ANALYSIS AND DESIGN OF BORDER IRRIGATION SYSTEMS

Transcription

1 ANALYSIS AND DESIGN OF BORDER IRRIGATION SYSTEMS D. Zerihun, C. A. Sanchez, K. L. Farrell-Poe, M. Yitayew ABSTRACT. Application efficiency ( ) is the primary criterion for border irrigation design and management. The objective of this study is to analyze the behavior of the application efficiency function of border irrigation with respect to border length (L) and unit inlet flow rate (q o ), given a target minimum application depth. The results show that the application efficiency function is unimodal with respect to L and q o. Optimality conditions are derived for both the (L) and (q o ) functions, based on which simple rules that reduce the design and management procedure into a series of one-dimensional optimization problems with respect to q o are developed. The proposed procedure has a variable bounding step in which the feasible ranges of L and q o are determined. This is then followed by a step wherein alternative approximate optimum values of (q o ) are calculated for each of the feasible values of L. Finally, the optimal (q o ) is selected from the available alternatives based on sensitivity analysis and other locally pertinent practical criteria. In addition, the advantages and limitations of advance-phase and post-advance-phase inflow cutoff options and their effects on system design and management are discussed. The distance-based (advance-phase) inflow cutoff option offers two main advantages over post-advance-phase cutoff: operational convenience, and a lower degree of sensitivity of design and management prescriptions to inaccuracies in inflow measurements and to non-uniformities in the distribution of inlet flow over the width of the border. However, the results of the study also show that, depending on the parameter set, there exist limiting conditions that preclude the applicability of the distance-based cutoff criterion in border irrigation management. Even when the distance-based inflow cutoff criterion is feasible, the corresponding design and management scenario can be sub-optimal, in which case a near-optimum operation scenario can be realized only with post-advance-phase inflow cutoff. Keywords. Border irrigation, Design, Management, Optimum application efficiency. Border irrigation is widely used to irrigate closegrowing crops that are susceptible to stem and/or crown injuries when exposed to prolonged inundation. Properly designed and managed border strips can apply irrigation water at high levels of efficiency and uniformity and with minimal adverse effects to the environment. The objective of border irrigation design is to maximize a measure of merit (performance criterion) while minimizing some undesirable consequences. The performance criterion could be economic or physical. In either case, mathematical models are used as design and management tools to relate the selected performance criterion with the decision variables. Widely used and relatively well tested surface irrigation mathematical models, such as SRFR (Strelkoff et al., 1998), Article was submitted for review in October 4; approved for publication by the Soil & Water Division of ASABE in July 5. Presented at the ASAE Annual Meeting as Paper No The authors are Dawit Zerihun, Former Assistant Research Scientist, Department of Soil, Water, and Environmental Sciences, University of Arizona, Tucson, Arizona; Charles A. Sanchez, Professor and Director, Department of Soil, Water, and Environmental Sciences and Yuma Agricultural Center, University of Arizona, Yuma, Arizona; Kathryn L. Farrell-Poe, ASABE Member Engineer, Associate Professor and Water Resources Extension Specialist, Department of Agricultural and Biosystems Engineering and Yuma Agricultural Center, University of Arizona, Yuma, Arizona; and Muluneh Yitayew, ASABE Member Engineer, Professor, Department of Agricultural and Biosystems Engineering University of Arizona, Tucson, Arizona. Corresponding author: Charles A. Sanchez, Professor and Director, Department of Soil, Water, and Environmental Sciences and Yuma Agricultural Center, University of Arizona, 149 7th Way, Yuma, AZ 85364; phone: ; fax: ; sanchez@ag.arizona.edu. can accurately simulate processes in irrigation borders by using either the zero-inertia or the kinematic-wave models, depending on the border bed slope. While SRFR is well suited to solving problems that involve system evaluation, its lack of optimal search capability limits its utility as a design and management tool. Simplified solutions that relate border irrigation performance indices with dimensionless variables were developed based on the zero-inertia model (Yitayew and Fangmeier, 1984; Strelkoff and Shatanawi, 1985; El Hakim et al., 1988). Site and irrigation specific charts and equations that relate performance indices with pertinent independent variables were proposed by Reddy (198), Shatanawi and Strelkoff (1984), and Holzapfel et al. (1986). Yitayew and Fangmeier (1985) used dimensionless curves (Yitayew and Fangmeier, 1984) to develop a procedure for the design of the reuse system of border strip irrigation. The dimensionless solutions of Strelkoff and Shatanawi (1985) form the basis for a border irrigation system design and management program, called BORDER, developed by Strelkoff et al. (1996). For infiltration events that can adequately be modeled using the single-term Kostiakov and the NRCS equations, BORDER can be a useful design and management tool. However, in cases where more general infiltration functions are most appropriate, in order to accurately describe the infiltration process, BORDER cannot be used as a design and management aid. Optimal design approaches that use economic cost/benefit criteria as the objective function were proposed for border irrigation systems by Reddy and Clyma (1981) and Holzapfel and Marino (1987). However, these approaches are not widely used, mainly because they are data intensive and they involve relatively complex solution techniques. Design and Transactions of the ASAE Vol. 48(5): American Society of Agricultural Engineers ISSN

