Effect of Scale - Transport and Fate of Chemicals in Soils: Principles and Applications - page 108

reasonable rem ai ns open. Although both distance from a source x and mean
travel distance x share a common dimension of length, these two terms are
quite different according to their physical meanings. Conceptually, disper-
sivities obtained under different mean travel distance and different distance
are not comp a rable in terms of length scale, which is the quantitative value
of either x or x . The reason is that the underlying assumptions for the estima-
tion of dispersivity with respect to distance and those with respect to mean
travel distance are actually exclusive to each other as discussed above.
When comparisons among dispersivities are made, values corresponding
to different scales are often from different media, that is, porous media and
fractures, instead of the same medium. This is understandable because of
difficulties in obtaining dispersivity information from the same medium
with scales varying from tens of centimeters to hundreds of meters. Beyond
the comparison of dispersivity is the regression of dispersivity versus scale.
Based on dispersivity data available in the literature, several studies have
been conducted to develop a functional relationship between dispersivity
values and associated scale based on regression analysis (Arya et al., 1988;
Neuman, 1990; Xu and Eckstein, 1995). Furthermore, a universal dispersiv-
ity value for a specific scale is determined based on the function obtained.
Mathematically, this kind of regression analysis is feasible. However, rigor-
ous theoretical development is needed in support of the regression analysis.
Curre nt ly, several models are available to describe the relationship between
α and x . For example, Zhou and Selim (2002) developed a model to describe
time-dependent dispersivity. In addition, others developed stochastic models
that describe dispersivity-mean travel distance based on heterogeneity of the
hydraulic conductivity in porous media (Neuman and Zhang, 1990; Zhang
and Neuman, 1990). These models are commonly used to support regression
of measured α versus different scales representing several distances and
mean travel distances over different sites (Neuman, 1990; Wheatcraft and
Tyler, 1988). To justify the regression of data from different sites, however, a
separate theory must be developed. In fact, both fractal and stochastic mod-
els describe an instantaneous relationship between dispersivity and time or
mean travel distance. Dispersivity takes a distinct value for a specific time.
The measured α available in the literature, however, is an apparent or aver-
age α over the period up to the time at which the dispersivity is estimated.
Thus, the estimated dispersivity under such condition is not actually appli-
cable to the models discussed above.
Discrepancies regarding use of the terms scale and scale effects are mainly
caused by the ambiguous definition of the term scale. In fact, we are dealing
with four types of relations between dispersivity and time and/or distance
rather than one universal dispersivity-scale relationship.
1. The first is a time-averaged dispersivity versus time (or mean travel
distance). Here apparent dispersivity is estimated at different dis-
crete times or mean travel distances. When plotted, dispersivities