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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
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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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