Danny Thorne

1
Lattice Boltzmann Method for the Elder Problem
Danny Thorne thorned_at_fiu.edu http//www.fiu.edu/
thorned
Mike Sukop sukopm_at_fiu.edu http//www.fiu.edu/suk
opm
Department of Earth Sciences
CMWR 2004
CMWR 2004
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Motivation
Thorne Sukop

• Ground water modeling of highly non-linear
  coupled equations that govern solute
  transport/density-driven flow problems are
  challenging.
• Lattice Boltzmann models (LBMs) can simulate
  many fluid dynamics problems that traditionally
  required more complex approaches.
• We use a multicomponent LBM to address the
  classic solute-induced buoyancy Elder problem,
  which has been widely adopted as a benchmark
  among density-dependent flow modelers.
• This work represents first steps in the
  development of LBMs for application to ground
  water problems (e.g. salt-water intrusion in
  coastal regions).

3
Elder Problem
Thorne Sukop
C0
E/H4 L/H2
Temperature-induced buoyancy
Solute-induced buoyancy
H
C1
Elder, J. W. (1967) J. Fluid Mech. 27 (3)
609-623 Voss, C. I., W. R. Souza (1987) Wat.
Resour. Res. 23, 1851-1866
4
LBM Basics I
Thorne Sukop
Lattice Unit, lu

• Notes
• Based on statistical mechanics and kinetic
  theory.
• Solute and buoyancy will be covered in later
  slides.

Unfold
5
LBM Basics I
Thorne Sukop
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Illustration
Thorne Sukop
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Streaming Step
Thorne Sukop
( f ? f )
8
Collision Step
Thorne Sukop
f
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Collision Step
Thorne Sukop
f eq
10
Collision Step
Thorne Sukop
t -1( f eq - f )
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Collision Step
Thorne Sukop
f f t -1( f eq - f )
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LBM Basics II
Thorne Sukop
Wolf-Gladrow, D. A., Lattice-Gas Cellular
Automata and Lattice Boltzmann Models An
Introduction, Springer, Berlin, 2000,
308pp. Succi, S., The Lattice Boltzmann Equation
for Fluid Dynamics and Beyond, Clarendon Press,
Oxford, 2001, 288pp.
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LBM Basics III
Thorne Sukop
Bounce-back no-slip wall boundary
Time t Initial configuration. Lattice node
adjacent to wall boundary before streaming step.
f l u i d
s o l i d
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LBM Basics III
Thorne Sukop
Bounce-back no-slip wall boundary
Time t After streaming. Directional densities
stream into the solid (for temporary storage).
f l u i d
s o l i d
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LBM Basics III
Thorne Sukop
Bounce-back no-slip wall boundary
Time t Bounce back. Temporarily stored densities
reflect back toward the fluid lattice node.
f l u i d
s o l i d
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LBM Basics III
Thorne Sukop
Bounce-back no-slip wall boundary
Time tDt After streaming. Densities stream back
into the fluid pointing in the direction exactly
opposite their initial direction.
f l u i d
s o l i d
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Body Forces
Thorne Sukop
Body forces such as gravity are implemented as an
addition to the velocity vector with f from
the last time step.
Martys, N. S. and H. Chen (1996) Phys. Rev. E 53,
743
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Solute Transport
Thorne Sukop

• Solute is simulated by a second distribution fs
  called the solute component or s-component.
• This corresponds to the fluid distribution
  function except with a simpler equilibrium
  distribution
• Concentration //
  Analogous to fluid density.
• Diffusion coefficient
  // Analogous to viscosity.
• Modified body force

Drives buoyant flow.
Yoshino, M. and T. Inamuro (2003) Int. J. Numer.
Meth. Fluids 43, 183.
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Porous Media
Thorne Sukop
Requires prohibitively large lattices to
transcend pore scale!
Dardis, O. and J. McCloskey (1998) Phys. Rev. E
57, 4834-4837 Dardis, O. and J. McCloskey (1998)
Geophys. Res. Let. 25, 1471-1474
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Porous Media
Thorne Sukop
Consider the traditional collision step as a
second intermediate step after streaming, denoted
by f
Then the porous medium step has the form
where a is the index of the direction opposite
ea.
Permeability
ns0, free-fluid. ns1, no-flow.
a
a

