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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. Davit
Article
  • 282 Downloads

Abstract

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.

Keywords

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

\(\epsilon \)

Porosity

\(\mu _{0}\)

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

\(k_{0}\)

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 \(\sqrt{8 k_{0}^{\text {XP}} / \epsilon }\)

Notes

Acknowledgements

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.

References

  1. Adams, D., Matheson, A.: Computation of dense random packings of hard spheres. J. Chem. Phys. 56(5), 1989–1994 (1972)CrossRefGoogle Scholar
  2. Agarwal, U., Dutta, A., Mashelkar, R.: Migration of macromolecules under flow: the physical origin and engineering implications. Chem. Eng. Sci. 49(11), 1693–1717 (1994)CrossRefGoogle Scholar
  3. Amundarain, J., Castro, L., Rojas, M., Siquier, S., Ramírez, N., Müller, A., Sáez, A.: Solutions of xanthan gum/guar gum mixtures: shear rheology, porous media flow, and solids transport in annular flow. Rheol. Acta 48(5), 491–498 (2009)CrossRefGoogle Scholar
  4. Aubert, J., Tirrell, M.: Effective viscosity of dilute polymer solutions near confining boundaries. J. Chem. Phys. 77(1), 553–561 (1982)CrossRefGoogle Scholar
  5. Ausserre, D., Hervet, H., Rondelez, F.: Concentration dependence of the interfacial depletion layer thickness for polymer solutions in contact with nonadsorbing walls. Macromolecules 19(1), 85–88 (1986)CrossRefGoogle Scholar
  6. Auvray, L.: Solutions de macromolécules rigides: effets de paroi, de confinement et d’orientation par un écoulement. J. Phys. 42(1), 79–95 (1981)CrossRefGoogle Scholar
  7. Barnes, H.A.: A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure. J. Nonnewton. Fluid Mech. 56(3), 221–251 (1995)CrossRefGoogle Scholar
  8. Bird, R., Carreau, P.: A nonlinear viscoelastic model for polymer solutions and melts. Chem. Eng. Sci. 23(5), 427–434 (1968)CrossRefGoogle Scholar
  9. Bird, R., Armstrong, R., Hassager, O., Curtiss, C.: Dynamics of Polymeric Liquids. Vol. 2: Kinetic Theory, vol. 2. Wiley, New York (1977)Google Scholar
  10. Blake, T.: Slip between a liquid and a solid: DM Tolstoi’s (1952) theory reconsidered. Colloids Surf. 47, 135–145 (1990)CrossRefGoogle Scholar
  11. Brochard, F., De Gennes, P.: Shear-dependent slippage at a polymer/solid interface. Langmuir 8(12), 3033–3037 (1992)CrossRefGoogle Scholar
  12. Chauveteau, G.: Rodlike polymer solution flow through fine pores: influence of pore size on rheological behavior. J. Rheol. 26(2), 111–142 (1982)CrossRefGoogle Scholar
  13. Chauveteau, G.: Concentration dependence of the effective viscosity of polymer solutions in small pores with repulsive or attractive walls. J. Colloid Interface Sci. 100, 41–54 (1984)CrossRefGoogle Scholar
  14. Chauveteau, G., Kohler, B.: Influence of microgels in polysaccharide solutions on their flow behavior through porous media. Soc. Pet. Eng. J. 24(03), 361–368 (1984)CrossRefGoogle Scholar
  15. Churaev, N., Sobolev, V., Somov, A.: Slippage of liquids over lyophobic solid surfaces. J. Colloid Interface Sci. 97(2), 574–581 (1984)CrossRefGoogle Scholar
  16. Cohen, Y., Metzner, A.: Apparent slip flow of polymer solutions. J. Rheol. 29(1), 67–102 (1985)CrossRefGoogle Scholar
  17. Cuenca, A., Bodiguel, H.: Submicron flow of polymer solutions: slippage reduction due to confinement. Phys. Rev. Lett. 110(10), 108,304 (2013)Google Scholar
  18. De Gennes, P.: Scaling Concepts in Polymer Physics. Cornell University Press, New York (1979)Google Scholar
