1887
Volume 24, Issue 2
  • ISSN: 1354-0793
  • E-ISSN:

Abstract

The variogram is a key parameter for geostatistical modelling. Inferring a stable variogram model from widely spaced well data is a longstanding challenge due to an often unreliable experimental horizontal variogram. The main aim of this paper is to improve the horizontal variogram inference in the presence of limited data by quantifying variogram uncertainty and reducing this uncertainty with secondary data. A new approach of variogram uncertainty is presented by computing the number of independent variogram pairs (degrees of freedom) for each lag. A methodology to improve the horizontal variogram uncertainty is developed considering the horizontal variogram of the seismic data and the vertical well variogram since these variograms are well defined in most cases. Seismic data provide constraints on the horizontal variogram of the well data. The constraints are inferred from the covariance between the well and seismic data. The vertical variogram of the well data can be scaled to scenarios of the horizontal variogram. Improved horizontal variogram realizations honouring the correlation between lags are attained by merging variogram distributions for each lag distance considering the constraints from the horizontal seismic variogram. A realistic case study is presented.

Loading

Article metrics loading...

/content/journals/10.1144/petgeo2016-161
2017-07-27
2024-04-18
Loading full text...

Full text loading...

References

  1. Blachman, N.M.
    1989. On combining target-location ellipses. IEEE Transactions on Aerospace and Electronic Systems, 25, 284–287.
    [Google Scholar]
  2. Bretherton, C.S., Widmann, M., Dymnikov, V.P., Wallace, J.M. & Bladé, I.
    1999. The effective number of spatial degrees of freedom of a time-varying field. Journal of Climate, 12, 1990–2009.
    [Google Scholar]
  3. Chiles, J.P. & Delfiner, P.
    1999. Geostatistics: Modeling Spatial Uncertainty. John Wiley & Sons, Chichester.
    [Google Scholar]
  4. Cressie, N.
    1985. Fitting variogram models by weighted least squares. Mathematical Geology, 17, 563–586.
    [Google Scholar]
  5. Davis, J.E.
    2007. Combining Error Ellipses. Technical Report, Massachusetts Institute of Technology, Cambridge, MA.
    [Google Scholar]
  6. Davis, M.W.
    1987. Production of conditional simulations via the LU triangular decomposition of the covariance matrix. Mathematical Geology, 19, 91–98.
    [Google Scholar]
  7. Deutsch, C.V. & Journel, A.G.
    1998. GSLIB: Geostatistical Software Library and User's Guide.2nd edn. Oxford University Press, Oxford.
    [Google Scholar]
  8. Frykman, P. & Deutsch, C.V.
    2002. Practical application of geostatistical scaling laws for data integration. Petrophysics, 43, 153–171.
    [Google Scholar]
  9. Gantmacher, F. & Krein, N.
    1950. Oscillation Matrices and Small Vibrations of Mechanical Systems. AMS Chelsea Publishing, Providence, RI.
    [Google Scholar]
  10. Genton, M.G.
    1998. Variogram fitting by generalized least squares using an explicit formula for the covariance structure. Mathematical Geology, 30, 323–345.
    [Google Scholar]
  11. Gringarten, E. & Deutsch, C.V.
    2001. Teacher's aide variogram interpretation and modeling. Mathematical Geology, 33, 507–534.
    [Google Scholar]
  12. Khan, K.D. & Deutsch, C.V.
    2016. Practical incorporation of multivariate parameter uncertainty in geostatistical resource modeling. Natural Resources Research, 25, 51–70.
    [Google Scholar]
  13. Kupfersberger, H. & Deutsch, C.V.
    1999. Methodology for integrating analog geologic data in 3-d variogram modeling. American Association of Petroleum Geologists Bulletin, 83, 1262–1278.
    [Google Scholar]
  14. Kupfersberger, H., Deutsch, C.V. & Journel, A.G.
    1998. Deriving constraints on small-scale variograms due to variograms of large-scale data. Mathematical Geology, 30, 837–852.
    [Google Scholar]
  15. Marchant, B.P. & Lark, R.M.
    2004. Estimating variogram uncertainty. Mathematical Geology, 36, 867–898.
    [Google Scholar]
  16. Matheron, G.
    1965. Les variables régionalisées et leur estimation. Masson et Cie, Paris.
    [Google Scholar]
  17. Meddaugh, W.S., Champenoy, N., Osterloh, W. & Tang, H.
    2011. Reservoir forecast optimism – Impact of geostatistics, reservoir modeling, heterogeneity, and uncertainty. Paper presented at the SPE Annual Technical Conference and Exhibition, 30 October–2 November 2011, Denver, Colorado, USA.
    [Google Scholar]
  18. Orechovesky, J.R.
    1996. Single Source Error Ellipse Combination. Master's thesis, Naval Postgraduate School, Monterey, CA.
    [Google Scholar]
  19. Ortiz, J. & Deutsch, C.V.
    2000. Calculation of the Uncertainty in the Variogram for More Realistic Reservoir Forecasting. Technical Report, CCG Annual Report 2-109. Centre for Computational Geostatistics, University of Alberta. Retrieved from http://www.ccgalberta.com/resources/reports/
    [Google Scholar]
  20. 2002. Calculation of uncertainty in the variogram. Mathematical Geology, 34, 169–183.
    [Google Scholar]
  21. Pardo-Igúzquiza, E. & Dowd, P.
    2001. Variance–covariance matrix of the experimental variogram: Assessing variogram uncertainty. Mathematical Geology, 33, 397–419.
    [Google Scholar]
  22. Pyrcz, M.J. & Deutsch, C.V.
    2014. Geostatistical Reservoir Modeling. Second Edition. Oxford University Press, Oxford.
    [Google Scholar]
  23. Roecker, J.
    1991. On combining multidimensional target location ellipsoids. IEEE Transactions on Aerospace Electronic Systems, 27, 175–177.
    [Google Scholar]
  24. Vandebril, R., Van Barel, M. & Mastronardi, N.
    2008. Matrix Computations and Semiseparable Matrices: Linear Systems, Volume 1. Johns Hopkins University Press, Baltimore, MD.
    [Google Scholar]
  25. Wang, J. & Dou, Q.
    2010. Integration of 3D seismic attributes into stochastic reservoir models using iterative vertical resolution modeling methodology. Paper presented at the SPE Western Regional Meeting, 27–29 May 2010, Anaheim, California, USA.
    [Google Scholar]
http://instance.metastore.ingenta.com/content/journals/10.1144/petgeo2016-161
Loading
/content/journals/10.1144/petgeo2016-161
Loading

Data & Media loading...

  • Article Type: Research Article

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error