flow, transport and chemistry in porous media : numerical … · 2017. 11. 1. · flow, transport...
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Flow, transport and chemistry in porous media : numericalmethods for coupled problems
Michel [email protected]
with L. Amir, J. Jaffre, A. Sboui, A. Taakili
Institut National de Recherche en Informatique et Automatique
EPFLMay 7, 2008
Partially funded by Cnrs GDR MOMAS
M. Kern (INRIA) Subsurface flow and transport 1 / 40
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1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Reactive transportSingle species with sorption (joint work with A. Taakili)Multi-species equilibrium chemistry
M. Kern (INRIA) Subsurface flow and transport 2 / 40
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Outline
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Reactive transportSingle species with sorption (joint work with A. Taakili)Multi-species equilibrium chemistry
M. Kern (INRIA) Subsurface flow and transport 3 / 40
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Nuclear waste storage (1)
Assess safety of deep geological nuclear waste storage (clay layer)
Long term simulation of radionuclide transport (one million years)
Wide variation of scales : from package (meter) to regional (kilometers)
Geochemistry: large number of species
Strong government regulation
Main actors : , ,
Research in mathematical and numerical modeling is conducted in the
CNRS MOMAS group (Director A. Ern).
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Nuclear waste storage (2)
Present choice in France: a sedimentary geological formation (in the Meuseregion)
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A 3D far field model
Used as a benchmark, similar to Andra safety model
Blown-up 30 times vertically
DifficultiesDistorted geometry (horizontal ≈40 km, vertical 700 m)
Strong heterogenities(permeability varies by 8 orders ofmagnitude)
General hexahedral mesh
Simulation over 500 000 years
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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
SHPCO project (funded byANR) High PerformanceSimulation of CO2sequestration
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Outline
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Reactive transportSingle species with sorption (joint work with A. Taakili)Multi-species equilibrium chemistry
M. Kern (INRIA) Subsurface flow and transport 8 / 40
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Outline
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Reactive transportSingle species with sorption (joint work with A. Taakili)Multi-species equilibrium chemistry
M. Kern (INRIA) Subsurface flow and transport 9 / 40
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Darcy’s law and mixed finite element
Flow equations
q =−K ∇h Darcy’s law h piezometric head
∇ ·q = 0 incompressibility q Darcy velocity
Mixed finite elements
Approximate both head and velocity
Continuous flux across element faces
Locally mass conservative
Allows full diffusion tensor
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A composite mixed finite element for hexahedra (1)
The problemStandard convergence theory not valid for RTN space over general (deformed)hexahedra, pressure space does not containt constant functions (T. Russell)
Kuznetsov, Repin (2003): construct macroelement on a hexahedron bysubdividing it into 5 tetrahedra
2
z
B
E
D
F
C
GH
x
y
A
35
6
4
1
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A composite mixed finite element for hexahedra (2)
FeaturesSame DOFs as before (average pressure in each element, flux acrosseach face)
Standadrd error estimates shown to apply: optimal order error (underregularity assumption).
Construction
Wh =
vh ∈ H(div,Ω); vh|Ti∈ RTN0(Ti), i = 1, · · · ,5,
divvh const. on H, vh ·n constant on faces of H .
vhH is uniquely defined by its normal components across the 6 faces, so Wh
contains constants.
A. Sboui’s PhD thesis, Sboui, Jaffr , Roberts to appear in SIAM J. SCi. Comp.
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A composite mixed finite element for hexahedra (3)
Simulation for 3D far field benchmark model, horizontal cros section ofmodulus of velocity
RTN FE New FE
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Outline
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Reactive transportSingle species with sorption (joint work with A. Taakili)Multi-species equilibrium chemistry
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Transport model
Convection–diffusion equation
ω∂c∂ t−div(Dgradc)
dispersion+ div(uc)
advection+ ωλc = f
c: concentration [mol/l]
ω : 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]
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Solution by operator splitting
Advection stepExplicit, finite volumes / discontinuous Galerkin
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)
First order method
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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. Hav, J. Houot, MK, A. Semin, O. Saouli
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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)
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Outline
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Reactive transportSingle species with sorption (joint work with A. Taakili)Multi-species equilibrium chemistry
M. Kern (INRIA) Subsurface flow and transport 19 / 40
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Chemical phenomena
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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.
