Optimum Spacing for Agriculture Drainage Espacement optimum des drains pour la production agricole

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1 Scientific registration n o : 56 Syposiu n o : 9 Presentation: poster Optiu Spacing for Agriculture Drainage Espaceent optiu es rains pour la prouction agricole ARROYO Victor (1), GARCIA Nahun (1), NAMUCHE Roolfo (1) (1) Instituto Mexicano e TecnologÌa el Agua, Paseo Cuauhnahuac, 853. C.P. 655 Jiutepec, Mor. Mexico Abstract Agricultural prouction has been irectly relate to grounwater levels. Seasonal variations an irrigate volues ake it ifficult to eterine the tie relation between a given water table an prouction. There is sufficient fiel ata to establish epirical relationships between grounwater an crop yiel. Criteria were propose to calculate optiu rain spacing as a function of average grounwater epth an of syste installation costs. Spacing was foun to optiize the relationship between percent prouction per unit space between rains an the percent cost for piping an installation. The resulting oel consiers the ain physical paraeters incluing hyraulic recharge an conuctivity, rain raius an epth, an spacing. The etho is illustrate using fiel ata taken fro Irrigation District 76, Mexico. The results showe that a relative ecrease in yiel is accopanie by a large ecrease in investent. Introuction That agricultural prouction is irectly relate to grounwater levels is supporte in the literature (ILRI, 1977; Martinez, 199; UNA, 1969). Although seasonal variations an irrigate volues ake it ifficult to quantify this relationship, it is generally accepte that there is a goo association between prouction an average grounwater epth (ILRI, 1979). The authors propose the evelopent of criteria to calculate optiu rain spacing as a function of average grounwater epth an of syste installation costs. This optial spacing axiizes the relation between the percent prouction per unit space between rains an the percent cost for piping an installation. The oel prouce consiers the ain physical paraeters affecting the calculation of spacing such as the hyraulic recharge an conuctivity, rain raius an epth, epth of ipereable stratu, an, of course, spacing. Data fro a plot in Irrigation District 76, El Carrizo, are presente to illustrate the propose etho. 1

2 Average Yiel Between Drains Developent of Moel Herbert (Miles an Kititto, 1989) propose Eq. (1) to escribe the shape of the water table between two rains uner constant flow, K h + q x = q L x + K h (1) h is calculate fro h = ql 4 K ln(.5 r ) + H π The epth of the water table fro a given point x is euce fro: () h = + P - qlx + Kh - q x K Equation 3, cobine with an expression that relates yiel with grounwater epth perits the calculation of the crop yiel for a given cross section between two rains. The iscrete functions that relate these factors are eterine in the fiel. They shoul be expresse as continuous function for inclusion in a atheatical oel. Base on the foregoing, a relation between percentage crop yiel, R(%), representing the yiel R for a epth h, an the axiu yiel, R o, for an optiu grounwater epth, h o, is obtaine: R(%) = R R 1 (4) If there is a quaratic function that correlates grounwater epth an crop yiel, By substituting Eq. (5) in Eq. (4), (3) R = a 1 + b 1 h + c 1 h (5) R(%) = a1 R + b1 R h + c1 R h (6) an by reefining Eq. (6) can be expresse as a = a1 R ; b = b1 R ; c = c1 R (7) R(%) = a + b h + c h (8) where h is obtaine using Eq. (3) an the coefficients a, b, an c are eterine using correlation techniques.

3 Substituting Eq. (3) in Eq. (8), an expression is obtaine to eterine the percentage crop yiel R(%) for a given section as a function of the esign paraeters use in the rain spacing calculations qlx + Kh - q x qlx + Kh - q x R(%) = a + b + P - + c + P - (9) K K By integrating Eq. (9) between x L, an iviing by L, the average percentage yiel R(%) 1per unit space between rains is R(%) = a+ b( + P ) + c( + P ) + cq L 6K + c h b + c( + P ) 1/ K L h + 4 K h + q L ql arcsin 1/ L K 4 q q L + 4qK h Equation 1 shoul be use within the valiity range of ajuste Eq. (8). When a single polynoial has not been properly ajuste, that is when the correlation coefficient is low, a set of quaratic polynoials is recoene to plot iscrete functions (Buren an Faires, 1985). The cost of the piping an installation for a given istance between rains (L) can be expresse in percent C(%) with respect to the syste cost for a istance (L) to obtain the axiu yiel. This expression can be approxiate by a lineal relation, When Eq. (1) is ivie by Eq. (11), C(%) = L L (1) 1 (11) R(%) C(%) = L + Lc h 3 α β L h L γ arcsin( ω) + L δ ε h arcsin( ω) (1) 1 L where a a + b( + P ) + c( + p )² = β = b + c( + P ) ε = K γ = 1L [ ] q b + c( + P ) 4 K L c q δ = 6 K L [ b + c( + P )] 1 ql have been reefine. ϖ = q L γ = ql ² q [ b + c( + P )] 4 K L 3

