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Research Article Turk J Agric For 36 (2012) 729-737 © TÜBİTAK doi:10.3906/tar-1202-74 Estimating wetting front coordinates under surface trickle irrigation Ahad MOLAVI1,*, Aliashraf SADRADDINI2, Amir Hossein NAZEMI2, Ahmad FAKHERI FARD2 1 Department of Soil and Water Sciences, Tabriz Branch, Islamic Azad University, Tabriz – IRAN 2 Department of Water Engineering, Faculty of Agriculture, Tabriz University, Tabriz – IRAN Received: 28.02.2012 ● Accepted: 30.04.2012 Abstract: In this study, wetting front or wetted bulb coordinates in soil under surface trickle irrigation were measured for 1 loam soil and 2 sandy loam soils with 2 different emitter discharges of 2 and 4 L h–1 by using the trenching method. A model is presented for estimating wetted bulb coordinates with a function of emitter discharge, water application time, average variation in volumetric water content, and saturated hydraulic conductivity of soil. For calculating the distance of the maximum wet surface, relationships are presented based on saturated hydraulic conductivity and water application time. By comparison of measured values of wetting front coordinates, the presented model shows good reliability. The goodness of fit ratio and root mean square error of the model were 0.82 and 17.85 mm, respectively. The model for predicting surface trickle irrigation wetting front coordinates can be applicable for the emitter with 2 and 4 L h–1 discharges. Key words: Incomplete sphere, saturated hydraulic conductivity, wetted bulb, wetting front coordinates Introduction Trickle irrigation is considered to be an appropriate method for areas of limited water resources due to high efficiency of water use under good management. For design and management of a trickle irrigation system, the shape of the wetted bulb should be known, and it can be predicted by solving numerical equations governing flow (Bristow et al. 2000). One of the basic factors in the design of trickle irrigation systems is the availability of information about soil texture (Philip 1984; Cote et al. 2003). To increase water use and nutrition efficiencies in trickle irrigation, there should be uniformity between emitter distances, emitter discharges, and soil moisture profile, as well as the duration of water application (Thorburn et al. 2003b). Sezen et al. (2006) found that having information about the wetted bulb of soil is necessary for trickle irrigation system design. One effective method to optimize trickle irrigation system design is using numerical simulation, resulting in moisture distribution in the soil (Schmitz et al. 2002). Thorburn et al. (2003a) used the equation of Philip (1984) to obtain wetted bulb dimensions in surface and subsurface trickle irrigation. Sepaskhah and Chitsaz (2004) studied the analysis of Green and Ampt (1911) to determine the wet radius and depth of surface trickle irrigation. Lazarovitch et al. (2007) studied the characteristics * E-mail: ahad.molavi@gmail.com 729 Estimating wetting front coordinates under surface trickle irrigation of wetted soil volume under surface and subsurface trickle irrigation. Researchers of the estimation of the dimensions of the wetted bulb in a number of methods have used Richards’ equation from 1931. This equation requires many inputs (Chu 1994). Main reservoir Preliminary reservoir The purpose of this study was to provide a new and simple model considering the wetted bulb under conditions of an incomplete sphere. The estimation of wetting front coordinates will be possible with minimum soil hydraulic parameters in the presented model. Figure 1. The layout of the experimental setup. Theory Materials and methods Field experiments were performed in 3 different locations of the Tabriz suburbs, namely Khalatpooshan, Arpadarasi, and Karkaj. Physical properties of the soils are presented in Table 1. Experiments Several emitters with spacing of 1 m were installed on lateral pipes of 16 mm in diameter on the experimental soils. These laterals were connected with a 200-L water reservoir through a main line. To decrease the turbulence in the reservoir, the water was first conveyed into a preliminary reservoir before entering the main reservoir. The main reservoir was equipped with a spillway to supply a constant head in emitters during the tests (Figure 1). The emitter discharges were set at 2 rates of 2 and 4 L h–1. The volumes of applied