Transport in Porous Media

, Volume 123, Issue 1, pp 45–99 | Cite as

Review of Steady-State Two-Phase Flow in Porous Media: Independent Variables, Universal Energy Efficiency Map, Critical Flow Conditions, Effective Characterization of Flow and Pore Network

  • Marios S. ValavanidesEmail author


In many applications of two-phase flow in porous media, a wetting phase is used to displace through a network of pore conduits as much as possible of a non-wetting phase, residing in situ. The energy efficiency of this physical process may be assessed by the ratio of the flow rate of the non-wetting phase over the total mechanical power externally provided and irreversibly dissipated within the process. Fractional flow analysis, extensive simulations implementing the DeProF mechanistic model, as well as a recent retrospective examination of laboratory studies have revealed universal systematic trends of the energy efficiency in terms of the actual independent variables of the process, namely the capillary number, Ca, and the flow rate ratio, r. These trends can be cast into an energy efficiency map over the (Ca, r) domain of independent variables. The map is universal for all types of non-wetting/wetting phase porous medium systems. It demarcates the efficiency of steady-state two-phase flow processes in terms of pertinent system parameters. The map can be used as a tool for designing more efficient processes, as well as for the normative characterization of two-phase flows, as to the predominance of capillary or viscous effects. This concept is based on the existence of a unique locus of critical flow conditions, for which the energy efficiency takes locally maximum values. The locus shape depends on the physicochemical characteristics of the non-wetting phase/wetting phase/porous medium system, and it shows a significant mutation as the externally imposed flow conditions change the type of flow, from capillary- to viscosity-dominated. The locus can be approached by an S-type functional form in terms of the capillary number and the system properties (viscosity ratio, wettability, pore network geometry, etc.), suggesting that formative criteria can be derived for flow characterization in any system. A new, extended definition of the capillary number is also proposed that effectively takes into account the critical properties of all the system constituents. When loci of critical flow conditions pertaining to processes with different viscosity ratio in the same pore network, are expressed in terms of this true-to-mechanism capillary number, they collapse into a unique locus. In this context, a new methodology for the effective characterization of pore networks is proposed.


Two-phase flow Porous media Steady-state Energy efficiency Critical flow conditions 

List of symbols

Subscripts, Superscripts and Embelishments


A tilde embelishment indicates a dimensional variable

No tilde embelishment denotes a dimensionless variable


An asterisk (as superscript) indicates a value corresponding to critical flow conditions


Subscript indicating a non-wetting phase variable


Subscript indicating a wetting phase variable

Physical Variables—Latin Letters


Capillary number


Energy utilization factor, or, energy efficiency


Fractional flow

\( \tilde{k} \)

Absolute permeability of the porous medium


Relative permeability

\( \tilde{p} \)

Macroscopic pressure


Non-wetting/wetting flow rate ratio


Wetting phase saturation

\( \tilde{q} \)

Flow rate

\( U \)

Superficial velocity, or, flow intensity

\( x \)

Reduced macroscopic pressure gradient


Vector containing the geometrical and topological parameters of the pore network

\( \tilde{z} \)

Coordinate length along the macroscopic flow direction

Physical Variables—Greek Letters

\( \tilde{\gamma }_{nw} \)

Interfacial tension between the wetting and the non-wetting phase

\( \theta_{0}^{a} \)

Static contact angle for advancing non-wetting/wetting phase meniscus

\( \theta_{0}^{r} \)

Static contact angle for receding non-wetting/wetting phase meniscus

\( \kappa \)

Non-wetting/wetting phase viscosity ratio

\( \lambda \)

Non-wetting/wetting phase mobility ratio

\( \tilde{\mu } \)

Dynamic viscosity



Critical flow condition(s)


Non-wetting phase


Non-wetting phase/wetting phase/porous medium


Wetting phase



Parts of this research work were carried out within the ImproDeProF project, financially supported by The European Union (European Social Fund) and Greek national resources in the frameworks of the ‘Archimedes III: Funding of Research Groups in TEI of Athens’ (MlS 379389) a RTD project of the ‘Education and Lifelong Learning’ Operational Program. The present work was also partly supported by the Research Council of Norway through its Centres of Excellence funding scheme, Project Number 262644. The author thanks Prof. Gregory Kamvyssas (TEI Western Greece), Dr. Christos Tsakiroglou (ICE/HT-FORTH), Prof. Marios Ioannides (Univ. of Waterloo), Dr. Zubair Kalam (ADCO), Professors Alex Hansen, Dick Bedeaux, Signe Kjelstrup (NTNU), Prof. Knut Jørgen Maløy (University of Oslo), Dr. Olga Vizika (IFP EN), Prof. Apostolos Kantzas (University of Calgary), Prof. Majid Hassanizadeh (Utrecht Univ.) and Prof. Michael Celia (Princeton Univ.) for stimulating discussions on topics of this work (in chronological order).


