Geoscience Reference
In-Depth Information
“mobile-immobile model” (van Genuchten
et al.
, 1989) and in rivers as the “transient storage
model” (Bencala and Walters, 1983). The general form is:
∂C
∂t
=
D
∂
2
C
v
∂C
−
∂x
−
˛
(
C
−
C
s
)
(B11.1.5)
∂x
2
ˇ
dC
s
dt
=
˛
(
C
−
C
s
)
(B11.1.6)
where
C
s
is the concentration in the immobile or transient storage,
˛
is the mass exchange
coefficient between the main flow and the storage zone and
ˇ
is the ratio between the
volumes of the storage zone and the main flow. This also has a linear analytical solu-
tion for predictions beyond the mixing length (see, for example, Worman, 2000; De Smedt
et al.
, 2005).
The ADE can also be solved numerically, using similar methods to those described in Box 5.3.
Care must be taken in this case, however, so as not to introduce too much
numerical disper-
sion
into the solution due to the numerical approximations to the partial differentials. This
might cause unrealistic dispersion in the predictions of transport relative to that expected for
a given value of the dispersion coefficient. It can also induce oscillations in the solution in
the vicinity of sharp concentration gradients. There is an extensive literature about the so-
lution of such equations, which generally require higher order numerical approximations to
be used (see, for example, Pinder and Gray, 1977). Fortunately, with the greater computer
power now available, this is not such a problem and in real applications the problem of
defining the appropriate sources and boundary conditions for the transport may still be much
more significant.
There is an alternative dispersion model that has been derived directly from data analysis
of tracer concentrations in rivers and soils. This is the aggregated dead zone (ADZ) model
(Beer and Young, 1983; Wallis
et al.
, 1989; Green
et al.
, 1994). It is also a linear model and
takes the form of the simple transfer functions described in Section 4.3 and Box 4.1. In the
ADZ model, advection is treated as a pure time delay and dispersion by routing through one
or more linear stores. It generally produces very good fits to available tracer data, includ-
ing the long tails that are often a feature of observed tracer curves. If both input and output
concentrations to the flow domain can be considered well mixed, for example in a river
reach where tracer has been added some way upstream of the input to the reach, then only
a single linear store (a first-order transfer function) should be necessary. In the intial stages
of mixing, or where there is mixing with a high volume storage, a higher order model might
be required.
The dispersion in the ADZ model is controlled by the mean residence time of the linear
store (or stores). In the first-order case, under steady flow conditions, this has a very simple
intepretation. The ADZ model is best understood in terms of simple “lag and route” concepts.
In any transport problem, there is a lag between the time at which the solute concentration
starts to rise at the start of a reach and when it starts to rise at the end of the reach. This, in
the ADZ methodology, is called the advective time delay,
. It is expected to decrease with
increasing discharge in the reach. The dispersion of the input plume in the reach is modelled as
a linear transfer function. In the simplest first-order case, the transfer function is equivalent to
a linear store with a mean residence time,
T
. The mean travel time in the reach is then
(
+
T
)
.
Given a steady discharge,
Q
, the volume of this effective mixing store is then
V
e
=
Q
∗
T
.The
total volume in the reach involved in the transport process is
V
∗
[
+
T
]. The ratio
V
e
/V
is
called the “dispersive fraction” (
DF
). It can also be defined in terms of the adective travel time
and mean residence time as
T/
(
+
T
)
.
In some analyses of river tracing experiments it has been shown that a higher order trans-
fer function can give a slightly better fit to the observed tracer experiments (e.g. Wallis
et
al.
, 1989; Young and Wallis, 1993). This suggests that there might be different mechanisms
affecting the dispersion with rather longer mean residence times than in the bulk flow and its
