Irrigation System Designing - Practices of Irrigation and On-farm Water Management - page 86

3.4.2 Mathematical Description of Water Flow in Furrow
Irrigation System

Different approaches to simulate surface flow and infiltration in furrows have been
developed by the researchers. Hall ( 1956 ) and Davis ( 1961 ) based their mod-
els on the solution of the mass conservation equation under the hypothesis of
normal depth of flow. Other authors, such as Katapodes and Strelkoff ( 1977 ) ,
Walker and Humphreys ( 1983 ) , Wallender and Rayej ( 1990 ) , and Schmitz and
Seus ( 1992 ) based their models on the numerical solution of the partial differen-
tial equations of mass and momentum in open-channel flow applied to borders and
furrows. Levien and de Souza ( 1987 ) presented algebraic model simulating furrow
irrigation.
In furrow system, as infiltration occurs along the furrow, flow with constant
inflow discharge is unsteady and gradually varied. The velocity of water advanc-
ing on the surface of the furrow is normally higher than the infiltration velocity
(infiltration rate). Thus, water moving into and inside the soil at each cross-section
can be considered two-dimensional and occurring perpendicular to the direction of
the flow on the furrow.
The water flow in a furrow is similar to the flow in an open porous channel ini-
tially dry. Therefore, the mathematical formulation describing water flow in a furrow
should take into account the wave propagation in the furrow during the advance
phase, the flow discharge variation during the supply and the recession phases, and
the movement of water penetration and redistribution in the soil.

3.4.2.1 Unsteady Gradually Varied Surface Water Flow

Unsteady gradually varied flow can be described by the partial differential equations
of Saint Venant ( 1871 ) :

Continuity equation

∂ Q

∂ y + ∂

A
∂ t + ∂

I

t =

0 (3.12)

∂

Dynamic equation

Q 2
Ag

1

g ∂

Q
∂

∂
∂

+ ∂

P
∂ t + y +

D − AS 0 = 0 (3.13)

y

with D = AS f

n 2 Q |

A 2 R 4 / 3

| Q

S 0 = (3.14)