Agriculture Reference
In-Depth Information
σ
) exists with respect to a dis-
On the contrary, a variance of travel time (
t
σ
from a BTC as follows:
tance
x
. For example, one can estimate
t
∞
2
σ=
∫ −
∫
(
ttctdt
ctdt
)
()
2
0
t
∞
()
0
where
c
()
is
the solute concentration change over time at a fixed distance
x
(BTC), and
t
is the mean travel time and can be estimated as follows:
∞
=
∫
∫
tc tdt
ctdt
()
()
0
t
∞
0
σ
is associated with time (
t
) or mean travel distance (
x
), whereas
the variance of travel time
Therefore,
x
σ
is associated w
it
h distance (
x
).
A linear relationship between
t
σ
and
t
or
x
implies that a constant α for
all times up to the maximum time under consideration could be estimated
based on experimentally measured solute concentration profiles. Such an
estimate of α applies to all distances from the source. Conversely, a disper-
sivity α estimated at a certain distance based on BTCs means a constant dis-
persivity is needed to described these BTCs at that specific distance for all
times under consideration. If one should express the estimated dispersivity
from a BTC at a certain distance in the variance-time (mean travel distance)
format, it should be a straight line with a slope of 2α. Using Equation (4.3) to
reconstruct
x
σ
s
i
mply means that the obtained dispersivity α only applies
to that specific
x
or time
t
. Therefore, to reconstruct variance based on α
and distance
x
at which the dispersivity is estimated violates the assumption
based on which the dispersivity is obtained. Clearly the examples of confu-
sion described above stem from the ambiguous definition of scale.
Dispersivities measured at different distances from
a
given source are often
compared with dispersivities measured at different
x
or time
t
(Pickens and
Grisak, 1981a; Arya et al., 1988; Gelhar, Welty, and Rehfeldt, 1992; Neuman,
1990). In addition, variation in dispersivities is almost always attributed to
the scale under which it is estimated. The heterogeneity or type of forma-
tion of the geological systems is often ignored. The other problem is that the
integrity of transport processes is often ignored when a regression model is
applied to dispersivities estimated based on different transport processes
and from different media.
Dispersivity has been estimated in both laboratory and field studies because
of its importance to the governing CDE. A comparison is often made among
measured dispersivity values for different scales to support the finding that
dispersivity is scale dependent. No matter whether the dispersivities were
estimated for a distance from a source or for a certain time in terms of mean
travel distance, they were compared in terms of a quantitative index: scale
(Gelhar, Welty, and Rehfeldt, 1992; Neuman, 1990). As discussed above, the
definition of the term scale is not clear. Therefore, whether this comparison is
x
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