water during each test by an emitter were, in total, 4 or 8 L. Coordinates of the wetting front for different times were measured from emission points by trenching. If the wetted bulb shape resulting from an emitter according to Figure 2 is considered in the form of an incomplete sphere, noting triangles ABO and ABC, we can express the following relationships. Cos ({) = R.Sin (b) –d R hmax (1) 2 R2 = H2 + Rh max Sin2(φ) (2) 2 R2 = d2 + Rh max Cos2(φ) + 2 (3) 2 2d. Rh max Cos(φ) + Rh max Sin2(φ) 2 2 R2 = d2 + Rh max + 2d. Rh max Cos(φ) (4) By combining Eqs. (1) and (4), the direct relationships between β and R will be as follows: 2 R2 = Rh max + 2d. R Sin(β)–d2, (5) Table 1. Physical properties of experimental soils. Sand Clay Silt (%) (%) (%) Khalatpooshan 70 12 18 Sandy loam Arpadarasi 44 24 32 Loam Karkaj 71 8 21 Sandy loam Experiment sites Soil texture θs* θ0* ρb* (%) (%) (g cm ) (m day–1) 38 10 1.62 0.3878 43.5 14.2 1.53 0.1874 37 7 1.58 0.5106 *ρb = bulk density (g cm–3); θ0 and θs = initial and saturated soil water content [L3 L–3], respectively. 730 Ks –3 A. MOLAVI, A. SADRADDINI, A. H. NAZEMI, A. FAKHERI FARD or: 2 R = (Rh max – d2 Cos2(β))1/2 + Sin(β), (6) where R is the radial distance of the wetting front (m), d is the distance of the maximum wetted width to ground surface (m), Rhmax is maximum wetted width (m), and β is the angle between the soil surface and any radial distance R. Eq. (6) presents the relation of R with the angle between the radial distance and the soil surface (β), the maximum wetted width, and its position toward the ground (d). To use Eq. (6) for estimation of the wetting front coordinates of the wetted bulb, Rhmax and d should be known. If the wetted bulb resulting from the surface emitter is considered to form an incomplete sphere (Figure 2), the wetted bulb volume can be calculated by: 3 V = π(4 Rh max –(Rhmax–d)2 (2Rhmax+d))/3, (7) Emitter Soil surface d H β ϕ B R (8) where Δθ is the average change in volumetric water content in the soil (L3 L–3). From Eq. (8), Rhmax is obtained as follows: 2 Rh max = –d/2 = d1/3 + 1/2m1/3 . m (9) In this equation: m= 1.91Qt + d3 + Oi (10) Qt 2 0.955Qd 3 t 1/2 2 c 0.9128 c m . m + Oi Oi As is evident from Eqs. (9) and (10), Rhmax is the function of flow rate (Q), Δθ, d, and the time of water application (t): (11) σ d = α k s tγ A Figure 2. General hypothetical wetted bulb for incomplete sphere for constant surface emitter discharge 5 = G W  NV Qt , Oi Parameter d can be estimated for each discharge based on the following relationship: C Rh max r ^ 4Rh 3max – (Rh max –d) 2 (2Rh max + d)h /3 = Rhmax= f(Δ θ,d,t,Q). O Rh where V is the volume of the wetted bulb. Assuming that the average soil moisture content after irrigation of the θ0 in θv is increased, we have: (12) where d, ks, and t have already been defined, and α, γ, and σ are coefficients of the equation. By combining Eqs. (6), (9), and (10) and with consideration of Eq. (12), the following model can be used to calculate the radial distance of the wetting front from the center emitter at any time t of the start of irrigation: §§ G W  N V ¨¨ − G +   ¨ ¨ W  NV   § 4W § ¨¨ § 4W · W4 G W  N V ¨  ¨ G    + + +  ¨ ¸ W  NV ¨¨ ¨ ∆θ ¨¨ ∆θ © ∆θ ¹ © © 6LQ(β ) + ¨ ¨ ¨¨      ¨¨ §   § · ·¸ 4 G    W    4W 4W § · W N  ¨  V ¨ ¨  ¸ + G W  N V + ¨ ¨ ¸ + ¨ ¸ ¸¸ ¨ ¨ ¨¨ ∆θ ∆θ ¹ ∆θ © © ¹ ¹ ¨© © ©        · ¸ ¸ ¹ · ¸ ¸¸ ¹  · · ¸ +¸ ¸ ¸ ¸ ¸ ¸ ¸   ¸ − G W  N V &RV (β )¸ , ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¹ ¹  731 Estimating wetting front coordinates under surface trickle irrigation Results where Δθ is obtained as (Ben-Asher et al. 1986): Oi = is , 2 (14) where θs is the saturated volumetric water content of the soil. Other parameters have been described previously. Using the above equation at any angle β from the ground, the radial distance of the wetting front and the center emitter can be estimated, and using , wetting front coordinates can be estimated. It is clear that with regard to β = 0° and β = 90°, Eq. (13) will yield the vertical and horizontal advances, respectively. To evaluate the model, the root mean square error (RMSE) and goodness of fit ratio (R2) were used as follows: RMSE = c/ (E i –M i) 2 /n m , n 1/ 2 (15) i= 1 R = c/ (M i –M ) –/ (E i –M ) m n n 2 2 i= 1 2 i= 1 c/ (M i –M ) m , n 2 –1 (16) i= 1 where M is the average of the measured values of the wetting front radial distance, and M and E are the measured and calculated radial distances