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 KernMichel.Kern@inria.fr

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

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1 Motivations

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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

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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

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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

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1 Motivations

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1 Motivations

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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

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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

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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

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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)

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1 Motivations

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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

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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

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 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

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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

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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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1 Motivations

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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.

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According to speed of reaction

Slow reactions Irreversible, modeled using kinetic law

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According to speed of reactionSlow reactions Irreversible, modeled using kinetic 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 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

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Sorption processes

Several possible mechanismsSurface complexation Formation of bond between surface and aqueous

species, due to electrostatic interactions. Depends on surfacepotential.

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Sorption processes

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Sorption processes

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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 )

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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)

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1 Motivations

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1 Motivations

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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

P = 0.0 P = 0.0

k = 4.8 10-13

ρ=1000 3kg m/

100.00 200.00 300.00 400.00 500.00

100.00

100.0 200.0 300.0 400.0 500.0

Case 1(720 s) Case 2 (464 s) Case 3 (393 s)

Simulated concentration at t = 20 years for the Elder

benchmark

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Physical model

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,

∂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

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Elder model

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1 Motivations

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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

∂y i

∂ t= r y

i ,∂sj

∂ t= r s

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

∂y i

∂ t= r y

i ,∂sj

∂ t= r s

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

+ LC = 0

H(z)−(

C + FW

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

+ 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

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)

Flow and trasnsport of pollutants in the subsurface : coupled ...logoINRIA Flow and trasnsport of...

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