Simultaneous and Sequential Estimation of Optimal Placement and Controls of Wells With a Covariance Matrix Adaptation Algorithm

Abstract

Summary In this paper, we present both simultaneous and sequential algorithms for the joint optimization of well trajectories and their life-cycle controls. The trajectory of a well is parameterized in terms of six variables that define a straight line in three dimensions. In the simultaneous joint optimization algorithm, the set of controls of a well throughout the life cycle of the reservoir is constructed as a linear combination of the left singular vectors that correspond to the largest singular values of a specified temporal covariance matrix. This covariance matrix is used to impose a temporal correlation on the controls at each well. In this approach, well controls are parameterized in terms of a few optimization parameters to reduce the dimension of the joint optimization problem. Moreover, the imposed smoothness on the well controls will result in temporally smooth well controls. We use an implementation of the covariance matrix adaptation–evolution strategy (CMA-ES) optimization algorithm to solve the defined optimization problem. In the sequential optimization algorithm, first, the trajectories of the wells are optimized with the CMA-ES optimization algorithm whereas the controls of the wells are prespecified. After the optimum trajectories of the wells are obtained, the life-cycle production optimization step is performed to find the optimal well controls for the specified well trajectories. For the production optimization step, we compare the performance of three optimization algorithms that are the standard ensemble-based optimization algorithm (EnOpt), the standard CMA-ES algorithm, and a variant of the CMA-ES algorithm in which we set the initial covariance matrix equal to a prespecified covariance that imposes a temporal correlation on the controls of each well. The performance of the proposed algorithms is tested for the joint optimization of well trajectories and controls of injectors and producers for the PUNQ reservoir model. The proposed simultaneous well placement/well control optimization algorithm obtained better results than did the sequential optimization framework. The CMA-ES algorithm performed well for both well placement and production optimization purposes. Moreover, the CMA algorithm with a prespecified covariance that imposes a temporal correlation on the well controls obtained a higher net present value compared with EnOpt for the life-cycle production optimization step of the sequential framework.

A Stochastic Simplex Approximate Gradient (StoSAG) for optimization under uncertainty

“…In EnOpt, u is considered to be a random vector with a Gaussian distribution centered at u ℓ and covariance matrix , that is, at iteration ℓ , . This notation indicates that it is possible to generate a new covariance matrix, for example, by covariance matrix adaptation . For our purposes, however, it suffices to consider the covariance matrix as fixed so we assume throughout that at iteration ℓ .…”

Recent Developments in Closed-Loop Approaches for Real-Time Mining and Petroleum Extraction

“…There is also an increasing number of studies involving the joint optimization of well controls and locations (e.g. Bailey et al 2005, Isebor et al 2014bForouzanfar and Reynolds, 2014;Humphries and Haynes, 2015;Forouzanfar et al 2016). Finally there are some studies that address comprehensive field development planning optimization including, e.g., surface facilities, artificial lift options and drilling schedules (e.g.…”

Development Favorable Area and Productivity Potential Evaluation Method of a Tight Oil Reservoir

“…Common examples of gradient-based algorithms are the adjoint algorithm and simultaneous perturbation stochastic approximation [9,11,12]. Evolutionary algorithms, evolution strategy, and swarm algorithms are among the three commonly used derivative-free algorithms [13][14][15][16][17][18][19]. Different approaches have been proposed to optimize well placement, including statistical methods, reservoir engineering methods, and proxy models [10,20,21].…”