A furrow irrigation model to improve irrigation practices in the Gharb valley of Morocco

Agricultural Water Management 42 (1999) 65±80

A furrow irrigation model to improve irrigation practices in the Gharb valley of Morocco Jean Claude Mailhola,*, Morgan Priola, Mohamed Benalib a

Cemagref, French Institute of Agricultural and Environmental Research (Irrigation Division) 361, rue J.F. Breton, BP 5095, F34033, Montpellier, France b ORMVAG, BP 79, Kenitra, Morocco Accepted 13 January 1999

Abstract In developing countries, modernization of surface irrigation is the most common solution to water management problems in irrigated areas because it is well adapted to the socio-economical context. This solution was adopted in the Gharb area near Kenitra in Morocco where an experimental site was set up to obtain irrigation and drainage references. Meaningful improvements in irrigation efficiency and better crop yields have yet to result from the modernization effort. Different sources of heterogeneity affecting the infiltration process can hinder the improvement of irrigation efficiency even in a modernized furrow irrigation context. The respective impact of deterministic and stochastic heterogeneity sources on the advance-infiltration process is analyzed. Then, a model based on the spatial and temporal variability of infiltration is developed to simulate the impact of irrigation practices on water saving during an irrigation season. This work will later contribute to the elaboration of a modelling approach simulating fertilization and irrigation practices. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Furrow irrigation; Advance-in®ltration variability; Stochastic simulation

1. Introduction The introduction of sprinkler irrigation has been perceived to be failure in the Gharb area. High energy costs and maintenance problems are the main reasons. Consequently, surface irrigation was adopted in this area. Modernization was insured through field levelling using laser techniques and an improved system of water distribution (siphons, * Corresponding author. +33-04-67-04-63-00; fax: +33-04-67-63-57-95 E-mail address: [email protected] (J.C. Mailhol) 0378-3774/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 7 7 4 ( 9 9 ) 0 0 0 2 4 - 4

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floppy pipe and gated pipe). However, modernization alone cannot ensure significant water saving and nitrate leaching reduction in a scarce water resource context. Good land preparation and efficient watering methods can significantly improve the irrigation efficiency. Farmers often neglect these two aspects. Land preparation does not always takes place in favourable conditions (soil sometimes too wet or sometimes too dry). Tillage equipment particularly in heavy clay soils such as those of Gharb plain is often illadapted. The closed-end furrow (CEF) is commonly used due to a lack of ditch maintenance, but it is not always suitable particularly when the advance variability is high. Different sources of heterogeneity (especially concerning water inflow and soil infiltration) can hinder better irrigation efficiency at the plot scale. A modern system such as the gated pipe system can limit variation coefficients of discharge input (Cv(Qin)) to 5% when some elementary criterion of installation are respected (Trout, 1990). In contrast, this performance level is often difficult to reach with the floppy pipe system. An improvement in irrigation efficiency through inflow setting adjustments is often questionable owing to heterogeneous soil surface conditions mostly resulting from poor land preparations. The main objective of this study is to optimize irrigation efficiency at the plot scale during the irrigation season through the analyses of irrigation practices. Owing to spatial and temporal variability of soil properties, our objective is not easy to achieve. Wetting and flooding can induce soil structural changes (Collis-Georges and Greene, 1979; Kemper and Rosenau, 1984; Or, 1996), and as a result, the infiltration properties may vary from one irrigation event to the next even in similar soil moisture conditions. Soil capillarity decreases right from the first irrigation event to the next during the same season. At the same time soil compaction and cracking magnitude increase in heavy clay soils. Optimisation efficiency can be obtained by means of a modelling approach of the advance-infiltration process taking into account the spatial and temporal variability of the infiltration characteristics. Spatial variability of the infiltration is often governed by a random process whose origin can be attributed to cracking development, running furrows conditions (or ridging), inlet discharge variability etc. But sometimes, significant soil properties change (change of soil type or change in soil water content (SWT)) add a deterministic cause to the variability of the infiltration as we shall see later. Temporal variability of the infiltration characteristics is governed by two phenomena. The first is closely connected to SWC variation which is governed by the climatic demand. The second is due to the soil structure change through the irrigations. The latter, which is particularly significant in a surface irrigation context, cannot be reliably predicted (Or, 1996). Consequently the variability of the infiltration parameter, which is mainly affected by the soil structural change, cannot be modelled. In order to get to grips with these problems of variability, the study was conducted over two irrigation seasons under contrasting land preparation conditions (i.e. sugar-cane ridging and running furrows) and in different climatic conditions. Before building the model which will enable the irrigation practices to be simulated and optimized, it is necessary to identity the sources of heterogeneity and evaluate their respective level of

