Anomalous
Dispersion
Particle tracking velocimetry and image processing
for classical and non-classical dispersion theories
J. Ramirez, T. Weinstein, S. Harrington, K. Bardsley, Y. Wu, A. Cagnioncle and L. Donado
John Cushman, Monica Moroni and Natalie Kleinfenter
Anomalous Dispersion
OUTLINE
Introduction
1.
•
•
•
Mathematical Theory
2.
•
•
3.
4.
Group 6
Experiments
Objectives
Image Processing
Classical Dispersion
Generalized Dispersion
Data Analysis
Summary & Conclusions
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Anomalous Dispersion
Experimental Setup
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Anomalous Dispersion
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Objectives
Other Questions
• Determine if and
when dispersion
becomes Fickian
(classical) using
Particle Tracking
Velocimetry (PTV)
• Calculate the
generalized dispersion
coefficient
• At what scale is the
medium
homogeneous?
• At what scale is the
medium
heterogeneous?
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Anomalous Dispersion
Original Photos
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Anomalous Dispersion
Filtered Noise
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Anomalous Dispersion
Final Image for Analyses
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Anomalous Dispersion
Centroid Tracking
t3
t2
t1
toll
t0
Dmax
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Anomalous Dispersion
Trajectories
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Anomalous Dispersion
OUTLINE
Introduction
1.
•
•
•
Mathematical Theory
2.
•
•
3.
4.
Group 6
Experiments
Objectives
Image Processing
Classical Dispersion
Generalized Dispersion
Data Analysis
Summary & Conclusions
Summer School in Geophysical Porous Media
10
Anomalous Dispersion
Lagrangian Description of Dispersion
Posit ion and velocity at t ime t of a part icle init ially locat ed at x 0
X (t) = X (t; x 0 ) 2 ¡ µ R3 ;
dX (t)
V (t) =
:
dt
Displacement s:
Y (t) ´ X (t) ¡ X (0)
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Anomalous Dispersion
Two assumptions:
T he mot ion of t he par t icl e is dict at ed by t -indep endent
t r ansit ion pr obabilit ies f :
Z
P(X (t+ ¿) 2 A) =
f (X (t); x; ¿) dx; for all t > 0
A
Steady-state assumption
hV (t)i = hV i
Group 6
Stationarity (Homogeneity)
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Anomalous Dispersion
Lagrangian Description of Dispersion
Probability dist ribut ion of displacement s:
Z
t
Y (t) = X (t; x 0 ) ¡ x 0 =
V (s; x 0 ) ds
0
G(y; t) = P(Y (t) = y)
=
=
h±(y ¡ (X (t) ¡ X (0)))i
Z
f (x 0 ; X (t); t) P(X (0) = x 0 ) dx 0
¡
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Anomalous Dispersion
Dispersion: Classical Approach
Y (t) = hY i + Y 0(t);
V (t) = hV i + V 0(t)
Z
t
Y 0(t) =
V 0(s; x 0 ) ds
hY (t)i = hV i t;
0
dh(Y 0(t)) 2 i
i
dt
¿
=
=
À
d
(Y 0(t)) 2 = 2 hY 0(t)V 0(t)i
i
i
dt i
¿Z
À
t
2
V 0(s)V 0(t) ds
i
Z
=
2
i
0
t
Z
hV 0(s)V 0(t)i ds = 2
i
i
0
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0
t
CV 0 (s) ds
i
14
Anomalous Dispersion
dh(Y 0(t)) 2 i
i
= 2
dt
Z
t
CV 0 (s) ds
i
0
as t ! 0:
h(Y 0(t)) 2 i ¼ htV 0(0) ¢tV 0(0)i = hV 02 i t 2
i
as t !
i
1 :
dh(Y 0(t)) 2 i
i
¼2
dt
Group 6
i
Z
0
Fickian
1
CV 0 (s) ds ´ 2D i i
i
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Anomalous Dispersion
Non-Classical Approach: Generalized Dispersion
Coefficient
Dispersion \ at scale" k:
dA(t)
= i L A(t);
dt
A k (t) = ei k ¢X ( t ) ;
^ t)
G(k;
i L = V ¢r
x
=
D
E D
³
´ ¤E
WHAT DA…?:
ei k ¢X ( t ) e¡ i k ¢X ( 0) = ei k ¢X ( t ) ei k ¢X ( 0)
´
CA (t)
k
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Anomalous Dispersion
From project ion operat or t heory: t here exist s a
m em or y ker nel K A such t hat ,
k
dCA
(t)¡
k = i hL A(0); A(0) ¤ i C
A
dt
k
Z
t
K^ A (¿) CA (t¡ ¿) d¿
0
k
k
Z
^
t
dG(k; t)
^ t) ¡
^ t ¡ ¿) d¿
= i k ¢hV i G(k;
K^ A (¿) G(k;
dt
k
0
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Anomalous Dispersion
Generalized dispersion coe± cient :
k ¢D^ (k; ¿) ¢k = K^ A (¿)
k
Let’s check it out…
Z
^
t
dG(k; t)
^
^
= i k¢hV i G(k; t)¡
[k¢D^ (k; ¿)¢k] G(k;
t¡ ¿) d¿
dt
0
Z Z
t
dG(x; t)
= ¡ hV i ¢r G(x; t)+ r ¢
D (y; ¿)¢r G(x¡ y; t¡ ¿) dy d¿
dt
0
¡
