Geoscience Reference
In-Depth Information
2
∂
∂
=
C
t
∂
∂
C
z
−
∂
∂
C
z
l
D
l
v
l
(5.9)
e
2
Suppose that at
t
= 0 we add a tracer with concentration
C
0
and steady water velocity
v
to a soil column with
C
l
(
z
) = 0. The general analytical solution of Eq. (
5.9
) for such
initial and boundary conditions is (Radcliffe and Simunek,
2010
):
Czt
(,)
=
l
( )
2
2
2
z t
−
)
vt
D
1
2
vt
z
Dt
−
1
2
vz
D
vt
D
vz
D
−−
z t
Dt
C
exp
−
+
erfc
− + +
1
exp
erfc
0
π
4
Dt
4
4
e
e
e
e
e
e
e
(5.10)
Question 5.6:
Take the transport experiment of
Figure 5.2
with
v
= -2 cm d
-1
and
L
dis
=
10 cm. Calculate with Eq. (
5.10
) the solute concentrations at
z
= -100 cm for
t
= 30, 40,
50, 60 and 80 days. Compare your results to
Figure 5.5
. (Hint: Use Excel to calculate the
various terms of Eq. (
5.10
). Note that Excel cannot calculate the complementary error
function of a negative argument. In that case use erfc(-
x
) = 1 + erf(
x
)).
We might monitor the chloride concentration at the outlow end
z
=
L
of the col-
umn. The plot of outlow concentration versus time (
Figure 5.5
) is called an out-
low curve, or a “breakthrough” curve (representing the solute breaking through the
outlow end). The centre of each of the solute fronts, drawn for different values of
the dispersion length
L
dis
, arrives at the outlow end of the column at the same time
T
res
=
L
/
v
= 100 / 2 = 50 d, called the breakthrough time (
Figure 5.5
). When dis-
persion is neglected (
L
dis
= 0), all the solutes move at identical velocity
v
, and the
front arrives as one discontinuous jump to the inal concentration
C
0
at
t = T
res
. This
model, in which dispersion is neglected, is called the “piston low” model of solute
movement (Jury et al.,
1991
). The effect of dispersion on the breakthrough curve
is to cause some early and late arrival of solutes with respect to the breakthrough
time. This deviation is due to dispersion and a small amount of diffusion ahead and
behind the front moving at velocity
v
and becomes more pronounced as
L
dis
and thus
D
e
becomes larger.
According to the piston low model, the incoming solute replaces the water
initially present in the soil, or, equivalently, pushes this water ahead of the solute
front like a piston. Thus one may calculate solute concentration with the piston
low model by estimating how long it will take to replace the water between the
point of entry and the inal location. For example, we want to calculate the time
required to transport nitrate (a mobile ion) from the bottom of the root zone to
groundwater
L
= 10 m below. The average water content of the subsoil
θ
= 0.15
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