Mathematical Geosciences

, Volume 48, Issue 4, pp 399–417 | Cite as

A Theoretical look at Ensemble-Based Optimization in Reservoir Management

  • Andreas S. Stordal
  • Slawomir P. Szklarz
  • Olwijn Leeuwenburgh
Article
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Abstract

Ensemble-based optimization has recently received great attention as a potentially powerful technique for life-cycle production optimization, which is a crucial element of reservoir management. Recent publications have increased both the number of applications and the theoretical understanding of the algorithm. However, there is still ample room for further development since most of the theory is based on strong assumptions. Here, the mathematics (or statistics) of Ensemble Optimization is studied, and it is shown that the algorithm is a special case of an already well-defined natural evolution strategy known as Gaussian Mutation. A natural description of uncertainty in reservoir management arises from the use of an ensemble of history-matched geological realizations. A logical step is therefore to incorporate this uncertainty description in robust life-cycle production optimization through the expected objective function value. The expected value is approximated with the mean over all geological realizations. It is shown that the frequently advocated strategy of applying a different control sample to each reservoir realization delivers an unbiased estimate of the gradient of the expected objective function. However, this procedure is more variance prone than the deterministic strategy of applying the entire ensemble of perturbed control samples to each reservoir model realization. In order to reduce the variance of the gradient estimate, an importance sampling algorithm is proposed and tested on a toy problem with increasing dimensionality.

Keywords

Ensemble optimization Production optimization Robust optimization Natural evolution Gaussian mutation 
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Notes

Acknowledgments

The first author acknowledges the Research Council of Norway and the industrial participants, ConocoPhillips Skandinavia AS, BP Norge AS, Det Norske Oljeselskap AS, Eni Norge AS, Maersk Oil Norway AS, DONG Energy AS, Denmark, Statoil Petroleum AS, GDF SUEZ E&P NORGE AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall Norge AS, of The National IOR Centre of Norway for financial support.

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Copyright information

© International Association for Mathematical Geosciences 2015

Authors and Affiliations

  • Andreas S. Stordal
    • 1
  • Slawomir P. Szklarz
    • 2
  • Olwijn Leeuwenburgh
    • 3
  1. 1.International Research Institute of StavangerStavangerNorway
  2. 2.Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands
  3. 3.TNOUtrechtThe Netherlands

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