Two-Phase Flow Numerical Modeling : Application to a Geological - - PowerPoint PPT Presentation

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-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 Chavant Chavant Chavant Chavant EDF R&D EDF R&D EDF R&D EDF R&D

Two-Phase Flow Numerical Modeling : Application to a Geological Nuclear Waste Disposal

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 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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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal

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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

• f 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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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal

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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)

( )

( )

O H H c Q F F Div m t

c c g c l c 2 2,

= = + + ∂ ∂

A classical two-phase flow model

l g l c

P P S f P − = = ) (

( )

( )

       − ∇ − = − ∇ − = g g

g g g g g r g l l l l l r l

P S k k F P S k k F ρ µ ρ µ ) ( . ) ( .

H H g

• l

H H l

K P M

2 2 2

≤ ρ

g g H g H g O H g O H g

C F F F ∇ − = +

2 2 2 2

ρ ρ

l l H l H l O H l O H l

C F F F ∇ − = +

2 2 2 2

ρ ρ

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-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
• f equations and front shape)

( ) ( )

Q P S k k Div t S

l l l l r l l

= ∇ + ∂ ∂ ) ) ( . ( µ φρ

• Ex. for relative permeabilities

( )

( )

2 / 1

1 1

m m l l l rel

S S k − − =

λ / 1 +

=

B l A l l rel

S S k

3 l l rel

S k =

(Van Genuchten) (Brooks&Corey) (cubic)

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal

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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 years 500 years t(year) 6.25 mol/year 0.5 mol/year Pl = 5.5 Mpa Q=0 Q=0 Pl = 4.2 Mpa

Modeling of a disposal cell for an intermediate level long-lived waste :

100 m

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal

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COUPLEX-GAZ I : caracteristic curves – Van Genuchten Mualem model

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)

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 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

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 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)

wres m n r c wres l

S P P S S +         +         − = 1 1

( )(

)

m m we we g rel

S S k

2 / 1

1 1 − − =

( )

( )

2 / 1

1 1

m m we we l rel

S S k − − =

Singularities for S = 1 :

) ( = ∂ ∂ Pc Pc S ∞ = ∂ ∂

w w g r

S S k ) ( ∞ = ∂ ∂

w w w r

S S k ) (

2nd

• rder polynomial C1

Interpolation for S > Smax (ex. Smax = 0,99)

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal

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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 m2 6 Mpa 0,1 clearance 0,8 Mpa 4,4 Mpa 3 Mpa Pcinit 10-15 m2 0,2 Primary package 10-19 m2 0,6 Concrete of package 10-18 m2 0,7 Filler concrete K Sinit

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-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 =

l

S ˆ ˆ

2 2

≤ − = ⇒ =

l g H l

• l

H H g

P P Pc M K P ρ

H H g

• l

H H l

K P M

2 2 2

= ρ

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-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 20 40 60 80 100 120 Y (m) Pc (Pa) 240 500 5000 10000 50000 500000

Time (years)

Saturation X = 103m

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 50 55 60 65 70 75 Y (m) S 240 500 5000 10000 50000 500000 Time (year)

Cox DZ FZ Conc. package gap Conc.

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-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 pressure remains constant 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 50 100 Y (m) Pg (Pa)

240 500 1000 5000 7000 10000 18000 38000 50000 500000

Times (years)

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal

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L

σ

K

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

( )

f u = ∇ Λ ∇ − .

∫ ∑

= −

∈ K K

f F

K

ε σ σ ,

σ

σ σ σ

d u F

K K

. .

,

∫

∇ Λ ≈ n

with

REF : Eymard R., Gallouët T., Herbin R. « A new finite volume scheme for anisotropic diffusion problems on general grids : convergence analysis » (CRAS 2007 – vol 344 – num 6)

( )

              − − ∇ +         − Λ =

σ σ σ σ σ σ σ

α α

K K K K K D K K K K K

d u d u u m F x x n

,

Coercivity term We remove what we added To determine ( ) ( )

∑

∈

−         − − = ∇

K

K K K K K K K D

u u d m u

ε σ σ σ σ σ σ

α x x n M .

( ) ( ) ( )

∑

∈ −

        − ⊗ − − − ⊗ =

K K

K K K K K K

d m

ε σ σ σ σ σ σ σ

α x x x x x x n M 1

( )

( )

∑

′ ′ ′

− =

σ σ σ σ σ K K K

u u F

, ,

C

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-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

( ) ( ) ( )

,

= − − − ∆

∑∑

′ ′ ′ − σ σ σ σ σ σ p K p K p p K p K K

u u k m m t A C

( ) ( ) ( ) ( )

, ,

= − + −

∑∑ ∑∑

′ ′ ′ ′ ′ ′ σ σ σ σ σ σ σ σ σ σ p L p L p K p K

u u u u C C

( ) ( )

p L p p p K p p p K

u k k u k k F if = = ≤

σ σ σ

else

,

K L

σ

( )

S p l ,

( )

g l

p p ,

( )

g l

p p ,

( )

g l

p p ,

( )

g l

p p ,

( )

S p e,

( )

g e g e

F F p p

σ σ ,

, ,

( )

g e g e

F F p p

σ σ ,

, ,

( )

g e g e

F F p p

σ σ ,

, ,

( )

g e g e

F F p p

σ σ ,

, ,

FE structure

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-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 (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 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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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-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)

5220 5230 Total nb of Newton iterations

3 12 Average nb of iterations per time step 26 24 Total CPU Times 10 43 Max nb of iterations per time step HFV FE Promising method (sushis developments)…

DoF normaly required

( )

S p e,

( )

g e g e

F F p p

σ σ ,

, ,

( )

g e g e

F F p p

σ σ ,

, ,

( )

g e g e

F F p p

σ σ ,

, ,

( )

g e g e

F F p p

σ σ ,

, ,

DoF actually used (FE structure)

( )

S p l ,

( )

g l

p p ,

( )

g l

p p ,

( )

g l

p p ,

( )

g l

p p ,

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal

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A 3D application : Couplex 2 (1/3)

Lenght of modulus : Ly= 100m Width of the modulus : 30m

15 mol/year/cell

QH2 (mol/year/cell)

10 000 4500

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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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal

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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

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal

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A 3D application : Couplex 2 (3/3)

Gas pressure evolution Saturation evolution

• 1

1 2 3 4 5 6 7 8 9 0,00001 0,001 0,1 10 1000 100000 time (years) Pgaz (MPa)

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 100000 time (years) S A B C D

A B C D concrete cox bentonite

4500 yrs

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Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal

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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-lived 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)