Advertisement

Computational Geosciences

, Volume 17, Issue 6, pp 991–1013 | Cite as

Waterflooding optimization in uncertain geological scenarios

  • Andrea Capolei
  • Eka Suwartadi
  • Bjarne Foss
  • John Bagterp JørgensenEmail author
ORIGINAL PAPER

Abstract

In conventional waterflooding of an oil field, feedback based optimal control technologies may enable higher oil recovery than with a conventional reactive strategy in which producers are closed based on water breakthrough. To compensate for the inherent geological uncertainties in an oil field, robust optimization has been suggested to improve and robustify optimal control strategies. In robust optimization of an oil reservoir, the water injection and production borehole pressures (bhp) are computed such that the predicted net present value (NPV) of an ensemble of permeability field realizations is maximized. In this paper, we both consider an open-loop optimization scenario, with no feedback, and a closed-loop optimization scenario. The closed-loop scenario is implemented in a moving horizon manner and feedback is obtained using an ensemble Kalman filter for estimation of the permeability field from the production data. For open-loop implementations, previous test case studies presented in the literature, show that a traditional robust optimization strategy (RO) gives a higher expected NPV with lower NPV standard deviation than a conventional reactive strategy. We present and study a test case where the opposite happen: The reactive strategy gives a higher expected NPV with a lower NPV standard deviation than the RO strategy. To improve the RO strategy, we propose a modified robust optimization strategy (modified RO) that can shut in uneconomical producer wells. This strategy inherits the features of both the reactive and the RO strategy. Simulations reveal that the modified RO strategy results in operations with larger returns and less risk than the reactive strategy, the RO strategy, and the certainty equivalent strategy. The returns are measured by the expected NPV and the risk is measured by the standard deviation of the NPV. In closed-loop optimization, we investigate and compare the performance of the RO strategy, the reactive strategy, and the certainty equivalent strategy. The certainty equivalent strategy is based on a single realization of the permeability field. It uses the mean of the ensemble as its permeability field. Simulations reveal that the RO strategy and the certainty equivalent strategy give a higher NPV compared to the reactive strategy. Surprisingly, the RO strategy and the certainty equivalent strategy give similar NPVs. Consequently, the certainty equivalent strategy is preferable in the closed-loop situation as it requires significantly less computational resources than the robust optimization strategy. The similarity of the certainty equivalent and the robust optimization based strategies for the closed-loop situation challenges the intuition of most reservoir engineers. Feedback reduces the uncertainty and this is the reason for the similar performance of the two strategies.

Keywords

Robust optimization Ensemble Kalman filter Oil reservoir Production optimization Automatic history matching 

Mathematics Subject Classifications (2010)

