Page 221 - Soil and water contamination, 2nd edition

208 Soil and Water Contamination

The mass balance for a system in which both advection and dispersion occurs is simply a
combination of the advection equation with the dispersion equation, which gives the
advection–dispersion equation :

C C 2 C
u x D x (11.28)
t x x 2
Actually, this one-dimensional Equation (11.28) is the simplest form of the advection –
dispersion equation and describes longitudinal dispersion (in the direction of the water

flow). The dispersion coefficient D is therefore also referred to as the longitudinal dispersion
x
coefficient. The full three-dimensional advection–dispersion equation reads:
C C 2 C 2 C 2 C
u x D x D y D z (11.29)
t x x 2 y 2 z 2
where x refers to the average direction of water flow and D and D refer to the respective
y z
dispersion laterally and vertically perpendicular to the average flow direction, also referred
to as the transverse dispersion coefficients. Advection and dispersion can also be defined in
relation to a Cartesian coordinate system :

C C C C 2 C 2 C 2 C
u x u y u z D x D y D z (11.30)
t x y z x 2 y 2 z 2
In this case, the dispersion coefficient s D , D , and D are defined in relation to a grid
x y z
system. The three-dimensional version of the advection –dispersion equation is used when a
pollutant disperses from a single point into a large three-dimensional water body, such as
in groundwater, a deep lake or an estuary. For rivers , the two-dimensional version is usually
used at a local scale (for instance, to model the dispersal of sediments and pollutants over
floodplain areas) and the one-dimensional version is usually used for long stretches of rivers.
In the next section, the one-dimensional advection–dispersion equation is further explored to
model longitudinal dispersion in a river or groundwater.

11.3.3 Longitudinal dispersion
Longitudinal dispersion is the process of mixing in the direction of the average water flow,
so we need to consider both advection and dispersion. The analytical solution of the one-
dimensional advection–dispersion equation for a pulse injection is similar to that of the
2
2
dispersion equation, but in the right-hand side of Equation (11.24) the x replaced by (x-u ) :
x
M ( x u 2 / 4 D
C ) t , x ( e ) t x t x (11.31)
2 A D t
x
The centre of the mass travels at a velocity u and while the mass is travelling downstream,
x
the Gaussian curve becomes broader as a result of longitudinal dispersion . This can be
illustrated by the development of a spreading cloud moving downstream after a pulse release
of a tracer into a river (see Figure 11.6).
In groundwater, dispersion occurs in a similar manner, as can be illustrated with a simple
column experiment (Figure 11.7). The column is filled with a permeable sediment and the
test begins with a continuous inflow of a conservative tracer at a relative concentration of
C/C = 1 added across the entire cross-section at the inflow end. Figure 11.7 shows how
0
the relative concentration of the tracer varies with time at both ends of the column. The

10/1/2013 6:44:55 PM
Soil and Water.indd 220 10/1/2013 6:44:55 PM
Soil and Water.indd 220