Post on 26-Dec-2015
transcript
1
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
2
Outline
CTRW: methods and applications
Conclusions
Modeling contaminant migrationin heterogeneous materials
3
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, …
4
[Kirchner et al., Nature 2000] Chloride transport in catchments.
Unexpectedly long retention times
Cause: complex (fractal) streams
An example
5
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.
6
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
7
CTRW transport equation
')',(2
)'(),(2
22
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) ?
8
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
9
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?
10
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
11
Developments
Advection and radioactive decay
Macroscopic interfaces
Asymptotic equations
Breakthrough curves
CTRW Monte Carlo
12
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
13
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*
14
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
15
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
16
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
17
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
18
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
19
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
20
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
1
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
21
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)
22
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
23
Fractional derivatives
ttt )1(
)1(,0
Definition in direct (t) space:
Definition in Laplace transformed (u) space:
Example: fractional derivative of a power
24
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
25
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
26
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
27
Standard vs. linear CTRW
t
x
(x): how far
w(t): how long
Linear CTRW
28
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
29
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’’
30
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)
31
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
32
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
33
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
1
Both behaviors observed in different physical contexts
(t)
t
CTRW
Our model