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Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
1
Sylvie Granet Sylvie Granet Sylvie Granet Sylvie Granet –––– Clément Clément Clément Clément ChavantChavantChavantChavantEDF R&DEDF R&DEDF R&DEDF R&D
Two-Phase Flow Numerical Modeling : Application to a Geological Nuclear
Waste Disposal
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
2
OUTLINE
�Industrial context : Nuclear waste storage
�A classical two-phase flow model
�An example of application : modeling of a disposal cell for an intermediate level long-lived waste (Benchmark Couplex Gaz 1 submitted by Andra)
� Presentation of the problem� Numerical methods :
• Classical FE Scheme : Discretization and results• Hybrid Finite Volume method : overview and first results
�A first 3D study : modeling of a modulus of High level long-lived waste (Benchmark Couplex Gaz 2 submitted by Andra)
� Results� Perspectives
�Conclusions
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
3
Industrial context : Underground waste storage
�A fully coupled THMC problem on a complex geometry !
�Main reasons of two-phase flow modelling :
� Presence of initially unsaturated media (plugs, sealing …)
� Ventilation of the galleries� Thermal drying� Hydrogen production due to corrosion
of steel components (containers, casing)
�Hydraulic specificities :
� High level of capillary pressure (50 Mpa) and high level of gas pressure (due to corrosion)
� Saturation closed to one in the geological media
Scheme for an underground mined repository
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
4
�2 components (H2 and H2O) in 2 phases (liquid and gas)�Capillary relation �Mass conservation for water and hydrogen :
�Transport equations : • Darcy’s law for each phase
• Diffusion law linking component velocities in each mixture (Fick’s Law)
• Dissolution (Henry’s Law)
• Vaporization (equilibrium equation)
( ) ( ) OHHcQFFDivmt c
cg
clc 22,==++
∂∂
A classical two-phase flow model
lglc PPSfP −== )(
( )
( )
−∇−=
−∇−=
g
g
ggg
ggr
g
lll
llr
l
PSkk
F
PSkk
F
ρµ
ρµ
)(.
)(.
H
Hg
olH
Hl
K
P
M
2
2
2
≤ρ
ggHg
Hg
OHg
OHg CF
FF∇−=+
2
2
2
2
ρρ llHl
Hl
OHl
OHl CF
FF ∇−=+2
2
2
2
ρρ
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
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Main modelisation difficulties
� Injection of gas in a saturated porous media• Geological media initially saturated : what is the initial
concentration of hydrogen or air in the liquid ?• High level of gas pressure
�Presence of multiple barriers• Very high level of heterogeneities of the different materials• Initial level of saturation very dependant of the material
�Huge non linearities • Capillary and Relative permeabilities functions (influencing type
of equations and front shape)
( ) ( ) QPSkk
Divt
Sl
l
llrll =∇+
∂∂
))(.
(µ
φρ
Ex. for relative permeabilities
( )( )2/111mm
lllrel SSk −−=
λ/1+= Bl
Al
lrel SSk
3l
lrel Sk =
(Van Genuchten)
(Brooks&Corey)
(cubic)
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
6
�Anisotropic problem (in the clay KH ≠Kv )
�Total hydrogen Flux for each primary package :
COUPLEX-GAZ I (Andra 2007): Exercice definition(1/2)
QH2
10 000 years500 years t(year)
6.25 mol/year
0.5 mol/year
Pl = 5.5 Mpa
Q=0Q=0
Pl = 4.2 Mpa
Modeling of a disposal cell for an intermediate level long-lived waste :
100 m
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
7
COUPLEX-GAZ I : caracteristic curves – Van Genuchten Mualem model
0
5000
10000
15000
20000
25000
30000
0,980 0,985 0,990 0,995 1,000
S
f(S)
f’(Smax) = P’(Smax)
S(Pc)
0
0,10,2
0,30,4
0,5
0,60,7
0,80,9
1
0,00E+00 2,00E+08 4,00E+08 6,00E+08 8,00E+08 1,00E+09
Pc (Pa)
S
package
pack. concrete
clerance
filler concrete
FZ
DZ
Cox
relative permeabilities
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,2 0,4 0,6 0,8 1
krelw (pack.)
krelw (pack. concrete)
krelw (gap)
krelw (fil. concrete)
Krelw (FZ)
krelw (DZ)
krelw (Cox)
krelgz(pack.)
