A Theoretical look at Ensemble-Based Optimization in Reservoir Management
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A Theoretical look at Ensemble-Based Optimization in Reservoir Management Andreas S. Stordal1 · Slawomir P. Szklarz2 · Olwijn Leeuwenburgh3
Received: 18 June 2014 / Accepted: 7 May 2015 / Published online: 18 June 2015 © International Association for Mathematical Geosciences 2015
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.
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Andreas S. Stordal [email protected]
1
International Research Institute of Stavanger, P.O. Box 8046, 4068 Stavanger, Norway
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Institute of Applied Mathematics, Delft University of Technology, 2600 GA Delft, The Netherlands
3
TNO, PO Box 800015, 3508 TA Utrecht, The Netherlands
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Math Geosci (2016) 48:399–417
Keywords Ensemble optimization · Production optimization · Robust optimization · Natural evolution · Gaussian mutation
1 Introduction Reservoir management could be defined as the collection of activities aimed at making decisions on field development including operating strategies and placement of wells. In order to make these decisions with confidence, one has to be able to make reliable predictions of the consequences of such decisions, and to quantify the associated risks. The basis for this is a good description of the reservoir, as captured by a numerical flow model. This model should be able to reproduce the past production history with acceptable accuracy given the uncertainty in the data (which is achieved through the process of history matching). If this condition is met, the model is generally assumed to be capable of capturing the future behavior of
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