Geoscience Reference
In-Depth Information
Figure 11.8
A representation of particle velocities chosen from an exponential distribution for each layer in a
way consistent with the nonlinear profile of hydraulic conductivity at G ardsj on (after Davies et al., 2011, with
kind permission of John Wiley and Sons).
package has a particle tracking component to predict transport (see Appendix A). Often these particles
are advected only with the local mean pore water velocity, but dispersion can also be added by a random
velcoity component chosen from a specified distribution around the mean.
The multiple interacting pathways (MIPs) model of Davies
et al.
(2011) is different in that the particle
formulation is chosen to represent both the flow and the transport. The representation of celerity effects
must therefore result from the movement of the water particles as the slope wets and dries. The initial
implementation of the model assumes that flow in the unsaturated zone is vertical and in the saturated
zone is parallel to the surface slope (the kinematic assumption). This was illustrated in Figure 9.4. The
step equation for each particle at each time step in the saturated zone can then be expressed as:
x
i,t
+
t
s
=
x
i,t
+
v
tan
βt
s
(11.7)
where
t
s
is the time step,
x
is horizontal distance,
v
is a randomly chosen Lagrangian velocity defined for
a unit hydraulic gradient and
β
is the local slope angle. The choice of velocity distribution is therefore
an important consideration in setting up the model. It was found for the case at Gardsj on that this could
be done in a way compatible with the field information on the strongly nonlinear profile of hydraulic
conductivity. The velocity distribution at each level in the profile should integrate to be consistent with
the conductivity at that level. By assuming an exponential velocity distribution for each level and that the
increase in conductivity is a result of an increase in larger pores, functional relationships were developed
between porosity, layer conductivity and velocity distribution (Figure 11.8).
Applying these relationships without further calibration resulted in the predicted hydrograph for the
Gardsj on slope in Figure 11.9a. Reproducing the tracer concentrations proved to be much more difficult,
but thinking directly about velocities within the MIPs framework turned out to be valuable. The tracer was
input to the system at a depth of 60 cm. 96% of the tracer was recovered at the outlet, so there was little
uncertainty about the tracer mass balance in this case. If it was allowed to continue to move at the 60 cm