krelgz(pack.concrete)
krelgz(gap)
krelgz(fil. concrete)
krelgz(FZ)
krelgz(DZ)
krelgz(Cox)
wresmn
r
c
wresl S
P
P
SS +
+
−=
1
1( )( ) mm
wewegrel SSk
2/111 −−=( )( )2/111mm
wewelrel SSk −−=
Singularities for S = 1 :
0)( =
∂∂
Pc
PcS∞=
∂∂
w
wgr
S
Sk )( ∞=∂
∂
w
wwr
S
Sk )(
2nd order polynomial C1 Interpolation for S > Smax (ex. Smax = 0,99)
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
8
COUPLEX-GAZ I : Exercice definition (2/2)
� Actually, we consider a small gas pressure : Pg= 1atm (corresponding to a initial concentration of hydrogen in liquid)
�Couplex’s initial conditions : high contrast of saturation and capillary pressure
• In the clay (healthy, disturbed or fractured) :
S = 1
Hydrostatic liquid pressure
10-12 m26 Mpa0,1clearance
0,8 Mpa
4,4 Mpa
3 Mpa
Pcinit
10-15 m20,2Primary package
10-19 m20,6Concrete of package
10-18 m20,7Filler concrete
KSinit
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
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COUPLEX-GAZ I : Numerical method (FE)
� Couplex Gaz modelisation’s tool (www.code-aster.org)
� Choice of the unknowns : Pc and Pg• In our formulation, we write
• In saturated area :
Variable transformation :
� A Classical Finite Element method :
• Finite Elements with Q1 elements• Lumping of the mass matrix :
Non diagonal mass matrix => maximum principle not verified => Oscillations
Integration points at the vertex of the elements
• Time discretization = Implicite Euler• Newton method for non linear resolution
1=lS
0ˆˆ 2
2
≤−=⇒= lgHlol
H
Hg PPPc
M
KP ρ
H
Hg
olH
Hl
K
P
M
2
2
2
=ρ
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
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COUPLEX-GAZ I : Gas saturation and capillary pressure profiles – X = 103 m
�Complete saturation at 60 000 years !
�Vertical cross section
�3 steps :
1- capillary equilibrium (t<200 years)
2- Small desaturation by gas production (t< 10 000 years)
3- The gas disappears graduately
Capillary pressure X = 103m
-6,00E+06
-4,00E+06
-2,00E+06
0,00E+00
2,00E+06
4,00E+06
6,00E+06
8,00E+06
0 20 40 60 80 100 120
Y (m)
Pc
(Pa)
0
240
500
5000
10000
50000
500000
Time (years)
Saturation X = 103m
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
50 55 60 65 70 75Y (m)
S
0
240
500
5000
10000
50000
500000
Time (year)
Cox DZ FZ
Conc.
package gap
Conc.
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
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COUPLEX-GAZ I : Gas pressure – X = 103 m
�Vertical cross section
Maximal Gas Pressure of 6,75 Mpa at 10 000 years– The p ressure remainsconstant in the engineered area
Gas pressure X = 103m
0,00E+00
1,00E+06
2,00E+06
3,00E+06
4,00E+06
5,00E+06
6,00E+06
7,00E+06
8,00E+06
0 50 100Y (m)
Pg
(Pa)
024050010005000700010000180003800050000500000
Times (years)
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
12
LσK
COUPLEX-GAZ I : Using of a Hybrid Finite Volume scheme (1/4)
�On an elliptic problem
� Volumic integration :
� Flow calculation
�Computation of discrete flow :
� Flow
� Consistence
� finally
( ) fu =∇Λ∇− .
∫∑ =−∈
KK fFKεσ
σ, σσ σσ duF KK .., ∫ ∇Λ≈ nwith
REF : Eymard R., Gallouët T., Herbin R. « A new finite volum e scheme for anisotropic diffusion problems on general grids : convergence analysis » ( CRAS 2007 – vol 344 – num 6)
( )
−−∇+
−Λ=σ
σσ
σ
σσσ αα
K
KKKKD
K
KKKK d
ud
uumF
xxn,
Coercivity term We remove what we addedTo determine
( ) ( )∑∈
−
−−=∇K
KK
KKKKKD uu
dmu
εσσ
σ
σσσ α xx
nM .
