1 fractional dynamics in underground contaminant transport: introduction and applications andrea...
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Fractional dynamics in underground
contaminant transport:introduction and
applications
Andrea Zoia
Current affiliation: CEA/SaclayDEN/DM2S/SFME/LSET
Past affiliation: Politecnico di Milano and MIT
MOMAS - November 4-5th 2008
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Outline
CTRW: methods and applications
Conclusions
Modeling contaminant migrationin heterogeneous materials
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Transport in porous media
Highly complex velocity spectrum
ANOMALOUS (non-Fickian) transport: <x2>~t
Relevance in contaminant migration Early arrival times (): leakage from repositories
Late runoff times (): environmental remediation
Porous media are in general heterogeneous
Multiple scales: grain size, water content, preferential flow streams, …
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[Kirchner et al., Nature 2000] Chloride transport in catchments.
Unexpectedly long retention times
Cause: complex (fractal) streams
An example
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Continuous Time Random Walk
t
xx0
Main assumption: particles follow stochastic trajectories in {x,t} Waiting times distributed as w(t) Jump lengths distributed as (x)
Berkowitz et al., Rev. Geophysics 2006.
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CTRW transport equation P(x,t) = probability of finding a particle in x at time t =
= normalized contaminant particle concentration
P depends on w(t) and (x): flow & material properties
)()(1
)()(1),( 0
kuw
kP
u
uwukP
Probability/mass balance (Chapman-Kolmogorov equation) Fourier and Laplace transformed spaces: xk, tu, P(x,t)P(k,u)
Assume: (x) with finite std and mean
‘Typical’ scale for space displacements
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CTRW transport equation
')',(2
)'(),(2
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dttxPxx
ttMtxPt
Rewrite in direct {x,t} space (FPK):
Heterogeneous materials: broad flow spectrum multiple time scales w(t) ~ t , 0<<2, power-law decay
M(t-t’) ~ 1/(t-t’): dependence on the past history
Homogeneous materials: narrow flow spectrum single time scale w(t) ~ exp(-t/) M(t-t’) ~ (t-t’) : memoryless = ADE
)(1
)()(
uw
uuwuM
Memory kernel M(u): w(t) ?
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Asymptotic behavior
),(2
),(2
221 txP
xxtxP
t t
Fractional Advection-Dispersion Equation (FADE) Fractional derivative in time ‘Fractional dynamics’
The asymptotic transport equation becomes:
Analytical contaminant concentration profile P(x,t)
)(1)()( 11 uoucucuwttw
Long time behavior: u0
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Long jumps
(x)~|x| , 0<<2, power law decay
),(),( txPx
txPt
The asymptotic equation is
Fractional derivative in space
Physical meaning: large displacements Application: fracture networks?
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Monte Carlo simulation
CTRW: stochastic framework for particle transport Natural environment for Monte Carlo method
Simulate “random walkers” sampling from w(t) and (x) Rules of particle dynamics
Describe both normal and anomalous transport
Advantage: Understanding microscopic dynamics link with macroscopic
equations
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Developments
Advection and radioactive decay
Macroscopic interfaces
Asymptotic equations
Breakthrough curves
CTRW Monte Carlo
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1. Asymptotic equations Fractional ADE allow for analytical solutions However, FADE require approximations Questions:
How relevant are approximations? What about pre-asymptotic regime (close to the
source)?
FADE good approximation of CTRW Asymptotic regime rapidly attained
Quantitative assessment via Monte Carlo
Exact CTRW .
Asymptotic FADE _
P(x,t)
x
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If 1 (time) or 2 (space): FADE bad approximation
1. Asymptotic equations
Exact
CTRW .Asymptotic FADE _
P(x,t)
x
FADE* _
P(x,t)
x
New transport equations including higher-order corrections: FADE* Monte Carlo validation of FADE*
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2. Advection
How to model advection within CTRW? x x+vt (Galilei invariance) <(x)>= (bias: preferential jump direction)
Water flow: main source of hazard in contaminant migration
Fickian diffusion: equivalent approaches (v = /<t>) Center of mass: (t) ~ t
Spread: (t) ~ t
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(t)~t
x
P(x,t)
2(t)~t
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2. Advection
Even simple physical mechanisms must be reconsidered in presence of anomalous diffusion
Anomalous diffusion (FADE): intrinsically distinct approaches
x x+vt
<(x)>=
P(x,t)
xx
P(x,t)
Contaminant migration
2(t)~t
(t)~t
2(t)~t2
(t)~t
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2. Radioactive decay
Coupling advection-dispersion with radioactive decay
),(),(2
2
txPx
vx
DtxPt
Normal
diffusion:Advection-dispersion
),(1
),(),(2
2
txPtxPx
vx
DtxPt
… & decay
),(),(2
21 txP
xv
xDtxP
t t
Anomalou
s diffusion:Advection-dispersion
),(1
),(),(2
2/1/ txPtxP
xv
xDeetxP
tt
tt
… & decay
/),(),( tetxPtxP
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3. Walking across an interface Multiple traversed materials, different physical properties
{,,}1 {,,}2
Set of properties {1}
Set of properties {2}
Two-layered medium
Stepwise changes
Interface
What happens to particles when crossing the interface?
