A Study of Forward Combustion in a Radial System Bounded by Permeable Media
Authors: Thomas G.W. 
Published: December 2007

Abstract
A mathematical model of forward combustion in an oil reservoir is treated in this paper. The model describes a radial system having a vertical section of essentially infinite thickness, all of which is permeable to gas flow. Combustion, however, is presumed initiated over a limited thickness of the total vertical section. In the interval supporting combustion, the mechanisms of radial conduction, convection and heat generation are taken into account. Above and below the burning interval, heat transport in the radial direction is by conduction and convection. Vertical heat losses from the ignited interval are accounted for by conduction alone.A general solution is presented for the temperature distribution caused by radial movement of the combustion front. The results show that no feedback of heat occurs into the ignited interval when convection and conduction are acting in the bounding media. Peak temperatures are also 5 to 10 per cent higher than in the case where heat transport in the bounding media is by conduction alone. We arbitrarily define vertical coverage to be that fraction of the total ignited interval which is at 600F above ambient, or greater, at any given time. The radial distance at which the vertical coverage becomes zero is the propagation range of the combustion front. It was found that an increase in vertical coverage results when the oxygen concentration, fuel concentration or gas-injection rate is increased. Moreover, the combustion front can be propagated 10 to 15 per cent further than in the case where only conduction is acting above and below the ignited interval.

Introduction
In the theoretical treatment of forward combustion in a radial system, one of the problems encountered is the determination of the transient temperature distributions caused by an expanding cylindrical heat source. Bailey and Larkin and Ramey simultaneously presented analytical solutions to the problem assuming heat transport by conduction alone. In a subsequent publication, Bailey and Larkin included the effects of both conduction and convection while treating linear and radial models. In this latter work, however, vertical heat losses were largely neglected. Selig and Couch dealt with a radial model in which both conduction and convection were acting. Only a limiting case involving vertical heat losses was considered, however. Namely, temperatures on the boundary of the bed of interest were set equal to zero. Solutions thus obtained were representative of a system having a maximum vertical heat flux. Chu recently treated a more general case in which a permeable bed was considered bounded by impermeable media. Conduction and convection took place within the bed, and only conduction outside of the bed. The effects of vertical heat losses were included in his study. Solutions were obtained by numerical techniques. This paper is an extension of the theoretical work of other authors pertaining to forward combustion in a radial system. In particular, a mathematical model of the process is treated in which heat generation occurs over a small vertical interval of a larger permeable section. In the interval supporting heat generation, and above and below this interval, the mechanisms of radial conduction and convection are also presumed acting. Heat losses from the ignited interval are accounted for by vertical conduction. An analytical solution for the temperature distribution caused by radial movement of the burning front is presented. The effects of certain process variables are indicated and comparisons with Chu's results are made.

THEORY
To render the mechanism of forward combustion tractable to mathematical treatment, we idealize the problem to the extent of assuming continuous reservoir media possessing homogeneous and isotropic properties. The following additional assumptions are implicit in this analysis.The thermal parameters, i.e., heat capacities, thermal conductivities and thermal diffusivities are invariant with temperature and pressure. Moreover, the bounding media possess the same thermal properties as the bed of interest.The temperatures of the porous media and its contained fluids at any point and at any time are equal.The reaction rate between the oxidant gas and the fuel is infinite. The assumption implies that the incoming oxygen concentration instantaneously goes to zero within an infinitesimal distance, i.e., the width of the combustion zone is negligible.The rate of gas injection is constant and corresponds to the average rate throughout the lifetime of the project.The fuel concentration is constant throughout the volume of rock swept out by the burning zone.There is complete burnoff of fuel. This assumption demands that the rate of propagation of the burning front equals the rate of fuel burnoff.
JPT
P. 1145^