90C30 49M37 49J20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brouwer, D., Nævdal, G., Jansen, J.: Improved reservoir management through optimal control and continuous model updating. In: SPE Annual Technical Conference and Exhibition, Houston (2004)Google Scholar
  2. 2.
    Brouwer, D.R., Jansen, J.D.: Dynamic optimization of waterflooding with smart wells using optimal control theory. SPE J. 9, 391–402 (2004)Google Scholar
  3. 3.
    Sarma, P., Durlofsky, L., Aziz, K.: Efficient closed-loop production optimization under uncertainty. In: SPE Europec/EAGE Annual Conference, Madrid (2005)Google Scholar
  4. 4.
    Nævdal, G., Brouwer, D.R., Jansen, J.-D.: Waterflooding using closed-loop control. Comput. Geosci. 10, 37–60 (2006)CrossRefGoogle Scholar
  5. 5.
    Jansen, J.-D., Bosgra, O.H., Van den Hof, P.M.J.: Model-based control of multiphase flow in subsurface oil reservoirs. J. Process Control 18, 846–855 (2008)CrossRefGoogle Scholar
  6. 6.
    Jansen, J.D., et al.: Closed-loop reservoir management. In: 2009 SPE Reservoir Simulation Symposium, SPE 119098 The Woodlands, Texas (2009)Google Scholar
  7. 7.
    Lorentzen, R.J., Shafieirad, A., Nævdal, G.: Closed loop reservoir management using the ensemble Kalman filter and sequential quadratic programming. In: 2009 SPE Reservoir Simulation Symposium, SPE 119101. The Woodlands, Texas (2009)Google Scholar
  8. 8.
    Foss, B., Jensen, J.P.: Performance analysis for closed-loop reservoir management. SPE J. 16, 183–190 (2011)Google Scholar
  9. 9.
    Capolei, A., Stenby, E.H., Jørgensen, J.B.: High order adjoint derivatives using esdirk methods for oil reservoir production optimization. In: ECMOR XIII 13th European Conference on the Mathematics of Oil Recovery (2012)Google Scholar
  10. 10.
    Van den Hof, P.M.J., Jansen, J.D., Heemink, A.: Recent developments in model-based optimization and control of subsurface flow in oil reservoirs. In: Proceedings of the 2012 IFAC Workshop on Automatic Control in Offshore Oil and Gas Production, pp 189–200, Trondheim (2012)Google Scholar
  11. 11.
    Capolei, A., Völcker, C., Frydendall, J., Jørgensen, J.B.: Oil reservoir production optimization using single shooting and ESDIRK methods. In: Proceedings of the 2012 IFAC Workshop on Automatic Control in Offshore Oil and Gas Production, pp 286–291, Trondheim (2012)Google Scholar
  12. 12.
    Foss, B.: Process control in conventional oil and gas fields - challenges and opportunities. Control. Eng. Pract. 20, 1058–1064 (2012)CrossRefGoogle Scholar
  13. 13.
    Rawlings, J.B., Mayne, D.Q.: Model predictive control: Theory and design. Nob Hill Publishing, Madison (2009)Google Scholar
  14. 14.
    Grüne, L., Pannek, J.: Nonlinear model predictive control theory and algorithms. Springer, London (2011)CrossRefGoogle Scholar
  15. 15.
    Grötschel, M., Krumke, S.O., Rambau, J. (eds.): Online optimization of large scale systems. Springer, Heidelberg (2001)Google Scholar
  16. 16.
    Allgöwer, F., Zheng, A. (eds.): Nonlinear Model Predictive Control, vol. 26 Progress in Systems and Control Theory. Birkhäuser, Basel (2000)Google Scholar
  17. 17.
    Findeisen, R., Allgöwer, F., Biegler, L.T. (eds.) : Assessment and future directions of nonlinear model predictive control. Lecture Notes in Control and Information Sciences, vol. 358, Springer, Heidelberg (2007)Google Scholar
  18. 18.
    Magni, L., Raimondo, D.M., Allgöwer, F. (eds.) : Nonlinear model predictive control. Towards New Challenging Applications. Lecture Notes in Control and Information Sciences, vol. 384, Springer, Heidelberg (2009)Google Scholar
  19. 19.
    Lazar, M., Allgöwer, F. (eds.) : 4th IFAC Nonlinear Model Predictive Control Conference (NMPC’12). IFAC Noordwijkerhout, NL (2012)Google Scholar
  20. 20.
    Evensen, G.: Data Assimilation: The Ensemble Kalman Filter, 2nd edn. Springer (2009)Google Scholar
  21. 21.
    Biegler, L.T., Ghattas, O., Heinkenschloss, M., van Bloemen Waanders, B. (eds.): Large-Scale PDE-Constrained Optimization. Springer (2003)Google Scholar
  22. 22.
