Transcript Scale-Dependent Dispersivities and The Fractional

Continuous Time Random
Walk Model
Primary Sources:
Berkowitz, B, G. Kosakowski, G. Margolin, and H. Scher,
Application of continuuous time random walk theory to tracer test
measurents in fractured and heterogeneous porous media, Ground
Water 39, 593 - 604, 2001.
Berkowitz, B. and H. Scher, On characterization of anomalous
dispersion in porous and fractured media, Wat. Resour. Res. 31,
1461 - 1466, 1995.
Mike Sukop/FIU
Introduction
Continuous Time Random Walk (CTRW)
models
 Semiconductors [Scher and Lax, 1973]
 Solute transport problems [Berkowitz and
Scher, 1995]
2
Introduction
Like FADE, CTRW solute particles move
along various paths and encounter
spatially varying velocities
The particle spatial transitions (direction
and distance given by displacement vector
s) in time t represented by a joint
probability density function y(s,t)
Estimation of this function is central to
application of the CTRW model
3
Introduction
The functional form y(s,t) ~ t-1-b (b > 0) is
of particular interest [Berkowitz et al, 2001]
b characterizes the nature and magnitude of
the dispersive processes
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Ranges of b
b ≥ 2 is reported to be “…equivalent to
the ADE…”
For b ≥ 2, the link between the dispersivity
(a = D/v) in the ADE and CTRW
dimensionless bb is bb = a/L
b between 1 and 2 reflects moderate nonFickian behavior
0 < b < 1 indicates strong ‘anomalous’
behavior
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Fitting Routines/Procedures
 http://www.weizmann.ac.il/ESER/People/Brian/CTRW/
 Three parameters (b, C, and C1) are involved. For the
breakthrough curves in time, the fitting routines return
b, T, and r, which, for 1 < b < 2, are related to C and C1
1
as follows:
T  LC1
b
1
C
r
L
C1
1
b
 L is the distance from the source
 Inverting these equations gives C and C1, which can
then be used to compute the breakthrough curves at
different locations. Thus C and C1 should be constants
for a ‘stationary’ porous medium
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Fits
1.0
Data
CTRW Fit
ADE Fit
C/C0
0.8
0.6
11 cm
17 cm
23 cm
0.4
0.2
0.0
0
20
40
60
80
100
Time (min)
120
140
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Fits
Length
CTRW Parameters
(cm)
T
b
r
C
11
44.1
1.68
0.11
4.01
1.92
17
68.5
1.72
0.092
4.03
2.09
23
94.2
1.73
0.084
4.10
2.23
C1
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Conclusions
CTRW models fit breakthrough curves
better
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