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Anomalous Dispersion
?
Est imat ing
K A = k ¢D^ (k; ¿) ¢k
k
Volt erra equat ion for K A
k
Z
^
^
t
2G
@
@
G
K^ A (t) = ¡
(k; t) ¡
K^ A (t)
(k; t ¡ ¿) d¿
@t2
@t
k
k
0
D
^ t) = ei k ¢X ( t ) e¡
G(k;
Group 6
E
i k ¢X ( 0)
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From
data
19
Anomalous Dispersion
OUTLINE
Introduction
1.
•
•
•
Mathematical Theory
2.
•
•
3.
4.
Group 6
Experiments
Objectives
Image Processing
Classical Dispersion
Generalized Dispersion
Data Analysis
Summary & Conclusions
Summer School in Geophysical Porous Media
20
Anomalous Dispersion
Velocity Correlation coefficients in the transverse
and longitudinal directions for two different mean velocities
1.2
1
Transverse
Longitudinal
0.8
Transverse
0.6
Longitudinal
0.4
0.2
0
-0.2
1
3
5
7
9
11
13
15
17
19
21
-0.4
-0.6
Time lag (s)
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Anomalous Dispersion
Mean Velocity
hV (t)i
Transverse direction
t
0
-0.01 0
5
10
15
20
25
-0.02
-0.03
-0.04
-0.05
v4
v2
-0.06
-0.07
Group 6
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Anomalous Dispersion
Correlation of velocity
hV (t)V (t + ¿)i
Transverse direction
0.12
0.10
0.08
V4
0.06
V2
0.04
¿
0.02
0.00
-0.02 0
5
10
15
20
-0.04
-0.06
Group 6
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Anomalous Dispersion
Displacement variance
$\la
(Y_z'(t))^2
\ra}
100 $
Longitudinal direction
10
t
1
1
0.1
Group 6
10
V2
V4
100
Variance of displacement
Summer School in Geophysical Porous Media
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Anomalous Dispersion
Intermediate scattering function
Transverse direction
^ ¿))
Re( G(k;
k=
2¼
d
increases
1.2
1
0.8
0.6
0.4
0.2
0
¿
1
Group 6
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19
d=0.05
d=0.1
d=0.4
d=0.8
d=1.4
d=2.0
d=6.0
d=9.0
d=13.0
d=19.0
Summer School in Geophysical Porous Media
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Anomalous Dispersion
Intermediate scattering function
^ ¿))
I m( G(k;
Transverse direction
0.08
0.06
0.04
0.02
¿
0
-0.02
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19
-0.04
-0.06
d=0.05
d=6.0
Group 6
d=0.1
d=6.0
d=0.4
d=9.0
d=0.8
d=13.0
d=1.4
d=19.0
d=2.0
Summer School in Geophysical Porous Media
d=2.0
d=2.0
26
Anomalous Dispersion
Intermediate scattering function
^ ¿))
Re( G(k;
Longitudinal direction
1.5
1
0.5
¿
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19
-0.5
-1
d=0.05
d=2.0
Group 6
d=0.1
d=6.0
d=0.4
d=9.0
d=0.8
d=13.0
Summer School in Geophysical Porous Media
d=1.4
d=19.0
27
Anomalous Dispersion
Intermediate scattering function
^ ¿))
I m( G(k;
Longitudinal direction
1
0.8
0.6
0.4
0.2
¿
0
-0.2
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19
-0.4
-0.6
d=0.05
Group 6
d=0.1
d=0.4
d=0.8
d=1.4
d=2.0
d=6.0
Summer School in Geophysical Porous Media
d=9.0
k=
d=13.0
d=19.0
2¼
d
28
Anomalous Dispersion
Generalized dispersion coefficient
D^ x (k; ¿)
Transversal direction
0.12
0.1
0.08
0.06
0.04
¿
0.02
0
-0.02
0
4
8
12
16
20
-0.04
-0.06
-0.08
k=
-0.1
d=0.05
d=0.4
d=1.4
d=6.0
d=13.0
2¼
d
Generalized dispersion tensor should equal to the
velocity covariance in the Fickian limit.
Group 6
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Anomalous Dispersion
Generalized dispersion tensor
longitudinal direction
D^ z (k; ¿)
k=
2¼
d
increases
¿
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Anomalous Dispersion
OUTLINE
Introduction
1.
•
•
•
Mathematical Theory
2.
•
•
3.
4.
Group 6
Experiments
Objectives
Image Processing
Classical Dispersion
Generalized Dispersion
Data Analysis
Summary & Conclusions
Summer School in Geophysical Porous Media
31
Anomalous Dispersion
Summary & Conclusions
• Examined a mathematical theory aimed at describing
non-Fickian (anomalous) dispersion.
• Analyzed experimental data to examine the mean,
variances, classical and non-classical measures of
dispersion.
• From that analysis, we concluded that on the observed
spatial scales the transport is anomalous even though the
medium is homogeneous.
• Other measures of dispersion are needed to describe
anomalous dispersion.
Group 6
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Anomalous Dispersion
5. References

