flow and trasnsport of pollutants in the subsurface : coupled ...logoinria flow and trasnsport of...
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Flow and trasnsport of pollutants in the subsurface :coupled models and numerical methods
Michel [email protected]
Institut National de Recherche en Informatique et Automatique
IBM Recherche et Innovation Campus DayParis
September 18, 2008
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 1 / 34
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Outline
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Coupled modelsDensisty driven flowReactive transport
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 2 / 34
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Plan
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Coupled modelsDensisty driven flowReactive transport
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 3 / 34
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Contaminant transport
Underground water22% of all natural water resources
51% of all drinking water
37% of agricultural water
Possible contamination ofgroundwater by industrialwaste
Microbial remediation
Variant : saltwaterintrusion : coupling to flow
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 4 / 34
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Nuclear waste storage
Assess safety of deep geological nuclear waste storage (clay layer)
Long term simulation of radionuclide transport
Wide variation of scales : from package (meter) to regional (kilometers)
Geochemistry: large number of species
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 5 / 34
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CO2 sequestration
Sleipner project, Norway
Long term capture of CO2
in saline aquifer
Simulation to understandCO2 migration throughsalt
Coupling of liquid and gasphase, reactive transport
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 6 / 34
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Plan
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Coupled modelsDensisty driven flowReactive transport
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 7 / 34
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Plan
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Coupled modelsDensisty driven flowReactive transport
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 8 / 34
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Flow : Darcy’s law
Henry Philibert Gaspard Darcy, (1803-1858) French engineer
Darcy’s law
Q = AK∆hL
Q flow (m3/s)
K Hydraulic conductivity (m/s)
h Piezometric head (m)(h = p/ρg + z)
Modern, differential version q =−K ∇h, q Darcy velocity
Flow equations
∇ ·q = 0 incompressibility
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 9 / 34
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Flow : Darcy’s law
Henry Philibert Gaspard Darcy, (1803-1858) French engineer
Darcy’s law
Q = AK∆hL
Q flow (m3/s)
K Hydraulic conductivity (m/s)
h Piezometric head (m)(h = p/ρg + z)
Modern, differential version q =−K ∇h, q Darcy velocity
Flow equations
∇ ·q = 0 incompressibility
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 9 / 34
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Mixed finite elements
Approximate both head and Darcy velocity
Locally mass conservative
Flux is continuous across element faces
Allows full diffusion tensor
Raviart-Thomas space
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 10 / 34
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Domain decomposition
Flow around nuclear wastestorage area
Computed by domaindecomposition (Robin–Robin)
Subdomain code in C++ (LifeV),interface solver in Ocaml
Parallelism in OcamlP3l (skeletonbased)
F. Clément, V. Martin (thesis), P.Weis (INRIA, Estime)
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 11 / 34
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Plan
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Coupled modelsDensisty driven flowReactive transport
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 12 / 34
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Physics of advection–dispersion
Convection Transport by velocity field
Diffusion motion due to concentration gradient
Dispersion due microscopic velocity heterogeneity
Reaction between species, interaction with host matrix
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 13 / 34
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Transport model
Convection–diffusion equation
ω∂c∂ t−div(Dgradc)
dispersion+ div(uc)
advection+ ωλc = f
c: concentration [mol/l]
omega: porosity (–)
λ radioactive decay [s−1]
u Darcy velocity [m/s]
Dispersion tensor
D = deI + |u|[α lE(u) + α t(I−E(u))], Eij(u) =uiuj
|u|
α l ,α t dispersicity coeff. [m], de molecular diffusion [m/s2]
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 14 / 34
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Solution by operator splitting
Advection stepExplicit, finite volumes / discontinuous Galerkine
Locally mass conservative
Keeps sharp fronts
Small numerical diffusion
Allows unstructured meshes
CFL condition: usesub–time–steps
Dispersion stepLike flow equation (time dependant): mixed finite elements (implicit)
Order 1 method
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 15 / 34
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Example: transport around an obstacle
MoMaS benchmark for reactive transport. Here transport only
Head and velocity
Concentration at t = 25
J. B. Apoung, P. Havé, J. Houot, MK, A. Semin
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 16 / 34
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Transport around a nuclear waste storage site
GdR MoMaS benchmark, Andra model
Concentration at 130 000 years Concentration at 460 000 years
A. Sboui, E. Marchand (INRIA, Estime)
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 17 / 34
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Plan
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Coupled modelsDensisty driven flowReactive transport
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 18 / 34
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Classification of chemical reactions
According to nature of reactionHomogeneous In the same phase (aqueous, gaseous, ...)