  1. Aggelopoulos, C.A., Tsakiroglou, C.D.: The effect of micro-heterogeneity and capillary number on capillary pressure and relative permeability curves of soils. Geoderma 148, 25–34 (2008). CrossRefGoogle Scholar
  2. Allen, B., Stacey, B.C., Bar-Yam, Y.: Multiscale information theory and the marginal utility of information. Entropy 19(6), 273 (2017). CrossRefGoogle Scholar
  3. Alvarado, V., Manrique, E.: Enhanced oil recovery: an update review. Energies 3(9), 1529–1575 (2010). CrossRefGoogle Scholar
  4. American Petroleum Institute: Recommended Practice RP 40 “Recommended Practices for Core Analysis”, 2nd edn. American Petroleum Institute, Washington (1998)Google Scholar
  5. Aris, R.: Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover, Illinois (1962)Google Scholar
  6. Armstrong, R.T., McClure, J.E., Berrill, M.A., Rücker, M., Schlüter, S., Berg, S.: Beyond Darcy’s law: the role of phase topology and ganglion dynamics for two-fluid flow. Phys. Rev. E 94, 043113 (2016). CrossRefGoogle Scholar
  7. Aursjo, O., Erpelding, M., Tallakstad, K.T., Flekkøy, E.G., Hansen, A., Maloy, K.J.: Film flow dominated simultaneous flow of two viscous incompressible fluids through a porous medium. Front. Phys. 2(63), 1–9 (2014). Google Scholar
  8. Avraam, D.G., Payatakes, A.C.: Flow regimes and relative permeabilities during steady-state two-phase flow in porous media. J. Fluid Mech. 293, 207–236 (1995). CrossRefGoogle Scholar
  9. Avraam, D.G., Payatakes, A.C.: Flow mechanisms, relative permeabilities and coupling effects in steady-state two-phase flow in porous media. Case of strong wettability. Ind. Eng. Chem. Res. 38(3), 778–786 (1999). CrossRefGoogle Scholar
  10. Bakke, J.O.H: Reservoir monitoring and up-scaling. PoreLab Group Kick-Off Meeting and 1st International Workshop, Oslo, Norway, 6–8 Sept (2017)Google Scholar
  11. Bazylak, A.: Liquid water visualization in PEM fuel cells: a review. Int. J. Hydrogen Energy 34(9), 3845–3857 (2009). CrossRefGoogle Scholar
  12. Bejan, A.: Convection Heat Transfer. Wiley, Hoboken (2013). ISBN 978-0-470-90037-6CrossRefGoogle Scholar
  13. Bentsen, R.G.: Interfacial coupling in vertical, two-phase flow through porous media. J. Pet. Sci. Technol. 23, 1341–1380 (2005)CrossRefGoogle Scholar
  14. Berg, C.F.: Permeability description by characteristic length, tortuosity, constriction and porosity. Transp. Porous Media 103, 381–400 (2014). CrossRefGoogle Scholar
  15. Berg, S., Ott, H., Klapp, S.A., Schwing, A., Neiteler, R., Brussee, N., Makurat, A., Leu, L., Enzmann, F., Schwarz, J.-O., Wolf, F., Kersten, M., Irvine, S., Stampanoni, M.: Real-time 3D imaging of Haines jumps in porous media flow. Proc. Natl. Acad. Sci. 110(10), 3755–3759 (2013). CrossRefGoogle Scholar
  16. Burnside, N.M., Naylor, M.: Review and implications of relative permeability of CO2/brine systems and residual trapping of CO2. Int. J. Greenhouse Gas Control 23, 1–11 (2014)CrossRefGoogle Scholar
  17. Celia, M.A.: Geological storage of captured carbon dioxide as a large-scale carbon mitigation option. Water Resour. Res. 53, 3527–3533 (2017). CrossRefGoogle Scholar
  18. Charpentier, J.-C.: In the frame of globalization and sustainability, process intensification, a path to the future of chemical and process engineering (molecules into money). Chem. Eng. J. 134, 84–92 (2007). CrossRefGoogle Scholar
  19. Clayton, S.: Keynote lecture. In: Rock & Fluid Physics: Academic and Industrial Perspectives Conference. Shell Technology Centre, Amsterdam, NL, Sep 15–18 (2014)Google Scholar
