A Linear Model And a Related Very Stable Numerical Method For Thermal Secondary Oil Recovery
Authors: Adler Gyorgy 
Published: March 2010

Abstract
This paper describes a linear mathematical model of the underground combustion process. The principal features of the model are:a large number of component in the system (oxygen, inert gas, two different ail components, combustion gas, the vapours of the liquids and the solutions of the combustion gas in the liquids), all together twelve components;the inclusion of complex physical-chemical processes (combustion of the oil components and they vapours, evaporation of the liquid, solution of the combustion gas in the liquids and their effects on the viscosities, heat conduction):a new approach to obtain a highly stable numerical method, based on:the use of more equations than the number of unknowns, exploiting the stability features of each equation,a new numerical treatment of the reaction term, for obtaining increased stability andthe use of a new, unconditionally stable, finite difference scheme for degenerating parabolic differential equations (including, first-order equations) [see ref. (4)].
The complexity of the system is also characterized by the fact that it involves 163 physical-chemical parameters, not counting the input data related to the numerical work itself.
The results of some "numerical tests" designed to check the consistency of the model are presented. The calculations show the following features to be in agreement, with laboratory experiments:because the wall of the reactor tube is supposed to be insulated against heat losses, it is difficult to find conditions which assure a heat front of constant temperature;if the ignition is successful and a heat front is obtained, it moves with constant velocity under constant injection conditions;water injection has a stabilizing effect on the heat front and increases its velocity.

INTRODUCTION
THE PRACTICAL AND THEORETICAL problems of secondary oil recovery by in-situ combustion are well known. Because the actual three-dimensional underground combustion process is extremely complicated, much effort has been concentrated on experimental and theoretical studies of linear laboratory models. The main difficulty in the theoretical work is mathematical, namely to find a sufficiently stable numerical method which permits the use of a large time step in the computation, so that the computer time needed for the computation will be practically feasible. The order of magnitude of this time step is the only limit which restricts the possible complexity of the mathematical model, because otherwise it is not difficult to "write down as general and complicated differential equations as one likes.
The main purpose of this paper is to present a numerical method which is stable enough to allow the consideration of a very general mathematical model possessing qualitatively new features. No attempt has been made to compare laboratory data with the results obtained by computation. We are convinced that the 163 physical-chemical input data involved are sufficient to fit the model to any laboratory experiment. Nevertheless, some results of a qualitative nature, showing important features of the combustion process, are included in the conclusions.