Skip to main content
SpringerLink
Log in

Uncertainty quantification of reservoir performance predictions using a stochastic optimization algorithm

  • Original Paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

The simultaneous perturbation stochastic approximation (SPSA) algorithm is modified to obtain a stochastic Gaussian search direction (SGSD) algorithm for automatic history matching. The search direction in the SGSD algorithm is obtained by simultaneously perturbing the reservoir model with unconditional realizations from a Gaussian distribution. This search direction has two nice properties: (1) it is always downhill in the vicinity of the current iterate and (2) the expectation of the stochastic search direction is an approximate quasi-Newton direction with a prior covariance matrix used as the approximate inverse Hessian matrix. For Gaussian reservoir models, we argue and demonstrate that the SGSD algorithm may generate more geologically realistic reservoir description than is obtained with the original SPSA algorithm. It is shown that the SGSD algorithm represents an approximation of the gradual deformation method but its search direction has the desirable properties that are lacking in all gradual deformation methods. The SGSD algorithm is successfully applied to the well-known PUNQ-S3 test case to generate a maximum a posteriori estimate and for uncertainty quantification of reservoir performance predictions using the randomized maximum likelihood method.

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. Aanonsen, S.I., Nævdal, G., Oliver, D.S., Reynolds, A.C., Vallès, B.: Review paper: ensemble Kalman filter to petroleum engineering. SPE J. 14(3), 393–412 (2009)

    Google Scholar 

  2. Bangerth, W., Klie, H. Wheeler, M. Stoffa, P., Sen, M.: On optimization algorithm for the reservoir oil well placement problem. Comput. Geosci. 10(3), 303–319 (2006)

    Article  MATH  Google Scholar 

  3. Barker, J.W., Cuypers, M., Holden, L.: Quantifying uncertainty in production forecasts: another look at the PUNQ-S3 problem. SPE J. 6(4), 433–441 (2001)

    Google Scholar 

  4. Bianco, A., Cominelli, A., Dovera, L., Nævdal, G., Vallès, B.: History matching and production forecast uncertainty by means of the ensemble Kalman filter: a real field application. In: SPE107161 in Proceedings of EAGE/EUROPEC Conference and Exhibition, London, UK (2007)

  5. Evensen, G.: Data Assimilation: The Ensemble Kalman Filter. Springer, Berlin (2007)

    MATH  Google Scholar 

  6. Evensen, G., Hove, J., Meisingset, H.C., Reiso, E., Seim, K.S., Espelid, Ø: Using the EnKF for assisted history matching of a North Sea reservoir. In: SPE-106184 in Proceedings of the SPE Reservoir Simulation Symposium (2007)

  7. Floris, F.J.T., Bush, M.D., Cuypers, M., Roggero, F., Syversveen, A.-R.: Methods for quantifying the uncertainty of production forecasts: a comparative study. Pet. Geosci. 7(SUPP), 87–96 (2001)

    Article  Google Scholar 

  8. Gao, G.: Data integration and uncertainty evaluation for large scale automatic history matching problems. Ph.D. thesis, University of Tulsa (2005)

  9. Gao, G., Reynolds, A.C.: An improved implementation of the LBFGS algorithm for automatic history matching. SPE J. 11(1), 5–17 (2006)

    Google Scholar 

  10. Gao, G., Zafari, M., Reynolds, A.C.: Quantifying uncertainty for the PUNQ-S3 problem in a Bayesian setting with RML and EnKF. SPE J. 11(4), 506–515 (2006)

    Google Scholar 

  11. Gao, G., Li, G., Reynolds, A.C.: A stochastic optimization algorithm for automatic history matching. SPE J. 12(2), 196–208 (2007)

    Google Scholar 

  12. Haugen, V., Natvik, L.-J., Evensen, G., Berg, A., Flornes, K., Nævdal, G.: History matching using the ensemble Kalman filter on a North Sea field case. In: SPE-102430 in Proceedings of the SPE Annual Technical Conference and Exhibition (2006)

  13. Hu, L.Y.: Gradual deformation and iterative calibration of Gaussian-related stochastic models. Math. Geol. 32(1), 87–108 (2000)

    Article  Google Scholar 

  14. Hu, L.Y., Blanc, G., Noetinger, B.: Gradual deformation and iterative calibration of sequential stochastic simulations. Math. Geol. 33(4), 475–489 (2001)

