Geoscience Reference
In-Depth Information
with (10.132). Moreover, in hilly terrain, torrential streams tend to be shallow, and they
usually have no effect on the groundwater flow in the adjoining hillslopes, so that it is
safe to assume that
D
c
=
0. Hence, instead of (10.50), for an initially saturated sloping
aquifer the boundary conditions can be formulated as follows
η
=
0
x
=
0
t
≥
0
α
∂η
∂
(10.133)
x
+
αη
=
=
≥
η
0
cos
sin
0
x
B
x
t
0
η
=
≤
≤
=
D
0
x
B
x
t
0
For the present two-dimensional case of flow down the slope, and in the absence of lateral
inflow, with
I
=
0, the linearized form of the governing equation (10.31) (i.e. (10.90))
can be written as
2
∂η
∂
k
0
η
0
cos
α
∂
η
k
0
sin
α
∂η
∂
t
=
x
2
+
(10.134)
n
e
∂
n
e
x
Notice again that this equation is in the form of the linear advective diffusion equation,
which was already encountered in the diffusion approach of open channel flow (cf.
Equations (5.88) and (5.92)). In the present case of a sloping aquifer the hydraulic
diffusivity is not simply (10.89), but it contains the slope effect, or
k
0
η
0
cos
α
D
h
=
(10.135)
n
e
In addition (10.134) contains a hydraulic (groundwater) advectivity
k
0
sin
α
c
h
=−
(10.136)
n
e
By analogy with flood wave propagation in open channels, a disturbance of the water
table height
in a sloping aquifer can be visualized as undergoing two types of changes.
The first is a deformation of its shape, which is governed by the diffusivity (10.135); the
second is a displacement of this disturbance down the slope, whose rate of propagation
is given by the advectivity (10.136).
η
Similarity considerations
The boundary conditions (10.133) and the form of the governing differential equation
(10.134) suggest that the variables be scaled as follows (cf. Equation (10.94))
x
+
=
B
x
t
+
=
k
0
η
0
cos
x
/
α/
n
e
B
x
t
(10.137)
η
+
=
η/
D
Equation (10.134) becomes in terms of these variables
2
∂η
+
∂
=
∂
η
+
Hi
∂η
+
∂
+
(10.138)
t
+
∂
x
2
x
+
+
