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Waterflooding optimization in uncertain geological scenarios

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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.

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References

  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)

  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. Sarma, P., Durlofsky, L., Aziz, K.: Efficient closed-loop production optimization under uncertainty. In: SPE Europec/EAGE Annual Conference, Madrid (2005)

  4. Nævdal, G., Brouwer, D.R., Jansen, J.-D.: Waterflooding using closed-loop control. Comput. Geosci. 10, 37–60 (2006)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  6. Jansen, J.D., et al.: Closed-loop reservoir management. In: 2009 SPE Reservoir Simulation Symposium, SPE 119098 The Woodlands, Texas (2009)

  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)

  8. Foss, B., Jensen, J.P.: Performance analysis for closed-loop reservoir management. SPE J. 16, 183–190 (2011)

    Google Scholar 

  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)

  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)

  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)

  12. Foss, B.: Process control in conventional oil and gas fields - challenges and opportunities. Control. Eng. Pract. 20, 1058–1064 (2012)

    Article  Google Scholar 

  13. Rawlings, J.B., Mayne, D.Q.: Model predictive control: Theory and design. Nob Hill Publishing, Madison (2009)

    Google Scholar 

  14. Grüne, L., Pannek, J.: Nonlinear model predictive control theory and algorithms. Springer, London (2011)

    Book  Google Scholar 

  15. Grötschel, M., Krumke, S.O., Rambau, J. (eds.): Online optimization of large scale systems. Springer, Heidelberg (2001)

    Google Scholar 

  16. Allgöwer, F., Zheng, A. (eds.): Nonlinear Model Predictive Control, vol. 26 Progress in Systems and Control Theory. Birkhäuser, Basel (2000)

    Google Scholar 

  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)

  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)

  19. Lazar, M., Allgöwer, F. (eds.) : 4th IFAC Nonlinear Model Predictive Control Conference (NMPC’12). IFAC Noordwijkerhout, NL (2012)

  20. Evensen, G.: Data Assimilation: The Ensemble Kalman Filter, 2nd edn. Springer (2009)

  21. Biegler, L.T., Ghattas, O., Heinkenschloss, M., van Bloemen Waanders, B. (eds.): Large-Scale PDE-Constrained Optimization. Springer (2003)

  22. Biegler, L.T., Ghattas, O., Heinkenschloss, M., Keyes, D., van Bloemen Waanders, B. (eds.) : Real-Time PDE-Constrained Optimization SIAM (2007)

  23. Markowitz, H.: Portfolio selection. J. Finance 7, 77–91 (1952)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. Chen, C., Wang, Y., Li, G., Reynolds, A.C.: Closed-loop reservoir management on the Brugge test case. Comput. Geosci. 14, 691–703 (2010)

    Article  Google Scholar 

  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. Evensen, G.: The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53, 342–367 (2003)

    Google Scholar 

  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. Ehrendorfer, M.: A review of issues in ensemble-based Kalman filtering. Meteorol. Z. 16, 795–818 (2007)

    Article  Google Scholar 

  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. Simon, D.: Optimal State EstimationKalman, H\(_{\infty }\), and Nonlinear Approaches. Wiley, Hoboken (2006)

    Chapter  Google Scholar 

  34. Rawlings, J.B., Bakshi, B.R.: Particle filtering and moving horizon estimation. Comput. Chem. Eng. 30, 1529–1541 (2006)

    Article  Google Scholar 

  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. 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)

  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)

  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)

  39. Chen, Y., Oliver, D.S., Zhang, D.: Efficient ensemble-based closed-loop production optimization. SPE J. 14, 634–645 (2009)

    Google Scholar 

  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. Lie, K.A., et al.: Open source matlab implementation of consistent discretisations on complex grids. Comput. Geosci. 16, 297–322 (2012)

    Article  Google Scholar 

  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. 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 

  44. Schlegel, M., Stockmann, K., Binder, T., Marquardt, W.: Dynamic optimization using adaptive control vector parameterization. Comput. Chem. Eng. 29, 1731–1751 (2005)

    Article  Google Scholar 

  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)

  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)

  47. Biegler, L.T.: Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation. Comput. Chem. Eng. 8, 243–248 (1984)

    Article  Google Scholar 

  48. Jansen, J.: Adjoint-based optimization of multi-phase flow through porous media - A review. Comput. Fluids 46, 40–51 (2011)

    Article  Google Scholar 

  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)

  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)

  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)

  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)

  53. MATLAB. version 7.13.0.564 (R2011b) (The MathWorks Inc., Natick, Massachusetts, 2011)

  54. Liu, Y.: Using the snesim program for multiple-point statistical simulation. Comput. Geosci. 32, 1544–1563 (2006)

    Article  Google Scholar 

  55. Schölkopf, B., Smola, A., Müller, K.-R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput. 10, 1299–1319 (1998)

    Article  Google Scholar 

  56. Kalman, R.E.: A new approach to linear filtering and predictions problems. J. Basic Eng. 82, 35–45 (1960)

    Article  Google Scholar 

  57. Kailath, T., Sayed, A.H., Hassibi, B.: Linear Estimation. Prentice Hall (2000)

  58. Jazwinski, A.H.: Stochastic Processes and Filtering Theory. Academic Press (1970)

  59. Burgers, G., van Leeuwen, P.J., Evensen, G.: Analysis scheme in the ensemble Kalman filter. Mon. Weather Rev. 126, 1719–1724 (1998)

    Article  Google Scholar 

  60. Wen, X.H., Chen, W.: Real-time reservoir model updating using ensemble Kalman filter. In: SPE Reservoir Simulation Symposium. The Woodlands, Texas (2005)

  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. Chen, Z. Reservoir Simulation Mathematical Techniques in Oil Recovery. SIAM Philadelphia (2007)

  63. Aziz, K., Durlofsky, L., Tchelepi, H.: Notes on petroleum reservoir simulation. Department of Petroleum Engineering School of Earth Sciences, Stanford University (2005)

  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)

  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 

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Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science and Center for Energy Resources Engineering, Technical University of Denmark (DTU), 2800 Kgs., Lyngby, Denmark

    Andrea Capolei & John Bagterp Jørgensen

  2. Department of Engineering Cybernetics, Norwegian University of Science and Technology (NTNU), 7491, Trondheim, Norway

    Eka Suwartadi & Bjarne Foss

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  1. Andrea Capolei
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  2. Eka Suwartadi
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  3. Bjarne Foss
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  4. John Bagterp Jørgensen
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Correspondence to John Bagterp Jørgensen.

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Capolei, A., Suwartadi, E., Foss, B. et al. Waterflooding optimization in uncertain geological scenarios. Comput Geosci 17, 991–1013 (2013). https://doi.org/10.1007/s10596-013-9371-1

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