    Article  MATH  Google Scholar 

  15. Kitanidis, P.K.: Parameter uncertainty in estimation of spatial functions: Bayesian estimation. Water Resour. Res. 22(4), 499–507 (1986)

    Article  Google Scholar 

  16. Le Ravalec, M., Hu, L.Y., Noetinger, B.: Stochastic reservoir modeling constrained to dynamic data: local calibration and inference of the structural parameters. In: SPE-56556 in Proceedings of the SPE Annual Technical Conference and Exhibition, Houston, 3–6 October 1999

  17. Le Ravalec, M., Noeinger, B.: Optimization with the gradual deformation method. Math. Geol. 34(2), 125–142 (2002)

    Article  MATH  Google Scholar 

  18. Li, R., Reynolds, A.C., Oliver, D.S.: Sensitivity coefficients for three-phase flow history matching. J. Can. Pet. Technol. 42(4), 70–77 (2003)

    Google Scholar 

  19. Nævdal, G., Johnsen, L.M., Aanonsen, S.I., Vefring, E.H.: Reservoir monitoring and continuous model updating using ensemble Kalman Filter. SPE J. 10(1), 66–74 (2005)

    Google Scholar 

  20. Oliver, D.S., He, N., Reynolds, A.C.: Conditioning permeability fields to pressure data. In: European Conference for the Mathematics of Oil Recovery V (1996)

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

    Book  Google Scholar 

  22. Reynolds, A.C., He, N., Oliver, D.S.: Reducing uncertainty in geostatistical description with well testing pressure data. In: Schatzinger, R. A., Jordan J. F. (eds) Reservoir Characterization: Recent Advances, pp. 149–162. American Association of Petroleum Geologists, Tulsa (1999)

    Google Scholar 

  23. Reynolds, A.C., Zafari, M., Li, G.: Iterative forms of the ensemble Kalman Filter. In: Proceedings of 10th European Conference on the Mathematics of Oil Recovery, Amsterdam, 4–7 September 2006

  24. Rodrigues, J.R.P.: Calculating derivatives for automatic history matching. Comput. Geosci. 10, 119–136 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Spall, J.C.: A stochastic approximation technique for generating maximum likelihood parameter estimates. In: Proceedings of the American Control Conference, pp 1161–1167, Minneapolis, MN, 10–12 June 1987

  26. Spall, J.C.: Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans. Automat. Contr. 37, 332–341 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  27. Spall, J.C.: Implementation of the simultaneous perturbation algorithm for stochastic optimization. IEEE Trans. Aerosp. Electron. Syst. 34, 817–823 (1998)

    Article  Google Scholar 

  28. Tarantola, A.: Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation. Elsevier, Amsterdam (1987)

    MATH  Google Scholar 

  29. Thulin, K., Li, G., Aanonsen, S.I., Reynolds, A.C.: Estimation of initial fluid contacts by assimilation of production data with EnKF. In: SPE-109975 in Proceedings of the SPE Annual Technical Conference and Exhibition (2007)

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

    Google Scholar 

  31. Wang, Y., Li, G., Reynolds, A.C.: Estimation of depths of fluid contacts by history matching using iterative ensemble Kalman smoothers. SPE J. 15(2), 509–525 (2010)

    Google Scholar 

  32. Zhang, F., Reynolds, A.C.: Optimization algorithms for automatic history matching of production data. In: Proceedings of the European Conference on the Mathematics of Oil Recovery (2002)

  33. Zhang, F., Skjervheim, J.A., Reynolds, A.C., Oliver, D.S.: Automatic history matching in a Bayesian framework: example applications. SPE Res. Eval. Eng. 8(3), 214–223 (2005)

    Google Scholar 

  34. Zhao, Y., Reynolds, A.C., Li, G.: Generating facies maps by assimilating production data and seismic data with the ensemble Kalman filter. In: SPE-113990 in Proceedings of the 2008 SPE Improved Oil Recovery Symposium, Tulsa, 20–23 April 2008

Download references

Author information

Authors and Affiliations

  1. Department of Petroleum Engineering, University of Tulsa, 800 S. Tucker Dr., Tulsa, OK, 74104, USA

    Gaoming Li & Albert C. Reynolds

Authors
  1. Gaoming Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Albert C. Reynolds
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gaoming Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, G., Reynolds, A.C. Uncertainty quantification of reservoir performance predictions using a stochastic optimization algorithm. Comput Geosci 15, 451–462 (2011). https://doi.org/10.1007/s10596-010-9214-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-010-9214-2

Keywords

Search

Navigation