Transcript Anomalous diffusion in radioactive contaminant migration

Andrea Zoia
Fractional dynamics
in underground
contaminant transport:
introduction and applications
Current affiliation: CEA/Saclay
DEN/DM2S/SFME/LSET
Past affiliation: Politecnico di Milano and MIT
MOMAS - November 4-5th 2008
1
Outline

Modeling contaminant migration
in heterogeneous materials

CTRW: methods and applications

Conclusions
2
Transport in porous media

Porous media are in general heterogeneous

Multiple scales: grain size, water content, preferential flow streams, …

Highly complex velocity spectrum

ANOMALOUS (non-Fickian) transport: <x2>~tg

Relevance in contaminant migration

Early arrival times (g>1): leakage from repositories

Late runoff times (g<1): environmental remediation
3
An example

[Kirchner et al., Nature 2000] Chloride transport in catchments.
 Unexpectedly long retention times
 Cause: complex (fractal) streams
4
Continuous Time Random Walk

Main assumption: particles follow stochastic trajectories in {x,t}

Waiting times distributed as w(t)

Jump lengths distributed as l(x)
Berkowitz et al., Rev. Geophysics 2006.
t
x0
x
5
CTRW transport equation

P(x,t) = probability of finding a particle in x at time t =
= normalized contaminant particle concentration

Probability/mass balance (Chapman-Kolmogorov equation)

Fourier and Laplace transformed spaces: xk, tu, P(x,t)P(k,u)
P(k , u ) 
P0 (k )
1  w(u )
u 1  w(u )l (k )

P depends on w(t) and l(x): flow & material properties

Assume: l(x) with finite std s and mean m
‘Typical’ scale for space displacements
6
CTRW transport equation

Rewrite in direct {x,t} space (FPK):
s 2  2


P( x, t )   M (t  t ' ) 
 m  P( x, t ' )dt'
2
t
2

x
x 




Memory kernel M(u):
M (u ) 
uw(u )
1  w(u )
w(t) ?
Heterogeneous materials: broad flow spectrum  multiple time scales

w(t) ~ ta1 , 0<a<2, power-law decay

M(t-t’) ~ 1/(t-t’)1a : dependence on the past history
Homogeneous materials: narrow flow spectrum  single time scale t

w(t) ~ exp(-t/t)

M(t-t’) ~ d(t-t’) : memoryless = ADE
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Asymptotic behavior
w(t )  t 1a  w(u)  1  ca ua  c1u  o(u)

Long time behavior: u0

The asymptotic transport equation becomes:
2

2

1a  s
P ( x, t )   t 
 m  P ( x, t )
2
t
x 
 2 x

Fractional Advection-Dispersion Equation (FADE)


Fractional derivative in time  ‘Fractional dynamics’
Analytical contaminant concentration profile P(x,t)
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Long jumps

l(x)~|x|b1 , 0<b<2, power law decay

Physical meaning: large displacements


Application: fracture networks?
The asymptotic equation is

b
P ( x, t ) 
P ( x, t )
b
t
x

Fractional derivative in space
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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 l(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
CTRW
Asymptotic
equations
Macroscopic
interfaces
Monte Carlo
Breakthrough
curves
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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

P(x,t)
Quantitative assessment via Monte Carlo
Exact CTRW .
Asymptotic FADE _
12
x
1. Asymptotic equations

If a  1 (time) or b  2 (space): FADE bad approximation

New transport equations including higher-order corrections: FADE*
Monte Carlo validation of FADE*

P(x,t)
P(x,t)
x
Asymptotic FADE _
x
Exact CTRW .
FADE* _
13
2. Advection

Water flow: main source of hazard in contaminant migration

How to model advection within CTRW?
 x  x+vt (Galilei invariance)


<l(x)>=m (bias: preferential jump direction)
Fickian diffusion: equivalent approaches (v = m/<t>)


Center of mass: z(t) ~ t
Spread:
S2(t)
~t
1
P(x,t)
0.9
0.8
0.7
0.6
S2(t)~t
0.5
0.4
0.3
z(t)~t
0.2
x
0.1
0
0
1
2
3
4
5
6
7
8
9
10
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2. Advection

Anomalous diffusion (FADE): intrinsically distinct approaches
P(x,t)
P(x,t)
x  x+vt
<l(x)>=m
Contaminant migration
S2(t)~ta
S2(t)~t2a
z(t)~t
z(t)~ta
x

x
Even simple physical mechanisms must be reconsidered
in presence of anomalous diffusion
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2. Radioactive decay

Coupling advection-dispersion with radioactive decay

Normal
diffusion:
 2


P( x, t )   D 2  v  P( x, t )
t
x 
 x
Advection-dispersion
 2


1
P( x, t )   D 2  v  P( x, t )  P( x, t )
t
x 
t
 x

Anomalous
diffusion:
P( x, t )  P( x, t )et /t

2

1a 
P( x, t )   t  D 2  v  P( x, t )
t
x 
 x
… & decay
Advection-dispersion

2

1
t / t 1a t / t 
P( x, t )  e  t e  D 2  v  P( x, t )  P( x, t )
t
x 
t
 x
… & decay
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3. Walking across an interface

Multiple traversed materials, different physical properties
?
{s,m,a}1
{s,m,a}2
Set of properties {1}
Set of properties {2}
 Two-layered medium

Stepwise changes
x
Interface

What happens to particles when crossing the interface?
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3. Walking across an interface
“Physics-based” Monte Carlo sampling rules
Analytical boundary conditions at the interface

