Continuous Outflow Variation along Irrigation Laterals: Effect of the Number of Outlets

Previous continuous-uniform outlet discharge approaches for the hydraulic analysis of irrigation laterals are generally valid for large (theoretically) infinite number of outlets. For a finite number of outlets, however, these approaches may lead to errors in hydraulic computation. A new continuous-uniform outflow approach that takes into account the effect of the number of outlets on the lateral hydraulics is presented. A new analytical equation describing the energy line shape along uniform sprinkle and trickle irrigation laterals and manifolds is developed. The effect of ground slope and velocity head on hydraulic computation is also considered. The method is however restricted by the simplified assumption of equal outlet discharge. An alternate improved analytical method considering the effect of non-uniform outflow distribution along the lateral is also included. Analytical expressions for determining the inlet pressure head and global statistical parameters characterizing the outflow distribution (Christiansen uniformity coefficient, pressure head variation) are developed for design and evaluation purposes. Comparison tests with an accurate numerical stepwise method indicated that the proposed simplified approach is more accurate than other previous works particularly when the number of outlets is relatively small. The improved method is the most accurate method for all cases examined even for low levels of uniformity.     

Analysis of Residential Irrigation Distribution Uniformity

Irrigation has become commonplace for residential homeowners desiring high quality landscapes in Florida. The goal of this project was to document irrigation system uniformity in Central Florida and to quantify distribution uniformity of residential sprinkler equipment under controlled conditions. The catch-can testing procedure used was a modified version of both the American Society of Agricultural Engineers standard and Florida Mobile Irrigation Laboratory (MIL) procedures. The modified version included a larger sample size to ensure complete sample collection over the entire irrigated area. The standard MIL procedure may overestimate the uniformity for residential systems. From the tests on residential irrigation systems, the average low quarter distribution uniformity (DUlq) value was calculated as 0.45. Rotary sprinklers resulted in significantly higher DUlq compared to fixed pattern spray heads with 0.49 compared to 0.41, respectively. From uniformity tests performed on rotor and spray heads under ideal conditions, rotor heads had more uniform distributions than the spray heads of 0.55 compared to 0.49, respectively. Spray heads had better uniformity when fixed quarter circle nozzles were used as opposed to adjustable nozzles. Both residential irrigation system and controlled tests resulted in (DUlq) at the low end of industry guidelines. Residential irrigation system uniformity can be improved by minimizing the occurrence of low pressure in the irrigation system and by ensuring proper spacing is used in design and installation.     

Optimal and Postirrigation Volume Balance Infiltration Parameter Estimates for Basin Irrigation

Engineering analysis of surface irrigation systems is predicated on reasonably accurate estimates of a field’s infiltration properties. Optimal estimation methods pose multiple volume balance equations at various stages of an irrigation event and are assumed to produce the most accurate results among volume balance based procedures. They have the disadvantage of requiring surface volume determinations, which may be difficult to obtain in practice under many field conditions. This study contrasts infiltration solutions from optimal and a simpler postirrigation volume balance method and examines the implications of those solutions on the performance of management strategies with zero-slope and low-gradient basins. With those types of systems, there is little benefit in using optimization over postirrigation volume balance due to the nonuniqueness of solutions and uncertainties of inputs required by the estimation procedures. In addition, system hydraulic characteristics mitigate the insensitivity of the distribution uniformity to reasonable variations in infiltration characteristics from those assumed in the analysis. For the type of systems considered here, management can be optimized based on time needed to infiltrate a target depth, even if the infiltration function parameters are uncertain.     

