Geoscience Reference
In-Depth Information
Unit response and response to an arbitrary input
Equation (10.143) represents the outflow rate following complete saturation of the hillslope
aquifer. Therefore it is the unit response, that is Green's function, or the instantaneous unit
hydrograph for a sloping aquifer. Again, the input applied over the whole aquifer (in two
dimensions, that is per unit length of stream channel), causing the response (10.143), is
(
n
e
DB
x
)
δ
(
t
); per unit area of ground surface this is (
n
e
D
)
δ
(
t
). Therefore, the response of a
linearized hydraulic hillslope aquifer to a delta function input, that is the unit response, is
z
n
[1
−
2 cos(
z
n
)exp(Hi
/
2)]exp[
−
z
n
+
Hi
2
/
4
t
+
∞
−
2
k
0
η
0
cos
α
(
n
e
B
x
)
z
n
+
2
u
(
t
)
=
Hi
2
/
+
/
4
Hi
n
=
1
,
2
,
3
...
(10.146)
This means that, just like Equations (10.113)-(10.116) for the horizontal case, this solution
can also accommodate arbitrary inputs such as infiltration of precipitation or snowmelt
and leakage through the bottom bed, by convolution by means of (10.118). As before, this
arbitrary input per unit ground surface area can be taken as
I
=
I
(
t
), with dimensions [L
/
T],
applied uniformly over the aquifer, i.e. independently of
x
. However, because of the slope
some caution is called for; if
I
represents an input per unit horizontal area (as would be the
case for rainfall), in (10.118) it should be replaced by (
I
cos
α
).
The observations regarding the unit response function for the position of the water table,
made below Equation (10.118) for horizontal aquifers, are equally applicable to the present
case of sloping aquifers.
Example 10.5
Consider the same situation as described previously in Example 10.3. This problem concerns
the formulation of the outflow rate from a hillslope aquifer after a long-duration steady
precipitation event has ceased. The application of Equation (10.118) can follow along the
same steps as outlined in (10.122) and (10.123). If the steady input rate per unit horizontal
area is given by
I
c
, the input rate used in the convolution integral should be (
I
c
cos
α
); with
the unit response (10.146), this integral yields the following result
z
n
[1
−
2 cos(
z
n
)exp(Hi
/
2)]exp
−
z
n
+
Hi
2
/
4
t
+
∞
q
(
t
)
=−
2
B
x
I
c
cos
α
z
n
+
Hi
2
/
4
+
Hi
/
2
z
n
+
Hi
2
/
4
n
=
1
,
2
,
3
...
(10.147)
Observe that Equation (10.125) represents the special case of (10.147) for a horizontal
aquifer with
α
=
0. Equation (10.147) is illustrated in Figure 10.28, for different values of
Hi; to allow easy comparison, as before in (10.126) the flow rate is scaled with its initial
value, namely as
q
+
=
q
/
(
I
c
B
x
cos
α
).
Example 10.6
Consider in this example the same input sequence as given in (10.127). By means of the
unit response (10.146), the calculations can be carried out in the same way as in Example
10.4. As an illustration, the aquifer outflow rate for the case when
t
>
t
3
can be written as
