Nonisothermal Multiphase Flow in Pipelines under Nonequilibrium Conditions

1. Nonisothermal Multiphase Flow in Pipelines under Nonequilibrium Conditions Guillermo Michel and Faruk Civan

2. Michel, G., “Modeling of Multiphase Flow in Wells under Nonisothermal and Nonequilibrium Conditions” , M.S. Thesis, November 2007, University of Oklahoma • Michel, G., Civan, F., “Modeling Nonisothermal Rapid Multiphase Flow in Wells under Nonequilibrium Conditions”, SPE Production and Operations Journal, May 2008 • Michel, G., Civan, F., “Modeling Rapid Multiphase Flow in Wells and Pipelines under Nonequilibrium and Nonisothermal Conditions”, SPE 107958, presented at the 2007 Rocky Mountain Oil & Gas Technology Symposium, Denver, Colorado,16-18 April 2007 • Michel, G., Civan, F., “Modeling Nonisothermal Rapid Multiphase Flow in Wells under Nonequilibrium Conditions”, SPE 102231, presented at the 2006 SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 24–27 September 2006.

3. Outline • Description of the Problem • Transport Phenomena in Producing Wells • The non-equilibrium condition • Estimating the relaxation in gas phase separation • Predicting the liquid hold up with correlations • Predicting the liquid hold up with a slip ratio formulation • Application • Pressure gradient • Temperature gradient • Dryness gradient • Relaxation Time • Summary

4. What is the problem of interest? The difference in the Velocity of the Phases indicates the occurrence of slippage The Fluid Expansion causes a change in Temperature The Heat Dissipation causes a drop in Temperature

5. The reservoir fluid is considered as a multiphase system A Homogeneous Area-averaged Model is adopted The infinitesimal element for the model is the cross-sectional area The system properties are defined as the average over a cross-sectional area. How can it be modeled?

6. Homogenous Area-Averaged Model Mass Conservation • Constant cross-sectional area • No gain or loss of mass Momentum Conservation Energy Conservation

7. Homogeneous Model at Steady-State • Unknown variables : density, velocity, pressure and temperature • A fourth equation is needed to give closure to the system • All remaining properties are estimated from correlations Momentum Mass Energy

8. The Nonequilibrium Condition • The slippage of the liquids is caused by the buoyancy of gas bubbles and gravitational pull exerted to the liquid droplets • This result in a higher mobility of the gas phase which tends to travel faster than the liquid phase • The slippage of the liquids causes a difference in velocities at which the phases are flowing • Mixture properties can not be calculated by using the formulations for ideal mixtures

9. The Nonequilibrium Condition • The following inequalities applies to the non-equilibrium condition Mixture Velocity Mixture Density Volumetric Fraction of the liquid phases Volumetric Fraction of the gas phase

10. Relaxation Time of Gas Phase Separation • The law of mass conservation for the gas phase is applied to the previously defined model for attaining its closure. • The mass transfer from the liquids to the gas is not assumed instantaneously • The mass generation of the gas phase is estimated by considering a relaxation in the time for the gas phase separation from the liquids

11. Relaxation Time of Gas Phase Separation • The quality or dryness at equilibrium conditions is obtain by phases are flowing at the same velocity • The relaxation in time can be characterized for reservoir fluids flowing at steady-state

12. Liquid Holdup Modeling • A constitutive equation defines the mixture density in order to estimate the pressure gradient • This constitutive equation is applied to the previously defined model for attaining its closure. • The mixture density is estimated by averaging using the volumetric fractions of the gas phase (void fraction) and the liquid phases (liquid holdup) • The mixture velocity is set equal to the volumetric flux even though the flow is at non-equilibrium conditions.

13. Liquid Holdup Modeling • The most intuitive parameters in the dimensional analysis are the fractional flow of the gas phase and the liquids. • The non-slip density of the mixture is calculated by using the fractional flow of the phases • The superficial velocity of the phases, the density of the phases, the liquid superficial tension, the pipe diameter and the gravitational force are the most common parameters used in the dimensionless analysis of the liquid hold up prediction

14. Bubble Slug Annular Mist Predicting the Liquid Holdup • Various correlations over dimensionless parameters were developed for identifying the flow-pattern of the phases • This flow-patterns are usually classified as: bubble, slug, annular and mist flow • Each flow-pattern uses a specific correlation for predicting the liquid holdup • The prediction of the liquid holdup is discontinuous when a change in flow-pattern occurs

15. Slip Ratio Formulations • It is desired to utilize a parameter capable of characterizing the density, velocity and liquid holdup of the mixture simultaneously • The slip ratio proves to be a parameter with such capability • The slip ratio is defined as the ratio of the actual velocity of the gas phase to the actual velocity of the liquid phases • The liquid hold up is expressed in terms of the slip ratio • The void fraction is expressed in terms of the liquid hold up

16. Slip Ratio Formulations • The quality or dryness of the mixture is directly related to the slip ratio • The mixture density formulation can be rearranged to be related to the slip ratio • The mixture velocity formulation can be rearranged to be similarly related • Note that if the value of the slip ratio is equal to the unit then the flow is at equilibrium

17. This value has to be estimated or measured Proposed Liquid Holdup Model • The slip ratio is estimated by a quadratic interpolation where the non-slip density of the mixture is the independent variable • The mixture is considered at equilibrium when the mixture is either a saturated gas or liquid • The mixture density is equal to the density of the saturated phase n

18. Proposed Liquid Holdup Model • Advantages • The need of correlations for liquid holdup prediction is avoided • The prediction is continuous in the saturated/under-saturated transitions • The prediction is continuous for all transitions of flow type • Disadvantages • The slip ratio at the inlet needs to be measured or estimated. • It was only tested for oil wells : Bubble and Slug flow.

19. Pressure Gradient Two-phase Flow of Light Oil

20. Pressure Gradient Two-phase Flow of Heavy Oil

21. Pressure Gradient Three-phase Flow of Heavy Oil

22. Temperature Gradient Two-phase Flow of Light Oil

23. Temperature Gradient Two-phase Flow of Heavy Oil

24. Temperature Gradient Three-phase Flow of Heavy Oil

25. Dryness Gradient and Relaxation TimeTwo-phase Flow of Light Oil

26. Dryness Gradient and Relaxation TimeTwo-phase Flow of Heavy Oil

27. Dryness Gradient and Relaxation TimeThree-phase Flow of Heavy Oil

28. Summary • The proposed approach predicts a continuously varying liquid-holdup by interpolating the slip ratio. • The heat dissipation to the surroundings ,and fluid expansion, and energy loss by friction cause a non-linear temperature drop. • The upward motion of reservoir fluids in producing wells can be successfully modeled by the developed homogenous model which has been closured with the proposed model for liquid holdup prediction. • The relaxation time of gas separation proved to be an adequate property for characterizing the deviation form the equilibrium for reservoir fluids. • The homogenous area-averaged model can be closured using the conservation law for the gas phase and the relaxation time of gas separation from the liquid phases.