2 management procedures based on physical performance criterion have a minimal data requirement and are amenable to simpler solution techniques. Both economic and environmental rationales suggest that, among the physical performance indices, application efficiency is the primary surface irrigation systems design and management criterion (Zerihun et al., 1). Widely used border irrigation system design approaches, such as those of Hart et al. (198) and Walker and Skogerboe (1987), use irrigation performance as the design criterion. Nonetheless, these procedures, generally, emphasize the development of a feasible-yet-satisfactory, instead of optimal, design. This article presents analyses of the application efficiency function of border irrigation systems. The type of border considered here is a graded and free-draining border without cross-slope and with no furrows. Soil and surface roughness are assumed homogeneous throughout the border, and inlet flow rate is considered to be uniformly distributed over the border width. The analyses show that the application efficiency ( ) of a border irrigation system is unimodal with respect to length and unit inlet flow rate. Based on these results, optimality conditions are derived for the (L) and (q o ) functions. The advantages and limitations of advancephase and post-advance-phase inflow cutoff options and their effects on design and management are discussed. Finally, the article proposes a simple design and management procedure for graded, free-draining border irrigation systems. DESIGN AND MANAGEMENT CRITERIA AND VARIABLES Considering the type of border described above, the performance of a border irrigation event can be evaluated using three different indices: efficiency (application efficiency, [%]), adequacy (water requirement efficiency, E r [%]), and uniformity (distribution uniformity, D u [ ])., E r, and D u can be expressed as (Zerihun et al., 1997): L Lov Zdx Zdx + Z rlov 1 (1) tco Cd q odt L Lov Zdx Zdx Z L + r ov Er 1 () ZrL Z min DU Zav Z minl L Zdx where L border strip length (m), L ov length of the border reach over which the infiltrated amount equals or exceeds Z r (m), C d unit conversion factor (1 3 m 3 /L), q o unit inlet flow rate (L/min/m), t co cutoff time (min), Z infiltrated amount (m 3 /m), Z r net irrigation requirement (m 3 /m), Z min minimum infiltrated amount (m 3 /m), and Z av average infiltrated amount (m 3 /m). Economic and environmental rationales suggest that application efficiency is the primary performance criterion in the design of surface irrigation systems (e.g., Zerihun et al., 1). With as the performance criterion, the border irrigation design problem can be posed as: (3) Max Ea St. Zr Z DUmin and C ( L,,tco) min( L,,tco), DU( L,,tco), ( ) i L,,tco where DU min minimum acceptable level of distribution uniformity ( ), and C i represents a set of constraints that can be categorized as variable bounds, conservation-like, and management related. A complete list of these constraints is given by Zerihun et al. (1999). Note that the first constraint imposes a restriction on the minimum cumulative infiltration and target E r. The constraints can be implicitly embedded within the hydraulic simulation model or explicitly enforced by the optimization algorithm, depending on whether a physically based model or explicit empirical functions are used to evaluate the terms in the constraint functions. In this study, a simulation model is used to evaluate the terms in the constraint functions; hence, most of the constraints need not be enforced explicitly. For reasons of simplicity, the only constraint that is explicitly considered in the current analysis is the requirement on Z min. Note that the above formulation considers unit inlet flow rate (q o ), border length (L), and cutoff time (t co ) as design variables. While distance-based cutoff criterion is widely used in border irrigation management, cutoff distance can always be expressed in terms of an equivalent cutoff time. Thus, a time-based inflow cutoff criterion is the more general of the two and is used here. The determination of border width is an important element of the physical design of irrigation borders. However, the study presented here is based on a one-dimensional flow analysis; hence, border width is selected as a function of available flow rate at the field supply channel, field width, width of available machinery, topography, top soil depth, and preferred aspect ratio. To the extent that width is determined on the basis of considerations that are not explicitly related to performance, it is not considered as a design variable here. For practical design and management purposes, the solution of equation 4 can be reduced to the solution of a series of one-dimensional problems (Zerihun et al., 1), simplifying the problem significantly. In subsequent sections, the (L) and (q o ) functions are analyzed separately to establish the existence/absence of convexity and unimodality. Based on the results of the analyses, simple equations that can be used to calculate approximate optimal length and unit inlet flow rate are developed. APPLICATION EFFICIENCY AS A FUNCTION OF BORDER LENGTH Given the net irrigation requirement (Z r ), target water requirement efficiency (E rt ), and unit inlet flow rate (q o ), the application efficiency, (L), can be given as: L Ea ( L) CL (5) tco( L) where C L E rt Z r /q o. At a stationary point, where d (L)/dL, the following holds: (4) 175 TRANSACTIONS OF THE ASAE

3 dtco ( L) tco( L) (6) dl L At a stationary point: d Ea ( L) C LL y L (7) dl [ tco ( L)] where y L d t co (L)/dL (Zerihun et al., 1). Since C L, L, and t co (L) are all positive quantities over the entire range of L, the (L) function is concave at a stationary point, and the stationary point represents a maximum for: y L > (8) Given a parameter set and q o combination and a requirement that Z min Z r, intuitive reasoning and experience with simulation results show that t co (L) is an increasing convex function of length. A power function of the following form can be used to relate t co and L (figs. 1a through 1e): ψ t co ψ1l + ψ3 (9) where 1 (min/m ), ( ), and 3 (min) are empirical curve-fitting parameters. Note that if > 1, then equation 8 holds and (L) is concave at a stationary point. In order to determine the domain of, simulation experiments were performed using SRFR (Strelkoff et al., 1998). The combinations of unit inlet flow rate and the parameter set (i.e., bed