Modeling Two-Dimensional Infiltration from Irrigation Furrows

Numerical simulation of the two-dimensional (2D) infiltration process during furrow irrigation requires considerable computational effort, which can be reduced by analytical modeling. This paper deals with the further development of the semianalytical infiltration model FURINF (furrow infiltration). Considering the varying impact of gravity and furrow geometry, the new approach models the impact of furrow geometry on infiltration progress using a transient geometric shape factor as a function of infiltration time and furrow geometry. FURINF portrays 2D infiltration from the wetted furrow perimeter by a series of one-dimensional (1D) infiltration computations that are performed in this paper on the basis of an analytical as well as a numerical solution of the 1D Richards equation. Comparing the FURINF results provided by the analytical and numerical 1D infiltration model confirmed the adequacy and reliability of the robust and simple analytical approach, which only requires soil parameters provided by rather simple measurements. The results and performances of the analytical FURINF model (FURINF-A) are compared within the frame of a sensitivity and error analysis with the outcome of the numerical subsurface flow model HYDRUS-2D considering three different soils.     

Comparison of HYDRUS-2D Simulations of Drip Irrigation with Experimental Observations

Realizing the full potential of drip irrigation technology requires optimizing the operational parameters that are available to irrigators, such as the frequency, rate, and duration of water application and the placement of drip tubing. Numerical simulation is a fast and inexpensive approach to studying optimal management practices. Unfortunately, little work has been done to investigate the accuracy of numerical simulations, leading some to question the usefulness of simulation as a research and design tool. In this study, we compare HYDRUS-2D simulations of drip irrigation with experimental data. A Hanford sandy loam soil was irrigated using thin-walled drip tubing installed at a depth of 6 cm. Three trials (20, 40, and 60 L?m?1 applied water) were carried out. At the end of each irrigation and approximately 24 h later, the water content distribution in the soil was determined by gravimetric sampling. The HYDRUS-2D predictions of the water content distribution are found to be in very good agreement with the data. The results support the use of HYDRUS-2D as a tool for investigating and designing drip irrigation management practices.     

Hydraulic Analysis and Direct Design of Multiple Outlets Pipelines Laid on Flat and Sloping Lands

Adequate hydraulic analysis of a multiple outlets pipelines is very important for the design and evaluation of irrigation systems. In this paper, an analytical direct design procedure for a single multiple outlets pipelines is presented. The proposed equations, taking into consideration the influence of local energy loss, are suitable for designing laterals and manifolds in both trickle and sprinkler irrigation systems, and can be applied for various types of outlet, different flow regimes, and uniform line slope ranges. In this analytical procedure, for any desired uniformity level and given design slope range with remaining known parameters, the pipe diameter and the pipe length can then be directly designed. For any desired uniformity level, the procedure also provides an opportunity to evaluate the influence of local energy loss, as well as the influence of different uniform line slopes on the pipe geometric characteristics (pipe size and length), and on the corresponding hydraulic variables (operating inlet pressure head, downstream end pressure head, and total energy loss). Comparison test with a revised step-by-step numerical method for various slope combinations indicated that the presented methodology produces sufficiently accurate results for various design cases in both trickle and sprinkler lateral design. The methodology is simple, easy to apply, and useful for hydraulic analysis and direct design of a multiple outlets pipelines in irrigation subunits.     

Unfrozen Water in Freezing and Thawing Soils: Kinetics and Correlation

The experimental data indicate that water in freezing or thawing wet soils undergoes a gradual phase change. The mathematical forms of reported correlations of the unfrozen water content of soils varies from researcher to researcher. An improved theoretical treatment of the gradual freezing∕thawing process provides insight into the mechanism of the gradual phase change of water in wet soils and a proper means of correlating experimental measurements. Direct numerical solution of the resulting ordinary differential equation may provide accurate prediction of the unfrozen water content of wet soils, when accurate soil property data are available. Two approximate analytical solutions are derived using average thermal properties of the soil constituents. The simpler of these analytical solutions is applied to various data and shown to be sufficient for all practical purposes. The analysis presented in this article demonstrates that the unfrozen water content can be adequately correlated by means of an exponential decay function shifted by the amount of the water adsorbed over the soil grains, which cannot freeze.     

