Summary
The popularity of intelligent wells (I-wells), which provide layer-by-layer monitoring and control capability of production and injection, is growing. However, the number of available techniques for optimal control of I-wells is limited (Sarma et al. 2006; Alghareeb et al. 2009; Almeida et al. 2010; Grebenkin and Davies 2012). Currently, most of the I-wells that are equipped with interval control valves (ICVs) are operated to enhance the current production and to resolve problems associated with breakthrough of the unfavorable phase. This reactive strategy is unlikely to deliver the long-term optimum production. On the other side, the proactive-control strategy of I-wells, with its ambition to provide the optimum control for the entire well's production life, has the potential to maximize the cumulative oil production. This strategy, however, results in a high-dimensional, nonlinear, and constrained optimization problem.
This study provides guidelines on selecting a suitable proactive optimization approach, by use of state-of-the-art stochastic gradient-approximation algorithms. A suitable optimization approach increases the practicality of proactive optimization for real field models under uncertain operational and subsurface conditions. We evaluate the simultaneous-perturbation stochastic approximation (SPSA) method (Spall 1992) and the ensemble-based optimization (EnOpt) method (Chen et al. 2009). In addition, we present a new derivation of the EnOpt by use of the concept of directional derivatives.
The numerical results show that both SPSA and EnOpt methods can provide a fast solution to a large-scale and multiple I-well proactive optimization problem. A criterion for tuning the algorithms is proposed and the performance of both methods is compared for several test cases. The used methodology for estimating the gradient is shown to affect the application area of each algorithm. SPSA provides a rough estimate of the gradient and performs better in search environments, characterized by several local optima, especially with a large ensemble size. EnOpt was found to provide a smoother estimation of the gradient, resulting in a more-robust algorithm to the choice of the tuning parameters, and a better performance with a small ensemble size. Moreover, the final optimum operation obtained by EnOpt is smoother. Finally, the obtained criteria are used to perform proactive optimization of ICVs in a real field.
Solutions of the equation of helmholtz in an angle with vanishing directional derivatives along each side. II