  20. Cobos, S., Carvalho, M.S., Alvarado, V.: Flow of oil–water emulsions through a constricted capillary. Int. J. Multiph. Flow 35, 507–515 (2009). CrossRefGoogle Scholar
  21. Constantinides, G.N., Payatakes, A.C.: Effects of precursor wetting films in immiscible displacement through porous media. Transp. Porous Media 38, 291–317 (2000). CrossRefGoogle Scholar
  22. Cushman, J.H.: The physics of fluids in hierarchical porous media: angstroms to miles. In: Theory and Applications of Transport in Porous Media, vol. 10. Kluwer. ISBN 0792347420, 467 pp (1997)Google Scholar
  23. Datta, S.S., Ramakrishnan, T.S., Weitz, D.A.: Mobilization of a trapped non-wetting fluid from a three dimensional porous medium. Phys. Fluids (2014). Google Scholar
  24. Doshi, V., del Maestro, A., Clark, A.: “2016 Oil and Gas Trends” Industry Perspectives, Strategyand, PwC report. Accessed 16 Sep 2017
  25. Fusseis, F., Xiao, X., Schrank, C., De Carlo, F.: A brief guide to synchrotron radiation-based microtomography in (structural) geology and rock mechanics. J. Struct. Geol. 65, 1–16 (2014). CrossRefGoogle Scholar
  26. Georgiadis, A., Berg, S., Makurat, A., Maitland, G., Ott, H.: Pore-scale microcomputed-tomography imaging: nonwetting-phase cluster-size distribution during drainage and imbibitions. Phys. Rev. E 88(033002), 1–9 (2013). Google Scholar
  27. Greco, R.: Soil water content inverse profiling from single TDR waveforms. J. Hydrol. 317, 325–339 (2006). CrossRefGoogle Scholar
  28. Guillen, V.R., Carvalho, M.S., Alvarado, V.: Pore scale and macroscopic displacement mechanisms in emulsion flooding. Transp. Porous Med. 94, 197–206 (2012a). CrossRefGoogle Scholar
  29. Guillen, V.-R., Romero, M.-I., da Silveira, Marcio, Carvalho, M.-S., Alvarado, V.: Capillary-driven mobility control in macro emulsion flow in porous media. Int. J. Multiph. Flow 43, 62–65 (2012b). CrossRefGoogle Scholar
  30. Hinkley, R.E., Dias, M.M., Payatakes, A.C.: On the motion of oil ganglia in porous media. Physicochem. Hydrodyn. 8(2), 185–211 (1987)Google Scholar
  31. Honarpour, M., Koederitz, L., Harvey, A.H.: Relative Permeability of Petroleum Reservoirs. CRC Press, Boca Raton (1986). ISBN 0-8493-5739-XGoogle Scholar
  32. Hsu, C.-T.: Dynamic modeling of convective heat transfer in porous media. In: Vafai, K. (ed.) Handbook of Porous Media. CRC Press. ISBN 0-8247-2747-9 (2005)Google Scholar
  33. 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(7), 677–688 (2000). CrossRefGoogle Scholar
  34. Khan, F.I., Husain, T., Hejazi, R.: An overview and analysis of site remediation technologies. J. Environ. Manage. 71, 95–122 (2004)CrossRefGoogle Scholar
  35. Kjarstad, J., Johnsson, F.: Resources and future supply of oil. Energy Policy 37, 441–464 (2009). CrossRefGoogle Scholar
  36. Knudsen, H.A., Hansen, A.: Relation between pressure and fractional flow in two-phase flow in porous media. Phys. Rev. E 65(056310), 1–10 (2002). Google Scholar
  37. Krummel, A.T., Datta, S.S., Münster, S., Weitz, D.A.: Visualizing multiphase flow and trapped fluid configurations in a model three-dimensional porous medium. AIChE J. 59, 1022–1029 (2013). CrossRefGoogle Scholar
  38. Lake, L.W.: Enhanced Oil Recovery. Prentice-Hall, Englewood Cliffs (1989)Google Scholar
  39. Langnes, G.L., Robertson, J.O., Mehdizadeh, A., Torabzadeh, J.: Waterflooding. In: Donaldson, E.C., Chilingarian, G.V., Yen, T.F. (eds.) Enhanced oil recovery I, fundamentals and analyses (developments in petroleum science; 17A), pp. 260–262. Elsevier Science Publishers B.V. ISBN 0444-42206-4 (1985)Google Scholar
  40. Lenormand, R., Touboul, E., Zarcone, C.: Numerical models and experiments on immiscible displacements in porous media. J. Fluid Mech. 189, 165–187 (1988)CrossRefGoogle Scholar
