two-phase flow numerical modeling : application to a

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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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


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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


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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


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�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


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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)


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�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


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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)


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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


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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


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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.


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COUPLEX-GAZ I : Gas pressure – X = 103 m


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)


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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


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�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 ,,,




FE structure


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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


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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) :


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 ,





DoF actually used (FE structure)

( )Sp l ,( )gl pp ,

( )gl pp ,

( )gl pp ,

( )gl pp ,


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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 :


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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


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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


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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)

two-phase flow numerical modeling : application to a

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