Skip to main content

Advertisement

SpringerLink
Log in

Multiple-point geostatistical modeling based on the cross-correlation functions

  • Original Paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

An important issue in reservoir modeling is accurate generation of complex structures. The problem is difficult because the connectivity of the flow paths must be preserved. Multiple-point geostatistics is one of the most effective methods that can model the spatial patterns of geological structures, which is based on an informative geological training image that contains the variability, connectivity, and structural properties of a reservoir. Several pixel- and pattern-based methods have been developed in the past. In particular, pattern-based algorithms have become popular due to their ability for honoring the connectivity and geological features of a reservoir. But a shortcoming of such methods is that they require a massive data base, which make them highly memory- and CPU-intensive. In this paper, we propose a novel methodology for which there is no need to construct pattern data base and small data event. A new function for the similarity of the generated pattern and the training image, based on a cross-correlation (CC) function, is proposed that can be used with both categorical and continuous training images. We combine the CC function with an overlap strategy and a new approach, adaptive recursive template splitting along a raster path, in order to develop an algorithm, which we call cross-correlation simulation (CCSIM), for generation of the realizations of a reservoir with accurate conditioning and continuity. The performance of CCSIM is tested for a variety of training images. The results, when compared with those of the previous methods, indicate significant improvement in the CPU and memory requirements.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aitokhuehi, I., Durlofsky, L.J.: Optimizing the performance of smart wells in complex reservoirs using continuously updated geological models. J. Pet. Sci. Eng. 48, 254–264 (2005)

    Article  Google Scholar 

  2. Arpat, B.: Sequential simulation with patterns. Ph.D. thesis, Stanford University (2005)

  3. Arpat, B., Caers, J.: Stochastic simulation with patterns. Math. Geol. 39, 177–203 (2007)

    Article  Google Scholar 

  4. Borgman, L., Taheri, M., Hagan, R.: Three-dimensional frequency-domain simulations of geological variables. In: Verly, G., et al. (eds.) Geostatistics for Natural Resources Characterization, pp. 517–541. Reidel, Dordrecht (1984)

    Google Scholar 

  5. Brunelli, R., Poggio, T.: Face recognition: features versus templates. IEEE Trans. Pattern Anal. Mach. Intell. 15, 1042–1052 (1993)

    Article  Google Scholar 

  6. Caers, J., Park, K.A.: Distance-based representation of reservoir uncertainty: the Metric EnKF. In: Proceedings of the 11th European Conference on the Mathematics of Oil Recovery, Bergen, Norway (2008)

  7. Caers, J.: Petroleum Geostatistics. Society of Petroleum Engineers, Richardson (2005)

    Google Scholar 

  8. Caers, J., Journel, A.G.: Stochastic reservoir simulation using neural networks trained on outcrop data. SPE paper 49026 (1998)

  9. Caers, J., Srinivasan, S., Journel, A.G.: Geostatistical quantification of geological information for a fluvial-type North Sea reservoir. SPE paper 56655 (1999)

  10. Caers, J., Strebelle, S., Payrazyan, K.: Stochastic Integration of Seismic and Geological Scenarios: A Submarine Channel Saga, pp. 192–196. The Leading Edge (2003)

  11. Chiles, J.P., Delfiner, P.: Geostatistics: Modeling Spatial Uncertainty. Wiley, New York (1999)

    Book  MATH  Google Scholar 

  12. Chu, J., Journel, A.G.: Conditional fBm simulation with dual kriging. In: Dimitrakopoulos, R. (ed.) Geostatistics for the Next Century, pp. 407–421. Kluwer, Dordrecht (1993)

    Google Scholar 

  13. Chugunova, T., Hu, L.: Multiple-point simulations constrained by continuous auxiliary data. Math. Geosci. 40, 133–146 (2008)

    Article  MATH  Google Scholar 

  14. Daly, C.: Higher order models using entropy, Markov random fields and sequential simulation. In: Leuangthong, O., Deutsch, C.V. (eds.) Geostatistics, pp. 215–224. Springer, Berlin (2004)

    Google Scholar 

  15. Daly, C., Knudby, C.: Multipoint statistics in reservoir modelling and in computer vision. In: Petroleum Geostatistics 2007, Cascais, Portugal (2007)

  16. Davis, M.: Production of conditional simulations via the LU decomposition of the covariance matrix. Math. Geol. 19, 91–98 (1987)

    Article  Google Scholar 

  17. Deutsch, C.V.: Annealing techniques applied to reservoir modeling and integration of geological and engineering (well test) data. Ph.D. thesis, Stanford University (1992)

