Transport in Porous Media

, Volume 107, Issue 3, pp 871–905 | Cite as

Geostatistical Simulation and Reconstruction of Porous Media by a Cross-Correlation Function and Integration of Hard and Soft Data

  • Pejman Tahmasebi
  • Muhammad SahimiEmail author


A new method is proposed for geostatistical simulation and reconstruction of porous media by integrating hard (quantitative) and soft (qualitative) data with a newly developed method of reconstruction. The reconstruction method is based on a cross-correlation function that we recently proposed and contains global multiple-point information about the porous medium under study, which is referred to cross-correlation-based simulation (CCSIM). The porous medium to be reconstructed is represented by a reference image (RI). Some of the information contained in the RI is represented by a training image (TI). In unconditional simulation, only the TI is used to reconstruct the RI, without honoring any particular data. If some soft data, such as a seismic image, and hard data are also available, they are integrated with the TI and conditional CCSIM method in order to reconstruct the RI, by honoring the hard data exactly. To illustrate the method, several two- and three-dimensional porous media are simulated and reconstructed, and the results are compared with those provided by the RI, as well as those generated by the traditional two-point geostatistical simulation, namely the co-sequential Gaussian simulation. To quantify the accuracy of the simulations and reconstruction, several statistical properties of the porous media, such as their porosity distribution, variograms, and long-range connectivity, as well as two-phase flow of oil and water through them, are computed. Excellent agreement is demonstrated between the results computed with the simulated model and those obtained with the RI.


Cross-correlation function Training image Data integration  Conditional simulation 



Work at USC was supported in part by the RPSEA Consortium.


