Irina Ginzburg

Cemagref, Antony, France

Lattice Boltzmann formulations for modeling variablysaturated flow in anisotropic heterogeneous soils

DYNAS WORKSHOP 2004

Bild 1

Advocated properties of the Lattice Boltzmann methods

� Each conservation law is exact as related to a microscopicquantity conserved by the collision operator

� Uniform analysis/implementation in 1D, 2D and 3D

� Simplicity and locality: neither initial guess nor globallinear/nonlinear systems of equations are needed

� Computational efforts per one evolution step increase onlylinearly with the space resolution

� Free equilibrium and collision components may improve for higherorder accuracy and enhance the stability

� On a regular grid, any shape boundaries may be fitted accurately

Bild 2

Outlook

– LB techniques to generic advection and anisotropic dispersionequations (AADE)

(1) Expanded equilibrium functions

(2) Anisotropic collisions

– Variably saturated flow modeling

(1) Equilibrium formulations for Richards’ equation

(2) Numerical examples in homogeneous and layered media onuniform and anisotropic grids

Bild 3

Velocities

�

Cq: D2Q9 and D3Q15 Models

� 0 � 0 �

��� 1 � 0 �

� 0 � � 1 �

� � 1 � � 1 �0 1

2

3

4

56

7 8

��� ���� ���� ���

� ��������� ��� ���

��� ���������

��� ���

� 0 � 0 � 0 �

��� 1 � 0 � 0 �

� 0 � � 1 � 0 �

� 0 � 0 � � 1 �

� � 1 � � 1 � � 1 �

z

x

y

3 1

2

6

5

04

78

9

12

14

10

11

13

��� ���

Also, D2Q5 � D3Q7 � D3Q13 � D3Q19 � D3Q27 � D4Q25 ��� � �

Bild 4

Generalized Lattice Boltzmann (GLBE) method

� Conservative equilibriumproperty:For any vector en conserved bythe evolution equation

en� � � � f ne.

��� 0 � theequilibrium contains the wholeprojection on en:

�

fn� � f � en ��� � feq.

� en �

� Evolution equation (following D.d’Humières, 1992)

fq �� r �

Cq � t 1 ��� fq �� r � t � ∑Q� 1

k� 0 λk ��

fk �

fkeq.

� ekq Qmq

� vector in population space f� � fq � , q� 0 � � � � Q 1

� � � f ne.� ∑Q� 1k� 0 λk �

�

fk �

fkeq.

� ek� Basis vectors and eigenvalues: ek and λk, 2 � λk � 0

� Projection into momentum space: f� ∑Q� 1q� 0

�

fkek,

�

fk� � f � ek ��

f� ek

� ek � 2

� f ne.� f f eq. , f eq. is the equilibrium function

� Qm

�� r � t � is a source term

Bild 5

Equilibrium distribution

� f eq.q � f eq. �

q f eq.�

q

f eq. �

q � f eq. �

q̄

f eq.�

q � f eq.�

q̄�

Cq� �

Cq̄

� f eq.� � � t� qJq � , Jq��

J��

Cq

(1) Jα� f� Cα, α� 1 � � � � � Dconserved momentum forhydrodynamic equations

(2) D-dimensional vector

�

J:external “advection vector”for AADE

� Mass conservative equilibrium function, s� ∑Q� 1q� 0 fq:

f eq. �

0 ∑Q� 1q� 1 f eq. �

q � s

f eq.q � tqD̄eq.

� s � , q �� 0

� D̄eq.

� s � is arbitrary “diffusion” function� “Isotropic” weights (Qian et al. 1992):

∑Q� 1q� 1 t� qCqαCqβ� δαβ � t� q� f ��

�

Cq�

2

�

� Expanded Equilibrium (E-model) for AADE:anisotropic weights tq

Bild 6

Orthogonal integer “hydrodynamic basis” of DdQq models:

� Basis vectors are obtained byconsequent multiplication onthe velocity components Cqαand orthogonalization