• Notes
• Probabilistic bounce-back.
• Fractured/heterogeneous porous media.
• Dispersion/diffusion under PM? // Future work

x
xeaDt
Dardis, O. and J. McCloskey (1998) Phys. Rev. E
57, 4834-4837 Dardis, O. and J. McCloskey (1998)
Geophys. Res. Let. 25, 1471-1474
21
Elder Problem
Thorne Sukop
// Controlling parameter
Elder, J. W. (1967) J. Fluid Mech. 27 (3), 609-623
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Elder Problem
Thorne Sukop
// Controlling parameter
Elder, J. W. (1967) J. Fluid Mech. 27 (3), 609-623
23
Results
Thorne Sukop
Year 1
Year 2
60
20
Year 10
Year 4
60
60
20
20
Year 15
Year 20
20
60
20
60

• Notes
• No fully accepted results (computer or lab).
• Maybe no unique solution.

Elder, J. W. (1967) J. Fluid Mech. 27 (1),
29-48 Elder, J. W. (1967) J. Fluid Mech. 27 (3),
609-623 Woods, J. A., et al. (2003) Wat. Resour.
Res. 39, 1158-1169
24
Results
Thorne Sukop
80
60
80
Year 1
Year 2
40
60
40
20
20
80
Year 4
Year 10
80
60
60
40
20
40
20
Year 15
Year 20
80
80
80
80
60
60
40
40
20
20
Frolkovic, P., H. De Schepper (2001) Adv. Wat.
Res. 24, 63-72
25
Results (year 15)
Thorne Sukop
Year 15
20
60
Year 15
80
80
80
60
40
20
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Conclusions
Thorne Sukop

• LBMs appear to be a highly viable tool for
  modeling non-linear coupled density-dependent
  flow problems.
• Our results compare favorably with Elder's
  classic results and with modern state-of-the-art
  results, such as those found in Frolkovic and De
  Schepper (2001), though no definitive solution
  exists.
• LBMs have certain numerical advantages over
  conventional methods
• Non-linear/coupled solute/density flow is
  handled naturally
• No approximations of derivatives
• No mesh spacing related to grid resolution
• Inconsistency between pressure and density
  approximations, artificial velocities (Voss
  Souza, 1987)

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Future Work
Thorne Sukop

• Dispersion under the Dardis McCloskey (1998)
  porous media method?
• Higher resolution (develop parallel LBM code).
• Other classic buoyancy benchmarks
• Henry
• HYDROCOIN
• Ghyben-Herzberg
• Etc.
• 3D

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CMWR 2004
CMWR 2004
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References
1. D.A. Wolf-Gladrow, Lattice-Gas Cellular
Automata and Lattice Boltzmann Models An
Introduction, Springer, Berlin, 2000, 308pp. 2.
S. Succi, The Lattice Boltzmann Equation for
Fluid Dynamics and Beyond, Clarendon Press,
Oxford, 2001, 288pp. 3. M. Yoshino, T. Inamuro,
Int. J. Numer. Meth. Fluids 43 (2003) 183. 4. O.
Dardis, J. McCloskey, Phys. Rev. E 57 (4) (1998)
4834-4837 April 1998. 5. O. Dardis, J. McCloskey,
Geophysical Research Letters, Vol. 25, No. 9,
Pages 1471-1474, May 1, 1998. 6. N.S. Martys, H.
Chen, Phys. Rev. E 53 (1996) 743. 7. J. W. Elder,
J. Fluid Mech., Vol. 27, Part 1, Pages 29-48,
1967. 8. J. W. Elder, J. Fluid Mech., Vol. 27,
Part 3, Pages 609-623, 1967. 9. P. Frolkovic, H.
De Schepper, Adv. Water Res. 24 (2001) 63-72. 10.
C. I. Voss, W. R. Souza, Water Resources
Research, Vol. 23, No. 10, Pages 1851-1866,
October 1987. 11. C. M. Oldenburg, K.
Pruess,Water Resources Research, Vol. 31, No. 2,
Pages 289-302, February 1995. 12. W. Guo, C. D.
Langevin, User's Guide to SEAWAT A Computer
Program for Simulationof Three-Dimensional
Variable-Density Ground-Water Flow, U.S.
GeologicalSurvey, Techniques of Water-Resources
Investigations 6-A7, Tallahassee,
Florida,2002. 13. O. Kolditz, R. Ratke, H. G.
Diersch, W. Zielke, Advances in Water Resources,
Vol.21, No. 1, Pages 27-46, 1998. 14. J. A.
Woods, M. D. Teubner, C. T. Simmons, K. A.
Narayan, Water Resources Research,Vol. 39, No. 6,
1158-1169, 2003.