Examples: Acid base, oxydo–reduction
Heterogeneous Involve different phasesExamples: Sorption, precipitation / dissolution, ...
According to speed of reaction
Slow reactions Irreversible, modeled using kinetic law
Fast reactions Reversible, modeled using equilibrium
Depends on relative speed of reactions and transport.
In this talk: Equilibrium reactions, with sorption.
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 19 / 34
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Classification of chemical reactions
According to nature of reactionHomogeneous In the same phase (aqueous, gaseous, ...)
Examples: Acid base, oxydo–reduction
Heterogeneous Involve different phasesExamples: Sorption, precipitation / dissolution, ...
According to speed of reaction
Slow reactions Irreversible, modeled using kinetic law
Fast reactions Reversible, modeled using equilibrium
Depends on relative speed of reactions and transport.
In this talk: Equilibrium reactions, with sorption.
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 19 / 34
![Page 22: Flow and trasnsport of pollutants in the subsurface : coupled ...logoINRIA Flow and trasnsport of pollutants in the subsurface : coupled models and numerical methods Michel Kern Michel.Kern@inria.fr](https://reader034.vdocuments.site/reader034/viewer/2022050609/5faff10e8746a5205106d421/html5/thumbnails/22.jpg)
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Classification of chemical reactions
According to nature of reactionHomogeneous In the same phase (aqueous, gaseous, ...)
Examples: Acid base, oxydo–reduction
Heterogeneous Involve different phasesExamples: Sorption, precipitation / dissolution, ...
According to speed of reactionSlow reactions Irreversible, modeled using kinetic law
Fast reactions Reversible, modeled using equilibrium
Depends on relative speed of reactions and transport.
In this talk: Equilibrium reactions, with sorption.
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 19 / 34
![Page 23: Flow and trasnsport of pollutants in the subsurface : coupled ...logoINRIA Flow and trasnsport of pollutants in the subsurface : coupled models and numerical methods Michel Kern Michel.Kern@inria.fr](https://reader034.vdocuments.site/reader034/viewer/2022050609/5faff10e8746a5205106d421/html5/thumbnails/23.jpg)
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Classification of chemical reactions
According to nature of reactionHomogeneous In the same phase (aqueous, gaseous, ...)
Examples: Acid base, oxydo–reduction
Heterogeneous Involve different phasesExamples: Sorption, precipitation / dissolution, ...
According to speed of reactionSlow reactions Irreversible, modeled using kinetic law
Fast reactions Reversible, modeled using equilibrium
Depends on relative speed of reactions and transport.
In this talk: Equilibrium reactions, with sorption.
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 19 / 34
![Page 24: Flow and trasnsport of pollutants in the subsurface : coupled ...logoINRIA Flow and trasnsport of pollutants in the subsurface : coupled models and numerical methods Michel Kern Michel.Kern@inria.fr](https://reader034.vdocuments.site/reader034/viewer/2022050609/5faff10e8746a5205106d421/html5/thumbnails/24.jpg)
logoINRIA
Classification of chemical reactions
According to nature of reactionHomogeneous In the same phase (aqueous, gaseous, ...)
Examples: Acid base, oxydo–reduction
Heterogeneous Involve different phasesExamples: Sorption, precipitation / dissolution, ...
According to speed of reactionSlow reactions Irreversible, modeled using kinetic law
Fast reactions Reversible, modeled using equilibrium
Depends on relative speed of reactions and transport.
In this talk: Equilibrium reactions, with sorption.
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 19 / 34
![Page 25: Flow and trasnsport of pollutants in the subsurface : coupled ...logoINRIA Flow and trasnsport of pollutants in the subsurface : coupled models and numerical methods Michel Kern Michel.Kern@inria.fr](https://reader034.vdocuments.site/reader034/viewer/2022050609/5faff10e8746a5205106d421/html5/thumbnails/25.jpg)
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Classification of chemical reactions
According to nature of reactionHomogeneous In the same phase (aqueous, gaseous, ...)
Examples: Acid base, oxydo–reduction
Heterogeneous Involve different phasesExamples: Sorption, precipitation / dissolution, ...
According to speed of reactionSlow reactions Irreversible, modeled using kinetic law
Fast reactions Reversible, modeled using equilibrium
Depends on relative speed of reactions and transport.
In this talk: Equilibrium reactions, with sorption.
