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Computational Geosciences

, Volume 16, Issue 2, pp 499–517 | Cite as

Nonlinear output constraints handling for production optimization of oil reservoirs

  • Eka SuwartadiEmail author
  • Stein Krogstad
  • Bjarne Foss
Original Paper

Abstract

Adjoint-based gradient computations for oil reservoirs have been increasingly used in closed-loop reservoir management optimizations. Most constraints in the optimizations are for the control input, which may either be bound constraints or equality constraints. This paper addresses output constraints for both state and control variables. We propose to use a (interior) barrier function approach, where the output constraints are added as a barrier term to the objective function. As we assume there always exist feasible initial control inputs, the method maintains the feasibility of the constraints. Three case examples are presented. The results show that the proposed method is able to preserve the computational efficiency of the adjoint methods.

Keywords

Adjoint method Output constraints Production optimization Oil reservoirs 

Mathematics Subject Classifications (2010)

90C30 49M37 49J20 

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References

  1. 1.
    Chen, Y., Oliver, D.S., Zhang, D.: Efficient ensemble-based closed-loop production optimization. SPE J. 14(4), 634–645 (2008)Google Scholar
  2. 2.
    Bryson, A., Ho, Y.: Applied Optimal Control. Hemisphere, Washington, D.C. (1975)Google Scholar
  3. 3.
    Mehra, R., Davis, R.: A Generalized gradient method for optimal control problems with inequality constraints and singular arch. IEEE Trans. Automat. Contr. 17, 69–79 (1972)zbMATHCrossRefGoogle Scholar
  4. 4.
    Hargraves, C., Paris, S.: Direct trajectory optimization using nonlinear programming and collocation. J. Guid. Control Dyn. 10(4), 338–342 (1987)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bloss, K.F., Biegler, L.T., Schiesser, W.E.: Dynamics process optimization through adjoint formulations and constraint aggregation. Ind. Eng. Chem. Res. 38(2), 421–432 (1999)CrossRefGoogle Scholar
  6. 6.
    Becerra, V.M.: Solving optimal control problems with state constraints using nonlinear programming and simulation tools. IEEE Trans. Ed. 43(3), 377–384 (2004)Google Scholar
  7. 7.
    Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York (1968)zbMATHGoogle Scholar
  8. 8.
    Forsgren, A., Gill, P.E., Wright, M.H.: Interior methods for nonlinear optimization. SIAM Rev. 44(4), 525–597 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kreisselmeier, G., Steinhauser, R.: Systematic Control Design by Optimizing a Vector Performance Index. IFAC Symposium on CADS, Zurich (1979)Google Scholar
  10. 10.
    Griewank, A., Korzec, M.: Approximating Jacobians by the TR2 formula. Proc. Appl. Math. Mech. 5, 791–792 (2005)CrossRefGoogle Scholar
  11. 11.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, New York (2009)Google Scholar
  12. 12.
    Virnovsky, G.A.: Waterflooding strategy design using optimal control theory. In: Proceeding of 6th. European IOR-symp. Stavanger, Norway (1991)Google Scholar
  13. 13.
    Zakirov, I.S., Aanonsen, S.I., Zakirov, E.S., Palatnik, B.M.: Optimizing reservoir performance by automatic allocation of well rates. In: Proceeding of 5th. ECMOR Leoben, Austria (1996)Google Scholar
  14. 14.
    Jansen, J.D.: Adjoint-based optimization of multi-phase flow through porous media—a review. Comput. Fluids 46, 40–51 (2011). doi: 10.1016/j.compfluid.2010.09.039 zbMATHCrossRefGoogle Scholar
  15. 15.
    Wu, Z.: Conditioning geostatistical models to two-phase flow production Data. Ph.D. thesis, University of Tulsa (1999)Google Scholar
  16. 16.
    Rommelse, J.R.: Data assimilation in reservoir management. Ph.D. thesis, TU Delft (2009)Google Scholar