Moroni M, Cushman JH
Statistical mechanics with three-dimensional particle tracking velocimetry experiments in the study of anomalous
dispersion. II. Experiments
PHYSICS OF FLUIDS 13 (1): 81-91 JAN 2001

Cushman JH, Moroni M
Statistical mechanics with three-dimensional particle tracking velocimetry experiments in the study of anomalous
dispersion. I. Theory
PHYSICS OF FLUIDS 13 (1): 75-80 JAN 2001

Moroni M, Cushman JH, Cenedese A
A 3D-PTV two-projection study of pre-asymptotic dispersion in porous media which are heterogeneous on the
bench scale
INTERNATIONAL JOURNAL OF ENGINEERING SCIENCE 41 (3-5): 337-370 FEB-MAR 2003

Monica Moroni, Natalie Kleinfelter and John H. Cushman
Analysis of dispersion in porous media via matched-index particle tracking velocimetry experiments • ARTICLE
ADVANCES IN WATER RESOURCES. In Press, Corrected Proof, Available online 31 March 2006

IGUCHI K
STARTING METHOD FOR SOLVING NONLINEAR VOLTERRA INTEGRAL-EQUATIONS OF OF SECOND
KIND
COMMUNICATIONS OF THE ACM 15 (6): 460& 1972
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Anomalous Dispersion
Questions??
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