Analytical Modeling of Oil Recovery by Steam Injection: Part 2-Asymptotic and Approximate Solutions (include associated papers 10943 and 10951 )

This article studies the development of asymptotic and approximate solutions for the growth of the steam zone in steam injection processes in one-dimensional reservoirs at constant injection rates. These solutions generally are derived by using integral balances which include heat losses to the surroundings and the hot liquid zone. In this way, the effects of preheating caused by heat transport in the hot liquid zone ahead of the steam front are accounted for completely. At the beginning of injection, the advance of the front is well described by the Marx-Langenheim (ML) model, provided that the injection rates are sufficiently high. At longer times, deviations occur and a criterion is developed in terms of a single heat transfer dimensionless parameter, R, that defines the time interval of applicability of the ML model. The asymptotic behavior at large times depends solely on a dimensionless parameter, F, defined as the ratio of the latent to the total heat injected. It is shown that the final dimensionless expression does not depend on R (i.e., on the injection rates) although the time taken to reach the asymptotic state is influenced significantly by R. An approximate analytical solution that reduces to the respective asymptotic expressions at small and large times is obtained under conditions of high injection rates (R »1). The solution is shown to give a better approximation to the steam-zone growth rate for intermediate and large times than the approximate expressions developed by Marx and Langenheim, Mandl and Volek (MV), and Myhill and Stegemeier (MS). For a wider range of operating conditions, including low injection rates (i.e., for R between 1 and), an approximate numerical solution based on a quasisteady state approximation is presented. The proposed solution requiring very modest computation is expected to give reliable results under a variety of operating conditions. Introduction In a previous paper we dealt with the derivation of upper bounds for the volume of the steam zone in one-, two-, or three-dimensional reservoirs. The resulting expressions incorporate minimal information regarding heat transfer in the hot liquid zone and find applications in setting an upper estimate to oil recovery at constant or variable injection rates. To obtain more precise results concerning the steam zone growth, an alternative approach is initiated involving a detailed description of heat transfer in the hot liquid zone. The subject of heat transfer by convection, conduction, and lateral heat losses in the region ahead of a moving condensation front has been discussed separately in another paper. Here we make use of the results obtained in that paper to derive approximate solutions to the volume of the steam zone as a function of time. The relative importance of including preheating effects in the hot liquid zone and the surroundings when calculating the performance of a steam drive is demonstrated by comparing the solutions obtained against simple approximate expressions developed by Marx and Langenheim, and subsequently revised by Mandl and Volek, and Myhill and Stegemeier. From the comparison with exact results, the range of validity of the previous approximations can be delineated. SPEJ P. 179^