� Maximum degrees of freedomfor hydrodynamic models

� Basis consists from

� Q 1 �� 2 1 even vectors and

� Q 1 �� 2 odd vectors

mass e0 � 1 �

momentum Cα � Cqα � α� 1 � � � � � d �

part of kinetic energy p � e � � Qc2q

b2 � � c2q� �

�

Cq�

2

viscous stress tensor p � xx � � DC2qx

c2q �

viscous stress tensor p � ww � � C2qy

C2qz � � d� 3

viscous stress tensor p � αβ � � CqαCqβ � α �� β �

energy flux q � αβ � � � b1c2q

b3 � Cqα �

energy flux q � α � � � C2qγ

C2qβ � Cqα �

third order momentum m � xyz � � CqxCqyCqz � d� 3 �

kinetic energy square π � e � � a1 � Qc4q

b6 � a2 � Qc2q

b2 � �

kinetic energy square π � xx � � � B1c2q

B2 � � dC2qx

c2q � �

kinetic energy square π � ww � � � B3c2q

B4 � � C2qy

C2qz � �

Bild 7

Macroscopic equation is second orderapproximation of the exact conservation relation:1 � � � � f� f eq. ��� 0

� Expanded equilibrium E-model +eigenvalues λα for velocity vectors � Cqα :

∂ts ∇ � �

J� ∂αΛ � λα � ∂βKαβ Sm

Λ � λ ��� �

12

1λ � , 2 � λ � 0

Kαβ � D̄eq.

� s � �� ∑Q� 1q� 1 CqαCqβ f eq. �

q

Sm� ∑Q� 1q� 0 Qm

q

� Second-order tensor of the numerical diffusion,Λ � λ � sUαUβ,

�

J� s�

Ucan be removed with expanded projections:f eq. �

q � f eq. �

q 12UαUβt� q � 3CqαCqβ δαβ �

Bild 8

Anisotropic collision: L-model for AADE

� BGK:

eq� δq � q� 0 � � � � � Q 1

Momentum space=populationspace, � f � eq ��� fq, λq� λ

� Even link basis vectors:eq

� � δq δq̄ � λ �

q

f �

q � � f � eq ���

12 � fq fq̄ �

� Odd link basis vectors:eq

� � δq δq̄ � λ� q

f� q � � f � eq ���

12 � fq fq̄ �

– “Link” evolution equation:

fq �� r �

Cq � t 1 ��� fq λ �

q � f �

q

f eq. �

q � λ� q � f

�

q

f eq.�

q � Qmq

� r �

Cq � r

� r �

Cq

� �

�

��

– Mass conservation: λ �

q� λe � q� 0 � � � � � � Q 1 �� 2

– Even for isotropic equilibrium:

∂ts ∇��

J� ∂α � αβ∂βD̄eq.

� s � Sm

� αβ are linear combination of eigenvalue functions Λ � λ

�

q �

Bild 9

Contour plots s � 10� 3 of 2D Gauss distribution:

� αβ� ��

U � � kT δαβ � kL kT � uαuβ � , uα� Uα

���

U �

,

�

J��

Us

Optimal solutions annihilate O � ε3 � (convective) or O � ε4 � (diffusion) errors

Pure advection kT� kL� 0:Optimal advection solution

Λ � λoptD � Λ � λ

opte ���

112

-40 -30 -20 -10 0 10 20 30 40X

-30

-20

-10

0

10

20

30

Z

theory: T=0, T=100, T=200T=100T=200

Pure convection with optimal advection solution

Pure advection kT� kL� 0:Optimal diffusion solution

Λ � λoptD � Λ � λ

opte ���

16

-40 -30 -20 -10 0 10 20 30 40X

-30

-20

-10

0

10

20

30

Z

theory: T=0, T=100, T=200T=100T=200

Pure convection with optimal diffusion solution

Anisotropic case

kL � kT� 220

Angle π4

-40 -30 -20 -10 0 10 20 30 40X

-20

-10

0

10

20

30

40

Z

theory: T=0, T=100, T=200T=100T=200

Dispersion test with optimal convection solutionAngle θ=π/4, k

L/k

T=2

20

Bild 10

Formulations to Richards’ Equation

� mixed, θ� h based∂tθ ∇� K � θ ���

a� � 1z� ∇� K � θ ���

a∇h

� moisture content, θ based∂tθ ∇� K � θ ���

a� � 1z� ∇� K � θ ���

ah� θ∇θ� pressure head, h basedθ� h∂th ∇� K � h ��

a� � 1z� ∇� K � h ��

a∇h

� Transformations of the diffusion term (e.g.,Kirchhoff Integraltransform with flux balancing correction for layered media)