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impact on the advance process which itself reflects the infiltration conditions (Renault and Wallender, 1992). This study is a part of an irrigation and drainage project which aims at improving water management in the Gharb area in Morocco. 2. Materials and methods 2.1. The experimental field context The experimental field consists of three sugar-cane plots in heavy clay soil (%clay > 65%) that have been laser levelled (S0 ˆ 0.2%). They are irrigated using, respectively, gated pipe (Plot P1), floppy pipe (Plot P2) and siphons (Plot P3). The first two plots are 230 m long and 100 m wide. The third plot is 175 m long and 150 m wide. The inter furrow spacing is Es ˆ 1.5 m. P1 and P2 are drained at 0.9 m and 1.3 depth, respectively whereas P3 in not drained (the water table [NaCl] ˆ 12 g/l, fluctuates between 1.5 and 2 m during the irrigation season). Drains are perpendicular to furrows with a 20 m spacing. Siphons are supplied by an earthen canal with a flow rate of 22 l/s and gated pipe and floppy pipes have a maximum flow rate up to 38 l/s. Water supply to plots is controlled using constriction water meters. Due to the inadequate functioning of draining ditches the irrigation is conducted using the CEF, the method commonly applied in the Gharb area. Sugar-cane was planted on 15 September 1995. After planting, P3 was irrigated three times and P1 and P2 twice. Sugar-cane ridging was late (end of May 1996) so that land preparation was not conducted in favourable conditions. The soil being too dry and compacted by the flooding of January 1996, it was irrigated in order to break it up. Unfortunately the running of furrows was particularly bad on some parts of the plots where the soil was still too wet (P2 in particular). As a result furrow beds are generally chaotic (a micro-basin generally preceded a big lump) in 1996. Sugar-cane was harvested in September 1996 (to obtain plant cuttings). In 1997 sugar-cane ridging (beginning of May) was conducted in good soil conditions in contrast with that of 1996. As a result, the shape and the bottom of the furrows were more regular during this irrigation season. Neutron access tubes of 1.5 m length were installed just before the beginning of the field tests of 1996 at upstream and downstream end of each sub-plots. In order to analyze the impact of the CEF practice on the infiltration conditions along the furrows, other neutron access tubes at 25 m interval from middle plot to downstream were installed on the first sub-plot of P3. They were laid between the top of the furrow and the sugar cane line to minimize the risk of preferential infiltration due to cracks. Advance stations allowing easy monitoring of a substantial number of furrows were installed in 1996. Drainage is collected in a underground room and monitored using an automatic gauging system. Three field tests were summarily carried out on P3 during irrigation of planting. During the 1996 irrigation season two field tests were conducted from mid July (around two weeks after the second water application of the season) to the end of August and three from 15 June (also after 2nd water application) to the end of July during 1997 irrigation season. They consist of monitoring the advance process of the 25 furrows of P1, 23 of P2

