Agriculture Reference
In-Depth Information
Dk
−
k
terms of a power series of
xxx
(
−
)
with
k
=
0, 1, 2,
…
, then, we may reach
σ
:
the following expression for
x
1
−
DD
2
αε
x
0
c
2
σ=
(4.45)
x
1
−
DD
−
2
1(
−
DD
− αε
1)
x
0
c
The above equation shows that the relationshi
p
between the variance of
travel distance
σ
and the mean travel distance
x
is rather
co
mplex if
D
is
greater than 1. Approximately,
x
σ
grows proportionally to
x
raised to the
power of
D
. If
D
reduces to 1, the a
bo
ve equation reduces to the nonfractal
case, that is,
x
σ
grows linearly with
x
. On the other hand, if
D
is equal to 2,
Equation 4.45 becomes
x
2
σ=
α
ε−α
2
x
0
2
(4.46)
x
2
c
0
Obviously, Equation 4.46 is consistent with the Mercado model (Pickens
and Grisak, 1981a). Since
σ
is always positive, the denominator of the right
side of Equation 4.46 must be greater than zero. This restriction implies that
the fractal cutoff limit, if
D
= 2, must satisfy the following:
x
ε>α
2
(4.47)
c
0
Based on fractal geometry, Equation 4.47 also casts restriction on the travel
distance
x
. That is, the shortest apparent straight-line travel distance must at
least have the length of the fractal cutoff limit in order to use it to represent
the actual length of stream lines with a fractal dimension other than 1. If we
assume the apparent travel distance follows a normal distribution, then the
shortest apparent straight-line travel distance can be estimated by:
x
min
=−σ
x
3
(4.48)
x
where
x
min
stands for the shortest straig
ht
-line travel distance after time
t
with t
h
e mean travel distance given by
x
. Similarly, the mean travel dis-
tance
x
in Equation 4.45 must be large enough so that the second term in the
denominator of the right side of the equation is less than
1.
That is, to use
Equation 4.45 to show the time dependence of dispersivity,
x
must abide by:
−
−−
1(2)
D
(2
α
)
0
x
>
(4.49)
ε
(
) (
)
D
12
D
c
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