Abstract
Spatially distributed phenomena typically do not exhibit Gaussian behavior, and consequently methods constrained to traditional two-point covariance statistics cannot correctly represent the spatial connectivity of such phenomena. This necessitates the development of multiple-point statistics (MPS)-based algorithms. However, due to the sparse data available to infer these MPS statistics, one needs to have a training image (TI) to accurately model higher-order statistics. Training images are usually inferred from outcrops and/or conceptual models and are subject to uncertainty that have to be accounted for in MPS algorithms.
In this study, we propose a new method for ranking different sets of TIs corresponding to different geological scenarios. These set of TIs represent the associated uncertainty and contain features of different shapes (channels, ellipses, fracture, etc.) as well as different sizes and orientation. We analyze the polyspectra of the different TIs (power spectrum and bispectrum) to distinguish between different TIs. We show that object size and orientation can be inferred from the power spectrum, while the object shapes can be inferred from the bispectrum. Therefore, the combination of power spectrum and bispectrum can be used as an identifier for each TI.
We then infer the power spectrum and bispectrum from the available conditioning data. Since the data is scattered and sparse, we use a nonuniform fast Fourier transform (NUFFT) method based on a basis pursuit algorithm to estimate the Fourier transform of the scattered data. We then use the Fourier transform to calculate the power spectrum and the bispectrum. Then, the identifier features are calculated from the higher-order spectra. Finally, we choose the TI with the closest identifier to that of conditioning data as the representative TI.
We implement the proposed algorithm on different geologic systems such as channelized reservoir, fractured reservoirs, and models with elliptic objects with different sizes and orientations and examine its performance. We study the sensitivity of the algorithm to the available conditioning data. We show that the algorithm performs well with very sparse conditioning data. This algorithm can address one of the main issues pertaining to MPS algorithms, which is ensuring the consistency between the training image and conditioning data in order to develop robust models that have improved predictive ability. In case MPS-based simulations are performed accounting for the uncertainty in TI, the method can be used to rank the prior TIs so as to yield robust estimates for uncertainty.
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Elahi Naraghi, M., Srinivasan, S. (2017). Robust MPS-Based Modeling via Spectral Analysis. In: Gómez-Hernández, J., Rodrigo-Ilarri, J., Rodrigo-Clavero, M., Cassiraga, E., Vargas-Guzmán, J. (eds) Geostatistics Valencia 2016. Quantitative Geology and Geostatistics, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-319-46819-8_34
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