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Numerical Investigation of the Apparent Viscosity Dependence on Darcy Velocity During the Flow of Shear-Thinning Fluids in Porous Media

  • Antonio Rodríguez de CastroEmail author
  • Mehrez Agnaou
Article
  • 39 Downloads

Abstract

The viscosity exhibited by shear-thinning fluids within the interstices of a porous medium differs depending on pore dimensions and injection velocity. Therefore, predicting the macroscopic value of viscosity required as input to Darcy’s law is challenging and needs accurate identification of the characteristic microscopic dimensions dominating global pressure losses. The most common approach consists of defining an apparent “in situ” shear rate which can be used in the bulk constitutive equation of the fluid to predict viscosity during flow through the porous medium. The dependence of this apparent shear rate on Darcy velocity has traditionally been assumed to be linear, which is appropriate in the case of Newtonian fluids and power-law fluids. However, yield stress and plateau viscosities that can potentially affect such dependence are not captured by power-law model, so the linear assumption may lead to inaccurate viscosity predictions. For this reason, a set of two-dimensional (2D) flow problems were considered and solved numerically to assess the effects of the shear rheology model, the pore size distribution and the microstructural complexity on the value of the apparent shear rate. In order to facilitate the analysis, the microscopic features of all the investigated porous media were well characterized through pore network modelling. The present results prove the inability of traditional approaches to predict the macroscopic viscous pressure losses generated during the creeping flow of widely used Herschel–Bulkley and Carreau fluids. In particular, the existence of a yield stress or a plateau viscosity induces significant deviations from linearity in the relationship between apparent shear rate and Darcy velocity. The importance of these deviations is, in turn, shown to be highly affected by the dispersion of the pore size distribution and the degree of shear thinning. Moreover, the reasons for such observations are discussed using an analytical approach.

Keywords

Shear-thinning fluids Yield stress Numerical simulations Apparent viscosity 2D porous media 

List of Symbols

Roman Letters

a

Consistency of a given power-law fluid (Pa sn)

b

Flow behaviour index of a given power-law fluid

c

Power-law index of a given Carreau fluid

h

Hydraulic aperture of a bundle of rectangular channels (m)

hi

Aperture classes of the non-uniform channels in the bundle of rectangular channels model (m)

k

Consistency of a given Herschel–Bulkley fluid (Pa sn)

K

Intrinsic permeability of a given porous medium (m2)

\( m_{1} \), \( m_{2} \)

Means of the distributions in a weighted sum of two normal laws (μm)

n

Flow index of a given Herschel–Bulkley fluid

N

Number of (ui, \( \nabla P_{i} \)) data obtained in a given numerical experiment

Nc

Number of capillaries in a bundle of cylindrical capillaries

p(hi)

Probability of a given aperture class hi

p(ri)

Probability of a given radius class ri

pj(ri)

Relative contribution of a given pore class ri to the total flow rate of a yield stress fluid at a given pressure gradient \( \nabla P_{j} \)

pv(ri)

Probability in terms of volume of a given pore radius class in a porous medium

pw(ri)

Relative contribution of a given pore class ri to the total flow rate of water at a given pressure gradient \( \nabla P_{\text{j}} \)

P

Pressure (Pa)

\( q\left( {\nabla P,\;r} \right) \)

Volumetric flow rate of a Herschel–Bulkley fluid through a capillary of radius \( r \) as a function of \( \nabla P \) (m3/s)

r

Radius of a cylindrical pore (m)

ri

Pore radius classes in a bundle of cylindrical capillaries (m)

rmin

Smallest pore radius class in a bundle of cylindrical capillaries (m)

rmax

Largest pore radius class in a bundle of cylindrical capillaries (m)

\( \bar{r} \)

Hydraulic radius of a bundle of cylindrical capillaries (m)

R

Radius of the circular cross-sectional area of the bundle of cylindrical capillaries (m)

u

Darcy velocity (m/s)

