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Boolean Spectral Analysis in Categorical Reservoir Modeling

Abstract

This work introduces a new method for simulating facies distribution with two categories based on Fourier analysis of Boolean functions. According to this method, two categories of facies distributed along vertical wells are encoded as Boolean functions taking two values. The subsequent simulation process is divided into three consecutive steps. First, Boolean functions of the well data are decomposed into a binary version of a Fourier series. Decomposition coefficients are then simulated over the two-dimensional area as stationary random fields. Finally, synthetic data in the interwell space are reconstructed from simulated coefficients. The described method was implemented experimentally in software and tested on a case of a real oil field and on a case of a synthetic oil field model. Simulations on the synthetic model were used to test the performance of the method for two different bases in the Fourier expansion (Walsh functions and Haar wavelets). The simulation results were compared to those obtained on the same synthetic model via the classical sequential indicator simulation. It was shown that, for both bases, the new method reproduces statistical parameters of the well data better than sequential indicator simulation.

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Notes

  1. 1.

    The smoother a function is, the faster the rate of decay of the coefficients is in its Fourier–Legendre series (Wang and Xiang 2012). The same is true for the decomposition with respect to a sufficiently regular wavelet basis (Mallat 1999).

  2. 2.

    Alternatively, as by the original paper of Baykov et al. (2010), covariance functions \(K_j\) could be estimated non-parametrically via the periodogram-based techniques. Sadly, this approach works well in practice only when one has a large amount of data, which rarely happens in the study of petroleum reservoirs, or requires substantial hand tuning. The parametric covariance function estimation for the spectral approach is discussed in the recent work Ismagilov et al. (2020a).

  3. 3.

    Actually, Walsh functions do induce a vertical stationarity assumption, but in a very irregular way. This is not the classical stationarity with respect to the translation on the real line, but it is the stationarity with respect to the bitwise xor operation acting on infinite binary representations of numbers in the [0, 1) interval. This cannot be easily spotted with the naked eye, though. The stationarity property is due to the fact that Walsh functions are precisely the characters of the dyadic group \(\{0,1\}^\infty \) under Pontryagin duality (Folland 2016).

  4. 4.

    In practice, for Walsh functions, the adaptive basis often coincides with the first M functions in Walsh ordering. Intuitively, this shows that this ordering closely reflects the importance of basis functions and that the importance of basis functions does not depend heavily on the particular data.

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Acknowledgements

The authors are grateful to both anonymous referees for stimulating advice that helped to clarify the representation of the results.

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Correspondence to Niyaz Ismagilov.

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Research was partially supported by Russian Science Foundation Grant 19-71-30002.

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Ismagilov, N., Borovitskiy, V., Lifshits, M. et al. Boolean Spectral Analysis in Categorical Reservoir Modeling. Math Geosci 53, 305–324 (2021). https://doi.org/10.1007/s11004-021-09919-z

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Keywords

  • Boolean functions
  • Spectral analysis
  • Reservoir modeling
  • Categorical simulation