The Injection of a Hot Liquid into a Porous Medium
Authors: Cheppelear J.E. , Volek C.W. 
Published: December 2007

Abstract
The injection of a hot liquid into initially cool porous media, saturated with the same liquid and porous media, saturated with the same liquid and surrounded by two impermeable but beat conducting media (cap and base rock), bas been studied both experimentally and theoretically. The temperature dependence of the viscosity was included in the theoretical model, but it was assumed that the specific heats and densities of the various materials were independent of the temperature. Solutions to the theoretical model were approximated by numerical methods.
Both theoretical and experimental results indicate that centerline temperatures are significantly higher than boundary temperatures. Comparison of experimental and theoretical results with a cold/hot viscosity ratio of 19:1 were in reasonable agreement. Theoretical calculations show, that the effect of the temperature dependence of viscosity was very significant at ratios of 100:1 to 1000:1, which are typical of those that occur when injecting hot water to flood heavy oil reservoirs.

Introduction
We consider the problem of prediction of fluid flow and temperature distribution in an initially cold-fluid-filled reservoir on the injection of the same hot liquid by the use of mathematical and physical models. The results reported are for a physical models. The results reported are for a two-dimensional rectangular section of the reservoir, as shown in Fig. 1. The injection and withdrawal faces are assumed to be equipotentials for fluid flow. The ultimate purpose of such models would be to predict hot-water injection performance. However, we note that in the work presented here, one of the most significant aspects of the problem - the instabilities resulting from two-phase, water and oil flow - is not included.
We will not give a historical review, but refer instead to the paper of Spillette and Nielsen, which contains a rather complete bibliography and critical discussion. The physical problem is exactly the same as Spillette and Nielsen, except for certain simplifications in our assumptions. The mathematical details are somewhat different, and we will present the details of our method here. The mass flow equation, which is elliptic in character, is handled by successive overrelaxation. The heat flow difficulty arose in the over-all heat balance due to by a straightforward explicit approximation. Some difficulty arose in the over-all heat balance due to small errors in the solution of the mass flow equation, and we feel that a different formulation of the heat flow equation would be desirable for future work.
In addition to the mathematical solutions for the temperature distributions in a porous medium due to the injection of hot liquids, experimental data are presented to check the validity of these solutions. presented to check the validity of these solutions. A schematic diagram of the model is shown in Fig. 2.
Certain qualitative physical conclusions were obtained from our numerical and experimental results These are:Assuming high (infinite) conductivity normal to the bedding plane in the reservoir is a poor approximation, and may lead to overestimates of the total heat losses (to cap and base rock) of as much as 50 percent.More heat is retained in the reservoir (per unit of heat injected) for higher viscosity changes.Any particular temperature isotherm moves more rapidly along the centerline for higher viscosity changes. Consequently, the approximation of temperature independent viscosity is not suitable for obtaining quantitatively correct results.

MATHEMATICAL MODEL
Our mathematical model is that of a reservoir of thickness 2h. The problem is idealized from that of a linear hot-water drive, and we imagine that the input face is sufficiently far away from the injection wells that the stream lines enter it normally. Similarly the production wells are far enough from the outflow face that the flow lines leave it normally.
SPEJ
P. 100