Greek Letters

\( \alpha \)

Shift factor relating \( \dot{\gamma }_{\text{pm}} \) to u

\( \alpha_{\text{N}} \)

Shift factor for the injection of a Newtonian fluid

\( \dot{\gamma } \)

Shear rate (s−1)

\( \dot{\gamma }_{\text{pm}} \)

Apparent shear rate for the flow of a complex fluid through a porous medium (s−1)

\( \dot{\gamma }_{\text{w}} \)

Wall shear rate in a circular channel for the flow of a yield stress fluid (s−1)

\( \dot{\gamma }_{{{\text{w}},{\text{Newtonian}}}} \)

Wall shear rate in a circular channel for Newtonian flow (s−1)

ε

Porosity of a given porous medium

\( \lambda \)

Time constant (also known as relaxation time) of a given Carreau fluid (s)

\( \mu \)

Shear viscosity (Pa s)

\( \mu_{0} \)

Lower Newtonian plateau viscosity of a given Carreau fluid (Pa s)

\( \mu_{\infty } \)

Upper Newtonian plateau viscosity of a given Carreau fluid (Pa s)

\( \mu_{ \rm max } \)

Maximum viscosity value allowed in the direct numerical computations

\( \mu_{\text{pm}} \)

Apparent “in situ” shear viscosity of a given non-Newtonian fluid (Pa s)

\( \sigma_{1} \), \( \sigma_{2} \)

Standard deviations of the distributions in a weighted sum of two normal laws (μm)

\( \tau \)

Shear stress (Pa)

\( \tau_{0} \)

Yield stress of a given Herschel–Bulkley fluid (Pa)

\( \tau_{\text{w}} \)

Wall shear stress (Pa)

