Overview of Wellbore Fluid Flow and Heat Transfer Modelling with Applications in the Oil Industry


equation should be added for each component and, therefore, the solution procedure becomes more difficult. The above equations with appropriate equations of state to estimate fluid properties, a set of multiphase flow equations to estimate the in situ fluid volume fractions, a heat loss equation to couple the energy and formation heat transfer equations due to heat loss from the wellbore to the surrounding medium and finally appropriate initial and boundary conditions make a set of constitutive equations that after discretisation (see Figure 1) can be solved with appropriate numerical techniques, typically finite difference or finite volume.


For the reader who is interested in more details of the mathematical formulation and solution procedure, paper CIPC-0052


is recommended.


In that paper a numerical non-isothermal two-phase wellbore flow model has been developed and tested against both field data and the prediction of other models to compute the wellbore fluid temperature, pressure, density and velocity profiles in steam injection wells. The model couples mass, momentum and energy balance equations and provides all the necessary data in the well with respect to depth and time for a pre-determined surface condition. A drift–flux model has been used to capture the slip phenomenon between the phases and the time- and depth-dependent overall heat transfer coefficient has been incorporated into the model to capture the heat loss to the surroundings. The wellbore system has sequentially been solved with a triple-iterative procedure and the formation system has been solved in a fully implicit manner. While the model was designed for steam injection wells, it is expected that with some minor modifications it can be extended to modelling the injection


1. 2.


Lin D, Transient Fluid Flow and Heat Transfer in Petroleum Production Systems, PhD Dissertation, The University of North Dakota, December 2001.


Bahonar M, Azaiez J, Chen Z, A Semi-unsteady State Wellbore Steam/Water Flow Model for the Prediction of Sandface Conditions in Steam Injection Wells, paper CIPC-005, presented in the Canadian International Petroleum Conference (CIPC), Calgary, Alberta, Canada, June 2009.


3.


Chen Z, Novotny RJ, Accurate Prediction Wellbore Transient Temperature Profile under Multiple Temperature Gradients: Finite Difference Approach and Case History. Paper SPE 84583, presented in the SPE Annual Technical Conference and Exhibition, 5–8 October, Denver, Colorado, 2003.


4. 5. 6. 7.


Izgec B, Transient Fluid and Heat Flow Modeling in Coupled Wellbore/Reservoir Systems, PhD Dissertation, Texas A&M University, May 2008.


Livescu S, Durlofsky LJ, Aziz K, Ginestra JCS, A Fully-Coupled Thermal Multiphase Wellbore Flow Model for Use in Reservoir Simulation, J Petrol Sci Eng, 2010;(71)3–4:138–46.


Wang X, Modeling Coupled Transient Transport of Mass, Momentum and Energy in Wellbore/Reservoir Systems, PhD Dissertation, The University of North Dakota, 1996.


Pourafshary P, A Compositional Wellbore/Reservoir Simulator to Model Multiphase Flow and Temperature Distribution, PhD Dissertation, The University of Texas at Austin, December 2007.


8. 9.


Hasan AR, Kabir CS, Fluid Flow and Heat Transfer in Wellbores, SPE Book, Richardson, Texas, 2002.


Livescu S, Durlofsky LJ, Aziz K, Ginestra JC, A Fully-Coupled Thermal Multiphase Wellbore Flow Model for Use in Reservoir Simulation, J Petrol Sci Eng, Fourth International Symposium on Hydrocarbons and Chemistry, 2010;71(3–4):138–46.


10. Schulumberger M, Doll GH, Temperature measurements in oil wells, J Inst Pet Technologist, 1937;23:159.


11. Nowak TJ, The estimation of water injection profiles from temperature surveys, Petroleum Transactions, AIME, 1953;198:203–12.


12. Bird JM, Interpretation of temperature logs in water- and gas-injection wells and gas-producing wells, Drilling Prod Prac, 1954:187–95.


of other fluids. The Society of Petroleum Engineers (SPE) book entitled Fluid Flow and Heat Transfer in Wellbores by Hasan and Kabir (2002)8 and most of the SPE papers written by the same authors,29,30,35,37–40 well as paper SPE 113215 by Livescu et al. (2008)33


as are excellent


additional references in wellbore flow modelling. Conclusion


Although many models have been proposed since Ramey’s first model, still wellbore simulators, specifically those coupled with reservoir simulators, are suffering from issues related to computational speed and stability. Active research is currently conducted by many groups to attempt to solve these issues. Most multiphase flow models currently used in wellbore simulators are either completely based on correlations or semi-mechanistic models. Consequently, they are applicable only in the range of operation conditions for which they were derived and out of these ranges they may lead to large errors. Coupling between the wellbore and surrounding formation via heat loss is another issue with wellbore simulators on the market. Furthermore, current heat loss models from the wellbore to its surroundings are usually assumed to be at the steady state and many assumptions used in model formulation should be further examined.


The desire of the petroleum industry is to couple reservoir, wellbore and surface facilities all together with sufficiently fast speed in computation, stable properties in solution and good accuracy in results. It is this desire that drives many investigators worldwide to perform research on wellbore flow modelling. n


13. Lesem IB, Greytok F, Marotta F, McKetta JJ, A method of calculating the distribution of temperature in flowing gas wells, Petroleum Transactions, 1957;210:169–76.