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 19 / 34
![Page 26: Flow and trasnsport of pollutants in the subsurface : coupled ...logoINRIA Flow and trasnsport of pollutants in the subsurface : coupled models and numerical methods Michel Kern Michel.Kern@inria.fr](https://reader034.vdocuments.site/reader034/viewer/2022050609/5faff10e8746a5205106d421/html5/thumbnails/26.jpg)
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Sorption processes
DefinitionSorption is the accumulation of a fluid on a solid at the fluid–solid interface.
Main mechanism for exchanges between dissolved species and solid surfaces.
Several possible mechanisms
Surface complexation Formation of bond between surface and aqueousspecies, due to electrostatic interactions. Depends on surfacepotential.
Ion exchange Ions are exchanged between sorption sites on the surface.Depends on Cationic Exchange Capacity.
Can be modeled as mass action law
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 20 / 34
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Sorption processes
DefinitionSorption is the accumulation of a fluid on a solid at the fluid–solid interface.
Main mechanism for exchanges between dissolved species and solid surfaces.
Several possible mechanismsSurface complexation Formation of bond between surface and aqueous
species, due to electrostatic interactions. Depends on surfacepotential.
Ion exchange Ions are exchanged between sorption sites on the surface.Depends on Cationic Exchange Capacity.
Can be modeled as mass action law
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 20 / 34
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Sorption processes
DefinitionSorption is the accumulation of a fluid on a solid at the fluid–solid interface.
Main mechanism for exchanges between dissolved species and solid surfaces.
Several possible mechanismsSurface complexation Formation of bond between surface and aqueous
species, due to electrostatic interactions. Depends on surfacepotential.
Ion exchange Ions are exchanged between sorption sites on the surface.Depends on Cationic Exchange Capacity.
Can be modeled as mass action law
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 20 / 34
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Sorption processes
DefinitionSorption is the accumulation of a fluid on a solid at the fluid–solid interface.
Main mechanism for exchanges between dissolved species and solid surfaces.
Several possible mechanismsSurface complexation Formation of bond between surface and aqueous
species, due to electrostatic interactions. Depends on surfacepotential.
Ion exchange Ions are exchanged between sorption sites on the surface.Depends on Cationic Exchange Capacity.
Can be modeled as mass action law
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 20 / 34
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The chemical problem
System of non-linear equations
c + ST x + AT y = T ,
c̄ + BT x̄ = W ,
}Mass conservation
logx = S logc + logKx ,
log x̄ = A logc + B log c̄ + logKy .
}Mass action law
Dissolved total: C = c + ST x , Fixed total: F = AT x̄ .
Role of chemical modelGiven totals T (and W , known), split into mobile and immobile totalconcentrations.
C = Φ(T ), F = Ψ(T )
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 21 / 34
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Numerical solution of chemical problem
Take concentration logarithms as main unknownsUse globalized Newton’s method (line search, trust region).
Ion exchange: 6 species, 4 components (vary initial guess)
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 22 / 34
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Plan
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Coupled modelsDensisty driven flowReactive transport
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 23 / 34
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Plan
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Coupled modelsDensisty driven flowReactive transport
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 24 / 34
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Motivations
Seawater intrusion can threaten drinking water resevoir
Synthetic model (Elder): fingering instabilityTHE ELDER BENCHMARK (Elder, 1966, 1967)
ρ=1200 3kg m/
0.0 100.0 200.0 300.0 400.0 500.0 600.0
0.0
50.0
100.0
150.0
P = 0.0 P = 0.0
k = 4.8 10-13
m2
ρ=1000 3kg m/
100.00 200.00 300.00 400.00 500.00
50.00
100.00
100.0 200.0 300.0 400.0 500.0
50.0
100.0
Case 1(720 s) Case 2 (464 s) Case 3 (393 s)
Simulated concentration at t = 20 years for the Elder
benchmark
22
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Physical model
Flow
Mass conservation for fluid∂ (ρω)
∂ t+ ∇.
(ερ~V
)= ρQS,
Generalized Darcy’s law ε~V =− 1µ
K (∇P + ρg~nz) , ,
Equation of state ρ = ρ0 +∂ρ
∂CmCm, ρ0 = 1000,
∂ρ
∂Cm= 200.
TransportSalt mass conservation
ερ∂Cm
∂ t+ ερ~V .∇Cm = ∇.
