Effect of Scale - Transport and Fate of Chemicals in Soils: Principles and Applications - page 110

characterized by the so-called dispersivity-scale relationship if one defines
scale as length of column or distance from source.

4.2 Case Studies
As pointed out earlier, distance-dependent dispersivity has been used rather
than time-dependent dispersivity in terms of dispersivity-mean travel dis-
tance relationship. In the pre vi ous section, we emphasized the differences
between mean travel distance x and distance from source x . Such differences
have been often ignored. Giv e n an expression for dispersivity-mean travel
distance, for example, α= , we inves ti gated the effect on BTCs and sol-

ute concentration profiles if the bar from x is removed and thus yields a dis-
persivity-distance relationship, that is,

0.1 x

α= . We used the i ni te-difference
approach to solve the CDE with a time-dependent (in terms of x ) or distance-
dependent dispersivity. We focused on the differences in BTCs and concen-
tration profiles resulting from different dispersivity models. We also propose
a procedure for obtaining a distance-dependent dispersivity model.
To compare the differences in transport processes in media with time-
dependent dispersivity and distance-dependent dispersivity, we have to
solve the CDE with time-dependent dispersivity or distance-dependent dis-
persivity. An analytical solution is available for several dispersivity-distance
models. Yates (1990) suggested an analytical solution for one-dimensional
transport in heterogeneous porous media with a linear distance-dependent
dispersion function. Su (1995) developed a similar solution for media with
a linearly distance-dependent dispersivity. Yates (1992) gave an analytical
solution for one-dimensional transport in porous media with an exponen-
tial dispersion function. Logan (1996) extended the work of Yates (1990, 1992)
and developed an analytical solution for transport in porous media with an
exponential dispersion function and decay. Transport in porous media with
time-dependent dispersion function has been studied (Zou, Xia, and Koussi,
1996). Generally, dispersivity is ex pressed directly as a function of time t
instead of mean travel distance x . As far as we know, analytical solutions
for trans p ort in porous media with time-dependent dispersion function in
terms of x are not available. Although analytical solutions for some cases are
available, they were not often used because they are difficult to use. Besides,
numerical evaluation is often necessary to compute the analytical solution.
In this investigation, we solved the CDE with time-dependent dispersivity
or distance-dependent dispersivity using numerical methods. As an illustra-
tion, we chose the linear or power law form as a model for time-dependent
dispersivity. The advantage of using a power law model lies in that it will
either recover a linear model or reduce to a constant (homogeneous case) if
we set the exponent term to proper values.

0.1 x