Effect of Scale - Transport and Fate of Chemicals in Soils: Principles and Applications - page 115

a distance-dependent dispersivity. Comparison between the finite-differ-
ence equations only is not enough to confirm the difference between the
two processes described by these two different governing equations. One
needs to show differences in BTCs as well as solute distribution along spatial
coordinate or flow direction to achieve a generalized conclusion. For com-
parison, both governing equations subject to the same initial and boundary
conditions were considered here. Because of the complexity of the difference
equations, it is difficult to assess convergence conditions. The time and space
increments were based on the governing equation with a constant dispersiv-
ity, which in our case equals the coefficient a in the dispersivity function. The
assessment of the convergence of numerical approximation was achieved
through mass balance calculations as well as the magnitude and oscillation
of the resulting numerical solutions for solute concentration.
The parameters used for our simulations are given in Table 4.1. Similar
parameter values were selected for both cases. The only difference lies in
that the variable mean travel distance for the time-dependent dispersivity
is replaced with distance from source to generate the distance-dependent
dispersivity. Two different column lengths were considered. One is 50 cm in
length, the other 100 cm. A longer pulse length is used for the 100-cm col-
umn to obtain comparable BTCs. BTCs as well as distribution profiles were
compared for a solute tracer under different scenarios.
Simulated BTCs with linearly time-dependent and distance-dependent α
are shown in Figure 4.1 for 50- and 100-cm soil columns. The BTCs from
either time-dependent or distance-dependent α appear somewhat similar.
Nevertheless, several distinct features are apparent. Based on our simula-
tions, the column length showed modest influence in the relative relation-
ship between BTCs of the media with a time-dependent dispersivity and
those with a distance-dependent dispersivity. For both long and short col-
umns, distance-dependent dispersivity resulted in earlier arrival of the BTC
than the time-dependent counterpart. Both BTCs exhibited similar leading

TABLE 4.1
Parameters Used for Simulations of Time-Dependent and Distance-Dependent
Dispersivities (α)

Time-Dependent
Dispersivity

Distance-Dependent
Dispersivity Parameter

Moisture content (cm 3 /cm 3 ) 0.40 0.40

Column length (cm) (short/long) 50.0/100.0 50.0/100.0

Water flux rate (cm/h) 5.0 5.0

Initial concentration (mg/L) 0.0 0.0

Concentration in input pulse (mg/L) 10.0 10.0

Pulse duration (hour) ( L = 50 cm/100 cm) 2.0/16.0 2.0/16.0

Dispersivity α (cm) 0.5 x 0.5 x