  17. 17.
    Zandvliet, M.J., Handels, M., Van Essen, G.M., Brouwer, D.R., Jansen, J.D.: Adjoint-based well placement optimization under production constraints. SPE J. 13(4), 392–399 (2008)Google Scholar
  18. 18.
    Montleau, P. de., Cominelli, A. , Neylon, K., Rowan, D., Pallister, I., Tesaker, O., Nygard, I.: Production optimization under constraints using adjoint gradient. In: Proceedings of ECMOR X-10th European Conference on the Mathematics of Oil Recovery Number A041. EAGE, Amsterdam, The Netherlands (2006)Google Scholar
  19. 19.
    Sarma, P., Chen, W.H., Durlofsky, L.J., Aziz, K.: Production optimization with adjoint models under nonlinear control-state path inequality constraints. SPEREE, 11(2), 326–339 (2008). Paper SPE 99959CrossRefGoogle Scholar
  20. 20.
    Kraaijevanger, J.F.B.M., Egberts, P.J.P., Valstar, J.R., Buurman, H.W.: Optimal waterflood design using the adjoint method. Paper SPE 105764 presented at 2007 SPE RSS, Houston, TX, USA, 26–28 Feb (2007)Google Scholar
  21. 21.
    Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Science, New York (1979)Google Scholar
  22. 22.
    Van Essen, G.M., Van den Hof, P.M.J., Jansen, J.D.: Hierarchical long-term and short-term production optimization. SPE J. 16(1), 191–199 (2011)Google Scholar
  23. 23.
    Chen, C., Li, G., Reynolds, A.: Robust constrained optimization of short and long-term NPV for closed-loop reservoir management. Paper SPE 141314 presented at 2011 SPE RSS, Houston, TX, USA, 21–23 Feb 2011Google Scholar
  24. 24.
    Griewank, A., Walther, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM, Philadelphia. ISBN 978-0-898716-59-7 (2008)Google Scholar
  25. 25.
    Suwartadi, E., Krogstad, S., Foss, B.: A Lagrangian-barrier function for adjoint state constraints optimization of oil reservoirs water flooding. In: IEEE Conference on Proceeding of 2010 49th, pp. 3884–3889 (2010). doi: 10.1109/CDC.2010.5717749
  26. 26.
    Jittorntrum, K., Osborne, M.R.: A modified barrier function method with improved rate of convergence for degenerate problems. J. Austral. Math. Soc, Series B 21, 305–329Google Scholar
  27. 27.
    Byrd, R.H., Nocedal, J., Waltz, R.A.: KNITRO: an integrated package for nonlinear optimization. In: Di Pillo, G., Roma, M. (eds.) Large-Scale Nonlinear Optimization. Springer, USA (2006)Google Scholar
  28. 28.
    Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. SIAM, Philadelphia (2000)zbMATHCrossRefGoogle Scholar
  29. 29.
    Steihaug, T.: The conjugate gradient method and trust regions in large scale optimization. SIAM J. Numer. Anal. 20, 626–637 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Conn, A.R., Gould, N.I.M., Toint, P.L.: A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Math. Comput. 66(217), 216–288 (1997)MathSciNetGoogle Scholar
  31. 31.
    Lie, K.-A., Krogstad, S., Ligaarden, I.S., Natvig, L.R., Nilsen, H.M., Skafltestad, B.: Open-source MATLAB implementation of consistent discretisations on complex grids. Comput. Geosci. (2011). doi: 10.1007/s10596-011-9244-4 Google Scholar
  32. 32.
    Christie, M.A., Blunt, M.J.: Tenth SPE comparative solution project: a comparative of upscaling technique. SPE Reserv. Evalu. Eng. 4, 308–317 (2001)Google Scholar
  33. 33.
    Rwechungura, R., Suwartadi, E., Dadashpour, M., Kleppe, J., Foss, B.: The Norne field case—a unique comparative case study. Paper SPE 127538 presented at 2010 SPE Intelligent Energy Conference and Exhibition, Utrecht, The Netherlands, 23–25 March 2010Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Engineering CyberneticsNorwegian University Science and Technology (NTNU)TrondheimNorway
  2. 2.Department of Applied MathematicsSINTEF ICTOsloNorway

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