Irrigation System Designing - Practices of Irrigation and On-farm Water Management - page 87

where Q is the flow rate (L 3 /T), A is the cross-sectional flow area (L 2 ), I is the
infiltrated volume per unit of length (L 3 /L) in the furrow, y is the distance along
the furrow in the direction of flow (L), g is the acceleration of gravity (L/T 2 ), t is
the time (T), P is the pressure force per specific weight of the water (L 3 ), S f is the
friction slope (L/L), S 0 is the slope of the furrow bottom (L/L), R is the hydraulic
radius (L), n is the Manning's roughness coefficient, and D is the drag force per
specific weight of water per unit length (L 3 ).
An approximate solution to Eqs. ( 3.12 ) and ( 3.13 ) can be generated by rewriting
these equations in the integral form and numerically integrating in the plane using
weighted averages. The integral can be defined in either Lagrangian or Eulerian
coordinate systems.
According to the latter approach, the discrimination of the Eqs. ( 3.12 ) and ( 3.13 )
yields the following pair of non-linear algebraic equations (Tabuada et al., 1995 ) :

Q t k + 1 )

( Q t + 1
k

Q t + 1
k +

)( Q t k −

RC = −

1 )

+ (1 − t

( A t k − A t + 1

k

)( A t k + 1 − A t + 1

k

+ ) + (1 −

1 )]

y

+

(3.15)

1 )

I t + 1
k

I t + 1
k

( I t k − )( I t k + 1 − + ) + (1 − y = 0

+

and

( Q t + 1
k

)( Q t + 1
k

Q t + 1
k

1

RD =

g [

ϕ ) + (1 − ϕ −

1 )]

y

+

Q 2

Ag 1 + 1

Q 2

Ag 1 + 0

1

) Q 2

Ag t k −

Q 2

Ag t k + 1

+ − + (1 − t

k k +

P t + 1
k 1

P t k + 1 − ) P t k −

P t + 1
k

+ − + ( 1 t

+

) D t k + 1 D t + 1

k

) D t + 1
k

D t k + − ϕ + (1 − ϕ

1 +

(1 − ) ϕ (1 − )(1 − y t

+

A t + 1
k

) A t + 1
k

A t k + ) A t k + 1 ] + S 0 [ + (1 −

1 +

(1 − (1 − )(1 − y t

+

= 0

(3.16)

which are the residuals of continuity (RC) and dynamic (RD).
Equations (3.15) and (3.16) are expressions of finite differences which allow an
approximate solution.

3.4.3 Some Relevant Terminologies

3.4.3.1 Intake Rate

The rate at which water is absorbed by the soil of furrow is termed as intake rate.
It varies with time. Initially, water is absorbed by the soil at a higher rate and
then decreases over time. The fairly constant intake rate is termed as basic intake
rate.