A general method to select representative models for decision making and optimization under uncertainty

The optimization of subsurface flow processes under geological uncertainty technically requires flow simulation to be performed over a large set of geological realizations for each function evaluation at every iteration of the optimizer. Because flow simulation over many permeability realizations (only permeability is considered to be uncertain in this study) may entail excessive computation, simulations are often performed for only a subset of 'representative' realizations. It is however challenging to identify a representative subset that provides flow statistics in close agreement with those from the full set, especially when the decision parameters (e.g., time-varying well pressures, well locations) are unknown a priori, as they are in optimization problems. In this work, we introduce a general framework, based on clustering, for selecting a representative subset of realizations for use in simulations involving 'new' sets of decision parameters. Prior to clustering, each realization is represented by a low-dimensional feature vector that contains a combination of permeability-based and flow-based quantities. Calculation of flow-based features requires the specification of a (base) flow problem and simulation over the full set of realizations. Permeability information is captured concisely through use of principal component analysis. By computing the difference between the flow response for the subset and the full set, we quantify the performance of various realization-selection methods. The impact of different weightings for flow and permeability information in the cluster-based selection procedure is assessed for a range of examples involving different types of decision parameters. These decision parameters are generated either randomly, in a manner that is consistent with the solutions proposed in global stochastic optimization procedures such as GA and PSO, or through perturbation around a base case, consistent with the solutions considered in pattern search optimization. We find that flow-based clustering is preferable for problems involving new well settings (e.g., time-varying well bottom-hole pressures) or small changes in well configuration, while both permeability-based and flow-based clustering provide similar results for (new) random multiwell configurations. We also investigate the use of efficient tracer-type simulations for obtaining flow-based features and demonstrate that this treatment performs nearly as well as full-physics simulations for the cases considered. The various procedures are applied to select realizations for use in production optimization under uncertainty, which greatly accelerates the optimization computations. Optimization performance is shown to be consistent with the realization-selection results for cases involving new decision parameters.