Application of the Time-Dependent Overall Heat-Transfer Coefficient Concept to Heat-Transfer Problems in Porous Media

Prediction of temperature distribution behavior during thermal recovery processes is necessary for engineering, evaluation of field operations. Such a prediction can be used in the case of hot- and cold-water injection into a reservoir and also applies in some other thermal recovery processes, such as in-situ combustion and steam-flooding. The mathematical formulas discussed involve the concept of a time-dependent overall heat-transfer coefficient. In the first portion of the paper, we discuss two new analytical solutions that describe the temperature distribution in linear and radial reservoirs in the case of hot- and cold-water injection. A comparison with published laboratory hot-water injection data demonstrates the validity of the solution for linear geometry. Since the new analytical model considers the heat conduction in addition to convection and heat loss. It describes the thermal behavior in a more general form than does Lauwerier's model. These two models are compared also. The application of the time-dependent overall heat-transfer coefficient concept to the thermal behavior of the steam plateau portion of the in-situ combustion process is discussed in the second pail of the paper. The result is fairly satisfactory. Introduction Since the mid-1950's, many models describing the temperature behavior and thermal efficiency of fluid injection into porous media have been formulated and solved analytically. In particular, increasing demands for thermal oil recovery processes and heat extraction processes from geothermal fields have led researchers to develop these models. With advancements in numerical solution techniques, it also has become possible to obtain solutions to problems that could not have been solved before. Heat-transfer models in porous media consider three heat-transfer mechanisms: thermal conduction, convective transfer between fluid and solid matrix, and energy transfer resulting from fluid flow. The conductive heat transfer describes the thermal conduction in the direction of flow. Convective heat transfer is accounted for by the assumption of thermal equilibrium between the porous medium and its contained fluids. The heat loss caused by fluid injection plays an important role. Keeping the loss low is a primary concern for the efficiency of thermal recovery processes. Although the heat is transferred by a combination of both conduction and convection, in earlier formulations of the energy balance of flow in porous media, the heat loss generally has been treated as either a convective or conductive heat-transfer mechanism. A constant overall heat-transfer coefficient, U, can be used to describe the heat transfer from the system to the adjacent strata: q = UA (T-Ti)........................................(1) This has proved a reasonable approximation for the heat-loss mechanisms occurring in nonadiabatic laboratory tube experiments. Eq. 1 describes the heat loss in a convective form. Heat loss in conductive form can be written as Tq=k A ---- ........................................(2)adj y However, such formulation of heat loss leads to a two-dimensional energy balance equation for one-dimensional flow geometry or a three-dimensional energy balance equation for two-dimensional flow geometry. The solutions of such energy-balance equations become difficult. Lauwerier used the conductive form of heat loss in his model and developed an analytical solution for temperature propagation in a linear flow geometry. He assumed all infinite thermal conductivity in the vertical direction within permeable sand. The reservoir and surrounding formation thermal conductivities in the horizontal direction, however, were neglected. Later, in 1959, Rubinshtein developed it more sophisticated analytical model for heat flow in porous media. His model removed the restrictions in dealing with the thermal conductivities in Lauwerier's model. SPEJ P. 107^