  18. Deutsch, C.V., Journel, A.G.: GSLIB: Geostatistical Software Library and User’s Guide, 2nd edn. Oxford University Press, Oxford (1998)

    Google Scholar 

  19. Deutsch, C.V., Wang, L.: Hierarchical object-based geostatistical modeling of fluvial reservoirs. Paper SPE 36514 (1996)

  20. Dimitrakopoulos, R., Mustapha, H., Gloaguen, E.: High-order statistics of spatial random fields: exploring spatial cumulants for modelling complex, non-Gaussian and non-linear phenomena. Math. Geosci. 42, 65–100 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dimitrakopoulos, R., Luo, X.: Generalized sequential Gaussian simulation on group size and screen-effect approximations for large field simulations. Math. Geol. 36, 567–591 (2004)

    Article  MathSciNet  Google Scholar 

  22. Di Stefano, L., Mattoccia, S., Tombari, F.: ZNCC-based template matching using bounded partial correlation. Pattern Recogn. Lett. 26, 2129–2134 (2005)

    Article  Google Scholar 

  23. El Ouassini, A., Saucier, A., Marcotte, D., Favis, B.: A patchwork approach to stochastic simulation: a route towards the analysis of morphology in multiphase systems. Chaos, Solitons Fractals 36, 418–436 (2008)

    Article  MathSciNet  Google Scholar 

  24. Farmer, C.: Numerical rocks. In: King, P.R. (ed.) The Mathematical Generation of Reservoir Geology, pp. 22–33. Clarendon, Oxford (1990)

    Google Scholar 

  25. Feyen, L., Caers, J.: Quantifying geological uncertainty for flow and transport modelling in multi-modal heterogeneous formations. Adv. Water Resour. 29, 912–929 (2006)

    Article  Google Scholar 

  26. Gloaguen, E., Dimitrakopoulos, R.: Two-dimensional conditional simulations based on the wavelet decomposition of training images. Math. Geosci. 41, 679–701 (2009)

    Article  MATH  Google Scholar 

  27. Goovaerts, P.: Geostatistics for Natural Resources Evaluation. Oxford University Press, Oxford (1997)

    Google Scholar 

  28. Goshtasby, A., Gage, S.H., Bartholic, J.F.: A two-stage cross-correlation approach to template matching. IEEE Trans. Pattern Anal. Mach. Intell. 6, 374–378 (1984)

    Article  Google Scholar 

  29. Guardiano, F., Srivastava, M.: Multivariate geostatistics: beyond bivariate moments. In: Geostatistics - Troia, pp. 133–144. Kluwer Academic, Dordrecht (1993)

    Chapter  Google Scholar 

  30. Haldorsen, H.H., Lake, L.W.: A new approach to shale management in field-scale models. Soc. Pet. Eng. J. 24, 447–452 (1984)

    Google Scholar 

  31. Hamzehpour, H., Rasaei, M.R., Sahimi, M.: Development of optimal models of porous media by combining static and dynamic data: the permeability and porosity distributions. Phys. Rev. E 75, 056311/1–056311/17 (2007)

  32. Harding, A., Strebelle, S., Levy, M.: Reservoir facies modeling: new advances in mps. In: Proceedings of GEOSTAT 2004, Banff, Canada (2004)

  33. Hoffman, B.T., Caers, J.: History matching by jointly perturbing local facies proportions and their spatial distribution: application to a North Sea reservoir. J. Pet. Sci. Eng. 57, 257–272 (2007)

    Article  Google Scholar 

  34. Holden, L., Hauge, R., Skare, O., Skorstad, A.: Modeling of fluvial reservoirs with object models. Math. Geol. 30, 473–496 (1998)

    Article  Google Scholar 

  35. Honarkhah, M., Caers, J.: Stochastic simulation of patterns using distance-based pattern modeling. Math. Geosci. 42, 487–517 (2010)

    Article  MATH  Google Scholar 

  36. Höppner, F., Klawonn, F.: Compensation of translational displacement in time series clustering using cross correlation. In: Proceedings of the 8th International Symposium on Intelligent Data Analysis, Advances in Intelligent Data Analysis VIII (2009)

  37. Huysmans, M., Dassargues, A.: Application of multiple-point geostatistics on modelling groundwater flow and transport in a crossbedded aquifer. Paper presented at GeoENV2008: 7th international conference on geostatistics for environmental applications, Southampton, 8–10 Sept 2008