  1. Adler, P.M., Jacquin, C.G., Quiblier, J.A.: Flow in simulated porous media. Int. J. Multiph. Flow 16(4), 691–712 (1990)CrossRefGoogle Scholar
  2. Almeida, A.S., Journel, A.G.: Joint simulation of multiple variables with a Markov-type coregionalization model. Math. Geol. 26(5), 565–588 (1994)CrossRefGoogle Scholar
  3. Arpat, G.B., Caers, J.: Conditional simulation with patterns. Math. Geol. 39(2), 177–203 (2007)CrossRefGoogle Scholar
  4. Bakke, S., Øren, P.E.: 3-D pore-scale modelling of sandstones and flow simulations in the pore networks. SPE J. Richardson 2, 136–149 (1997)CrossRefGoogle Scholar
  5. Bekri, S., Xu, K., Yousefian, F., Adler, P.M., Thovert, J.-F., Muller, J., Iden, K., Psyllos, A., Stubos, A.K., Ioannidis, M.A.: Pore geometry and transport properties in North Sea chalk. J. Petrol. Sci. Eng. 25(3), 107–134 (2000)CrossRefGoogle Scholar
  6. Biswal, B., Øren, P.E., Held, R.J., Bakke, S., Hilfer, R.: Stochastic multiscale model for carbonate rocks. Phys. Rev. E 75(6), 061303 (2007)CrossRefGoogle Scholar
  7. Biswal, B., Hilfer, R.: Microstructure analysis of reconstructed porous media. Phys. A Stat. Mech. Appl. 266(1), 307–311 (1999)CrossRefGoogle Scholar
  8. Biswal, B., Manwart, C., Hilfer, R., Bakke, S., Øren, P.E.: Quantitative analysis of experimental and synthetic microstructures for sedimentary rock. Phys. A Stat. Mech. Appl. 273(3), 452–475 (1999)CrossRefGoogle Scholar
  9. Bryant, S., Blunt, M.: Prediction of relative permeability in simple porous media. Phys. Rev. A 46(4), 2004 (1992)CrossRefGoogle Scholar
  10. Bryant, S.L., King, P.R., Mellor, D.W.: Network model evaluation of permeability and spatial correlation in a real random sphere packing. Transp. Porous Media 11(1), 53–70 (1993)CrossRefGoogle Scholar
  11. Bryant, S.L., Mellor, D.W., Cade, C.A.: Physically representative network models of transport in porous media. AIChE J. 39(3), 387–396 (1993)CrossRefGoogle Scholar
  12. Bryant, S., Raikes, S.: Prediction of elastic-wave velocities in sandstones using structural models. Geophysics 60(2), 437–446 (1995)CrossRefGoogle Scholar
  13. Coelho, D., Thovert, J.F., Adler, P.M.: Geometrical and transport properties of random packings of spheres and aspherical particles. Phys. Rev. E 55(2), 1959 (1997)CrossRefGoogle Scholar
  14. Coker, D.A., Torquato, S.: Extraction of morphological quantities from a digitized medium. J. Appl. Phys. 77(12), 6087–6099 (1995)CrossRefGoogle Scholar
  15. Coker, D.A., Torquato, S., Dunsmuir, J.H.: Morphology and physical properties of Fontainebleau sandstone via a tomographic analysis. J. Geophys. Res. Solid Earth (1978–2012) 101(B8), 17497–17506 (1996)CrossRefGoogle Scholar
  16. Coles, M.E., Hazlett, R.D., Muegge, E.L., Jones, K.W., Andrews, B., Dowd, B., Siddons, P., Peskin, A., Spanne, P., Soll, W.E.: Developments in synchrotron X-ray microtomography with applications to flow in porous media. SPE Reservoir Evaluat. Eng. 1(4), 288–296 (1998)CrossRefGoogle Scholar
  17. Deutsch, C.V., Journel, A.G.: GSLIB: Geostatistical Software Library and User’s Guide, 2nd edn. Oxford University Press, Oxford (1998)Google Scholar
  18. Doyen, P., Guidish, T., & de Buyl, M. (1989). Monte Carlo simulation of lithology from seismic data in a channel-sand reservoir. In: 1st European Conference on the Mathematics of Oil Recovery Google Scholar
  19. Dunsmuir, JH., Ferguson, SR., D’Amico, KL., Stokes, JP.: X-ray microtomography: a new tool for the characterization of porous media, SPE paper 22860. In: Proc 66th Annual Technical Conf. and Exhibition Soc. Petroleum Engineers, Dallas, October 6–9 (1991)Google Scholar
  20. Fredrich, J.T.: 3D imaging of porous media using laser scanning confocal microscopy with application to microscale transport processes. Phys. Chem. Earth Part A Solid Earth Geod. 24(7), 551–561 (1999)CrossRefGoogle Scholar
  21. Goovaerts, P.: Geostatistics for Natural Resources Evaluation. Oxford University Press, Oxford (1997)Google Scholar
  22. Hajizadeh, A., Safekordi, A., Farhadpour, F.A.: A multiple-point statistics algorithm for 3D pore space reconstruction from 2D images. Adv. Water Resour. 34(10), 1256–1267 (2011)CrossRefGoogle Scholar
  23. 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(5), 056311 (2007)CrossRefGoogle Scholar
