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Pore-Scale Simulation of Shear Thinning Fluid Flow Using Lattice Boltzmann Method

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Abstract

The present work attempts to identify the roles of flow and geometric variables on the scaling factor which is a necessary parameter for modeling the apparent viscosity of non-Newtonian fluid in porous media. While idealizing the porous media microstructure as arrays of circular and square cylinders, the present study uses multi-relaxation time lattice Boltzmann method to conduct pore-scale simulation of shear thinning non-Newtonian fluid flow. Variation in the size and inclusion ratio of the solid cylinders generates wide range of porous media with varying porosity and permeability. The present study also used stochastic reconstruction technique to generate realistic, random porous microstructures. For each case, pore-scale fluid flow simulation enables the calculation of equivalent viscosity based on the computed shear rate within the pores. It is observed that the scaling factor has strong dependence on porosity, permeability, tortuosity and the percolation threshold, while approaching the maximum value at the percolation threshold porosity. The present investigation quantifies and proposes meaningful correlations between the scaling factor and the macroscopic properties of the porous media.

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Abbreviations

c :

Lattice speed

C :

Consistency constant

\(\mathbf{e}, e_i \) :

Discretized lattice velocity vector and its components

\(\dot{e}_{\alpha \beta } \) :

Strain rate tensor

f :

Vectors of lattice Boltzmann distribution function

K :

Permeability \((\hbox {m}^{2})\)

1 / m :

Critical shear rate (1/s)

n :

Power law index

\(\mathbf{m}\) :

Vector of distribution function in moment space

\(\mathbf{M}\) :

Transformation matrix

P :

Pressure (Pa)

q :

Volumetric flow rate (m\(^{3}\)/s)

S :

Relaxation matrix

t :

Time (s)

T :

Tortuosity

u, v :

Velocity components (m/s)

u :

Macroscopic velocity vector

v :

Temporal velocity vector

x :

Position vector

X, Y :

Cartesian coordinate directions (m)

\(\alpha \) :

Scaling factor

\(\dot{\gamma }\) :

Shear rate (1/s)

\(\dot{\gamma }_m \) :

Porous medium shear rate (1/s)

\({\varepsilon } \) :

Porosity

\(\varepsilon _p \) :

Porosity at percolation threshold

\(\tau _s \) :

Shear stress (Pa)

\(\mu \) :

Bulk viscosity of fluid (Pa s)

\(\mu _\infty \) :

Bulk viscosity at infinite shear rate (Pa s)

\(\mu _0 \) :

Bulk viscosity at zero shear rate (Pa s)

\(\mu _\mathrm{pm} \) :

Porous medium equivalent viscosity (Pa s)

\(\tau \) :

Non-dimensional relaxation time

\(\nu \) :

Kinematic viscosity \((\hbox {m}^{2}/\hbox {s})\)

\(\rho \) :

Density \((\hbox {kg/m}^{3})\)

References

  • Aharonov, E., Rothman, D.H.: Non-Newtonian flow (through porous media): a lattice-Boltzmann method. Geophys. Res. Lett. 20, 679–682 (1993)

    Article  Google Scholar 

  • Artoli, A.M., Sequeira, A.: Mesoscopic simulations of unsteady shear-thinning flows. In: Computational Science–ICCS 2006. Springer, pp. 78–85 (2006)

  • Arumuga, P.D., Kumar, G.V., Dass, A.K.: Lattice Boltzmann simulation of flow over a circular cylinder at moderate Reynolds numbers. Therm. Sci. 18, 1235–1246 (2014)

    Article  Google Scholar 

  • Ashrafizaadeh, M., Bakhshaei, H.: A comparison of non-Newtonian models for lattice Boltzmann blood flow simulations. Comput. Math. Appl. 58, 1045–1054 (2009)

    Article  Google Scholar 

  • Bernsdorf, J., Wang, D.: Non-Newtonian blood flow simulation in cerebral aneurysms. Comput. Math. Appl. 58, 1024–1029 (2009)

    Article  Google Scholar 

  • Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511 (1954)

    Article  Google Scholar 

  • Boek, E.S., Chin, J., Coveney, P.V.: Lattice Boltzmann simulation of the flow of non-Newtonian fluids in porous media. Int. J. Mod. Phys. B 17, 99–102 (2003)

    Article  Google Scholar 

  • Boyd, J., Buick, J.M., Green, S.: Analysis of the Casson and Carreau–Yasuda non-Newtonian blood models in steady and oscillatory flows using the lattice Boltzmann method. Phys. Fluids 19, 093103 (2007)

    Article  Google Scholar 

  • Breuer, M., Bernsdorf, J., Zeiser, T., Durst, F.: Accurate computations of the laminar flow past a square cylinder based on two different methods: lattice-Boltzmann and finite-volume. Int. J. Heat Fluid Flow 21, 186–196 (2000)

    Article  Google Scholar 

  • Cannella, W., Huh, C., Seright, R.: Prediction of xanthan rheology in porous media. In: SPE annual technical conference and exhibition, Society of Petroleum Engineers, (1988)

