the low-dimensional model of the density-driven flow in...
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The low-dimensional modelof the density-driven flowin fractured porous media:
Verification
Dr. Dmitry Logashenko, Dr. Alfio Grillo, S. Stichel, Dr. MichaelHeisig
Steinbeis Research Center 936, Olbronn (Dr. M. Heisig)and Goethe Center for Scientific Computing, University of Frankfurt a.M.
(Prof. Dr. G. Wittum)
Lepzig, April 2010
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 1 / 17
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1 Density driven flow
2 Flow and transport in the porous medium with fractures
3 Discretization
4 Verification
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 2 / 17
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Density driven flow
Variables: ω (mass fraction of the salt),p (hydrodynamic pressure).
System of the PDEs:
∂t(φρω) +∇ · (ρωq− ρD∇ω) = 0∂t(φρ) +∇ · (ρq) = 0.
}q = −K
µ(∇p − ρg),
where ρ = ρ(ω).
+ boundary conditions
+ initial conditions
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 3 / 17
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Fractures
The coefficient of the PDEs depend on the geometrical point. Forexample: The coefficient depend on the subdomain.
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 4 / 17
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Fractures
The coefficient of the PDEs depend on the geometrical point. Forexample: The coefficient depend on the subdomain.
Fractures: Very thin subdomains (thin layers or clefts filled withporous medium).
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 4 / 17
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Fractures
The coefficient of the PDEs depend on the geometrical point. Forexample: The coefficient depend on the subdomain.
Fractures (approximation): Hypersurfaces + PDEs with thereduced dimensionality.
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 4 / 17
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Fractures
The coefficient of the PDEs depend on the geometrical point. Forexample: The coefficient depend on the subdomain.
Fractures (approximation): Hypersurfaces + PDEs with thereduced dimensionality.
full-dim. profiles 7→ averaged quantities
changes in the geometry 7→ changes in the mass
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 4 / 17
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Fractures
The coefficient of the PDEs depend on the geometrical point. Forexample: The coefficient depend on the subdomain.
Fractures (approximation): Hypersurfaces + PDEs with thereduced dimensionality.
full-dim. profiles 7→ averaged quantities
changes in the geometry 7→ changes in the mass
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 4 / 17
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Fractures as low-dimensional objects
Bulk medium
Fracture
Bulk medium
c
Microscopically: Variables are continuous functions.
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 5 / 17
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Fractures as low-dimensional objects
Bulk medium
Fracture
Bulk medium
c
cf
Macroscopically: Averaged values in the fracture
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 5 / 17
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Fractures as low-dimensional objects
Bulk medium
Fracture
Bulk medium
c
cb
ca
cf
Macroscopically: Discontinuities at the fractures
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 5 / 17
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Fractures as low-dimensional objects
Bulk medium
Fracture
Bulk medium
c
cb
ca
cfε
Macroscopically: Ready to “shrink”
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 5 / 17
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Fractures as low-dimensional objects
Bulk medium
Bulk medium
c
cb
ca
ε cf
Representation: The fracture is a hypersurface.The physical thickness ε 6= 0 of the fracture
is an additional coefficient of the PDEs.
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 5 / 17
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Density driven flow in a fracture (“original model”)
Variables: ω (mass fraction of the salt),p (hydrodynamic pressure).
System of the PDEs:
∂t(φρω) +∇ · (ρωq− ρD∇ω) = 0∂t(φρ) +∇ · (ρq) = 0.
}q = −K
µ(∇p − ρg),
where ρ = ρ(ω) =ρwaterρbrine
ρbrine − (ρbrine − ρwater)ω.
+ boundary conditions
+ initial conditions
Averaging: Products!
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 6 / 17
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Density driven flow in a fracture (“original model”)
Variables: ω (mass fraction of the salt),p (hydrodynamic pressure).
System of the PDEs:
∂t(φρω) +∇ · (ρωq− ρD∇ω) = 0∂t(φρ) +∇ · (ρq) = 0.
}q = −K
µ(∇p − ρg),
where ρ = ρ(ω) =ρwaterρbrine
ρbrine − (ρbrine − ρwater)ω.
+ boundary conditions
+ initial conditions
Averaging: Products!
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 6 / 17
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New unknown functions (“new model”)
Instead of ω, use concentration c :
ρ(c) = ρwater +ρbrine − ρwater
ρbrineMsaltc =: α + βc .
New formulation of the model:
∂t(φc) +∇ · (cq− D∇c) = 0
∇ · (αq + βD∇c) = 0.
}q = −K
µ(∇p − ρg),
where D(c) := Dα
α + βc.
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 7 / 17
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PDEs in the fracture
In the fracture Σ:
(φΣcf )t +∇Σ · (cf qΣ − Df ,Σ∇Σcf ) + 1ε (cq⊥ − D⊥δc)
∣∣∣ab
= 0
∇Σ · (αqΣ + βDf ,Σ∇Σcf ) + 1ε (αq⊥ + βD⊥δc)
∣∣∣ab
= 0
where
(δc)|a :=ca − cfε/2
, (δp)|a :=pa − pfε/2
qΣ = −KΣ
µf(∇Σpf − ρf gΣ)
q⊥ = −K⊥µf
(δp − (ρ− ρf )g⊥)
Df ,Σ :=⟨
D⟩≈ Df (1− β
αc).
