Reservoir Engineering - General - Fosterton Field – An Unusual Problem of Bottom Water Coning and...

One of the most complicated and potentially one of the most promising secondary recovery methods is that of underground combustion. A number of field tests1,2,3 have been performed recently, apparently indicating that the underground combustion process is technically feasible. Relatively little has been published concerning laboratory or theoretical investigations of the underground combustion process. A few results of laboratory tube runs have been published4,5 and these data are valuable in understanding the process. The tube run data also emphasize the complexity of the thermal recovery process. Under various conditions, problems in heat transfer, multiphase fluid flow and combustion rates arise. For example, at low temperatures the combustion process proceeds so slowly that the front cannot be propagated. It is a problem in heat transfer to predict what injection rate of air is necessary to sustain combustion in a particular reservoir. Certain fluid flow aspects of underground combustion have been studied by Wilson, et a13. In the development of some of the flow equations, assumptions of adiabatic combustion were made. Vogel and Krueger1 have devised an analog model of the heat conduction process for a constant temperature cylindrical source moving radially outward in a homogeneous conducting medium. Jenkins and Ramey7, in a discussion of the Vogel and Krueger paper, present an analytical solution of a heat conduction problem related to problems considered by Vogel and Krueger. In the present paper an idealized model of the heat conduction problem is considered, which is more general than the model of Vogel and Krueger or Jenkins and Ramey. In particular, initial well heating, vertical heat losses and arbitrary frontal velocities are included. Equations for the temperature are obtained in terms of integrals. These integrals have been evaluated numerically for the case of frontal movement corresponding to a constant air injection rate and the case of constant frontal velocity. These results are presented graphically. From these graphs one may estimate conditions under which nearly adiabatic temperatures are obtained. The application of these results requires some knowledge of reservoir thermal properties, combustion rates and residual fuel content. Assuming no vertical losses it is possible to obtain explicit evaluations of these integrals in the linear con- stant velocity case and in the radial case corresponding to a constant air injection rate. PRESENTATION OF RESULTS The geometry of the radial system is illustrated in Fig. 1 where the reservoir is represented as a slab of thickness h. By making an energy balance for an infinitesimal volume, one obtains the following equation. T= x T+F(z.r).........(1) where F (z,r) =HWv1/pc (r-r1), if z h/2 = 0, if z >h/2 . In Eq. 1, V- represents the Laplacian operator and 8 is the Dirac delta function. Eq. 1 is the well-known diff-sion equation with a source term and may be applied to a linear system. For example, if the source is moving in the direction of the x axis, then the argument of the delta function is x - 1,. In deriving Eq. 1 the following approximations were made concerning the actual combustion process: (1) all heat transfer is by conduction in a homogeneous medium, (2) the formations bounding the reservoir have the same thermal properties as the reservoir, (3) the combustion fuel is consumed instantaneously as the front arrives, (4) there is a constant fuel concentration