Geoscience Reference
In-Depth Information
(b)
in this form implies that the aquifer has a uniform drainable porosity that may however
depend on
t
;
(c)
in this form implies that the aquifer has a constant and uniform hydraulic conductivity;
(d)
yields Laplace's equation under steady flow conditions;
(e)
is based on the assumption that the specific flux is not a function of
x
;
(f)
is applicable only when the horizontal length scale of the aquifer is much larger than the
vertical scale.
10.17
Multiple choice. Indicate which of the following statements are correct. The recession curve
of base flow, or drought flow, as a function of time observed at a streamflow gauging
station:
(a)
is relatively (i.e. as compared with storm runoff) insensitive to the temporal storm and
rainfall pattern over the basin;
(b)
depends primarily on the characteristics of the effluent ground water aquifers in the basin;
(c)
is often plotted as a straight line on log-log paper for engineering applications;
(d)
can sometimes be used to separate the amount of storm runoff due to a given storm, from
the observed hydrograph;
(e)
may conceivably be affected by the rate of evapotranspiration from the basin.
10.18
Derive the solution (10.7) from Laplace's equation (10.5) for the boundary conditions (10.6). Hint.
Use the method of separation of variables in a manner analogous to that leading to the solution
(10.105).
10.19
Derive the expression (10.9) for the steady outflow rate from a saturated unconfined riparian
aquifer, from the solution (10.7). Use either the first or the second of (10.8).
10.20
Show how Equation (10.72) is obtained from (10.70).
10.21
Determine the ratio of the second over the first term in the scaled outflow rate (10.112) obtained
in the linear solution. For what value of the scaled time
t
+
, does it become smaller than
1%?
10.22
Consider an extensive unconfined aquifer on a horizontal impermeable layer bounded by a straight
open channel (similar to Figure 10.20, with
B
=∞
); after the channel and the aquifer have both
been dry (empty) for a very long time, this channel is suddenly (to alleviate flooding elsewhere)
at
t
9
D
. The flow in the aquifer is assumed to be governed by
Boussinesq's equation (10.30). (a) State three boundary conditions for the Boussinesq equation
(one of which is an initial condition), describing this situation. (b) Suggest a method of reducing
the partial differential equation to an ordinary differential equation, which is permitted by these
boundary conditions. (c) Give the functional relationship between
x
and
t
(except for one or more
undetermined constants) for a given specific value of
η.
In other words, if the solution of the
problem
η
=
η
(
x
,
t
) is known and if
η
is given a certain value, what is the remaining relationship
x
=
x
(
t
)? Note. Do not try to find that solution; just assume that it is known and use it as such.
=
0 filled up to a level
D
c
=
0
.
10.23
In Example 10.4, expressions are presented for
q
=
q
(
t
), over the time intervals
t
1
<
t
<
t
2
and
t
>
t
3
in Equation (10.129) and (10.131), respectively. Derive the expression for the time
interval
t
2
<
t
<
t
3
.
