1887

Abstract

Summary

Ensemble methods are remarkably powerful for quantifying geological uncertainty. However, robust optimization of a cost function for a problem in which uncertainty is characterized by a large ensemble size can be computationally demanding. In a straightforward approach, the computation of expected net present value (NPV) requires many expensive simulations. Several techniques (e.g., model selection, coarsening) have been proposed to reduce the cost but generally lead to a less accurate optimization. To reduce the amount of computation without sacrificing accuracy, we developed a fast and effective approach for computing the expected NPV by using only the reservoir mean model with a bias correction factor. At each iteration of the optimization procedure, we only require one additional simulation in the mean model with a different set of controls to obtain an initial approximate value through which the bias will be corrected with a multiplicative correction factor. Information from individual simulations with distinct controls and model realizations can be used to estimate the correction factor for different controls. The effectiveness of various bias-corrected methods is illustrated by the application of the drilling-order problem in the synthetic REEK Field model. Compared with the average NPV, the results show that the average error of estimated expected NPV from the mean model is reduced from -9% to 0.56% by estimating the bias correction factor. Distance-based localization with an appropriate taper length can further improve the accuracy of estimation. By adding a regularization term with a tuning parameter associated with the variance of the correction factor, the sensitivity of the estimates to the taper length is reduced such that the regularized estimate is potentially more accurate for a wider range of taper lengths. In previous work, we proposed a nonparametric online-learning methodology (learned heuristic search) to efficiently compute a sequence of drilling wells that is optimal or near-optimal. In this work, we apply the learned heuristic search (LHS) to the reservoir mean model with bias correction to optimize the drilling sequence and show that it leads to the same solution as the LHS with the average NPV. Moreover, we investigate the possibility of optimizing the first few wells without finding an entire drilling sequence. Our results show that LHS can optimize complete drilling sequences or only the first few wells at a reduced cost.

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2020-09-14
2021-09-20
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References

