Time Series Modeling for Predicting Spatially Variable Infiltration

Furrow irrigation performance is affected by spatial variability of infiltration; thus, its characterization is essential for accurate performance evaluation. A time-series modeling approach was used to characterize the spatial and temporal correlation structure of field-measured infiltration data. The time-series analysis indicated minimal spatial correlation of infiltration characteristics along the furrow for the space intervals used (5 m). The time-series analysis led to a strategy by which relationships between infiltration characteristics at short- and long-intake opportunity times for a limited number of infiltration tests could be used to “extend” infiltration information gathered by doing shorter duration tests at other furrow locations.     

Furrow Irrigation Performance under Spatially Varying Conditions

A zero-inertia furrow irrigation model with specified space solution was used to investigate the effects of variability in furrow inflow rate and spatial variability in infiltration, geometry, and roughness on end-of-furrow advance, average infiltrated depth, and Christiansen's and distribution uniformities. Extensive field-gathered infiltration, geometry, and roughness data were used as input to the zero-inertia model. Simulations were performed on a single furrow as well as fieldwide. Variable furrow inflow was incorporated into the fieldwide analysis. Model simulations were evaluated to determine the importance to irrigation performance of variability in each input variable. Variability of furrow physical characteristics, in decreasing order of their relative impact on furrow irrigation performance, were furrow inflow rate, infiltration, geometry, and roughness for the conditions studied.     

Economics of Furrow Irrigation under Partial Infiltration Information

The effects of partial infiltration and furrow geometry information on furrow irrigation design and economic return to water were quantified on a single furrow (reference furrow) and field-wide (10-furrow set) basis using a kinematic-wave furrow irrigation model in conjunction with an economic optimization model. A furrow sampled at 10 locations was assumed to represent the actual field condition. Subsamples were randomly drawn from the 10 samples and return to water was maximized. These suboptimal designs were applied to the actual furrows and monetary loss due to lack of information was simulated. The monetary loss was less for furrow irrigation designs having high inflow rates ($0.38∕furrow, $17∕ha) than for the low inflow rates ($2.27∕furrow, $100∕ha). Average loss decreased from $31∕ha ($0.71∕furrow) to $0∕ha in the case of the reference furrow, and from $1.0∕furrow ($44∕ha) to $0.3∕furrow ($13∕ha) in the case of the 10-furrow set for the samples sizes of 1 and 10, respectively.     

Simulation Model for Level Furrows. I: Analysis of Field Experiments

The level-furrow irrigation system consists of furrowing a level basin. In level furrows, irrigation proceeds just like in level basins: the field is flooded from one point and water spreads to irrigate each furrow. Several writers have reported that this irrigation system has the potential to conserve water as compared to level-basin irrigation. However, no comparative studies on the performance of both irrigation systems are available, and the simulation of level furrows has not been attempted. In this work, two field experiments are reported. Both of them were performed in the same soil and in the same conditions. In the first experiment, infiltration was estimated for a series of furrow irrigation discharges and for a level basin. In the second experiment, a level furrow irrigation event was evaluated. A simulated level basin irrigation event in the level furrow experimental field required six times more time and water to complete advance. Infiltration equations including the irrigation discharge or the wetted perimeter as independent variables were proposed for the experimental furrow conditions. Application of a furrow simulation model to the level–furrow experiment resulted in an underestimation of the time of advance. To overcome this problem, a simulation model for level furrows was developed and is presented in a companion paper. The reported field experiments were used to validate the model, which was applied (in a companion paper) to explore adequate conditions for level furrow irrigation performance.     

Physically Based Model for Simulating Flow in Furrow Irrigation. I: Model Development

Researchers developed several mathematical models for simulating furrow irrigation using the Saint-Venant equations. Most of these irrigation models use numerical techniques to solve these equations, which in general, require extensive programming and computational skills. Moreover, several of these models consider uniform soil and use empirical equations for modeling infiltration. In this article, a physically based furrow irrigation model was presented for simulating flow in irrigated furrows under both uniform and layered soils. The model consisted of an overland flow and an infiltration module that are modeled using analytical solution of the zero-inertia and the Green and Ampt [one-dimensional and two-dimensional infiltration equations) equations, respectively. Furthermore, the infiltration was also modeled using the Kostiakov-Lewis infiltration equation. The model considered all possible furrow shapes and included graphical user interface. The developed model was evaluated using the field data and the model performance was discussed in the second part of the article.     