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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 21 / 40
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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.
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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
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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
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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
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Modeling general equilibrium models
General chemical reactions : Ns species, Nr reactions
Ns
∑j=1
νijY j 0, i = 1, . . . ,Nr
νij stoichiometric coefficients. Matrix equation νY = 0
Assumptionν has full rank : Rankν = Nr .
Basis for null-space of ν hasdimensions Nc = Ns−Nr .
Partition ν =(G N
), B ∈ RNr×Nr invertible, N ∈ RNc×Nr . Let H =−G−1N
General solution of νY = 0: Y =
(xc
),x = Hc. c ∈ RNc , x ∈ RNr .
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Modeling general equilibrium models
General chemical reactions : Ns species, Nr reactions
Ns
∑j=1
νijY j 0, i = 1, . . . ,Nr
νij stoichiometric coefficients. Matrix equation νY = 0
Assumptionν has full rank : Rankν = Nr .
Basis for null-space of ν hasdimensions Nc = Ns−Nr .
Partition ν =(G N
), B ∈ RNr×Nr invertible, N ∈ RNc×Nr . Let H =−G−1N
General solution of νY = 0: Y =
(xc
),x = Hc. c ∈ RNc , x ∈ RNr .
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Modeling general equilibrium models
General chemical reactions : Ns species, Nr reactions
Ns
∑j=1
νijY j 0, i = 1, . . . ,Nr
νij stoichiometric coefficients. Matrix equation νY = 0
Assumptionν has full rank : Rankν = Nr .
Basis for null-space of ν hasdimensions Nc = Ns−Nr .
Partition ν =(G N
), B ∈ RNr×Nr invertible, N ∈ RNc×Nr . Let H =−G−1N
General solution of νY = 0: Y =
(xc
),x = Hc. c ∈ RNc , x ∈ RNr .
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The chemical problem
c (resp c) mobile (resp. fixed) primary speciesx (resp x) mobile (resp. fixed) secondary species
System of non-linear equations
c + ST x + AT x = T ,
c + BT x = W ,
Mass conservation
logx = S logc + logK ,
log x = A logc + B log c + log K .
Mass action law
Role of chemical modelGiven T (and W , known), split into mobile C and fixed F concentrations.
C = c + ST x = Φ(T ), F = AT x = Ψ(T )
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Outline
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Reactive transportSingle species with sorption (joint work with A. Taakili)Multi-species equilibrium chemistry
M. Kern (INRIA) Subsurface flow and transport 25 / 40
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Outline
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Reactive transportSingle species with sorption (joint work with A. Taakili)Multi-species equilibrium chemistry
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Sorption models
One species reacts with rock matrix, description by a sorption isotherm :v = Ψ(u).u aqueous concentration, v fixed “concentration”
Common isothermsLinear v = Kd u
Langmuir v =kf σ0u
kf u + kb
Freundlich v = γu1/p (p > 1 possible)
Coupled model
ω∂u∂ t
+ ω∂v∂ t
+ Lu = 0, L adv. diff operator
v = Ψ(u).
Mathematical, numerical analysis: van Duijn, Knabner, Barrett,Kacur, Frolkovic
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Formulations of coupled system
After space and time discretization,
Coupled formulation F(u,v) = 0 with
F
(uv
)=
((M + ∆L)u + Mv + b
v−Ψ(u)
)Eliminate v F1(u) = (M + ∆tL)u + MΨ(u)−bn
Eliminate u F2(u) = v−Ψ((M + ∆L)−1(b−Mv)
)Jacobian for coupled formulation,with D = diag(Ψ′(u1), . . . ,Ψ′(uN))
J =
(M + ∆L M−D I
)J2 = I + D(M + ∆tL)−1M is Schur complement of J
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Solution by Newton–Krylov
Solve the linear system by an iterative method (GMRES)
Requires only jacobian matrix by vector products.