  • Chen, L., Kang, Q., Dai, Z., Viswanathan, H.S., Tao, W.: Permeability prediction of shale matrix reconstructed using the elementary building block model. Fuel 160, 346–356 (2015)

    Article  Google Scholar 

  • Chen, L., Kang, Q., Viswanathan, H.S., Tao, W.Q.: Pore-scale study of dissolution-induced changes in hydrologic properties of rocks with binary minerals. Water Resour. Res. 50, 9343–9365 (2014)

    Article  Google Scholar 

  • Chen, S., Doolen, G.D.: Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–364 (1998)

    Article  Google Scholar 

  • Coveney, P., Maillet, J.-B., Wilson, J., Fowler, P., Al-Mushadani, O., Boghosian, B.: Lattice-gas simulations of ternary amphiphilic fluid flow in porous media. Int. J. Mod. Phys. C 9, 1479–1490 (1998)

    Article  Google Scholar 

  • d’Humieres, D.: Generalized lattice-Boltzmann equations. Prog. Astronaut. Aeronaut. 159, 450–450 (1994)

    Google Scholar 

  • Gabbanelli, S., Drazer, G., Koplik, J.: Lattice Boltzmann method for non-Newtonian (power-law) fluids. Phys. Rev. E 72, 046312 (2005)

    Article  Google Scholar 

  • Gebart, B.: Permeability of unidirectional reinforcements for RTM. J. Compos. Mater. 26, 1100–1133 (1992)

    Article  Google Scholar 

  • Hayes, R., Afacan, A., Boulanger, B., Shenoy, A.: Modelling the flow of power law fluids in a packed bed using a volume-averaged equation of motion. Transp. Porous Media 23, 175–196 (1996)

    Article  Google Scholar 

  • He, X., Zou, Q., Luo, L.-S., Dembo, M.: Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model. J. Stat. Phys. 87, 115–136 (1997)

    Article  Google Scholar 

  • Higuera, F., Succi, S., Benzi, R.: Lattice gas dynamics with enhanced collisions. EPL (Europhys. Lett.) 9, 345 (1989)

    Article  Google Scholar 

  • Jeong, N., Choi, D.H., Lin, C.-L.: Prediction of Darcy–Forchheimer drag for micro-porous structures of complex geometry using the lattice Boltzmann method. J. Micromech. Microeng. 16, 2240 (2006)

    Article  Google Scholar 

  • Jithin, M., Kumar, N., Das, MK., De, A.: Estimation of permeability of porous material using pore scale LBM simulations. In: Fluid Mechanics and Fluid Power–Contemporary Research. Springer, pp. 1381–1388 (2017)

  • Kehrwald, D.: Lattice Boltzmann simulation of shear-thinning fluids. J. Stat. Phys. 121, 223–237 (2005)

    Article  Google Scholar 

  • Koelman, J.: A simple lattice Boltzmann scheme for Navier–Stokes fluid flow. EPL (Europhys. Lett.) 15, 603 (1991)

    Article  Google Scholar 

  • Lallemand, P., Luo, L.-S.: Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E 61, 6546 (2000)

    Article  Google Scholar 

  • Lee, S., Yang, J.: Modeling of Darcy–Forchheimer drag for fluid flow across a bank of circular cylinders. Int. J. Heat Mass Transf. 40, 3149–3155 (1997)

    Article  Google Scholar 

  • Liu, Q., He, Y.-L., Li, D., Li, Q.: Non-orthogonal multiple-relaxation-time lattice Boltzmann method for incompressible thermal flows. Int. J. Heat Mass Transf. 102, 1334–1344 (2016)

    Article  Google Scholar 

  • Liu, Q., He, Y.-L., Li, Q., Tao, W.-Q.: A multiple-relaxation-time lattice Boltzmann model for convection heat transfer in porous media. Int. J. Heat Mass Transf. 73, 761–775 (2014)

    Article  Google Scholar 

  • Liu, S., Masliyah, J.H.: Non-linear flows in porous media. J. Nonnewton. Fluid Mech. 86, 229–252 (1999)

    Article  Google Scholar 

  • Lopez, X., Valvatne, P.H., Blunt, M.J.: Predictive network modeling of single-phase non-Newtonian flow in porous media. J. Colloid Interface Sci. 264, 256–265 (2003)

    Article  Google Scholar 

  • McNamara, G.R., Zanetti, G.: Use of the Boltzmann equation to simulate lattice-gas automata. Phys. Rev. Lett. 61, 2332 (1988)

    Article  Google Scholar 

  • Meakin, P.: Fractals, Scaling and Growth Far from Equilibrium. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  • Ohta, M., Nakamura, T., Yoshida, Y., Matsukuma, Y.: Lattice Boltzmann simulations of viscoplastic fluid flows through complex flow channels. J. Nonnewton. Fluid Mech. 166, 404–412 (2011)

