Use of reduced-order models in well control optimization

Many aspects of reservoir management can be expected to benefit from the application of computational optimization procedures. The focus of this review paper is on well control optimization, which entails the determination of well settings, such as flow rates or bottom hole pressures, that maximize a particular objective function. As is the case with most reservoir-related optimizations, this problem is in general computationally demanding since function evaluations require reservoir simulation runs. Here we describe reduced-order modeling procedures, which act to accelerate these simulation runs, and discuss their use within the context of well control optimization. The techniques considered apply proper orthogonal decomposition (POD), which enables the representation of reservoir states (e.g., pressure and saturation in every grid block) in terms of a highly reduced set of variables. Two basic approaches are described-the direct application of POD-based reduction at each Newton iteration, and a trajectory piecewise linearization (POD-TPWL) procedure that applies POD to a linearized representation of the governing equations. Both procedures require one or more pre-processing 'training' simulation runs using the original full-order model. The use of both gradient-based optimization methods (including adjoint procedures) and direct search approaches with reduced-order models is described. Several concepts relevant to the general topic, including adjoint formulations and controllability, are also reviewed. Numerical results are presented for both approaches. In particular, the POD-TPWL procedure is applied to a computationally demanding bi-objective optimization problem, where it is shown to provide reasonable accuracy and a high degree of speedup.