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Decision-theoretic sensitivity analysis for reservoir development under uncertainty using multilevel quasi-Monte Carlo methods

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Abstract

At various stages of petroleum reservoir development, we encounter a large degree of geological uncertainty under which a rational decision has to be made. In order to identify which parameter or group of parameters significantly affects the output of a decision model, we investigate decision-theoretic sensitivity analysis and its computational issues in this paper. In particular, we employ the so-called expected value of partial perfect information (EVPPI) as a sensitivity index and apply multilevel Monte Carlo (MLMC) methods to efficient estimation of EVPPI. In a recent paper by Giles and Goda, an antithetic MLMC estimator for EVPPI is proposed and its variance analysis is conducted under some assumptions on a decision model. In this paper, for an improvement on the performance of the MLMC estimator, we incorporate randomized quasi-Monte Carlo methods within the inner sampling, which results in an multilevel quasi-Monte Carlo (MLQMC) estimator. We apply both the antithetic MLMC and MLQMC estimators to a simple waterflooding decision problem under uncertainty on absolute permeability and relative permeability curves. Through numerical experiments, we compare the performances of the MLMC and MLQMC estimators and confirm a significant advantage of the MLQMC estimator.

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References

  1. Bickel, J.E., Gibson, R.L., McVay, D.A., Pickering, S., Waggoner, J.: Quantifying the reliability and value of 3D land seismic. SPE Res. Eval. Eng. 11, 832–841 (2008)

    Google Scholar 

  2. Bratvold, R.B., Bickel, J.E., Lohne, H.P.: Value of information in the oil and gas industry: past, present, and future. SPE Res. Eval. Eng. 12, 630–638 (2009)

    Google Scholar 

  3. Brennan, A., Kharroubi, S., O’Hagan, A., Chilcott, J.: Calculating partial expected value of perfect information via Monte Carlo sampling algorithms. Med. Decis. Making 27, 448–470 (2007)

    Article  Google Scholar 

  4. Burkholder, D.L., Davis, B., Gundy, R.F.: Integral inequalities for convex functions of operators on martingales. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 223–240. University of California Press, Berkeley (1972)

  5. Dick, J.: Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands. Ann. Stat. 39, 1372–1398 (2011)

    Article  Google Scholar 

  6. Dick, J., Pillichshammer, F.: Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  7. Giles, M.: Multilevel Monte Carlo path simulation. Oper. Res. 56, 607–617 (2008)

    Article  Google Scholar 

  8. Giles, M.: Multilevel Monte Carlo methods. Acta Numer. 24, 259–328 (2015)

    Article  Google Scholar 

  9. Giles, M.B., Goda, T.: Decision-making under uncertainty: using MLMC for efficient estimation of EVPPI. arXiv:1708.05531 (2017)

  10. Giles, M., Szpruch, L.: Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation. Ann. Appl. Prob. 24, 1585–1620 (2014)

    Article  Google Scholar 

  11. Goda, T., Dick, J.: Construction of interlaced scrambled polynomial lattice rules of arbitrary high order. Found. Comput. Math. 15, 1245–1278 (2015)

    Article  Google Scholar 

  12. Heath, A., Manolopoulou, I., Baio, G.: Estimating the expected value of partial perfect information in health economic evaluations using integrated nested Laplace approximation. Stat. Med. 35, 4264–4280 (2016)

    Article  Google Scholar 

  13. Heinrich, S.: Monte Carlo complexity of global solution of integral equations. J. Complexity 14, 151–175 (1998)

    Article  Google Scholar 

  14. Howard, R.A.: Information value theory. IEEE Trans. Sys. Sci. Cybern. 2, 22–26 (1966)

    Article  Google Scholar 

  15. Matoušek, J.: On the L 2 discrepancy for anchored boxes. J. Complexity 14, 527–556 (1998)

    Article  Google Scholar 

  16. Nakayasu, M., Goda, T., Tanaka, K., Sato, K.: Evaluating the value of single-point data in heterogeneous reservoirs with the expectation-maximization algorithm. SPE Econ. Manage. 8, 1–10 (2016)

    Article  Google Scholar 

  17. Niederreiter, H.: Random number generation and quasi-Monte Carlo methods. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63, SIAM, Philadelphia (1992)

  18. Oakley, J.E.: Decision-theoretic sensitivity analysis for complex computer models. Technometrics 51, 121–129 (2009)

    Article  Google Scholar 

  19. Oakley, J.E., Brennan, A., Tappenden, P., Chilcott, J.: Simulation sample sizes for Monte Carlo partial EVPI calculations. J. Health Econ. 29, 468–477 (2010)

    Article  Google Scholar 

  20. Owen, A.B.: Randomly permuted (t, m, s)-nets and (t, s)-sequences. In: Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P. J.-S. Shiue, Eds.), pp. 299–317, Springer, New York (1995)

  21. Owen, A.B.: Scrambling Sobol’ and Niederreiter-Xing points. J. Complexity 14, 466–489 (1998)

    Article  Google Scholar 

  22. Owen, A.B.: Local antithetic sampling with scrambled nets. Ann. Stat. 25, 2319–2343 (2008)

    Article  Google Scholar 

  23. Sloan, I.H., Joe, S.: Lattice methods for multiple integration. Oxford University Press, Oxford (1994)

    Google Scholar 

  24. Strong, M., Oakley, J.E., Brennan, A.: Estimating multiparameter partial expected value of perfect information from a probabilistic sensitivity analysis sample: a nonparametric regression approach. Med. Decis. Making 34, 311–326 (2014)

    Article  Google Scholar 

  25. Welton, N.J., Ades, A.E., Caldwell, D. M., Peters, T.J.: Research prioritization based on expected value of partial perfect information: a case-study on interventions to increase uptake of breast cancer screening. J. R. Stat. Soc. Series A 171, 807–841 (2008)

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank Jotaro Tomoeda, Eiji Fujisawa, Hirobumi Shimano, and Makoto Ogushi of JX NOEX for helpful discussions and comments, and CMG Ltd. for the software license. The first named author would like to thank Prof. Micheal B. Giles of the University of Oxford and Dr. Howard Thom of the University of Bristol for useful discussions.

Funding

This work is financially supported by JX Nippon Oil and Gas Exploration Corporation (JX NOEX). The work of the first named author is supported by JSPS Grant-in-Aid for Young Scientists No.15K20964 and Arai Science and Technology Foundation.

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Authors and Affiliations

  1. School of Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

    Takashi Goda, Daisuke Murakami, Kei Tanaka & Kozo Sato

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  1. Takashi Goda
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  2. Daisuke Murakami
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  3. Kei Tanaka
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  4. Kozo Sato
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Correspondence to Takashi Goda.

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Goda, T., Murakami, D., Tanaka, K. et al. Decision-theoretic sensitivity analysis for reservoir development under uncertainty using multilevel quasi-Monte Carlo methods. Comput Geosci 22, 1009–1020 (2018). https://doi.org/10.1007/s10596-018-9735-7

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  • DOI: https://doi.org/10.1007/s10596-018-9735-7

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