  38. Jensen, J.L., Lake, L.W., Corbett, P.W.M., Goggin, D.J.: Statistics for Petroleum Engineers and Geoscientists, 2nd edn. Prentice Hall, Upper Saddle River (2000)

    Google Scholar 

  39. Journel, A.G.: Geostatistics for conditional simulation of ore bodies. Econ. Geol. 69, 673–687 (1974)

    Article  Google Scholar 

  40. Journel, A.G.: Geostatistics: roadblocks and challenges. In: Soares, A. (ed.) Geostatistics—Troia, vol. 1, pp. 213-224. Kluwer Academic, Dordrecht (1992)

    Google Scholar 

  41. Journel, A.G.: Combining knowledge from diverse sources: an alternative to traditional data independence hypotheses. Math. Geol. 34, 573 (2002)

    Article  MATH  Google Scholar 

  42. Journel, A.G., Alabert, F.G.: Non-Gaussian data expansion in the earth sciences. Terra Nova 1, 123–134 (1989)

    Article  Google Scholar 

  43. Journel, A.G., Deutsch, C.V.: Entropy and spatial disorder. Math. Geol. 25, 329–355 (1993)

    Article  Google Scholar 

  44. Journel, A.G., Huijbregts, C.J.: Mining Geostatistics. Academic, New York (1978)

    Google Scholar 

  45. Kjønsberg, H., Kolbjørnsen, O.: Markov mesh simulations with data conditioning through indicator kriging. In: Proceedings of the 8th International Geostatistics Congress, Santiago, Chile (2008)

  46. Kleingeld, W.J., Lantuejoul, C., Prins, C.F., Thurston, M.L.: The conditional simulation of a Cox process with application to deposits with discrete particles. In: Baafi, E.Y., Schofield, N.A. (eds.) Geostatistics Wollongong 96, pp. 683–694. Kluwer Academic, Dordrecht (1997)

    Chapter  Google Scholar 

  47. Knuth, D.: Art of Computer Programming, 3rd edn. Addison-Wesley, Reading (1997)

    Google Scholar 

  48. Lewis, J.P.: Fast template matching. Vision Interface, pp. 120–123 (1995, Fast normalized cross-correlation, unpublished extended version of the same paper)

  49. Liu, Y., Harding, A., Abriel, W., Strebelle, S.: Multiple-point simulation integrating wells, three-dimensional seismic data, and geology. AAPG Bull. 88, 905–921 (2004)

    Article  Google Scholar 

  50. Lyster, S., Deutsch, C.V.: MPS simulation in a Gibbs sampler algorithm. In: Proceedings of the 8th International Geostatistics Congress, Santiago, Chile (2008)

  51. Mariethoz, G., Renard, P., Straubhaar, J.: The direct sampling method to perform multiple-point simulations. Water Resour. Res. 46, 2008WR007621 (2010)

    Google Scholar 

  52. Matheron, G.: The intrinsic random functions and their applications. Adv. Appl. Probab. 5, 439–68 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  53. Mattoccia, S., Tombari, F., Di Stefano, L.: Reliable rejection of mismatching candidates for efficient ZNCC template matching. In: Image Processing. ICIP 2008. 15th IEEE International, pp. 849–852 (2008)

  54. Ooi, J., Rao, K.: New insights into correlation-based template matching. In: Proceedings of SPIE, V., Applications of Artificial Intelligence IX, p. 1468. Orlando (1991)

  55. Ortiz, J.M., Deutsch, C.V.: Indicator simulation accounting for multiple-point statistics. Math. Geol. 36, 545–565 (2004)

    Article  Google Scholar 

  56. Ortiz, J.M., Emery, X.: Integrating multiple point statistics into sequential simulation algorithms. In: Leuangthong, O., Deutsch, C.V. (eds.) Geostatistics Banff 2004, pp. 969–978. Springer, Berlin (2005)

    Chapter  Google Scholar 

  57. Park, K.A., Schiedt, C., Caers, J.: Simultaneous conditioning of multiple non-Gaussian geostatistical models to highly nonlinear data using distances in kernel space. In: Proceedings of the 8th International Geostatistical Congress, Santiago, Chile (2008)

  58. Parra, A., Ortiz, J.M.: Conditional multiple-point simulation with a texture synthesis algorithm. In: IAMG 2009 Conference, Stanford University (2009)

  59. Remy, N., Boucher, A., Wu, J.: Applied Geostatistics with SGeMS: A User’s Guide. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  60. Renard, P.: Stochastic hydrogeology: what professionals really need? Ground Water 45, 531–541 (2007)

