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Numerical solution and convergence analysis of steam injection in heavy oil reservoirs

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Abstract

In this paper, the numerical methods for solving the problem of steam injection in the heavy oil reservoirs are presented. We consider a 3-dimensional model of 3-phase flow, oil, water, and steam, with the effect of 3-phase relative permeability. Interphase mass transfer of water and steam is considered; oil is assumed nonvolatile. We apply the simultaneous solution approach to solve the corresponding nonlinear discretized partial differential equation in the fully implicit form. The convergence of finite difference scheme is proved by the Rosinger theorem. The heuristic Jacobian-Free-Newton-Krylov (HJFNK) method is proposed for solving the system of algebraic equations. The result of this proposed numerical method is well compared with some experimental results. Our numerical results show that the first iteration of the full approximation scheme (FAS) provides a good initial guess for the Newton method. Therefore, we propose a new hybrid-FAS-HJFNK method while there is no steam in the reservoir. The numerical results show that the hybrid-FAS-HJFNK method converges faster than the HJFNK method.

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Change history

  • 15 September 2018

    The original publication of this manuscript which appeared online last 10 August, 2018 contains errors. There were several mistakes that appeared in the text body, Algorithms 1 and 2, Fig. 1 and Equation 16.

References

  1. Shutler, N.D.: Numerical, three-phase simulation of the linear steamflood process. Soc. Pet. Eng. J. 9(02), 232–246 (1969)

    Article  Google Scholar 

  2. Shutler, N.D.: Numerical three-phase model of the two-dimensional steamflood process. Soc. Pet. Eng. J. 10 (04), 405–417 (1970)

    Article  Google Scholar 

  3. Coats, K.H., George, W.D., Chu, C., Marcum, B.E.: Three-dimensional simulation of steamflooding. Soc. Pet. Eng. J. 14(06), 573–592 (1974)

    Article  Google Scholar 

  4. Ferrer, J., Farouq Ali, S.M.: A three-phase, two-dimensional compositional thermal simulator for steam injection processes. Journal of Canadian Petroleum Technology 16(01), 78–90 (1977)

  5. Coats, K.H.: Simulation of steamflooding with distillation and solution gas. Soc. Pet. Eng. J. 16(05), 235–247 (1976)

    Article  Google Scholar 

  6. Coats, K.H.: In-situ combustion model. Soc. Pet. Eng. J. 20(06), 533–554 (1980)

    Article  Google Scholar 

  7. Trottenberg, U., Oosterlee, C.W., Schuller, A.: Multigrid. Academic press, Cambridge (2000)

    Google Scholar 

  8. Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31(138), 333–390 (1977)

    Article  Google Scholar 

  9. Hackbusch, W.: Multi-grid methods and applications, vol. 4 of Springer series in computational mathematics. Springer-Verlag, Berlin (1985)

    Google Scholar 

  10. Fogwell, T.W., Brakhagen, F.: Multigrid Methods for the solution of porous media multiphase flow equations. In: Nonlinear hyperbolic equations—theory, computation methods, and applications. Springer, pp. 139–148 (1989)

  11. Teigland, R., Fladmark, G.: Cell-centered multigrid methods in porous media flow. In: multigrid methods III. Springer, pp. 365–376 (1991)

  12. Collins, D., Mourits, F.: Multigrid methods applied to near-well modelling in reservoir simulation. In: ECMOR III-3rd European Conference on the Mathematics of Oil Recovery (1992)

  13. Molenaar, J.: Multigrid Methods for Fully Implicit Oil Reservoir Simulation. Delft University of Technology, Faculty of Technical Mathematics and Informatics (1995)

  14. Russell, T.F., Wheeler, M.F.: Finite element and finite difference methods for continuous flows in porous media. Math. Reserv. Simul. 1, 35–106 (1983)

    Article  Google Scholar 

  15. Rosinger, E.E.: Stability and convergence for non-linear difference schemes are equivalent. IMA J. Appl. Math. 26(2), 143–149 (1980)

    Article  Google Scholar 

  16. Knoll, D.A., Keyes, D.E.: Jacobian-free Newton–Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193(2), 357–397 (2004)

    Article  Google Scholar 

  17. Saad, Y.: Iterative methods for sparse linear systems. SIAM, Bangkok (2003)

    Book  Google Scholar 

  18. Feng, C., Shu, S., Xu, J., Zhang, C.-S.: Numerical study of geometric multigrid methods on CPU-GPU heterogeneous computers. Adv. Appl. Math. Mech. 6(01), 1–23 (2014)

    Article  Google Scholar 

  19. Geveler, M., Ribbrock, D., Göddeke, D., Zajac, P., Turek, S.: Efficient Finite Element Geometric Multigrid Solvers for Unstructured Grids on GPUs. In: Second International Conference on Parallel, Distributed, Grid and Cloud Computing for Engineering (2011)

  20. Williams, S., Kalamkar, D.D., Singh, A., Deshpande, A.M., Van Straalen, B., Smelyanskiy, M., Oliker, L.: Optimization of Geometric Multigrid for Emerging Multi-And Manycore Processors. In: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis. IEEE Computer Society Press (2012)

  21. Pethiyagoda, R., McCue, S.W., Moroney, T.J., Back, J.M.: Jacobian-free Newton–Krylov methods with GPU acceleration for computing nonlinear ship wave patterns. J. Comput. Phys. 269, 297–313 (2014)

    Article  Google Scholar 

  22. Stroia, I., Itu, L., Nitã, C., Lazãr, L., Suciu, C.: GPU accelerated geometric multigrid method: performance comparison on recent NVIDIA architectures. In: 2015 19th International Conference on System Theory, Control and Computing (ICSTCC). IEEE (2015)

