The purpose of history matching is to achieve geological realizations calibrated to the historical performance of the reservoir. For complex geological structures it is usually intractable to run tens of thousands of full reservoir simulation to trace the most probable geological model. Hence the inadequacy of the history-matching results frequently leads to poor estimation of the true model and high uncertainty in production forecasting. Reduced-order modeling procedures, which have been applied in many application areas including reservoir simulation, represent a promising means for constructing efficient surrogate models. Nonlinear dimensionality reduction techniques allow for encapsulating the high-resolution complex geological description of reservoir into a low-dimensional subspace, which significantly reduces number of unknowns and provides an efficient way to construct a proxy model based on the the reduced-dimension parameters.
Polynomial Chaos Expansions (PCE) is a powerful tool to quantify uncertainty in dynamical system when there is probabilistic uncertainty in the system parameters. In reservoir simulation it has been shown to be more accurate and efficient compared to traditional experimental design (ED). PCEs have a significant advantage over other response surfaces as the convergence to the true probability distribution is proved when the order of the PCE is increased. Accordingly PCE proxy can be used as the pseudo-simulator to represent the surface responses of the uncertain variables. When the objective and constraints of a reservoir model is described by multivariate polynomial functions, there are very efficient algorithms to compute the global solutions. We have developed a workflow at which incorporates PCE to find the global minimum of the misfit surface and assess the uncertainty associated with. The accuracy of the PCE proxy increases with the additional trial runs of the reservoir simulator.
We conduct a two dimensional synthetic case study of a fluvial channel as well as a real field example to demonstrate the effectiveness of this approach. Kernel Principal Component Analysis (KPCA) is used to parameterize the complex geological structure. The study has revealed useful reservoir information and delivered more reliable production forecasts.
PCE-based history match enhances the quality and efficiency of the estimation of the most probable geological model and improve the confidence interval of production forecasts.