Computational Geosciences

, Volume 22, Issue 6, pp 1433–1444 | Cite as

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

  • H. Hajinezhad
  • Ali R. SoheiliEmail author
  • Mohammad R. Rasaei
  • F. Toutounian
Original Paper


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.


Steam injection Cell-centered finite difference Fully implicit JFNK method GMRES method Simultaneous solution FAS method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Shutler, N.D.: Numerical, three-phase simulation of the linear steamflood process. Soc. Pet. Eng. J. 9(02), 232–246 (1969)CrossRefGoogle Scholar
  2. 2.
    Shutler, N.D.: Numerical three-phase model of the two-dimensional steamflood process. Soc. Pet. Eng. J. 10 (04), 405–417 (1970)CrossRefGoogle Scholar
  3. 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)CrossRefGoogle Scholar
  4. 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)Google Scholar
  5. 5.
    Coats, K.H.: Simulation of steamflooding with distillation and solution gas. Soc. Pet. Eng. J. 16(05), 235–247 (1976)CrossRefGoogle Scholar
  6. 6.
    Coats, K.H.: In-situ combustion model. Soc. Pet. Eng. J. 20(06), 533–554 (1980)CrossRefGoogle Scholar
  7. 7.
    Trottenberg, U., Oosterlee, C.W., Schuller, A.: Multigrid. Academic press, Cambridge (2000)Google Scholar
  8. 8.
    Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31(138), 333–390 (1977)CrossRefGoogle Scholar
  9. 9.
    Hackbusch, W.: Multi-grid methods and applications, vol. 4 of Springer series in computational mathematics. Springer-Verlag, Berlin (1985)Google Scholar
  10. 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)Google Scholar
  11. 11.
    Teigland, R., Fladmark, G.: Cell-centered multigrid methods in porous media flow. In: multigrid methods III. Springer, pp. 365–376 (1991)Google Scholar
  12. 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)Google Scholar
  13. 13.
    Molenaar, J.: Multigrid Methods for Fully Implicit Oil Reservoir Simulation. Delft University of Technology, Faculty of Technical Mathematics and Informatics (1995)Google Scholar
  14. 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)CrossRefGoogle Scholar
  15. 15.
    Rosinger, E.E.: Stability and convergence for non-linear difference schemes are equivalent. IMA J. Appl. Math. 26(2), 143–149 (1980)CrossRefGoogle Scholar
  16. 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)CrossRefGoogle Scholar
  17. 17.
    Saad, Y.: Iterative methods for sparse linear systems. SIAM, Bangkok (2003)CrossRefGoogle Scholar
  18. 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)CrossRefGoogle Scholar
  19. 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)Google Scholar
  20. 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)Google Scholar
  21. 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)CrossRefGoogle Scholar
  22. 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)Google Scholar
  23. 23.
    Chen, Z., Huan, G., Ma, Y.: Computational methods for multiphase flows in porous media, vol. 2. Siam, Bangkok (2006)Google Scholar
  24. 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)Google Scholar
  25. 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)Google Scholar
  26. 26.
    Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Science Publisher, London (1979)Google Scholar
  27. 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)CrossRefGoogle Scholar
  28. 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)CrossRefGoogle Scholar
  29. 29.
    Yao, S.C.: Fluid mechanics and heat transfer in steam injection wells. University of Tulsa, Tulsa (1985)Google Scholar
  30. 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)CrossRefGoogle Scholar
  31. 31.
    Sarathi, P.: Thermal numerical simulator for laboratory evaluation of steamflood oil recovery. National Inst for Petroleum and Energy Research, Bartlesville (1991)Google Scholar
  32. 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)Google Scholar
  33. 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)CrossRefGoogle Scholar
  34. 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)CrossRefGoogle Scholar
  35. 35.
    Brown, P.N., Saad, Y.: Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput. 11(3), 450–481 (1990)CrossRefGoogle Scholar
  36. 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)CrossRefGoogle Scholar
  37. 37.
    Heyouni, M.: Newton generalized Hessenberg method for solving nonlinear systems of equations. Numer. Algorithm. 21(1-4), 225–246 (1999)CrossRefGoogle Scholar
  38. 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)Google Scholar
  39. 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)CrossRefGoogle Scholar
  40. 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)CrossRefGoogle Scholar
  41. 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)CrossRefGoogle Scholar
  42. 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. 43.
    Beggs, H.D., Robinson, J.: Estimating the viscosity of crude oil systems. J. Pet. Technol. 27(09), 1140–1141 (1975)CrossRefGoogle Scholar
  44. 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

Copyright information

© Springer Nature Switzerland AG 2018
corrected publication September/2018

Authors and Affiliations

  • H. Hajinezhad
    • 1
  • Ali R. Soheili
    • 1
    Email author
  • Mohammad R. Rasaei
    • 2
  • F. Toutounian
    • 1
  1. 1.Department of Applied Mathematics, Faculty of Mathematical SciencesFerdowsi University of MashhadMashhadIran
  2. 2.Institute of Petroleum Engineering, College of EngineeringTehran UniversityTehranIran

Personalised recommendations