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and 15 of P3 (except P3 2nd test of 1997 where 30 furrows were monitored at a time). Inflow of each furrow is controlled using Parshall flumes installed when front advance is sufficiently far from the upstream end in order to avoid disturbing the advance process. Three free draining furrows (FDF) are maintained in order to validate model calibrations of stabilized infiltration rate (on advance trajectory) and model predictions of infiltration by monitoring runoff (using Parshall flumes). Before each sub-plot watering, and 48 h later, the SWC profile was measured. Cutoff time tco of the different water applications vary from 180 to 420 min and regarding discharge values, most irrigation tests use an inlet discharge of around 1.5 l/s per furrow. But lower discharge values near 1 l/s and 0.7 l/s were used on P1 and P3 in 1997, respectively. Over the whole irrigation season, water supply was not as high in 1997 as it was in 1996 due to a lower discharge inlet and/or because of shorter cutoff times as shown in Table 1. At last a complete meteorological station exists on the experimental site which enabled us, in particular, to obtain daily evapotranspiration reference (ET0) values. 2.2. Identification of heterogeneity sources As mentioned above, inflow variability and soil conditions are the main sources of heterogeneity in furrow irrigation. Inflow variability can be characterized by the variation coefficient Cv(Qin). Under our field conditions Cv(Qin) varies from 5% to 35%. On one occasion we kept the variation coefficient to 8% on 25 furrows of the gated pipe system (although it is possible with this system to reach 5% as reported by Trout, 1990) whereas 5% was obtained on a P3 sub-plot (15 siphons). With the floppy pipe system significant efforts were required to attain a level close to 15%. For the latter, Cv(Qin) values greater than 35% can be obtained when no care is devoted to the system setting. Soil heterogeneity is due to the crack development process and tillage operations (inducing clean or no-clean till furrows conditions during the Spring sugar-cane ridging and running furrows). They both characterize the random component of soil variability whereas the impact of the CEF practice characterizes the deterministic component, assuming a homogeneous soil structure at the plot scale. For a given advance station of a plot, the correlation level (R2) between advance time and discharge inlet (of the different furrows) can be considered as an indicator of land preparation quality. A significant difference in the quality of correlation was observed between 1996 and 1997 irrigation seasons under similar soil moisture conditions. This difference is attributed to the contrasting forms of land preparation (sugar-cane ridging and running furrows) of the two seasons the causes of which have been previously evoked. Since infiltration is a function of opportunity time, discharge variability should have a more significant impact on water application depth variability in clean till furrow conditions than in no-clean till furrow conditions, where Cv(Qin) must reach 20% at least to induce a significant value of R2. This Cv(Qin) value can be considered as the threshold value under which soil surface status masks inflow variability under the no-clean tilled furrow conditions of 1996. Soil samples analysis show that soil texture is fairly constant. Thus, we assume that the stochastic origin of infiltration is due essentially to the crack phenomenon whose

Table 1 Field test results Bup (l/m)

Cv (%)

Bdn (l/m)

Cv (%)

Qin (l/s)

Cv (%)

0.06 0.07 0.07 0.07 0.06 0.05 0.05

15 17 19 19 16 16 16

1.58 1.62 1.63 1.57 1.44 1.48 1.45

12 13 35 21 9 8 5

0.06 0.06 0.06 0.05 0.05 0.06 0.07

12 12 20 20 12 12 15

1.57 1.40 1.53 1.58 1.41 1.44 1.45

0.05 0.06 0.05

7 10 8

0.05 0.05 0.05 0.05 0.04 0.05 0.06 0.05 0.05

TL (min)

Cv (%)

Nj

MET (mm)

200 235 230 245 116 98 174

14 15 37 18 12 8 6

10 12 10 12 9 9 11

60 72 60 72 52 52 65

87 95 79 95 99 99 100

65 95 95 85 65 67 66

8 8 29 24 12 17 15

293 222 256 201 105 157 143

6 9 24 24 16 6 14

18 12 16 12 11 15 15

91 57 69 53 40 55 62

87 95 95 63 66 60 67

102 95 95 87 55 79 65

1.43 1.54 1.23

9 15 21

279 208 186

10 19 11

21 17 19

97 75 85

55 70 63

84 101 67

7 9 11 5 5

1.41 1.33 1.50 1.23 0.68

12 8 15 13 15

171 176 142 157 131

6 11 13 10 13

10 11 9 11 8

47 52 42 52 37

59 44 52 52 80

55 53 48 58 38

9 5 7 8

0.94 1.52 0.74 1.33

10 16 15 17

243 184 316 93

10 17 12 14

11 13 16 11

47 55 69 47

55 53 38 42

44 50 63 61

PWAD (mm)

