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

An Algorithm for 3D Pore Space Reconstruction from a 2D Image Using Sequential Simulation and Gradual Deformation with the Probability Perturbation Sampler


The construction of a faithful 3D pore space model of a porous medium that could reproduce the macroscopic behavior of that medium is of great interest in various fields including medicine, material science, hydrology and petroleum engineering. A computationally efficient algorithm is developed that uses the probability perturbation method and sequential multiple-point statistics simulations to generate 3D stochastic and equiprobable representations of random porous media when only a 2D thin section image is available. By employing the probability perturbation method as a gradual deformation technique, the pore patterns of a single 2D image are deformed to generate a series of 2D stochastically simulated images. The 3D pore structure is then generated by simply stacking the 2D-simulated images. The quality of the 3D reconstruction is critically dependent on the rate of deformation and a simple general procedure for choosing this parameter is presented. Various criteria such as porosity, two-point auto-correlation function, multiple-point connectivity function, local percolation probability, absolute permeability obtained by lattice-Boltzmann method (LBM), formation factor and two-phase relative permeability calculations are used to validate the results. The method is tested on two random porous solids; Berea Sandstone and synthetic Silica, for which directly measured 3D micro-CT images are available. The stochastically reconstructed 3D pore space preserves the low- and high-order spatial statistics, the macroscopic flow properties and the microstructure of the 3D micro-CT images.

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


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

  2. Bakke S., Oren P.E.: 3-D pore-scale modeling of sandstones and flow simulations in the pore networks. SPE J. 2, 136–149 (1997)

  3. Caers J.: History matching under a training image-based geological model constraint: Soc. Petrol. Eng., SPE paper No. 74716, 218–226 (2003)

  4. Caers J.: Comparing the gradual deformation with the probability perturbation method for solving inverse problems. Math. Geol. (2007). doi:10.1007/s11004-006-9064-6

  5. Caers J., Hoffman B.T.: The probability perturbation method: an alternative Bayesian approach to history matching. Math. Geol. 38, 81–100 (2006)

  6. Caers J., Zhang T.: Multiple-point geostatistics: a quantitative vehicle for integration geologic analogs into multiple reservoir model, Integration of outcrop and modern analog data in reservoir models. AAPG Memoir 80, 383–394 (2004)

  7. Caers J., Hoffman B.T., Strebelle S., Wen X.-H.: Probabilistic integration of geological information, seismic and production data. The Leading Edge 25, 240–244 (2006)

  8. Chen S., Doolen G.D.: Lattice Boltzmann method for fluid flows. Ann. Rev. Fluid Mech. 30, 329–364 (1998)

  9. Coles M.E.: Developments in synchrotron X-ray micro-tomography with applications to flow in porous media. SPE Reserv. Eval. Eng. 1, 288–296 (1998)

  10. Deutsch C.V.: Reservoir modeling with publicly available software. Comput. Geosci. 25, 355–363 (1999)

  11. Dong, H., Touati, M., Blunt, M.J.: Pore network modeling: analysis of pore size distribution of Arabian core samples. SPE 105156 (2007)

  12. Dunsmuir J.H. et al.: X-ray microtomography. A new tool for the characterization of porous media. Proceedings of 1991 SPE Annual Technical Conference, Dallas, Texas, SPE 22860, 423–430 (1991)

  13. Ferreol B., Rothman D.H.: Lattice-Boltzmann simulations of flow-through fontainebleau sandstone. Transp. Porous Media 20, 3–20 (1995)

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

  15. Grunau D., Chen S.Y., Eggert K.: A lattice Boltzmann model for multiphase fluid-flows. Phys. Fluids A-Fluid Dyn. 5, 2557–2562 (1993)

  16. Hajizadeh A., Safekordi A., Farhadpour F.A.: A multiple-point statistics algorithm for 3D pore space reconstruction from 2D images. Adv. Wat. Resour. 34, 1256–1267 (2011)/ doi:10.1016/j.advwatres.2011.06.003

