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vances after release. So the validity of determining a scale based solely on
x
is questionable.
The term scale is frequently misrepresented because it often refers to the
mean travel distance as well as the distance from the source. However, this is
some
w
hat misleading and causes confusion. One sourc
e
of confusion is that
both
x
and
x
are used interchangeably. In other words,
x
can be replaced by
x
and vise versa. This type of interchange occurs quite often in the literature.
For example, E
qu
ation [16] in Wheatcraft and Tyler (1988) is a relationship
between α and
x
. However, when this relationship was cited by Su (1995), it
is converted to a relationship between α and
x
(see also Yates, 1990, 1992). We
recognize that a relationship between dispersivity α and the space coordinate
x
may exist. However, we want to emphasize here that one cannot derive
a
relationship between α and
x
on the basis of a relationship between α and
x
.
Another example of confusion is the reconstruction of the var
i
ance of travel
distance
σ
based on dispersivity α at a distance
x
rather than
x
. Pickens and
Grisak (1981b) reconstructed the variance-mean travel distance relationship
based on dispersivity values measured at different distances by Peaudecerf
and Sauty (1978). Sudicky and Cherry (1979) plotted reconstructed variances
based on dispersivity from BTCs at different distances and those estimated
from snapshots at different times in one graph to discuss the scale effects.
The reconstruction of variance can be described as follows. Based on the
assumption of homogeneous media, the variance of travel distance increases
linearly with time such that
x
σ=
2
Dt
(4.1)
x
where
D
is the dispersion coefficient.
D
can be expressed as:
Dv
=α
(4.2)
where
v
is the pore water velocity. We thus have
σ=α
(4.3)
2
x
x
The
relationship given by Equation 4.3 is often used to estimate α given
both
x
and
σ
. However, we found that this relationship is also employed
to reconstruct
x
σ
(Pickens and Grisak, 1981b; Sudicky and Cherry, 1979).
A dispersivity α measured at distance
x
is substituted into Equation (4.3).
Therefore, the actual equation implemented reads:
x
σ=α
2
x
(4.4)
x
Thus, implicitly, an interchange fr
o
m
x
to
x
is carried out. It should be
pointed out that
σ
is related onl
y
to
x
and not
x
. In other words, for a given
time
t
or mean travel distance
x
, one can compute
x
σ
given α is known.
x
However, no variance of travel distance
σ
exists for a given distance
x
. The
reason is that one needs a set or a collection of points to estimate a variance.
x
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