Couplex-Gas Benchmark

Transcription

Couplex-Gas Benchmark
Florian Caro
CEA Saclay
Couplex-Gaz workshop, 4th and 5th april 2007
Outline
1. Couplex Gas case 1.a
Hypothesis
Numerical scheme
Numerical simulation vs analytical cases
Gas pressure time evolution
Mesh convergence
Model degeneration (Sl → 1)
2. Couplex Gas case 1.b
Hypothesis
Numerical scheme
Gas pressure time evolution
Mesh convergence
Model degeneration (Sl → 1)
CEA Saclay DEN/DM2S/SFME
Laboratoire de Modélisation des Transferts en Milieu Solide
p. 2
Two Phase Flow Model (Hypotesis)
1. Mass conservation for each phase
2. Darcy’s flow
3. Capillary pressure Pc = Pg − Pl
4. Isothermal flow
5. Gas phase : compressible (perfect gas)
6. Liquid phase : incompressible
˛
˛
˛ ρl − ρ0l ˛
˛ ≤ 4 × 10−3 .
˛
0
˛
˛ ρ
l
7. Porosity variations negligible
˛
˛
˛ Φ − Φ0 ˛
˛ ≤ 4 × 10−2 ,
˛
˛ Φ0 ˛
8. Van Genuchten model for relative permeabilities and capillary pressure
p. 3
Mathematical model
M (Sl , Pg )∂t
"
Sl
Pg
#
+ div
A(Sl , Pg )
"
∇Sl
∇Pg
#!
=
"
Ql /ρl
rT Qg
#
+B
where M ∈ R2×2 , A ∈ R4×4 and B ∈ R2 (λα = krα /µα ) :
M (Sl , Pg ) = Φ
A(Sl , Pg ) = k
1
−Pg
0
(1 − Sl )
−λl dPc /dSl Id
0
!
,
λl Id
λg Pg Id
!
0
“
” 1
div kλl ρl g∇z
@
“
” A
B=−
div kλg Pg ρg g∇z
p. 4
Numerical Scheme
MPFA Scheme dor the spatial discretization (Aaavatsmark and al.,
98’)
Ql
+ F (Sl , Pg )z,
ρl
>
>
: −ΦPg ∂t Sl + Φ(1 − Sl )∂t Pg − E(Sl , Pg )Pg = RT Qg + H(Sl , Pg )z,
M
8
>
>
<
Φ∂t Sl − B(Sl )Sl − D(Sl )Pg =
where B, D, E, F, H = spatial discretization matrix
Implicit fully coupled Euler scheme for the time discretization
8
n+1
n
n
S
−
S
Q
>
l
n+1
n+1
n+1
n+1
n
n
l
l
>
>
=
)P
−
D(S
)S
−
B(S
)z,
,
P
Φ
+
F
(S
g
g
l
l
l
l
>
>
∆t
ρl
>
>
<
n+1
n+1
− Pgn
− Sln
n+1 Pg
n Sl
+ Φ(1 − Sl )
−ΦPg
>
>
∆t
∆t
>
>
>
>
RT n
>
:
−E(Sln+1 , Pgn+1 )Pgn+1 =
Qg + H(Sln , Pgn )z.
M
Fixed point method for the non linear system resolution
p. 5
1D Validation : Analytical Solution Definition
Domain : (−1, 1)
Dirichlet limit conditions
Absolut discontinuous permeability
k(x) =
(
k1 in (−1, 0)
k2 in (0, 1)
Liquid saturation and gas pressure
8
>
>
>
>
>
>
<
Sl (x, t) =
8
”
1“
−t/τ
>
>
sin(πx)
>
< 2 1 + βe
“
”
>
>
1
>
−t/τ
:
>
1 + βe
sin(πk1 /k2 x)(1 − x)
>
>
2
>
>
>
: P (x, t) = Cste in (−1, 1)
g
in (−1, 0)
in (0.1)
Definition of limit conditions, initial conditions and source terms
using this definition of Sl and Pg
p. 6
1D Validation : Analytical and Approximate Liquid Saturation
Time step : ∆t = 31 × 103 s
Final time : T = 31 × 106 s
Number of cells : N = 200
Saturation
0.60
0.58
0.56
0.54
0.52
0.50
0.48
0.46
0.44
Exact
0.42
Approximate
0.40
−1.00
−0.80
−0.60
−0.40
−0.20
0.00
0.20
0.40
0.60
0.80
1.00
x
p. 7
1D Validation : Mesh Convergence
Final time : T = 31 × 106 s ≃ 1 year (τ ≃ 6 months)
N ∈ {100, 200, 500, 1000, 2000}
Relative L2 error
εN =
k(Sl , Pg )exa (T ) − (Sl , Pg )app (T )kL2
,
k(Sl , Pg )exa (T )kL2
−5
log(Relative L2 error)
−6
−7
−8
−9
−10
−11
−8.0
−7.5
−7.0
Time step = 31*10^4 s
−6.5
−6.0
−5.5
−5.0
−4.5
log(1/N)
p. 8
N ∈ {100, 200, 500, 1000, 2000}
−5
−6
−7
−8
−9
−10
−11
−8.0
−7.5
−7.0
−6.5
−6.0
−5.5
−5.0
−4.5
log(1/N)
p. 8
1D Validation : Time Convergence
∆t ∈ {31 × 10i ; i = 5, 1}
−3
−4
−5
−6
−7
−8
−9
−10
−11
8
9
10
Number of cells = 2000
11
12
13
14
15
log(dt)
p. 9
2D Validation : Analytical Solution Definition
