We perform the viscosity-independent Stokes flow simulations in regular sphere packings using the two-relaxation-times (TRT) lattice Boltzmann method (LBM) with the simple bounce-back (BB) rule. Our special discretization procedure reduces the scatter in integral quantities, such as drag force, and quantifies the solution convergence error. We assume transition to linear (−1) convergence rate for different sets of TRT parameters and use this assumption to provide a simple extrapolation scheme. After establishing the accurate reference values of drag for a wide range of porosities, 0.26–0.78, we show a ten-fold decrease in the drag error using the suggested extrapolations. This error decrease allows the simple LBM/BB scheme to reach an accuracy of the high-order interpolated boundary schemes. The suggested extrapolation approach is straightforward to apply in porous media, whose pore space can be discretized at several resolutions.
REFERENCES
1.
B.
Ferréol
and D. H.
Rothman
, “Lattice-Boltzmann simulations of flow through Fontainebleau sandstone
,” Transp. Porous Media
20
, 3
–20
(1995
).2.
G.
Jin
, T. W.
Patzek
, and D.
Silin
, “SPE 90084-MS: Direct prediction of the absolute permeability of unconsolidated and consolidated reservoir rock
,” in SPE Annual Technical Conference and Exhibition
(SPE
, Houston, TX
, 2004
).3.
T.
Ramstad
, N.
Idowu
, C.
Nardi
, and P.-E.
Øren
, “Relative permeability calculations from two-phase flow simulations directly on digital images of porous rocks
,” Transp. Porous Media
94
, 487
–504
(2012
).4.
G.
Jin
, T. W.
Patzek
, and D. B.
Silin
, “Modeling the impact of rock formation history on the evolution of absolute permeability
,” J. Pet. Sci. Eng.
100
, 153
–161
(2012
).5.
D. B.
Silin
, G.
Jin
, and T. W.
Patzek
, “Robust determination of the pore-space morphology in sedimentary rocks
,” J. Pet. Technol.
56
, 69
–70
(2004
).6.
D. B.
Silin
and T.
Patzek
, “Pore space morphology analysis using maximal inscribed spheres
,” Physica A
371
, 336
–360
(2006
).7.
D.
Silin
, L.
Tomutsa
, S. M.
Benson
, and T. W.
Patzek
, “Microtomography and pore-scale modeling of two-phase fluid distribution
,” Transp. Porous Media
86
, 495
–515
(2011
).8.
E.
Jettestuen
, J. O.
Helland
, and M.
Prodanović
, “A level set method for simulating capillary-controlled displacements at the pore scale with nonzero contact angles
,” Water Resour. Res.
49
, 4645
–4661
(2013
), https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/wrcr.20334.9.
S.
Khirevich
, I.
Ginzburg
, and U.
Tallarek
, “Coarse- and fine-grid numerical behavior of MRT/TRT Lattice-Boltzmann schemes in regular and random sphere packings
,” J. Comput. Phys.
281
, 708
–742
(2015
).10.
P. L.
Bhatnagar
, E. P.
Gross
, and M.
Krook
, “A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems
,” Phys. Rev.
22
, 511
–525
(1954
).11.
Y.
Qian
, D.
D’Humières
, and P.
Lallemand
, “Lattice BGK models for Navier-Stokes equation
,” Europhys. Lett.
17
, 479
–484
(1992
).12.
H.
Chen
, S.
Chen
, and W. H.
Matthaeus
, “Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method
,” Phys. Rev. A
45
, R5339
–R5342
(1992
).13.
I.
Ginzbourg
and P. M.
Adler
, “Boundary flow condition analysis for the three-dimensional lattice Boltzmann model
,” J. Phys. II
4
, 191
–214
(1994
).14.
I.
Ginzburg
and D.
D’Humières
, “Multireflection boundary conditions for lattice Boltzmann models
,” Phys. Rev. E
68
, 066614
(2003
).15.
C.
Pan
, L.-S.
Luo
, and C. T.
Miller
, “An evaluation of lattice Boltzmann schemes for porous medium flow simulation
,” Comput. Fluids
35
, 898
–909
(2006
).16.
D.
D’Humières
, “Generalized Lattice-Boltzmann equations
,” Prog. Astronaut. Aeronaut.
159
, 450
–458
(1992
).17.
I.
Ginzburg
, F.
Verhaeghe
, and D.
D’Humières
, “Proceedings of eighteenth international symposium on rarefied gas dynamics
,” Commun. Comput. Phys.
3
, 427
–478
(2008
).18.
I.
Ginzburg
, “Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation
,” Adv. Water Res.
28
, 1171
–1195
(2005
).19.
D.
D’Humières
and I.
Ginzburg
, “Viscosity independent numerical errors for lattice Boltzmann models: From recurrence equations to ‘magic’ collision numbers
,” Comput. Math. Appl.
58
, 823
–840
(2009
).20.
L.
Talon
, D.
Bauer
, N.
Gland
, S.
Youssef
, H.
Auradou
, and I.
Ginzburg
, “Assessment of the two relaxation time Lattice-Boltzmann scheme to simulate Stokes flow in porous media
,” Water Resour. Res.
48
, W04526
(2012
).21.
A. S.
Sangani
and A.
