Mathematical Modeling of Steam Assisted Gravity Drainage

Abstract

Abstract A mathematical model for gravity drainage in heavy-oil reservoirs and tar sands during steam injection in linear geometry is proposed. The mathematical model is based on the experimental observations that the steam zone shape is an inverted triangle with the vertex fixed at the bottom production well. Both temperature and asphaltene content dependence of viscosity of the drained heavy oil is considered. The developed model has been validated using experimental data presented in the literature. The heavy oil production rate conforms well to previously published data covering a wide range of heavy oils and sands for gravity drainage. Introduction Gravity drainage of heavy oils is of considerable interest to the oil industry. Because heavy oils are very viscous and, thus, almost immobile, a recovery mechanism is required that lowers the viscosity of the material to the point where it can flow easily to a production well. Conventional thermal processes, such as cyclic steam injection and steam assisted gravity drainage (SAGD) are based on thermal viscosity reduction1. Cyclic steam injection incorporates a drive enhancement from thermal expansion. On the other hand, SAGD is based on horizontal wells and maximizing the use of gravity forces2. In the ideal SAGD process, a growing steam chamber forms around the horizontal injector and steam flows continuously to the perimeter of the chamber where it condenses and heats the surrounding oil. Effective initial heating of the cold oil is important for the formation of the steam chamber in gravity drainage processes3. Heat is transferred by conduction, convection, and by the latent heat of steam. The heated oil drains to a horizontal production well located at the base of the reservoir just below the injection well. Based on the aforementioned concepts, Butler et al. (1981)4 derived Eqn. (1) assuming that the steam pressure is constant in the steam chamber, only steam flows in the steam chamber, oil saturation is residual, and heat transfer ahead of the steam chamber to cold oil is only by conduction. One physical analogy of the above process is that of a reservoir where an electric heating element is placed horizontally above a parallel horizontal producing well.Equation 1 In this equation, L is the length of the well, f is porosity, ?So is the difference between initial oil saturation and residual oil saturation, k is the effective permeability for the flow of oil, g is the acceleration due to gravity, a is the thermal diffusivity, h is the reservoir height, m is a constant between 3 and 55,6, and ?s is the kinematic viscosity of oil at steam temperature. The temperature profile is assumed to be time independent and decline exponentially with distance from the interface.Equation 2 In this equation U is the local velocity of the steam zone interface and e is the coordinate perpendicular to steam oil interface. There are three major consequences of this theory. First, steam chamber growth is necessary for oil production, and oil production occurs so long as steam is injected. Second, oil rate increases as the steam temperature increases. Third, at a given steam temperature, the oil with the lowest viscosity (usually the highest °API gravity) exhibits the greatest production response. One major problem with the aforementioned model was that the observed experimental oil rates reported in the literature was of the same order as that predicted by Butler's model but slightly lower. Butler1 associated the deviation of the observed rate to the factors that were not recognized in the derivation of the equation such as a change in the effective steam chamber height (i.e. it becomes lower than the reservoir height) because of depletion and because some heat is used to cause the lateral transfer of the draining oil to the fixed well.