slope, surface roughness, and infiltration) used were selected such that a broad range of irrigation conditions was taken into account (table 1). Equation 9 was then fitted to the t co (L) data obtained using simulation experiments, and the regression results are summarized in table and figure 1. Figure 1a represents an irrigation scenario that occurs in a border strip with a low bed slope and on a high intake rate soil with a very high surface roughness. Figure 1f, on the other hand, represents an irrigation scenario at the opposite end of the spectrum, where infiltration rate and surface roughness are very low and bed slope is steep. Figures 1b through 1e represent irrigation scenarios that could be described as physically realistic. Note that most physically realistic irrigation scenarios fall between the two extreme bounds represented by data sets 1 and 6 (table 1, figs. 1a and 1f). Moreover, figures 1a, 1b, and 1f represent irrigation management scenarios where inflow cutoff occurred after completion of the advance phase, and figures 1c through 1e represent conditions in which the inflow is cutoff in the course of the advance phase. The results summarized in figure 1 show that in all the cases considered, regardless of the inflow cutoff option used, cutoff time remains a monotonic increasing power function of border length: > 1 (table ). It then follows that the right side of equation 7 is less than zero at a stationary point. Hence, a stationary point on (L) represents a maximum point. The absence of a local minimum automatically precludes the existence of multiple local maxima. Therefore, the stationary point on the (L) function is a global maximum, and the (L) function is unimodal. MAXIMUM APPLICATION EFFICIENCY AS A FUNCTION OF BORDER LENGTH Combining the first-order optimality condition (eq. 6) and the power-law expression for t co (L) (eq. 9) yields an expression for the approximate optimal length (L opt ): 1 ψ3 ψ L 1( 1) opt (1) ψ ψ t co Eq. 9 4 t co Eq t co Eq (1a) (1b) (1c) t co Eq t co Eq t co Eq (1d) (1e) 4 6 (1f) Figure 1. Cutoff time (t co ) and advance time to the downstream end (t a ) as a function of border length: (a) data set 1, (b) data set, (c) data set 3, (d) data set 4, (e) data set 5, and (f) data set 6. Vol. 48(5):

4 Table 1. Data sets used in figures 1,, 5, and 6. Parameter Units Data Set 1 Data Set Data Set 3 [a] Data Set 4 [a] Data Set 5 Data Set 6 q o L/min/m 9. [b] 18. [b] 15. [b] 15. [b] 4. [b] 3. [b] L m 4. [c] 5. [c]. [c]. [c] 15. [c] 1. [c] S o n m 1/ k mm/h a a f o mm/h W m Z r mm [a] Data sets 3 and 4 are the same except that the required application depths (Z r ) are different. Due to differences in Z r, applicable cutoff options are also different: for data set 3, inflow cutoff occurred during the advance phase (figs. 1c and c); for data set 4, inflow cutoff occurred in the post-advance phase (figs. 1d and d). [b] Unit inlet flow rates used to generate figures a through f. [c] Border lengths used to generate figures 1a through 1f. The parameters of equation 9 ( 1,, and 3 ) can be estimated using the three-point method (Zerihun et al., 1): where tco1 ψ1 ψ ψ (1 r ) L L 1 t 1 ln co tco3 ψ L ln 1 L ψ ψ3 tco ( L1 ) ψ1l1 tco1 tco( L1 ) tco( L ) tco3 tco( L ) tco( L3) rl L3 / L1 (11) L rl L1 L3 / rl (1) and L 3 < L < L 1. Given a parameter set and unit inlet flow rate (q o ), the following procedure can be used to determine the parameters of equation 1: (1) determine L 1 as the length of a block that is irrigated as a unit (discussion on how to determine L 1 is presented in the design section of this article); () select the minimum acceptable length (L 3 ) based on operational and economic considerations; (3) determine r L using equation 1; (4) determine L ; (5) determine t co (L 1 ), t co (L ), and t co (L 3 ) using a simulation model, such that Z min Z r in each case; and (6) calculate 1,, and 3 using equation 11. APPLICATION EFFICIENCY AS A FUNCTION OF UNIT INLET FLOW RATE Given a parameter set, the net irrigation requirement (Z r ), target water requirement efficiency (E rt ), and border length (L), the application efficiency, (q o ), can be expressed as: Cq ( ) (13) tco( ) where C q E rt Z r L. At a stationary point, where d (q o )/ dq o, the following holds: dt co( ) tco( ) (14) d At a stationary point: d Ea ( ) d C q [ t co ( )] t co ( ) yq 3 [ tco( )] q o (15) where y q d t co (q o )/dq o (Zerihun et al., 1). Since C q, q o, and t co (q o ) are all positive quantities over the entire range of q o, the (q o ) function is concave at a stationary point, and the stationary point represents a maximum for: Table. Summary of regression results. Parameter Units Data Set 1 Data Set Data Set 3 Data Set 4 Data Set 5 Data Set 6 Cutoff Time as a Function of Length ψ 1 min/m ψ ψ ψ 3 min r Cutoff Time as a Function of Unit Inlet Flow Rate β 1 min β+1 /L β 9,97,635 4,747,46.,9,93.8 5,593, ,917,565 1,43,56 β β 3 min r TRANSACTIONS OF THE ASAE

5 tco( ) y q > (16) Given a parameter set and L combination and a requirement that Z min Z r, intuitive reasoning and experience with the results of surface irrigation simulations suggest that t co (q o ) is a decreasing convex function of flow rate (fig. ). A power function of the following form can be used to relate t co with unit inlet flow rate (q o ): β t ( ) co β1 + β3 (17) where 1 (min +1 /L ), ( ), and 3 (min) are empirical curve-fitting parameters. Using equation 17 and the first-order optimality condition (eq. 14), it can be shown that at a stationary point: tco( ) yq (1 β) (18) Comparing equations 18 and 16 shows that a stationary point on the (q o ) function represents a maximum for < 1. In order to determine the domain of, simulation experiments were performed using five data sets that cover a wide range of irrigation conditions using SRFR (Strelkoff et al., 1998). The combinations of border length and the parameter set (i.e., bed slope, surface roughness, and infiltration) used were selected such that a broad range of irrigation conditions was taken into account (table 1). Equation 17 was then fitted to the t co (q o ) data obtained using simulation experiments, and the regression results are summarized in table and figure. Figure a represents an irrigation scenario that occurs in a border strip with a low bed slope