(ερD(~V )∇Cm
),
Dispersion tensor D(~V ) = DmI + (αL−αT )~V ⊗~V∣∣~V ∣∣ + αT
∣∣~V ∣∣ IM. Kern (INRIA) Subsurface flow and transport IBM Campus Day 26 / 34
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Distributed implementation
Coupling between flow and transport :
Flowcode
Transport code
Controller
velocity
scalarsscalars
concentration
Use Corba for coupling componentsJoint work with J. Erhel, Ph. Ackerer, Ch. Perez, M. Mancip
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 27 / 34
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Elder model
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 28 / 34
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Plan
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Coupled modelsDensisty driven flowReactive transport
M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 29 / 34
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The coupled system
Transport for each species (same dispersion tensor for all species)
∂x i
∂ t+ L(x i) = r x
i ,∂c j
∂ t+ L(c j) = r c
j ,
∂y i
∂ t= r y
i ,∂sj
∂ t= r s
j ,
Eliminate (unknown) reaction rates by using conservation laws (T = C + F )
∂T ic
∂ t+ L(C ic) = 0, ic = 1, . . . ,Nc
T icix = C ic
ix + F icix ic = 1, ..,Nc and ix = 1, ..,Nx
F ix = Ψ(T ix ) ix = 1, . . . ,Nx .
Number of transport equations reduced from Nx + Ny to Nc + Ns
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The coupled system
Transport for each species (same dispersion tensor for all species)
∂x i
∂ t+ L(x i) = r x
i ,∂c j
∂ t+ L(c j) = r c
j ,
∂y i
∂ t= r y
i ,∂sj
∂ t= r s
j ,
Eliminate (unknown) reaction rates by using conservation laws (T = C + F )
∂T ic
∂ t+ L(C ic) = 0, ic = 1, . . . ,Nc
T icix = C ic
ix + F icix ic = 1, ..,Nc and ix = 1, ..,Nx
F ix = Ψ(T ix ) ix = 1, . . . ,Nx .
Number of transport equations reduced from Nx + Ny to Nc + Ns
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Coupling formulations and algorithms(1)
CC formulation, explicit chemistry
dCdt
+dFdt
+ LC = 0
H(z)−(
C + FW
)= 0
F −F(z) = 0.
+ Explicit Jacobian
+ Chemistry function, nochemical solve
− Intrusive approach (chemistrynot a black box)
− Precipitation not easy to include
Coupled system is index 1 DAE
Mdydt
+ f (y) = 0
Use standard DAE softwareJ. Erhel, C. de Dieuleveult (Andra thesis)
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Coupling formulations and algorithms(2)
TC formulation, implicit chemistrydTdt
+ LC = 0
T −C−F = 0
F −Ψ(T ) = 0
+ Non-intrusive approach(chemistry as black box)
+ Precipitation can (probably) beincluded
− One chemical solve for eachfunction evaluation
Cn+1−Cn
∆t+
F n+1−F n
∆t+ L(Cn+1) = 0
T n+1 = Cn+1 + F n+1
F n+1 = Ψ(T n+1)
Solve by Newton’s method
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Solution by Newton–Krylov
Structure of Jacobian matrix
f ′(C,T ,F) =
(I + ∆tL) 0 I−I I −I0 −Ψ′(T ) I
Solve the linear system by an iterativemethod (GMRES)
Require only jacobian matrix by vectorproducts.
Inexact NewtonApproximation of the Newton’s direction ‖f ′(xk )d + f (xk )‖ ≤ η‖f (xk )‖Choice of the forcing term η?
Keep quadratic convergence (locally)Avoid oversolving the linear system
η = γ‖f (xk )‖2/‖f (xk−1)‖2 (Kelley, Eisenstat and Walker)
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Solution by Newton–Krylov
Structure of Jacobian matrix
f ′(C,T ,F) =
(I + ∆tL) 0 I−I I −I0 −Ψ′(T ) I
Solve the linear system by an iterativemethod (GMRES)
Require only jacobian matrix by vectorproducts.
Inexact NewtonApproximation of the Newton’s direction ‖f ′(xk )d + f (xk )‖ ≤ η‖f (xk )‖Choice of the forcing term η?
Keep quadratic convergence (locally)Avoid oversolving the linear system
η = γ‖f (xk )‖2/‖f (xk−1)‖2 (Kelley, Eisenstat and Walker)M. Kern (INRIA) Subsurface flow and transport IBM Campus Day 33 / 34
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MoMaS reactive transport benchmark
Numerically difficult tets case, 12 chemical species (J. Carrayrou)Concentration of species 2 at t = 50, t = 1000,t = 2000, t = 5010.
C. de Dieuleveult (Andra Thesis, INRIA, Sage)
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