Journal of Irrigation and Drainage Engineering

To support a sensitivity analysis in the framework of catchment modeling, three potential evapotranspiration $\left({\mathrm{ET}}_p\right)$ scenarios were generated by means of two Food and Agriculture Organization (FAO) approaches, namely, the FAO-24 and the FAO-56 approaches. The crop ${\mathrm{ET}}_p$ was estimated as a function of the reference evapotranspiration $\left({\mathrm{ET}}_o\right)$ by means of the $k_c-{\mathrm{ET}}_o$ approach. Scenario A was generated with the standard FAO-24 approach; Scenario B considered also the FAO-24 approach, but with some nonstandard parameters. Scenario C considered the standard FAO-56 approach. The ${\mathrm{ET}}_o$ data were compared to point-scale ${\mathrm{ET}}_o$ constraints. The annual cumulative value of ${\mathrm{ET}}_o$ from Scenario A was on average approximately 200 mm larger than the values from Scenarios B and C. The research revealed similar ${\mathrm{ET}}_o$ estimates for Scenarios B and C. The research also assessed the performance of the angstrom approach for estimating incoming solar radiation $\left(R_s\right)\text{.}$ In this context, a set of angstrom coefficients was derived by means of an optimization process that considered available $R_s$ data.

This paper shows how a sequential irrigation schedule for a tertiary unit can be interpreted as a single machine scheduling problem with earliness, tardiness, and a common deadline. An integer program solution is presented for this irrigation scheduling problem. Two different models are presented to reflect different management options at the tertiary level. The first model allows jobs to be scheduled noncontiguously. In the second model only contiguous jobs are allowed. The second model has three submodels reflecting the various ways in which contiguous jobs can be scheduled over a fixed interval. Earlier work in determining unit costs of earliness/tardiness is reviewed and an alternative improved method is suggested. The models presented in this paper are applied to a tertiary unit with 16 users, both as a single interval and multi-interval irrigation scheduling problem. An alternative integer program is also presented which although computationally more efficient can only be used for single period scheduling problems. The models developed in this paper can be used to solve small scheduling problems and also to calibrate the heuristics as presented in the companion paper.

A sequential irrigation schedule that honors user demands of duration and minimizes earliness and tardiness is interpreted as a single-machine schedule with earliness and tardiness costs and a common deadline (or fixed interval). A heuristic solution is presented for this irrigation scheduling problem. Four models are presented to reflect the different methods in which an irrigation system at the tertiary unit level may be operated, the first model permits jobs to be noncontiguous, i.e., idle time between jobs is permitted, whereas the others permit contiguous jobs only. The heuristic is tested extensively and the solution quality is compared with either an optimum solution from an integer program or the best available solution obtained from an integer program within allocated computation time. The heuristic is computationally efficient for all models presented, however for schedules with noncontiguous jobs, or where idle time is inserted before and after a contiguous set of jobs, solution quality deteriorates. The work brings the science of single scheduling from operations research into irrigation scheduling and suggests areas for further development.

A new class of downstream water-level feedback controllers is proposed that can vary from a series of individual proportional-integral (PI) controllers (each gate adjusted based on one water level) to fully centralized controllers (each gate adjusted based on all water levels) that include the effects of lag time. The controller design method uses discrete-time state-feedback control with a quadratic penalty function, physically based states, and no state estimation. A simple, linear model of canal pool response, the integrator-delay model, is used to define the state transitions. All controllers within this class are tuned for the entire canal using optimization techniques. This avoids the tedious task of manually tuning simple controllers. The relative performance of the various controllers within this class can be directly compared without simulation, since the same objective function is used to tune each controller. An example is provided which suggests that the fully centralized controller will perform better than a series of local controllers. However, reasonably good performance can be obtained for some intermediate PI controllers that pass information to one additional check structure upstream and downstream. This should limit some of the difficulties reported for full optimal controllers where all check structures respond to water-level errors in all pools (e.g., saturation of inputs). The results of simulation studies of these controllers are provided in a companion paper.

In a companion paper, a class of downstream-water-level feedback canal controllers was described. Within this class, a particular controller is chosen by selecting which controller coefficients to optimize (tune), the remaining coefficients being set to zero. These controllers range from a series of simple proportional-integral (PI) controllers to a single centralized controller that considers lag times. In this paper, several controllers within this class were tuned with the same quadratic performance criteria (i.e., identical penalty functions for optimization). The resulting controllers were then tested through unsteady-flow simulation with the ASCE canal automation test cases for canal 1. Differences between canal and gate properties, as simulated and as assumed for tuning, reduced controller performance in terms of both water-level errors and gate movements. The test case restrictions placed on minimum gate movement caused water levels to oscillate around their set points. This resulted in steady-state errors and much more gate movement (hunting). More centralized controllers handle unscheduled flow changes better than a series of local PI controllers. Controllers that explicitly account for pool wave travel times did not improve control as much as expected. Sending control actions within a given pool to upstream pools improved performance, but caused oscillations in some cases, unless control signals were also sent downstream. A good compromise between controller performance and complexity is provided by controllers that pass feedback from a given water level to the check structure at the upstream end of its pool (i.e., that used for downstream control of an individual pool) and to all upstream and one downstream check structures.