?
x
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3. Walking across an interface
“Physics-based” Monte Carlo sampling rules
Linking Monte Carlo parameters with equations coefficients
Case study: normal and anomalous diffusion (no advection)
Analytical boundary conditions at the interface
),(),()(),()(2
)(),(
0),()0,(),(
uxPuxMxuxMxx
xuxJ
uxJx
xPuxuP
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3. Walking across an interface
Fickian diffusion
layer1 layer2
P(x,t)
x Interface
Anomalous
Interface
P(x,t)
xExperimental
results
Key feature: local particle velocity
layer1 layer2
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4. Breakthrough curves Transport in finite regions
A
Injection
x0
0 1 2 3 4 5 6 7 8 9 10
x 105
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
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Breakthrough curve (t)
t
Outflow
The properties of (t) depend on the eigenvalues/eigenfunctions of the transport operator in the region [A,B]
Physical relevance: delay between leakage and contamination
Experimentally accessible
B
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4. Breakthrough curves Time-fractional dynamics: transport operator = Laplacian
)()(2
2
xxx
2
2
x
Well-known formalism
Space-fractional: transport operator = Fractional Laplacian
)()( xxx
|| x
Open problem…
Numerical and analytical characterization of eigenvalues/eigenfunctions
x
x
(t)
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Conclusions
Current and future work:
link between model and experiments (BEETI: DPC, CEA/Saclay)
Transport of dense contaminant plumes: interacting particles.
Nonlinear CTRW?
Strongly heterogeneous and/or unsaturated media:
comparison with other models: MIM, MRTM…
Sorption/desorption within CTRW: different time scales?
Contaminant migration within CTRW model
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Fractional derivatives
ttt )1(
)1(,0
Definition in direct (t) space:
Definition in Laplace transformed (u) space:
Example: fractional derivative of a power
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Generalized lattice Master Equation
''
),'()',(),(),'(),(ss
tsCssrtsCssrtsCt
' 0' 0
')','()','(')',()','(),(s
t
s
t
dttspttssdttspttsstspt
sus
usuus
),(1
),(),(
Master Equation
Normalized particle concentration
Transition ratesMass conservation at each lattice site
s
Ensemble average on possible rates realizations:
),( ts Stochastic description of traversed medium
Assumptions:
lattice continuum )()(),( stwts
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Chapman-Kolmogorov Equation
P(x,t) = normalized concentration (pdf “being” in x at time t)
Source terms
(t) = probability of not having moved
p(x,t) = pdf “just arriving” in x at time t
Contributions from the past history
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Higher-order corrections to FDE
)(1)()( 11 uoucucuwttw
)(1)()( 222
1kokckckxx
FDE: u0
FDE: k0
),(),(),( 12
21 txP
tqtxP
xtxP
t tt
),(),(),(2
2
txPx
qtxPx
txPt
Fourier and Laplace transforms, including second order contributions
Transport equations in direct space, including second order contributions
FDE
FDE
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Standard vs. linear CTRW
t
x
(x): how far
w(t): how long
Linear CTRW
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3. Walking across an interface
“Physics-based” Monte Carlo sampling rules
Sample a random jump:
t~w(t) and x~(x)
Start in a given layer
The walker lands in the same layer
The walker crosses the interface
“Reuse” the remaining portion of the jump in the other layer
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Re-sampling at the interface
xx’,t’ v’=x’/t’
t=x/v’
x’ = -1(Rx), t’ = W-1(Rt) Rx = (x), Rt = W(t)
1 2
x = -1(Rx) - -1(Rx), t = W-1(Rt) - W-
1(Rt)
,w ,w
x,t
v’
v’’
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3. Walking across an interface
Analytical boundary conditions at the interface
),(),()(),()(2
)(),(
0),()0,(),(
uxPuxMxuxMxx
xuxJ
uxJx
xPuxuP
JJ Mass conservation:
Concentration ratio at the interface: PuMPuM )()( P(x,u)
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Local particle velocity
Normal diffusion: M(u)=1/
Equal velocities: ()+=()-
Anomalous diffusion: M(u)=u1-/
Equal velocities: ()+=()- and +=-
(x/t)-
(x/t)+
Different concentrations at the
interface
Equal concentrations at
the interface
(x/t)-
(x/t)+Monte Carlo simulation:
Local velocity: v=x/t
PuMPuM )()(
Boundary conditions
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5. Fractured porous media
Experimental NMR measures [Kimmich, 2002] Fractal streams (preferential water flow)
Anomalous transport
Develop a physical model Geometry of paths
df
Schramm-Loewner Evolution
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5. Fractured porous media
Compare our model to analogous CTRW approach [Berkowitz et al., 1998] Identical spread <x2>~t( depending on df)
Discrepancies in the breakthrough curves
Anomalous diffusion is not universal There exist many possible realizations and descriptions
0 5 10 15 20 25 30 35 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
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Both behaviors observed in different physical contexts
(t)
t
CTRW
Our model