Transport in Porous Media

, Volume 126, Issue 2, pp 411–429 | Cite as

Lattice Boltzmann Simulation of Immiscible Displacement in Porous Media: Viscous Fingering in a Shear-Thinning Fluid

  • Menghao Wang
  • Youming XiongEmail author
  • Liming Liu
  • Geng Peng
  • Zheng Zhang


In this work, we investigate immiscible displacement in porous media with the displaced fluid being shear-thinning. We focus on the influence the heterogeneous viscosity field in the shear-thinning fluid brings on viscous fingering, which has received little attention in the existing researches. Lattice Boltzmann simulations of immiscible displacement with a power law model implementation in the displaced fluid are conducted. The lattice Boltzmann algorithm is validated against Newtonian and non-Newtonian flows in a channel. The effects of the shear-thinning property and the viscosity heterogeneity on viscous fingering are considered in the simulations. The results show that with stronger shear-thinning property (lower power law exponent n), there is stronger viscosity heterogeneity in the displaced fluid, and the viscous fingering shows weaker instability. The influence of a heterogeneous viscosity field on viscous fingering is dominated by the viscosity in the low-viscosity regions, while the high-viscosity regions show little influence. The influence of the local viscosity on viscous fingering is dependent upon the local shear rate. A concept of ‘effective field viscosity’ is introduced to quantitatively characterize a heterogeneous viscosity field. A shear rate weighted averaging algorithm is proposed to calculate the effective field viscosity from a heterogeneous viscosity field. The algorithm is tested in several cases and shows good performance to represent the influence of the heterogeneous viscosity field.


Immiscible displacement Viscous fingering Shear-thinning fluid Heterogeneous viscosity field Lattice Boltzmann method 

List of Symbols


Dimensionless interfacial length


Dimensionless lattice time

\(\mathbf e \)

Velocity vectors

\(\mathbf F \)

Surface tension

\(\mathbf G \)

Body force

\(\mathbf u \)

Velocity vector (lu/ts)

\(\mathbf x \)

Coordinates of lattice nodes

\(\theta \)

Static contact angle (\(^{\circ }\))


Lattice speed (lu/ts)


Index distribution function


Pressure distribution function


Width of simulation field (lu)


Interfacial length (lu)


Lattice length unit


Viscosity ratio (\(\mu _\mathrm{d}/\mu _\mathrm{in}\))


Power law exponent




Shear strain tensor


Lattice time


Lattice time step


Local velocity (lu/ts)


Weighting coefficients

Greek Letters

\(\gamma \)

Shear rate (/ts)

\(\kappa \)

Parameter to control surface tension magnitude

\(\mu \)

Dynamic viscosity

\(\nu \)

Kinematic viscosity

\(\phi \)

Index function

\(\varPi \)

Momentum flux tensor

\(\rho \)


\(\tau \)

Relaxation factor



Equilibrium state



Displaced fluid


Effective field value


Directions in the D2Q9 model


Invading fluid


Local node value


Mean value


Maximum value


Minimum value




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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Menghao Wang
    • 1
  • Youming Xiong
    • 1
    Email author
  • Liming Liu
    • 1
  • Geng Peng
    • 1
  • Zheng Zhang
    • 1
  1. 1.State Key Laboratory of Oil and Gas Reservoir Geology and ExploitationSouthwest Petroleum UniversityChengduChina

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