Abstract
The saturation distribution of steam, water, and oil within the steam zone in a steam-injection process at constant injection rates is examined. It is shown theoretically that for typical values of injection parameters the oil saturation in the steam zone rapidly parameters the oil saturation in the steam zone rapidly reaches its residual value at steam zone conditions. This result, which corroborates previous experimental evidence, is a consequence of the relatively fast changes in phase saturations compared with the rate of the advance of the steam front. Explicit expressions for the steam saturation distribution are obtained. It is shown that the average steam saturation is a slightly decreasing function of time and approaches a limiting value that is a nearly constant fraction of the steam saturation at the injection point. This result provides theoretical justification for the assumption of constant average steam saturation in steam-injection calculations.
Introduction
Steam injection has emerged over the past years as one of the most efficient methods to recover oil from medium- to heavy-oil reservoirs. Since the inception of the process a variety of laboratory, field, and mathematical investigations have studied the process mechanisms and performance prediction. The mathematical studies range from detailed, highly sophisticated, but generally expensive numerical simulators to simplified, inexpensive, overall prediction schemes that are best suited for quick engineering-type calculations. Analytical studies on steamdrive have focused primarily on the description of the growth rate of me primarily on the description of the growth rate of me steam zone. Such models range from simple energy-balance considerations presented by Marx and Langenheim and modified later by Mandl and Volek and Myhill and Stegemeier to a detailed modeling of the heat transfer in the reservoir and the surrounding formation proposed by Yortsos and Gavalas. The obtained growth equation subsequently is combined with a Buckley-Leverett type of oil displacement in the liquid zone to provide an estimate of the oil recovery.
With the exception of the study by Shutler and Boberg most of the previous investigators have not treated the fluid flow phenomena inside the steam zone, proceeding instead with the Assumption of constant proceeding instead with the Assumption of constant values for the average saturations for the oil, water, and steam phases. This approximation is essential for the decoupling of fluid flow and heat transfer calculations and allows for explicit solutions of the energy balances. Partial experimental support for this hypothesis has been Partial experimental support for this hypothesis has been provided by Willman et al. among others, who observed provided by Willman et al. among others, who observed in laboratory experiments that the oil saturation left behind the steam front reaches a constant value considered to be its residual value at steam zone conditions. This paper describes the fluid flow and the resulting saturation distributions inside the steam zone. Our objectives are to test the assumption of constant average saturations and to provide theoretical support for the experimental evidence of residual oil saturation in the steam zone. For simplicity, only one-dimensional (1D) geometries (thin reservoirs) are examined. The formulation in the text and the results obtained pertain to systems with negligible distillation of oil by steam. As shown in Appendix A, the effect of steam distillation in the saturation distribution of the gas phase is negligible, provided certain conditions are met. In contrast to the provided certain conditions are met. In contrast to the technique of Shutter and Boberg, the following model incorporates in the fluid flow description the steam condensation induced by the heat losses to the surrounding formations, thus extending the theory of immiscible displacement to processes involving phase condensation.
Mathematical formulation
We proceed by assuming ID, linear or cylindrical geometries, a constant temperature over the steam zone, Ts, and negligible distillation of oil by steam. A typical case when distillation is important is examined in Appendix A. Three immiscible and incompressible phases (steam, liquid water, and oil) flow inside the steam zone. In view of the steam condensation resulting from the lateral heat losses to the over- and under burdens, the respective mass balances read
...............(1)
...............(2)
and
....................(3)
SPEJ
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Development of new oil/water partitioning tracers for the determination of residual oil saturation in the inter-well region of water-flooded reservoirs