69

Nj: interval in days between two water applications; PWAD: previous WAD; *: with Cutback; P1 and P2 sub-plots: 0.945 ha; P3 sub-plot: 0.405 ha.

WAD (mm)

J.C. Mailhol et al. / Agricultural Water Management 42 (1999) 65±80

3rd Irrigation (field test nb 1) from 15/7/1996 P11 74 16 40 18 P12 99 15 35 19 P21 74 19 45 20 P22 81 14 48 17 P31 44 16 42 16 P32 36 17 32 17 P33* 71 13 65 13 6th Irrigation (Field test nb 2) from 19/8/1996 P11 120 11 70 14 P12* 75 9 41 14 P21 90 22 68 16 P22 70 18 50 20 P31* 44 7 30 15 P32 71 17 50 19 P33* 58 16 40 18 3rd Irrigation (Field test nb 1) from 17/6/1997 P11 97 13 61 14 P21 77 9 52 14 P32 75 14 50 13 5th Irrigation (Field test nb 2) from 8/7/1997 P11* 60 13 27 23 57 10 25 21 P12* P21 50 14 29 12 P22 48 14 20 14 P31,2 25 18 15 27 6th Irrigation (Field test nb 3) from 22/7/1997 P11 55 9 22 25 P21 63 18 40 20 P31,2 89 14 40 27 P33* 40 17 23 25

Cs Cv (l/m/min) (%)

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Fig. 1. Advance velocity of average furrows under the 3rd irrigation conditions of 1997 on the different plots.

magnitude may vary along and between furrows (Mailhol and Gonzalez, 1993). In noclean till furrow we have to add the storage in the micro basins preceding lumps (devoted to infiltration). Deterministic origin of infiltration variation is due to CEF practice as attested by the graphic of Fig. 1 which represents an evolution example of the front advance velocity from the upstream to downstream end of the mean furrow of P1 (Sub-plot 1) P2 and P3 (Sub-plot 1) for 3rd irrigation of 1997. We observe that velocity decreases slowly from x ˆ 0 until stabilization at approximately x ˆ 160 m for P1 (idem for P2) and 100 m for P3 then sharply increases until furrow end. This change in advance front velocity is due to changes in infiltration condition as proved by the initial soil moisture evolution profile from the upstream to the downstream end. Fig. 2 which plots this phenomena along with attested field observations clearly show that crack phenomenon is less pronounced at the downstream end of the plot than at the upstream one. In addition, flooding occurring at the end of the plot (the plot can be pounded for 24 h) induces soil structural changes surely greater than those induced by wetting of furrows length (the upper part of the furrows is not pounded). Maximal water depths stored at furrow end vary from 15 to

Fig. 2. Soil moisture profile before (^) and after (&) 3rd irrigation in 1997 on Sub-plot 1 of P3.

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18 cm. That value, multiplied by S0 ˆ 0.002 m/m gives approximately a dike close to 75 m in length. 2.3. An operative modelling to simulate the advance-infiltration process on cracking soil The relevance of the cumulative infiltration: I …l=m† ˆ B ‡ Cs t

