Numerical Model for Thermal Processes
Authors: Weinstein H.G. , Wheeler J.A. , Wood E.G. 
Published: December 2007

Abstract
The numerical model developed in this paper gives fast, reliable, and physically meaningful results for steam stimulation and other thermal processes. The paper describes a numerical processes. The paper describes a numerical technique that simulates two-dimensional heat transfer and one-dimensional, three-dimensional heat in a multilayered reservoir. Interphase mass transfer accounts for steam condensation, solution gas drive, distillation, and solvent extraction. The mathematical development and the various features of the model are presented. Computer calculations for several test problems show the model agrees closely with analytical solutions, laboratory experiments, and field data. The major advantages of this thermal reservoir model over previous models are increased speed and reliability with little sacrifice of physical significance. These advantages stem from an implicit-pressure/explicit-saturation formulation, simultaneous solution of pressure and energy equations, and improved techniques for invoking phase constraints and calculating mass-transfer phase constraints and calculating mass-transfer terms. The model is a valuable tool for studying steam stimulation and other thermal processes. This has been demonstrated in many studies under varying reservoir conditions. Because of its speed, the model also can be used to investigate the effects of various parameters on the different thermal processes. Moreover, the model can be extended easily to treat two-and three-dimensional configurations and to include other phenomena such as gravity, capillary pressure, temperature dependence of relative pressure, temperature dependence of relative permeability, and in situ combustion. permeability, and in situ combustion

Introduction
Steam injection stimulates producing wells by providing localized heating in the vicinity of the providing localized heating in the vicinity of the wellbore. The use of steam injection has received considerable attention throughout the oil industry since the early 1960's. Both medium and heavy oil reservoirs are candidates for steam injection as a means of stimulating oil recovery. Currently, steam stimulation is being applied on a commercial scale in California and Venezuela. Because of the high costs associated with injecting steam in the field, it has been necessary to develop computation schemes that simulate the steam-injection process. Both analytical and numerical solutions have been used. In general, the analytical techniques remove many of the physical processes by invoking too many simplifying processes by invoking too many simplifying assumptions. On the other hand, the numerical methods incorporate most of the physical processes, but suffer from reliability problems and excessive computation-time requirements. Thus, there has been a definite need to develop an improved calculation procedure for the steam-stimulation process. procedure for the steam-stimulation process. This paper describes a numerical procedure for simulating coupled energy and mass transport in a multilayered reservoir. The model treats linear flow of three fluid phases and linear convective energy transport, but heat conduction is two-dimensional over a vertical cross-section spanning the oil sand and adjacent strata. Mathematical development of several computational features and physical phenomena is detailed in the paper. A major feature phenomena is detailed in the paper. A major feature is the simultaneous solution of mass and energy equations. Steam condensation/evaporation effects are discussed, and a new solution gas drive model is developed. Treatment of condensation terms and phase constraints for both these processes is phase constraints for both these processes is outlined. To facilitate solution when injection and production rates are high or when a vapor phase is production rates are high or when a vapor phase is present near the wellbore, semi-implicit mobilities present near the wellbore, semi-implicit mobilities are also introduced. The numerical procedure has been applied successfully to a number of problems. It has accurately matched experimental and analytical results; and in both single and multicycle steam-stimulation treatments, it has accurately matched field data. In addition to steam stimulation, the model has been tested on other reservoir engineering problems involving thermal operation. In all studies, problems involving thermal operation. In all studies, computation time was small.
SPEJ
p. 65