Summary
Reservoir modeling and simulation are subject to significant uncertainty, which usually arises from heterogeneity of the geological formation and deficiency of measured data. Uncertainty quantification, thus, plays an important role in reservoir simulation. In order to perform accurate uncertainty analysis, a large number of simulations are often required. However, it is usually prohibitive to do so because even a single simulation of practical large-scale simulation models may be quite time consuming. Therefore, efficient approaches for uncertainty quantification are a necessity. The experimental-design (ED) method is applied widely in the petroleum industry for assessing uncertainties in reservoir production and economic appraisal. However, a key disadvantage of this approach is that it does not take into account the full probability-density functions (PDFs) of the input random parameters consistently—that is, the full PDFs are not used for sampling and design but used only during post-processing, and there is an inherent assumption that the distributions of these parameters are uniform (during sampling), which is rarely the case in reality. In this paper, we propose an approach to deal with arbitrary input probability distributions using the probabilistic-collocation method (PCM). Orthogonal polynomials for arbitrary distributions are first constructed numerically, and then PCM is used for uncertainty propagation. As a result, PCM can be applied efficiently for any arbitrary numerical or analytical distribution of the input parameters. It can be shown that PCM provides optimal convergence rates for linear models, whereas no such guarantees are provided by ED. The approach is also applicable to discrete distributions. PCM and ED are compared on a few synthetic and realistic reservoir models. Different types of PDFs are considered for a number of reservoir parameters. Results indicate that, while the computational efforts are greatly reduced compared to Monte Carlo (MC) simulation, PCM is able to accurately quantify uncertainty of various reservoir performance parameters. Results also reveal that PCM is more robust, more accurate, and more efficient than ED for uncertainty analysis.
A comparison of the discrete cosine and wavelet transforms for hydrologic model input data reduction
