Transport in Porous Media

, Volume 88, Issue 1, pp 133–148 | Cite as

Comparison of Two-Phase Darcy’s Law with a Thermodynamically Consistent Approach

  • Jennifer Niessner
  • Steffen BergEmail author
  • S. Majid Hassanizadeh
Open Access


The extended Darcy’s law is a commonly used equation for the description of immiscible two-phase flow in porous media. It dates back to the 1940s and is essentially an empirical relationship. According to the extended Darcy’s law, pressure gradient and gravity are the only driving forces for the flow of each fluid. Within the last two decades, more advanced and physically based descriptions for multiphase flow in porous media have been developed. In this work, the extended Darcy’s law is compared to a thermodynamically consistent approach which explicitly takes the important role of phase interfaces into account, both as entities and as parameters. In this theoretically derived approach, forces related to capillarity and interfaces appear as driving/resisting forces, in addition to the traditional terms. It turns out that the extended Darcy’s law and the thermodynamically based approach are compatible if either (i) relative permeabilities are a function of saturation only, but capillary pressure is a function of saturation and specific interfacial area or (ii) relative permeabilities are a function of saturation and saturation gradients. Theoretical considerations suggest that the former alternative is only valid in case of reversible displacement while in the general case (irreversible displacement), the latter alternative is relevant.


Porous media Two-phase flow Interfacial area Thermodynamically consistent approach 

List of Symbols

Latin Variables


Specific interfacial area of αβ-interface [m−1]


Gravity [m s−2]


Relative permeability of phase α [−]


Pressure of phase α [Pa]


Capillary pressure [Pa]


Time [s]

\({\underline{v}_{\alpha}, \underline{v}_{\alpha\beta}}\)

Velocity of phase α or αβ-interface, respectively [m s−1]


Area [m2]


Net efficiency [−]


Specific Helmholtz free energy [m2 s−1]


Intrinsic permeability tensor [m2]


Length [m]


External source or sink of phase α [s−1]


Volumetric flux [m3 s−1]


Resistance tensor [Pa s m−2]


Saturation of phase α [−]


Temperature of phase α [K]


Bulk volume [m3]


Work [J]

Greek Symbols




Macroscopic interfacial tension [Pa m]


Dynamic viscosity of phase α [Pa s]


Porosity [−]


Density of phase α [kg m−3]


Pore-scale interfacial tension of αβ-interface [Pa m]


Areal mass density of αβ-interface [kg m−2]


Contact angle [rad]














The authors acknowledge A. W. Cense, J. G. Maas, W. Scherpenisse, P. Doe, and J. Jennings from Shell for their helpful discussions. J. Niessner and S.M. Hassanizadeh are members of the International Research Training Group NUPUS, financed by the German Research Foundation (DFG) and The Netherlands Organisation for Scientific Research (NWO), and thank the DFG (GRK 1398) and NWO/ALW for their support.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Jennifer Niessner
    • 1
  • Steffen Berg
    • 2
    Email author
  • S. Majid Hassanizadeh
    • 3
  1. 1.Institute of Hydraulic EngineeringUniversity of StuttgartStuttgartGermany
  2. 2.Shell International Exploration and Production B.V.Rijswijk (ZH)The Netherlands
  3. 3.Department of Earth Sciences, Faculty of GeosciencesUtrecht UniversityUtrechtThe Netherlands

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