and on a high intake rate soil with a very high surface roughness. Figure f, on the other hand, represents an irrigation scenario at the opposite end of the spectrum, where infiltration rate and surface roughness are very low and bed slope is steep. Figures b through e represent irrigation scenarios that can be described as realistic. Note that most physically realistic irrigation scenarios fall between the two extreme bounds represented by data sets 1 and 6 (table 1, figs. a and f). Moreover, figures a, c, and e represent irrigation management scenarios where inflow cutoff occurred during the advance phase, and figures b, d, and f represent conditions in which inflow is cutoff in the post-advance phase. The results summarized in figure show that in all the irrigation scenarios considered, regardless of the inflow cutoff option used, cutoff time remains a monotonic decreasing power function of unit inlet flow rate: < 1 (table ). This shows that equation 16 holds at a stationary point on the (q o ) function. Consequently, a stationary point on (q o ) represents a maximum point. The absence of a local minimum automatically precludes the existence of multiple local maxima. Therefore, the stationary point on the (q o ) function is a global maximum, and the (q o ) function is unimodal. MAXIMUM APPLICATION EFFICIENCY AS A FUNCTION OF UNIT INLET FLOW RATE Combining the first-order optimality condition (eq. 14) and the power-law expression for t co (q o ) (eq. 17) yields an expression for an approximate optimal unit inlet flow rate (q opt ): 1 β3 β q opt 1( 1) (19) β β + The parameters of equation 17 ( 1,, and 3 ) can be estimated using the three-point method: t co Eq t co Eq t co Eq (a) (b) (c) t co Eq t co Eq t co Eq (d) 1 3 (e) (f) Figure. Cutoff time (t co ) and advance time to the downstream end (t a ) as a function of unit inlet flow rate: (a) data set 1, (b) data set, (c) data set 3, (d) data set 4, (e) data set 5, and (f) data set 6. Vol. 48(5):

6 where rq t β 1 1 co β β (1 rq ) 1 t 1 ln co tco3 β q ln 1 o β β3 tco( 1) β1 1 tco1 tco ( 1) tco( ) tco3 tco( ) tco ( 3) 3 / 1 () q q o o rq q 3 o1 (1) rq and q o3 < q o < q o1. Given a parameter set and border length, the following procedure can be used to determine 1,, and 3 : (1) q o1 can be taken as the maximum non-erosive unit inlet flow rate; () determine the minimum unit inlet flow rate (q o3 ) as the minimum unit inlet flow rate that can reach the downstream end of the border or the minimum unit inlet flow rate required for adequate spread, whichever is greater; (3) determine r q using equation 1; (4) determine q o using equation 1; (5) determine t(q o1 ), t(q o ), and t co (q o3 ) using a simulation model, such that Z min Z r in each case; and (6) determine 1,, and 3 using equation. EVALUATION OF OPTIMUM LENGTH AND FLOW RATE EQUATIONS Six test problems (table 3) were used in the evaluation of the approximate optimality conditions (eqs. 1 and 19). The surface irrigation simulation model, SRFR (Strelkoff et al., 1998), was used in the analysis. Data sets 7 through 9 (table 3) were used to test the optimality condition derived for L, (eq. 1) and data sets 1 through 1 (table 3) were used to test the optimality conditions derived for q o (eq. 19). The approximate optimum solutions, [L opt, (L opt )] and [q opt, (q opt )], were calculated using the procedures outlined above (fig. 3, table 4). The actual optimum solutions, [L opt, (L opt )] and [q opt, (q opt )], were determined based on repeated runs of SRFR (fig. 3). Note that all the approximate optimum solutions, (L opt ) and (q opt ), are within three percentage points of the actual optimum values (fig. 3, table 4). Given the imprecision involved in the determination of the system parameters and numerical errors, the results are satisfactory for practical design purposes. Note that figures 3a, 3b, 3d, and 3e represent irrigation management scenarios in which inflow cutoff occurred during the advance phase, and figures 3c and 3f represent conditions where the inflow is cutoff in the post-advance phase. INFLOW CUTOFF OPTIONS: ADVANTAGES, LIMITATIONS, AND EFFECTS ON DESIGN AND MANAGEMENT INFLOW CUTOFF OPTIONS AND ADVANTAGES In irrigation borders, inflow cutoff can occur in the course of the advance phase or at the end of the wetting phase. Advance-phase cutoff offers some practical advantages over post-advance cutoff. Wattenburger and Clyma (1989a, 1989b) and Clemmens (1998) observed that level basin designs that use distance-based inflow cutoff criterion (i.e., advance-phase inflow cutoff) are less sensitive to wide variations in decision variables and system parameters. Clemmens (1998) stated that design decisions based on distance-based cutoff criterion are more transferable to irrigators and allow basin designs to be adapted to local practices. Experience with simulation experiments shows that similar observations can be made with regard to the sensitivity of the (q o ) function of border irrigation systems when inflow cutoff occurs during the advance phase (e.g., fig. 4a). Advance-phase cutoff has the effect of dampening the influence that changes in q o can have on the runoff fraction (R f ) over a large interval of q o (fig. 4a). As a result, becomes nearly insensitive to changes in q o over a relatively wide range (a 3% increase in q o resulted only in a 4.5% change in, fig. 4a). As can be seen from figures 3a through 3c, the inflow cutoff option used does not have a significant effect on the sensitivity of to changes in L. In general, the (L) function is distinctly unimodal and attains its peak value where R f approximately equals D f, regardless of the cutoff option used (figs. 3a through 3c, 4c, and 4d). On the other hand, (q o ) may not necessarily attain its maximum value within physically realistic ranges of q o, if inflow cutoff is to occur in the course of the advance phase (figs. 3e, 4a, and 4b). Even in the cases where inflow cutoff occurs during the advance phase, the preceding theoretical observation on the Table 3. Data sets used in figures 3 through 6. Data Sets Used to Test Equation 1 Data Sets Used to Test Equation 19 Parameter Units Data Set 7 Data Set 8 Data Set 9 Data Set 1 Data Set 11 Data Set 1 L m q o L/min/m S o n m 1/ k mm/h a a f o mm/h W m Z r mm TRANSACTIONS OF THE ASAE