  1. Alexanderian, A., Petra, N., Stadler, G. and Ghattas, O.
    [2017] Mean-Variance Risk-Averse Optimal Control of Systems Governed by PDEs with Random Parameter Fields Using Quadratic Approximations. SIAM/ASA Journal on Uncertainty Quantification, 5(1), 1166–1192.
    [Google Scholar]
  2. Barros, E.G.D., Maciel, S., Moraes, R.J. and Fonseca, R.M.
    [2018] Automated clustering based scenario reduction to accelerate robust life-cycle optimization. In: ECMOR XVI-16th European Conference on the Mathematics of Oil Recovery. European Association of Geoscientists & Engineers, Barcelona, Spain.
    [Google Scholar]
  3. Beyer, H.G. and Sendhoff, B.
    [2007] Robust optimization — A comprehensive survey. Computer Methods in Applied Mechanics and Engineering, 196(33), 3190–3218.
    [Google Scholar]
  4. Cardoso, M.A. and Durlofsky, L.J.
    [2010a] Linearized reduced-order models for subsurface flow simulation. J. Comput. Phys., 229(3), 681–700.
    [Google Scholar]
  5. [2010b] Use of Reduced-Order Modeling Procedures for Production Optimization. SPE Journal, 15(2), 426–435.
    [Google Scholar]
  6. Chen, C., Wang, Y., Li, G. and Reynolds, A.C.
    [2010] Closed-loop reservoir management on the Brugge test case. Computational Geosciences, 14(4), 691–703.
    [Google Scholar]
  7. Chen, P., Villa, U. and Ghattas, O.
    [2019] Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty. Journal of Computational Physics, 385, 163–186.
    [Google Scholar]
  8. Chen, Y., Oliver, D.S. and Zhang, D.
    [2009] Efficient Ensemble-Based Closed-Loop Production Optimization. SPE Journal, 14(4), 634–645.
    [Google Scholar]
  9. Darlington, J., Pantelides, C.C., Rustem, B. and Tanyi, B.A.
    [1999] An algorithm for constrained nonlinear optimization under uncertainty. Automatica, 35(2), 217–228.
    [Google Scholar]
  10. [2000] Decreasing the sensitivity of open-loop optimal solutions in decision making under uncertainty. European Journal of Operational Research, 121(2), 343–362.
    [Google Scholar]
  11. Denney, D.
    [2010] Pros and cons of applying a proxy model as a substitute for full reservoir simulations. Journal of Petroleum Technology, 62(7), 634–645.
    [Google Scholar]
  12. van Doren, Jorn F. M. and Markovinović, R. and Jansen, J.D.
    [2006] Reduced-order optimal control of water flooding using proper orthogonal decomposition. Computational Geosciences, 10(1), 137–158.
    [Google Scholar]
  13. Durlofsky, L.J.
    [2005] Upscaling and gridding of fine scale geological models for flow simulation. In: 8th International Forum on Reservoir Simulation Iles Borromees.
    [Google Scholar]
  14. Fonseca, R.M., Chen, B., Jansen, J.D. and Reynolds, A.C.
    [2017] A Stochastic Simplex Approximate Gradient (StoSAG) for Optimization Under Uncertainty. International Journal for Numerical Methods in Engineering, 109(13), 1756–1776.
    [Google Scholar]
  15. Gaspar, A.T.F.S., MuÃśoz Mazo, E. and Schiozer, D.
    [2011] Case Study of the Structure of the Process for Production Strategy Selection. International Journal of Modeling and Simulation for the Petroleum Industry, 4.
    [Google Scholar]
  16. Gaspari, G. and Cohn, S.E.
    [1999] Construction of correlation functions in two and three dimensions. Quarterly Journal of the Royal Meteorological Society, 723–757.
    [Google Scholar]
  17. Hamming, R.
    [1980] Coding and Information Theory. Prentice-Hall.
    [Google Scholar]
  18. Hanea, R.G., Casanova, P., Hustoft, L., Bratvold, R.B., Nair, R., Hewson, C., Leeuwenburgh, O. and Fonseca, R.M.
    [2017] Drill and learn: A decision making workflow to quantify value of learning. In: SPE Reservoir Simulation Conference. Society of Petroleum Engineers.
    [Google Scholar]
  19. Hart, P.E., Nilsson, N.J. and Raphael, B.
    [1968] A formal basis for the heuristic determination of minimum cost paths. IEEE Transactions on Systems Science and Cybernetics, 4(2), 100–107.
    [Google Scholar]
  20. Jansen, J.D. and Durlofsky, L.J.
    [2016] Use of reduced-order models in well control optimization. Optimization and Engineering, 105–132.
    [Google Scholar]
  21. Jaro, M.A.
    [1989] Advances in Record-Linkage Methodology as Applied to Matching the 1985 Census of Tampa, Florida. Journal of the American Statistical Association, 84(406), 414–420.
    [Google Scholar]
  22. Jesmani, M., Jafarpour, B., Bellout, M.C. and Foss, B.
    [2020] A reduced random sampling strategy for fast robust well placement optimization. Journal of Petroleum Science and Engineering, 184, 106414.
    [Google Scholar]
  23. Kish, L.
    [1965] Survey sampling. John Wiley and Sons, Inc.
    [Google Scholar]
  24. Korf, R.E.
    [1990] Depth-limited search for real-time problem solving. Real-Time Systems, 2, 7–24.
    [Google Scholar]
  25. Lamas, L.F., Botechia, V.E., Schiozer, D.J. and Delshad, M.
    [2017] Optimization for drilling schedule of wells in the development of heavy oil reservoirs. Brazilian Journal of Petroleum and Gas, 11(3).
    [Google Scholar]
  26. Leeuwenburgh, O., Chitu, A.G., Nair, R., Egberts, P.J.P., Ghazaryan, L., Feng, T. and Hustoft, L.
    [2016] Ensemble-based methods for well drilling sequence and time optimization under uncertainty. In: ECMOR XV-15th European Conference on the Mathematics of Oil Recovery.
    [Google Scholar]
  27. Levenshtein, V.I.
    [1966] Binary codes capable of correcting deletions, insertions and reversals. Soviet Physics Doklady, 10(8), 707–710.
    [Google Scholar]
  28. Rahim, S. and Li, Z.
    [2015] Well Placement Optimization with Geological Uncertainty Reduction. IFAC-PapersOnLine, 48(8), 57–62. 9th IFAC Symposium on Advanced Control of Chemical Processes ADCHEM 2015.
    [Google Scholar]
  29. Ronald, S.
    [1998] More distance functions for order-based encodings. In: 1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence. 558–563.
    [Google Scholar]
  30. Schiavinotto, T. and Stützle, T.
    [2007] A review of metrics on permutations for search landscape analysis. Computers & Operations Research, 34(10), 3143–3153.
    [Google Scholar]
  31. Silva, V.L.S., Emerick, A.A., Couto, P. and Alves, J.L.D.
    [2017] History matching and production optimization under uncertainties—Application of closed-loop reservoir management. Journal of Petroleum Science and Engineering, 157, 860–874.
    [Google Scholar]
  32. Van Essen, G., Zandvliet, M., Van den Hof, P., Bosgra, O. and Jansen, J.D.
    [2009] Robust Waterflooding Optimization of Multiple Geological Scenarios. SPE Journal, 14(1), 202–210.
    [Google Scholar]
  33. Wang, L. and Oliver, D.S.
    [2019] Efficient Optimization of Well Drilling Sequence with Learned Heuristics. SPE Journal, 24(5), 2111–2134.
    [Google Scholar]
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