Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

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


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.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27


  1. Adler, P.M., Jacquin, C.G., Quiblier, J.A.: Flow in simulated porous media. Int. J. Multiph. Flow 16(4), 691–712 (1990)

  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)

  3. Arpat, G.B., Caers, J.: Conditional simulation with patterns. Math. Geol. 39(2), 177–203 (2007)

  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)

  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)

  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)

  7. Biswal, B., Hilfer, R.: Microstructure analysis of reconstructed porous media. Phys. A Stat. Mech. Appl. 266(1), 307–311 (1999)

  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)

  9. Bryant, S., Blunt, M.: Prediction of relative permeability in simple porous media. Phys. Rev. A 46(4), 2004 (1992)

  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)

  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)

  12. Bryant, S., Raikes, S.: Prediction of elastic-wave velocities in sandstones using structural models. Geophysics 60(2), 437–446 (1995)

  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)

  14. Coker, D.A., Torquato, S.: Extraction of morphological quantities from a digitized medium. J. Appl. Phys. 77(12), 6087–6099 (1995)

  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)

  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)

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

  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

  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)

  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)

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

  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)

  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)

  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)

  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)

  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)

  27. Honarkhah, M., Caers, J.: Stochastic simulation of patterns using distance-based pattern modeling. Math. Geosci. 42(5), 487–517 (2010)

  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)

  29. Isaaks, E.: The Application of Monte Carlo Methods to the Analysis of Spatially Correlated Data, Ph.D. Thesis, Stanford University, Stanford, California (1990)

  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)

  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)

  32. Kirkpatrick, S., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)

  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)

  34. Krishnan, S., Journel, A.G.: Spatial connectivity: from variograms to multiple-point measures. Math. Geol. 35(8), 915–925 (2003)

  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)

  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)

  37. Levitz, P.: Off-lattice reconstruction of porous media: critical evaluation, geometrical confinement and molecular transport. Adv. Colloid Interf. Sci. 76, 71–106 (1998)

  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)

  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)

  40. Lu, B., Torquato, S.: Nearest-surface distribution functions for polydispersed particle systems. Phys. Rev. A 45(8), 5530 (1992)

  41. Manwart, C., Torquato, S., Hilfer, R.: Stochastic reconstruction of sandstones. Phys. Rev. E 62(1), 893 (2000)

  42. Mariethoz, G., Renard, P., Straubhaar, J.: The direct sampling method to perform multiple-point geostatistical simulations. Water Resour. Res. 46(11), (2010)

  43. Okabe, H., Blunt, M.J.: Prediction of permeability for porous media reconstructed using multiple-point statistics. Phys. Rev. E 70(6), 066135 (2004)

  44. Okabe, H., Blunt, M.J.: Pore space reconstruction using multiple-point statistics. J. Petrol. Sci. Eng. 46(1), 121–137 (2005)

  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)

  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)

  47. Øren, P.E., Bakke, S., Arntzen, O.J.: Extending predictive capabilities to network models. SPE J. RICHARDSON 3, 324–336 (1998)

  48. Quiblier, J.A.: A new three-dimensional modeling technique for studying porous media. J. Colloid Interf. Sci. 98(1), 84–102 (1984)

  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)

  50. Reeves, C.R., Rowe, J.E.: Genetic Algorithms Principle and Perspectives: A Guide to GA Theory. Kluwer Academic, Dordrecht (2003)

  51. Roberts, A.P.: Statistical reconstruction of three-dimensional porous media from two-dimensional images. Phys. Rev. E 56(3), 3203 (1997)

  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)

  53. Roberts, J.N., Schwartz, L.M.: Grain consolidation and electrical conductivity in porous media. Phys. Rev. B 31(9), 5990 (1985)

  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)

  55. Sahimi, M.: Heterogeneous Materials. Springer, New York (2003)

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

  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)

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

  59. Tahmasebi, P., Hezarkhani, A., Sahimi, M.: Multiple-point geostatistical modeling based on the cross-correlation functions. Comput. Geosci. 16(3), 779–797 (2012)

  60. Tahmasebi, P., Sahimi, M.: Reconstruction of three-dimensional porous media using a single thin section. Phys. Rev. E 85(6), 066709 (2012)

  61. Tahmasebi, P., Sahimi, M.: Cross-correlation function for accurate reconstruction of heterogeneous media. Phys. Rev. Lett. 110(7), 078002 (2013)

  62. Tahmasebi, P., Sahimi, M. (2015). Reconstruction of nonstationary disordered materials and media: watershed transform and cross-correlation function. Phys. Rev. E (in press)

  63. Tahmasebi, P., Sahimi, M., In: K. Vafai (ed.) Handbook of Porous Media, 3rd ed., CRC Press (to be published, 2015)

  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)

  65. Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties, vol. 16. Springer, New York (2002)

  66. Torquato, S., Lu, B.: Chord-length distribution function for two-phase random media. Phys. Rev. E 47(4), 2950 (1993)

  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)

  68. Xu, W., Tran, T, Srivastava, R., Journel, A.: Integration seismic data in reservoir modeling: the collocated cokriging alternative. SPE paper 24742, (1992)

  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)

  70. Zachary, C.E., Torquato, S.: Improved reconstructions of random media using dilation and erosion processes. Phys. Rev. E 84(5), 056102 (2011)

  71. Zhang, T., Switzer, P., Journel, A.: Filter-based classification of training image patterns for spatial simulation. Mathematical Geology 38(1), 63–80 (2006)

Download references


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

Author information

Correspondence to Muhammad Sahimi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tahmasebi, P., Sahimi, M. Geostatistical Simulation and Reconstruction of Porous Media by a Cross-Correlation Function and Integration of Hard and Soft Data. Transp Porous Med 107, 871–905 (2015). https://doi.org/10.1007/s11242-015-0471-3

Download citation


  • Cross-correlation function
  • Training image
  • Data integration
  • Conditional simulation