Efficient and Robust Uncertainty Quantification in Reservoir Simulation with Polynomial Chaos Expansions and Non-intrusive Spectral Projection

Abstract

Abstract Polynomial Chaos Expansions (PCE) as applied with the Probabilistic Collocation Method (PCM) has been shown to be a promising approach for uncertainty quantification in reservoir simulation. In particular, 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 they theoretically guarantee convergence to the true distribution (of the output random variable of interest, such as cumulative oil production) as the order of the PCE and number of simulations used to calculate the PCE coefficients is increased. However, there are many issues with PCM that limit the practical applicability of the method to large scale problems. A key issue with the PCM approach is that the number of simulations required to create the PCE is directly dependent on the number of terms in the PCE, and this increases exponentially with PCE order and number of random variables. We propose the application of another related approach that also uses PCEs called Non-Intrusive Spectral Projection (NISP) that essentially eliminates many of the key problems with PCM. The main difference with PCM is that instead of using a collocation or regression approach to estimate the PCE coefficients, a sampling approach is used in NISP. This makes the number of simulations almost independent of the number of random variables and PCE order, and as a result is much more efficient for large-scale problems. Furthermore, statistically rigorous error estimates on the PCE coefficients are also provided with this approach. We also propose a robust approach for eliminating PCE coefficients that have been estimated incorrectly, thereby improving the accuracy and robustness of NISP. The accuracy and efficiency of the approach is demonstrated through various examples.