Geoscience Reference
In-Depth Information
where
v
x
is the Darcian velocity (flux per unit cross-sectional area of saturated soil) measured with respect
to the downslope distance
x
(measured along the slope),
K
s
is the saturated hydraulic conductivity of
the soil (for the moment, assumed to be constant with depth of saturation) and sin
β
is the slope angle.
The kinematic wave approximation was first applied to saturated subsurface stormflow by Henderson
and Wooding (1964). Later, Beven (1981) showed that, at least for steeper slopes and high hydraulic
conductivities, it could be a useful approximation to a more complete description of shallow saturated
flow over an impermeable layer on a hillslope (see Box 5.7). This work was later extended to include
delays associated with the propagation of a wetting front into the soil before recharge starts and different
profiles of hydraulic conductivity with depth (Beven and Germann, 1982). The kinematic wave equation
is a better approximation if hydraulic conductivity increases with depth of saturation, as is the case in
many soils due to the increased macroporosity expected near the surface (Kirkby, 1988).
For the constant hydraulic conductivity case, an examination of the wave velocity is interesting. As
shown in Box 5.7, for saturated subsurface flow, the wave velocity is given by:
c
=
K
s
sin
α/ε
(5.13)
where
ε
is a storage coefficient representing the “effective” difference between soil water content and
saturation in the region above the water table. As for the
a
=
1 case for surface flow, if all the three
variables controlling the wave velocity were constant,
c
would be constant, but in practice
ε
is likely to
vary in magnitude both with depth of saturation and distance downslope. For wet soils,
ε
may be very
small. In this case, comparing the expression for
c
with that for the Darcy velocity
v
x
, the wave velocity
c
may be very much faster than the Darcy velocity. That is to say that disturbances to the flow, such as
new input of recharge, must propagate downslope faster than the Darcy velocity of the flow. The effects
of recharge will also propagate downslope faster than mean pore water velocity, i.e. the average flow
velocity through the part of the cross-section that is pore space rather than solids. This is given by
v
p
=
K
s
sin
α/θ
s
(5.14)
where
θ
s
is the porosity. The wave velocity will be faster than
v
p
since
ε
will always be smaller than
θ
s
.
This is one explanation why storm hydrographs that have an important contribution of subsurface
flow tend to show a high proportion of “old” water, even at peak flows (see Section 1.5). The effects
of recharge to the saturated zone move downslope with the wave velocity, which is faster than the pore
water velocity. The effect is that the discharge downslope rises more quickly than water can flow from
significant distances upslope, so that some of the water which flows out of the slope must be displaced
from storage in what was the unsaturated zone above the water table prior to the event. The same analysis
holds for more complex descriptions of subsurface flows, but the comparison of the
v
x
,v
p
and
c
velocities
in the kinematic wave description illustrates the effect quite nicely.
The THALES and TOPOG models (Grayson
et al.
, 1992a, 1995; Vertessy
et al.
, 1993; Vertessy and
Elsenbeer, 1999; Chirico
et al.
, 2003) both use the kinematic wave approximation on one-dimensional
sequences of hillslope elements representing a catchment. Both are based on the TAPES-C digital terrain
analysis package (see Section 3.7). THALES allows that each element may have different infiltration
characteristics (using either the Green-Ampt or Smith-Parlange infiltration models described in Box 5.2),
vertical flow in the unsaturated zone assuming a unit hydraulic gradient and using the Brooks-Corey
soil moisture characteristics (Box 5.4), and downslope saturated zone flow using the one dimensional
kinematic wave approximation (Box 5.7). The use of THALES in an application to the Walnut Gulch
catchment is described in Section 5.6. The TOPOG-Dynamic model, developed by CSIRO in Australia,
uses an analytical solution of the Richards equation to describe vertical flow in the unsaturated zone and
an explicit kinematic wave solution for the lateral saturated zone fluxes (Vertessy
et al.
, 1993). Later
versions include plant growth and carbon balance components for ecohydrological modelling of the
effects of land use change (Dawes
et al.
, 1997).
