Accurate and efficient uncertainty quantification of subsurface fluid flow via the probabilistic collocation method

Uncertainty quantification of subsurface flow has recently attracted a significant amount of attention. The uncertainty can result from the combination of the formation heterogeneity and the incomplete knowledge of its properties. Traditional flow simulations treat the geological formation deterministic, thus resulting in deterministic predictions. However, taking uncertainty into account necessitates a stochastic description of the formation properties and hence stochastic approaches to flow simulations.;This dissertation explores an efficient approach, i.e., the probabilistic collocation method (PCM) for uncertainty quantification of flow in random porous media. In this approach, the dependent random variables are represented by employing the orthogonal polynomial functions (polynomial chaos expansions) as the bases of the random space. Utilizing the collocation technique in the random space directly results in a set of independent simulations. This independence feature of this stochastic approach allows us to directly employ existing flow simulators. Random fields such as heterogeneous permeability (or porosity) fields are parameterized using some dimension reduction techniques such as the Karhunen-Loeve expansion. Applications to both single-phase and multiphase flows are performed. Other than the typical log-normal probability distribution, other non-Gaussian probability distributions are also considered for characterizing the formation properties that are treated as random fields. Conditional simulations of both Gaussian and non-Gaussian random fields when measurements of hydraulic conductivity are available are discussed in this dissertation.;This dissertation also considers uncertainty analysis of petroleum reservoir simulations, where the uncertain parameters have arbitrary probability distributions. The experimental design methods are widely used for uncertainty analysis in the oil industry. However, the traditional experimental design methods usually have an inherent assumption of uniform distributions for the random inputs and they do not take into account the full probability density functions (PDFs) of the input random parameters consistently. In this dissertation, orthogonal polynomials for arbitrary PDFs are constructed numerically and the PCM is utilized for uncertainty propagation. Each collocation point captures a different weight of the random variables for a given PDF and thus yields optimal approximation of the responses. Various studied cases reveal that, while the computational efforts are greatly reduced compared to Monte Carlo simulation, the PCM is able to accurately quantify uncertainty of the reservoir performance. Results also reveal that the PCM is more robust, accurate, and efficient than experimental design methods for uncertainty analysis.