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Flow Regimes and Models
The process of surface irrigation combines the hydraulics of surface flow in the furrows or over the irrigated land with the infiltration of water into the soil profile. The flow is unsteady and varies spatially. The flowat a given section in the irrigated field changes over time and depends upon the soil infiltration behaviour. Performance necessarily depends on the combination of surface flow and soil infiltration characteristics.
The Equations describing the hydraulics of surface irrigation are the continuity and momentum equation.These equations are known as the St.Venant equation.In general, the continuity equation expressing the conservation of mass, can be written as:
(31.1)
The momentum equation expressing the dynamic equilibrium of the flow process is:
(31.2)
Where, y  Depth of flow (m)
t Time from beginning of irrigation (sec)
v  Velocity of flow as f (x, t) (m/s)
x  Distance along the furrow length (m)
I  Infiltrations rate as f (x, t) (m/s)
g  Acceleration due to gravity (m/s²)
S_{o} Longitudinal slope of furrow (m/m)
S_{f} Slope of energy grade line (friction slope) in (m/m)
A  Crosssectional area as f (x, t) (m²)
Q  The discharge (m³/s)
These equations are firstorder nonlinear partial differential Eq. without a known closedform solution. Appropriate conversion or approximations of these equations are required. So, several mathematical simulation models (Full hydrodynamic, zeroinertia, kinematicwave and volumebalance) have been developed, however, among them volume balance models are more commonly used for design. The volume balance models consider only the continuity eq. (31.1) and ignore the momentum eq. (31.2).
Empirical Infiltration Equation
Infiltration rate affects surface flow as well as performance of irrigation. Several expressions have been proposed for expressing infiltration rate as a function of elapsed time.
31.1.1 The Lewis KostiakovEquation
A widely used empirical expression, for design of surface irrigation system, was originally proposed by Lewis (1937) but was erroneously attributed to Kostiakov.
Fig.31.1.Infiltration rate and Cumulative infiltration vs. elapsed time.
Cumulative: Z = kt^{a } (31.3)
(31.4)
Where, Z= the cumulative depth of infiltration or the volume of water per unit soil surface area
t = elapsed time
k and a = empirical parameters
i = the infiltration rate
Disadvantages of the original Lewis Kostiakov Equation
Infiltration rate never becomes zero instead it reaches a steady state or constant rate condition after a long time. Therefore, the above Eq. was modified to reflect the steady state infiltration rate which may occur during surface irrigation system with longer set times.
31.1.2 Modified KostiakovEquation
The later problem can be fixed by adding a parameter representing a final infiltration rate (constant infiltration rate) to the previous Eq. (31.1) and (31.2). So the equation becomes:
Z = kt^{a} + f_{o} t (31.5)
(31.6)
31.2 Volume Balance Concept
Volume balance equation
(31.7)
Fig. 31.2.Approximation of subsurface and surface profiles during volumebalance.
Also assume that advance characteristics follow a power function
x = pt^{r } (31.8)
Power Advance Volume Balance Model
Using power advance, Elliott and Walker (1982) gave the following solution to the volume balance considering Modified Kostiakoveq:
(31.9)
Where, Q_{0} = inflow rate, m³/min
A_{0 }= cross sectional area of flow inlet, m²
x = the advance distance, m
t= the advance time to distance x since beginning of irrigation, min
k and a= coefficients of modified Kostikov’s Eq.
f_{0}= basic infiltration rate, m³/m/min
p and r =empirical parameters of advance curve
σ_{y} =surface storage factor and generally it has a value of 0.77
σzis defined as:
(31.10)
31.3 Advance Time Determination
Water will be distributed within a surfaceirrigated field nonuniformly due to the differential time required for water to cover the field. To account for these differences in the design procedures, it is necessary to calculate the advance trajectory (curve).
It is first necessary to describe the flow cross section using two of the following functions:
A = a_{1} y^{a2} (31.11)
And
WP = b_{1}y^{b2 } (31.12)
Or as a simpler substitute,
AR^{0.67 }= p_{1}A^{p}^{2 } (31.13)
Where,
A= cross sectional flow area, m²
R = hydraulic radius, m
y = flow depth, m
WP= wetted perimeter, m
a_{1}, a_{2}, b_{1}, b_{2}, p_{1},p_{2} = empirical shape coefficients
For border and basin systems,a_{1}, a_{2}, b_{1} and p_{1} are equal to 1. The value of b_{2} is 0.0 and p_{2 }is 3.3333.
The next step is to determine the crosssectional flow area at the field inlet . For sloping fields, this can be accomplished with the Manning Eq. as follows:
(31.14)
Where,
Q_{0} = Field inlet discharge, m³/min/unit width
n = Manning roughness coefficient
S_{0} = field slope
The design input data required at this point are , field length (L), S, nand . This information can be used to solve the volume balance Eq. for the time of advance,
(31.15)
Where,
Computation Steps of Advance Time
(31.16)
The halflength (0.5L) is substituted for ‘L’ and ‘ ’ for ‘ ’.
Volume balance Eq. is used with half length (0.5L) to find an appropriate value of
If the difference between the two values is less than 0.0001 (error criterion), the procedure for finding is concluded.
If not then is replaced with and steps 36 are repeated until the prescribed error criterion is satisfied.
31.4 Intake Opportunity Time Determination
The basic mathematical model of infiltration utilized in the intake opportunity time determination is the Kostiakov Lewis relation:
(31.18)
Where, Z required infiltrated volume per unit length, m³/m
(per furrow or per unit width are implied)
t The design intake opportunity time, minute
a The constant exponent
k  The constant coefficient, m^{3}/mina/m of length
f_{o}  the basic intake rate, m^{3}/min/m of length
In order to express intake as a depth of application, Z must be divided by the unit width. For furrows, the unit width is the furrow spacing, w, while for borders and basins it is 1.0. Values of k, a, f_{o} and w along with the volume per unit length required to refill the root zone, Z_{req}, are design input data.
The volume balance design procedure requires that the intake opportunity time associated with be known. This time, represented by can be obtained from modified Kostiakov Eq. using the NewtonRaphson procedure.
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