Transport in Porous Media

, Volume 92, Issue 1, pp 187–212 | Cite as

A Semi-Analytical Model for Two Phase Immiscible Flow in Porous Media Honouring Capillary Pressure

  • F. Hussain
  • Y. CinarEmail author
  • P. Bedrikovetsky


This article describes a semi-analytical model for two-phase immiscible flow in porous media. The model incorporates the effect of capillary pressure gradient on fluid displacement. It also includes a correction to the capillarity-free Buckley–Leverett saturation profile for the stabilized-zone around the displacement front and the end-effects near the core outlet. The model is valid for both drainage and imbibition oil–water displacements in porous media with different wettability conditions. A stepwise procedure is presented to derive relative permeabilities from coreflood displacements using the proposed semi-analytical model. The procedure can be utilized for both before and after breakthrough data and hence is capable to generate a continuous relative permeability curve unlike other analytical/semi-analytical approaches. The model predictions are compared with numerical simulations and laboratory experiments. The comparison shows that the model predictions for drainage process agree well with the numerical simulations for different capillary numbers, whereas there is mismatch between the relative permeability derived using the Johnson–Bossler–Naumann (JBN) method and the simulations. The coreflood experiments carried out on a Berea sandstone core suggest that the proposed model works better than the JBN method for a drainage process in strongly wet rocks. Both methods give similar results for imbibition processes.


Two phase flow Porous medium Semi-analytical model Coreflood Relative permeability 

List of Symbols


Velocity of the shock front


Fractional flow of water phase

\({{f}_{\rm s}^{{\prime}}}\)

Derivative of fractional flow w.r.t. saturation


Fraction of the water-wet rock surface


Leverett function




Oil relative permeability


Maximum oil relative permeability


Water relative permeability


Maximum water relative permeability


Core length


Exponent for Corey-type power law




Capillary pressure


Water phase saturation


Initial water saturation


Irreducible water saturation


Maximum water saturation, equals to unity minus residual/critical oil saturation




Dimensionless time, p.v. injection


Pore volumes produced


Pore volumes of oil produced


Pore volumes of water produced




Linear coordinate, from injectors towards producers


Dimensionless linear coordinate


Total mobility of the oil-water fluid


Potential for capillary forces


Capillary-viscous ratios


Self-sharpening large scale (slow) coordinate


Travelling wave fast coordinate near to the displacement front


Fast coordinate near to the core outlet




Fluid viscosity


Interfacial tension


Contact angle

Subscripts and Superscripts

W, O

Water, oil


Initial (of water saturation)


Boundary value on the injector (saturations, flux)






Stabilized zone


End effect




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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.University of New South WalesSydneyAustralia
  2. 2.University of AdelaideAdelaideAustralia

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