Geoscience Reference
In-Depth Information
Box 11.1 Representing Advection and Dispersion
The concentrations of any solute or tracer moving through the catchment system with the flow
of water are subject to
advection
and
dispersion
. Advection is controlled by the local mean
velocity of the flow, dispersion by the distribution of velocities around the mean value. Similar
theory is used to describe advection and dispersion for both surface and subsurface flows.
By analogy with molecular diffusion of concentration gradients in a solution, dispersion is
often described by the product of the gradient of concentration of a tracer or solute in the water,
C
, and a scaling constant called the dispersion coefficient,
D
. This is a simple representation of
the expectation that the steeper the concentration gradient, then the greater the net dispersive
flux of solute. If there is no concentration gradient (the tracer is completely and homogeneously
mixed in the water), there is no net dispersive flux even if the water is moving with a velocity
distribution. This first-order gradient relationship is here known as Fick's Law:
J
x
=−
D
dC
dx
(B11.1.1)
where
J
x
is the mass flux in direction
x
. The dimensions of
J
x
are [ML
−1
T
−1
] and of
C
are [ML
−3
],
so that
D
must have the dimensions [L
2
T
−1
]. Similar equations are used for heat transport in
response to a gradient of temperature and, as shown in Chapter 5, for the transport of water in
response to a gradient of hydraulic potential as Darcy's law.
If Fick's law is combined with a mass balance equation for the tracer or solute (this can be
derived using the control volume approach explained in Box 2.3):
∂C
∂t
=−
∂J
∂x
−
v
∂C
∂x
−
R
(
t
)
(B11.1.2)
where
v
is the mean flow velocity and
R
(
t
) is a general source or sink term that will be zero
for a fully conservative tracer. If
D
can be assumed constant and
R
(
t
) can be assumed to be
proportional to
C
, then subsituting for
J
from Equation (B11.1.1) gives
∂C
∂t
=
D
∂
2
C
v
∂C
−
∂x
−
˛C
(B11.1.3)
∂x
2
This is the one dimensional “advection-dispersion equation” (ADE). The first term on the
RHS represents the dispersive flux of solute; the second term, the advective flux; and the last
term, a loss proportional to concentration. The ADE is a partial differential equation in which
the dispersion coefficient
D
is, in general, different for different flow directions. This is often
represented in two or three dimensions, again for both surface and subsurface flows, in terms of
the “longitudinal dispersion coefficient” in the direction of the mean flow and the “transverse
dispersion coefficient” orthogonal to the direction of mean flow.
As a quick exercise, show how the ADE can be derived using the control volume method of
Box 2.3.
It can be shown that, for a conservative tracer, the ADE has an analytical solution for an
impulse input. For the one-dimensional case of only longitudinal dispersion in the direction
of the mean velocity and
˛
= 0, then:
2
A
(
Dt
)
0
.
5
exp
vt
)
2
4
Dt
M
(
x
−
C
(
x, t
)
=
−
(B11.1.4)
where
M
is the mass of solute added,
A
is the cross-sectional area involved in the flow and
is the constant pi (3.141592 ...).Thisisthesame form as that used to describe the normal
or Gaussian distribution in statistics (Figure B11.1.1). It implies that the impulse input will
gradually disperse as it moves through the system with a Gaussian distribution in the space