7 65 (L opt ) 6. % (L opt ) 61.7 % 7 (L opt ) 67.1 % and (L opt1 ) 67 % 63 (L opt ) (L opt ) 6.1 % (%) (%) 49 4 (%) 49 4 L opt L oopt 191 m L L opt 1 m opt 18 m (3a) L opt 35 m L opt 37 m (3b) (3c) 7 (q opt ) 66.5 % (q opt ) 67. % 7 (q opt ) 66.5 % 7 (q opt ) 58.9 % (q opt ) 66. % (q opt ) 6.5 % 6 (%) 5 (%) 55 (%) q opt 1 L/min/m q opt 11 L/min/m 45 q opt 396 L/min/m q opt 4 L/min/m 4 q opt 63 L/min/m q opt 79.9 L/min/m (3d) (3e) (3f) Figure 3. Application efficiency as a function of border length: (a) data set 7 (advance-phase cutoff), (b) data set 8 (advance-phase cutoff), and (c) data set 9 (post-advance-phase cutoff); and application efficiency as a function of unit inlet flow rate: (d) data set 1 (both advance-phase and post-advancephase cutoff are used), (e) data set 11 (advance-phase cutoff), and (f) data set 1 (post-advance-phase cutoff). unimodality of (q o ) is valid. The fact that t co (q o ) is a decreasing convex function (fig., table ), irrespective of the cutoff option used, confirms the general validity of the optimality condition derived above (eq. 19). However, the very low sensitivity of (q o ) over a wide range of q o, when inflow cutoff occurs during the advance phase, means that a distinct maximum could not be attained within realistic ranges of q o. In which case, the optimality condition developed above is still applicable, but the optimum q o, calculated as such, may not be the theoretical optimum. It could, instead, be a value close to the maximum feasible unit inlet flow rate (fig. 3e). The very low sensitivity of (q o ) over a wide range of q o, when inflow cutoff occurs during the advance phase, is a desirable property because errors in flow measurements or non-uniform distribution of inlet flow rate across the border can have minimal impact on the reliability of design and management prescriptions. In general, whenever it is feasible, and when near-optimum management scenarios are achievable, border design and management can preferably be based on distance-based cutoff criterion (i.e., advance-phase inflow cutoff option). However, advance-phase inflow cutoff is feasible only if the combination of system parameters and variables is such that the crop root zone reservoir can be replenished to the extent desired (say Z min Z r ), even when inflow cutoff occurs prior to, or at, the completion of advance. INFLOW CUTOFF OPTIONS AND LIMITATIONS Given a unit inlet flow rate and a parameter set (infiltration parameters, the Manning roughness coefficient, Z r, and bed slope), there exists a minimum threshold border length (L t ) below which a border strip becomes too short to be operated under the distance-based inflow cutoff criterion and still meet the requirement that Z min Z r. In other words, Unit Inlet Flow Rate (q o ) ψ 1 (min/m ψ ) ψ 3 ( ) Table 4. Results used in figure 3. ψ 3 (min) L opt (m) t co [a] (min) (L opt ) (%) L opt (m) (L opt ) (%) 18 L/min/m L/min/m L/min/m Border Length (L) β 1 (min β+1 /L) β ( ) β 3 (min) q opt (L/min/m) t co [a] (min) (q opt ) (%) q opt (L/min/m) (q opt ) (%) 34 m 6,83, m 5, m [a] t co cutoff time corresponding to L opt. Vol. 48(5):

8 6 6 (%), D f (%), and R f (%) 4 R f D f (4a) 4 6 (4a) (%), D f (%), and R f (%) 4 R f D f (4b) 6 6 (%), D f (%), and R f (%) 4 R f D f (%), D f (%), and R f (%) 4 R f D f (4c) (4d) (4d) Figure 4. Application efficiency ( ), deep percolation fraction (D f ), and runoff fraction (R f ) expressed as a function of unit inlet flow rate: (a) data set 11 (advance-phase cutoff), and (b) data set 1 (post-advance-phase cutoff); and as a function of border length: (c) data set 7 (advance-phase cutoff), and (d) data set 9 (post-advance-phase cutoff). for L < L t, the duration of the advance phase becomes too short for the surface storage volume to be sufficiently large to replenish the root zone in full (figs. 5a and 5b). For any given unit inlet flow rate and a parameter set mix, the corresponding L t can be defined as the border length that yields an infiltration profile in which Z min Z r for R 1%, where R (inflow) cutoff distance expressed as a percentage of the border length (figs. 5a and 5b). Note that L t is a dynamic quantity that changes in the course of an irrigation season with changes in irrigation parameters. A similar observation can be made with respect to border unit inlet flow rate (q o ). Given a combination of a field parameter set and border length, the corresponding threshold unit inlet flow rate value (q ot ) can be defined as the minimum q o below which the requirement Z min Z r cannot be met if the border strip is to be operated under the distance-based inflow cutoff criterion (fig. 5c). Although Z r can be reduced to overcome this problem, reducing Z r has its own problems. Lowering Z r means opting for a lighter irrigation, which in turn leads to more frequent irrigation. This may not always be compatible with the high dose, low frequency nature of surface-irrigated systems. In addition, it is important to recognize that even though Z r can be adjusted to achieve a feasible irrigation scenario with advance-phase cutoff, such a scenario may correspond to a sub-optimal solution that is inferior to the solution that can be obtained if postadvance-phase cutoff is used. There exist irrigation scenarios that have two threshold unit inlet flow rates, q ot1 and q ot, where the interval q ot1 < q o < q ot represents the range of q o in which distance-based cutoff is feasible (fig. 5d). However, outside this range (i.e., in the ranges q o < q ot1 and q ot < q o ), only the post-advance inflow cutoff option is feasible. In the range q o < q ot1, the surface storage volume at the end of the advance phase is not sufficiently large to replenish the root zone in full; hence, inflow cutoff needs to occur after completion of the advance phase. On the other hand, as q o increases, the duration of the advance phase [t a (q o, L)] progressively shortens, and eventually as q o approaches q ot, t a (q o, L) becomes shorter than the duration of the recession phase. This causes the location of Z min to shift from the downstream end to the inlet end of the border, at which point q o passes a threshold with respect to its effect on the cutoff time. The t co needed to meet the requirement Z min Z r at the upstream end of the border becomes virtually insensitive to further increases in q o (fig. 6a, eq. A.1 in the Appendix). In contrast to t co (q o ), which remains nearly constant with further increases in q o, advance time to the downstream end, t a (q o, L), continues to decline at a relatively higher rate (fig. 6a). Eventually, as q o exceeds q ot, t a (q o, L) falls below t co (q o ) and continues to do so with further increases in q o, making distance-based cutoff criterion inapplicable in the range q ot < q o. The insensitivity of the t co (q o ) function, in the range where Z min occurs at the inlet end of the border, can be explained 1758 TRANSACTIONS OF THE ASAE