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 8 / 17
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PDEs in the fracture
In the fracture Σ:
(φΣcf )t +∇Σ · (cf qΣ − Df ,Σ∇Σcf ) + 1ε (cq⊥ − D⊥δc)
∣∣∣ab
= 0
∇Σ · (αqΣ + βDf ,Σ∇Σcf ) + 1ε (αq⊥ + βD⊥δc)
∣∣∣ab
= 0
where
(δc)|a :=ca − cfε/2
, (δp)|a :=pa − pfε/2
qΣ = −KΣ
µf(∇Σpf − ρf gΣ)
q⊥ = −K⊥µf
(δp − (ρ− ρf )g⊥)
Df ,Σ :=⟨
D⟩≈ Df (1− β
αc).
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 8 / 17
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Boundary conditions at the fracture
At the sides (a) and (b) of the fracture Σ:
(q− D∇c) · n = cq⊥ − D⊥δc
(αq + βD∇c) · n = αq⊥ + βD⊥δc.
n: the unit normal to Σ
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 9 / 17
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Gravitation in the fracture
q⊥ = −K⊥µf
(δp − (ρ− ρf )g⊥)
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 10 / 17
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Gravitation in the fracture
q⊥ = −K⊥µf
(δp − ρg⊥)
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 10 / 17
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Discretization: Degrees of Freedom
B1
B2 B3
B4
B5 B6
B7
x1
x2
x3=x4x5=x6=x7
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 11 / 17
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Discretization: Degrees of Freedom
B1
B2 B3
B4
B5 B6
B7
x1
x2
x3=x4x5=x6=x7
ch,i ≈ limBi3x→xi
c(x), ph,i ≈ limBi3x→xi
p(x)
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 11 / 17
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Discretization: Degrees of Freedom
B1
B2 B3
B4
B5 B6
B7
x1
x2=x101
x3=x4=x102x5=x6=x7=x103
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 11 / 17
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Discretization: Numerics
In the bulk-medium DOFs: Full-dimensional vertex-centeredFE-FV-method.
For the fracture DOFs: The vertex-centered FE-FV-method ind − 1 dimensions.
Interactions between the fractures and the bulk medium:Finite differences as formulated in the model.
Solvers for the discretized equations: The Newton-methodwith the BiCGStab-iteration. Preconditioner: The geometricalMG-method with the ILU smoothers.
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 12 / 17
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Discretization: Numerics
In the bulk-medium DOFs: Full-dimensional vertex-centeredFE-FV-method.
For the fracture DOFs: The vertex-centered FE-FV-method ind − 1 dimensions.
Interactions between the fractures and the bulk medium:Finite differences as formulated in the model.
Solvers for the discretized equations: The Newton-methodwith the BiCGStab-iteration. Preconditioner: The geometricalMG-method with the ILU smoothers.
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 12 / 17
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Discretization: Numerics
In the bulk-medium DOFs: Full-dimensional vertex-centeredFE-FV-method.
For the fracture DOFs: The vertex-centered FE-FV-method ind − 1 dimensions.
Interactions between the fractures and the bulk medium:Finite differences as formulated in the model.
Solvers for the discretized equations: The Newton-methodwith the BiCGStab-iteration. Preconditioner: The geometricalMG-method with the ILU smoothers.
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 12 / 17
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Discretization: Numerics
In the bulk-medium DOFs: Full-dimensional vertex-centeredFE-FV-method.
For the fracture DOFs: The vertex-centered FE-FV-method ind − 1 dimensions.
Interactions between the fractures and the bulk medium:Finite differences as formulated in the model.
Solvers for the discretized equations: The Newton-methodwith the BiCGStab-iteration. Preconditioner: The geometricalMG-method with the ILU smoothers.
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 12 / 17
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Henry Problem with a fracture: Solution
(Low-dimensional representation of the fracture)
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 13 / 17
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Henry Problem with a fracture: Solution
(Full-dimensional representation of the fracture)
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 13 / 17
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Error in the fracture
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1000 2000 3000 4000 5000 6000 0
0.0005
0.001
0.0015
0.002
time t [min]
Full-dimensionalLow-dimensional
Abs. error
non-
dimen
siona
l con
cent
ratio
n c f /!
pBabsolute error
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 14 / 17
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Error in the fracture
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0 1000 2000 3000 4000 5000 6000
abso
lute
erro
r
time t [min]
GL-low: 5, GL-full: 7GL-low: 5, GL-full: 6GL-low: 5, GL-full: 5GL-low: 6, GL-full: 7GL-low: 6, GL-full: 6GL-low: 6, GL-full: 5GL-low: 7, GL-full: 7GL-low: 7, GL-full: 6GL-low: 7, GL-full: 5
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 14 / 17
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Error of the jump
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0 1000 2000 3000 4000 5000 6000 0
1e-05
2e-05
3e-05
4e-05
5e-05
time t [min]
Full-dimensionalLow-dimensional
Abs. error
jump (!
(2) - !
(1) ) absolute error
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 15 / 17
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Henry Problem with a fracture: Very thick fracture
(Low-dimensional representation of the fracture)
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 16 / 17
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Henry Problem with a fracture: Very thick fracture
(Full-dimensional representation of the fracture)
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 16 / 17
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Summary:
The model for the density-driven flow in fractured porousmedium with the low-dimensional reprentation of the fractureswas presented.
Verification by the comparison with the full-dimensionalcomputations.
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 17 / 17
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Summary:
The model for the density-driven flow in fractured porousmedium with the low-dimensional reprentation of the fractureswas presented.
Verification by the comparison with the full-dimensionalcomputations.
We thank
Dr. M. Lampe (G-CSC)
S. Reiter (G-CSC)
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 17 / 17
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Summary:
The model for the density-driven flow in fractured porousmedium with the low-dimensional reprentation of the fractureswas presented.
Verification by the comparison with the full-dimensionalcomputations.
Thank you for your attention!
Logashenko, Grillo, Stichel, Heisig Fractures with d3f 17 / 17