Geoscience Reference
In-Depth Information
6
OVERLAND FLOW
This type of flow, also variously called sheet flow or shallow flow, is likely to occur in the
initial stages of surface runoff. It is usually observed on surfaces with low permeability
and in areas with a saturated soil profile and with the water table close to the surface. Over-
land flow has been one of the central problems in urban hydrology and it has been the sub-
ject of much research. Interest in this phenomenon has been largely the result of its rele-
vance in the design of small engineering structures for roads, highways, airports and other
urban and industrial settings and also in the design of some surface irrigation systems.
6.1
THE STANDARD FORMULATION
The main objective in the analysis is usually the determination of the flow at the down-
stream end of a sloping plane for a known lateral inflow or outflow, for example, owing to
rainfall, irrigation and/or infiltration. This type of situation is sketched in Figure 6.1. In
general, flow of water with a free surface over a plane bed, with a constant slope
S
0
and
with a length
L
, receiving a uniform, but possibly unsteady lateral inflow
i
i
(
t
), whose
velocity component in the direction of flow is negligible, can be described by the shallow
water equations (5.13) and (5.22); for the present purpose these can be rewritten here as
∂
=
h
∂
∂
t
+
(
Vh
)
−
i
=
0
(6.1)
∂
x
for the conservation of mass, and
∂
g
∂
S
0
V
∂
V
∂
V
h
iV
h
=
t
+
x
+
x
+
S
f
−
+
0
(6.2)
∂
∂
for the conservation of momentum. The dependent variables are the vertically averaged
velocity
V
and the height
h
of the water surface above the bed;
g
is the acceleration of
gravity and
S
f
is the friction slope. As the lateral inflow rate may consist of rainfall
P
(or irrigation), and infiltration
f
, one has
i
f
.
When the plane is initially dry, the essential features of the problem can be captured
by the following boundary conditions
=
P
−
0
≤
x
≤
L
,
t
=
0
,
V
=
0
,
h
=
0
and
(6.3)
x
=
0
t
>
0
V
=
0
,
h
=
0
Although analytical solutions can be obtained for certain special conditions (see
Brutsaert, 1968), the complete solution can be obtained only by numerical methods.