Notes

References

  1. Balhoff, M., Sanchez-Rivera, D., Kwok, A., Mehmani, Y., Prodanovic, M.: Numerical algorithms for network modeling of yield stress and other non-Newtonian fluids in porous media. Transp. Porous Media 93, 363–379 (2012)CrossRefGoogle Scholar
  2. Ball, J.T., Pitts, M.J.: Effect of varying polyacrylamide molecular weight on tertiary oil recovery from porous media of varying permeability. In: SPE Enhanced Oil Recovery Symposium, 15–18 April, Tulsa, Oklahoma (1984)Google Scholar
  3. Benmouffok-Benbelkacem, G., Caton, F., Baravian, C., Skali-Lami, S.: Non-linear viscoelasticity and temporal behavior of typical yield stress fluids. Carbopol, xanthan and ketchup. Rheol Acta 49, 305–314 (2010)CrossRefGoogle Scholar
  4. Berg, S., van Wunnik, J.: Shear rate determination from pore-scale flow fields. Transp. Porous Media 117(2), 229–246 (2017)CrossRefGoogle Scholar
  5. Broniarz-Press, L., Agacinski, P., Rozanski, J.: Shear-thinning fluids flow in fixed and fluidized beds. Int. J. Multiph. Flow 33, 675–689 (2007)CrossRefGoogle Scholar
  6. Carnali, J.O.: A dispersed anisotropic phase as the origin of the weak-gel properties of aqueous xanthan gum. J. Appl. Polym. Sci. 43(5), 929–941 (1991)CrossRefGoogle Scholar
  7. Carreau, P.J.: Rheological equations from molecular network theories. Trans. Soc. Rheol. 16, 99–127 (1972)CrossRefGoogle Scholar
  8. Chauveteau, G.: Rodlike polymer solution flow through fine pores: influence of pore size on rheological behavior. J. Rheol. 26, 111 (1982)CrossRefGoogle Scholar
  9. Chauveteau, G., Zaitoun, A.: Basic rheological behavior of xanthan polysaccharide solutions in porous media: effects of pore size and polymer concentration. In: European Symposium on Enhanced Oil Recovery, Bournemouth, England (1981)Google Scholar
  10. Chevalier, T., Talon, L.: Generalization of Darcy’s law for Bingham fluids in porous media: from flow-field statistics to the flow-rate regimes. Phys. Rev. E 91, 023011 (2015)CrossRefGoogle Scholar
  11. Chhabra, R.P., Richardson, J.F.: Non-Newtonian flow and applied rheology: engineering applications. Butterworth-Heinemann/Elsevier, Boston/Amsterdam (2008)Google Scholar
  12. Chhabra, R.P., Srinivas, B.K.: Non-Newtonian (purely viscous) fluid flow through packed beads: effect of particle shape. Powder Technol. 67, 15–19 (1991)CrossRefGoogle Scholar
  13. Chhabra, R.P., Comiti, J., Machac, I.: Flow of non-Newtonian fluids in fixed and fluidised beds. Chem. Eng. Sci. 56, 1–27 (2001)CrossRefGoogle Scholar
  14. Christopher, R.H., Middleman, S.: Power-law flow through a packed tube. Ind. Eng. Chem. Fundam. 4(4), 422–426 (1965)CrossRefGoogle Scholar
  15. Comba, S., Dalmazzo, D., Santagata, E., Sethi, R.: Rheological characterization of xanthan suspensions of nanoscale iron for injection in porous media. J. Hazard. Mater. 185, 598–605 (2011)CrossRefGoogle Scholar
  16. COMSOL Multiphysics Version 5.3. COMSOL AB, Stockholm, Sweden. www.comsol.com (2017)
  17. Coussot, P.: Rheometry of Pastes, Suspensions, and Granular Materials. Applications in Industry and Environment. Wiley, New York (2005)CrossRefGoogle Scholar
  18. Coussot, P.: Yield stress fluid flows: a review of experimental data. J. Nonnewton. Fluid Mech. 211, 31–49 (2014)CrossRefGoogle Scholar
  19. Darcy, H.P.G.: Les fontaines publiques de la ville de Dijon, pp. 590–594. Libraire des Corps Imperiaux des Ponts et Chaussées et des Mines, Paris (1856)Google Scholar
  20. Delshad, M., Kim, D.H., Magbagbeola, O.A., Huh, C., Pope, G.A., Tarahom, F.: Mechanistic interpretation and utilization of viscoelastic behavior of polymer solutions for improved polymer-flood efficiency. In: SPE 113620 (2008)Google Scholar
