Geoscience Reference
In-Depth Information
takes over, the bulk of the water left in the aquifer tends to move down more as a translatory
wave, causing the appearance of a “pause” in the hydrograph during the transition between
the two regimes. The occurrence of this pause is related to the time when the height
η
+
of
the water table approaches zero at the divide, where
x
+
=
1. The rate at which the water
table height approaches zero at the divide, like the strength of the pause illustrated in Figure
10.27, depends mainly on the magnitude of the hillslope flow number Hi. Actually, it can be
seen that Equation (10.141) predicts that at the divide
η
+
can never become zero, and that
with time it can only approach it exponentially. This is somewhat counterintuitive, because
the hydraulic groundwater approach is based on the assumption that the water table is a
true free surface and thus a sharp interface; therefore physically, there is no reason why
such a sharp interface
η
+
(
x
+
,
t
+
) would not be able to become 0 at
x
+
=
1, and it would be
expected that after this occurs, the point of
η
+
=
0 would slide down along the bottom of
the aquifer from
x
+
=
0. Originally, the fact that (10.141) does
not predict such a sequence of events was thought to be caused by the shortcomings of the
hydraulic approach with the Boussinesq equation; it was recently (Stagnitti
et al
., 2004)
shown, however, that this is not the case, and that it is in fact the result of the linearization
of that equation. This should not be surprising, as it is well known that solutions of the
linear diffusion equation usually do not exhibit sharp fronts, but rather long exponential
tails; other examples of this or similar features can be found in applications of the linear
diffusion equation in open channel flow (Equation (5.95) and Figure 5.9) and in infiltration
(Equation (9.56) and Figure 9.10). Nevertheless, in the present context, the inability of
Equation (10.141) to allow the water table height
η
+
=
0 to move down along the bottom
of the aquifer past
x
+
=
1, may not be a crucial issue in hillslope hydrology. In real aquifers,
the falling water table is not a sharp drying front, and the flow is more closely described by
the Richards equation, than by the Boussinesq equation. This means that the solution of the
linearized formulation, viz. Equation (10.141), with its asymptotic approach to zero, may
not necessarily provide a worse approximation than the sharp interface description. Still,
regardless of these shortcomings, the analysis of the linearized problem shows how with
increasing Hi, the diffusive aspects of the phenomenon gradually become less important,
and perhaps even irrelevant in the description of hillslope flows in actual catchments; this
suggests that for large values of Hi, say in excess of 10, it may be preferable to use the
considerably simpler kinematic approach outlined below in Section 10.5.
The short time limit of Equation (10.143) can be shown (Brutsaert, 1994) to be simply
1 in the direction of
x
+
=
q
=−
(
k
0
η
0
n
e
cos
α/π
)
1
/
2
Dt
−
1
/
2
(10.144)
This is Equation (10.59) with (10.61), as expected, with a value of the constant
a
given by
a
=
[4
η
0
cos
α/
(
π
D
)]
1
/
2
(10.145)
Actually, Equation (10.144) is the exact solution of (10.138) without the second term on
the right, subject to (10.133), in which
B
→∞
and sin
α
=
0. This means that after a
sudden cessation of the water supply (e.g. from rainfall) at the soil surface, the outflow
proceeds initially as a diffusion phenomenon in an infinitely long aquifer. This is not unex-
pected. Recall that a similar type of behavior with initial
t
−
1
/
2
dependency occurs in other
phenomena described by the advection diffusion equation as well; one example, covered
in Chapter 9, is vertical infiltration of ponded water into a dry soil profile. Note also, that
Equation (10.144) with (10.145) for
α
=
0 is the solution for drainage from an infinitely
long horizontal aquifer proposed by Edelman (cited by Kraijenhoff, 1966).
