Noname manuscript No. (will be inserted by the editor) Well Control Optimization using Derivative-Free Algorithms and a Multiscale Approach Xiang Wang · Ronald D. Haynes · Qihong Feng Received: date / Accepted: date Abstract Smart well technologies, which allow remote tion (PSO) and covariance matrix adaptation evolution control of well and production processes, make the prob- strategy (CMA-ES). These three algorithms encompass lem of determining optimal control strategies a timely the breadth of available black{box optimization strate- endeavour. In this paper, we use numerical optimiza- gies: deterministic local search, stochastic global search tion algorithms and a multiscale approach in order to and stochastic local search. In addition, we hybridize find an optimal well management strategy over the life the three derivative-free algorithms with a multiscale of the reservoir. Optimality is measured in terms of the regularization approach. Starting with a reasonably small values of the net present value objective function. The number of control steps, the control intervals are sub- large number of well rates for each control step make sequently refined during the optimization. Results for the optimization problem more difficult and at a high experiments studied indicate that CMA-ES performs risk of achieving a suboptimal solution. Moreover, the best among the three algorithms in solving both small optimal number of adjustments is not known a priori. and large scale problems. When hybridized with a mul- Adjusting well controls too frequently will increase un- tiscale regularization approach, the ability to find the necessary well management and operation cost, and an optimal solution is further enhanced, with the perfor- excessively low number of control adjustments may not mance of GPS improving the most. Topics affecting the be enough to obtain a good yield. We investigate three performance of the multiscale approach are discussed derivative-free optimization algorithms, chosen for their in this paper, including the effect of control frequency robust and parallel nature, to determine optimal well on the well control problem. The parameter settings control strategies. The algorithms chosen include gener- for GPS, PSO, and CMA-ES, within the multiscale ap- alized pattern search (GPS), particle swarm optimiza- proach are considered. Keywords Well Control · Production Optimization · X. WANG Derivative-Free Algorithms · Multiscale Approach Department of Petroleum Engineering, China University of Petroleum (Huadong), Qingdao, Shandong, China Department of Mathematics & Statistics, Memorial Univer- 1 Introduction arXiv:1509.04693v1 [math.OC] 15 Sep 2015 sity, St. John's, NL, Canada Tel.: +86.132.8085.6026 E-mail: [email protected] Determining the well production and injection rates is R.D. Haynes of paramount importance in modern reservoir develop- Department of Mathematics & Statistics, Memorial Univer- ment. The decision is difficult since the optimal rates sity, St. John's, NL, Canada depend on the heterogeneity of the rock and liquids, Tel.: +1.709.864.8825 the well placements and other parameters. Indeed, these Fax: +1.709.864.3010 E-mail: [email protected] properties and input parameters are coupled in a highly nonlinear fashion. Moreover, the optimal production Q. Feng Department of Petroleum Engineering, China University of and injection rates are usually not constant throughout Petroleum (Huadong), Qingdao, Shandong, China the life cycle of reservoir. The oil saturation distribu- E-mail: [email protected] tion changes during the well injection and production 2 Xiang Wang et al. processes. This will then affect the optimal production dition to local convergence. The performance of MCS and injection rate for each well. for well placement and control optimization is discussed Well control planning can be formulated as an op- in [43]. timization problem, using economic or cumulative oil The optimization algorithms mentioned above can production as the objective function. The well rates or be further classified as either stochastic or determinis- bottom hole pressures at different times are the opti- tic. Stochastic methods use information from the pre- mization variables. Many optimization algorithms have vious iteration and a random element to generate new been investigated to solve such problems. These algo- search points. The random component of the algorithm rithms can be broadly placed in two categories: derivative- makes it more likely to avoid local optima, but it may based algorithms and derivative-free algorithms [25]. also make the control of solution quality difficult, espe- Derivative-based or gradient-based algorithms, take cially with a limited computational budget. The stochas- advantage of the gradient information to guide their tic algorithms from the above list are MADS, CMA-ES, search. This type of algorithm, commonly used in well GA, PSO, and DE. Deterministic methods have no ran- control optimization, includes steepest ascent [42], con- dom element. For a given problem, deterministic meth- jugate gradient [2], and sequential quadratic program- ods will give the same results for each trial. GPS, HJDS, ming methods [25]. Gradients of the objective function and MCS are examples of deterministic algorithms. may be calculated by using an adjoint equation. This is All the above mentioned algorithms have been used an invasive approach, requiring a detailed knowledge in well control optimization and/or well placement opti- of mathematics inside the reservoir simulator [25,8]. mization problems. The performance of the algorithms Other ways to approximate the gradients include meth- are problem-dependent. Some of the algorithms, like ods such as finite difference perturbation [2], or the si- GPS (1960s), GA (1960s), and PSO (1990s), have been multaneous perturbation stochastic approximation [42]. around for decades, and have been used in petroleum These algorithms assume a certain degree of smooth- industrial problems for a relatively long time. People ness of the objective function with respect to the opti- has accumulated a great deal of experience through mization variables. Derivative-based algorithms are po- case studies. CMA-ES, developed in 2000s [19], was tentially very quick to converge but sometimes fall into first used in petroleum related optimization problems local optimal. only in 2012 [7]. Though CMA-ES showed great perfor- Derivative-free algorithms can be subdivided into mance in solving well placement optimization problem local search methods and global search methods. Lo- [7]; to date there has been little work to apply it in well cal derivative-free algorithms include generalized pat- control problems. tern search (GPS) [23], mesh adaptive direct search Although many optimization algorithms have been (MADS) [25], Hooke-Jeeves direct search (HJDS) [25], used, well control optimization is still challenging prob- ensemble-based optimization (EnOpt) [32], covariance lem and an active area of research. The number of op- matrix adaptation evolution strategy (CMA-ES) [7,31], timization variables is large in many real{life scenar- and so on. These methods have strong ability to find ac- ios. The required number of function evaluations will curate optima in a local space, but may face with some rise sharply with the increase of the number of vari- difficulties in finding global optima, especially when a ables. A single function evaluation requires one reser- good initial guess is not available. Global derivative- voir simulation which is often very demanding in terms free algorithms search through the entire space and of CPU time. The non-convex, non-smooth and multi- provide techniques to avoid being trapped in local op- modal objective surface further increases the optimiza- tima. Examples of global search algorithms include ge- tion difficulty. netic algorithms (GAs) [3], particle swarm optimization It is difficult to find a reasonable frequency for well (PSO) [26], and differential evolution (DE) [39,9]. Al- control; an excessively low number of control adjust- though these algorithms are robust and easy to use, ments may not truly optimize oil recovery. Adjusting they often require more function evaluations than lo- each well control too frequently imposes an unrealistic cal search and derivative-based algorithms to converge. control burden on operations, increasing the total well However, most of these algorithms parallelize naturally management cost. Moreover, imposing a high number and easily, which make their efficiency satisfactory [10]. of control adjustments increases the complexity of the Recently, some hybridization of these techniques such optimization problem so much that there is a high risk as PSO-MADS[23,26,22], multilevel coordinate search of optimization algorithms becoming trapped at local (MCS) [24], etc., are developed and applied in well optima and hence missing the optimal startegy [38]. placement and/or well control optimization problems. Multiscale regularization approaches are developed to These methods provide global search capabilities in ad- address these problems. The main idea of the multi- Multiscale Approach 3 scale approach is to start the optimization process with • The effects of control frequency on well control op- a very coarse control frequency (and thus, with a small timization, including the effects on the production number of control variables) and refine successively. and the effects on the control strategy. The solution at the coarse-scale is used as the initial • The performance