  21. Dimitriou, C.J., McKinley, G.H.: A comprehensive constitutive law for waxy crude oil: a thixotropic yield stress fluid. Soft Matter 10, 6619–6644 (2014)CrossRefGoogle Scholar
  22. Economides, M.J., Nolte, K.G.: Reservoir Stimulation, 3rd edn. Wiley, New York (2000)Google Scholar
  23. Gogarty, W.B.: Rheological properties of pseudoplastic fluids in porous media. Soc. Petrol. Eng. J. 7(2), 149–160 (1967)CrossRefGoogle Scholar
  24. Gostick, J.T.: Versatile and efficient pore network extraction method using marker-based watershed segmentation. Phys. Rev. E 96(2), 023307 (2017)CrossRefGoogle Scholar
  25. Hernández-Espriú, A., Sánchez-León, E., Martínez-Santos, P., Torres, L.G.: Remediation of a diesel-contaminated soil from a pipeline accidental spill: enhanced biodegradation and soil washing processes using natural gums and surfactants. J. Soils Sediments 13, 152–165 (2013)CrossRefGoogle Scholar
  26. Herschel, W.H., Bulkley, R.: Konsistenzmessungen von Gummi-Benzollösungen. Kolloid-Zeitschrift 39, 291 (1926)CrossRefGoogle Scholar
  27. Hirasaki, G.J., Pope, G.A.: Analysis of factors influencing mobility and adsorption in the flow of polymer solutions through porous media. Soc. Petrol. Eng. 14, 337–346 (1974)CrossRefGoogle Scholar
  28. Khodja, M.: Les fluides de forage: étude des performances et considérations environnementales. Ph.D. thesis, Institut National Polytechnique de Toulouse (2008)Google Scholar
  29. Kozeny, J.: Ueber kapillare Leitung des Wassers im Boden. Sitzungsber Akad. Wiss. 136(2), 271–306 (1927)Google Scholar
  30. Lake, L.W.: Enhanced Oil Recovery. Prentice-Hall Inc, Englewood Cliffs (1989)Google Scholar
  31. Lavrov, A.: Non-Newtonian fluid flow in rough-walled fractures: a brief review. In: Proceedings of ISRM SINOROCK 2013, 18–20 June, Shanghai, China (2013)Google Scholar
  32. Lebedev, M., Pervukhina, M., Mikhaltsevitch, V., Dance, T., Bilenko, O., Gurevich, B.: An experimental study of acoustic responses on the injection of supercritical CO2 into sandstones from the Otway Basin. Geophysics 78(4), D293–D306 (2013)CrossRefGoogle Scholar
  33. López, X.: Ph.D. Thesis, Department of Earth Science and Engineering Petroleum Engineering and Rock Mechanics Group, Imperial College, London (2004)Google Scholar
  34. López, X., Valvatne, P.H., Blunt, M.J.: Predictive network modeling of single-phase non-Newtonian flow in porous media. J. Colloid Interface Sci. 264(1), 256–265 (2003)CrossRefGoogle Scholar
  35. Machac, I., Cakl, J., Comiti, J., Sabiri, N.E.: Flow of non-Newtonian fluids through fixed beds of particles: comparison of two models. Chem. Eng. Process. 37, 169–176 (1998)CrossRefGoogle Scholar
  36. Macosko, C.W.: Rheology: Principles, Measurements and Applications. Wiley (1994)Google Scholar
  37. Malvault, G., Ahmadi, A., Omari, A.: Numerical simulation of yield stress fluid flow in capillary bundles: influence of the form and the axial variation in the cross section. Transp. Porous Media 120(2), 255–270 (2017)CrossRefGoogle Scholar
  38. Matyka, M., Khalili, A., Koza, Z.: Tortuosity-porosity relation in porous media flow. Phys. Rev. E 78, 026306 (2008)CrossRefGoogle Scholar
  39. Minagawa, H., Nishikawa, Y., Ikeda, I., Miyazaki, K., Takahara, N., Sakamoto, Y., Komai, T., Narita, H.: Characterization of sand sediment by pore size distribution and permeability using proton nuclear magnetic resonance measurement. J. Geophys. Res. 113, B07210 (2008)CrossRefGoogle Scholar