Fertigation in Furrows and Level Furrow Systems. II: Field Experiments, Model Calibration, and Practical Applications

Furrow fertigation can be an interesting practice when compared to traditional overland fertilizer application. In the first paper of this series, a model for furrow fertigation was presented. The simulation model combined overland water flow (Saint-Venant equations), solute transport (advection-dispersion), and infiltration. Particular attention was paid to the treatment of junctions present in level furrow systems. In this paper, the proposed model is validated using five furrow fertigation evaluations differing in irrigation discharge, fertilizer application timing, and furrow geometry. Model parameters for infiltration and roughness were estimated using error minimization techniques. The error norm was based on observed and simulated values of advance time, flow depth, and fertilizer concentration. Model parameters could be adequately predicted from just one discharge experiment, although the use of more experiments resulted in decreased error. The validated model was applied to the simulation of a level furrow system from the literature. The model adequately reproduced irrigation advance and flow depth. Fertigation events differing in application timing were simulated to identify conditions leading to adequate fertilizer uniformity.     

Simulation Model for Level Furrows. II: Description, Validation, and Application

In a companion paper, experimental evidence was elaborated to confirm that in particular circumstances the performance of level-furrow irrigation can exceed that of level-basin irrigation. The application of a single furrow simulation model to an irrigation event in a level-furrow field resulted in large estimation errors. To overcome them, the development and validation of a numerical model of level-furrow irrigation is reported in this work. The model is based on the interconnection of a number of one-dimensional channels. The individual channels are connected using confluence or bifurcation points. Furrow infiltration is modelled through a Kostiakov infiltration equation including the furrow discharge as an independent variable. The proposed model is validated using the experimental level furrow evaluation presented in the companion paper. Finally, the model is applied to explore the conditions in which level furrow irrigation can outperform level basin irrigation. The proposed model stands as a valuable tool in the design and management of level furrow irrigation systems.     

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.     

Rotation Compatibility Approach to Moment Redistribution for Design and Rating of Steel I-Girder Bridges

Recent research has culminated in the development of moment redistribution design and rating procedures based on a “rotation compatibility” procedure. The key aspects of the rotation compatibility method are presented herein along with the resulting series of simple equations that may be used for both design and rating of straight continuous-span steel I-girders. This procedure has several advantages over the previous moment redistribution procedures. Most significantly, the rotation compatibility method provides a rational basis for removing the current restrictions on girder geometries permissible for use with moment redistribution provisions. Thus, sections that are more slender and/or have greater unbraced lengths, compared to previous inelastic procedures, may be considered. This is particularly beneficial for incorporating inelastic methods into rating specifications because many existing bridges have geometries such that they have previously been outside the scope of applicability of inelastic procedures. A second key advantage of the rotation compatibility procedure is that maximum allowable redistribution moments are specifically computed, which justifies the use of higher levels of moment redistribution and consequently greater design economy in some cases.     

Physically Based Model for Simulating Flow in Furrow Irrigation. II: Model Evaluation

In this study, the physically based furrow irrigation model presented in Part 1 was evaluated using the experimental data collected from a field plot consisting of three 40-m-long free-drained furrows of parabolic shape and having a top width of 0.30 m, a depth of 0.15 m, and a slope of 0.5%. The irrigation experiments were carried out with a constant inflow of 0.2–0.5ls?1 and 0.3–0.7ls?1 on bare and cropped fields, respectively. The field data pertaining to furrow cross section, advance and recession times, water depth and velocity, and runoff rate and volume were collected from the irrigation experiments. The model performance was studied for simulating advance, recession, infiltration, and runoff using two–dimensional (2D) Fok, one–dimensional (1D) Green-Ampt, and KL infiltration functions (Part 1) by estimating the root mean square error and index of agreement (Ia). For all the irrigations performed under bare and cropped furrow conditions, the model slightly overpredicted the advance time and runoff rate and slightly underpredicted the recession time and infiltration using the 2D Fok, 1D Green-Ampt, and KL infiltration functions. Furthermore, simulated results were closer to their observed counterparts for 2D Fok infiltration function than for 1D Green-Ampt and KL infiltration functions. Sensitivity analysis was carried out to study the effect of ±5 and ±10% change in 2D or 1D infiltration parameters (i.e., Ks and Sav) and KL infiltration parameters (i.e., Ak and Bk) on outputs. Ks is the most sensitive parameter for predicting advance time, infiltration, and runoff followed by Sav. In the KL infiltration parameter, Bk is the most sensitive parameter for estimating advance and recession times, infiltration, and runoff. The test results of the model suggested that the model can be used as a tool for designing and managing furrow irrigation.     

