A new hierarchical multiscale optimization method: Gradient and non-gradient approaches for waterflooding optimization

This work proposes new optimization techniques, either gradient-based or derivative-free, to improve the performance of the production optimization step in the closed-loop reservoir management framework in terms of computational cost and the ultimate objective function result. The concept of closed-loop reservoir management has been largely applied in the reservoir engineering literature for the last decade. Composed of two main steps, data assimilation and optimal well control, closed-loop reservoir management seeks to attain optimum decisions on how to develop a field, considering the best knowledge of the reservoir's uncertainty. Although both steps in closed-loop reservoir management are important, this work focuses on techniques to solve the life-cycle production optimization problem.;Typically, methods based on the gradient of the objective function are more efficient in terms of both the computational cost and the ultimate objective function value. However, the limited capability for computing gradients in commercial reservoir simulators and the difficulty in implementation of adjoint codes lead to a growing interest in optimization algorithms which do not require accurate gradient information. Thus, we propose a method based on function models that vaguely mimic Newton's method, but since neither the gradient nor the Hessian of the objective function are available, a quadratic interpolation model is built from a set of interpolation points and then minimized using a trust region method. Our methodology is inspired by the ideas underlying the New Unconstrained Optimization Algorithm NEWUOA proposed by Powell [88] and extends the algorithm of Zhao et al. [122]. We propose and implement features to control and maintain the quality of the geometric aspect of the interpolation set and the quadratic model, which are neglected in the work of Zhao et al. [122]. We also derive a methodology which does not use any gradient approximation but still is based on a quadratic interpolation model and, unlike NEWUOA, improves the objective function from the first iteration onwards. Our methodology has been tested with success for two problems: the well-known Rosenbrock test function and a synthetic channelized reservoir.;Life-cycle optimization problems present two key aspects related to the dimensionality of a production optimization problem and the practical preference for temporal smoothness in the controls at each well. In this work, we propose to tackle both aspects, dimensionality and smoothness, by using new multiscale techniques. We propose a multiscale estimation technique for adaptively selecting the lengths of control steps as the overall optimization proceeds, without necessarily requiring correlation between well controls to provide smoothness. The well controls can be coarsened or refined, which allows the method to automatically defines the appropriated level of refinement. Two refining strategies are implemented: first, the controls are evenly split into a predetermined number of new controls; secondly, we refine based on the so-called refinement indicators, as presented by Chavent and Bissell [21], Ben Ameur et al. [11], and Lien et al. [68].In addition, our proposed methods can be applied to either gradient-based or derivative-free methods.;We have successively applied our multiscale techniques to synthetic reservoir problems and to a real field case and compared against other approaches presented in the literature, considering problems with only bounds on well controls, or, in addition to the bound constraints, we also consider linear inequality constraints. Our multiscale technique based on the refinement indicators has also been successfully applied to production optimization problems under geological uncertainty for two problems with an ensemble-based representation of the uncertainty where the expected value of the net present value is maximized.