(1)

to conceptually represent the infiltration process in cracking soil has been justified by Mailhol and Gonzalez (1993), where B (l/m) is the quantity of water per unit of furrow length. It is nearly instantaneously trapped by cracks and/or surface micro-basins. Parameter Cs (l/m/min) is the stabilized infiltration rate. This infiltration equation induces the exact advance solution:    Q ÿtCs 1 ÿ exp (2) x…t† ˆ A0  ‡ B Cs B and Cs are calibrated on the advance trajectory whereas A0, the wetted cross-section at the upstream end, is estimated by the Manning±Strickler equation for a trapezoidal standard cross-section and , the water line shape coefficient, is settled to 0.8. Eq. (2) simulates well the advance process on cracking soils when calibration is conducted on homogeneous furrow lengths. The infiltration equation only has two parameters which offer some advantages such as the use of a spatial approach and the possibility of finding more significant relationships from cause to effect as we further shall see. Moreover, calibration is more efficient when the selected infiltration equation induces an exact advance solution (Mailhol et al., 1997). The basic model was developed for FDF (Mailhol and Gonzalez, 1993). Different options, allowing irrigation practices to be tested, were added such as the CEF option described in Mailhol et al. (1997). In order to take into account the deterministic aspect of infiltration variation, a last option was further added. It is summarized in the following paragraph. Using Laplace transform properties, Renault and Wallender (1994) have proposed a analytic solution for advance velocity (by resolving the water balance equation WBE i.e. the renowned Lewis and Milne equation) in case of heterogeneous furrow infiltration conditions using Horton's infiltration. Since the asymptotic behavior of Horton's equation …Z ˆ bf…1 ÿ exp…ÿt†g ‡ f0 t† to Eq. (1) is particularly well pronounced in heavy clay soil context, this analytic solution could be directly applied to simulate the advanceinfiltration process from upstream to downstream end of plots. Such a proposition appears justified as that advance solution derived from Horton's equation simulates the observed advance process as well as Eq. (2) when great  values (  2) are used (with b ˆ B and f0 ˆ Cs). The same mathematical procedure as Renault and Wallender was used in the case of Eq. (1). But unlike them, it was not conducted using the derivative expression of water balance equation (which leads to the elimination of the B term) but directly on the WBE as proposed in the article of Mailhol and Gonzalez (1993). In the case of two heterogeneous furrow lengths, model calibration consists of using the method based on the optimization technique evoked in Mailhol (1992) and analyzed

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in Mailhol et al. (1997) to calibrate, for a given furrow i, Biup and Csi on the upstream part, (from x ˆ 0 to x ˆ 150 m for P1 and P2 and x ˆ 0±100 m for P3 as justified by Fig. 1) and Bidn manually (a single parameter calibration) on the downstream part (from 160 to 230 m for P1 and P2 and from 100 to 175 m for P3), assuming Cs is spatially constant. 2.4. Temporal and spatial variability of the infiltration parameters of the model Averaged value of parameter Cs decreases from 0.25 l/m/min (planting irrigations of September±October 1995) to a stabilized value of 0.05 l/m/min obtained during the third irrigation of 1996. During the planting irrigations Cs equals infiltration rate at time t ˆ TL (TL: advance time, the time required to reach the end of the plot) rather than stabilized infiltration rate due to the substantiality of capillary forces. At this period, Cs is considered as time dependent. The average value of 0.05 l/m/min derived from advance calibration during the first field experiment of June 1996 match well those obtained from outlet discharge stabilization. To date from this period, capillary forces become very low due to soil compaction. So the cracking phenomenon inducing non-darcyan flow conditions legitimizes the choice of the conceptual infiltration model. In the case of permanent crop such as sugar-cane we can accept Cs inter-annual stability because the impact of 1997 spring ridging on a possible Cs increase was not observed. Due to B magnitude, as we shall see later, the advance process is quite insensitive to Cs parameter variation. For instance, a 30% Cs variation around the central value 0.05 l/m/min, gives less than 2% advance time variation only. Owing to its low impact on the advance process (in contrast to that of B) and accordingly to its low effect on the water application depth estimation (cutoff time tco is generally lower than twice advance time TL) Cs will be assumed constant from the beginning of the first irrigation season (that follows the planting irrigations) with an average value near to 0.05 l/m/min. In addition, due to the homogeneity of the soil structure context, Cs will be assumed constant with space. The value of B is related to crack size which can be linked to the water plant consumption between subsequent water applications. This later under full irrigated conditions, can be assimilated to maximal evapotranspiration magnitude MET (MET ˆ Kc ET0, Kc ˆ crop coefficient). So, the temporal variability of B should be greatly governed by MET. As shown by Childs et al. (1993) soil variability causes greater infiltration compared with intake opportunity time variability for all irrigation. That is still more relevant in case of cracking soils due to cumulative infiltration shape. In our soil context, the origin of infiltration variation from upstream to downstream end is due to the practice of CEF as explained above. Having assumed Cs constant with time and space for a given furrow i, two heterogeneous furrow lengths are characterized by the Biup and Bidn infiltration parameters. Averaged B …B ˆ 1=Nf Bi where i ˆ 1; Nf furrows of a given sub plot) and that resulting from calibration conducted on the mean furrow (that obtained from averaged time advance with averaged inflow values as input discharge) are not significantly different due to the linear shape of the infiltration equation. Consequently, B value (Biup or Bidn ) of a given plot for a given water application will signify as well the averaged Bi

