Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

A Multi-Scale Approach to Model Two-Phase Flow in Heterogeneous Porous Media

  • 483 Accesses

  • 11 Citations


The immiscible displacement of a wetting fluid by a non-wetting one in heterogeneous porous media is modeled using a multi-scale network-type analysis: (1) The pressure-controlled immiscible displacement of water by oil in pore-and-throat networks (1st length scale ~ 1 mm) is simulated as a capillary-driven process. (2) The pressure-controlled immiscible displacement in uncorrelated cubic lattices (2nd length scale ~ 1 cm) is simulated as a site percolation process governed by capillary and gravity forces. At this scale, each node represents a network of the previous scale. (3) The rate-controlled immiscible displacement of water by oil in cubic networks (3rd length scale ~ 10 cm), where each node represents a lattice of the previous scale, is simulated by accounting for capillary, gravity, and viscous forces. The multi-scale approach along with the information concerning the pore structure properties of the porous medium can be employed to determine the transient responses of the pressure drop and axial distribution of water saturation, and estimate the effective (up-scaled) relative permeability functions. The method is demonstrated with application to data of highly heterogeneous soils.

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


  1. Abriola L.M., Pinder G.F.: A multiphase approach to the modeling of porous media contamination by organic compounds. 2. Numerical simulation. Water Resour. Res. 21, 19–26 (1985)

  2. Aggelopoulos C.A., Tsakiroglou C.D.: Quantifying soil heterogeneity from solute dispersion experiments. Geoderma 146, 412–424 (2008)

  3. Aggelopoulos C.A., Tsakiroglou C.D.: A multi-flow path approach to model immiscible displacement in undisturbed heterogeneous soil columns. J. Contam. Hydrol. 105, 146–160 (2009)

  4. Bravo M.C., Mariela Araujo M., Lago M.E.: Pore network modeling of two-phase flow in a liquid–(disconnected) gas system. Physica A 375, 1–17 (2007)

  5. Chueh C.C., Secanell M., Bangerth W., Djilali N.: Multi-level adaptive simulation of transient two-phase flow in heterogeneous porous media. Comput. Fluids 39, 1585–1596 (2010)

  6. Inoue M., Simunek J., Shiozawa S., Hopmans J.W.: Simultaneous estimation of soil hydraulic and solute transport parameters from transient infiltration experiments. Adv. Water Resour. 23, 677–688 (2000)

  7. Ioannidis M.A., Chatzis I., Dullien F.A.L.: Macroscopic percolation model of immiscible displacement: effects of buoyancy and spatial structure. Water Resour. Res. 23, 3297–3310 (1996)

  8. Joekar-Niasar V., Hassanizadeh S.M., Dahle H.K.: Non-equilibrium effects in capillary an interfacial area in two-phase flow: dynamic pore-network modeling. J. Fluid Mech. 655, 38–71 (2010)

  9. Kueper B.H., Frind E.O.: Two-phase flow in heterogeneous porous media. 2. Model application. Water Resour. Res. 27, 1059–1070 (1991)

  10. Laroche C., Vizika O.: Two-phase flow properties prediction from small-scale data using pore network modeling. Transp. Porous Media 61, 77–91 (2005)

  11. Lee S.H., Zhou H., Tchelepi H.A.: Adaptive multiscale finite-volume method for nonlinear multiphase transport in heterogeneous formations. J. Comput. Phys. 228, 9036–9058 (2009)

  12. Markicevic B., Bazylad A., Djilali N.: Determination of transport parameters for multiphase flow in porous gas diffusion electrodes using a capillary network model. J. Power Sour. 171, 706–717 (2007)

  13. Neuweiler I., Papafotiou A., Class H., Helmig R.: Estimation of effective parameters for a two-phase flow problem in non-Gaussian heterogeneous porous media. J. Contam. Hydrol. 120-121, 141–156 (2011)

  14. Papafotiou A., Helmig R., Schaap J., Lehmann P., Kaestner A., Flühler H., Neuweiler I., Hassanein R., Ahrenholz B., Tölke J., Peters A., Durner W.: From the pore scale to the lab scale: 3-D lab experiment and numerical simulation of drainage in heterogeneous porous media. Adv. Wat. Res. 31, 1253–1268 (2008)

  15. Ramstad T., Hansen A., Oren P.E.: Flux-dependent percolation transition in immiscible two-phase flows in porous media. Phys. Rev. E 79, 036310 (2009)

  16. Ryazanov A.V., van Dijke M.I.J., Sorbie K.S.: Two-Phase Pore-Network Modelling: Existence of Oil Layers During Water Invasion. Transp. Porous Media 80, 79–99 (2009)

  17. Singh M., Mohanty K.K.: Dynamic modeling of drainage through three-dimensional porous materials. Chem. Eng. Sci. 58, 1–18 (2003)

  18. Teodorovich E.V., Spesivtsev P.E., Noetinger B.: A stochastic approach to the two-phase displacement problem in heterogeneous porous media. Transp. Porous Media 87, 151–177 (2011)

  19. Tsakiroglou C.D.: A method to calculate the multiphase flow properties of heterogeneous porous media by using network simulations. AIChE J. 57, 2618–2628 (2011)

  20. Tsakiroglou C.D., Ioannidis M.A.: Dual-porosity modeling of the pore structure and transport properties of a contaminated soil. Eur. J. Soil Sci. 59, 744–761 (2008)

  21. Tsakiroglou C.D., Sygouni V., Aggelopoulos C.A.: A dynamic network-type simulator to investigate the multiphase flow properties of heterogeneous soils. Vadose Zone J. 9, 285–294 (2010)

  22. Valavanides M.S., Payatakes A.C.: True-to-mechanism model of steady-state two-phase flow in porous media, using decomposition into prototype flows. Adv. Water Res. 24, 385–407 (2001)

  23. Yortsos Y.C., Satik C., Bacri J.-C., Salin D.: Large-scale percolation theory of drainage. Transp. Porous Media 10, 171–195 (1993)

Download references

Author information

Correspondence to Christos D. Tsakiroglou.

Additional information

This article is dedicated to the memory of Professor Alkiviades Payatakes (deceased on 29 November 2009) who taught us the principles and disciplines of scientific research, and inspired us to conduct systematic and hierarchical studies toward the development of true-to-the mechanism numerical models.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Tsakiroglou, C.D. A Multi-Scale Approach to Model Two-Phase Flow in Heterogeneous Porous Media. Transp Porous Med 94, 525–536 (2012).

Download citation


  • Multi-scale models
  • Two-phase flow
  • Heterogeneous media
  • Relative permeability
  • Capillary pressure
  • Network analysis