Geoscience Reference
In-Depth Information
Fig. 10.17 Examples of scaled water table
height
η/
B
as a function of scaled
distance
x
+
=
x
/
B
from the
drainage channel for different
values of the scaled recharge rate
I
+
=
I
c
/
k
0
and water level
D
c
+
=
D
c
/
B
in the open channel,
calculated with Equation (10.37) for
the conditions shown in Figure
10.16. Curve 1 represents
D
c
+
=
0
.
1 and
I
+
=
0
.
1; curves 2
and 3 represent
D
c
+
=
0, and
I
+
=
0.4
0.3
η
/B
1
2
0.2
3
0.1
0
1 and 0.01, respectively.
Each curve is a quarter ellipse.
0
.
−
0.2
0
0.2
0.4
0.6
0.8
1
x
+
evaporation from the water table or seepage through the bed, or any combination of
several of these rates. The following boundary conditions
η
=
D
c
x
=
0
(10.34)
∂η
∂
x
=
0
x
=
B
indicate, as before in Equations (10.2) and (10.6), that the water level in the channel
is
D
c
, and on account of (10.27) that the divide represents an impermeable boundary.
Integrating (10.33) twice, one obtains for the height of the water table
I
c
k
0
(2
Bx
2
x
2
)
D
c
η
=
−
+
(10.35)
Actually, this result can also be derived directly without formal recourse to the Boussinesq
equation, by simply observing that at any point
x
according to (10.27) the rate of flow
through the area
η
is [
−
η
k
0
(
∂η/∂
x
)]; this equals the rate of recharge at the surface, which
is given by [
x
)], and solution of this equality yields Equation (10.35).
Equation (10.35) can be generalized, to facilitate comparison with experimental
results and other theoretical approaches, by scaling the variables as follows
−
I
c
(
B
−
B
,
x
B
,
D
c
B
I
c
k
0
η
+
=
x
+
=
D
c
+
=
and
I
+
=
(10.36)
This transforms (10.35) into
η
+
=
I
+
2
x
+
−
+
D
c
+
1
/
2
x
2
+
(10.37)
This result is illustrated in Figure 10.17 for a few examples.
Application of Equation (10.35) to
x
=
B
, where the water table has its maximal
height
η
=
η
max
, yields immediately
I
c
η
D
c
k
0
B
2
2
max
=
−
(10.38)
Equation (10.38) was originally intended as a design equation, to determine the spacing
(2
B
) of drainage ditches or underground pipe drainage systems of agricultural lands;
