Geoscience Reference
In-Depth Information
Darcy's law, with a hydraulic head gradient (
D
c2
−
D
c1
)
/
B
, and an average thickness
of the flow zone given by (
D
c2
+
2; this suggests that when
D
and
D
c
have nearly
the same value, a good approximation should be
D
c1
)
/
2. The second case is
encountered when the depth in the channel is negligible so that
D
c
=
η
0
=
(
D
+
D
c
)
/
0, and
D
is the
only remaining parameter that can be used to characterize the average thickness of the
flow zone. For such situations it is convenient to put
η
0
=
pD
(10.93)
in which
p
is a constant adjustment parameter to compensate for the linearization. The
linearized solution
a
)
1
/
2
for the short-time unsteady outflow rate by Edelman,
mentioned earlier in Section 10.3.4, shows that it can simulate the exact result (10.63)
provided
p
=
(4
p
/π
=
0.3465. This suggests that in the initial stages
p
probably lies in the vicinity
of 1
4 is applicable only for small to intermediate
times, at most; for larger times, as the water table height
/
3. However, this value of
p
=
0
.
3
∼
0
.
η
continues to decrease, the
η
0
is likely to become smaller as well.
optimal value of
10.4.2
Flow from a horizontal aquifer
Consider again the standard case of outflow from an initially saturated aquifer, after the
cessation of rainfall or recharge, as described by boundary conditions (10.50). In the
linearized system the governing differential equation is now (10.88).
Similarity considerations
As before in (10.66), it is convenient to scale the variables. The form of (10.88) and of
(10.50) suggest that this be done as follows
x
+
=
/
x
B
t
+
=
[
k
0
η
0
/
(
n
e
B
2
)]
t
(10.94)
η
+
=
(
η
−
D
c
)
/
(
D
−
D
c
)
These scaled variables allow the differential equation (10.88) to be written as
∂η
+
∂
=
∂
2
η
+
(10.95)
x
2
+
t
+
∂
Similarly, the boundary conditions (10.50) can be expressed in terms of the scaled
variables as follows
η
+
=
0
x
+
=
0
t
+
≥
0
∂η
+
∂
(10.96)
=
0
x
+
=
1
t
+
≥
0
x
+
η
+
=
1
0
≤
x
+
≤
1
t
+
=
0
