Agriculture Reference
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400
300
200
100
0
0
100
200
300
400
500
Time (day)
FIGURE 4.6
Comparison among different models: variance versus time. Data from Cape Cod field experi-
ments (Garabedian et al., 1991).
4.4 Fractals
Another alternative to describe a time-dependent dispersivity in terms of
mean travel distance
x
using fractal models has been proposed. Zhou and
Selim (2003) proposed a modified fractal as follows. If one assumes a Fickian
process occurs when a particle moves along the fractal tortuous streamline,
the variance of travel distance at a certain time
t
is given by:
2
σ=α
2
x
(4.43)
x
0
fr
where
α
is a constant, time-independent true dispersivity, and
x
fr
is the
mean path length of all possible fractal streamlines traveled by solute par-
ticles till time
t
. Taking expressions on both sides of Equation 4.14 and sub-
stituting it in the above equation yields:
σ=αε
−
2
1
DD
2
Ex
[
]
(4.44)
x
0
c
where
Ex
[]
stands for the mean value of
x
D
. In fact,
Ex
[]
is the
D
th-order
moment of the travel distance. The difficulty her
e
lie
s i
n that
D
is a fractional
number rather than an integer. If we write
x
D
as
xxx
D
[
−−
and expand it in
(
)]
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