9 1 R 1 R R (%), Zmin (mm), and zr (mm) R (%), Zmin (mm), and zr (mm) 9 8 Z r 7 L L t Z min (5a) 1 9 R 8 7 Z r 6 Z min q o q ot 5 R (%), Zmin (mm), and zr (mm) R (%), Zmin (mm), and zr (mm) 8 Z r Z min L L t (5b) R q o q ot1 q ot 7 Z r 6 Z min (5c) (5d) Figure 5. Cutoff ratio (R), minimum infiltrated depth (Z min ), and required depth (Z r ) expressed as a function of border length: (a) data set 7, and (b) data set ; and as a function of unit inlet flow rate: (c) data set 11, and (d) data set 1. using an equation that relates cutoff time (t co ) with the required intake opportunity time, req (Z r ), and the duration of the depletion phase (t dep ): τ req ( Z r ) tco + tdep () As can be seen from figure 6a, t dep is virtually insensitive to changes in q o in the range where Z min occurs at the inlet end of the border. In addition, for a given Z r and infiltration parameter set, req (Z r ) is a constant. Thus, equation shows that if the requirement Z min Z r is to be met, then t co (q o ) also needs to be nearly constant. Note that t co can also be insensitive to changes in L when border lengths are very short (figs. 1c and 1d). Here as well, it is the combined effect of a constant req (Z r ) and a nearly insensitive t dep (L) (fig. 6b) that renders t co nearly insensitive to changes in L for short borders. In addition, it can be seen from figures 5a and 5b that if L is increased beyond L t the cutoff ratio decreases steadily. However, if L becomes excessively high, then the consequent progressive steepening of the advance curve and the final infiltration profile near the downstream end of the border make the cutoff distance very sensitive to changes in L. As a result, the inflow cutoff distance begins to grow at a faster rate than L; hence, R begins to back up (fig. 5b). Depending on the range of L considered in the analysis, R may back up to 1% (fig. 5b). This suggests that a second threshold border length may exist. In general, the question of a second threshold border length is pertinent only when extremely long borders are considered (fig. 5b). Such border lengths are physically unrealistic, and hence the issue of a second threshold border length is of no practical design and management significance. INFLOW CUTOFF OPTIONS AND DESIGN AND MANAGEMENT IMPLICATIONS Based on the preceding discussion, the following inferences are drawn: (1) regardless of the cutoff option used the (L) and (q o ) functions are unimodal; however, when inflow cutoff occurs during the advance phase, could be nearly insensitive to changes in q o, and as a result, the maximum may not be attained within physically realistic ranges of q o ; () there exist limiting conditions, which are dependent on the field parameter set, that preclude the applicability of the distance-based cutoff criterion in border irrigation management; and (3) even when distance-based inflow cutoff criterion is feasible, the corresponding design and management scenario could be sub-optimal, in which case, a near-optimal operation scenario can be realized only with post-advance-phase cutoff. DESIGN AND MANAGEMENT APPLICATIONS Considering t co, L, and q o as the three border irrigation design and management variables, the border irrigation design and management procedure can be simplified substantially. Given a parameter set and the condition Z min Z r, t co is a function of q o and L. Consequently, t co cannot be treated as an independent variable in itself. This leaves only q o and L as the design variables. In addition, for a given field length and parameter set combination, the set of all practically realistic values of L is limited to a couple of known alternatives. For each feasible value of L, can therefore be optimized with respect to q o. In which case, the two-dimensional problem is reduced to a series of onedimensional optimization problems. The following optimal design procedure is proposed: Vol. 48(5):

10 4 Relative sensitivity of t co, t a, and t dep ( ) 3 1 t dep t a t co Change in unit inlet flow rate from q ot [see Figure 5d] (%) (6a) 4 Relative sensitivity of t co, t a, and t dep ( ) 3 1 t a t co t dep Change in border length from L t [see Figure 5b] (%) (6b) Figure 6. Relative sensitivity of cutoff time (t co ), advance time to the downstream end (t a ), and depletion time (t dep ) as a function of: (a) unit inlet flow rate (data set 1), and (b) border length (data set ). 1. Establish the feasible range of L and q o. 1a. Determine the maximum non-erosive unit inlet flow rate (q max ) and the minimum unit inlet flow rate required to ensure adequate spread (q min ). Any appropriate set of equations can be used to determine q max and q min. For example, using the empirical equations of Hart et al. (198), q max (in L/min/m) for non-sod-forming crops, such as alfalfa and small grains, is: 1.59 q max 3/ 4 (3) So 176 TRANSACTIONS OF THE ASAE