Development of Simplified Solutions for Modeling Recession in Basins

In irrigation basins the decrease in the gradient of the water-surface elevation following inflow cutoff often leads to reduced rate of convergence, increased computational time, and reduced robustness of the numerical solutions of the recession phase. As the water surface levels off, the underlying physical problem simplifies, thus allowing the use of highly accurate yet simple alternate solutions to the full-numerical solution of the zero-inertia equations. For level basins, the simplification involves treating the stream as a static pool, in which water level only falls in response to infiltration. Graded basins may require partitioning the stream into a flowing and static pool, before water-surface eventually levels off over the entire stream length. Implementation of these solutions enhances computational efficiency and robustness of surface irrigation models without a concomitant loss of accuracy. This paper discusses numerical problems related to the recession phase computation in basins and proposes simplified and robust, yet highly accurate solutions. A comparison of the recession trajectories and final infiltration profiles predicted by the full-numerical solution of the zero-inertia equations, obtained by using double-precision floating-point arithmetic, and the simplified alternate solutions, which is robust enough to be implemented over a range of hardware–software capabilities, show that the two approaches yield essentially identical results. Finally, the general validity of the proposed solutions is tested by comparing predictions of recession trajectories and infiltration profiles with those obtained using a surface irrigation hydraulic model, SRFR.     

Determining Operating Inlet Pressure Head Incorporating Uniformity Parameters for Multioutlet Plastic Pipelines

An important objective of a microirrigation system is to determine the proper operating inlet pressure head, ensuring the desired level of water application uniformity as well as the allowable pressure head variation along the multioutlet pipeline. This paper offers, simple, direct, but sufficiently accurate, relationships incorporating different microirrigation uniformity parameters, such as Christiansen uniformity coefficient, coefficient of variation of emitter discharge, and allowable pressure head variation, to determine the operating inlet pressure head (i.e., pressure head and outflow profiles) for multioutlet plastic pipelines. In this analysis some mathematical expressions were deduced to relate three uniformity parameters; then the operating inlet pressure head is simply reformulated by taking into account a multiplying factor α to the required average outlet pressure head, in terms of three uniformity parameters. Resulting, the influence of different uniform pipe slopes on the water application uniformity and the operating inlet pressure head for various emitter discharge exponents, was evaluated. In addition, to cover various design combinations an extensive comparison between the proposed equations and those of the previous studies was also presented. Examination of the results from this research indicated that, the performance of the proposed technique is sufficient in comparison to those of the recent analytical and numerical procedures.     

Two-Dimensional Basin Flow with Irregular Bottom Configuration

Two-dimensional flow from a point or line source is simulated in an irrigated basin with a nonlevel soil surface; the goal is to predict the distribution uniformity of infiltrated depths. The zero-inertia approximation to the equations of motion allows computation in both wet and dry areas. A fully implicit, nonlinear finite-difference scheme is developed for the solution, but practical numerical considerations suggest local linearization instead. Both isotropic and anisotropic resistance to flow are considered. Results in basins with irregular bottom configurations and small inflows show stream advance confined to the lowest elevations in the basin.     

Evapotranspiration of Full-, Deficit-Irrigated, and Dryland Cotton on the Northern Texas High Plains

Cotton (Gossypium hirsutum L.) is beginning to be produced on the Northern Texas High Plains as a lower water-requiring crop while producing an acceptable profit. Cotton is a warm season, perennial species produced like an annual yet it requires a delicate balance of water and water deficit controls to most effectively produce high yields in this thermally limited environment. This study measured the water use of cotton in fully irrigated, deficiently irrigated, and dryland regimes in a Northern Texas High Plains environment using precision weighing lysimeters in 2000 and 2001. A lateral-move sprinkler system was used to irrigate the fields. The water use data were used to develop crop coefficient data and compared with the FAO-56 method for estimating crop water use. Cotton yield, water use, and water use efficiency was found to be as good in this region as other more noted cotton regions. FAO-56 evapotranspiration prediction procedures performed better for the more fully irrigated treatments in this environment.     

