Stochastic Collocation Methods for Efficient and Accurate Quantification of Uncertainty in Multiphase Reservoir Simulations

Abstract

Abstract Reservoir simulation is subject to uncertainties, which may stem from inaccurate and imprecise measurements or inadequate characterization of spatially or temporally varying medium properties. Such uncertainties render the model parameters random and the equations describing flow and transport in the media stochastic. There exist several stochastic approaches for quantifying uncertainties, which, however, are either incompatible with existing deterministic simulators or are too demanding computationally. In this study, an alternative approach is proposed that is both accurate and efficient. In this approach, the uncertainty quantities such as permeability and porosity fields are represented by the Karhunen-Loeve expansions while the fluid saturations and reservoir production quantities are expressed by the polynomial chaos or Lagrange polynomial expansions. Two collocation-based methods, i.e. probabilistic collocation method (PCM) and Smolyak method are used to determine the coefficients of the polynomials expansions by solving for the fluid saturations and pressures at different collocation points via the original partial differential equations. This approach is non-intrusive in that it results in independent deterministic differential equations, which similar to the Monte Carlo method, can be implemented with existing codes or simulators. However, the required number of simulations in PCM or Smolyak method is much smaller than that in the Monte Carlo method. The approach is demonstrated with three-phase flow problems in heterogeneous reservoirs using Eclipse. The accuracy, efficiency, and compatibility of this approach are compared against Monte Carlo simulations. This study reveals that while their computational efforts are greatly reduced compared to Monte Carlo method, the stochastic collocation methods are able to accurately estimate the statistical moments and probability density functions of the fluid saturations, pressures, and production quantities of interest.