11 For well-established, dense sod crops, q max can be twice as large. From Hart et al. (198), the q min (in L/min/m) required to ensure adequate spread is:.357l So qmin (4) n where n is the Manning roughness coefficient and S o is border bed slope. Note that equations 3 and 4 are inapplicable to combinations of L, S o, and n that result in q max < q min. 1b. Specify the field parameter set, field length (L f ), and minimum acceptable border length (L min ). Selection of L min can be based on operational and economic considerations. 1c. Determine q min and check if the corresponding maximum possible advance distance (L max ) is less than or greater than L f. Calculate q min as a function of L f using equation 4. Using a simulation model (e.g., SRFR), check if the irrigation stream can advance to the downstream end of the border (for L L f and q o q min ). If the advance phase is completed, then L max > L f. If, on the other hand, the stream failed to reach the downstream end of the border, then L max < L f, and L max needs to be determined through repeated runs of a simulation model. 1d. Determine the maximum possible border length (L 1 ) that can be irrigated as a block: (1) L 1 L f if L f < L max, or (): L f L f L1 for L f Lmax int + 1 Lmax L f L1 L f int Lmax L f > int Lmax L f L f for int L max Lmax (5) if L max < L f, where int an operator that truncates the fraction part of the quotient of L f and L max. 1e. Determine the feasible set that contains wholenumber divisors of L 1 that are greater than or equal to L min. For example, if L 1 3 m and L min 75 m, then the set containing whole-number divisors of L 1 that are greater than or equal to L min is given as L fs {3, 15, 1, 75}.. For each element of L fs, determine (q opt ). 3. Perform a sensitivity analysis and select the scenario that provides the highest application efficiency and satisfies other locally pertinent practical requirements. 4. Determine border ridge height as a function of the flow depth at the inlet and with an allowance for freeboard. The flow depth at the inlet can be calculated using the Manning equation. 5. Determine border width taking into account available machinery width, field width, available field supply channel discharge, top soil depth, cross-slope, and preferred aspect ratio. The procedure presented above is for the design of a border irrigation system. Once an irrigation system is constructed, the length of the border is known, and hence system management is a function of q o only. A management problem can therefore be considered as a particular case of a design problem. Consequently, the design procedure proposed here can be directly applied to solve management problems by setting L constant. EXAMPLE DESIGN PROBLEM It is required to determine the combination of border length and flow rate that yields maximum application efficiency given specific field conditions. The field parameter set used in this example is: S o.8, n.1 m 1/6, k 15 mm/h a, a.3, f o 5 mm/h, and Z r 75 mm, where k, a, and f o are the coefficients and exponent of the modified Kostiakov-Lewis infiltration function. The procedure used to determine the optimal L q o combination is described below: 1. Establish the feasible range of L and q o. 1a. Considering an alfalfa crop, the maximum flow rate (q max ) calculated using equation 3 is 6 L/ min/m. This is an extremely high value to be considered realistic; hence, a lower value of 5 L/min/m is used as q max. Note that q max corresponds to q o1 in equations and 1 (table 5). 1b. The field length (L f ) is 4 m, and the minimum acceptable border length (L min ) is taken as 1 m. 1c. The minimum unit inlet flow rate (q min ) calculated using equation 4 for L L f, is 4 L/min/m. Using SRFR (Strelkoff et al., 1998), it can be shown that when q o q min 4 L/min/m, the irrigation stream can advance to a distance well beyond L f. Hence, L f < L max, and the maximum possible border length (L 1 ) is 4 m. 1d. The feasible set for L that contains a whole-number divisor of L 1 that is greater than or equal to L min is: L fs {4,, 1}. 1e. Determine q min for each element of L fs using equation 4. For L 4 m, q min 4 L/min/m; for L m, q min L/min/m; and for L 1 m, q min 1 L/min/m. Note that the q min values here correspond to q o3 in equations and 1 (table 5). Table 5. Results of design example. Parameter Units L 4 m L m L 1 m q o1 L/min/m q o L/min/m q o3 L/min/m t co (q o1 ) min t co (q o ) min t co (q o3 ) min β 1 min β+1 /L β β 3 min q opt L/min/m t [a] copt min (q opt ) % q opt L/min/m t [b] copt min (q opt ) % [a] t copt cutoff time corresponding to L opt. [b] t copt optimum cutoff time. Vol. 48(5):

12 1f. For each element of L fs calculate q o (q o1, q o3 ) using the procedure outlined above (eqs. and 1). The results are summarized in table 5.. For each element of L fs, determine (q opt ): a. For each element of L fs, three simulation runs (corresponding to q o1, q o, and q o3 ) are performed using SRFR, and the resulting cutoff times, t co (q o1 ), t co (q o ), and t co (q o3 ), are summarized in table 5. b. For each element of L fs, the parameters of equation 17 ( 1,, and 3 ) are calculated using equations and 1 (table 5). c. The approximate optimum unit inlet flow rate (q opt ) is calculated using equation 19, and the approximate maximum application efficiency, (q opt ), is calculated using equation 13 (table 5). d. For L 1 m, the approximate maximum application efficiency is 65.5%; for L m, it is 63.%; and for L 4 m, it is 6.9% (table 5). On the other hand, the maximum application efficiency values estimated through repeated simulation runs using SRFR are 66.5 % for L 1 m, 65.7% for L m, and 63.6% for L 4 m (table 5). As can be seen from table 5, q opt varies as a function of length, but the optimum application efficiency remains nearly unchanged. This result concurs with observations made by Zerihun et al. (1993) that the potential maximum application efficiency of surface irrigation systems is a function of the parameter set only. Consequently, from the point of view of maximization of application efficiency, all three design scenarios are equally valid. This implies that practical and economic considerations need to be taken into account in the selection of the best option among the three border lengths. For instance, in situations where flow regulation and measurement devices are of low accuracy, the design scenario with the least sensitivity to flow rate variation around the optimum (q o ) is recommended. 3. Analysis of the sensitivity of (q o ) around the optimum (fig. 7) shows that in a close vicinity of the optimum (between 5% and +15%), shows the least sensitivity to changes in q o for L 1 m, followed by L m, and then L 4 m. Hence, if flow regulation and measurement devices are of low accuracy, then L 1 m is the best option. However, final selection of the border length needs to take into account other local economic and operational considerations. 