Analytical Modeling of Oil Recovery by Steam Injection: Part I - Upper Bounds

Abstract

Abstract This paper deals with the derivation of upper bounds for the growth of the steam zone in steam injection processes for one- or multidimensional reservoirs at constant or variable injection rates. The bounds are derived from the integral balances describing a reservoir of arbitrary geometry by introducing lower bounds for the heat losses to the surrounding area and the hot liquid zone. In this way, the effect of preheating in the hot liquid zone is estimated to determine the recovery efficiency of a steam drive. The growth rate of a one-dimensional steam zone at variable injection rates is subject to two upper bounds resulting from the total thermal energy and the latent heat balances, respectively. Each of the bounds controls the rate of growth of the steam zone in a certain time interval, depending on the dominant mode of heat transfer in the hot liquid zone. At constant injection rates, the steam zone growth at large times is controlled by the bound based on the latent heat balance. This balance depends on a dimensionless parameter, F, defined as the ratio of the latent heat to the total heat injected. Based on the relative magnitude of F with respect to the critical value F= 2/pi, the region of validity of the Marx-Langenheim solution is delineated on a Ts vs. fs diagram. The Marx-Langenheim solution is satisfactory at large times when F greater than 2/pi and becomes less satisfactory as F assumes smaller values. Similar upper bounds are obtained for a two-dimensional steam drive (thin reservoirs). In three-dimensional reservoirs, on the other hand, bounds are derived only for a special form of displacement (separable front). These bounds depend on the models for the steam front shape, K can be determined in terms of the physical variables of the process. Introduction Injection of steam (steamflood or steam drive) is an important thermal recovery method that is applied on a commercial scale in many parts of the world. The main elements of continuous steam injection, as a displacement process, were analyzed thoroughly by experimental studies under both laboratory and field conditions. Along with laboratory and field tests, mathematical models are sought to aid in understanding and designing the process. The engineering evaluation of a steam drive often is based on a simplified mathematical description of reservoir heating by hot fluid injection presented by Marx and Langenheim and subsequently modified by Mandl and Volek. This theory was combined further with simple fluid flow considerations to determine the oil recovery rates in one-dimensional reservoirs. To account for the important effects of gravity override in three-dimensional geometries, Neuman and van Lookeren (following different approaches) derived simple analytical formulas for the calculation of the performance of a steamflood in three-dimensional reservoirs. An increasing number of investigators also have concentrated on the development of reliable numerical models. Three-phase numerical simulators were derived by Shutter for one- and two-dimensional flow; by Abdalla and Coats for two-dimensional flow; and by Coats et al., Coats, and Weinstein et al. for three-dimensional flow. The last two models also account for steam distillation of oil. SPEJ P. 162^