– K � L T� 1

� hydraulicconductivity, K� KrKs

– Ks � L T� 1

� saturated hydraulicconductivity, Ks� kρg� µ

– k � L2

� -permeability

– Kr � h �� krw dimensionlessrelative hydraulic conductivity

– k� a permeability tensor

�

a is dimensionless tensor

�

a� � in isotropic case

Bild 11

Convective part of Richards’ Equation

� Odd equilibrium part, f eq.�

q � t� qJq, Jq��

J��

Cq

matches the gravitation part for all formulations:�

J� K � θ � h ���

a� � 1z � ∇� K � θ � h ���

a� � 1z

� Saturated conductivity Ks plays the role of characteristic velocity� Darcy velocity, � u� K�

a

� ∇h �

1z �

is computed locally and equally for all formulations:

� u��

J �

DDiffusion flux:

�D� � � � 1

2 � �� f ne.� , � � � Cx � Cy � Cz �

Bild 12

LB Formulations to Richards’ Equation:

� Mass conservation:f eq. �

0

� s ∑Q� 1q� 1 f eq.

q

� Even equilibrium part of movingpopulations defines thediffusion term: f eq. �

q � tqD̄eq.� s �

� moisture content, θ based: s� θ, D̄eq.

� s �� f � θ �

Particular case: D̄eq.

� θ �� θL-Model: K � θ ���

ah� θ� L � Λ � λ � � , λ� λ � θ �

E-Model: K � θ � h

�

θ� L � Λ � λ � �

� mixed, θ� h based: s� θ, D̄eq.

� s �� h, h� h � θ �

L-Model: K � h ���

a� L � Λ � λ � � , λ� λ � h �

E-Model: K � h �� L � Λ � λ � �

� pressure head, h based: s� h, D̄eq.

� s ��� h∂th ∇� K � h ���

a� � 1z� ∇� K � h ���

a∇hSteady state only, ∂th� 0:

∇� K � h ���

a� � 1z� ∇� K � h ���

a∇h

Bild 13

Saturated zone with extended moisture content

� During the sub-iterations, thepopulations coming from theunsaturated points are fixed

� When local stationary massstate is reached with givenaccuracy, the propagation stepfrom the saturated into theunsaturated zone takes place

– The retention curve is linearly extrapolated beyond air entry valuehs� h � θs � :h � θ �� P � θ θs � hs � P� h� θ � θ� θs � � h � hs � θ � θs

– The continuous numerical variable, extended water content isθ� θs P� 1

� h hs �

– The Poisson equation is covered in pseudo incompressible form:∂tθ ∇� � u� qs,

� u� Ks�

a� � ∇h �

1z �

� Sub-steps in saturated points are used to reduce ∂tθ

Bild 14

Heavy rainfall episodes

physical axis parallel to LB axis

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6z, [

m]x, [m]

Vector fieldwater-table

geometry

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

z, [m]

x, [m]

Vector fieldwater-table

geometry

open surface parallel to LB axis

– No-flow condition except foropen surface

– Seepage face conditions onopen surface

– Multi-reflection boundaryconditions

Bild 15

Multi-reflection boundary condition

extension of Ginzburg & d’Humières, Phys. Rev. E, 68, 2003

– Boundary surface cuts at

� rb δq

�

Cq the link betweenboundary node� rb and anoutside one at� rb

�

Cq

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � �

� � � � � ���

����

��� ��

���

Cq

δq

�

Cq

δq

τ

� n

� rb 2

�

Cq

� rb �

Cq

� rb

� rb �

Cq

κ1� rb 12

�

Cq

� rb δq

�

Cq

�

κ̄� 2

�

κ� 1

�

κ̄� 1

�

κ0

� � ��

��

�fq̄ �

� rb � t 1 � � κ1 fq �� rb �

Cq � t 1 �

κ0 fq �� rb � t 1 �

κ� 1 fq �� rb �

Cq � t 1 �

κ̄� 1 fq̄ �� rb �

Cq � t 1 �

κ̄� 2 fq̄ �� rb 2

�

Cq � t 1 �

Fp c

q wq �� rb δq

�

Cq � t 1 �

� Closure relation for even and odd equilibrium partsbased on Chapman-Enskog and Taylor expansions

Bild 16

Reduced vertical velocity on open surface, uz� Ks

SAND on non-aligned grid

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4

u_z/

Ks

x, [m]

hs=0, FE hs=-9cm, LB, Nodehs=-9cm, LB, Wall

YLC and SAND, on aligned grid

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4

u_z/

Ks

x, [m]