  40. Molnar, I.: Uniform quartz—silver nanoparticle injection experiment. Retrieved December 23, 2018, from www.digitalrocksportal.org (2016)
  41. Morris, D.A., Johnson, A.I., Summary of hydrologic and physical properties of rock and soil materials, as analyzed by the Hydrologic Laboratory of US Geological Survey 1948–60. Water Supply Paper 1839-D, US Geological Survey, Washington, p 42 (1967)Google Scholar
  42. Nolan, G.T., Kavanagh, P.E.: The size distribution of interstices in random packings of spheres. Powder Technol. 78(3), 231–238 (1994)CrossRefGoogle Scholar
  43. Ouyang, L., Hill, A.D., Zhu, D.: Theoretical and numerical simulation of Herschel–Bulkley fluid flow in propped fractures. In: International Petroleum Technology Conference (2013)Google Scholar
  44. Owens, R.G., Phillips, T.N.: Computational Rheology. Imperial College Press, London (2002)CrossRefGoogle Scholar
  45. Pipe, C.J., Majmudar, T.S., McKinley, G.H.: High shear rate viscometry. Rheol. Acta 47, 621–642 (2008).  https://doi.org/10.1007/s00397-008-0268-1 CrossRefGoogle Scholar
  46. Rao, P.T., Chhabra, R.P.: Viscous non-Newtonian flow in packed beads: effects of column walls and particle size distribution. Powder Technol. 77, 171–176 (1993)CrossRefGoogle Scholar
  47. Rodríguez de Castro, A.: Extending Darcy’s law to the flow of yield stress fluids in packed beads: method and experiments. Adv. Water Resour 126, 55–64 (2019)CrossRefGoogle Scholar
  48. Rodríguez de Castro, A., Radilla, G.: Flow of yield and Carreau fluids through rough-walled rock fractures: prediction and experiments. Water Resour Res 53(7), 6197–6217 (2017)CrossRefGoogle Scholar
  49. Rodríguez de Castro, A., Omari, A., Ahmadi-Sénichault, A., Bruneau, D.: Toward a new method of porosimetry: principles and experiments. Transp. Porous Media 101(3), 349–364 (2014)CrossRefGoogle Scholar
  50. Rodríguez de Castro, A., Omari, A., Ahmadi-Sénichault, A., Savin, S., Madariaga, L.-F.: Characterizing porous media with the yield stress fluids porosimetry method. Transp. Porous Media 114(1), 213–233 (2016a)CrossRefGoogle Scholar
  51. Rodríguez de Castro, A., Oostrom, M., Shokri, N.: Effects of shear-thinning fluids on residual oil formation in microfluidic pore networks. J Colloid Interface Sci 472, 34–43 (2016b)CrossRefGoogle Scholar
  52. Roustaei, A., Chevalier, T., Talon, L., Frigaard, I.A.: Non-Darcy effects in fracture flows of a yield stress fluid. J. Fluid Mech. 805, 222–261 (2016)CrossRefGoogle Scholar
  53. Sabiri, N.-E., Comiti, J.: Pressure drop in non-Newtonian purely viscous fluid flow through porous media. Chem. Eng. Sci. 50, 1193–1201 (1995)CrossRefGoogle Scholar
  54. Saramito, P., Wachs, A.: Progress in numerical simulation of yield stress fluid flows. Rheol. Acta 56(3), 211–230 (2017)CrossRefGoogle Scholar
  55. Schenk, O., Gärtner, K.: Solving unsymmetric sparse systems of linear equations with PARDISO. Future Gener Comput Syst 20(3), 475–487 (2004)CrossRefGoogle Scholar
  56. Sheng, J.J.: Modern chemical enhanced oil recovery, theory and practice, Elsevier, Boston (2011)Google Scholar
  57. Shenoy, A.V.: Non-Newtonian fluid heat transfer in porous media. Adv. Heat Transf. 24, 101–190 (1994).  https://doi.org/10.1016/S0065-2717(08)70233-8 CrossRefGoogle Scholar
  58. Skauge, A., Zamani, N., Gausdal Jacobsen, J., Shaker Shiran, B., Al-Shakry, B., Tormod, S.: Polymer flow in porous media: relevance to enhanced oil recovery. Colloids Interfaces 2, 27 (2018)CrossRefGoogle Scholar
  59. Skelland, A.H.P.: Non-Newtonian Flow and Heat Transfer. Wiley, New York (1967)Google Scholar