Analytical solution :
Domain (−1, 1) × (−1, 1)
Dirichlet limit conditions
Continuous absolut permeability
Liquid saturation and gas pressure :
8
(1 − x2 (1 − y 2 )
1 + t/τ
>
>
S
(x,
t)
=
S
−
×
>
l
lim
<
d
1 + t2 /τ 2
„
«
2
2
>
1
(1 − x )(1 − y )
>
>
1−
: Pg (x, t) = P0 +
c
1 + t/τ
Definition of limit conditions, initial conditions and source terms
using this definition of Sl and Pg
p. 10
Final time : T = 31 × 106 s ≃ 1 year (τ = T )
Time step : ∆t = 31 × 103 s
Number of cells : N ∈ {1/20, 1/30, 1/40, 1/50, 1/100, 1/200}
Relative L2 errors for the gas pressure and liquid saturation
−2
−4
−6
−8
−10
−12
−14
−16
−5.5
−5.0
Liquid saturation
Gas pressure
−4.5
−4.0
−3.5
−3.0
−2.5
log(1/N)
p. 11
Couplex Gas 1a : Numerical Simulation Parameters
Mesh of about 19 000 cells (11 000 quadrangles and 8 000 triangles)
Number of time step : 6226
Numerical time simulation : about 8 hours
Computer : Cluster Linux 64bits, 4 Giga of RAM, Processor AMD
250
p. 12
Couplex Gas 1a : Mesh used
p. 13
Couplex Gas 1a : Gas Pressure profiles at x = 103 m (1/2)
x1.E2
SCAL
1.00
0.90
0.80
0.70
0.60
0.50
t = 5 000 years
0.40
t = 1 000 years
0.30
t = 500 years
0.20
t = 250 years
0.10
t = 0 years
0.00
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
x1.E2
ABS
Gas pressure (bar)
p. 14
Couplex Gas 1a : Gas Pressure profiles at x = 103 m (2/2)
x1.E2
SCAL
1.00
0.90
0.80
0.70
0.60
0.50
t = Tfinal
0.40
t = 40 000 years
0.30
t = 20 000 years
0.20
t = 10 000 years
0.10
t = 7 500 years
0.00
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
x1.E2
ABS
Gas pressure (bar)
p. 15
Mesh convergence
Time evolution of t 7→ kuN1 −uN2 kL2 (Ω) /kuN1 kL2 (Ω) , with N1 ≃ 23 000
cells, N2 ≃ 19 000 cells, u = Pg and u = Pl .
x1.E−2 L2 differencer
6.00
5.00
4.00
3.00
2.00
1.00
Pl L2 difference
Pg L2 difference
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
x1.E4
Time
p. 16
Residual saturation influence
Time evolution of t 7→ kuSres1 − usres2 kL2 (Ω) /kuSres1 kL2 (Ω) , with
Sres1
=
10−4 , Sres2
10−3 , u
=
=
Pg and u
=
Pl .
x1.E−2 L2 difference
4.50
4.00
3.50
3.00
2.50
2.00
1.50
1.00
Pl L2 difference
0.50
Pg L2 difference
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
x1.E4
Time
p. 17
Initial Residual saturation influence location
Relative L2 difference map between two numerical solutions computed with two initial residual saturation (respectively 10−3 and 10−4 ) at time t
=
1 000 years.
PG
> 2.36E−05
< 5.11E−02
4.23E−04
2.82E−03
5.21E−03
7.61E−03
1.00E−02
1.24E−02
1.48E−02
1.72E−02
1.96E−02
2.20E−02
2.44E−02
2.68E−02
2.92E−02
3.16E−02
3.40E−02
3.64E−02
3.88E−02
4.12E−02
4.36E−02
4.59E−02
4.83E−02
5.07E−02
Relative L2 error : t = 1000 years
p. 18
Two Phase Flow Model with Hydrogen Dissolution
Hypothesis :
1. Mass conservation for each phase
2. Mass conservation for dissolve and gas hydrogen
3. Darcy’s flow
4. Capillary pressure : Pc = Pg − Pl
5. Isothermal flow
6. Gas phase : compressible (perfect gas)
7. Liquid phase : incompressible
8. Porosity variations negligible
9. Van Genuchten model for closure laws
10. Equilibrium between liquid water and vapor with PgH2 O = P sat (T )
11. Equilibrium for dissolve hydrogen concentration with Henry’s law
12. Dalton’s law for partial pressure of each species
p. 19
Mathematical Model
M (Sl , PgH2 )∂t
"
Sl
PgH2
#
+div
A(Sl , PgH2 )
"
