MESA LAB
CTRW MATLAB Toolbox
Applied Fractional Calculus Workshop Series
Tomas Oppenheim
MESA(Mechatronics, Embedded Systems and Automation)LAB
School of Engineering,
University of California, Merced
E: [email protected] Phone: 310-853-9234
Lab: CAS Eng 820 (T: 228-4398)
June 16, 2014. Monday 4:00-6:00 PM
Applied Fractional Calculus Workshop Series @ MESA Lab @ UCMerced
MESA LAB
Outline
1.
2.
3.
4.
5.
Continuous Time Random Walk (CTRW) Background
CTRW MATLAB Basic Equations
1D Forward Modeling: Generating Temporal/Spatial Profiles
1D Inverse Problem: Fitting CTRW Solutions to Data Sets
Conclusions and Future Work
06/16/2014
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
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)
06/16/2014
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
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.”
 

p ( x, t )   ( x) (t )    (t  t ' )   w( x  x' ) p ( x' , t ' )dx'dt '
0


p ( x, t )  Probability of the particle being in point x at instant t.
t
06/16/2014
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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).
Porous Media
Dye Flowing Through Porous Media
Different velocities due
to heterogeneities in
Porous media
Use CTRW to model flow of non-Fickian (anomalous)
and Fickian (Gaussian) transport
http://www.newcastle.edu.au/research-and-innovation/centre/cgmm/research/thermal-transport-in-composites-and-porous-media
06/16/2014
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
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).
Gaussian or Anomalous Transport (PDF)?
Hu Sheng, YangQuan Chen, TianShuang Qiu. Fractional Processes and
Fractional-Order Signal Processing, Techniques and Applications. Springer (2012)
06/16/2014
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
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).
Generalized Master Equation (GME): Continuous transitions of tracer transport
c( s, t )
    ( s ' s, t  t ' )c( s, t ' )dt '     ( s  s ' , t  t ' )c( s, t ' )dt '
t
s' o
s' o
t
06/16/2014
t
c ( s, t ) 
Mean, ensemble-averaged, normalized concentration
(e.g. of dye)
 ( s, t ) 
Defined later
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
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).
CTRW Transport Equations: Using Laplace Transform, GME = CTRW
t
R ( s, t )    ( s  s ' , t  t ' ) R ( s ' , t ' )dt '
s' 0
06/16/2014
R ( s, t ) 
Probability/time for a walker to just arrive at site s at
time t
 ( s, t ) 
Probability/time for a displacement s with a
difference of arrival times of t
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
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).
CTRW Transport Equations: Using Laplace Transform, GME = CTRW
 ( s, t ) 
Basic PDF. Determines nature of transport (Fickian
or Non-Fickian)
Correspondence between GME and LT of GME
t
c( s, t )   (t  t ' ) R ( s, t ' )dt '
0
u~ ( s, u )
 ( s, u ) 
1 ~ (u )
~
06/16/2014
AFC Workshop Series @ MESALAB @ UCMerced
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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
2




~
~
~
~
uc ( x, u )  c0 ( x)   M (u ) 
c ( x, u )  D 2 c ( x, u )

x
x


~ (u )
~
“heart” of CTRW
M (u )  t u
~
1  (u )
 
formulation
Transport velocity
D  Generalized Diffusion Coefficient
06/16/2014
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
CTRW MATLAB Basic Equations
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s
guide (2013).
06/16/2014
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
1D Forward Modeling: Generating Temporal Profiles
The Options Variable:
06/16/2014
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
1D Forward Modeling: Generating Temporal Profiles
The Options Variable:
06/16/2014
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
1D Forward Modeling: Generating Temporal Profiles
The Options Variable:
?
06/16/2014
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
1D Forward Modeling: Generating Temporal Profiles
The Options Variable:
06/16/2014
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
1D Forward Modeling: Generating Temporal Profiles
“A concentration measurement in the field or laboratory
will 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.
06/16/2014
AFC Workshop Series @ MESALAB @ UCMerced
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1D Forward Modeling: Generating Temporal Profiles
Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical
ADE Equation
Assume: Porous Column, use ADE model
L  20cm
Ac  4cm 2
n  ( porosity )  0.3
Q  2.4cm 3 min 1
 v  Q /( ALn)  0.1 min 1
D  0.005 min 1
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s
guide (2013).
06/16/2014
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
1D Forward Modeling: Generating Temporal Profiles
Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical
ADE Equation
Matlab Commands:
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
06/16/2014
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
1D Forward Modeling: Generating Temporal Profiles
Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical
ADE Equation
Matlab Commands:
06/16/2014
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
1D Forward Modeling: Generating Temporal Profiles
Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical
ADE Equation
Resulting Temporal Profile:
06/16/2014
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
Outline
1.
2.
3.
4.
5.
Continuous Time Random Walk (CTRW) Background
CTRW MATLAB Basic Equations
1D Forward Modeling: Generating Temporal/ Spatial Profiles
1D Inverse Problem: Fitting CTRW Solutions to Data Sets
Conclusions and Future Work
06/16/2014
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
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
06/16/2014
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
Future Work
06/16/2014
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
AFC Workshop Series @ MESALAB @ UCMerced
MESA LAB
Future Work
06/16/2014
Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).
AFC Workshop Series @ MESALAB @ UCMerced