Well-Balanced Scheme between Flux and Source Terms for Computation of Shallow-Water Equations over Irregular Bathymetry

Although many numerical techniques such as approximate Riemann solvers can be used to analyze subcritical and supercritical flows modeled by hyperbolic-type shallow-water equations, there are some difficulties in practical applications due to the numerical unbalance between source and flux terms. In this study, a revised surface gradient method is proposed that balances source and flux terms. The new numerical model employs the MUSCL–Hancock scheme and the HLLC approximate Riemann solver. Several verifications are conducted, including analyses of transcritical steady-state flows, unsteady dam break flows on a wet and dry bed, and flows over an irregular bathymetry. The model consistently returns accurate and reasonable results comparable to those obtained through analytical methods and laboratory experiments. The revised surface gradient method may be a simple but robust numerical scheme appropriate for solving hyperbolic-type shallow-water equations over an irregular bathymetry.     

Analytical Expressions for Hydraulic Calculation of Trapezoidal Drip Irrigation Units

Analytical expressions have been developed relating water distribution indexes in trapezoidal drip irrigation units to design variables which define these units: lengths and diameters of pipes, emitter and lateral spacing, slopes, emitter flow equation parameters, and equivalent lengths characterizing local losses. The proposed expressions are founded in classical hydraulics. They are more accurate than predictions in irrigation practice and are easier to handle than the simulation models frequently proposed to irrigation technicians. Unit design and irrigation decision making and evaluation can thus be furthered. An example for the application of the proposed expressions is presented. First, lateral and submain diameters are determined for different shapes of irrigation units to achieve a given water application uniformity. The irrigation time to supply the desired irrigation depth is then calculated. Results are finally compared with values obtained by simulations that take into account hydraulic and manufacture variations in the unit.     

Modified Hazen–Williams and Darcy–Weisbach Equations for Friction and Local Head Losses along Irrigation Laterals

In previous analytical approaches, the direct calculation of friction loss along a lateral is usually based on empirical power-form flow resistance equations, such as the Hazen–Williams and Blasius equations. The more generalized Darcy–Weisbach resistance equation is not usually applied since its friction coefficient varies along the lateral. In this paper, initially, the Darcy–Weisbach and Hazen–Williams equations are systematically compared, leading to a correction form for the Hazen–Williams coefficient. In addition, a more accurate procedure assuming a power function form for the Darcy–Weisbach equation along irrigation laterals is also proposed. The systematic analysis of various typical flow pipe irrigation situations (e.g., sprinkler irrigation laterals of linear or radial-center pivot displacement, trickle irrigation laterals, and manifolds) indicates that the friction loss along laterals calculated using the Darcy–Weisbach equation closely follows a discharge-power form function. The two empirical parameters of the power function depend on the specific pipe characteristics as well as the specific range of discharge values along the lateral. The proposed analytical solution is extended to incorporate the local head loss, the velocity head variation, and the outflow nonuniformity along sprinkler and trickle irrigation laterals. The suggested direct computation solution is demonstrated in two application examples of sprinkler and trickle irrigation laterals and compared with accurate numerical solutions.     

A Paradigm Shift in Irrigation Management

In coming decades, irrigated agriculture will be called upon to produce up to two thirds of the increased food supply needed by an expanding world population. But the increasing dependence on irrigation will coincide with accelerating competition for water and rising concern about the environmental effects of irrigation. These converging pressures will force irrigators to reconsider what is perhaps the most fundamental precept of conventional irrigation practice; that crop water demands should be satisfied in order to achieve maximum crop yields per unit of land. Ultimately, irrigated agriculture will need to adopt a new management paradigm based on an economic objective—the maximization of net benefits—rather than the biological objective of maximizing yields. Irrigation to meet crop water demand is a relatively simple and clearly defined problem with a singular objective. Irrigation to maximize benefits is a substantially more complex and challenging problem. Identifying optimum irrigation strategies will require more detailed models of the relationships between applied water, crop production, and irrigation efficiency. Economic factors, particularly the opportunity costs of water, will need to be explicitly incorporated into the analysis. In some cases the analysis may involve multi-objective optimization. The increased complexity of the analysis will necessitate the use of more sophisticated analytical tools. This paper examines the underlying logic of this alternative approach to irrigation management, explores the factors that will compel its adoption, and examines its economic and environmental implications. Two important concerns, sustainability and risk, are discussed in some depth. Operational practices for implementing the new approach are contrasted with current, conventional irrigation practices. Some of the analytical tools that might be employed in the search for optimum irrigation strategies are reviewed. Finally, the limited and largely intuitive efforts that have already been made to implement this new paradigm are discussed.     