Approximate analytical expressions were obtained that relate uniformity indices of water distribution in rectangular drip irrigation units as a function of the variables that define that unit: lengths and diameters of laterals and submain, spacing of emitters and laterals, ground slopes, parameters of the emitter discharge equation, and equivalent lengths characterizing local losses. The proposed expressions offer greater precision than might be needed in irrigation practice. They do not require iterative calculations and improve the procedures normally used. They may be useful in the design of drip irrigation units and in their evaluation and management. An example of their application is offered. The proposed relationships simplify studies of the sensitivity of variables involved in optimum hydraulic design. Users are thus allowed a rational understanding of their influence, improving that which can be gained from computer programs. Graphs obtained with the mentioned expressions are also offered. They can be of interest, although their use is not treated specifically in this paper.

Prediction of selenium (Se) mass distribution and Se speciation in wetlands is desired for a comprehensive assessment of the capability of vegetated wetlands to remove Se from agricultural drainage waters prior to impoundment in evaporation basins. A mathematical model was developed to describe Se transformations and transport in Tulare Lake Drainage District’s pilot flow-through wetland cells in California having dimensions of 15 m width and 76 m length. In the model, the wetland cell is subdivided into five subcells and each subcell consists of 10 internal compartments and Se can be transferred between the compartments by physical and chemical processes. Physical processes include water movement, plant litter drop, and physical material breakdown. Chemical processes include Se reduction from selenate (SeVI) to selenite (SeIV), and further reduction to elemental selenium (Se0) and organic Se (Se-II). In the chemical processes, the Se transformation reaction was assumed to obey first-order kinetics and the Arrhenius equation for temperature dependency. A total of 33 ordinary differential equations were written to describe all the processes within and between the internal compartments. A Fortran program with a numeric subroutine using the fourth order Runge–Kutta method followed by the four-step Adams–Bashforth–Moulton predictor-corrector method was written to solve the equations simultaneously. The daily rate model was successfully calibrated with data from a test plot planted to Smooth cordgrass and validated with data from test plots planted to Rabbitsfoot grass, Saltgrass, and Cattail. This model provided seasonal variations in Se mass distribution and Se speciation in different compartments by considering seasonal changes in water temperature, Se volatilization rate, and evapotranspiration. Sensitivity analyses on physical parameters such as water inflow rate, water depth, or longitudinal length of a wetland showed that decreasing the inflow rate or increasing the water depth or increasing the longitudinal length of a wetland cell can increase the accumulation of Se in compartments of a wetland cell and reduce the total mass of Se in the outflow water. Such information will be useful for developing engineering design criteria for constructed flow-through wetlands.

The influence of spatial variability of saturated hydraulic conductivity $\left(K\right)$ on drain spacing $\left(L\right)$ calculations, based on Kostyakov and Hooghoudt steady state equations, was studied in plots with $L$ equal to 10, 20, and 40 m. The drain discharge rate $\left(q\right)$ and the midpoint water-table height above the drain were measured. Then, the $K$ values required to obtain the actual drain spacing by means of the equations were found by trial and error. When using the Hooghoudt equation different values of the depth of the “impermeable” layer were also assumed. For the plots with $L$ equal to 10 and 20 m the use of any central tendency estimator of $K$ practically would have produced the actual drain spacing. However, for the plot with $L$ equal to 40 m, none of these estimators would have produced this drain spacing. It is recommended to increase the number of $K$ measurements to at least one per hectare and to use $q$ equal to 0.01 m/d for humid tropical areas with similar climatic, soil and agronomic conditions.

A simplified analytical solution that takes into account the effect of the emitter discharge exponent on the hydraulic computations of tapered microirrigation laterals, is presented. The hydraulic analysis is evaluated based on the spatially variable discharge function approach. A simple power equation was used to express distribution of the variable outflow delivered from the each emitter along the lateral. An analytical solution is developed for the case of a linear relationship between the emitter discharge and pressure head, namely, the emitter discharge exponent equals to unique, $y=1.0\text{.}$ In this procedure, the analytical derivations can be applied for uphill, downhill, and zero slope conditions. Results are comparable to those obtained from the literature.

The steady flow in open channels, when the depth of the flow varies gradually with distance, is governed by the classic gradually-varied-flow equation. The solution of this ordinary differential equation allows the tracing of the longitudinal profiles of the water surface of the flow. In this note, a relation obtained by direct integration is proposed for a wide rectangular channel, when Manning’s formula is used for the computation of the energy slope. Then the profiles for subcritical and supercritical flow in a mild and steep channel are presented and a comparison with the Bresse solution, relative to the same channels, is carried out.

The effect of water surface slope in the lateral direction on flow over side weirs was studied. Water surface elevation on the weir plane was expressed by a parameter ψ based upon the hydraulic profile on the channel axis. Two different relationships of ψ as a function of the Froude number were used to calculate side weir discharges. Results were compared with the experimental data. It was shown that better results are obtained when transition conditions of $\psi =1$ at the ends of the side weirs with no lateral surface slope are taken into account. However the effect of water surface slope in lateral direction is of secondary importance as compared to the angle of the deflected water jet along the side weir.