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Geostatistical Seismic Inversion with Direct Sequential Simulation and Co-simulation with Multi-local Distribution Functions

Abstract

Stochastic sequential simulation is a common modelling technique used in Earth sciences and an integral part of iterative geostatistical seismic inversion methodologies. Traditional stochastic sequential simulation techniques based on bi-point statistics assume, for the entire study area, stationarity of the spatial continuity pattern and a single probability distribution function, as revealed by a single variogram model and inferred from the available experimental data, respectively. In this paper, the traditional direct sequential simulation algorithm is extended to handle non-stationary natural phenomena. The proposed stochastic sequential simulation algorithm can take into consideration multiple regionalized spatial continuity patterns and probability distribution functions, depending on the spatial location of the grid node to be simulated. This work shows the application and discusses the benefits of the proposed stochastic sequential simulation as part of an iterative geostatistical seismic inversion methodology in two distinct geological environments in which non-stationarity behaviour can be assessed by the simultaneous interpretation of the available well-log and seismic reflection data. The results show that the elastic models generated by the proposed stochastic sequential simulation are able to reproduce simultaneously the regional and global variogram models and target distribution functions relative to the average volume of each sub-region. When used as part of a geostatistical seismic inversion procedure, the retrieved inverse models are more geologically realistic, since they incorporate the knowledge of the subsurface geology as provided, for example, by seismic and well-log data interpretation.

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References

  • Azevedo L, Nunes R, Soares A, Mundin EC, Neto GS (2015) Integration of well data into geostatistical seismic amplitude variation with angle inversion for facies estimation. Geophysics 80(6):M113–M128

    Article  Google Scholar 

  • Bortoli LJ, Alabert F, Haas A, Journel AG (1992) Constraining stochastic images to seismic data. In: Soares A (ed) Geostatistics Tróia, vol 1. Kluwer Academic Publishers, The Netherlands, pp 325–337

    Google Scholar 

  • Caeiro MH, Soares A, Demyanov V, Christie M (2011) Optimization of a geostatistical non-stationary model in history matching. In: Marschallinger R, Zobl F (eds) Mathematical geosciences at the crossroads of theory and practice—Proceedings IAMG 2011 conference. COGeo, Salzburg, pp 68–82

    Google Scholar 

  • Deutsch C, Journel AG (1998) GSLIB: Geostatistical Software Library and Users’ Guide. Oxford University Press, Oxford

  • Doyen PM (2007) Seismic reservoir characterization. EAGE, Vienna

  • González EF, Mukerji T, Mavko G (2008) Seismic inversion combining rock physics and multiple-point geostatistics. Geophysics 73(1):R11–R21. doi:10.1190/1.2803748

    Article  Google Scholar 

  • Haas A, Dubrule O (1994) Geostatistical inversion-a sequential method of stochastic reservoir modelling constrained by seismie data. First Break 12(11):561–569

    Google Scholar 

  • Horta A, Soares A (2010) Direct sequential co-simulation with joint probability distributions. Math Geosci 42(3):269–292. doi:10.1007/s11004-010-9265-x

    Article  Google Scholar 

  • Horta A, Caeiro M, Nunes R, Soares A (2009) Simulation of continuous variables at meander structures: application to contaminated sediments of a lagoon. In: Atkinson P, Lloyd C (eds) geoENV VII–Geostatistics for environmental applications. Quantitative geology and geostatistics. Springer, The Netherlands, pp 161–172

    Google Scholar 

  • Mariethoz G, Renard P, Straubhaar J (2010) The direct sampling method to perform multiple-point geostatistical simulations. Water Resour Res 46(11):1–14. doi:10.1029/2008WR007621

    Google Scholar 

  • Nunes R, Almeida JA (2010) Parallelization of sequential gaussian, indicator and direct simulation algorithms. Comput Geosci 36(8):1042–1052. doi:10.1016/j.cageo.2010.03.005

    Article  Google Scholar 

  • Nunes R, Soares A, Neto GS, Dillon L, Guerreiro L, Caetano H, Maciel C, Leon F (2012) Geostatistical Inversion of Prestack Seismic Data. In: Ninth International Geostatistics Congress. Oslo, Norway, pp 1–8

  • Soares A (2001) Direct sequential simulation and cosimulation. Math Geol 33(8):911–926

    Article  Google Scholar 

  • Soares A, Diet JD, Guerreiro L (2007) Stochastic Inversion with a global perturbation method. EAGE Petroleum Geostatistics, Cascais, Portugal, pp 10–14

  • Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34(1):1–21

    Article  Google Scholar 

  • Wu J, Boucher A, Zhang T (2008) A SGeMS code for pattern simulation of continuous and categorical variables: FILTERSIM. Comput Geosci 34(12):1863–1876

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Galp E&P and ENMC for providing the data shown in this paper; Schlumberger for the donation of Petrel\(^{\circledR }\) and CGG for the donation of Hampson-Russell. We would also like to express our gratitude to CERENA for supporting this work. The authors would also like to acknowledge the comments of the anonymous reviewer which contributed to improving the quality of the original paper.

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Correspondence to Leonardo Azevedo.

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Nunes, R., Soares, A., Azevedo, L. et al. Geostatistical Seismic Inversion with Direct Sequential Simulation and Co-simulation with Multi-local Distribution Functions. Math Geosci 49, 583–601 (2017). https://doi.org/10.1007/s11004-016-9651-0

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  • DOI: https://doi.org/10.1007/s11004-016-9651-0

Keywords

  • Non-stationarity
  • Stochastic sequential simulation
  • Subsurface characterization
  • Stochastic inversion