  24. Hamzehpour, H., Sahimi, M.: Development of optimal models of porous media by combining static and dynamic data: the porosity distribution. Phys. Rev. E 74(2), 026308 (2006)CrossRefGoogle Scholar
  25. Hidajat, I., Rastogi, A., Singh, M., Mohanty, K. K.: Transport properties of porous media from thin-sections. In: SPE Latin American and Caribbean Petroleum Engineering Conference. Society of Petroleum Engineers (2001)Google Scholar
  26. Holt, R.M., Fjaer, E., Torsaeter, O., Bakke, S.: Petrophysical laboratory measurements for basin and reservoir evaluation. Marine Petrol. Geol. 13(4), 383–391 (1996)CrossRefGoogle Scholar
  27. Honarkhah, M., Caers, J.: Stochastic simulation of patterns using distance-based pattern modeling. Math. Geosci. 42(5), 487–517 (2010)CrossRefGoogle Scholar
  28. Ioannidis, M.A., Chatzis, I.: On the geometry and topology of 3D stochastic porous media. J Colloid Interf. Sci. 229(2), 323–334 (2000)CrossRefGoogle Scholar
  29. Isaaks, E.: The Application of Monte Carlo Methods to the Analysis of Spatially Correlated Data, Ph.D. Thesis, Stanford University, Stanford, California (1990)Google Scholar
  30. Jasti, J.K., Jesion, G., Feldkamp, L.: Microscopic imaging of porous media with X-ray computer tomography. SPE Form. Eval. 8(3), 189–193 (1993)CrossRefGoogle Scholar
  31. Jiao, Y., Stillinger, F.H., Torquato, S.: A superior descriptor of random textures and its predictive capacity. Proc. Natl. Acad. Sci. 106(42), 17634–17639 (2009)CrossRefGoogle Scholar
  32. Kirkpatrick, S., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)CrossRefGoogle Scholar
  33. Knackstedt, M.A., Sheppard, A.P., Sahimi, M.: Pore network modelling of two-phase flow in porous rock: the effect of correlated heterogeneity. Adv. Water Resour. 24(3), 257–277 (2001)CrossRefGoogle Scholar
  34. Krishnan, S., Journel, A.G.: Spatial connectivity: from variograms to multiple-point measures. Math. Geol. 35(8), 915–925 (2003)CrossRefGoogle Scholar
  35. Latham, J.P., Lu, Y., Munjiza, A.: A random method for simulating loose packs of angular particles using tetrahedra. Geotechnique 51(10), 871–879 (2001)CrossRefGoogle Scholar
  36. Latham, J.P., Munjiza, A., Lu, Y.: On the prediction of void porosity and packing of rock particulates. Powder Technol. 125(1), 10–27 (2002)CrossRefGoogle Scholar
  37. Levitz, P.: Off-lattice reconstruction of porous media: critical evaluation, geometrical confinement and molecular transport. Adv. Colloid Interf. Sci. 76, 71–106 (1998)CrossRefGoogle Scholar
  38. Liang, Z.R., Fernandes, C.P., Magnani, F.S., Philippi, P.C.: A reconstruction technique for three-dimensional porous media using image analysis and Fourier transforms. J. Petrol. Sci. Eng. 21(3), 273–283 (1998)CrossRefGoogle Scholar
  39. Liang, Z.R., Philippi, P.C., Fernandes, C.P., Magnani, F.S.: Prediction of permeability from the skeleton of three-dimensional pore structure. SPE Reservoir Evaluat. Eng. 2(2), 161–168 (1999)CrossRefGoogle Scholar
  40. Lu, B., Torquato, S.: Nearest-surface distribution functions for polydispersed particle systems. Phys. Rev. A 45(8), 5530 (1992)CrossRefGoogle Scholar
  41. Manwart, C., Torquato, S., Hilfer, R.: Stochastic reconstruction of sandstones. Phys. Rev. E 62(1), 893 (2000)CrossRefGoogle Scholar
  42. Mariethoz, G., Renard, P., Straubhaar, J.: The direct sampling method to perform multiple-point geostatistical simulations. Water Resour. Res. 46(11), (2010)Google Scholar
  43. Okabe, H., Blunt, M.J.: Prediction of permeability for porous media reconstructed using multiple-point statistics. Phys. Rev. E 70(6), 066135 (2004)CrossRefGoogle Scholar
  44. Okabe, H., Blunt, M.J.: Pore space reconstruction using multiple-point statistics. J. Petrol. Sci. Eng. 46(1), 121–137 (2005)CrossRefGoogle Scholar
  45. Øren, P.E., Bakke, S.: Process based reconstruction of sandstones and prediction of transport properties. Transp. Porous Media 46(2–3), 311–343 (2002)CrossRefGoogle Scholar
  46. Øren, P.E., Bakke, S.: Reconstruction of Berea sandstone and pore-scale modelling of wettability effects. J. Petrol. Sci. Eng. 39(3), 177–199 (2003)CrossRefGoogle Scholar
  47. Øren, P.E., Bakke, S., Arntzen, O.J.: Extending predictive capabilities to network models. SPE J. RICHARDSON 3, 324–336 (1998)CrossRefGoogle Scholar
  48. Quiblier, J.A.: A new three-dimensional modeling technique for studying porous media. J. Colloid Interf. Sci. 98(1), 84–102 (1984)CrossRefGoogle Scholar
