Agriculture Reference
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The exponent λ is introduced by Zhang (2000) to describe transport pro-
cesses in which the travel time of solute may increase with depth nonlin-
early. For the ETFM by Liu and Dane (1996), the relationships are
(
)
2
a
21
−
a
2
Et z
Et l
(, )
(,)
z
l
Var( ,)
Var( ,)
tz
tl
z
l
CV tz
CV tl
(, )
(,)
l
z
=
=
=
(4.33)
2
The value of parameter
a
in the above equation lies in the range between
0.5 and 1 (
≤≤
). Based on the observation of the above relationships
for different transfer function models, Zhang (2000) proposed a generalized
relationship of means, variances, and CVs of the travel time at depth
z
and
l
for a GTF:
0.5
a
1
λ
λ
2(
λ−λ
)
2
Et z
Et l
(, )
(,)
z
l
1
Var( ,)
Var( ,)
tz
tl
z
l
2
CV tz
CV tl
(, )
(,)
l
z
12
=
=
=
(4.34)
2
where λ
1
and λ
2
are parameters of the time moments.
The dispersivity can be estimated based on the CV at depth
z
and is given by:
z
2
CV(, )
2
α=
tz
(4.35)
Substituting the CV for the GTF (Equation 4.34) into the above equation gives:
α∝
+λ−λ
z
12(
)
21
(4.36)
λ−λ=−
,
dispersivity is constant with the distance (CDE). Otherwise, Equation 4.36
describes a distance-dependent dispersivity.
If the two parameters λ
1
and λ
2
in Equation 4.36 satisfy
0.5
2
1
4.3.2 Fractional CDE in Porous Media
An alternative to the scale approach discussed above is that of the fractional
convection-dispersion equation (FCDE) to describe anomalous transport
phenomena in aquifers (Benson et al. 2000a). Cushman and Ginn (2000)
showed that the FCDE is a special case of the convolution-Fickian nonlo-
cal advection-dispersion equation (ADE) proposed earlier by Cushman and
Ginn (1993).
The one-dimensional symmetrical FCDE reads:
α
α
∂
∂
=−
∂
c
t
c
x
1
2
∂
∂
c
∂
∂−
c
x
v
+
D
+
(4.37)
α
α
∂
x
(
)
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