Geoscience Reference
In-Depth Information
In (10.138) Hi can be called the groundwater hillslope flow number, which is defined
as
B
x
tan
α
Hi
=
(10.139)
η
0
Equation (10.138) is in the form of the advection diffusion equation; because it is dimen-
sionless it has a unit diffusivity. From a comparison between Equations (10.95) and
(10.138), it can be seen that the dimensionless parameter Hi, represents the relative mag-
nitude of the slope term, that is the effect of gravity, versus the diffusion term. This ratio
increases with the slope
D
of the aquifer. For large
values of Hi the diffusion term, i.e. the first term on the right of Equation (10.138), is
negligible, and the kinematic flow approximation (see Section 10.5) is valid. For small
values of Hi (i.e. a small slope
α
and with the “shallowness”
B
x
/
α
/
B
x
)) the
problem can be treated as one of horizontal flow, and the solution can be approximated
by (10.106).
and/or a relatively deep aquifer with large
D
Solution
The boundary conditions (10.133) can be written in dimensionless form
η
+
=
0
x
+
=
0
t
+
≥
0
∂η
+
∂
x
+
(10.140)
+
Hi
η
+
=
0
x
+
=
1
t
+
≥
0
η
+
=
1
≤
x
+
≤
1
t
+
=
0
0
The solution of (10.138), subject to (10.140) can be obtained by means of the Laplace
transform. This can be shown (Brutsaert, 1994) to be
η
+
∞
2
z
n
[exp(
−
Hi
/
2)
−
2cos(
z
n
)]sin(
z
n
x
+
)exp
−
z
n
+
Hi
2
/
4
t
+
+
(1
−
x
+
)Hi
/
2
z
n
+
Hi
2
/
4
+
Hi
/
2
n
=
1
,
2
,
3
...
(10.141)
In (10.141)
z
n
is the
n
th root of
z
tan(
z
)
=
(10.142)
−
/
Hi
2
which has an infinity of roots, say
z
1
,
z
2
,...
z
n
,...;Equation (10.142) arises in many
problems and its roots
z
n
have been tabulated (see Carslaw and Jaeger, 1959, p. 492;
Abramowitz and Stegun, 1964, p. 224). Note that for very small values of Hi, one has
z
n
=
[(2
n
−
1)
π/
2] which is in agreement with (10.101) for the horizontal case. For large
values of Hi, the roots approach
z
n
=
(
n
π
).
Outflow rate
The outflow rate can be obtained by applying (10.132) to (10.141) at
x
=
0. This yields
z
n
[1
−
2 cos(
z
n
)exp(Hi
/
2)] exp
−
z
n
+
Hi
2
/
4
t
+
∞
q
+
=−
2
z
n
+
2
(10.143)
Hi
2
/
4
+
Hi
/
n
=
1
,
2
,
3
...
