CTRW MATLAB Toolbox Applied Fractional Calculus Workshop Series

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CTRW MATLAB ToolboxApplied Fractional Calculus Workshop Series

Tomas OppenheimMESA (Mechatronics, Embedded Systems and Automation)LAB

School of Engineering,University of California, Merced

E: [email protected] Phone: 310-853-9234Lab: CAS Eng 820 (T: 228-4398)

June 16, 2014. Monday 4:00-6:00 PMApplied Fractional Calculus Workshop Series @ MESA Lab @ UCMerced

mailto:[email protected]

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Outline

1. Continuous Time Random Walk (CTRW) Background

2. CTRW MATLAB Basic Equations

3. 1D Forward Modeling: Generating Temporal/Spatial Profiles

4. 1D Inverse Problem: Fitting CTRW Solutions to Data Sets

5. Conclusions and Future Work

AFC Workshop Series @ MESALAB @ UCMerced06/16/2014

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CTRW Background


Based on Book: Klages et.al. Anomalous Transport. Foundations and Applications. Wiley (2008).

“CTRW’s are rather good and general phenomenological models of diffusion, including processes of anomalous transport.”

PHYSICAL REVIEW E 84, 051139 (2011)

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CTRW Background


Based on Book: Klages et.al. Anomalous Transport. Foundations and Applications. Wiley (2008).

“CTRW’s are rather good and general phenomenological models of diffusion, including processes of anomalous transport.”

'')','()'()'()()(),(0

dtdxtxpxxwtttxtxpt

Probability of the particle being in point x at instant t.

),( txp

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CTRW Background


Based on paper: Berkowitz et. al. Modeling Non-Fickian Transport in Geological Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006).

http://www.newcastle.edu.au/research-and-innovation/centre/cgmm/research/thermal-transport-in-composites-and-porous-media

Porous Media

Dye Flowing Through Porous Media

Use CTRW to model flow of non-Fickian (anomalous) and Fickian (Gaussian) transport

Different velocities due to heterogeneities inPorous media

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CTRW Background


Based on paper: Berkowitz et. al. Modeling Non-Fickoan Transport in Geoligical Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006).

Hu Sheng, YangQuan Chen, TianShuang Qiu. Fractional Processes and Fractional-Order Signal Processing, Techniques and Applications. Springer (2012)

Gaussian or Anomalous Transport (PDF)?

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CTRW Background



''

')',()','(')',()','(),(

s

t

os

t

o

dttscttssdttscttsst

tsc

),( tsc Mean, ensemble-averaged, normalized concentration (e.g. of dye)

Generalized Master Equation (GME): Continuous transitions of tracer transport

),( ts Defined later

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CTRW Background



' 0

')','()','(),(s

t

dttsRttsstsR

),( tsR Probability/time for a walker to just arrive at site s at time t

CTRW Transport Equations: Using Laplace Transform, GME = CTRW

),( ts Probability/time for a displacement s with a difference of arrival times of t

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CTRW Background



),( ts Basic PDF. Determines nature of transport (Fickian or Non-Fickian)

t

dttsRtttsc0

')',()'(),(

Correspondence between GME and LT of GME

)(~1

),(~),(

~

u

usuus

CTRW Transport Equations: Using Laplace Transform, GME = CTRW

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CTRW MATLAB Basic Equations


Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).

Laplace Transform of CTRW

Transport velocity

Generalized Diffusion Coefficient

“heart” of CTRW formulation)(~1

)(~)(

~

),(~),(~)(~

)(),(~2

2

0

u

uutuM

uxcx

Duxcx

uMxcuxcu

D

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CTRW MATLAB Basic Equations



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1D Forward Modeling: Generating Temporal Profiles

AFC Workshop Series @ MESALAB @ UCMerced06/16/2014Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).

The Options Variable:

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?

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“A concentration measurement in the field or laboratorywill be most closely related to either the resident or the flux concentration. The former describes all solute particles present at some point while the latter describes solute particles that have moved irrevocably past some point.” Zhang et al. Geophysical Research Letters. (33) 2006.

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1

1

13

2

min005.0

min1.0)/(

min4.2

3.0)(

4

20

D

ALnQv

cmQ

porosityn

cmA

cmL

c

Assume: Porous Column, use ADE model

1D Forward Modeling: Generating Temporal ProfilesEx: Breakthrough Curve/Concentration Profile (temporal profile) for classical ADE Equation

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Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical ADE Equation

Matlab Commands:

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Matlab Commands:

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Resulting Temporal Profile:

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Outline

1. Continuous Time Random Walk (CTRW) Background

2. CTRW MATLAB Basic Equations

3. 1D Forward Modeling: Generating Temporal/ Spatial Profiles

4. 1D Inverse Problem: Fitting CTRW Solutions to Data Sets

5. Conclusions and Future Work


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Conclusions


1. CTRW is good model for Fickian and Non-Fickian transport – “CTRW’s are rather good and general phenomenological models of diffusion, including processes of anomalous transport.”

2. MATLAB toolbox available with detailed instruction manual to generate temporal/spatial concentration profiles and model data using either TPM, ADE, or ETA model

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Future Work


CTRW MATLAB Toolbox Applied Fractional Calculus Workshop Series

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