Effect of Scale - Transport and Fate of Chemicals in Soils: Principles and Applications - page 109

may increase with time as reported by several studies (Freyberg,
1986; Sudicky, Cherry, and Frind, 1983). Though dispersivity is esti-
mated according to a snapshot at a certain time t , such α represents
an average value for the time period up to the time t.
2. The second is a time-dependent dispersivity as described by fractal
and stochastic models (Zhou and Selim, 2002; Neuman, 1990; Arya,
1988). This relationship reveals that α is a continuous function of
time. For each time t , one has an instantaneous value of α for that
specific time.
3. The third is α as a continuous function with distance from source.
This type of relationship is not actually observed experimentally or
developed theo r etically. Rather it is derived by replacing the mean
travel distance x with x distance from source in the second case (Su,
1 9 95; Yates, 1990, 1992). Removal of the bar from mean travel distance
( x ) changes the relationship to a dispersivity-distance relationship.
One may argue that the first type of relationship is a description of
dispersivity versus scale and this derived relationship should hold
in terms of scale (distance from source is also a scale). As discussed
above, the distance from the source and mean travel distance are
not interchangeable. If dispersivity can vary with distance from the
source according to such a trend, theoretical support is needed to
support such an expression.
4. The fourth type of dispersivity versus scale is a distance-averaged
α versus the length scale in consideration (Zhang, Huang, and
Xiang, 1994; Butters and Jury, 1989; Burns, 1996; Khan and Jury,
1990; Pang and Close, 1999). In this case, dispersivity for differ-
ent column lengths or distances between observation wells and
an injection well is obtained through fitting of observed BTCs. In
some cases, dispersivity values were found to increase with the
length scale under consideration, that is, the column length or the
distance between the observation and injection wells. What such
a relationship means is that a constant dispersivity for the whole
column is needed to predict the breakthrough process. However,
for each column length, a different (constant) dispersivity must be
used. Thus, one cannot conclude from this observation that disper-
sivity is distance dependent. Actually, in the fourth case, for each
individual length, the basic assumption is that the dispersivity is
constant along the whole column up to the length considered. The
dispersivity value could thus be considered as an average or appar-
ent value.
Rather than one so-called generic dispersivity-scale relationship, we
emphasize here four distinct dispersivity-time or dispersivity-distance rela-
tionships. Among these four relationships, only the fourth could possibly be