Hydraulic Analysis of Multidiameter Center-Pivot Sprinkler Laterals

Analytical equations for direct hydraulic analysis of a multidiameter center-pivot lateral with and without an end gun were developed. The pressure head profile along the multidiameter center-pivot lateral is described by simple analytical functions. The analysis is based on both continuous outlet and discrete outlet approaches. Friction losses can be calculated using the Darcy–Weisbach or the Hazen–Williams formulas. The proposed equations simplify important practical applications such as the economic design and evaluation of a multidiameter center-pivot system. A comparison test with a numerical stepwise method indicates that the proposed analytical approach is sufficiently accurate for practical applications.     

Multiple correlation: Exact power and sample size calculations.

This article discusses power and sample size calculations for observational studies in which the values of the independent variables cannot be fixed in advance but are themselves outcomes of the study. It reviews the mathematical framework applicable when a multivariate normal distribution can be assumed and describes a method for calculating exact power and sample sizes using a series expansion for the distribution of the multiple correlation coefficient. A table of exact sample sizes for level .05 tests is provided. Approximations to the exact power are discussed, most notably those of J. Cohen (1977). A rigorous justification of Cohen's approximations is given. Comparisons with exact answers show that the approximations are quite accurate in many situations of practical interest. More extensive tables and a computer program for exact calculations can be obtained from the authors. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Offtake Sensitivity, Operation Effectiveness, and Performance of Irrigation System

Analytical relationships between the control of canal water depth, the sensitivity of irrigation delivery structures, and the resulting internal performance are established at the system level. One system sensitivity indicator is derived for both adequacy and efficiency, and two for equity (coefficient of variation and Theil information index). The level of precision which reflects the effectiveness in controlling water depth is defined as a permissible variation of water depth at the cross-regulator (±ΔHR) about the target. The degree of influence exercised by the cross-regulator on offtakes is accounted for through an influence factor between zero and one. The behavior of three different irrigation systems in Sri Lanka and Pakistan is studied with both analytical system indicators and numerical hydraulic simulations. It shows good agreement for a range of precision between 0.02 and 0.2 m. These global system indicators can be used to define the precision level required to achieve a given performance, to estimate actual performance from recorded precision at regulators, and to diminish the system sensitivity, improving the performance for a given precision. Practical operating policies can be inferred from sensitivity information of irrigation systems without the necessity of a complex irrigation operation model.     

Numerical Error in Weighting Function-Based Unsteady Friction Models for Pipe Transients

The accurate simulation of pressure transients in pipelines and pipe networks is becoming increasingly important in water engineering. Applications such as inverse transient analysis for condition assessment, leak detection, and pipe roughness calibration require accurate modeling of transients for longer simulation periods that, in many situations, requires improved modeling of unsteady frictional behavior. In addition, the numerical algorithm used for unsteady friction should be highly efficient, as inverse analysis requires the transient model to be run many times. A popular model of unsteady friction that is applicable to a short-duration transient event type is the weighting function-based type, as first derived by Zielke in 1968. Approximation of the weighting function with a sum of exponential terms allows for a considerable increase in computation speed using recursive algorithms. A neglected topic in the application of such models is evaluation of numerical error. This paper presents a discussion and quantification of the numerical errors that occur when using weighting function-based models for the simulation of unsteady friction in pipe transients. Comparisons of numerical error arising from approximations are made in the Fourier domain where exact solutions can be determined. Additionally, the relative importance of error in unsteady friction modeling and unsteady friction itself in the context of general simulation is discussed.