  41. Lenormand, R.: Liquids in porous media. J. Phys.: Condens. Matter. 2, 79–88. (1990)
  42. Maitland, G.: Towards a low carbon fossil fuel future with gas & CCS—challenges and opportunities. In: Invited Talk, Rock & Fluid Physics: Academic and Industrial Perspectives Conference. Shell Technology Centre. Amsterdam, NL, Sep 15–18 (2014)Google Scholar
  43. Markicevic, B., Djilali, N.: Two-scale modeling in porous media: relative permeability predictions. Phys. Fluids 18(033101), 1–13 (2006)Google Scholar
  44. Müller, N.: Supercritical CO2-brine relative permeability experiments in reservoir rocks—literature review and recommendations. Transp. Porous Media 87, 367–383 (2011). CrossRefGoogle Scholar
  45. Naar, J., Wygal, R.J., Henderson, J.H.: Imbibition relative permeability in unconsolidated porous media. Soc. Petrol. Eng. J. 2, 13–17 (1962). CrossRefGoogle Scholar
  46. Niven, R.K.: Steady state of a dissipative flow-controlled system and the maximum entropy production principle. Phys. Rev. E 80(021113), 1–15 (2009). Google Scholar
  47. Nguyen, V.H., Sheppard, A.P., Knackstedt, M.A., Pinczewski, W.: The effect of displacement rate on imbibition relative permeability and residual saturation. J. Petrol. Sci. Eng. 52, 54–70 (2006). CrossRefGoogle Scholar
  48. Oughanem, R., Youssef, S., Bauer, D., Peysson, Y., Maire, E., Vizika, O.: A multi-scale investigation of pore structure impact on the mobilization of trapped oil by surfactant injection. Transp. Porous Media 109, 673–692 (2015). CrossRefGoogle Scholar
  49. Pak, T., Butler, I.B., Geiger, S., Dijke, M.I.J., Sorbie, K.S.: Droplet fragmentation: 3D imaging of a previously unidentified pore-scale process during multiphase flow in porous media. Proc. Natl. Acad. Sci. 112(7), 1947–1952 (2015). CrossRefGoogle Scholar
  50. Payatakes, A.C.: Dynamics of oil ganglia during immiscible displacement in water-wet porous media. Ann. Rev. Fluid Mech. 14, 365–393 (1982). CrossRefGoogle Scholar
  51. Payatakes, A.C., Constantinides, G.N., Valavanides, M.S.: Hierarchical theoretical models: an informal introduction. In: Dassios, G. et al. (eds.) Mathematical Methods in Scattering Theory and Biomedical Technology. Pitman Research Notes in Mathematics Series, No. 390, pp. 158–169. Addison Wesley Longman Ltd. ISBN 0582368049 (1998)Google Scholar
  52. Perez-Mercader, J. (2004). Coarse-graining, scaling, and hierarchies. In: Gell-Mann, M., Tsallis, C. (eds.) Nonextensive Entropy: Interdisciplinary Applications, Santa-Fe Institute Studies in the Science of Complexity. Oxford University Press. ISBN 0-19-515976-4Google Scholar
  53. Pinder, G.F., Celia, M.A.: Subsurface Hydrology. Wiley Interscience, Hoboken (2006). ISBN 978-0-471-74243-2CrossRefGoogle Scholar
  54. Ponce, R.V., Alvarado, V., Carvalho, M.S.: Water-alternating-macroemulsion reservoir simulation through capillary number-dependent modelling. J Braz. Soc. Mech. Sci. Eng. (2017). Google Scholar
  55. Ramakrishnan, T.S., Wasan, D.T.: Effect of capillary number on the relative permeability function for two-phase flow in porous media. Powder Technol. 48(2), 99–124 (1986). CrossRefGoogle Scholar
  56. Ramstad, T., Idowu, N., Nardi, C., Øren, P.-E.: Relative permeability calculations from two-phase flow simulations directly on digital images of porous rocks. Transp. Porous Med. 94, 487–504 (2012). CrossRefGoogle Scholar
  57. Reynolds, C.A., Menke, H., Andrew, M., Blunt, M.J., Krevor, S.: Dynamic fluid connectivity during steady-state multiphase flow in a sandstone. In: Proceedings of the National Academy of Sciences Early Edition, 1–6. (2017)
