Transport in Porous Media

, Volume 6, Issue 2, pp 115–142 | Cite as

Displacement of a Newtonian fluid by a non-Newtonian fluid in a porous medium

  • Y.-S. Wu
  • K. Pruess
  • P. A. Witherspoon


This paper presents an analytical Buckley-Leverett-type solution for one-dimensibnal immiscible displacement of a Newtonian fluid by a non-Newtonian fluid in porous media. The non-Newtonian fluid viscosity is assumed to be a function of the flow potential gradient and the non-Newtonian phase saturation. To apply this method to field problems a practical procedure has been developed which is based on the analytical solution and is similar to the graphic technique of Welge. Our solution can be regarded as an extension of the Buckley-Leverett method to Non-Newtonian fluids. The analytical result reveals how the saturation profile and the displacement efficiency are controlled not only by the relative permeabilities, as in the Buckley-Leverett solution, but also by the inherent complexities of the non-Newtonian fluid. Two examples of the application of the solution are given. One application is the verification of a numerical model, which has been developed for simulation of flow of immiscible non-Newtonian and Newtonian fluids in porous media. Excellent agreement between the numerical and analytical results has been obtained using a power-law non-Newtonian fluid. Another application is to examine the effects of non-Newtonian behavior on immiscible displacement of a Newtonian fluid by a power-law non-Newtonian fluid.

Key words

Non-Newtonian fluids Buckley-Leverett immiscible displacement power-law fluids rheological models Welge method fractional flow theory enhanced oil recovery 


Roman Letters


cross-sectional area (m2)


fractional flow of Newtonian phase


fractional flow of non-Newtonian phase


gravitational acceleration vector (m/s2)


magnitude of the gravitational acceleration (m/s2)


power law coefficient (Pa sn)


absolute permeability (m2)


relative permeability of non-Newtonian phase


power-law exponential index


cumulative displaced Newtonian fluid (m3)


pressure (Pa)


capillary pressure (Pa)


pressure of Newtonian phase (Pa)


pressure of non-Newtonian phase (Pa)


injection rate of non-Newtonian fluid (m3/s)


cumulative injection rate (m3)


saturation at moving front (m)


Newtonian phase saturation


irreducible Newtonian phase saturation


non-Newtonian saturation


connate non-Newtonian saturation

\(\bar S\)nn

average saturation of non-Newtonian phase in swept zone


distance from inlet, coordinate (m)


distance to shock saturation front (m)


distance of saturation Snn from the inlet (m)


time (s)


Darcy velocity (m/s)


total flux (m/s)


Darcy velocity of Newtonian phase (m/s)


Darcy velocity of non-Newtonian phase (m/s)

Greek Letters


angle between horizontal plane and flow direction


shear rate (s-1)


apparent viscosity (Pa s)


effective viscosity (Pa s n m1 - n)


viscosity of Newtonian fluid (Pa s)


equivalent viscosity of non-Newtonian fluid (Pa s)


density of Newtonian fluid (kg/m3)


density of non-Newtonian fluid (kg/m3)


shear stress (Pa)


porosity of porous media


flow potential (Pa)













relative to Newtonian phase


relative to non-Newtonian phase


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Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Y.-S. Wu
    • 1
  • K. Pruess
    • 1
  • P. A. Witherspoon
    • 1
  1. 1.Earth Sciences DivisionLawrence Berkeley LaboratoryBerkeleyUSA

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