## Governing Equations and Modeling

Flow Equations

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 flow at a given section in the irrigated field changes over time and depends upon the soil infiltration behavior. 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 equations [20]. In general, the continuity equation, expressing the conservation of mass, can be written as

d t d x where t is time (s), Q is the discharge (m3 s-1), x is the distance (m) along the flow direction, A is the flow cross-sectional area (m2), and I is the infiltration rate per unit length (m3 s-1 m-1).

The momentum equation, expressing the dynamic equilibrium of the flow process, is

g d x gA d x gA d t g d t d x where g is the gravitational acceleration (ms-2), So is the land (or furrow) slope (m m-1), Sf is the friction loss per unit length or friction slope (mm-1), v is the flow velocity (m s-1 ), and y is the flow depth (m).

These equations are first-order nonlinear partial differential equations without a known closed-form solution. Appropriate conversion or approximations of these equations are required. Several mathematical simulation models have been developed.

Infiltration Equations

Several infiltration equations are used in surface irrigation studies. Most common are the empirical Kostiakov equations,

where I is the infiltration rate per unit area (mm h-1), a and k are empirical parameters, f0 is the empirical final or steady-state infiltration rate (mmh-1), and t is the time of opportunity for infiltration (h). The latter is commonly used in furrow infiltration where cutoff times are long and infiltration tends to approach a steady rate. When initial preferential flow occurs, as is the case with swelling or cracking soils, an initial "instantaneous" infiltration amount must be added to the cumulative infiltration.

Other infiltration equations used are the empirical Horton equation,

where ยก3 is an empirical parameter, It is the initial infiltration rate (mmh-1), and f0 is the final infiltration rate (mm h-1); the semiempirical equation of Philips,

where S is soil sorptivity (mmh-2) and As is soil transmissibility (mmh-1); and the Green-Ampt equation,

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