Two Main Methods for Yield Line Analysis of Slabs

There is a controversy about whether the classical yield lines analysis methods are in fact different methods or simply different ways to develop basically the same method. In this paper two methods are proposed that, without invalidating previous ones, really correspond to the two different ways of performing yield line analysis and therefore facilitate a better comprehension of the general problem of the failure of slabs. These methods are the nearly abandoned “normal moment method” and a new “skew moment method.” In “normal moment method” only bending moments are supposed to act at yield lines. In “skew moment method,” twisting moments in addition to bending moments act along yield lines. The “normal moment method” is general only if yield patterns are “correct,” that is, they are composed by possible yield lines. If yield lines are “incorrect,” or not possible, yield line analysis can only be performed, in general, by means of “skew moment method.” As shown in this paper, many of the classical solutions of yield line analysis correspond to incorrect yield patterns. This work demonstrates that Johansen’s “nodal force theory”—or “equilibrium method”—and “work method” are only partial applications of “skew moment method.” This generalization of yield line analysis allows defining new equilibrium conditions not included in classical yield lines theory and permits obtaining more accurate solutions.     

Explicit Infiltration Function for Furrows

This study addresses infiltration from furrows or narrow channels. The basic approach is to develop the two-dimensional infiltration as a combination of the corresponding one-dimensional vertical and an edge effect. The idea is borrowed from previous applications for infiltration from disc and strip sources. The assumption is tested directly with numerical experiments using four representative soils and three furrow shapes (triangular, rectangular, and parabolic). The edge effect is the difference between the cumulative infiltration per unit of adjusted wetting perimeter and the corresponding one-dimensional infiltration. A general conclusion is that the edge effect is linearly related to time. In addition, it was observed that the two empirical parameters in the function used to relate the edge effect with time have narrow ranges and are related to soil hydraulic parameters, furrow shape, the boundary and initial conditions and additional geometric factors. The approach leads to a physically based infiltration function for irrigation furrows (or narrow channels) without the need to perform a fully two-dimensional simulation. Also, a simplified expression was found for the limiting steady-state case, which is analogous to Wooding’s equation for infiltration from a shallow pond.     

Water Flow through Cover Soils Using Modeling and Experimental Methods

The vertical flow of water in cover soils is simulated using published analytical and finite-element methods. The two methods gave virtually identical pressure head and water content profiles during steady infiltration of water in a multilayer soil cover and transient infiltration in a single-layer cover. The finite-element model was then used to simulate flow in two laboratory columns packed with multilayer soils and subjected to downward drainage and conditions of evaporation and no evaporation. The model adequately predicted transient pressure heads and water contents for the first 7.5 h of drainage in a till-sand layer without evaporation. Predictions at times equal to and greater than 3 days were not as good, probably due to the formation of discontinuous water pockets in the draining sand around the residual water content, which apparently produced “locked-in” or “static” nonequilibrium pressures. These pressures are not captured by existing methods used for estimating the unsaturated hydraulic conductivity–pressure function of soils. Further modeling showed that at times greater than 8 days, the flux from the column with evaporation was all in the vapor phase.     

Validation of a Predictive Form of Horton Infiltration for Simulating Furrow Irrigation

Due to spatially varying conditions the improvement of furrow irrigation efficiency should be sought not just for a limited number of furrows or for one specific irrigation event. A simplified predictive modeling approach of the averaged advance-infiltration process is proposed in this paper. Horton’s equation, derived from the asymptotic form of the Talsma-Parlange infiltration equation, allows us to use a predictive approach for the advance infiltration process by means of the exact solution of the Lewis and Milne water balance equation. The references to the works of White and Sully, for a surface point source, result in the use of parameters which characterize the hydraulic properties of the soil: Δθ (saturated water content minus initial water content); Ks (saturated conductivity); and λc (macroscopic capillary length). The physical meaning of parameters involved in the proposed modeling is attested using field experiments carried out in a loamy soil plot context. Assuming a same Δθ measured value before irrigation for the whole of a 30 furrow sample, the averaged values of λc and Ks obtained from calibration on the advance trajectory are comparable to those derived from local infiltration tests (disk permeameter and double ring methods). The applicability of the model is then extended to heavy clay soil where the parameters λc and Ks still agree with the values proposed in the literature. This paper can be considered as a contribution to the development of a tool for evaluating the impact of irrigation practices on the efficiency at the plot and cropping season scale.     

