Ensemble-based stochastic gradient methods, such as the ensemble optimization (EnOpt) method, the simplex gradient (SG) method, and the stochastic simplex approximate gradient (StoSAG) method, approximate the gradient of an objective function using an ensemble of perturbed control vectors. These methods are increasingly used in solving reservoir optimization problems because they are not only easy to parallelize and couple with any simulator but also computationally more efficient than the conventional finite-difference method for gradient calculations. In this work, we show that EnOpt may fail to achieve sufficient improvement of the objective function when the differences between the objective function values of perturbed control variables and their ensemble mean are large. On the basis of the comparison of EnOpt and SG, we propose a hybrid gradient of EnOpt and SG to save on the computational cost of SG. We also suggest practical ways to reduce the computational cost of EnOpt and StoSAG by approximating the objective function values of unperturbed control variables using the values of perturbed ones. We first demonstrate the performance of our improved ensemble schemes using a benchmark problem. Results show that the proposed gradients saved about 30–50% of the computational cost of the same optimization by using EnOpt, SG, and StoSAG. As a real application, we consider pressure management in carbon storage reservoirs, for which brine extraction wells need to be optimally placed to reduce reservoir pressure buildup while maximizing the net present value. Results show that our improved schemes reduce the computational cost significantly.
Bibliographical noteFunding Information:
Funding. HJ was supported by the National Research Foundation of Korea (NRF) under grant number 2018R1C1B5045260, and the Korea Institute of Geoscience and Mineral Resources(KIGAM) and the Ministry of Science, ICT and Future Planning of Korea under grant number GP2020-006. AS was supported by the U.S. Department of Energy, National Energy Technology Laboratory (NETL) under grant number DE-FE0026515. BM was supported by the National Research Foundation of Korea (NRF) under grant numbers 2018R1A6A1A08025520 and 2019R1C1C1002574.
© Copyright © 2020 Jeong, Sun, Jeon, Min and Jeong.
- active pressure management
- ensemble optimization
- hybrid simplex gradient
- simplex gradient
- stochastic gradient
- stochastic simplex approximate gradient