  58. Richards, F.J.: A flexible growth function for empirical use. J. Exp. Bot. 10(2), 290–300 (1959). CrossRefGoogle Scholar
  59. Ricketts, J.H., Head, G.: A five-parameter logistic equation for investigating asymmetry of curvature in baroreflex studies. Am. J. Physiol. Regul. Integr. Comp. Physiol. 277(2), R441–R454. (1999)
  60. Rücker, M., Berg, S., Armstrong, R.T., Georgiadis, A., Ott, H., Schwing, A., Neiteler, R., Brussee, N., Makurat, A., Leu, L., Wolf, M., Khan, F., Enzmann, F., Kersten, M.: From connected pathway flow to ganglion dynamics. Geophys. Res. Lett. 42, 3888–3894 (2015). CrossRefGoogle Scholar
  61. Rücker, M., Bartels, W.-B., Unsal1, E., Berg, S., Brussee, N., Coorn, A., Bonnin, A.: The formation of microemulsion at flow conditions in rock, SCA2017-041. In: 2017 International Symposium of the Society of Core Analysts, Vienna, Austria, 21 Aug–1 Sept. Accessed 13 Sep 2017
  62. Sinha, S., Hansen, A.: Effective rheology of immiscible two-phase flow in porous media. Europhys. Lett. 99(4), 1–6 (2012). CrossRefGoogle Scholar
  63. Sinha, S., Bender, A.T., Danczyk, M., Keepseagle, K., Prather, C.A., Bray, J.M., Thrane, L.W., Seymour, J.D., Codd, S.I., Hansen, A.: Effective rheology of two-phase flow in three-dimensional porous media: experiment and simulation. Transport in Porous Media, published online 13 June 2017, 1–18. (2017)
  64. Tallakstad, K.T., Knudsen, H.A., Ramstad, T., Løvoll, G., Maløy, K.J., Toussaint, R., Flekkøy, E.G.: Steady-state two-phase flow in porous media: statistics and transport properties. Phys. Rev. Lett. 102(074502), 1–4 (2009)Google Scholar
  65. Thomas, S.: Enhanced oil recovery—an overview. Oil Gas Sci. Technol. Rev. IFP 63(1), 9–19 (2008). CrossRefGoogle Scholar
  66. Tsai, T.M., Miksis, M.J.: Dynamics of a drop in a constricted capillary tube. J. Fluid Mech. 274, 197–217 (1994). CrossRefGoogle Scholar
  67. Tsakiroglou, C.D., Payatakes, A.C.: Characterization of the pore structure of reservoir rocks with the aid of serial sectioning analysis, mercury porosimetry and network simulation. Adv. Water Resour. 23, 773–789 (2000). CrossRefGoogle Scholar
  68. Tsakiroglou, C.D., Avraam, D.G., Payatakes, A.C.: Transient and steady-state relative permeabilities from two-phase flow experiments in planar pore networks. Adv. Water Resour. 30, 1981–1992 (2007). CrossRefGoogle Scholar
  69. Tsakiroglou, C.D., Aggelopoulos, C.A., Terzi, K., Avraam, D.G., Valavanides, M.S.: Steady-state two-phase relative permeability functions of porous media: a revisit. Int. J. Multiph. Flow 73, 34–42 (2015). CrossRefGoogle Scholar
  70. Unsal, E., Broens, M., Armstrong, R.T.: Pore scale dynamics of microemulsion formation. Langmuir 32, 7096–7108 (2016). CrossRefGoogle Scholar
  71. Valavanides, M.S.: Macroscopic theory of two-phase flow in porous media based on integration of pore scale phenomena. PhD Dissertation University of Patras, National Archive of PhD Theses—National Documentation Center (in Greek). Accessed 15 Sep 2017 (1998)
  72. Valavanides, M.S., Constantinides, G.N., Payatakes, A.C.: Mechanistic model of steady-state two-phase flow in porous media based on ganglion dynamics. Transp. Porous Media 30, 267–299 (1998). CrossRefGoogle Scholar
  73. Valavanides, M.S., Payatakes, A.C.: A true-to-mechanism model of steady-state two-phase flow in porous media, including the contribution of the motion of ganglia and droplets. In: Bentley, L.R., et al (eds.) Computational Methods in Water Resources XIII, vol. 1, pp. 239–243. A.A Balkema, Rotterdam. ISBN 9058091236 (2000)Google Scholar