  23. Chen, Z., Huan, G., Ma, Y.: Computational methods for multiphase flows in porous media, vol. 2. Siam, Bangkok (2006)

  24. Dale, E.I., van Dijke, M.I., Skauge, A.: Prediction of three-phase capillary pressure using a network model anchored to two-phase data. Modelling of immiscible WAG with emphasis on the effect of capillary pressure (2008)

  25. Abdalla, A., Coats, K.H: A three-phase, experimental and numerical simulation study of the steam flood process. In: Fall Meeting of the Society of Petroleum Engineers of AIME. Society of Petroleum Engineers (1971)

  26. Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Science Publisher, London (1979)

    Google Scholar 

  27. Fayers, F.J., Matthews, J.D.: Evaluation of normalized Stone’s methods for estimating three-phase relative permeabilities. Soc. Pet. Eng. J. 24(02), 224–232 (1984)

    Article  Google Scholar 

  28. Mozaffari, S., Nikookar, M., Ehsani, M.R., Sahranavard, L., Roayaie, E., Mohammadi, A.H.: Numerical modeling of steam injection in heavy oil reservoirs. Fuel 112, 185–192 (2013)

    Article  Google Scholar 

  29. Yao, S.C.: Fluid mechanics and heat transfer in steam injection wells. University of Tulsa, Tulsa (1985)

    Google Scholar 

  30. Tortike, W.S., Farouq Ali, S.M.: Saturated-steam-property functional correlations for fully implicit thermal reservoir simulation. SPE Reserv. Eng. 4(04), 471–474 (1989)

    Article  Google Scholar 

  31. Sarathi, P.: Thermal numerical simulator for laboratory evaluation of steamflood oil recovery. National Inst for Petroleum and Energy Research, Bartlesville (1991)

  32. Mohammadi, S., Ehsani, M.R., Nikookar, M., Sahranavard, L., Mohammadi, A.H.: Steam Injection Process in Fractured and Non-Fractured Heavy Oil Reservoirs: Comparison of Effective Parameters. In: Ambrosio, J. (ed.) Handbook on Oil Production Research. Nova Science Publishers, Inc. (2014)

  33. Shafiei, A., Zendehboudi, S., Dusseault, M., Chatzis, I.: Mathematical model for steamflooding naturally fractured carbonate reservoirs. Ind. Eng. Chem. Res. 52(23), 7993–8008 (2013)

    Article  Google Scholar 

  34. Mousseau, V., Knoll, D., Rider, W.: A multigrid Newton-Krylov solver for non-linear systems. Lect. Notes Comput. Sci. Eng. 14, 200–206 (2000)

    Article  Google Scholar 

  35. Brown, P.N., Saad, Y.: Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput. 11(3), 450–481 (1990)

    Article  Google Scholar 

  36. Chan, T.F., Jackson, K.R.: Nonlinearly preconditioned Krylov subspace methods for discrete Newton algorithms. SIAM J. Sci. Stat. Comput. 5(3), 533–542 (1984)

    Article  Google Scholar 

  37. Heyouni, M.: Newton generalized Hessenberg method for solving nonlinear systems of equations. Numer. Algorithm. 21(1-4), 225–246 (1999)

    Article  Google Scholar 

  38. Toutounian Mashhad, F., Rafiei, A.: A Comparison between Gmres and Global Gmres Methods for Solving Matrix Equations. Journal of Herbs, Spices & Medicinal Plants (2004)

  39. Willman, B.T., Valleroy, V.V., Runberg, G.W., Cornelius, A.J., Powers, L.W.: Laboratory studies of oil recovery by steam injection. J. Petrol. Tech. 13(07), 681–690 (1961)

    Article  Google Scholar 

  40. Brabazon, K.J., Hubbard, M.E., Jimack, P.K.: Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient. Comput. Math. Appl. 68(12, Part A), 1619–1634 (2014)

    Article  Google Scholar 

  41. Hackbusch, W.: Comparison of different multi-grid variants for nonlinear equations. ZAMM-J. Appl. Math. Mech./Z. Angew. Math. Mech. 72(2), 148–151 (1992)

    Article  Google Scholar 

  42. Stals, L.: Comparison of non-linear solvers for the solution of radiation transport equations. Electron. Trans. Numer. Anal. 15, 78–93 (2003)

    Google Scholar 

  43. Beggs, H.D., Robinson, J.: Estimating the viscosity of crude oil systems. J. Pet. Technol. 27(09), 1140–1141 (1975)

    Article  Google Scholar 

  44. Edreder, E.A., Rahuma, K.M.: Testing the performance of some dead oil viscosity correlations. Pet. Coal 54(4), 397–402 (2012)

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

    H. Hajinezhad, Ali R. Soheili & F. Toutounian

  2. Institute of Petroleum Engineering, College of Engineering, Tehran University, Tehran, Iran

    Mohammad R. Rasaei

Authors
  1. H. Hajinezhad
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  2. Ali R. Soheili
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  3. Mohammad R. Rasaei
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  4. F. Toutounian
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Correspondence to Ali R. Soheili.

Additional information

The original version of this article was revised: There were several mistakes that appeared in the text body, Algorithms 1 and 2, Fig. 1 and Equation 16 introduced during typesetting process. The Publisher apologizes for these errors.

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Hajinezhad, H., Soheili, A.R., Rasaei, M.R. et al. Numerical solution and convergence analysis of steam injection in heavy oil reservoirs. Comput Geosci 22, 1433–1444 (2018). https://doi.org/10.1007/s10596-018-9763-3

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