Acrivos
, “Slow flow through a periodic array of spheres
,” Int. J. Multiphase Flow
8
, 343
–360
(1982
).22.
A. A.
Zick
and G. M.
Homsy
, “Stokes flow through periodic arrays of spheres
,” J. Fluid Mech.
115
, 13
–26
(1982
).23.
R. E.
Larson
and J. J. L.
Higdon
, “A periodic grain consolidation model of porous media
,” Phys. Fluids
1
, 38
–46
(1989
).24.
S.
Khirevich
, A.
Daneyko
, A.
Höltzel
, A.
Seidel-Morgenstern
, and U.
Tallarek
, “Statistical analysis of packed beds, the origin of short-range disorder, and its impact on eddy dispersion
,” J. Chromatogr. A
1217
, 4713
–4722
(2010
).25.
T.
Atmakidis
and E. Y.
Kenig
, “Numerical analysis of residence time distribution in packed bed reactors with irregular particle arrangements
,” Chem. Prod. Process Model.
10
, 17
–26
(2015
).26.
X.
Frank
and P.
Perré
, “Droplet spreading on a porous surface: A lattice Boltzmann study
,” Phys. Fluids
24
, 042101
(2012
).27.
M.
Sbragaglia
and S.
Succi
, “Analytical calculation of slip flow in lattice Boltzmann models with kinetic boundary conditions
,” Phys. Fluids
17
, 093602
(2005
).28.
M. A. A.
Spaid
and J. F. R.
Phelan
, “Lattice Boltzmann methods for modeling microscale flow in fibrous porous media
,” Phys. Fluids
9
, 2468
–2474
(1997
).29.
D.
Maggiolo
, F.
Picano
, and M.
Guarnieri
, “Flow and dispersion in anisotropic porous media: A lattice-Boltzmann study
,” Phys. Fluids
28
, 102001
(2016
).30.
R. S.
Maier
and R. S.
Bernard
, “Lattice-Boltzmann accuracy in pore-scale flow simulation
,” J. Comput. Phys.
229
, 233
–255
(2010
).31.
M.
Bouzidi
, M.
Firdaouss
, and P.
Lallemand
, “Momentum transfer of a Boltzmann-lattice fluid with boundaries
,” Phys. Fluids
13
, 3452
–3459
(2001
).32.
M.
Junk
and Z.
Yang
, “Convergence of lattice Boltzmann methods for Stokes flows in periodic and bounded domains
,” Comput. Math. Appl.
55
, 1481
–1491
(2008
).33.
A. J. C.
Ladd
, “Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results
,” J. Fluid Mech.
271
, 311
–339
(1994
).34.
M. A.
van der Hoef
, R.
Beetstra
, and J. A. M.
Kuipers
, “Lattice-Boltzmann simulations of low-Reynolds-number flow past mono- and bidisperse arrays of spheres: Results for the permeability and drag force
,” J. Fluid Mech.
528
, 233
–254
(2005
).35.
Y.
Chen
, J. R.
Third
, and C. R.
Müller
, “A drag force correlation for approximately cubic particles constructed from identical spheres
,” Chem. Eng. Sci.
123
, 146
–154
(2015
).36.
B.
Chun
and A. J. C.
Ladd
, “Interpolated boundary condition for lattice Boltzmann simulations of flows in narrow gaps
,” Phys. Rev. E
75
, 066705
(2007
).37.
E.
Fattahi
, C.
Waluga
, B.
Wohlmuth
, U.
Rüde
, M.
Manhart
, and R.
Helmig
, “Lattice Boltzmann methods in porous media simulations: From laminar to turbulent flow
,” Comput. Fluids
140
, 247
–259
(2016
).38.
G.
Silva
and V.
Semiao
, “Consistent lattice Boltzmann modeling of low-speed isothermal flows at finite Knudsen numbers in slip-flow regime: Application to plane boundaries
,” Phys. Rev. E
96
, 013311
(2017
).39.
See http://www.khirevich.com/lbm for
S.
Khirevich
, “MATLAB code for discretization of regular sphere packings
;” accessed 15 August 2017.40.
R. S.
Maier
, D. M.
Kroll
, Y. E.
Kutsovsky
, H. T.
Davis
, and R. S.
Bernard
, “Simulation of flow through bead packs using the lattice Boltzmann method
,” Phys. Fluids
10
, 60
–74
(1998
).41.
C.
Manwart
, U.
Aaltosalmi
, A.
Koponen
, R.
Hilfer
, and J.
Timonen
, “Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media
,” Phys. Rev. E
66
, 016702
(2002
).42.
C.
Chukwudozie
and M.
Tyagi
, “Pore scale inertial flow simulations in 3-D smooth and rough sphere packs using lattice Boltzmann method
,” Chem. Eng. Sci.
59
, 4858
–4870
(2013
).43.
C. J.
Roy
, “Grid convergence error analysis for mixed-order numerical schemes
,” AIAA J.
41
, 595
–604
(2003
).44.
R. S.
Maier
, D. M.
Kroll
, R. S.
Bernard
, S. E.
Howington
, J. F.
Peters
, and H. T.
Davis
, “Pore-scale simulation of dispersion
,” Phys. Fluids
12
, 2065
–2079
(2000
).© 2018 Author(s).
2018
Author(s)
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