∇Sl
∇PgH2
#!
=
where M ∈ R2×2 , A ∈ R4×4 and B ∈ R2 :
Φ(ρl − ρg )
´
`
1
ΦMH2 HH2 − RT P
M=
A1,1
kr
= kρl l Pc′ Id,
µl
„
A2,2 = −k HH2
B1 = div
k
X
α
A1,2 = −k
α
krg
krl
1
+
×
µl
RT
µg
kr
ρ2α α g∇z
µα
X
!
,
«
MH2
Φ(1 − S) RT
´
`
1−S
ΦMH2 SHH2 + RT
ρα
krα
Id,
µα
A2,1
!
P
Qα
P α H2 +B
α Qα
!
,
krl ′
Pc P Id,
= kMH2 HH2
µl
MH2 P Id− DlH2 HH2 + DgH2
XgH2
1−
RT
!
MH2 Id.
„ „
«
«
krl
1 krg
B2 = div k HH2
ρg MH2 P g∇z .
ρl +
µl
RT µg
p. 20
Numerical Scheme
MPFA Scheme dor the spatial discretization
M (Sl , PgH2 )∂t
Sl
PgH2
!
− A(Sl , PgH2 )
Sl
PgH2
!
=Q
where M ∈ R2N ×2N and A ∈ R2N ×2N : spatial discretization matrix (N = number of cells of the mesh)
Implicit fully coupled Euler scheme for the time discretization
Fixed point method for the non linear system resolution of unknowns
` H ´n+1
n+1
and Pg 2
Sl
p. 21
Couplex Gas 1b : Gas Pressure profiles at x = 103 m (1/2)
x1.E2
SCAL
1.00
0.90
0.80
0.70
0.60
0.50
t = 5 000 years
0.40
t = 1 000 years
0.30
t = 500 years
0.20
t = 250 years
0.10
t = 0 years
0.00
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
x1.E2
ABS
Gas pressure
p. 22
Couplex Gas 1b : Gas Pressure profiles at x = 103 m (1/2)
x1.E2
SCAL
1.00
0.90
0.80
0.70
0.60
0.50
t = Tfinal
0.40
t = 40 000 years
0.30
t = 20 000 years
0.20
t = 10 000 years
0.10
t = 7 500 years
0.00
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
x1.E2
ABS
Gas pressure
p. 23
Mesh convergence
Time evolution of t 7→ kuN1 −uN2 kL2 (Ω) /kuN1 kL2 (Ω) , with N1 ≃ 23 000
cells, N2 ≃ 19 000 cells, u = Pg and u = Pl .
x1.E−2 L2 differencer
6.00
5.00
4.00
3.00
2.00
1.00
Pl L2 difference
Pg L2 difference
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
x1.E4
Time
p. 24
Initial Residual saturation influence
Time evolution of t 7→ kuSres1 − usres2 kL2 (Ω) /kuSres1 kL2 (Ω) , with
Sres1
=
10−4 , Sres2
10−3 , u
=
=
Pg and u
=
Pl .
x1.E−2 L2 difference
4.50
4.00
3.50
3.00
2.50
2.00
1.50
1.00
Pl L2 difference
0.50
Pg L2 difference
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
x1.E4
Time
p. 25
Residual saturation influence location
Relative L2 difference map between two numerical solutions computed with two initial residual saturation (respectively 10−3 and 10−4 ) at time t
=
1 000 years.
PG
> 1.50E−05
< 4.96E−02
4.02E−04
2.73E−03
5.05E−03
7.37E−03
9.70E−03
1.20E−02
1.43E−02
1.67E−02
1.90E−02
2.13E−02
2.36E−02
2.60E−02
2.83E−02
3.06E−02
3.29E−02
3.53E−02
3.76E−02
3.99E−02
4.22E−02
4.46E−02
4.69E−02
4.92E−02
Relative L2 error : t = 1000 years
p. 26
Some Questions
What is the validity domain for this two-phase flow model without
hydrogen dissolution (Couplex gas 1.a) ?
What’s about the validity of closure laws (relative permeabilities,
capillary pressure) when the saturation is closed to one ?
Existence of an entry pressure for the capillary pressure ?
How treat properly the transition from the two-phase flow model to
the one-phase flow model (residual saturation influence) ?
p. 27
What’s Next ?
Model validation with experimental data
Numerical treatment of the model degeneration when the saturation
goes to one
Developpment of a parallel 3D code to treat 3D test case with
meshes over million cells
Improvement of numerical scheme to treat high heterogneities
p. 28

Couplex-Gas Benchmark

Transcription

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