Modeling complex spatial patterns in reservoir models using high order spectra
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One of the most challenging issues in reservoir modeling The important goal of reservoir modeling is to generate a map of geologic attributes that can yield predictions of hydrocarbon production. Mostly, the primary source of information for creating such a map is borehole measurements, which are only available at sparse locations. Reservoir modeling constrained to the available data along the wells allows us to generate multiple realizations for the whole reservoir. In order for these realizations to yield robust estimates of the uncertainty in reservoir performance prediction, it is imperative that they exhibit connectivity characteristics that are typical for the geological system being modeled. Different algorithms have been developed to stochastically simulate reservoir properties using sparse measured data. In these methods, the spatial variability represented by the underlying joint distribution is in the form of the spatial covariance. The major drawback of traditional variogram-based modeling is that they are not able to reproduce complex spatial patterns. Multiple-point statistical algorithms, however, can reconstruct such curvilinear features. In this study, we study the link between the multiple point spatial pattern connectivity and the Fourier spectrum. This will allow us to infer statistical functions describing reservoir connectivity more efficiently. This can be further sub-divided into two approaches based on the availability of data and information. We also propose methods for selecting an optimum training image when there is ambiguity associated with it, and integrating non-stationary secondary information into the simulation framework. Then, we develop a simulation algorithm in Fourier domain. We will show that the amplitude of Fourier transform can be calculated directly from power spectrum (Fourier transform of covariance function). The phase identification can be achieved by either solving an optimization problem to match the available conditioning data or from higher order spectra such as bispectrum or trispectrum. Finally, we present a new framework for integrating dynamic data and performing history matching. We show how polyspectra affect the production behavior and therefore, we can use the production measurements at well location to identify amplitude and phase within the proposed framework.