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values as the average value of B resulting from mean furrow calibration. The Bi values, fit well a Gaussian distribution for a given plot. The motivation behind this study being the development of a model to simulate irrigation practices and their impact at the plot scale for the whole irrigation season, it is necessary to verify that a predictive approach for the infiltration can be proposed. 2.5. A predictive approach for infiltration estimation Owing to the complexity of physical phenomena involved in the furrow irrigation process, empirical modelling approaches still continue to be widely used to propose operative solutions to water management problems. Nevertheless, due to the lack of physical meaning of some parameters such as those governing infiltration, it is often impossible to identify significant links between external variables and the infiltration parameters. So, the simulation possibilities of numerous contrasted scenarios are limited. Childs et al. (1993) have evoked the possibility of infiltration prediction for future irrigations, having observed that infiltration (modelled using extended Kostiakov equation) of two subsequent irrigations were correlated (R2 ˆ 0.64). Since only one parameter is assumed to represent infiltration condition changes and because we are dealing with a permanent crop, one can expect to find reliable links between our sole infiltration parameter and external variables. An encouragement to that is provided by the significant correlation (R2 ˆ 0.833) between average Bup values and MET Bup ˆ 64 L n…MET† ÿ 198

(3)

This simple relation, established for the clean till furrow context of 1997 (under LAI conditions allowing a Kc value of 1 to be adopted) could be used provided that each water application fills the easily usable water reserve (EUWR). This first result militates in favour of finding reliable relationships between B and a soil dryness status indicator. Such a type of relationship can be accepted between average B values of upstream part of plots (Bup) and SWC value before each water application calculated using a simple water balance model. Just before each irrigation, SWC can be expressed as: SWC…t† ˆ SWC…t ÿ 1† ‡ Irrig…t ÿ 1† ÿ MET;

(4)

where Irrig(t ÿ 1) is the water supply to the root zone during the previous watering. About the first 60 cm soil depth seems to be primarily affected by plant water consumption as shown by neutron probe measurements, conducted before each water application (Fig. 2), and root profile density. Since maximal available water stored MAWS ˆ 170 mm/m (derived from Marcesse, 1967 method) maximal soil water capacity can be settled to MSWC ˆ 100 mm. In such a soil context the B value can be used to predict upstream infiltration is related to soil water depletion SWD which is estimated by means of: SWD…t† ˆ MSWC ÿ SWC…t†

(5)

The significance of Bup/SWD relationship: Bup ˆ 0:968 SWD ÿ 1:67

(6)

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Fig. 3. Relationship between parameter B (upstream part of plots) of infiltration and a soil water depletion factor.