YLC, hs=-2cm, FEYLC, hs=-2cm, LB

SAND, hs=0, FESAND, hs=0, LB

Relative ex-filtration fluxes

SCL on non-aligned grid

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10 12 14 16 18

Rel

ativ

e E

xfilt

ratio

n

time, [h]

FE, hs=0 LB: hs=0, it=80LB, hs=0, it=20

YLC on aligned grid

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6 7 8R

elat

ive

Exf

iltra

tion

time, [h]

FE, hs=-2cmLB, hs=-2cm

� Compared to finite elementsolutions ofBeaugendre et.al,CMWR 2004

� Rainfall intensity is qin� 0� 1Ks

for all soils

� FE grid with 280 nodes on theopen surface

� LB grid with 70 nodes on theopen surface

Bild 17

Layered Porous media: LB continuity conditions

Let i be the index of a soil layer perpendicular to some axis z

� Interface basic conditions:h � i �� h � i � 1 �

u � i �

z � u � i � 1 �

z

��

� r 12

�

Cq

� r

� r 12

�

Cq

interface

i 1 layer

i layer

�

z� Continuity of the odd component:

� Jq �� r� 1

2

�

Cq � �

λ� q

2 1 � f ne.�

q �� r� 1

2

�

Cq � �� 0 � Jq��

J��

Cq The sum overthe links cut by the interface corresponds to normal component ofthe diffusion flux, uz� Jz Dz

� Continuity of even equilibrium component midway between latticenodes: � f

eq. �

q �� r� 1

2

�

Cq � �

12∂q f eq. �

q �� r� 1

2

�

Cq � �� O � ε2

�

– Moisture content form: θ is continuous and h is discontinuous onthe interface.

– Mixed and pressure head forms: h is continuous and θ isdiscontinuous on the interface

Bild 18

Exact three-layered solution in saturated zone:h � i � � z �� h � i� 1 �

l ∂zh � i �� z Z � i� 1 � �

– BCM system: medium-grained sand in the middle, fine-grained sand at the top andat the bottom, Ks ratio 100

– VGM system: Sandy soil (n� 5) in the middle, Silty Clay Loam (n� 1 � 31) is at thetop and at the bottom, Ks ratio � 38

BCM

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

z, [m

]

Pressure head h, [m]

BCM, LB

soils

1 6 11 16 21 26Extended water content

BCM, LB

VGM

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

z, [m

]

Pressure head h, [m]

VGM, LB

soils

1 1.25 1.5Extended water content

VGM, LB

Retention curves

0

0.2

0.4

0.6

0.8

1

-2 -1.5 -1 -0.5 0

Effe

ctiv

e w

ater

con

tent

Pressure head h, [m]

BCM, fine-grained sandBCM, coarse-grained sand

VGM, SCLVGM, SAND, h_s=0

� Solution is fixed byinterface conditions andtop/bottom Dirichletconditions � Extended effective water content: θ̃� 1 � P � h� hs � , P � h� θ̃ � 1 �

Bild 19

Steady state solution in three-layered drainage tube

Semi-analytical approach:

– Implicit in time, Runge-Kuttamethod with (RKA) and without(RK) adaptive step-size controlfollowing Marinelli & Durnford,J. of Irrigation and DrainageEngineering, 1998

– Integration of Darcy law withenforcing interface conditions:h � Z � i � ��� h � Z � i� 1 � �

z� Z � i� 1 � �

h� i� 1 �

l

hK� i �

r � h

��

K� i �

r � h

��� R� i ��

dh� ,

R� � qin� Ks

� Phreatic surface h� 0 is instantaneously lowered from the top to the bottom.Constant flux �ur� � qin

�

1z� � 0 � 1Ks

�

1z is fixed at the top

� BCM system with Ks ratio equal to 10 (middle to top and to bottom).Air entry values,� 0 � 35 m (top/bottom) and� 0 � 15 m(middle)

– Minimum ∆x of the adaptive RKA � 10 � 7m, LB � 10 � 3m

0

0.5

1

1.5

2

-0.8 -0.6 -0.4 -0.2 0

z, [m

]

Pressure head h, [m]

DarcyLB, 600 cells

LB, 3000 cells

Mixed formulation

0

0.5

1

1.5

2

-0.8 -0.6 -0.4 -0.2 0

z, [m

]

Pressure head h, [m]