of the wetting front, respectively. Model sensitivity analysis We assume that each variable has different effects on the model results; therefore, it is necessary to assess the effects of parameters t, Q, d, and Δθ using sensitivity analysis before using the model. To do this, a base case was considered such that the radial distances at different values of angle β were estimated using the presented model and were then compared with those resulting from changes in the quantity of the parameters in 16 modes. To analyze the results of changes in input parameters from those of the base case, the RMSE was used as in Eq. (15), where M and E are estimated values of the radial distance from the model in the base case and the mode to increase or decrease the amount of input parameters, respectively. 732 As was mentioned in theory, based on saturated hydraulic conductivity and water application time for flow rates of 2 and 4 L h–1, for calculation of the position of the maximum wet width of the ground surface, the empirical relations are as follows. d = 4.5327ks0.03765 t0.5187, Q = 2L h–1 R2 = 0.93 (17) d = 1.9038ks–0.1282 t0.58248, Q = 4L h–1 R2 = 0.933 (18) Here, d is the position of the maximum wet width of the ground (mm), ks is the saturated hydraulic conductivity (mm s–1), and t is the water application time (min). The presented model sensitivity analysis was performed for the change in parameters, and also in a base case with values t = 120 min, Q = 2 L h–1, d = 44 mm, and Δθ = 0.2. Results are available in Figure 3 and Table 2. Table 2 and Figure 3 show that the presented model has less sensitivity to parameter d compared to other parameters. High model sensitivity in the mode for reducing quantities is related to parameter Δθ, and the mode for increasing quantities is equally more effective in 2 parameters, Q and t. Measured values of wetting front coordinates for 1 loamy (Arpadarasi region) and 2 sandy loam (Khalatpooshan and Karkaj regions) soils with 2 flow rates of 2 and 4 L h–1 were compared with those of the presented model from Eq. (13). The results are presented in Tables 3 and 4 and Figures 4–6. The graphs in Figure 4 show, for the sandy loam soil of Karkaj region, that the measured values of radial distance and those of the presented model are in good agreement with all applied water volumes in mode Q = 2 and 4 L h–1. In mode Q = 2 L h–1 and t = 240 min, agreement between the measured values of the wetting front and the estimated values is high; the RMSE and R2 values were estimated to be 11.506 mm and 0.976, respectively. Considering the graphs in Figure 5 (loamy soil), it is evident that for Q = 2 L h–1 with applications of both 4 and 8 L, the values of the presented model are 300 300 250 250 Radial distance (mm) Radial distance (mm) A. MOLAVI, A. SADRADDINI, A. H. NAZEMI, A. FAKHERI FARD 200 150 base 20% increase of d 20% decrease of d 80% increase of d 80% decrease of d 100 50 0 0 15 30 45 β 60 200 150 100 base 20% increase of Q 20% decrease of Q 80% increase of Q 80% decrease of Q 50 75 0 90 300 0 15 30 45 β 60 75 90 450 400 250 200 Radial distance (mm) Radial distance (mm) 350 150 base 20% increase of t 20% decrease of t 80% increase of t 80% decrease of t 100 50 0 0 15 30 45 60 75 300 250 200 base 20% increase of Δθ 20% decrease of Δθ 80% increase of Δθ 80% decrease of Δθ 150 100 50 0 90 0 15 β 30 45 60 75 90 β Figure 3. Radial distances resulting from base mode of presented model and different modes of the parameters. Table 2. RMSE of model for different modes of parameters. Parameters 20% increase 20% decrease 80% increase 80% decrease In all 4 modes d 4.08 3.91 17.54 3.91 17.54 Q 13.31 15.23 45.97 88.38 50.83 Δθ 12.53 16.4 37.82 150.73 78.38 t 13.31 15.23 45.97 88.38 50.83 733 Estimating wetting front coordinates under surface trickle irrigation Table 3. R2 and RMSE for Q = 4 L h–1 of model for sandy loam soil (Khalatpooshan). T (min) Point Measured 120 Presented model Measured 60 Presented model 1 2 3 4 5 6 7 240 232.43 250.59 240.83 266.27 291.10 300 235.9 258.93 268.38 281.11 293.66 304.17 307 200 213.77 206.15 209.34 208.08 199.24 230 192.4 198.34 207.23 220.73 228.43 236.54 239 RMSE (mm) R2 22.75 0.71 18.17 0.13 20.59 0.818 60, 120 Presented model Horizontal advance (mm) Horizontal advance (mm) 0 0 0 50 100 150 200 250 Vertical advance (mm) -100 -150 -200 -150 -200 -250 Measured -350 Presented model Q = 2 L h-1 t = 120 min Q = 2 L h-1 t = 240 min 0 50 100 Presented model Horizontal advance (mm) Horizontal advance (mm) 0 150 200 0 250 0 100 200 300 -50 -50 -100 -100 Vertical advance (mm) Vertical advance (mm) 300 -100 Measured -300 -150 -200 -150 -200 -250 Measured -250 Presented model Q = 4 L h-1 t = 60 min Measured -300 -350 Q = 4 L h-1 t = 120 min Figure 4. Measured and calculated wetting fronts in sandy loam soil (Karkaj region). 