Application of Variational Principles to Cap and Base Rock Heat Losses
Authors: Chase C.A. , O'Dell P.M. 
Published: December 2007

Abstract
Variational principles stated by, Biot have been applied to obtain a two-parameter (approximation for heat losses to cap and base rock from a reservoir undergoing thermal recovery. The approximation predicts heat losses to within a few percent of the predicts heat losses to within a few percent of the exact value when the beat losses result from one-dimensional conduction into cap and base rock in the direction normal to the reservoir boundary surfaces. Conduction in the longitudinal direction is neglected. Therefore, the approximate temperature distribution is valid only when the temperature gradient in this direction is small. But because the Peclet number (ratio of convective to conductive heat transport) is high in most reservoir thermal processes, the horizontal temperature gradient will processes, the horizontal temperature gradient will be small everywhere except in the vicinity of a thermal front, and the approximation will be valid. Comparison with a finite-difference solution in cap and base rock shows that reasonable accuracy is obtained when the Peclet number is 100 or greater. The variation solution has been incorporated into our thermal simulator and yields a considerable sailings in core storage. It is no longer necessary to store grid-block temperatures for cap and base rock nor to solve the finite-difference form of the energy balance in this region. Instead a system of two nonlinear ordinary differential equations must be solved for each grid block at the interface of the reservoir and the cap rock. In addition to savings in core storage, a reduction in computation time is achieved because fewer finite-difference grid blocks are needed.

Introduction
Heat losses to cap and base rock must be considered in modeling thermal processes in petroleum reservoirs. Since there is no mass petroleum reservoirs. Since there is no mass transport in the cap and base rock, the only mechanism for heat transfer is conduction. One of the most obvious ways of determining heat losses from the reservoir is to solve the energy equation in the cap- and base-rock region by finite differences. To do this, the reservoir finite-difference grid must be extended into the cap- and base-rock region. This can consume a good deal of computer core storage - at a time when all available core storage is needed to adequately model mass and energy transport in the reservoir region. Furthermore, since there is no mass transport in the cap and base rock, one would like to eliminate having to solve the conservation-of-mass equations in this region, but to do so requires a special computer code. Hence, a finite-difference solution can be costly. It does, however, have the advantage of generality in that a minimum of assumptions is involved in formulating the conservation equations. There are ways of calculating heat losses to cap and base rock other than by finite differences. However, for a method to be competitive with the finite-difference method, it must offer some advantage such as accuracy, reduced computer core storage, or lower computation time. One alternative to finite differences is the use of superposition to couple an analytic solution for the cap and base-rock temperature distribution with the finite-difference solution of the reservoir energy balance. But, during the course of the simulation, the superposition principle would necessitate having temperature data for all previous time steps for each grid block adjacent to the cap and base rock. This requires an appreciable amount of computer core storage, perhaps even more than would be required for a complete finite-difference solution. Hence, this method does not seem attractive. The use of variational principles appeared to offer the advantages of both reduced core storage and lower computation time and was therefore considered as a means of treating heat losses to cap and base rock. The advantage of the variational method is that a priori knowledge of the approximate shape of the temperature profile can be used to choose the functional form of the temperature distribution. The chosen functional form will contain several free parameters.
SPEJ
P. 200