14. Moss JT, White PD, How to calculate temperature profile in a water-injection well, Oil Gas J, 1959;57(11):174.


15. Kirkpatrick CV, Advances in gas-lift technology, Drilling Prod Prac, 1959:24–60.


16. Ramey HJ, Wellbore heat transmission, Paper SPE 96-PA, J Petroleum Tech, 1962;14(4):427–35.


17. Edwardson MJ, Girner HM, Parkison HR, et al., Calculation of formation temperature disturbances caused by mud circulation, Paper 124-PA, J Petroleum Tech, 1962;14(4):416–26.


18. Satter A, Heat losses during flow of steam down a wellbore, paper SPE 1071-PA, J Petroleum Tech, 1965;17(7):845–51.


19. Holst PH, Flock DL, Wellbore behavior during saturated steam injection, JCPT, 1966:184–93.


20. Willhite GP, Over-all heat transfer coefficients in steam and hot water injection wells, Paper SPE 1449-PA, J Petroleum Tech, 1967;19(5):607–15.


21. Pacheco EF, Farouq Ali SM, Wellbore heat losses and pressure drop in steam injection, SPE Paper 3428-PA, J Petroleum Tech, 1972;14(2):139–44.


22. Herrera JO, Birdwell BF, Hanzlik EJ, Wellbore Heat Losses in Deep Steam Injection Wells, S1-B Zone, Cat Canyon Field, Paper SPE 7117-MS, SPE California Regional Meeting, San Francisco, California, 1978.


23. Farouq Ali SM, A comprehensive wellbore stream/water flow model for steam injection and geothermal applications, paper SPE 7966-PA, SPE Journal, 1981;21(5):527–34.


24.


Fontanilla JP, Aziz K, Prediction of bottom-hole conditions for wet steam injection wells, J Can Petroleum Tech, 1982;21(2):82–8.


25. Yao SC, Fluid Mechanics and Heat Transfer in Steam Injection Wells, MSc Thesis, The University of Tulsa, 1985.


26. Sharma Y, Shoham O, Brill JP, Simulation of downhole heater phenomena in the production of wellbore fluids, paper SPE 16904-PA, SPE Prod Eng J, 1989;4(3):309–12.


27. Wu YS, Pruess K, An analytical solution for wellbore heat transmission in layered formations, SPE paper 17497-PA, SPE Reservoir Engineering, 1990;5(4):531–8.


28. Segar RK, Dotty DR, Schmidt Z, Predicting temperature


profiles in a flowing well, paper 19702-PA, SPE Prod Eng J, 1991;6(4):441–8.


29. Hasan AR, Kabir CS, Aspects of heat transfer during two-phase flow in wellbores, paper SPE 22948-PA, SPE Prod Facilities J, 1994;9(3):211–6.


30. Hasan AR, Kabir CS, Lin D, Analytic wellbore-temperature model for transient gas-well testing, paper SPE 84288-PA, SPE Reservoir Eval Eng J, 2005;8(3):240–7.


31. Hasan AR, Kabir CS, Wang X, A Robust Steady-state Model for Flowing-fluid Temperature in Complex Wells, paper SPE 109765-MS, presented at the SPE Annual Technical Conference and Exhibition, 11–14 November 2007, Anaheim, California, US, 2007.


32. Hagoort J, Ramey’s wellbore heat transmission revisited, paper SPE 87305-PA, SPE Journal, 2004;9(4):465–74.


33. Livescu S, Durlofsky LJ, Aziz K, Application of a New Fully-Coupled Thermal Multiphase Wellbore Flow Model, paper SPE 113215-MS, presented at the SPE/DOE Symposium on Improved Oil Recovery, Tulsa, Oklahoma, US, 20–23 April 2008.


34. Livescu S, Durlofsky LJ, Aziz K, A Semianalytical Thermal Multiphase Wellbore Flow Model for Use in Reservoir Simulation, SPE paper 115796-MS, presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, USA, 21–24 September 2008.


35. Hasan AR, Kabir CS, Sayarpour M, A Basic Approach to Wellbore Two-Phase Flow Modeling, paper SPE 109868-MS, presented at the SPE Annual Technical Conference and Exhibition, Anaheim, California, US, 11–14 November 2007.


36. Beggs DH, Brill JP, A study of two-phase flow in inclined pipes, paper 4007-PA, J Petroleum Tech, 1973;25(5):607–17.


37. Hasan AR, Kabir CS, A wellbore/reservoir simulator for testing gas wells in high-temperature reservoirs, SPE Formation Evaluation J, 1996;11(2):128–34.


38. Hasan AR, Kabir CS, Simplified wellbore flow modeling in gas/condensate systems, SPEPO, 2006;21(1):89–97.


39. Hasan AR, Kabir CS, A simple model for annular two-phase flow in wellbores, SPEPO, 2007;22(2):168–75.


40. Hasan AR, Kabir CS, Wang X, Development and application of a wellbore/reservoir simulator for testing oil wells, SPE Formation Eval J, 1997;12(3):182–8.


EXPLORATION & PRODUCTION – VOLUME 9 ISSUE 1


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