A Gravity Override Model of Steamdrive
Authors: Neuman C.H. 
Published: December 2007

Summary
Equations presented in this paper represent the steamdrive process assuming steam rises quickly to an impermeable overburden. Subsequent growth of the volume heated results from increase in both the area and thickness of the region heated to steam temperature. Equations derived predictthe rate of increase of steam zone thickness and areal extent,the volume of oil displaced from the steam zone and the heated reservoir beneath it,the reduced injection rate that will sustain steam after an area has been headted to steam temperature, andthe additional oil displaced after steam injection is stopped.
Predictions are compared to observed behavior of Chevron's Kern River (CA) 10-pattern pilot steamdrive.

Introduction
Marx and Langenheim1 presented a valid relationship between the rate of injection of a hot fluid and the area heated by the fluid. The usefulness of the resulting predictions is limited by the assumption that hot fluids heat each areal increment of the reservoir uniformly from top to bottom. Typical steamdrive performance is different from that predicted by Marx and Langenheim because the difference between densities of steam and liquids causes steam to rise to the top of a permeable interval. Because of this gravity override, heat losses are not to the confining impermeable beds but partly to the permeable reservoir beneath the steam zone.
Several models based on assumed or observed average steam zone thickness have been proposed. The model presented here enables prediction of steam zone thickness as well as the area covered by steam.

Conditions Assumed
Steam is assumed to rise to the top of a permeable reservoir in time negligible compared with that required to heat the total reservoir area. The horizontal pressure gradient in the steam zone is assumed to be much less than the vertical pressure gradient on liquids caused by the difference between densities of steam and liquids. Single values are assumed for both oil and water saturations throughout the steam zone. Fortunately, predictions depend only indirectly on water saturation, which seldom is known accurately.
Because the horizontal pressure gradient in the heated reservoir immediately below the steam zone equals the small horizontal pressure gradient in the steam zone, horizontal flow immediately below the steam interface is neglected.
Oil saturations in the heated reservoir beneath the steam are assumed to be the residual oil saturation to hot-waterflooding. The thickness of the interval flooded by hot water is represented by a single temperature that be estimated from laboratory data or field measurements. With this assumption, temperature-dependent relative permeabilities and flow properties are represented by a single temperature and a single oil saturation.
The portion of injected heat that leads to steam zone growth is assumed equal to the portion in steam vapor. Injected hot water, like that from steam condensation, is assumed to fall quickly to the reservoir beneath the steam zone, where it contributes to oil displacement but does not influence steam zone growth.

Predicted Steam Zone Growth and Oil Displacement
Equations representing the concepts described as follows are presented in the Appendices. The rate of steam condensation resulting from heat diffusion into the overburden is a function of the time since the areal increment considered was first heated (Appendix A, Eq. A-4). The volume rate at which water flows downward through the steam zone equals the rate of steam condensation at the overburden/reservoir interface.The volume rate (per unit area) of water flowing downward beneath the steam interface minus that in the steam zone equals the volume rate of water displaced by movement of the steam interface (Eq. A-5).The enthalpy (per unit area) carried by the downward flow of oil and water beneath the steam interface minus that carried by water flowing downward through the steam zone equals the rate of interface motion times the difference in enthalpy above and below the interface (Eq. A-7).Vertical displacement of the steam interface does not require a change in the temperature gradient just below the interface.