    Article  MathSciNet  Google Scholar 

  61. Ronayne, M., Gorelick, S., Caers, J.: Identifying discrete geologic structures that produce anomalous hydraulic response: an inverse modeling approach. Water Resour. Res. 44, W08426 (2008)

    Article  Google Scholar 

  62. Sahimi, M.: Large-scale porous media and wavelet transformations. Comput. Sci. Eng. 5 (4), 75–87 (2003)

    Article  Google Scholar 

  63. Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock, 2nd edn. Wiley-VCH, Weinheim (2011)

    Book  MATH  Google Scholar 

  64. Skorstad, A., Hauge, R., Holden, L.: Well conditioning in a fluvial reservoir model. Math. Geol. 31, 857–872 (1999)

    Article  Google Scholar 

  65. Srivastava, R.: An overview of stochastic methods for reservoir characterization. In: Chambers, R.L. (ed.) Stochastic Modeling and Geostatistics: Principles, Methods, and Case Studies, Computer applications, vol. 3, p. 380. AAPG, Tulsa (1994)

    Google Scholar 

  66. Straubhaar, J., Renard, P., Mariethoz, G., Froidevaux, R., Besson, O.: An improved parallel multiple-point algorithm using a list approach. Math. Geosci. 43, 305–328 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  67. Straubhaar, J., Walgenwitz, A., Renard, P., Froidevaux, R.: Optimization issues in 3D multipoint statistics simulation. Paper presented at Geostats 2008, Santiago, Chile (2008)

  68. Strebelle, S.: Conditional simulation of complex geological structures using multiple-point geostatistics. Math. Geol. 34, 1–22 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  69. Strebelle, S., Payrazyan, K., Caers, J.: Modeling of a deepwater turbidite reservoir conditional to seismic data using multiple-point geostatistics. SPE paper 77425 (2002)

  70. Suzuki, S., Caers, J.: A distance-based prior model parameterization for constraining solutions of spatial inverse problems. Math. Geosci. 40, 445–469 (2008)

    Article  MATH  Google Scholar 

  71. Suzuki, S., Caumon, G., Caers, J.: Dynamic data integration for structural modeling: model screening approach using a distance-based model parameterization. Comput. Geosci. 12, 105–119 (2008)

    Article  MATH  Google Scholar 

  72. Suzuki, S., Strebelle, S.: Real-time post-processing method to enhance multiple-point statistics simulation. Paper presented at Petroleum Geostatistics 2007, Cascais, Portugal (2007)

  73. Theodoridis, S., Koutroumbas, K.: Pattern Recognition & Matlab Intro: Pattern Recognition. Academic, New York (2007)

    Google Scholar 

  74. Tjelmeland, H.: Stochastic models in reservoir characterization and Markov random fields for compact objects. Doctoral dissertation, Norwegian University of Science and Technology, Trondheim (1996)

  75. Tjelmeland, H., Eidsvik, J.: Directional Metropolis–Hastings updates for posteriors with non linear likelihood. In: Leuangthong, O., Deutsch, C.V. (eds.) Geostatistics, pp. 95–104. Springer, Banff (2004)

    Google Scholar 

  76. Vargas-Guzmán, J.A.: Unbiased resource evaluations with kriging and stochastic models of heterogeneous rock properties. Nat. Resour. Res. 17, 245–254 (2008)

    Article  Google Scholar 

  77. Vargas-Guzmán, J.A.: The Kappa model of probability and higher-order rock sequences. Comput. Geosci 15, 661–671 (2011)

    Article  Google Scholar 

  78. Wu, J., Zhang, T., Journel, A.G.: A fast FILTERSIM simulation with score-based distance. Math. Geosci. 47, 773–788 (2008)

    Article  Google Scholar 

  79. Zhang, T., Switzer, P., Journel, A.G.: Filter-based classification of training image patterns for spatial simulation. Math. Geol. 38, 63–80 (2006)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mining, Metallurgy and Petroleum Engineering, Amir Kabir University of Technology, Tehran, Iran

    Pejman Tahmasebi & Ardeshir Hezarkhani

  2. Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, 90089-1211, Los Angeles, CA, USA

    Muhammad Sahimi

Authors
  1. Pejman Tahmasebi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ardeshir Hezarkhani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Muhammad Sahimi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Muhammad Sahimi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tahmasebi, P., Hezarkhani, A. & Sahimi, M. Multiple-point geostatistical modeling based on the cross-correlation functions. Comput Geosci 16, 779–797 (2012). https://doi.org/10.1007/s10596-012-9287-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-012-9287-1

Keywords

Search

Navigation