  19. De Gennes, P.: Polymer solutions near an interface. 1. Adsorption and depletion layers. Macromolecules 14(6), 1637–1644 (1981)CrossRefGoogle Scholar
  20. Fåhræus, R., Lindqvist, T.: The viscosity of the blood in narrow capillary tubes. Am. J. Physiol. Legacy Content 96(3), 562–568 (1931)Google Scholar
  21. Fletcher, A., et al.: Measurements of polysaccharide polymer properties in porous media. In: SPE International Symposium on Oilfield Chemistry. Society of Petroleum Engineers (1991)Google Scholar
  22. Gao, C.: Viscosity of partially hydrolyzed polyacrylamide under shearing and heat. J. Pet. Explor. Prod. Technol. 3(3), 203–206 (2013)CrossRefGoogle Scholar
  23. Gogarty, W.: Mobility control with polymer solutions. Soc. Pet. Eng. J. 7(2), 161–173 (1967)CrossRefGoogle Scholar
  24. González, J., Müller, A., Torres, M., Sáez, A.: The role of shear and elongation in the flow of solutions of semi-flexible polymers through porous media. Rheol. Acta 44(4), 396–405 (2005)CrossRefGoogle Scholar
  25. Groisman, A., Steinberg, V.: Elastic turbulence in a polymer solution flow. Nature 405(6782), 53–55 (2000)CrossRefGoogle Scholar
  26. Hatzikiriakos, S., Dealy, J.: Wall slip of molten high density polyethylene. I. Sliding plate rheometer studies. J. Rheol. 35(4), 497–523 (1991)CrossRefGoogle Scholar
  27. Huh, C., et al.: Polymer retention in porous media. In: SPE/DOE Enhanced Oil Recovery Symposium. Society of Petroleum Engineers (1990)Google Scholar
  28. Joanny, J., Leibler, L., De Gennes, P.: Effects of polymer solutions on colloid stability. J. Polym. Sci. Polym. Phys. Ed. 17(6), 1073–1084 (1979)CrossRefGoogle Scholar
  29. Joshi, Y., Lele, A., Mashelkar, R.: Slipping fluids: a unified transient network model. J. Nonnewton. Fluid Mech. 89(3), 303–335 (2000)CrossRefGoogle Scholar
  30. Kalyon, D.: Apparent slip and viscoplasticity of concentrated suspensions. J. Rheol. 49(3), 621–640 (2005)CrossRefGoogle Scholar
  31. Lasseux, D., Parada, F., Tapia, J., Goyeau, B.: A macroscopic model for slightly compressible gas slip-flow in homogeneous porous media. Phys. Fluids 26(5), 053,102 (2014)Google Scholar
  32. Lund, T., et al.: Polymer retention and inaccessible pore volume in north sea reservoir material. J. Pet. Sci. Eng. 7(1–2), 25–32 (1992)CrossRefGoogle Scholar
  33. Ma, H., Graham, M.: Theory of shear-induced migration in dilute polymer solutions near solid boundaries. Phys. Fluids 17(8), 083,103 (2005)Google Scholar
  34. Maerker, J., et al.: Shear degradation of partially hydrolyzed polyacrylamide solutions. Soc. Pet. Eng. J. 15(04), 311–322 (1975)CrossRefGoogle Scholar
  35. Morais, A., Seybold, H., Herrmann, H., Andrade Jr., J.: Non-newtonian fluid flow through three-dimensional disordered porous media. Phys. Rev. Lett. 103(19), 194502 (2009)CrossRefGoogle Scholar
  36. Morel, D., et al.: Polymer injection in deep offshore field: the dalia angola case. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2008)Google Scholar
  37. Morel, D., et al.: Dalia/camelia polymer injection in deep offshore field angola learnings and in situ polymer sampling results. In: SPE Asia Pacific Enhanced Oil Recovery Conference. Society of Petroleum Engineers (2015)Google Scholar
  38. Navier, C.L.: Mémoire sur les lois du mouvement des fluides. Mémoires de l’Académie Royale des Sciences de l’Institut de France 6, 389–440 (1823)Google Scholar
  39. Nordlund, M., Stanic, M., Kuczaj, A., Frederix, E., Geurts, B.: Improved piso algorithms for modeling density varying flow in conjugate fluid-porous domains. J. Comput. Phys. 306, 199–215 (2016)CrossRefGoogle Scholar