Toward Physically Based Estimation of Surface Irrigation Infiltration

Irrigation practitioners use empirical infiltration equations. Theoretical infiltration equations are currently not capable of capturing surface irrigation infiltration behavior, particularly during initial wetting. For a coarse textured soil, an example is shown where the Green-Ampt equation can be adjusted to match field “average” infiltration conditions by altering the soil’s physical properties. For finer textured soils, a time offset is proposed for adjusting the Green-Ampt equation to account for cracking and soil consolidation upon wetting. This results in a nonzero infiltration amount at time 0, a phenomenon commonly observed for infiltration of cracking soils. Applying this concept to the Philip equation (same as Modified Kostiakov equation with a = 1/2) suggests the addition of an offset parameter. A modification to the two-point method is presented for this equation with the aim to better fit mathematical parameter functions to infiltration data.     

Errors in Surface Irrigation Evaluation from Incorrect Model Assumptions

Some two-dozen methods have been proposed in the literature for estimating an infiltration function from field measurements. These methods vary in their data requirements and analytical rigor, however most assume some functional form of the infiltration equations. In this paper, if is shown that the form of infiltration and roughness equations can cause errors in the estimation of actual conditions. For example, assumptions regarding the influence of wetted perimeter on furrow infiltration can result in inappropriate infiltration equations and parameters. Also, the Manning n has been shown to vary with time during an irrigation event as the soil is smoothed by the flowing water. Thus estimates of Manning n based on the advance curve may vary substantially from those based on measured water depths. Inappropriate selection of equations or parameter values for infiltration or roughness can lead to unrealistic parameter values for the other. The estimated parameters from evaluation of a measured irrigation event usually give reasonable estimates of actual performance. However, extrapolation to future irrigation events, particularly with a different application depth or flow rate, can lead to inappropriate recommendations.     

Quick Method for Estimating Furrow Infiltration

This paper presents a simple and quick method for estimating furrow infiltration using a single advance point based on the volume balance equation. The furrow infiltration and water front advance along the furrow are assumed to follow the modified Kostiakov infiltration and power advance equations, respectively. The volume balance equation, including these equations, is simplified to a function containing two parameters, i.e., the exponents of power advance and Kostiakov infiltration equation (with a prior-known basic infiltration rate). These parameters are estimated by minimizing the function to zero using a quasi-Newton search algorithm, provided with Excel Solver. The estimated exponents are used to determine the Kostiakov infiltration parameters. The proposed one-point method is tested with seven independent furrow irrigation evaluation data sets and the estimated cumulative infiltration is compared with the observed counterparts. Performance of the proposed method was evaluated using the root-mean-square error and index of agreement (Ia). The results show that the proposed one-point method estimated cumulative infiltration is closer to the observed; the method performed as good as Valiantzas’ method. Shepard’s method did not perform well for the tested data sets. The algorithm and the results of the proposed method reveal that the proposed method can be used as a tool for quick estimation of furrow infiltration using a single advance point.     

Using HYDRUS-2D Simulation Model to Evaluate Wetted Soil Volume in Subsurface Drip Irrigation Systems

Drip irrigation is considered one of the most efficient irrigation systems. Alternatively to the traditional drip irrigation systems, laterals can be installed below the soil surface. Realizing the subsurface drip irrigation (SDI), which recently has been increasing in use as a consequence of advances in plastics technology, making SDI equipment more affordable and long lasting. Due to its potential high efficiency SDI may produce benefits, especially in places where water is a limited source. As the use of SDI is relatively new, a better understanding of the infiltration process around a buried point source can contribute to increased water use efficiency and consequently the success of drip irrigation system. In addition, proper design and management of such a system needs the judicious combination of drip spacing, discharge rates, irrigation duration and time interval between consecutive irrigations. To this aim, numerical models can represent a powerful tool to analyze the evolution of the wetting pattern during the distribution and redistribution processes, in order to explore SDI management strategies, to set up the duration of irrigation, and finally to optimize water use efficiency. In the paper the suitability of the HYDRUS-2D simulation model is verified, at the scale of a single emitter, on the basis of experimental observations, with the aim to assess the axis-symmetrical infiltration process consequent to subsurface drip irrigation. The model was then applied in order to evaluate the main dimensions of the wetted soil volume surrounding the emitter during irrigation as a function of time and initial soil water content. The investigation, carried out in a sandy-loam soil, showed the suitability of the model to well simulate infiltration processes around an emitter during irrigation. Model application allowed also, for the examined soil, to evaluate the emitter spacing accounting for the maximum soil depth to irrigate.     

