1 Fractional dynamics in underground contaminant transport: introduction and applications Andrea Zoia Current affiliation: CEA/Saclay DEN/DM2S/SFME/LSET Past affiliation: Politecnico di Milano and MIT MOMAS - November 4-5 th 2008

2 Outline CTRW: methods and applications Conclusions Modeling contaminant migration in heterogeneous materials

3 Transport in porous media Highly complex velocity spectrum ANOMALOUS (non-Fickian) transport: ~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 x x0x0 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 Probability/mass balance (Chapman-Kolmogorov equation)  Fourier and Laplace transformed spaces: x  k, t  u, P(x,t)  P(k,u) Assume: (x) with finite std  and mean  ‘Typical’ scale for space displacements

7 CTRW transport equation 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  Memory kernel M(u): w(t) ?

8 Asymptotic behavior Fractional Advection-Dispersion Equation (FADE)  Fractional derivative in time  ‘Fractional dynamics’ The asymptotic transport equation becomes: Analytical contaminant concentration profile P(x,t) Long time behavior: u  0

9 Long jumps (x)~|x| , 0<  <2, power law decay 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)  =  (bias: preferential jump direction) Water flow: main source of hazard in contaminant migration Fickian diffusion: equivalent approaches ( v =  / )  Center of mass:  (t) ~ t  Spread:   (t) ~ t  (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 =  P(x,t) x x Contaminant migration  2 (t)~t   (t)~t  2 (t)~t 2   (t)~t 

16 2. Radioactive decay Coupling advection-dispersion with radioactive decay  Normal diffusion: Advection-dispersion … & decay  Anomalous diffusion: Advection-dispersion … & decay

17 3. Walking across an interface Multiple traversed materials, different physical properties {,,}1{,,}1 {,,}2{,,}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

19 3. Walking across an interface Fickian diffusion layer 1 layer 2 P(x,t) x Interface Anomalous Interface P(x,t) x Experimental results Key feature: local particle velocity layer 1 layer 2

20 4. Breakthrough curves Transport in finite regions A Injection x0x0 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 Well-known formalism Space-fractional: transport operator = Fractional Laplacian 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  Definition in direct (t) space:  Definition in Laplace transformed (u) space:  Example: fractional derivative of a power

24 Generalized lattice Master Equation Master Equation Normalized particle concentration Transition rates Mass conservation at each lattice site s Ensemble average on possible rates realizations: Stochastic description of traversed medium Assumptions: lattice  continuum

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 FDE: u  0 FDE: k  0 Fourier and Laplace transforms, including second order contributions Transport equations in direct space, including second order contributions 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 xx x’,t’  v’=x’/t’  t=  x/v’ x’ =  -1 (R x ), t’ = W -1 (R t )   R x =  (  x),  R t = W(  t) 12  x =  -1 (R x ) -  -1 (  R x ),  t = W -1 (R t ) - W -1 (  R t ),w x,tx,t v’ v’’

30 3. Walking across an interface Analytical boundary conditions at the interface Mass conservation: Concentration ratio at the interface: P(x,u)

31 Local particle velocity Normal diffusion: M(u)=1/  Equal velocities: (  ) + =(  ) - Anomalous diffusion: M(u)=u 1-  /   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  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 dfdf Schramm-Loewner Evolution

33 5. Fractured porous media Compare our model to analogous CTRW approach [Berkowitz et al., 1998]  Identical spread ~t  (  depending on d f )  Discrepancies in the breakthrough curves Anomalous diffusion is not universal  There exist many possible realizations and descriptions Both behaviors observed in different physical contexts  (t) t CTRW Our model