    Biegler, L.T., Ghattas, O., Heinkenschloss, M., Keyes, D., van Bloemen Waanders, B. (eds.) : Real-Time PDE-Constrained Optimization SIAM (2007)Google Scholar
  23. 23.
    Markowitz, H.: Portfolio selection. J. Finance 7, 77–91 (1952)Google Scholar
  24. 24.
    Terwiesch, P., Ravemark, D., Schenker, B., Rippin, D.W.: Semi-batch process optimization under uncertainty: Theory and experiments. Comput. Chem. Eng. 22, 201–213 (1998)CrossRefGoogle Scholar
  25. 25.
    Srinivasana, B., Bonvina, D., Vissera, E., Palankib, S.: Dynamic optimization of batch processes: II role of measurements in handling uncertainty. Comp. Chem. Eng. 27, 27–44 (2003)CrossRefGoogle Scholar
  26. 26.
    Van Essen, G.M., Zandvliet, M.J., Van den Hof, P.M.J., Bosgra, O.H., Jansen, J.D.: Robust waterflooding optimazation of multiple geological scenarios. SPE J. 14, 202–210 (2009)Google Scholar
  27. 27.
    Chen, C., Wang, Y., Li, G., Reynolds, A.C.: Closed-loop reservoir management on the Brugge test case. Comput. Geosci. 14, 691–703 (2010)CrossRefGoogle Scholar
  28. 28.
    Peters, L., et al.: Results of the Brugge benchmark study for flooding optimization and history matching. SPE Reserv. Eval. Eng. 13, 391–405 (2010)Google Scholar
  29. 29.
    Evensen, G.: The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53, 342–367 (2003)Google Scholar
  30. 30.
    Wen, X.-H., Chen, W.H.: Some practical issues on real-time reservoir model updating using ensemble Kalman filter. SPE J. 12, 156–166 (2007)Google Scholar
  31. 31.
    Ehrendorfer, M.: A review of issues in ensemble-based Kalman filtering. Meteorol. Z. 16, 795–818 (2007)CrossRefGoogle Scholar
  32. 32.
    Aanonsen, S.I., Nævdal G, Oliver, D.S., Reynolds, A.C., Valls, B.: The ensemble Kalman filter in reservoir engineering-a review. SPE J. 14, 393–412 (2009)Google Scholar
  33. 33.
    Simon, D.: Optimal State EstimationKalman, H\(_{\infty }\), and Nonlinear Approaches. Wiley, Hoboken (2006)CrossRefGoogle Scholar
  34. 34.
    Rawlings, J.B., Bakshi, B.R.: Particle filtering and moving horizon estimation. Comput. Chem. Eng. 30, 1529–1541 (2006)CrossRefGoogle Scholar
  35. 35.
    Wen, X.-H., Chen, W.H.: Real-time reservoir model updating using ensemble Kalman filter with confirming option. SPE J. 11, 431–442 (2006)Google Scholar
  36. 36.
    Sarma, P., Chen, W. Preventing ensemble collapse and preserving geostatistical variability across the ensemble with the subspace enkf. In: ECMOR XIII-13th European Conference on the Mathematics of Oil Recovery. Biarritx, France (2012)Google Scholar
  37. 37.
    Sarma, P., Chen, W.H. Robust and efficient handling of model contraints with the kernal-based ensemble Kalman filter. In: Reservoir Simulation Symposium. The Woodlands, Texas (2011)Google Scholar
  38. 38.
    Sarma, P., Chen, W. Generalization of the ensemble Kalman filter using kernels for nongaussian random fields. In: SPE Reservoir Simulation Symposium. The Woodlands, Texas (2009)Google Scholar
  39. 39.
    Chen, Y., Oliver, D.S., Zhang, D.: Efficient ensemble-based closed-loop production optimization. SPE J. 14, 634–645 (2009)Google Scholar
  40. 40.
    Chen, Y., Oliver, D.S.: Ensemble-based closed-loop optimization applied to Brugge field. SPE Reserv. Eval. Eng. 13, 56–71 (2010)Google Scholar
  41. 41.
    Lie, K.A., et al.: Open source matlab implementation of consistent discretisations on complex grids. Comput. Geosci. 16, 297–322 (2012)CrossRefGoogle Scholar
  42. 42.
    Peaceman, D.W.: Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability. SPE J. 23(3), 531–543 (1983)Google Scholar
  43. 43.
    Suwartadi, E., Krogstad, S., Foss, B.: Nonlinear output constraints handling for production optimization of oil reservoirs. Comput. Geosci. 16, 499–517 (2012)CrossRefGoogle Scholar
  44. 44.
    Schlegel, M., Stockmann, K., Binder, T., Marquardt, W.: Dynamic optimization using adaptive control vector parameterization. Comput. Chem. Eng. 29, 1731–1751 (2005)CrossRefGoogle Scholar
  45. 45.