Used for CFD, shallow water, radiative transfer(Keyes, Knoll, JCP 04), and forreactive transport (Hammond et al., Adv. Wat. Res. 05)
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
Solve the linear system by an iterative method (GMRES)
Requires only jacobian matrix by vector products.
Used for CFD, shallow water, radiative transfer(Keyes, Knoll, JCP 04), and forreactive transport (Hammond et al., Adv. Wat. Res. 05)
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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Preconditioning
Essential for good linear performance
Difficult for matrix free formulation
Possible choices
Block diagonal P = diag(M + ∆tL, I),
Block Gauss Seidel “Physics based”, equivalent to (lienar) sequential method
P =
(M + ∆tL 0−D I
)Aproximate block factorization ?
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Numerical results
Geometry of MoMaS reactive transport benchmark (2D), LifeV + Kinsol(Sundials)
σ0 0.125 0.25 0.4 0.45 0.5nb nonlin iter 5 6 6 6 7
nb lin. iter 12 19 27 43 86
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Outline
1 Motivations
2 Basic models and methodsFlow modelTransport modelChemistry
3 Reactive transportSingle species with sorption (joint work with A. Taakili)Multi-species equilibrium chemistry
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The coupled system
Transport for each species (same dispersion tensor for all species)Eliminate (unknown) reaction rates by using conservation laws (T = C + F )
ω∂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 .
Coupling methodsIterative, based on fixed point (Yeh Tripathi ’89, Carrayrou et al. ’04)
Substitution, global (Saaltink ’98, Hammond et al. ’05)
Reduction method (Knabner, Kratle, ’06)
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Coupling formulations and algorithms(1)
CC formulation, explicit chemistryω
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
Kdydt
+ f (y) = 0
Use standard DAE softwareC. de Dieuleveult (Andra thesis), J. Erhel, MK (JCP ’09)
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Coupling formulations and algorithms(2)
TC formulation, implicit chemistryω
dTdt
+ 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 = (M + ∆tL)−1 (Cn + F n−F n+1)T n+1 = Cn+1 + F n+1
F n+1 = Ψ(T n+1)
Fixed point problem, can be solved by block Gauss Seidel or by Newton’smethod
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Solution by Newton–Krylov
Structure of Jacobian matrix
f ′(C,T ,F) =
I 0 (I + ∆tL)−1
−I I −I0 −Ψ′(T ) I
Transport independant for eachspecies
Chemistry independant for eachgrid cell
Find a good preconditioner ?Is block diagonal good enough ?
Physics based (cf Hammond et al.) ?M. Kern (INRIA) Subsurface flow and transport 36 / 40
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Example: ion exchange
Column experiment (Phreeqc ex. 11, Alliances ex. 3)
Column contains a solution with1mmolNa , 0.2mmolK and1.2mmolNO3. Inject solution with1.2mmolCaCl2. CEC = 1.110−3.
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Ion exchange example (ctd)
Snapshots at t = 35
Cl
Na
Ca
K
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MoMaS benchmark
Difficult test case, heterogenous medium, with complex chemistry (even for“easy” level).12 species, eq. constants vary by 45 orders of magnitude (1D, Matalb code,L.Amir’s thesis).
0 0.5 1 1.5 20
0.25
0.5
0.75
1
1.25
1.5
Space
xi c
once
ntra
tions
at t
=10
Global approach with NKM
x1−220x2−220x3−220x4−220x1−440x2−440x3−440x4−440x1−660x2−660x3−660x4−660x1−880x2−880x3−880x4−880
0 400 800 1200 1600 2000 2400 2800 32000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time
TD
3 to
tal d
isso
lved
con
cent
ratio
n at
x=2
.1
Global approach with NKM
TD3−220
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Conclusions – perspectives
Robust methods for solving flow and transport in porous media
Preliminary results for reactive transport
Newton–Krylov promising framework, implementation in progress
Move to two-phase (multiphase) flows (water and gas)
Transport in fractured media
For chemistry, take into account “real” phenomena (minerals, kinetics,...)
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