Model Order Reduction and Control Polynomial Approximation for Well-Control Production Optimization

Abstract The objective of this paper is to reduce the computational effort in reservoir flooding optimization problems by a combination of different optimization parametrization methods and model order reduction techniques. We compare three different parametrization methods that reduce the cardinality of the original infinite set of control-decision variables to a finite set. The three methods include a traditional piece-wise constant (PWC) approximation, a polynomial approximation by Chebyshev orthogonal polynomials and a piece-wise polynomial approximation by cubic Spline interpolation. The Proper Orthogonal Decomposition with Discrete Empirical Interpolation Method (POD-DEIM) accomplishes the reduced order modeling (ROM).. We compare a gradient-free global stochastic search approach and a gradient-based local search approach. We used Particle Swarm Optimization (PSO) as a gradient-free algorithm and Interior-Point Optimization (IPOPT) with L-BFGS method as a gradient-based algorithm. First, we compare the performances of the three parametrization methods solved by each optimizer, using fine scale simulations for an increasing level of parametrization refinement. Then, in the second part of this paper, we combine the parametrization methods with the reduced modeling workflow. For a given level of parametrization refinement, we compare the performance of each parametrization method coupled with POD-DEIM, and solved by each optimizer. In this part, we introduce an online training procedure, where the first optimization iteration is used to construct the snapshot matrix. The results demonstrate how refining the control approximation with more decision variables per well lead to better NPV values, but with a higher computational cost. The best NPV was achieved using the highest refining level with Chebyshev polynomial approximation. Both polynomial and piece-wise polynomial approximations served as better training sets for POD-DEIM leading to a more accurate and fast reduced model. With the strategy proposed, POD-DEIM showed the best optimization accuracy for Chebyshev polynomial with the gradient-free optimizer, thus permitting the use of the model reduction methodology for global-stochastic search methods. However, the gradient-based approach seems to consistently outperform the gradient-free approach in terms of NPV and number of iterations for the cases shown.

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