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

In this paper, we present both simultaneous and sequential algorithms for the joint optimization of well trajectories and their lifecycle 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.