    Capolei, A., Jørgensen, J.B. Solution of constrained optimal control problems using multiple shooting and esdirk methods. In: American Control Conference (ACC), 295–300 (2012)Google Scholar
  46. 46.
    Bock, H.G., Plitt, K.J. A multiple shooting algorithm for direct solution of optimal control problems. In: Proceedings 9th IFAC World Congress Budapest, pp. 243–247. Pergamon Press (1984)Google Scholar
  47. 47.
    Biegler, L.T.: Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation. Comput. Chem. Eng. 8, 243–248 (1984)CrossRefGoogle Scholar
  48. 48.
    Jansen, J.: Adjoint-based optimization of multi-phase flow through porous media - A review. Comput. Fluids 46, 40–51 (2011)CrossRefGoogle Scholar
  49. 49.
    Sarma, P., Aziz, K., Durlofsky, L.J. Implementation of adjoint solution for optimal control of smart wells. In: SPE Reservoir Simulation Symposium, 31 January-2 Feburary 2005, The Woodlands, Texas (2005)Google Scholar
  50. 50.
    Jørgensen, J.B. Adjoint sensitivity results for predictive control, state- and parameter-estimation with nonlinear models. In: Proceedings of the European Control Conference 2007, pp. 3649–3656. Kos, Greece (2007)Google Scholar
  51. 51.
    Völcker, C., Jørgensen, J.B., Stenby, E.H. Oil reservoir production optimization using optimal control. In: 50th IEEE Conference on Decision and Control and European Control Conference, 7937–7943 Orlando, Florida (2011)Google Scholar
  52. 52.
    Byrd, R.H., Nocedal, J., Waltz, R.A.: Knitro: An integrated package for nonlinear optimization. In: Large Scale Nonlinear Optimization, pp. 35–59 (2006)Google Scholar
  53. 53.
    MATLAB. version 7.13.0.564 (R2011b) (The MathWorks Inc., Natick, Massachusetts, 2011)Google Scholar
  54. 54.
    Liu, Y.: Using the snesim program for multiple-point statistical simulation. Comput. Geosci. 32, 1544–1563 (2006)CrossRefGoogle Scholar
  55. 55.
    Schölkopf, B., Smola, A., Müller, K.-R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput. 10, 1299–1319 (1998)CrossRefGoogle Scholar
  56. 56.
    Kalman, R.E.: A new approach to linear filtering and predictions problems. J. Basic Eng. 82, 35–45 (1960)CrossRefGoogle Scholar
  57. 57.
    Kailath, T., Sayed, A.H., Hassibi, B.: Linear Estimation. Prentice Hall (2000)Google Scholar
  58. 58.
    Jazwinski, A.H.: Stochastic Processes and Filtering Theory. Academic Press (1970)Google Scholar
  59. 59.
    Burgers, G., van Leeuwen, P.J., Evensen, G.: Analysis scheme in the ensemble Kalman filter. Mon. Weather Rev. 126, 1719–1724 (1998)CrossRefGoogle Scholar
  60. 60.
    Wen, X.H., Chen, W.: Real-time reservoir model updating using ensemble Kalman filter. In: SPE Reservoir Simulation Symposium. The Woodlands, Texas (2005)Google Scholar
  61. 61.
    Gu, Y., Oliver, D.S.: History matching of the punq-s3 reservoir model using the ensemble Kalman filter. SPE J. 10, 217–224 (2005)Google Scholar
  62. 62.
    Chen, Z. Reservoir Simulation Mathematical Techniques in Oil Recovery. SIAM Philadelphia (2007)Google Scholar
  63. 63.
    Aziz, K., Durlofsky, L., Tchelepi, H.: Notes on petroleum reservoir simulation. Department of Petroleum Engineering School of Earth Sciences, Stanford University (2005)Google Scholar
  64. 64.
    Völcker, C., Jørgensen, J.B., Thomsen, P.G., Stenby, E.H. Simulation of subsurface two-phase flow in an oil reservoir. In: Proceedings of the European Control Conference 2009, pp. 1221–1226. Budapest, Hungary (2009)Google Scholar
  65. 65.
    Dehdari, V., Oliver, D.S.: Sequential quadratic programming for solving constrained production optimization – case study from Brugge field. SPE J. 17, 874–884 (2012)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Andrea Capolei
    • 1
  • Eka Suwartadi
    • 2
  • Bjarne Foss
    • 2
  • John Bagterp Jørgensen
    • 1
    Email author
  1. 1.Department of Applied Mathematics and Computer Science and Center for Energy Resources EngineeringTechnical University of Denmark (DTU)LyngbyDenmark
  2. 2.Department of Engineering CyberneticsNorwegian University of Science and Technology (NTNU)TrondheimNorway

Personalised recommendations