  74. 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 Resour. 24(3–4), 385–407 (2001)CrossRefGoogle Scholar
  75. Valavanides, M.S., Payatakes, A.C.: Effects of pore network characteristics on steady-state two-phase flow based on a true-to-mechanism model (DeProF). In: SPE-78516-MS, 10th ADIPEC Abu Dhabi International Petroleum Exhibition & Conference, Abu Dhabi, UAE, October 13–16. (2002)
  76. Valavanides, M.S., Payatakes, A.C. (2003). Prediction of optimum operating conditions for steady-state two-phase flow in pore network systems using the DeProF true-to-mechanism theoretical model. In: SCA2003-18, 2003 International Symposium of the Society of Core Analysts, Pau, France, 21–25 Sep. Accessed 24 Aug 2015
  77. Valavanides, M.S., Payatakes, A.C.: Wetting film effects on steady-state two-phase flow in pore networks using the DeProF theoretical model. In: SPE-88713-MS, 11th ADIPEC Abu Dhabi International Petroleum Exhibition & Conference, Abu Dhabi, UAE, October 10–13. (2004)
  78. Valavanides, M.S.: Steady-state two-phase flow in porous media: review of progress in the development of the DeProF theory bridging pore- to statistical thermodynamics-scales. Oil Gas Sci. Technol. 67, 787–804 (2012). CrossRefGoogle Scholar
  79. Valavanides, M.S.: Transformer for steady-state relative permeability data. In: ImproDeProF Project. Accessed 10 Sep 2015 (2015)
  80. Valavanides, M.S., Totaj, E., Tsokopoulos, M.: Retrospective examination of relative permeability data on steady-state 2-phase flow in porous media transformation of rel-perm data (k ro, k rw) into operational efficiency data (f EU) ImproDeProF/Archimedes III, project internal report. (2015)
  81. Valavanides, M.S., Daras, T.: Definition and counting of configurational microstates in steady-state two-phase flows in pore networks. Entropy 18(054), 1–28 (2016). Google Scholar
  82. Valavanides, M.S., Totaj, E., Tsokopoulos, M.: Energy efficiency characteristics in steady-state relative permeability diagrams of two-phase flow in porous media. J. Petrol. Sci. Eng. 147, 181–201 (2016). CrossRefGoogle Scholar
  83. Valavanides, M.S.: Flow structure maps for two-phase flow in model pore networks. Predictions based on extensive, DeProF model simulations. Oil Gas Sci. Technol. (2017)
  84. Van de Merwe, W., Nicol, W.: Trickle flow hydrodynamic multiplicity: experimental observations and pore-scale capillary mechanism. Chem. Eng. Sci. 64, 1267–1284 (2009). CrossRefGoogle Scholar
  85. Vizika, O., Avraam, D.G., Payatakes, A.C.: On the role of viscosity ratio during low-capillary-number forced imbibition in porous media. J. Colloid Interface Sci. 165, 386–401 (1994)CrossRefGoogle Scholar
  86. Wyckoff, R.D., Botset, H.G.: The flow of gas-liquid mixtures through unconsolidated sands. Physics 7, 325–345 (1936). CrossRefGoogle Scholar
  87. Yiotis, A.G., Talon, L., Salin, D.: Blob population dynamics during immiscible two-phase flows in reconstructed porous media. Phys. Rev. E 87, 033001 (2013). CrossRefGoogle Scholar
  88. Youssef, S., Rosenberg, E., Deschamps, H., Oughanem, R., Maire, E., Mokso, R.: Oil ganglia dynamics in natural porous media during surfactant flooding captured by ultra-fast x-ray microtomography. In: SCA 2014-23, Symposium of the Society of Core Analysts, France, 11–18 Sep, 1–12 (2014)Google Scholar
  89. Zinchenko, A.Z., Davis, R.H.: Emulsion flow through a packed bed with multiple drop breakup. J. Fluid Mech. 725, 611–663 (2013). CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil EngineeringTEI AthensAthensGreece

Personalised recommendations