734 200 -300 -250 -300 100 -50 -50 Vertical advance (mm) 0 Presented model A. MOLAVI, A. SADRADDINI, A. H. NAZEMI, A. FAKHERI FARD Table 4. R2 and RMSE values of presented model. Sandy loam (Khalatpooshan) Sandy loam (Karkaj) Emitter discharge (L h–1) RMSE R2 R2 (mm) RMSE 0 RMSE R2 (mm) RMSE (mm) 0.982 9.66 0.931 10.43 0.879 13.25 0.921 11.22 4 0.976 12.07 0.818 20.59 0.475 30.52 0.787 22.39 2,4 0.976 11.506 0.836 16.89 0.65 23.53 0.82 17.85 Horizontal advance (mm) 50 100 150 200 0 250 0 50 100 150 200 250 300 -50 Horizontal advance (mm) Horizontal advance (mm) In all 3 soils 2 -50 -100 -150 -200 Measured 50 100 -150 -200 -250 Measured -300 Q = 2 L h-1 t = 240 min -350 150 Presented model Horizontal advance (mm) Horizontal advance (mm) 0 -100 Presented model Q = 2 L h-1 t = 120 min -250 0 R2 (mm) Horizontal advance (mm) 0 Loam 0 200 250 0 100 200 300 -50 Horizontal advance (mm) Horizontal advance (mm) -50 -100 -150 -200 Measured -250 Q = 4 L h-1 t = 60 min -100 -150 -200 -250 Measured -300 Presented model -350 Q = 4 L h-1 t = 120 min Presented model Figure 5. Measured and calculated wetting fronts in loam soil. 735 Estimating wetting front coordinates under surface trickle irrigation Horizontal advance (mm) 0 0 50 100 Horizontal advance (mm) 150 200 0 250 200 300 -100 Vertical advance (mm) -100 -150 -150 -200 -200 -250 Measured -250 Presented model Q = 4 L h-1 t = 60 min -300 Measured -300 Presented model Q = 4 L h-1 t = 120 min -350 Horizontal advance (mm) 0 0 50 100 150 Horizontal advance (mm) 200 0 250 0 100 200 300 -50 -50 -100 Vertical advance (mm) -100 Vertical advance (mm) 100 -50 -50 Vertical advance (mm) 0 -150 -150 -200 -200 Measured Presented model -250 Q = 2 L h-1 t = 120 min -300 -250 Measured -300 -350 Q = 2 L h-1 t = 240 min Presented model Figure 6. Measured and calculated wetting fronts in sandy loam soil (Khalatpooshan region). in high accordance with the measured data of the wetting front coordinates. However, for Q = 4 L h–1, the accordance is not as good as in the application mode of Q = 2 L h–1. In application mode Q = 4 L h–1 with 4 L of volume, less matching is seen, and the R2 and RMSE of the model in this soil are 0.65 and 23.53 mm, respectively. Measured radial distance (mm) 350 300 y = 0.8613x + 26.004 R2 = 0.8236 250 200 150 100 100 150 200 250 300 Estimated radial distance (mm) 350 Figure 7. Comparison of measured and estimated radial distances of wetting fronts in all 3 soils. 736 Results in the sandy loam soil (Khalatpooshan region) show good agreement between the measured data of the wetting front coordinates and those of the presented model for Q = 2 and 4 L h–1 for both application volumes of 4 and 8 L. In the application mode of Q = 2 L h–1 with a volume of 8 L, there is A. MOLAVI, A. SADRADDINI, A. H. NAZEMI, A. FAKHERI FARD particularly high accordance and R2 and RMSE are estimated to be 0.836 and 16.89 mm, respectively. The advantage of using the presented model as compared with other methods such as those of Schwartzman and Zur (1986), Li et al. (2004), Thabet and Zayani (2008), and Amin and Ekhmaj (2006) is that the presented model is a geophysical model that anticipates the profile of the wetting front, while the other mentioned methods are often experimental and can only estimate the width and depth of the wetting front. Discussion Results of these 3 soils showed that the measured values of wetting front coordinates for both Q = 2 and 4 L h–1 and for both 4 and 8 L of water volume had very good accordance with those of the presented model. The R2 and RMSE values of the model for these soil types were estimated to be 0.82 and 17.85 mm, respectively (Figure 7). The results in Table 2 show that the presented model in the sandy loam soils (Karkaj and Khalatpooshan) has higher R2 and lower RMSE values than in the loamy soil (Arpadarasi). Model inputs are the average change in volumetric water content, soil saturated hydraulic conductivity, flow rate of emitter, and water application time. 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