  40. Omari, A., Moan, M., Chauveteau, G.: Hydrodynamic behavior of semirigid polymer at a solid-liquid interface. J. Rheol. 33(1), 1–13 (1989)CrossRefGoogle Scholar
  41. Patankar, S.: Numerical heat transfer and fluid flow. Corp. New York, Washington, Series in computational methods in mechanics and thermal sciences. Hemisphere Pub (1980)Google Scholar
  42. Priezjev, N., Troian, S.: Molecular origin and dynamic behavior of slip in sheared polymer films. Phys. Rev. Lett. 92(1), 018,302 (2004)Google Scholar
  43. Rhie, C., Chow, W.: Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J. 21(11), 1525–1532 (1983)CrossRefGoogle Scholar
  44. Rouse Jr., P.E.: A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys. 21(7), 1272–1280 (1953)CrossRefGoogle Scholar
  45. Seright, R., et al.: Injectivity characteristics of eor polymers. In: SPE annual technical conference and exhibition, Society of Petroleum Engineers (2008)Google Scholar
  46. Seright, R., et al.: New insights into polymer rheology in porous media. SPE J. 16(1), 35–42 (2011)CrossRefGoogle Scholar
  47. Sherwood, J., Dusting, J., Kaliviotis, E., Balabani, S.: The effect of red blood cell aggregation on velocity and cell-depleted layer characteristics of blood in a bifurcating microchannel. Biomicrofluidics 6(2), 024,119 (2012)Google Scholar
  48. Skauge, T., Kvilhaug, O., Skauge, A.: Influence of polymer structural conformation and phase behaviour on in-situ viscosity. In: IOR 2015-18th European Symposium on Improved Oil Recovery (2015)Google Scholar
  49. Sochi, T.: Slip at fluid-solid interface. Polym. Rev. 51(4), 309–340 (2011)CrossRefGoogle Scholar
  50. Sorbie, K.: Depleted layer effects in polymer flow through porous media: I. Single capillary calculations. J. Colloid Interface Sci. 139(2), 299–314 (1990)CrossRefGoogle Scholar
  51. Sorbie, K.: Polymer-Improved Oil Recovery. Springer, New York (1991)CrossRefGoogle Scholar
  52. Sorbie, K., Huang, Y.: Rheological and transport effects in the flow of low-concentration xanthan solution through porous media. J. Colloid Interface Sci. 145(1), 74–89 (1991)CrossRefGoogle Scholar
  53. Sorbie, K., Huang, Y.: The effect of ph on the flow behavior of xanthan solution through porous media. J. Colloid Interface Sci. 149(2), 303–313 (1992)CrossRefGoogle Scholar
  54. Stavland, A., et al.: Polymer flooding-flow properties in porous media versus rheological parameters. In: SPE EUROPEC/EAGE Annual Conference and Exhibition. Society of Petroleum Engineers (2010)Google Scholar
  55. Tolstoi, D.: Mercury sliding on glass. Dokl. Akad. Nauk SSSR 85, 1329–1335 (1952)Google Scholar
  56. Uematsu, Y.: Nonlinear electro-osmosis of dilute non-adsorbing polymer solutions with low ionic strength. Soft Matter 11(37), 7402–7411 (2015)CrossRefGoogle Scholar
  57. UTCHEM Technical Documentation. University of Texas at Austin (2000)Google Scholar
  58. Weller, H., Tabor, G., Jasak, H., Fureby, C.: A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput. Phys. 12(6), 620–631 (1998)CrossRefGoogle Scholar
  59. Zami-Pierre, F., de Loubens, R., Quintard, M., Davit, Y.: Transition in the flow of power-law fluids through isotropic porous media. Phys. Rev. Lett. 117(7), 074,502 (2016)Google Scholar
  60. Zitha, P., Chauveteau, G., Zaitoun, A.: Permeability dependent propagation of polyacrylamides under near-wellbore flow conditions. In: SPE International Symposium on Oilfield Chemistry (1995)Google Scholar

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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