Efficient and Accurate Quantification of Uncertainty for Multiphase Flow With the Probabilistic Collocation Method

Original SPE manuscript received for review 24 December 2007. Revised manuscript received for review 15 October 2008. Paper (SPE 114802) peer approved 3 December 2008. Summary In this study, we explore an efficient and accurate method for uncertainty analysis of petroleum reservoir simulations. The essence of the approach is the combination of Karhunen-Loeve (KL) expansion and probabilistic collocation method. Monte Carlo (MC) simulation is the most common and straightforward approach for uncertainty quantification. It generates a large number of realizations of the underlying reservoir. Solving the multiple realizations leads to a large computational effort, especially for large-scale problems. We present an accurate and efficient alternative. In this approach, the underlying random fields, such as permeability and porosity are represented by the KL expansion and the resulting random fields (e.g., fluid saturations and pressures) or variables (e.g., hydrocarbon production) are expressed by the polynomial chaos expansions. The probabilistic collocation method (PCM) is used to determine the coefficients of the polynomial chaos expansions by solving for the fluid saturation and pressure fields via the original partial differential equations for selected sets of collocation points. This approach is nonintrusive because it results in independent deterministic differential equations, which, similar to the MC method, can be implemented with existing codes or simulators. However, the required number of simulations in the PCM is much less than that in the MC method. The approach is demonstrated with black-oil problems in heterogeneous reservoirs with the commercial Eclipse simulator. The accuracy, efficiency, and compatibility of this approach are compared against MC simulations. This study reveals that, while its computational efforts are greatly reduced compared to the MC method, the PCM is able to estimate accurately the statistical moments and probability density functions of the fluid saturations (and pressures) and the hydrocarbon production.