Sediment Transport Concepts - Sediment Transport in Irrigation Canals: A New Approach - page 90

These five equations form a non-linear partial differential system,
which cannot be solved analytically, but instead by a numerical method
(Cunge et al., 1980). These implicit equations are not independent; they
depend on each other. For instance, the water flow influences the rough-
ness coefficient and, vice versa, the sediment transport depends heavily
on the water flow.
Many mathematical models are based on the finite difference method
in which the equations are replaced by a set of discrete numerical
equations, which can be solved by one of the following methods:
• Uncoupled solution : firstly, the equations related to the water movement

are separated from the total set of equations. Next, these equations are
solved and the results are used to solve the sediment transport equa-
tion and the continuity equation for sediment transport. The uncoupled
solution can be used for long-term simulations that experience gradual
changes (Chuang et al., 1989);

• Coupled solution : the equations for the water movement and sediment

transport are solved simultaneously, which requires general boundary
conditions for the water flow and the sediment transport. In this way,
numerical oscillations or instabilities are reduced. This method is rec-
ommended for short-term simulations that experience rapid changes
(Chuang et al., 1989).

Although one-dimensional flow is rarely found in nature, in this
book the water flow in an irrigation canal will be considered to be one-
dimensional. The main assumptions for a one-dimensional flow are that
the main flow direction is along the canal axis ( x -direction), that the veloc-
ity is averaged over the cross section and that the water level perpendicular
to the flow direction is horizontal. Other assumptions are that the effect of
the boundary friction and turbulence are accounted for by the resistance
laws and that the curvature of the streamlines is small and has a negligi-
ble vertical acceleration. For these assumptions, the general equations for
one-dimensional flow can be described by the de Saint-Venant equations,
which read as (Cunge et al., 1980):

∂A
∂t +

∂Q
∂x =

0 Continuity equation (5.6)

and

v 2
C 2 R +

∂y
∂x +

∂z
∂x +

v
g

∂v
∂x +

1
g

∂v
∂t =

0 Dynamic equation (5.7)

The amount of water flowing into irrigation canals during the irrigation
season and moreover during the lifetime of the irrigation network is not
constant. Seasonal changes in crop water requirement, water supply and