain thermal recovery problems. The following presents the results of this study. presents the results of this study.

STATEMENT OF THE PROBLEM
Since the overall heat transfer coefficient depends neither on the type of heat source nor the temperature and geometry of flow, we can write the following equation for heat loss from the reservoir to the adjacent strata:
(2)