Effect of Scale - Transport and Fate of Chemicals in Soils: Principles and Applications - page 120

CDE predicts a linear increase of travel time variance with distance. With
an approach similar to that used by Mercado (1967), Jury and Roth (1990)
developed a stochastic-convective model. The stochastic-convective pro-
cess assumes that the solute moves at different velocities in isolated stream
tubes without lateral mixing. Use of log normal distribution of travel time
results in a convective log normal transfer function model (CLT). The CLT
model describes solute transport characterized by a quadratic increase in the
travel time variance with depth. However, the travel time variance is often
reported to increase nonlinearly with distance because of heterogeneity of
the media (Zhang, Huang, and Xiang, 1994). To account for the nonlinear-
ity in the relationship between travel time variance and distance or depth,
Liu and Dane (1996) proposed an extended transfer function model (ETFM).
They introduced an additional parameter to represent the degree of lateral
solute mixing. The ETFM can be considered as a transition between the CDE
and the CLT. Zhang (2000) also proposed an extended convective log normal
transfer function model (ECLT). Meanwhile, he attempted to unify all types
of transfer function model with a generalized transfer function model (GTF).
More important, Zhang (2000) showed that the distance-dependent disper-
sivity model could be associated with the parameters of the GTF.
The mean and variance of the travel time are two important characteris-
tic elements in transfer function theory. A third parameter related to scale
effects is the coefficient of variation (CV) of the travel time. The squared CV
at depth z is given by:

Var( ,)
(, )

tz
Et z

2

CV tz (, ) =

(4.29)

2

where Var( t,z ) and E ( t,z ) are the variance and mean of the travel time at depth
z . For the CDE, the means, variances, and CVs of the travel time at depth z
and l are related by:

2

Et z
Et l

(, )
(,)

z
l

Var( ,)
Var( ,)

tz
tl

z
l

CV tz
CV tl

(, )
(,)

l
z

= = =

(4.30)

2

For the CLT, similar relationships can be established as:

2

2

Et z
Et l

(, )
(,)

z
l

Var( ,)
Var( ,)

tz
tl

z
l

CV tz
CV tl

(, )
(,)

= = = 1

(4. 31)

2

For the ECLT (Zhang, 2000), the relationships are

λ 2 λ

2

Et z
Et l

(, )
(,)

z
l

Var( ,)
Var( ,)

tz
tl

z
l

CV tz
CV tl

(, )
(,)

= = = 1

(4.32)

2