Skip to main content
SpringerLink
Log in

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

  • ORIGINAL PAPER
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

An adjoint formulation for the gradient-based optimization of oil–gas compositional reservoir simulation problems is presented. The method is implemented within an automatic differentiation-based compositional flow simulator (Stanford’s Automatic Differentiation-based General Purpose Research Simulator, AD-GPRS). The development of adjoint procedures for general compositional problems is much more challenging than for oil–water problems due to the increased complexity of the code and the underlying physics. The treatment of nonlinear constraints, an example of which is a maximum gas rate specification in injection or production wells, when the control variables are well bottom-hole pressures, poses a particular challenge. Two approaches for handling these constraints are presented—a formal treatment within the optimizer and a simpler heuristic treatment in the forward model. The relationship between discrete and continuous adjoint formulations is also elucidated. Results for four example cases of increasing complexity are presented. Improvements in the objective function (cumulative oil produced) relative to reference solutions range from 4.2 to 11.6 %. The heuristic treatment of nonlinear constraints is shown to offer a cost-effective means for obtaining feasible solutions, which are, in some cases, better than those obtained using the formal constraint handling procedure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Asheim, H: Maximization of water sweep efficiency by controlling production and injection rates. In: SPE Paper 18365 Presented at the SPE European Petroleum Conference. London, 1988.

  2. Asouti, V.G., Zymaris, A.S., Papadimitriou, D.I., Giannakoglou, K.C.: Continuous and discrete adjoint approaches for aerodynamic shape optimization with low mach number preconditioning. Int. J. Numer. Methods Fluids. 57(10), 1485–1504 (2008).

    Article  Google Scholar 

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

    Google Scholar 

  4. Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999).

    Google Scholar 

  5. Brouwer, D.R., Jansen, J.D.: Dynamic optimization of water flooding with smart wells using optimal control theory. SPE J. 9(4), 391–402 (2004).

    Article  Google Scholar 

  6. Bryson, A., Ho, Y.: Applied Optimal Control. Taylor and Francis (Hemisphere), Levittown (2001).

    Google Scholar 

  7. Cao, H.: Development of Techniques for General Purpose Simulators. Ph.D. thesis, Stanford University, 2002.

  8. Chavent, G., Dupuy, M., Lemonnier, P.: History matching by use of optimal control theory. SPE J. 15, 74–86 (1975). doi:10.2118/4627-PA

    Google Scholar 

  9. Chen, W., Gavalas, G., Wasserman, M.: A new algorithm for automatic history matching. SPE J. 14(6), 593–608 (1974).

    Google Scholar 

  10. Chen, C., Li, G., Reynolds, A.C.: Robust constrained optimization of short- and long-term net present value for closed-loop reservoir management. SPE J. 17(3), 849–864 (2012). doi:10.2118/141314-PA

    Article  Google Scholar 

  11. Christie, M., Blunt, M.: Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reserv. Eval. Eng. 4(4), 308–317 (2001).

    Google Scholar 

  12. Coats, K.: An equation of state compositional model. SPE J. 20(5), 363–376 (1980).

    Google Scholar 

  13. De Montleau, P., Cominelli, A., Neylon, K., Rowan, D., Pallister, I., Tesaker, O., Nygard, I.: Production optimization under constraints using adjoint gradients. In: 10th European Conference on the Mathematics of Oil Recovery, 2006.

  14. Doublet, D., Aanonsen, S., Tai, X.: An efficient method for smart well production optimisation. J. Petrol. Sci. Eng. 69(1–2), 25–39 (2009).

    Article  Google Scholar 

  15. Echeverría Ciaurri, D., Isebor, O.J., Durlofsky, L.J.: Application of derivative-free methodologies to generally constrained oil production optimisation problems. Int. J. Math. Model. Numer. Optim. 2, 134–161 (2011).

    Google Scholar 

  16. Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 47(1), 99–131 (2005).

    Article  Google Scholar 

  17. Gunzburger, M.: Perspectives in Flow Control and Optimization. SIAM, 2003.

  18. Hager, W.W.: Runge–Kutta methods in optimal control and the transformed adjoint system. Numer. Math. 87(2), 247–282 (2000). doi:10.1007/s002110000178.

    Article  Google Scholar 

  19. Han, C., Wallis, J., Sarma, P., Li, G., Schrader, M.L., Chen, W.: Adaptation of the CPR preconditioner for efficient solution of the adjoint equation. SPE J. 18(2), 207–213 (2013). doi:10.2118/141300-PA.

    Article  Google Scholar 

  20. Jansen, J.D.: Adjoint-based optimization of multi-phase flow through porous media—a review. Comput. Fluids. 46(1), 40–51 (2011). doi:10.1016/j.compfluid.2010.09.039.

    Article  Google Scholar 

  21. Kraaijevanger, J., Egberts, P., Valstar, J., Buurman, H.: Optimal waterflood design using the adjoint method. In: Paper SPE 105764 Presented at the SPE Reservoir Simulation Symposium. Houston, 2007.

  22. Li, R., Reynolds, A., Oliver, D.: History matching of three-phase flow production data. SPE J. 8(4), 328–340 (2003). doi:10.2118/87336-PA.

    Google Scholar 

  23. Lien, M., Brouwer, D., Manseth, T., Jansen, J.D.: Multiscale regularization of flooding optimization for smart field management. SPE J. 13(2), 195–204 (2008).