is attested by Fig. 3 where Bup values are obtained from the averaged Biup values of each test filed during the 1997 irrigation season. Water supply is currently estimated assuming an irrigation efficiency of nearly 80%, however, the model will estimate the real efficiency in a later version. So, Irrig(t ÿ 1) (PAWD in Table 1) is multiplied by 80% whereas SWC(t ÿ 1) is assumed to be close to EUWR with EUWR ˆ 0.5 MSWC). Two parts of the plot being solicited by the same climatic demand, a very good correlation should be expected between Bup and Bdn assuming Cs is constant with space. In fact, because impounding length is not exactly the same for each furrow and that infiltration changes are progressive, correlation is not as good as expected. It is nevertheless significant. The correlation range between Bup and Bdn varies from r2 ˆ 0.60 to r2 ˆ 0.91 through the different field tests of 1997, whereas, the determination coefficient of the mean value Bup and Bdn relationship: Bdn ˆ 0:64 Bup ÿ 5:3

(7)

is r2 ˆ 0.79. Relationships such as those proposed by Eqs. (6) and (7) would permit to predict the values of Bi (using an adequate space step) along the whole length of furrow i. But we cannot assume that the cracking development process is exactly the same after each water application due to the low cross correlation between Bup values of two subsequent water applications (R2 ˆ 0.0571 on the example of irrigation number 5 and 6 of 1997 of first P1 sub-plot under similar initial soil moisture conditions). So a stochastic approach would appear to be a realistic way of simulating the variability of the advance-infiltration process at the plot scale. 2.6. A stochastic simulation of advance-infiltration variability process A stochastic simulation requires in addition to the mean values, standard deviation knowledge concerning the parameters. The relationship between average B values of 1997 and standard deviation is given by: SD…B† ˆ 0:12 B ‡ 1:0

(8)

It shows that the drier the soil, the greater the infiltration variability within the range of initial soil conditions met during our field tests. Despite its low significance (R2 ˆ 0.680)

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this relation is used to propose a SD value once an average Bup value is calculated by means of Eqs. (5) and (6) for Bdn. Discharge inlet variability depends on the distribution system performance and its conditions of use. We can assume that discharge variability is only due to random factors when the elementary hydraulic rules are complied to (which should be the case in a modernization context). Taking our own experience into account, the range of Cv(Qin) for the different irrigation system can vary from 5% to 10% for gated pipe and siphons and 15% to 20% for floppy pipes when extra care is given to setting. The stochastic approach is based on the Monte-Carlo simulation method for the infiltration parameters Bup, Bdn and discharge inlet Qin. The Monte-Carlo method which was widely used in the hydrologic studies (Freeze, 1975; Marchand, 1988; Ruelle, 1995) appears well adapted to attacking spatial variability problem in furrow irrigation. Scaling factors method is also widely used to treat of infiltration variability but it is best suited to infiltration processes which are modelled using a mechanistic approach i.e. involving parameters having a physical meaning and so obtainable by means of field measurements (Russo and Bresler, 1980; Boulier, 1985). N values zi of variable Z are drawn in a Gauss distribution to obtain zi ˆ z ‡ z Vi , where Vi values are generated by the RND function enabling random draws in a uniform law between 0 and 1. The N generated zi values, are a realization of the stochastic process. Mean value mz and SDz of that realization provide the estimation of z and z: : mz ˆ …1=N†zi ; SD2z ˆ ‰1=…N ÿ 1†Š…zi ÿ mz †. Accounting for correlation between two variables Z and T (such as Bup and Bdn) we have to generate doublets (Z,T) for N furrows by 2N draws values of random variable vij of V. For a given furrow, (zi, ti) is defined as: zi ˆ mz ‡ SDz vi1