DarcyRKA

RK, 600RK, 3000RK, 80000RK, 800000

“Semi-analytical” solution Bild 20

From orthorhombic discretization grid to cuboid computational grid

� In lattice units, ∆x� ∆y� ∆z� 1and ∆t� 1

� Let L� Llb

� L, U� U lb� U and

T� T lb

� T be ratios of thecharacteristic values for length,velocity and time variables(T� L� U , T lb� Llb

� U lb)

� Coordinate scaling: � x� � L � � � x,

� � diag � Lx � Ly � Lz �

� � T� 1

� � � � ��

J � T� 1

� ��

J

� Richards’ equation on anisotropic grid∂t� θ ∇� � K� � θ � K

� � � 1z� ∇� � K� � θ � Ka� � ∇h�

h� � LhK� � UKK� αβ� Ka

αβLα

Ka�

αβ� KaαβLαLβ

� For layered media, LB equations are divided by L � i �

z to keep thecontinuity of the normal physical velocity, u� z

� i �� Uu � i �

z :K� αβ � Ka

αβLα� Lz

Ka�

αβ � KaαβLαLβ� Lz

Tangential LB velocity is discontinuous on the boundaries:

u� τ

� i � 1 �� u� τ � i �� L � i �

z � L � i � 1 �

z

but the physical velocity should be continuous:u� τ

� i � 1 � L � i � 1 �

z � u� τ

� i � L � i �z

Bild 21

Layered media with uniform and anisotropic refining

No explicit enforcing of the interface conditions on the heterogeneousboundaries, Ks ratio 100

Equal columns of 132 cells are used in both cases

0

0.5

1

1.5

2

-0.8 -0.6 -0.4 -0.2 0

z, [m

]

Pressure head h, [m]

DarcyA=1

Uniform refining

0

0.5

1

1.5

2

-0.8 -0.6 -0.4 -0.2 0

z, [m

]

Pressure head h, [m]

DarcyA=5

L � 2 �

z � L � 1 �

z � 5, L � 3 �

z � L � 1 �

z

Bild 22

Steady-state unconfined flowcomputed with h formulation on anisotropic grids

Uniform refining in1002� 12 m2 box

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

Dis

tanc

e z,

[m]

Distance x, [m]

Uniform grid, 100x100 boxwater table

open_water level

Anisotropic non-uniform refiningL � 1 �

z � 0� 5, L � 2 �

z � 4, Lτ� 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

Dis

tanc

e z,

[m]

Distance x, [m]

Anisotropic grid I, 50x130 box, A=8A=1 MRT, A=8 L_II, A=8 TRT, A=3.666

open_water level

� Reference solution:Clement et al, J.Hydrol.,1996

� Boundary conditions:

(1) Top/bottom: No-flow withbounce-back reflection

(2) West: Hydrostatic,h z� 1 m withanti-bounce-back reflection

(3) East: Seepage abovez� 0� 2 m.

Bild 23

Solutions on anisotropic grids

Vertical velocity

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.0005 -0.00025 0

Dis

tan

ce z

, [m

]

Vertical velocity, [m]

A=1 MRT, A=3.666L_II, A=3.666 L_I, A=8 TRT, A=3.666

interface

Pressure head

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Dis

tan

ce z

, [m

]

Pressure head, [m]

A=1 MRT, A=8 L_I, A=8 TRT, A=3.666

interface

Horizontal velocity

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0005 0.001 0.0015 0.002

Dis

tan

ce z

, [m

]

Horizontal velocity, [m]

A=1 MRT, A=3.666L_II, A=3.666 L_I, A=8 TRT, A=3.666

interface

� u� τ

� 1 �� u� τ

� 2 �� � 3� 666 � 8 �

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Current and forthcoming work

� Discontinuous equilibrium/eigenvalue functions for anisotropicheterogeneous soils

� Stability analysis of LB schemes for AADE in function of all freecollision/equilibrium parameters

� Coupling of different formulations:

(1) Moisture content formulation is the most robust for imbibitionproblems but is suitable for homogeneous soils only

(2) Pressure head formulation is the most robust forheterogeneous soils but is suitable for steady state problemsonly

(3) Mixed formulation is the most general but rigorous stabilityconditions for expanded equilibrium functions are not yetestablished

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THANKS !!!

� DYNAS Colleaguesfrom CEMAGREF/ENPC/CERMICS/INRIA

� Cyril Kao and Jean-Philippe Carlierfrom Cemagref

� H. Beaugendre from Cermics, Enpc

� Dominique d’Humières from l’ENS, Paris

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