  49. Rasaei, M.R., Sahimi, M.: Upscaling and simulation of waterflooding in heterogeneous reservoirs using wavelet transformations: application to the SPE-10 model. Transport Porous Media 72(3), 311–338 (2008)CrossRefGoogle Scholar
  50. Reeves, C.R., Rowe, J.E.: Genetic Algorithms Principle and Perspectives: A Guide to GA Theory. Kluwer Academic, Dordrecht (2003)Google Scholar
  51. Roberts, A.P.: Statistical reconstruction of three-dimensional porous media from two-dimensional images. Phys. Rev. E 56(3), 3203 (1997)CrossRefGoogle Scholar
  52. Roberts, A.P., Torquato, S.: Chord-distribution functions of three-dimensional random media: approximate first-passage times of gaussian processes. Phys. Rev. E 59(5), 4953 (1999)CrossRefGoogle Scholar
  53. Roberts, J.N., Schwartz, L.M.: Grain consolidation and electrical conductivity in porous media. Phys. Rev. B 31(9), 5990 (1985)CrossRefGoogle Scholar
  54. Sahimi, M.: Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and simulated annealing. Rev. Modern Phys. 65(4), 1393 (1993)CrossRefGoogle Scholar
  55. Sahimi, M.: Heterogeneous Materials. Springer, New York (2003)Google Scholar
  56. Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock, 2nd edn. Wiley, Weinheim (2011)CrossRefGoogle Scholar
  57. Spanne, P., Thovert, J.F., Jacquin, C.J., Lindquist, W.B., Jones, K.W., Adler, P.M.: Synchrotron computed microtomography of porous media: topology and transports. Phys. Rev. Lett. 73(14), 2001 (1994)CrossRefGoogle Scholar
  58. Strebelle, S.: Conditional simulation of complex geological structures using multiple-point statistics. Math. Geol. 34(1), 1–21 (2002)CrossRefGoogle Scholar
  59. Tahmasebi, P., Hezarkhani, A., Sahimi, M.: Multiple-point geostatistical modeling based on the cross-correlation functions. Comput. Geosci. 16(3), 779–797 (2012)CrossRefGoogle Scholar
  60. Tahmasebi, P., Sahimi, M.: Reconstruction of three-dimensional porous media using a single thin section. Phys. Rev. E 85(6), 066709 (2012)CrossRefGoogle Scholar
  61. Tahmasebi, P., Sahimi, M.: Cross-correlation function for accurate reconstruction of heterogeneous media. Phys. Rev. Lett. 110(7), 078002 (2013)CrossRefGoogle Scholar
  62. Tahmasebi, P., Sahimi, M. (2015). Reconstruction of nonstationary disordered materials and media: watershed transform and cross-correlation function. Phys. Rev. E (in press)Google Scholar
  63. Tahmasebi, P., Sahimi, M., In: K. Vafai (ed.) Handbook of Porous Media, 3rd ed., CRC Press (to be published, 2015)Google Scholar
  64. Tahmasebi, P., Sahimi, M., Caers, J.: MS-CCSIM: accelerating pattern-based geostatistical simulation of categorical variables using a multi-scale search in Fourier space. Comput. Geosci. 67, 75–88 (2014)CrossRefGoogle Scholar
  65. Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties, vol. 16. Springer, New York (2002)Google Scholar
  66. Torquato, S., Lu, B.: Chord-length distribution function for two-phase random media. Phys. Rev. E 47(4), 2950 (1993)CrossRefGoogle Scholar
  67. Tsotsis, T.T., Patel, H., Najafi, B.F., Racherla, D., Knackstedt, M.A., Sahimi, M.: Overview of laboratory and modeling studies of carbon dioxide sequestration in coal beds. Ind. Eng. Chem. Res. 43(12), 2887–2901 (2004)CrossRefGoogle Scholar
  68. Xu, W., Tran, T, Srivastava, R., Journel, A.: Integration seismic data in reservoir modeling: the collocated cokriging alternative. SPE paper 24742, (1992)Google Scholar
  69. Yeong, C.L.Y., Torquato, S.: Reconstructing random media. II. Three-dimensional media from two-dimensional cuts. Phys. Rev. E 58(1), 224 (1998)CrossRefGoogle Scholar
  70. Zachary, C.E., Torquato, S.: Improved reconstructions of random media using dilation and erosion processes. Phys. Rev. E 84(5), 056102 (2011)CrossRefGoogle Scholar
  71. Zhang, T., Switzer, P., Journel, A.: Filter-based classification of training image patterns for spatial simulation. Mathematical Geology 38(1), 63–80 (2006)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Mining, Metallurgy, and Petroleum EngineeringAmir Kabir University of TechnologyTehranIran
  2. 2.Mork Family Department of Chemical Engineering and Materials ScienceUniversity of Southern CaliforniaLos AngelesUSA

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