Problems Of Heat Transfer In Porous Media
Authors: Atkinson P.G. , Ramey H.J. 
Published: December 2007

Abstract
This paper presents new mathematical models which can be used to study heat transfer behavior in fractured and non-fractured porous media. A general relationship between these and other similar models is discussed, with an emphasis on application to laboratory and field systems. One important general conclusion of the study was that simplified mathematical models were useful for finding the relative importance of different heat transfer mechanisms in a specific set of bench-scale experiments.

Introduction
There is a continuing interest in processes involving nonisothermal fluid transport in underground systems. Hot fluid injection for purposes of enhanced oil recovery has been of interest to reservoir engineers for many years. More recently the injection of cold fluids into hot reservoirs has also become important. The purpose of this process is to scavenge heat from a reservoir for use process is to scavenge heat from a reservoir for use in generating electricity, household heating, and process heating. process heating. A knowledge of a temperature front movement through an underground system is important in the design of either hot or cold fluid injection projects. This temperature front movement can be projects. This temperature front movement can be calculated using sophisticated computer models which are able to consider a wide range of physical phenomena. However, there is often an physical phenomena. However, there is often an uncertainty concerning important phenomena and phenomenological coefficients. Such models cannot phenomenological coefficients. Such models cannot always provide enough insight into the problem to justify the cost of developing and running their solutions. Alternatively, one can derive simplified mathematical models for which analytical solutions are available. Combinations of such models are made conveniently, and can be used to study the interaction of various heat transfer mechanisms.
This paper presents several models of heat transfer in fractured and non-fractured porous media. The models are used for the calculation of temperature distributions caused by nonisothermal fluid flow through a uniform porous bed or thin fracture. Several models of this kind have been described previously. This paper compares results from these models, and in addition, presents new results. Finally, the application of these simplified mathematical models to the nonisothermal liquid injection experiments of Arihara is shown.

BASIC ASSUMPTIONS
Most of the models described in this study require the assumption of a uniform and constant fluid flow field. This allows energy balance equations to be uncoupled from mass balance equations, simplifying mathematical solutions. Such assumptions imply that the effects of temperature varying viscosity and density on energy transfer are negligible for the physical processes under consideration. These have been processes under consideration. These have been shown to be reasonable assumptions for horizontal fluid displacement in fine-grained porous media at the laboratory scale and at the field-scale. These assumptions can also be shown to be reasonable for an idealized, narrow, horizontal fracture system.
A second basic assumption used in this paper is that of a uniform fluid temperature at any cross-section perpendicular to the direction of fluid flow. This is equivalent to assuming an infinite thermal conductivity perpendicular to fluid flow within the injection interval. This has been called the Lauwerier assumption.
Many authors have described the mechanisms by which thermal energy may be transported in underground fluid systems.