This mathematical model of the mechanics of flow of wet and saturated steam down the wellbore during steam injection takes into account the variation of steam temperature and pressure due to friction, as well as heat losses by radiation, conduction, and convection. It consists of two coupled nonlinear equations that are solved iteratively.
Chekalyuk et al., Moss and White, and Fokeev and Kepyrin, investigated wellbore heat losses during injection of a hot or cold fluid down the casing. Ramey made a comprehensive study of the injection of liquids and gases down casing or tubing. Squier et al. presented a complete analytical treatment of hot water injection down the wellbore. Satter extended Ramey's treatment to the case of steam injection. Huygen and Huitt, presented results of a theoretical and experimental treatment of wellbore heat losses during steam injection, and pointed out the importance of the radiation heat loss. Holst and Flock as well as Earlougher extended the above treatments by including, steam pressure calculations. Eickmeier et al. considered the early transient performance, using a finite-difference model. performance, using a finite-difference model. Apart from these studies, Leutwyler gave a comprehensive treatment of casing temperature behavior, and Willhite presented a complete calculation of the over-all heat presented a complete calculation of the over-all heat transfer coefficients.
The chief objective of our work was to formulate a comprehensive mathematical model of steam injection into a reservoir to include simultaneous calculation of steam pressure and quality. The model took into account the variation of steam temperature and pressure due to friction, as well as heat losses by pressure due to friction, as well as heat losses by radiation, conduction, and convection, and consisted of two coupled nonlinear differential equations that were solved iteratively. The model results were compared with those predicted by several of the above-mentioned approaches.
Steam at a constant injection pressure, constant mass flow rate, and constant quality (at the surface) is injected through the tubing into the wellbore. The complete system consists of the fluid, the tubing, the annular space containing low-pressure air, the casing, the cement, and the formation. We propose to compute the pressure and the quality of steam as functions of depth and time.
An energy balance and a momentum balance can be written for the flowing fluid as follows:
Energy balance
..........(1)
Momentum balance
..............................(2)
The term dQ is given by,...................(3)
where the rate of heat transfer to the formation is
JPT
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