4. Calculate border ridge height. Considering the option L 1 m and using the Manning equation, the normal depth at the inlet is 13.5 cm. Taking the freeboard as 1% of the normal depth, the ridge height can be given as 7 cm. 5. The actual calculation of border width in itself is very basic. However, the number of factors that need to be taken into account, as enumerated above, are many and their interrelationship is not as simple. This makes the determination of border width more of a subjective process, dominated by intuition rather than by mathematical rigor. For instance, if the available flow rate in the field supply channel is known, then a first estimate of the border width can be calculated as the ratio of the field supply channel discharge to the optimum unit inlet flow rate. This initial estimate needs to be revised such that the final border width is an integer divisor of the field width. Other factors that need to be considered are width of available farm machinery in relation to border width and preferred aspect ratio, if any, as related to adequate spread of water across the border L m Relative sensitivity of (q o ) L 4 m L 1 m Change in unit inlet flow rate from the optimum (%) Figure 7. Relative sensitivity plot around the optimum unit inlet flow rate for the three alternative lengths. 176 TRANSACTIONS OF THE ASAE

13 In addition, in situations where substantial land grading and shaping is involved, the border width should not exceed a maximum permissible width, which is a function of the topography and the topsoil depth. CONCLUSIONS AND SUMMARY Application efficiency is the primary criterion in border irrigation system design and management. The application efficiency function of border irrigation systems is unimodal with respect to length and unit inlet flow rate. Optimality conditions are derived for the (L) and (q o ) functions. Differences between the solutions obtained using the approximate optimality conditions derived here and the actual optimal solutions are less than three percentage points. Given the imprecision involved in the determination of the system parameters and numerical errors, the results are satisfactory for practical purposes. The advantages and limitations of advance-phase and post-advance-phase inflow cutoff options and their effects on design and management are discussed. Based on the optimality conditions derived here, simple design and management rules are developed. REFERENCES Clemmens, A. J Level basin design based on cutoff criteria. Irrig. and Drain. Systems 1(): El-Hakim, O., W. Clyma, and E. V. Richardson Performance functions of border irrigation systems. J. Irrig. and Drain. Eng., ASCE 114(1): Hart, W. E., H. G. Collins, G. Woodard, and A. J. Humphereys Chapter 13: Design and operation of surface irrigation systems. In Design and Operation of Farm Irrigation Systems. ASAE Monograph No. 3. M. E. Jensen, ed. St. Joseph, Mich.: ASAE. Holzapfel, E. A., M. A. Marino, and J. Chevez-Morales Surface irrigation optimization models. J. Irrig. and Drain. Eng., ASCE 14(3): Holzapfel, E. A., and M. A. Marino Surface-irrigation nonlinear optimization models. J. Irrig. and Drain. Eng., ASCE 113(3): Reddy, J. M Irrigation system improvement by simulation and optimization. PhD diss. Fort Collins, Colo.: Colorado State University. Reddy, J. M., and W. Clyma Optimal design of border irrigation systems. J. Irrig. and Drain. Div., ASCE 17(3): Shatanawi, M. R., and T. Strelkoff Management contours for border irrigation. J. Irrig. and Drain. Eng., ASCE 11(4): Strelkoff, T., and M. R. Shatanawi Normalized graphs of border irrigation performance. J. Irrig. and Drain. Eng., ASCE 11(4): Strelkoff, T. S., A. J. Clemmens, B. V. Schmidt, and E. J. Solsky BORDER: A design and management aid for sloping border irrigation systems. V. 1.. WCL Report No. 1. Phoenix, Ariz.: USDA-ARS, U.S. Water Conservation Laboratory. Strelkoff, T. S., A. J. Clemmens, and B. V. Schmidt SRFR: Computer program for simulating flow in surface irrigation: Borders-basins-furrows. V Phoenix, Ariz.: USDA-ARS, U.S. Water Conservation Laboratory. Walker, W. R., and G. V. Skogerboe Surface Irrigation Theory and Practice. Englewood Cliffs, N.J.: Prentice-Hall. Wattenburger, P. L., and W. Clyma. 1989a. Level basin design and management in the absence of water control: Part I. Evaluation of completion of advance irrigation. Trans. ASAE 3(3): Wattenburger, P. L., and W. Clyma. 1989b. Level basin design and management in the absence of water control: Part II. Design method for completion of advance irrigation. Trans. ASAE 3(3): Yitayew, M., and D. D. Fangmeier Dimensionless runoff curves for border irrigation. J. Irrig. and Drain. Eng., ASCE 11(): Yitayew, M., and D. D. Fangmeier Reuse system design for border irrigation. J. Irrig. and Drain. Eng., ASCE 111(): Zerihun, D., J. Feyen, J. M. Reddy, and G. Breinburg Design and management nomograph for furrow irrigation. Irrig. and Drain. Systems 7: Zerihun, D., J. Feyen, and J. M. Reddy Analysis of the sensitivity of furrow irrigation performance parameters. J. Irrig. and Drain. Eng., ASCE 1(1): Zerihun, D., Z. Wang, R. Suman, J. Feyen, and J. M. Reddy Analysis of surface irrigation performance terms and indices. Agric. Water Mgmt. 34(1): Zerihun, D., J. Feyen, J. M. Reddy, and Z. Wang Minimum cost design of furrow irrigation systems. Trans. ASAE 4(4): Zerihun, D., C. A. Sanchez, and K. L. Farrell-Poe. 1. Analysis and design of furrow irrigation systems. J. Irrig. and Drain. Eng., ASCE 17(3): APPENDIX: RELATIVE SENSITIVITY EQUATION The equation used to calculate relative sensitivity in figures 6 and 7 (e.g., Zerihun et al., 1996) is: fm m f RS o m 1 xm xo (A.1) where RS m is the relative sensitivity at the mth perturbation of x, x is the independent variable, x o is the reference value of the independent variable, f is a function whose sensitivity is being analyzed, f o f(x o ), and: and fm f ( xm 1 + xm ) f ( xm 1) xm xm xm 1 NOMENCLATURE (A.) a exponent of the modified Kostiakov-Lewis infiltration function C L constant expressed as E rt Z r /q o C q constant expressed as E rt Z r L application efficiency E r water requirement efficiency E rt target water requirement efficiency DU distribution uniformity DU min minimum acceptable level of distribution uniformity ( ) f o coefficient of the linear term of the modified Kostiakov-Lewis infiltration function Vol. 48(5):