Mathematical Geosciences

, Volume 48, Issue 4, pp 353–369 | Cite as

Change-of-Support Models on Irregular Grids for Geostatistical Simulation

  • Victor Zaytsev
  • Pierre Biver
  • Hans Wackernagel
  • Denis Allard
Article
  • 331 Downloads

Abstract

In many domains, numerical models are initialized with inputs defined on irregular grids. In petroleum reservoir engineering, they consist of a great variety of grid cells of different size and shape to enable fine-scale modeling in the vicinity of the wells and coarse modeling in less important regions. Geostatistical simulation algorithms, which are used to populate the cells of unstructured grids, often have to address the problem of transition from the small-scale statistical data stemming from laboratory cores analysis and seismic processing to the multiple larger scale geological supports. The reasonable generalization of the above-mentioned problem is integrating the point-support data to simulations on irregular supports. Classical geostatistical simulation methods for generating realizations of a stationary Gaussian random function cannot be applied to unstructured grids directly, because of the uneven supports. This article provides a critical review of existing geostatistical simulation methodologies for unstructured grids, including fine-scale simulations with upscaling and direct sequential simulation algorithms, and presents two different generalizations of the discrete Gaussian model for this purpose, thereby discussing the theoretical assumptions and the accuracy when implementing these models.

Keywords

Unstructured grids Geostatistics Direct sequential simulation Change of support Discrete Gaussian model 

References

  1. Brown G, Ferreira J, Lantuéjoul C (2008) Conditional simulation of a Cox process using multiple sample supports. In: Ortiz JM, Emery X (eds) 8th international geostatistics congress. Gecamin, Santiago, pp 459–468Google Scholar
  2. Chilès J-P (2014) Validity range of the discrete Gaussian change-of-support model and its variant. J South Afr Inst Min Metall 114:231–235Google Scholar
  3. Chilès J-P, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, 2nd edn. Wiley, HobokenCrossRefGoogle Scholar
  4. Cressie N, Wikle CK (2011) Statistics for spatio-temporal data. Wiley, HobokenGoogle Scholar
  5. de Fouquet C (1994) Reminders on the conditioning kriging. In: Armstrong M, Dowd PA (eds) Geostatistical simulations. Springer, Netherlands, pp 131–145CrossRefGoogle Scholar
  6. Emery X (2007) On some consistency conditions for geostatistical change-of-support models. Math Geol 39:205–223CrossRefGoogle Scholar
  7. Emery X (2009) Change-of-support models and computer programs for direct block-support simulation. Comput Geosci 35:2047–2056CrossRefGoogle Scholar
  8. Farmer C (2002) Upscaling: a review. Int J Numer Methods Fluids 40:63–78CrossRefGoogle Scholar
  9. Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, OxfordGoogle Scholar
  10. Journel AG (1993) Modeling uncertainty: some conceptual thoughts. In: Dimitrakopoulos R (ed) Geostatistics for the next century. Kluwer Academic Pub, Dordrecht, pp 30–43Google Scholar
  11. Lantuéjoul C (2002) Geostatistical simulation: models and algorithms. Springer, BerlinCrossRefGoogle Scholar
  12. Manchuk JG (2010) Geostatistical modeling of unstructured grids for flow simulation. Unpublished dissertation, doctoral, University of AlbertaGoogle Scholar
  13. Manchuk JG, Leuangthong O, Deutsch CV (2005) Direct geostatistical simulation on unstructured grids. In: Leuangthong O, Deutsch CV (eds) Geostatistics Banff 2004. Springer, Netherlands, pp 85–94CrossRefGoogle Scholar
  14. Matheron G (1976) A simple substitute for conditional expectation: the disjunctive kriging. In: Guarascio M, David M, Huijbregts C (eds) Advanced geostatistics in mining industry, pp 221–236Google Scholar
  15. Matheron G (1985) Change of support for diffusion-type random functions. J Int Assoc Math Geol 17:137–165CrossRefGoogle Scholar
  16. Matheron G (1989) Estimating and choosing. Springer, BerlinCrossRefGoogle Scholar
  17. Oz B, Deutsch CV, Tran T, Xie Y (2003) DSSIM-HR: a FORTRAN 90 program for direct sequential simulation with histogram reproduction. Comput Geosci 29(1):39–51CrossRefGoogle Scholar
  18. Rivoirard J (1994) Introduction to disjunctive kriging and non-linear geostatistics (spatial information systems). Oxford University Press, OxfordGoogle Scholar
  19. Robertson RK, Mueller UA, Bloom LM (2006) Direct sequential simulation with histogram reproduction: a comparison of algorithms. Comput Geosci 32:382–395CrossRefGoogle Scholar
  20. Shiryaev A (1996) Probability, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
  21. Soares A (2001) Direct sequential simulation and cosimulation. Math Geol 33:911–926CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2015

Authors and Affiliations

  • Victor Zaytsev
    • 1
    • 2
  • Pierre Biver
    • 1
  • Hans Wackernagel
    • 2
  • Denis Allard
    • 3
  1. 1.Geostatistics and Uncertainties ServiceTotal S.A.Pau CedexFrance
  2. 2.Equipe de Géostatistique, Centre de Géosciences, MINES ParisTechPSL Research UniversityFontainebleauFrance
  3. 3.Biostatistics and Spatial Processes (BioSP)INRAAvignon Cedex 9France

Personalised recommendations