Geoscience Reference
In-Depth Information
auger hole. It can also be derived by conformal mapping (Polubarinova-Kochina, 1952).
However, separation of variables in the present coordinate system is probably the most
straightforward. The result for the hydraulic head can be written as follows
cos
n
π
D
c
2
D
cos
n
π
z
2
D
∞
8
D
(
n
π
)
2
h
=
D
−
n
=
1
,
3
,
5
,...
×
cosh
n
π
(
B
−
x
)
2
D
cosh
n
π
B
2
D
(10.7)
Outflow rate
The rate of flow into the open channel or water body at
x
=
0, expressed as volume of water
per unit time and per unit length of channel (i.e. per unit width of aquifer normal to the
main direction of the flow in the aquifer), can be derived by applying Darcy's law to the
solution (10.7), to wit
∂
∂
D
B
h
∂
x
h
∂
z
q
=−
k
0
dz
or
q
=−
k
0
dx
(10.8)
x
=
0
z
=
D
0
0
which yield in either case
cos
n
π
D
c
2
D
tanh
n
π
B
2
D
∞
8
D
(
n
π
)
2
(
−
1)
(
n
−
1)
/
2
q
=−
k
0
(10.9)
n
=
1
,
3
,...
The minus sign in front of the right-hand side indicates that the outflow is in the negative
x
direction. To allow comparison with other solutions and with experimental data, it is once
again convenient to scale the variables and express the result in dimensionless terms. The
form of Equation (10.9) suggests immediately the following
D
c
+
=
D
c
/
D
B
+
=
B
/
D
(10.10)
q
+
=
q
/
(
k
0
D
)
Thus (10.9) assumes the form
cos
n
π
D
c
+
2
tanh
n
π
B
+
2
∞
8
(
n
π
(
−
1)
(
n
−
1)
/
2
q
+
=−
(10.11)
)
2
n
=
1
,
3
,...
In many situations of practical interest, the aquifer thickness
D
is much smaller than its
horizontal dimensions, or
B
+
→∞
and the water depth in the adjoining open channel
is very small compared to the aquifer thickness, so that
D
c
+
→
0. These two conditions
simplify (10.11) to the following
1
−
−
8
π
1
9
+
1
25
−···
q
+
=
=−
0
.
742 45
(10.12)
2
The significance of (10.12) is that it represents the maximal outflow rate from a fully
saturated shallow extensive aquifer into an empty channel. As an aside, the sum inside the
brackets is also known as Catalan's constant, which equals 0.915 965.