    Article  Google Scholar 

  24. Liu, W., Ramirez, W., Qi, Y.: Optimal control of steam flooding. SPE Adv. Technol. Ser. 1(2), 73–82 (1993).

    Google Scholar 

  25. Mehos, G., Ramirez, W.: Use of optimal control theory to optimize carbon dioxide miscible-flooding enhanced oil recovery. J. Petrol. Sci. Eng. 2(4), 247–260 (1989).

    Article  Google Scholar 

  26. Nadarajah, S., Jameson, A.: A comparison of the continuous and discrete adjoint approach to automatic aerodynamic optimization. In: AIAA 38th Aerospace Sciences Meeting and Exhibit. Reno, 2000.

  27. Nadarajah, S., Jameson, A.: Optimum shape design for unsteady flows with time-accurate continuous and discrete adjoint methods. AIAA J. 45(7), 1478–1491 (2007). doi:10.2514/1.24332.

    Article  Google Scholar 

  28. Petra, N., Stadler, G.: Model Variational Inverse Problems Governed by Partial Differential Equations. ICES Report 11-05, The Institute for Computational Engineering and Sciences. The University of Texas at Austin (2011)

  29. NTNU (IO Center): Center of Integrated Operations in Petroleum Industry. Website (2011). www.ipt.ntnu.no/%7Enorne.

  30. Oliver, D., Reynolds, A., Liu, N.: Inverse Theory for Petroleum Reservoir Characterization and History Matching. Cambridge University Press, Cambridge (2008).

    Book  Google Scholar 

  31. Ramirez, W.: Application of Optimal Control Theory to Enhanced Oil Recovery. Elsevier, Amsterdam (1987).

    Google Scholar 

  32. Sarma, P., Chen, W., Durlofsky, L.J., Aziz, K.: Production optimization with adjoint models under nonlinear control-state path inequality constraints. SPE Reserv. Eval. Eng. 11(2), 326–339 (2008).

    Google Scholar 

  33. Sarma, P., Durlofsky, L.J., Aziz, K., Chen, W.H.: Efficient real-time reservoir management using adjoint-based optimal control and model updating. Comput. Geosci. 10(1), 3–36 (2006). doi:10.1007/s10596-005-9009-z.

    Article  Google Scholar 

  34. Stengel, R.: Stochastic Optimal Control Theory and Application. Wiley, New York (1986).

    Google Scholar 

  35. Sudaryanto, B., Yortsos, Y.: Optimization of fluid front dynamics in porous media using rate control. Phys. Fluids. 12(7), 1656–1670 (2000).

    Article  Google Scholar 

  36. Suwartadi, E., Krogstad, S., Foss, B.: Nonlinear output constraints handling for production optimization of oil reservoirs. Comput. Geosci. 16, 499–517 (2012).

    Article  Google Scholar 

  37. Van Essen, G., Zandvliet, M., Van den Hof, P., Bosgra, O., Jansen, J.D.: Robust waterflooding optimization of multiple geological scenarios. SPE J. 14(1), 202–210 (2009). doi:10.2118/102913-PA.

    Google Scholar 

  38. Virnovski, G.: Waterflooding strategy design using optimal control theory. In: 6th European IOR Symposium, pp. 437–446. Stavanger, 1991.

  39. Voskov, D., Tchelepi, H.: Comparison of nonlinear formulations for two-phase multi-component EoS based simulation. J. Pet. Sci. Eng. 82–83, 101–111 (2012). doi:10.1016/j.petrol.2011.10.012.

    Article  Google Scholar 

  40. Voskov, D.V., Younis, R., Tchelepi, H.A.: Comparison of nonlinear formulations for isothermal compositional flow simulation. In: SPE Paper 118966 Presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA (2009)

    Google Scholar 

  41. Walther, A.: Automatic differentiation of explicit Runge–Kutta methods for optimal control. Comput. Optim. Appl. 36(1), 83–108 (2007). doi:10.1007/s10589-006-0397-3.

    Article  Google Scholar 

  42. Wang, C., Li, G., Reynolds, A.: Production optimization in closed-loop reservoir management. SPE J. 14(3), 506–523 (2009).

    Article  Google Scholar 

  43. Young, L., Stephenson, R.: A generalized compositional approach for reservoir simulation. J. Pet. Sci. Eng. 23(5), 727–742 (1983). doi:10.2118/10516-PA.

    Google Scholar 

  44. Younis, R., Aziz, K.: Parallel automatically differentiable data-types for next-generation simulator development. In: SPE Paper 106493 Presented at the SPE Reservoir Simulation Symposium. Houston (2007). doi:10.2118/106493-MS.

  45. Younis, R., Tchelepi, H., Aziz, K.: Adaptively localized continuation-Newton method—nonlinear solvers that converge all the time. SPE J. 15(2), 526–544 (2010). doi:10.2118/119147-PA.SPE-119147-PA.

    Article  Google Scholar 

Download references

Author information

Author notes

  1. Drosos Kourounis

    Present address: Institute of Computational Science, Faculty of Informatics, Università della Svizzera italiana, 6904, Lugano, Switzerland

Authors and Affiliations

  1. Department of Energy Resources Engineering, Stanford University, Stanford, CA, 94305-2220, USA

    Drosos Kourounis, Louis J. Durlofsky & Khalid Aziz

  2. Department of Geoscience and Engineering, Delft University of Technology, Delft, Netherlands

    Jan Dirk Jansen

Authors
  1. Drosos Kourounis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Louis J. Durlofsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jan Dirk Jansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Khalid Aziz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Drosos Kourounis.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kourounis, D., Durlofsky, L.J., Jansen, J.D. et al. Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow. Comput Geosci 18, 117–137 (2014). https://doi.org/10.1007/s10596-013-9385-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-013-9385-8

Keywords

Search

Navigation