The Use of the Least Squares Probabilistic Collocation Method in Decision Making in the Presence of Uncertainty for Chemical EOR Processes
Authors: Alkhatib A. M. , King P. R. 
Published: October 2014

Abstract
The Least Squares Monte Carlo method is a decision evaluation method that can capture the value of flexibility of a process. This method was shown to provide us with some insight into the effect of uncertainty on decision making and to help us capture the upside potential or mitigate the downside effects for a chemical EOR process. The method is a stochastic approximate dynamic programming approach to decision making. It is based on a forward simulation coupled with a recursive algorithm which produces the near-optimal policy. It relies on Monte Carlo simulation to produce convergent results. This incurs a significant computational requirement when using this method to evaluate decisions for reservoir engineering problems because this requires running many reservoir simulations.
The objective of this study was to enhance the performance of the Least Squares Monte Carlo method by improving the sampling method used to generate the technical uncertainties used in producing the production profiles. The probabilistic collocation method has been proven to be a robust and efficient uncertainty quantification method. It approximates the random input distributions using polynomial chaos expansions and produces a proxy polynomial for the output parameter requiring a limited number of model responses that is conditional on the number of random inputs and the order of the approximation desired. The resulting proxy can then be used to generate the different statistical moments with negligible computational requirement. By using the sampling methods of the probabilistic collocation method to approximate the sampling of the technical uncertainties, it is possible to significantly reduce the computational requirement of running the decision evaluation method. Thus we introduce the least square probabilistic collocation method.
Both methods are then applied to surfactant-polymer flooding problems using a number of stylized reservoir models. The technical uncertainties for the surfactant-polymer flooding process considered were the residual oil saturation to chemical flooding, surfactant and polymer adsorption and the viscosity multiplier of the polymer. The economic uncertainties considered were the oil price and the surfactant and polymer price. Both methods were applied using three reservoir case studies: a simple homogeneous model, the PUNQ-S3 model and a modified portion of the SPE10 model. The results show that using the sampling techniques of the probabilistic collocation method produced relatively accurate responses compared with the original method. For instance, it was possible to produce the same output for the modified SPE10 model by using the second order quadrature nodes, 81 realizations, rather than the 103 realizations used for the least squares Monte Carlo method, thus achieving an order of magnitude reduction in computational time.
Different possible enhancements were discussed in order to practically adapt the least squares probabilistic collocation method to more realistic and complex reservoir models. The application was extended to other chemical EOR processes such as alkaline-surfactant-polymer flooding and polymer flooding. Uncertainty in heterogeneity was also incorporated using Gaussian and multiple point statistically generated permeability fields. The results show that using the least squares probabilistic collocation method produced accurate results when compared with the least squares Monte Carlo method. This was applied to polymer, surfactant-polymer and alkaline-surfactant-polymer flooding case studies.