J ( x, u )  0
x
 s ( x) 

J ( x, u )   
s ( x ) M ( x, u )  m ( x ) M ( x, u )  P ( x, u )
2 x


uP( x, u )  P( x,0) 
Linking Monte Carlo parameters with equations coefficients

Case study: normal and anomalous diffusion (no advection)
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3. Walking across an interface
P(x,t)
layer1
layer2
Key feature: local particle velocity
Fickian diffusion
P(x,t)
layer1
layer2
Anomalous
x
Interface
Experimental results
x
Interface
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4. Breakthrough curves
1

Transport in finite regions
Breakthrough curve
r(t)
0.9
0.8
0.7
Injection
Outflow
0.6
0.5
B
A
x0
0.4
0.3
0.2
t
0.1
0
0
1
2
3
4
5
6
7
8
9
10
5
x 10

Physical relevance: delay between leakage and contamination

Experimentally accessible

The properties of r(t) depend on the eigenvalues/eigenfunctions of
the transport operator in the region [A,B]
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4. Breakthrough curves

Time-fractional dynamics: transport operator = Laplacian
2
 ( x)  l ( x)
x 2

Well-known formalism
Space-fractional: transport operator = Fractional Laplacian
b
x

2
x 2
b
 ( x)  l ( x)
b
 | x |b
Open problem…
Numerical and analytical characterization of eigenvalues/eigenfunctions
(x)
l
r(t)
b
b
x
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Conclusions

Contaminant migration within CTRW model

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?
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Fractional derivatives
 Definition in direct (t) space:
 Definition in Laplace transformed (u) space:
 Example: fractional derivative of a power
a0,t t b 
(1  b ) b a
t
(1  b  a )
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Generalized lattice Master Equation

C ( s, t )   r ( s' , s)C ( s, t )  r ( s, s' )C ( s' , t )
t
s'
s'
Normalized particle
concentration
Master Equation
Mass conservation at
each lattice site s
Transition rates
Ensemble average on possible rates realizations:

p( s, t )     ( s' s, t  t ' ) p( s, t ' )dt'    ( s's, t  t ' ) p( s' , t ' )dt'
t
s' 0
s' 0
t
 ( s, u ) 
t
u ( s, u )
1  s ( s, u )
Assumptions:
 ( s, t )
Stochastic description of
traversed medium
lattice  continuum
 ( s, t )  w(t )l ( s)
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Chapman-Kolmogorov Equation
p(x,t) = pdf “just arriving” in x at time t
Source terms
Contributions from the past history
(t) = probability of not having moved
P(x,t) = normalized concentration (pdf “being” in x at time t)
25
Higher-order corrections to FDE
FDE: u0
w(t )  t 1a  w(u)  1  ca ua  c1u  o(u)
l ( x)  x
1 b
b
 l ( k )  1  c b k  c2 k 2  o ( k 2 )
Fourier and Laplace transforms,
including second order contributions
FDE: k0
FDE
2

1a 
1a 
P( x, t )   t
P
(
x
,
t
)

q

P( x, t )
a t
t
x 2
t

b
2
P ( x, t ) 
P ( x, t )  q b 2 P ( x, t )
b
t
x
x
FDE
Transport equations in direct space,
including second order contributions
26
Standard vs. linear CTRW
t
Linear CTRW
 l(x): how far
 w(t): how long
x
27
3. Walking across an interface
“Physics-based” Monte Carlo sampling rules
Start in a given layer
Sample a random jump:
t~w(t) and x~l(x)
“Reuse” the remaining portion of
the jump in the other layer
The walker crosses the interface
The walker lands in the same layer
28
Re-sampling at the interface
1
l,w
2
l,w
v’’
v’
x’,t’  v’=x’/t’
Dx
Dx,Dt
Dt=Dx/v’
x’ = L-1(Rx), t’ = W-1(Rt)  DRx = L(Dx), DRt = W(Dt)
Dx = L-1(Rx) - L-1(DRx), Dt = W-1(Rt) - W-1(DRt)
29
3. Walking across an interface
Analytical boundary conditions at the interface

J ( x, u )  0
x
 s ( x) 

J ( x, u )   
s ( x ) M ( x, u )  m ( x ) M ( x, u )  P ( x, u )
2 x


uP( x, u )  P( x,0) 


Mass conservation:
J  J
Concentration ratio at the interface:
(sM (u)P)  (sM (u)P)
P(x,u)
30
Local particle velocity
(x/ta)+
Monte Carlo simulation:
(x/ta)-
(x/ta)+
(x/ta)-
Local velocity: v=x/ta
Different concentrations
at the interface
Equal concentrations
at the interface
Normal diffusion: M(u)=1/t
Equal velocities: (s/t)+=(s/t)Anomalous diffusion: M(u)=u1-a/ta
Boundary conditions
(sM (u)P)  (sM (u)P)
Equal velocities: (s/ta)+=(s/ta)- and a+=a31
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
32
5. Fractured porous media

Compare our model to analogous CTRW approach [Berkowitz et al., 1998]

Identical spread <x2>~tg (g depending on df)

Discrepancies in the breakthrough curves
1
r(t)
0.9
0.8

0.7
CTRW
0.6
Both behaviors observed in
different physical contexts
0.5
Our model
0.4
0.3
0.2
t
0.1
0

0
5
10
15
20
25
30
35
40
Anomalous diffusion is not universal

There exist many possible realizations and descriptions
33