    Article  Google Scholar 

  • Pan, C., Luo, L.-S., Miller, C.T.: An evaluation of lattice Boltzmann schemes for porous medium flow simulation. Comput. fluids 35, 898–909 (2006)

    Article  Google Scholar 

  • Pearson, J., Tardy, P.: Models for flow of non-Newtonian and complex fluids through porous media. J. Nonnewton. Fluid Mech. 102, 447–473 (2002)

    Article  Google Scholar 

  • Perrin, C.L., Tardy, P.M., Sorbie, K.S., Crawshaw, J.C.: Experimental and modeling study of Newtonian and non-Newtonian fluid flow in pore network micromodels. J. Colloid Interface Sci. 295, 542–550 (2006)

    Article  Google Scholar 

  • Psihogios, J., Kainourgiakis, M., Yiotis, A., Papaioannou, A.T., Stubos, A.: A lattice Boltzmann study of non-Newtonian flow in digitally reconstructed porous domains. Transp. Porous Media 70, 279–292 (2007)

    Article  Google Scholar 

  • Qian, Y., d’Humières, D., Lallemand, P.: Lattice BGK models for Navier–Stokes equation. EPL (Europhys. Lett.) 17, 479 (1992)

    Article  Google Scholar 

  • Rakotomalala, N., Salin, D., Watzky, P.: Simulations of viscous flows of complex fluids with a Bhatnagar, Gross, and Krook lattice gas. Phys. Fluids 8, 3200–3202 (1996)

    Article  Google Scholar 

  • Shabro, V., Torres-Verdín, C., Javadpour, F., Sepehrnoori, K.: Finite-difference approximation for fluid-flow simulation and calculation of permeability in porous media. Transp. Porous Media 94, 775–793 (2012)

    Article  Google Scholar 

  • Sochi, T.: Non-Newtonian flow in porous media. Polymer 51, 5007–5023 (2010)

  • Sorbie, K., Clifford, P., Jones, E.: The rheology of pseudoplastic fluids in porous media using network modeling. J. Colloid Interface Sci. 130, 508–534 (1989)

    Article  Google Scholar 

  • Succi, S.: The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond. Oxford University Press, Oxford (2001)

    Google Scholar 

  • Sullivan, S., Gladden, L., Johns, M.: Simulation of power-law fluid flow through porous media using lattice Boltzmann techniques. J. Nonnewton. Fluid Mech. 133, 91–98 (2006)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Wang, D., Bernsdorf, J.: Lattice Boltzmann simulation of steady non-Newtonian blood flow in a 3D generic stenosis case. Comput. Math. Appl. 58, 1030–1034 (2009)

    Article  Google Scholar 

  • Wang, M., Pan, N.: Prediction of effective physical properties of complex multiphase materials. Mater. Sci. Eng. R 63, 1–30 (2008)

    Article  Google Scholar 

  • Wang, M., Pan, N.: Elastic properties of multiphase composites with random microstructures. J. Comput. Phys. 228, 5978–5988 (2009)

    Article  Google Scholar 

  • Wang, M., Wang, J., Pan, N., Chen, S.: Mesoscopic predictions of the effective thermal conductivity for microscale random porous media. Phys. Rev. E 75, 036702 (2007)

    Article  Google Scholar 

  • Wang, M., Wang, J., Pan, N., Chen, S., He, J.: Three-dimensional effect on the effective thermal conductivity of porous media. J. Phys. D Appl. Phys. 40, 260 (2006)

    Article  Google Scholar 

  • Wang, M., Wang, X., Wang, J., Pan, N.: Grain size effects on effective thermal conductivity of porous materials with internal thermal contact resistance. J. Porous Media 16, 1043–1048 (2013)

    Article  Google Scholar 

  • Wang, Z., Jin, X., Wang, X., Sun, L., Wang, M.: Pore-scale geometry effects on gas permeability in shale. J. Nat. Gas Sci. Eng. 34, 948–957 (2016)

    Article  Google Scholar 

  • Wolf-Gladrow, D.A.: Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction. Springer, Berlin (2000)

    Book  Google Scholar 

  • Yu, D., Mei, R., Luo, L.-S., Shyy, W.: Viscous flow computations with the method of lattice Boltzmann equation. Prog. Aerosp. Sci. 39, 329–367 (2003)

    Article  Google Scholar 

  • Zou, Q., He, X.: On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Phys. Fluids 9, 1591–1598 (1997)

    Article  Google Scholar 

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Authors and Affiliations

  1. Energy Conversion and Storage Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, UP, 208016, India

    M. Jithin, Nimish Kumar & Malay K. Das

  2. Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, UP, 208016, India

    Ashoke De

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  1. M. Jithin
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Correspondence to Malay K. Das.

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Jithin, M., Kumar, N., De, A. et al. Pore-Scale Simulation of Shear Thinning Fluid Flow Using Lattice Boltzmann Method. Transp Porous Med 121, 753–782 (2018). https://doi.org/10.1007/s11242-017-0984-z

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