Effect of Scale - Transport and Fate of Chemicals in Soils: Principles and Applications - page 105

not with mean travel distance x , can one conclude that the system has a scale
effect? Or, can one ignore that the system has a scale effect because α does
not increase with distance from source x ? We are not able to answer these
questions because there is no unique concept of the scale effects.
4.1.1 Mean Travel Distance and Distance from Source
The above-mentioned confusion stems from the ambiguous meaning of the
term “scale.” When one refers to scale, one generally means the space scale
such as the length of a soil column, the dimension (length and width) of an
aquifer in a field transport experiment, or the area of the aquifer covered by
a monitoring instrument. It should be pointed out that scale is a physically
measurable quantitative property. In other words, a scale is a characteris-
tic index associated with transport processes. For laboratory experiments, a
scale could be the length of the soil column used. For field transport experi-
ments, the potential scale could thus be infinite. Analogous to laboratory
experiments, under certain circumstances, the distance between an obser-
vation well and an injection well could also be taken as a scale. In general,
a space coordinate could be treated as a scale as long as the origin is set at
the inlet of the soil column or the injection well. Obviously, scale is associ-
ated with distance from a source or a space coordinate and has nothing to
do with time. In other words, scale should depend only on distance b ut not
on time. Theref o re, it is not appropriate to define scale in terms of x . Mean
travel distance x is actually not a distan c e from source or a space coordinate
in the physical is e nse. On the contrary, x is a function of time or an expres-
sion for time. If x is taken as a scale, we encounter incorrectly the situation
that scale varies with time. Like a space coordinate, scale is an independent
variable and should not depend on any other variables. In laboratory experi-
ments, this concept is straightforward and easily understood because a soil
column length is the obvious parameter that can be associated with scale.
However, in field experiments, we have difficulties in u si ng this concept.
In fact, several researchers use the mean travel distance x as a scale indis-
criminately. Mean travel distance is theoretically where the solute front is at
a certain time t . If one considers x as a scale, it turns out that the scale for
a field experiment does not exist prior to t he application of a solute pulse.
Besides, because of the linear increase of x with time, scale is also a linear
function of time in this case. These two deductions definitely do not sound
correct. As mentioned above, scale is a characteristic parameter for a field site
and is determined for a given system, for example, an experimental setup,
regardless whether a transport experiment is actually conducted or when a
transport experiment is initiated. If we have observation wells at different
distances from an injection well, potentially we can say that this ex pe riment
is monitored at different scales. If the scale is defined based on x , one is
unable to predetermine the scale without p r ior knowledge of the schedule of
the experimental sampling scheme. What x tells us is how far a solute front