  17. Hazlett R.D.: Statistical characterization and stochastic modeling of pore networks in relation to fluid flow. Math. Geol. 29, 801–822 (1997)

  18. Hilfer R.: Review on scale dependent characterization of the microstructure of porous media. Transp. Porous Media 46, 373–390 (2002)

  19. Hoffman, B.T., Caers, J.: Geostatistical history matching using the regional probability perturbation method. SPE Annual Conference and Technical Exhibition, Denver, 5–8 Oct 2003 SPE 84409 (2003)

  20. Hoffman, B.T., Caers, J.: History matching with the regional probability perturbation method—applications to a North Sea reservoir In: Proceedings to the ECMOR IX, Cannes, 29 Aug–2 Sept 2004

  21. Hu L.-Y., Blanc G., Noetinger B.: Gradual deformation and iterative calibration of sequential simulations. Math. Geol. 33, 475–489 (2001)

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

  23. Journel A.: Combining knowledge from diverse information Sources: an alternative to Bayesian analysis. Math. Geol. 34, 573–596 (2002)

  24. Manwart C., Torquato S., Hilfer R.: Stochastic reconstruction of sandstones. Phys. Rev. E 62, 893–899 (2000)

  25. Manwart, C., Aaltosalmi, U., Koponen, A., Hilfer, R., Timonen, J.: Lattice- Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media. Phys. Rev. E, 66. (2002). doi:10.1103/PhysRevE.66.016702

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

  27. Okabe H., Blunt M.J.: Pore-space reconstruction using multiple-point statistics. J. Pet. Sci. Eng. 46, 121–137 (2004b)

  28. Okabe, H., Blunt, M.J.: Space reconstruction of vuggy carbonates using microtomography and multiple-point statistics. Wat. Resour. Res. 43 (2007). doi:10.1029/2006WR005680

  29. Oren P.E., Bakke S.: Process based reconstruction of sandstones and prediction of transport properties. Transp. Porous Media 46, 311–343 (2002)

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

  31. Spanne P. et al.: Synchrotron computed microtomography of porous-media-topology and transports. Phys. Rev. Lett. 73, 2001–2004 (1994)

  32. Stauffer D., Aharnony A.: Introduction to Percolation Theory, revised 2nd ed. Taylor & Francis, London (1994)

  33. Strebelle S.: Sequential Simulation Drawing Structures from Training Images Doctoral Dissertation. Stanford University, Stanford (2001)

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

  35. Sukop, M.C.: (2012)

  36. Sukop M.C., Thorne D.T. Jr.: Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers. Springer, New York (2006)

  37. Valvatne P.H., Blunt M.J.: Predictive pore-scale modeling of two-phase flow in mixed wet media. Wat. Resour. Res. 40, W07406 (2004). doi:10.1029/2003WR002627

  38. Wu K., Nunan N., Crawford J., Young I., Ritz K.: An efficient markov chain model for the simulation of heterogeneous soil structure. Soil Sci. Soc. Am. J. 68, 346–351 (2004)

  39. Wu K., Van Dijke M., Couples G., Jiang Z., Ma J., Sorbie K.: 3D stochastic modeling of heterogeneous porous media-applications to reservoir rocks. Transp. Porous Media 65, 443–467 (2006). doi:10.1007/s11242-006-0006-z

Download references

Author information

Correspondence to Alireza Hajizadeh.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hajizadeh, A., Farhadpour, Z. An Algorithm for 3D Pore Space Reconstruction from a 2D Image Using Sequential Simulation and Gradual Deformation with the Probability Perturbation Sampler. Transp Porous Med 94, 859–881 (2012).

Download citation


  • Porous media reconstruction
  • Micro-CT imaging
  • Pore network modeling
  • Multiple-point statistics
  • Probability perturbation method
  • Lattice-Boltzmann method