Geoscience Reference
In-Depth Information
10.4.4 Incorporation of capillary flow zone
Attempts have been made to incorporate certain features of the partly saturated flow above
the water table into the linearized hydraulic approach. In principle, such an approximation
can be used for any type of free surface formulation, but until now mainly the linear
Boussinesq equation has been considered. Examples of such studies are presented in the
journal articles by Pikul
et al
. (1974) and Parlange and Brutsaert (1987).
10.5
KINEMATIC WAVE IN SLOPING AQUIFERS:
A FOURTH APPROXIMATION
Equations (10.26) and (10.27) show how the flow is driven by a pressure gradient,
as manifested by the inclination of the water table with respect to the underlying bed
∂η/∂
. The
pressure gradient term results in diffusive transport, which appears as a second derivative
in the Boussinesq equation; the bed slope term results in advective transport. For large
values of the slope, and thus of the hillslope flow number Hi, the effect of advection
overwhelms the diffusion. This can also be seen in Equations (10.134) and (10.138). In
the kinematic wave approach, Hi is assumed to be sufficiently large that the pressure
gradient term, leading to the diffusive term, can be simply neglected; thus the hydraulic
gradient in (10.128) is assumed to be equal to the bed slope sin
x
, and also by gravity, as manifested by the magnitude of the bed slope sin
α
α
, and (10.29) reduces
to a first-order linear equation, as follows
∂η
∂
k
0
sin
α
∂η
∂
I
n
e
t
−
x
=
(10.150)
n
e
This approach was briefly introduced by Boussinesq (1877) for steep slopes; in the
simple case of outflow without recharge
I
, he pointed out that, because (10.150) is in
the form of a total derivative
d
dt
=
∂η
t
+
∂η
dx
dt
=
0
,
∂
∂
x
a water table height
η
travels down the slope at a speed
dx
dt
=
−
k
0
sin
α
c
k
=
(10.151)
n
e
Conversely, to an imaginary observer traveling down the slope at a speed given by
Equation (10.151), it would appear that the height of the water table
does not change
with time. This result is not unexpected, and it is analogous with open channel flow, in that
the advectivity of the diffusion equation (10.136) and the celerity of the kinematic wave
(10.151) are identical, or
c
k
=
η
c
h
. In contrast to the kinematic wave in open channel flow,
Boussinesq's result (10.151) has the following two features. First, it can be seen that
c
k
is
independent of
travel at the same speed, and the water
table maintains its original shape as it moves downhill. For example, a rectangular input
pulse of precipitation, which enters the aquifer instantaneously at the soil surface, will
flow out from the aquifer into the stream channel as a time-delayed rectangular output
pulse. Second, it could be argued that Equation (10.151) does not really describe a wave;
η
. This means that all values of
η
