Geoscience Reference
In-Depth Information
Figure B11.1.1
Solution of the advection-dispersion equation (Equation (B11.1.4)): (a) plotted against
distance in the flow direction; (b) plotted against time for a single point in space.
coordinate in the direction of flow. The further the travel of the tracer through the system, the
lower the peak concentration and the greater the spread but the cloud of tracer will continue
to have the form of a Gaussian distribution if the theory holds. It is also a linear solution,
so that twice the mass input in the pulse will result in a curve of the same form with twice
the concentration. For steady flow conditions, it can therefore be used as a linear transfer
function in the same way as the unit hydrograph is used in predicting runoff generation in
Section 2.2.
This form of this solution is quite revealing in that it implies that the dispersion is the result of
each tracer particle sampling all the various velocities in the distribution of the mean velocity
over time. With a large number of particles, and regardless of the underlying form of the actual
velocity, then the law of large numbers suggests that as all the velocities are sampled then
the resulting distribution will tend asymptotically towards a normal distribution. Before that is
the case, in the very early stages of the dispersion, the shape of the concentration curve will
be skewed in space because particles of tracer that start with fast velocities will not have had
time to sample the slow velocities. How long it will take for this initial mixing to be complete
depends on the structure of the flow domain and the accessibility of all the possible pathways.
The length scale of the initial mixing process is called the “mixing length”. Equation (B11.1.4)
is strictly valid only after this initial phase of mixing has been reached. Tracer experiments
have revealed that, in both surface and subsurface systems, the mixing length can be long,
sometimes very long, such that the dispersion coefficient that is calculated from the results
of an experiment can depend on the length scale of the experiment (e.g. the work of Gelhar
et al.
(1992) in groundwater systems, Beven
et al.
(1993) in soils and Green
et al.
(1994) in
rivers). Thus the skewness of the initial mixing phase can persist for long distances and long
times. However, in applying the ADE for prediction, it is often assumed that the initial mixing
is complete and the dispersion coefficient is constant or a simple function of velocity.
In fact, it is somewhat rare to measure the dispersion of a tracer as distributed in space at a
given time. Much more often the time distribution is measured as the cloud passes a particular
point in space. This depends on making only sequential measurements at a point rather than
simulataneous measurements at many points so is much easier. The resulting concentration
curve should then be expected to be skewed in time (even if it had the symmetric Gaussian
distribution in space), since particles arriving later will have been subjected to more dispersion
than those in the early part of the cloud. Because of incomplete mixing, however, that skewness
will be greater and will persist for longer, so that concentration curves remain skewed with
long tails.
The ADE can be modified to help simulate these skewed distributions by introducing
an exchange with an immobile store.
In soils and groundwaters,
this is known as the
