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with
D
> 1 instead of growing linearly with time
t
as commonly observed
(Weeks, 1995). Therefore, the diffusion process described by Equation 4.29
is actually anomalous. To be specific, if
σ
is proportional to
t
D
with
D
> 1,
the diffusion process is referred to as superdiffusion.
x
4.5 Simulations
To show the effects of the fractal dimension on solute spreading, we con-
ducted several simulations. First, we chose a value for the true constant dis-
persivity
α
. Here
α
= 0.1 m was used. Then, we determine the fractal cutoff
limit
ε
according to Equation 4.47. In our example,
ε
is set to be 0.3 m. Now,
we can calculate the variance of travel distance at different time (correspond-
ing to the mean travel distance
x
) given the fractal dimension of the stream
lines is known. It should be pointed out that
x
should meet the requirements
of Equation 4.48 and the calculated shortest travel distance must exceed the
lower fractal cutoff limit. With
x
and
σ
known, the solute distribution profile
can be simulated if we assume the travel distance follows a normal distribu-
tion. Three examples are shown in Figures 4.7, 4.8, and 4.9. The figures show
that enhanced diffusion occurs with the increase of the fractal dimension. If
x
1.0
D
= 1.0
0.8
1.1
1.2
1.5
0.6
0.4
0.2
0.0
0
4
8
12
16
20
Travel Distance (m)
FIGURE 4.7
Comparison of solute profiles for systems with different fractal dimension at mean travel dis-
tance of 10 meters.
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