A Two-Dimensional Analysis of Reservoir Heating by Steam Injection
Authors: Abdus Satter , Parrish David R. 
Published: December 2007

Abstract
The widely used Marx and Langenheim solution for reservoir heating by steam injection fails to account for the growth of the hot liquid zone ahead of the steam zone. Furthermore, that solution does not consider radial heat conduction both within and outside the reservoir and vertical conduction within the reservoir. In the present paper, a more realistic and generalized solution is provided by eliminating several restrictive assumptions of the ‘old theory'. However, fluid flow is not considered in this model. The partial-difference equations that describe the condensation within the steam zone and temperature distribution within the system have been solved by finite-difference schemes.
Calculated results are presented to show the effects of steam injection pressures ranging from 500 to 2,500 psia and rates, 120 and 240 lb/hr-ft, on the growth of the steam and hot liquid zones. A 50-ft thick reservoir with fixed thermal and physical characteristics was considered. Results show that heat losses from the reservoir into the surrounding rocks are not greatly different from those predicted by Marx and Langenheim. However, the heat distribution is markedly different. A sizable portion of the reservoir heat was contained in the hot liquid zone which grows indefinitely. This means that heat (warm water) could arrive at the producing wells sooner than predicted by the old theory. This is particularly true for low injection rate or high injection pressure. Curiously, for a given injection rate and pressure, the heat content of the hot liquid zone remains (except for early times) essentially a constant percentage of the cumulative heat injected.

INTRODUCTION
In 1959. Marx and Langenheim1 made a theoretical study of reservoir heating by hot fluid injection. Their solution has been widely used in the industry for the evaluation of the steam-drive process. This solution, however, is based upon an unrealistic assumption that the growth of the hot liquid zone ahead of the steam zone is negligible. Therefore, it cannot predict the arrival of warm water at the producing wells earlier than steam. Furthermore, in the so-called ‘old theory', radial heat conduction both within and outside the reservoir was neglected.
Willman et al.2 presented another analytical solution of the same problem. Their solution is comparable to the Marx-Langenheim solution and suffers from the same disadvantages.
Wilson and Root3 presented a numerical solution for reservoir heating by steam injection. While radial and vertical heat conduction both within and outside the reservoir were considered, their solution was provided essentially for the injection of a noncondensable fictitious hot fluid. The specific heat of the injected fluid was assumed to be equal to the difference between the enthalpy of steam and the enthalpy of water at the reservoir temperature divided by the difference in the two temperatures.
Baker4 carried out an experimental study of heat flow in steam flooding using a sand pack. 4 in. thick and 6 ft in diameter. The steam injection pressure was 2 to 5 psig and rates ranged from 22 to 299 lb/hr-ft. He showed that a significant portion of the injected heat was contained in the hot water zone. The theoretical steamed or heated volume, as calculated by the Marx and Langenheim method, fell between the experimental steamed and heated (including hot water) volumes.
Spillette5 made a critical review of the known analytical solutions dealing with heat transfer during hot water injection into a reservoir. These solutions are based upon many restrictive assumptions similar to the simplified solutions of the steam heating process. Spillette also presented a numerical solution for multidimensional heat transfer problems associated with hot water injection and demonstrated the utility and accuracy of the method.
Most mathematical models of steam and hot water recovery processes neglect fluid flow considerations.