Uncertainty Quantification in Reservoir Simulation Models with Polynomial Chaos Expansions: Smolyak Quadrature and Regression Method Approach

Polynomial chaos expansions (PCE) are used to analyze how uncertainty present in input parameters is propagated through the model to its output. The use of PCE enables the representation of the outcome of a given model as a polynomial, created by a function basis that depends on the probability distribution of the input variables and beyond that, the estimation of statistical properties such as the mean, standard deviation, percentiles and more rigorously, the entire probability distribution. In oil reservoir management, it is of great importance to determine the influence the parameters have in the behavior of the model response. In high dimensional problems the estimation of the expansion coefficients has a high computational cost due to the existence of an exponentially increasing relationship between the number of variables and the number of coefficients. Adding the inherent cost of simulating a real reservoir model, finding efficient solutions becomes absolutely necessary. The Smolyak or sparse quadrature is proposed, as well as the regression approach with experimental design, to approximate the expansion coefficients dealing with high dimensional cost. The accuracy of these techniques will be tested in a reservoir simulation model compo- sed of eleven uncertain parameters, and results will be compared to traditional Monte Carlo Simulation.