  60. Smit, G.J.F., du Plessis, J.P.: Pressure drop prediction of power law fluid through granular media. J. Non-Newtonian Fluid Mech. 72, 319–323 (1997)CrossRefGoogle Scholar
  61. Song, K.-W., Kim, Y.-S., Chang, G.S.: Rheology of concentrated xanthan gum solutions: steady shear flow behavior. Fibers Polym. 7(2), 129–138 (2006)CrossRefGoogle Scholar
  62. Sorbie, K.S., Clifford, P.J., Jones, R.W.: The rheology of pseudoplastic fluids in porous media using network modeling. J. Colloid Interface Sci. 130, 508–534 (1989)CrossRefGoogle Scholar
  63. Talon, L., Auradou, H., Hansen, A.: Effective rheology of Bingham fluids in a rough channel. Front. Phys. 2, 24 (2014)CrossRefGoogle Scholar
  64. Tiu, C., Zhou, J.Z.Q., Nicolae, G., Fang, T.N., Chhabra, R.P.: Flow of viscoelastic polymer solutions in mixed beds of particles. Can. J. Chem. Eng. 75, 843–850 (1997)CrossRefGoogle Scholar
  65. Tosco, T., Marchisio, D.L., Lince, F., Sethi, R.: Extension of the Darcy-Forchheimer law for shear-thinning fluids and validation via pore-scale flow simulations. Transp. Porous Media 96, 1–20 (2013).  https://doi.org/10.1007/s11242-012-0070-5 CrossRefGoogle Scholar
  66. Truex, M., Vermeul, V.R., Adamson, D.T., Oostrom, M., Zhong, L., Mackley, R.D., Fritz, B.G., Horner, J.A., Johnson, T.C., Thomle, J.N., Newcomer, D.R., Johnson, C.D., Rysz, M., Wietsma, T.W., Newell, C.J.: Field test of enhanced remedial amendment delivery using a shear-thinning fluid. Groundw. Monit. Remediat. 35, 34–45 (2015)CrossRefGoogle Scholar
  67. Van der Plas, B., Golombok, M.: Engineering performance of additives in water floods. J. Petrol. Sci. Eng. 135, 314–322 (2015)CrossRefGoogle Scholar
  68. Wang, D., Cheng, J., Yang, Q., Gong, W., Li, Q., Chen, F.: Viscous-elastic polymer can increase microscale displacement efficiency in cores. In: SPE 63227, Presented at the 2000 SPE Annual Technical Conference and Exhibition held in Dallas, Texas, 1–4 Oct (2000)Google Scholar
  69. Wever, D.A.Z., Picchioni, F., Broekhuis, A.A.: Polymers for enhanced oil recovery: a paradigm for structure–property relationship in aqueous solution. Prog. Polym. Sci. 36(11), 1558–1628 (2011)CrossRefGoogle Scholar
  70. Withcomb, P.J., Macosko, C.W.: Rheology of xanthan gum. J. Rheol. 22, 493 (1978)CrossRefGoogle Scholar
  71. Xiong, Q., Baychev, T.G., Jivkov, A.P.: Review of pore network modelling of porous media: experimental characterisations, network constructions and applications to reactive transport. J. Contam. Hydrol. 192, 101–117 (2016)CrossRefGoogle Scholar
  72. Zamani, N., Bondino, I., Kaufmann, R., Skauge, A.: Computation of polymer in situ rheology using direct numerical simulation. J. Petrol. Sci. Eng. 159, 92–102 (2017)CrossRefGoogle Scholar
  73. Zhang, L., Sun, H., Han, B., Peng, L., Ning, F., Jiang, G., Chehotkin, V.F.: Effect of shearing actions on the rheological properties and mesostructures of CMC, PVP and CMC + PVP aqueous solutions as simple water-based drilling fluids for gas hydrate drilling. J. Unconv. Oil Gas Resour. 14, 86–98 (2016)CrossRefGoogle Scholar
  74. Zhong, L., Oostrom, M., Wietsma, T.W., Covert, M.A.: Enhanced remedial amendment delivery through fluid viscosity modifications: experiments and numerical simulations. J. Contam. Hydrol. 101, 29–41 (2008)CrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Arts et Métiers ParisTechChâlons-en-ChampagneFrance
  2. 2.Laboratoire MSMP – EA7350Châlons-en-ChampagneFrance
  3. 3.Department of Chemical EngineeringUniversity of WaterlooWaterlooCanada

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