Evaluation of Water Retention Functions and Computer Program “Rosetta” in Predicting Soil Water Characteristics of Seasonally Impounded Shrink–Swell Soils

Soil water retention is a critical factor influencing irrigation decisions and hence agricultural crop yields. However, information on soil water retention characteristics (SWRC) is seldom available for irrigation planning, crop yield modeling, or hydrological simulations, especially for problematic soils, such as seasonally impounded shrink-swell soils. As large scale direct measurement of SWRC is not viable due to a number of reasons, researchers have developed pedotransfer functions (PTFs) to estimate SWRC from easily measured soil properties, such as texture, organic matter content, bulk density, etc. However, PTF applicability in locations other than those of data collection has been rarely reported. One of the most recent PTFs that has shown overall reasonable predictions in evaluation studies is Rosetta, a numerical code for estimating soil hydraulic parameters with hierarchical pedotransfer functions. Relatively, the development of large databases makes it one of the widely used PTFs. If validated for spatial application, it has immense use potential in countries like India, where data on soil hydraulic properties are seldom available, a deficiency that hampers better simulations in processes, like partitioning runoff and infiltration, assessing evapotranspiration, irrigation scheduling, etc. Rosetta is also relatively flexible allowing estimation of hydraulic properties from easily available minimum input of textural fractions. This study was conducted to evaluate (1) an applicability of four widely used soil water retention functions to describe SWRC; and (2) the computer program Rosetta for its validity. Statistical indices, i.e., root mean square error (RMSE), mean absolute error, maximum absolute error, and degree of agreement (d) were computed to evaluate “goodness-of-fit” of the four functions to the measured SWRC data. These indices were also used to compare measured SWRC with estimates of SWRC by Rosetta. For soil samples collected from 41 profiles, 175 SWRC were measured in the laboratory. The van Genuchten function fitted relatively better (RMSE = 0.052?m3?m?3) to SWRC of clay soils, whereas the Brooks–Corey (BC) function was better in expressing SWRC of clay loam and sandy clay loam soils with RMSE = 0.06 and 0.07?m3?m?3, respectively. Campbell and Cass–Hutson (CH) functions were of intermediate value. Worst performing functions were BC (clay soils), Campbell (clay loam), and CH (sandy clay loam) with corresponding RMSE = 0.059, 0.065, and 0.077?m3?m?3. Estimates of two important points on the SWRC curve, i.e., field capacity and permanent wilting point were predicted with relatively better accuracy for clay and sandy clay loam soils by all the four functions. RMSE and d ranged from 0.027?to?0.043?m3?m?3 and from 0.73 to 0.88 for clay soils. Corresponding values for sandy clay loam soils were 0.008?–0.019?m3?m?3, and 0.92–0.98. However, in clay loam soils, only two functions were found suitable. Estimates of SWRC obtained by applying hierarchical rules in Rosetta were reliable (RMSE<0.05?m3?m?3). Magnitude of average RMSE increased progressively in clay loam, clay and sandy clay loam soils (0.028<0.035<0.042?m3?m?3). The study established that SWRC of the “Haveli” soils could be estimated using generic PTF and thus information that is prerequisite in simulating hydrological processes occurring in seasonally impounded soils could be acquired.     

Infiltration of Water into Soil with Cracks

This paper presents the physical basis of the FRACTURE submodel for simulating infiltration of precipitation∕irrigation water into relatively dry, cracked, fine-textured soils. The FRACTURE submodel forms part of the HYDRUS-ET variably saturated flow∕transport model. Infiltration into the soil matrix is formally divided into two components: (1) Vertical infiltration through the soil surface; and (2) lateral infiltration via soil cracks. The first component is described and solved using the 1D Richards' equation. Excess water that does not infiltrate through the soil surface is either considered to be runoff, if no soil cracks are present, or routed into soil cracks from where it may laterally infiltrate into the soil matrix. Horizontal infiltration from soil cracks into the soil matrix is calculated using the Green-Ampt approach and incorporated as a positive source∕sink term Sf in the Richards' equation describing flow in the matrix. In addition to the hydraulic properties of the soil matrix, the FRACTURE submodel requires parameters characterizing the soil cracks, notably the specific crack length per surface area lc and the relationship between crack porosity Pc and the gravimetric soil water content w. An example problem shows that infiltration from soil cracks can be an important process affecting the soil water regime of cracked soils. A comparison with the more traditional approach, involving surface infiltration only, indicates important differences in the soil water content distribution during a rainfall∕irrigation event. This extension of the classical approach to include crack infiltration significantly improves the identification and prediction of the soil water regime.