( ) ( ) ( )∑∈
−
−⊗−−−⊗=
K
K KKK
KKK d
mεσ
σσσ
σσσα
xxxxxxnM 1
( ) ( )∑′
′′ −=σ
σσσσ KKK uuF ,, C
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
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COUPLEX-GAZ I : Using of a Hybrid Finite Volume scheme (2/4)
�Hybrid Finite Volume for two phase flow modeling
� Mass conservation for the two constituents
� Flow continuity
� Upstream flow
� Unkowns : • P,S at the center, • Pl and Pg at the interface
( ) ( ) ( ) 0, =−−−∆ ∑∑
′′′
−
σ σσσσσ
pK
pK
ppK
pK
K uukmmt
AC
( ) ( ) ( ) ( ) 0,, =−+− ∑∑∑∑′
′′′
′′σ σ
σσσσ σ
σσσpL
pL
pK
pK uuuu CC
( )
( )pL
pp
pK
pp
pK
ukk
ukk
Fif
=
=
≤
σ
σ
σ
else
0,
K Lσ
( )Sp l ,( )gl pp ,
( )gl pp ,
( )gl pp ,
( )gl pp ,
( )Sp e ,
( )gege FFpp σσ ,,,
( )gege FFpp σσ ,,,
( )gege FFpp σσ ,,,
( )gege FFpp σσ ,,,
FE structure
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
14
COUPLEX-GAZ I : Using of a Hybrid Finite Volume scheme (3/4)
�Vertical cross section
Gas Pressure(X=103m)
0,00E+00
1,00E+06
2,00E+06
3,00E+06
4,00E+06
5,00E+06
6,00E+06
7,00E+06
0 50 100
Y(m)
Pg(
Pa)
500 years
10000 years
100 000 years
500 000 years
1 million years
Not exactly the same case (no gravity and isotropic permeability), but closed results
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
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�In this test, Hybrid Finite Volume method allows us :
�A better initial saturation condition (S is an unknown instead of Pc)�Good performance (better matrix profile) :
COUPLEX-GAZ I : Using of a Hybrid Finite Volume scheme (4/4)
52205230Total nb of Newton iterations
312Average nb of iterations per time step
2624Total CPU Times
1043Max nb of iterations per time step
HFVFE
Promising method (sushis developments)…
DoF normaly required
( )Sp e ,
( )gege FFpp σσ ,,,
( )gege FFpp σσ ,,,
( )gege FFpp σσ ,,,
( )gege FFpp σσ ,,,
DoF actually used (FE structure)
( )Sp l ,( )gl pp ,
( )gl pp ,
( )gl pp ,
( )gl pp ,
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
16
A 3D application : Couplex 2 (1/3)
Lenght of modulus : Ly= 100mWidth of the modulus : 30m
15 mol/year/cell
QH2 (mol/year/cell)
10 0004500 t(year)
100 mol/year/cell
50 000
Gas production around each cell :
3D Modeling of a modulus of High level long-lived waste :
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
17
A 3D application : Couplex 2 (2/3)
Numerical scheme : Classical FE scheme (sequential computation)
Mesh : 115 000 elements – 160 000 nodes – 287 000 equations
Performances : CPU time : 100 hours for simulation of 500 000 years
�Initial conditions :
�In the geological media :
S=1; Hydrostatic liquid pressure
�In plugs and drifts :
S=0,7; Pg = 1atm
�Material datas :
�Mualem/Van Genuchten model
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
18
A 3D application : Couplex 2 (3/3)
Gas pressure evolution Saturation evolution
-1
0
1
2
3
4
5
6
7
8
9
0,00001 0,001 0,1 10 1000 100000
time (years)
Pga
z (M
Pa)
A
B
C
D
0,65
0,7
0,75
0,8
0,85
0,9
0,95
1
1,05
0,00001 0,001 0,1 10 1000 100000time (years)
S
A
B
C
D
A
BCD
concretecox
bentonite
4500 yrs
Scaling Up and Modeling for Transport and Flow in PorousMedia - October 13-16, Dubrovnik 2008Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
19
Conclusions
�Description of a coupled two-phase flow model :
� Model of gas transfer in porous media
� 2 phases, 2 constituents
� Diffusion in gas and liquid mixture
�Application to industrial studies (undeground waste storage modeling ):
�Benchmark Couplex 1 (2D - intermediate level long-liv ed waste )
� Treated with a classical FE method
� Partially treated with a promising Hybrid Finite volume method
�Benchmark Couplex 3 (3D - High level long-lived waste )
� First results obtained with a sequential FE method and a coarse mesh� To be continued with a parallelism strategy
� To be continued with Hybrid Finite volume method (Sushi Method – PHD O. Angelini)