Fractional dynamics in underground contaminant transport: introduction and applications

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Fractional dynamics in underground contaminant
transportintroduction and applications
Andrea Zoia

• Current affiliation CEA/Saclay
• DEN/DM2S/SFME/LSET
• Past affiliation Politecnico di Milano and MIT
• MOMAS - November 4-5th 2008

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Outline

• Modeling contaminant migration
• in heterogeneous materials

• CTRW methods and applications
• Conclusions

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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 ltx2gttg
• Relevance in contaminant migration
• Early arrival times (ggt1) leakage from
  repositories
• Late runoff times (glt1) environmental remediation

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An example

• Kirchner et al., Nature 2000 Chloride transport
  in catchments.

• Unexpectedly long retention times
• Cause complex (fractal) streams

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Continuous Time Random Walk
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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 l(x) flow material
  properties

• Assume l(x) with finite std s and mean m
• Typical scale for space displacements

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CTRW transport equation

• Heterogeneous materials broad flow spectrum ?
  multiple time scales
• w(t) t-a-1 , 0ltalt2, 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

• The asymptotic transport equation becomes

• 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 , 0ltblt2, power law decay

• Physical meaning large displacements
• Application fracture networks?

• The asymptotic equation is

• 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
Monte Carlo
CTRW
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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

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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

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2. Advection

• Water flow main source of hazard in contaminant
  migration

• How to model advection within CTRW?
• x ? xvt (Galilei invariance)
• ltl(x)gtm (bias preferential jump direction)

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2. Advection

• Anomalous diffusion (FADE) intrinsically
  distinct approaches

P(x,t)
P(x,t)
x ? xvt
ltl(x)gtm
Contaminant migration
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

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3. Walking across an interface

• Multiple traversed materials, different physical
  properties

• Two-layered medium
• Stepwise changes

s,m,a1
s,m,a2
x
Set of properties 1
Set of properties 2
Interface

• What happens to particles when crossing the
  interface?

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3. Walking across an interface
Physics-based Monte Carlo sampling rules

• Case study normal and anomalous diffusion (no
  advection)

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3. Walking across an interface
P(x,t)
layer1 layer2
Fickian diffusion
layer1 layer2
P(x,t)
Anomalous
x
Interface
Experimental results
Interface
x
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4. Breakthrough curves

• Transport in finite regions

Injection
A
B
x0

• 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

Well-known formalism
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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

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Generalized lattice Master Equation
Master Equation
Mass conservation at each lattice site s
Normalized particle concentration
Transition rates
Ensemble average on possible rates realizations
Stochastic description of traversed medium
Assumptions
lattice ? continuum
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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)
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Higher-order corrections to FDE
FDE u?0
Fourier and Laplace transforms, including second
order contributions
FDE k?0
FDE
Transport equations in direct space, including
second order contributions
FDE
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Standard vs. linear CTRW
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3. Walking across an interface
Physics-based Monte Carlo sampling rules
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Re-sampling at the interface
1
2
l,w
l,w
x,t ? vx/t
Dx
DtDx/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)
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3. Walking across an interface
Analytical boundary conditions at the interface
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Local particle velocity
Monte Carlo simulation Local velocity vx/ta
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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

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5. Fractured porous media

• Compare our model to analogous CTRW approach
  Berkowitz et al., 1998
• Identical spread ltx2gttg (g depending on df)
• Discrepancies in the breakthrough curves

• Anomalous diffusion is not universal
• There exist many possible realizations and
  descriptions