4 Applying traitional optiization techniques an ifferentiating, a relationship is obtaine to eterine the length L that axiizes Eq. (1), to prouce c α + λ β λ 1L [ L h + h ] ( L+ h ) (13) γ[ φ L + L arcsen( ω)] + 3δ L ε[ φ h + λ h arcsen( ω)] = where q φ = q² q² 1 q² Lq² + 8λqKh ql q² ( q² ) q ln.5 r λ = 4πK Equation (13) can be solve using an iterative nuerical etho, such as Newton-Raphson, Fixe-Point Iteration or bisection algorith (Buren an Faires, 1985). q ln.5 r λ = 4π K Percentage Yiel Functions-Grounwater Depth Figure 1 illustrates a set of percentage yiel functions with respect to grounwater epth. These relationships are vali for teperate areas an are generally applicable to all crops (UNA, 1969). In Fig. (1), the curves refer to ifferent soil classifications: 1. clay;. loa; 3. sany; 4, 5, & 6 ifferent proportions of san; 7, huus. Case Stuy The ata for the physical paraeters in this exaple were taken fro stuies of a plot in the El Carrizo Irrigation District. The soil was classifie as clay. Curve 1 in Fig. (1) for clay soil was ajuste to an exponential equation for the "wet ar" (ascening portion); a constant value of R(%)=1% for h>1.1 was assue for the "ry ar" (escening portion). Base on these values, the valiity range for the ajuste equation was h 1.1. Figure shows the secon egree equation that resulte. Fiel ata: Hyraulic conuctivity, K =.15/hr; target hyraulic pressure in central area, h =.4; rain raius, r =.5; recharge, q =.1/hr; epth of ipereable stratus = 3.5; rain epth, P = 1.4. Results The behavior of the grounwater level for a range of rain spacings is shown in Fig. (3). The optiu yiel (1%) is obtaine with a water table of 1.1 in the stuy area. This yiel was 4

5 obtaine with a rain spacing of less than 4 (Fig. 4). For a yiel of 8%, the return on investent, or percentage cost, ecreases alost 5% (Fig. 4). Fro Fig. (4), it ay be eterine that the optiu yiel/cost proportion (financial axiu) is obtaine with a spacing of 5 (fro Eq. (13) 49.77) with a yiel percentage of 8% an an investent of 5% of the axiu at a spacing of 4. It ay be conclue that a technically- an econoically-viable alternative for rain spacing is 5. Future growth coul conteplate placing intereiate rains that woul raise crop yiel to 1%. This ecision coul be ae later base on ata that coul justify a % increase in yiel through an aitional 1% investent with respect to the initial installation. An iportant isavantage in a rain spacing of 5 with respect to the 4 optiu is the tie require for rainage uner floo conitions (for this exaple estiate at 4 ays). This tie ay be iprove upon through the use of ole rain to increase the rainage rate in the superficial layer, an iprove aeration an soil porosity. Conclusions an Recoenations The technique escribe in this article ay be use to preict yiel ata for profitability stuies an take ecisions for investent projects an return on capital. While theoretically well-base, aitional stuies ust be ae of the prouction functions with respect to grounwater levels in all zones that require rainage. Notation C(%) = percentage cost of piping an installation H = water potential in rain [L] h = hyraulic pressure in a given section [L] h = water potential above rain [L] h = epth fro water table to given cross section [L] K = hyraulic conuctivity [L/T] q = unifor recharge (rain) [L/T] R(%) = percentage crop yiel R = crop yiel for a epth h r = rain raius [L] = istance fro ipereable stratu to rain[l] x = spatial coorinate [L] P = rain epth [L] L = istance between rains [L] L = istance between rains to obtain the axiu yiel [L] References Buren, L. an J. D. Faires An lisis NuÈrico Cap. 3. Grupo Eitorial Iberoaericano, C. MÈxico. MÈxico, 71 pp. International Institute for Lan Reclaation an Iproveent (ILRI) Drainage principles an applications. Publication 16. VOL.1. Wageningen Holana, pp MartÌnez-Beltran, JulÌan V Curso Internacional e Riego y Drenaje. Voluen 1. CENTER, Spain. 39 pp. Miles, J.C. an Kititto K "New Drain Flow Forula," J. Irrigation & Drainage Engineering. VOL.115, No., April, pp Universia Nacional Agraria - La Molina (UNA) II Curso Nacional e Drenaje e Tierras AgrÌcolas Too 1. CENDRET, La Molina, Lia, Peru, February 4 - March, pp..5/1-.5/1. Key wors: Drainage, econoical aspects, prouction, oelling. Mots clés : Drainage, aspects éconoiques, prouction, oélisation. 5

6 Figures Figure 1: Water table-crop yiel relation. Figure : Ajuste curve 1 for clay soil. Figure 3: Effect of rain spacing on water table. Figure 4: Crop yiel, investent, an yiel/investent ratio. 6