Effect of Scale - Transport and Fate of Chemicals in Soils: Principles and Applications - page 114

4.2.1.4 CDE with a Nonlinearly Distance-Dependent Dispersivity

If we remove the bar from x in Equation 4.21, we obtain the following non-
linearly distance-dependent dispersivity:

α= − xax D fr

1

()

(4.26)

4

2− D fr

where now a 4 is a constant with dimension L . Under this condition, the

dispersion coefficient D is given by:

D fr − 1

(4.27)

Dx () =α () xv = ax v

4

Accordingly, the governing equation now reads:

2

∂
∂

c
t

∂
∂ −− −

c
x

∂
∂

c
x

D − 1 D − 1

= avx 1( aD 1) x v

fr fr

(4.28) 4 4 fr

2

Equation 4.28 subject to initial and boundary conditions (Equations 4.17
through 4.20) was solved with the finite-difference method (see Appendix 4D).

4.2.1.5 Comparison of Models and Simulations

For the CDE with a time-dependent dispersivity, the magnitude of the disper-
sivity α increases with mean travel distance or time. Under this situation, the
dispersivity value remains constant over the entire spatial domain. In other
words, the entire medium is treated as a homogeneous system with a fixed con-
stant dispersivity value for each specific time. On the contrary, if one removes
the bar from the mean travel distance in the dispersivity-mean travel distance
relationship, the dispersive property of the medium is completely altered. As
a result, dispersivity becomes a function of distance from source instead of
mean travel distance or time. Under such conditions, the dispersivity is held
constant over the time being considered for any location but increases with
distance from the source where the solute is released. Therefore, the result-
ing parameter fields and thus the governing equations are quite different and
depend on whether the bar in mean travel distance is removed. An extra term
occurs for the distance-dependent dispersivity models in the governing equa-
tion to account for the dependency of dispersivity on distance.
Differences in the governing equations inevitably induce the differences
in the numerical scheme (finite difference equations, see the Appendices
for details). The dependence of dispersivity on time or distance is carried
over to the finite-difference approximations. Corresponding to time depen-
dence, index j , which indicates time domain discretization, occurs in the
finite-difference approximation for the governing equation with dispersiv-
ity as a function of mean travel distance. Accordingly, index i appears for