MESA LAB
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
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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
MESA LAB
CTRW Background
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014
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)
MESA LAB
CTRW Background
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014
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
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014
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
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014
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)?
MESA LAB
CTRW Background
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014
Based on paper: Berkowitz et. al. Modeling Non-Fickian Transport in Geological Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006).
''
')',()','(')',()','(),(
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
MESA LAB
CTRW Background
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014
Based on paper: Berkowitz et. al. Modeling Non-Fickian Transport in Geological Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006).
' 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
MESA LAB
CTRW Background
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014
Based on paper: Berkowitz et. al. Modeling Non-Fickian Transport in Geological Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006).
),( 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
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014
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
MESA LAB
CTRW MATLAB Basic Equations
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
MESA LAB
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:
MESA LAB
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:
MESA LAB
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:
?
MESA LAB
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:
MESA LAB
1D Forward Modeling: Generating Temporal Profiles
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014
“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.
MESA LAB
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
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
MESA LAB
1D Forward Modeling: Generating Temporal Profiles
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical ADE Equation
Matlab Commands:
MESA LAB
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).
Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical ADE Equation
Matlab Commands:
MESA LAB
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).
Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical ADE Equation
Resulting Temporal Profile:
MESA LAB
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
MESA LAB
Conclusions
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
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
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
MESA LAB
Future Work
AFC Workshop Series @ MESALAB @ UCMerced06/16/2014Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).