Transport in Porous Media

, Volume 119, Issue 3, pp 521–538 | Cite as

Polymer Flow Through Porous Media: Numerical Prediction of the Contribution of Slip to the Apparent Viscosity

  • F. Zami-Pierre
  • R. de Loubens
  • M. Quintard
  • Y. DavitEmail author


The flow of polymer solutions in porous media is often described using Darcy’s law with an apparent viscosity capturing the observed thinning or thickening effects. While the macroscale form is well accepted, the fundamentals of the pore-scale mechanisms, their link with the apparent viscosity, and their relative influence are still a matter of debate. Besides the complex effects associated with the rheology of the bulk fluid, the flow is also deeply influenced by the mechanisms occurring close to the solid/liquid interface, where polymer molecules can arrange and interact in a complex manner. In this paper, we focus on a repulsive mechanism, where polymer molecules are pushed away from the interface, yielding a so-called depletion layer in the vicinity of the wall. This depletion layer acts as a lubricating film that may be represented by an effective slip boundary condition. Here, our goal is to provide a simple mean to evaluate the contribution of this slip effect to the apparent viscosity. To do so, we solve the pore-scale flow numerically in idealized porous media with a slip length evaluated analytically in a tube. Besides its simplicity, the advantage of our approach is also that it captures relatively well the apparent viscosity obtained from core-flood experiments, using only a limited number of inputs. Therefore, it may be useful in many applications to rapidly estimate the influence of the depletion layer effect over the macroscale flow and its relative contribution compared to other phenomena, such as non-Newtonian effects.


Porous media Polymer Apparent slip Apparent viscosity 

List of symbols

\(\delta \)

Depletion layer thickness

\(\varDelta \mu _{\text {app}}^{\text {XP}}\)

Experimental apparent viscosity drop

\(\ell \)

Slip length

\(\epsilon \)


\(\mu _{0}\)

Newtonian plateau of the bulk viscosity

\(\mu _{\text {app}}\)

Apparent viscosity used in the Darcy’s law

\(\mu _{\text {app}}^{\text {CFD}}\)

Numerical apparent viscosity calculated on the generated packings

\(\mu _{\text {app}}^{\text {XP}}\)

Experimental apparent viscosity measured on the core-flood packings

\(\varrho \)

Viscosity ratio between the bulk and the depletion layer

\(A_{\beta \sigma }\)

Solid/liquid interface


Intrinsic permeability (without a depletion layer)

\(k_{0}^{\text {CFD}}\)

Numerical intrinsic permeability (calculated with a no-slip condition at \(A_{ \beta \sigma }\))

\(k_{0}^{\text {XP}}\)

Experimental intrinsic permeability (measured with the flow of a solvent)

\(R_{\text {eq}}\)

Equivalent radius, defined as \(\sqrt{8 k_{0}/ \epsilon }\)

\(R_{\text {eq}}^{\text {CFD}}\)

Numerical equivalent radius, defined as \(\sqrt{8 k_{0}^{\text {CFD}} / \epsilon }\)

\(R_{\text {eq}}^{\text {XP}}\)

Experimental equivalent radius, defined as \(\sqrt{8 k_{0}^{\text {XP}} / \epsilon }\)



We thank Total for the support of this study. This work was granted access to the HPC resources of CALMIP supercomputing center under the allocation 2016-1511.


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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Institut de Mécanique des Fluides de Toulouse (IMFT)Université de Toulouse, CNRS, INPT, UPSToulouseFrance
  2. 2.Total, CSTJFPauFrance

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