(9)

ti ˆ mt ‡ SDz vi1 SDt …1 ÿ r 2 †vi2

(10)

where SDt (1 ÿ r2) represents T variance non-associated to linear regression with Z (Snedecor and Cochran, 1971). Note that Eq. (10) only concerns parameters Bup and Bdn since B and Qin are non-correlated (R2 < 0.07 on the example of 3rd irrigation of P2). According to Monte-Carlo method, statistical characteristics conservation implies that N has to be high (N  500). 2.7. Validation tests of the stochastic advance-infiltration model Model validation becomes problematic when the number of furrows monitored at a time are too few. Consequently, observed variability derived from a limited number of furrows may be sensitively different (certainly lower) than that observed on a more substantial number of furrows. The 3rd irrigation in 1997 of P2 sub-plot 1 is chosen as an example of model application because of the substantial variability of the advance process. Eq. (3) (which provides in that case similar result as Eq. (6)) is used to obtain Bup ˆ 78 l/m with SD ˆ 10.5 Eq. (8). Eq. (7) gives Bdn ˆ 45 l/m with SD ˆ 6.4 Eq. (8). Using an averaged r2 ˆ 0.75 to account for Bup/Bdn correlation, the model simulates (N ˆ 500 draws) the averaged advance process and its variability (with Qin ˆ 1.53 l/s,

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Table 2 Example of advance time (min) monitoring under 3rd irrigation conditions of 1997 on P2 Sp 1 Furrow 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Qin (l/s) 25 m

50 m

75 m

100 m

125 m

150 m

175 m

200 m

230 m

1.5 1.6 1.3 0.92 1.3 1.14 1.3 1.43 1.3 1.53 1.68 1.8 1.8 1.47 1.5 1.8 1.7 1.8 1.8 1.56 1.8 1.56 1.56

16 16 17 26 21 26 28 20 22 23 21 21 21 21 21 15 18 19 19 18 19 21 23

34 33 42 44 49 59 55 46 50 51 44 42 35 43 43 35 40 40 38 40 41 45 53

53 54 69 96 77 94 84 73 78 79 65 60 56 67 66 57 61 59 57 59 62 69 80

76 74 96 137 107 122 116 101 107 110 103 85 79 91 90 83 84 83 80 81 84 95 108

99 93 121 167 138 156 147 131 134 136 128 107 102 115 111 105 105 103 101 102 107 119 135

118 112 150 209 170 199 174 165 162 162 156 132 125 141 138 129 128 125 121 121 127 141 160

136 131 173 240 193 239 200 188 190 193 184 156 150 163 160 149 147 141 136 137 146 166 184

151 146 195 271 212 262 220 206 208 213 212 166 163 183 176 164 163 153 150 151 166 183 198

174 170 218 309 238 300 244 228 228 242 219 181 178 210 195 181 182 172 167 168 193 201 220

Mean Cv (%)

1.54 15%

20 14

43 15

68 18

94 17

119 16

145 18

168 18

186 19

208 19

M Sim Cv Sim

1.54 15%

25 17

46 17

74 18

99 18

123 18

150 19

172 19

189 20

211 20

M sim: averaged simulation; Cv Sim: variation coef. simulated.

Cv ˆ 15%) in a very satisfactory manner as shown in the last columns of Table 2 which presents an example of time advance monitoring. A true validation would consist of model application on a field experiment out of the calibration context which is not entirely the case for the previous model application. This is not easy as infiltration conditions in 1996 were different from those of 1997 due to land preparation (Spring sugar-cane ridging). As explained above, B in 1996 results from both the cracking phenomenon and surface basin storage preceding a lump. Assuming that Eq. (6) is applicable under similar soil surface conditions to describe cracking impact in 1996 on B, the difference to match the expected B value using Eq. (6) should be attributed in part to runoff trapped in the micro basins devoted to infiltration. An estimation of this value is proposed using the case P3 sub-plot 3 of 1996 where MET ˆ 53 mm. An application of Eq. (6) would give Bup ˆ 53 l/m (WAD of irrigation 5th was 60 mm) for sub-plot 3 of P3, whereas, that obtained from calibration equals 71 l/m. Consequently an estimation of the mean stored surface volume value can be

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Fig. 4. Model validation on the advance process of averaged furrow of P1 Sub-plot 2 for 6th irrigation 1996.

proposed as MSV ˆ 18 l/m. Adopting this MSV value, a model validation is conducted under irrigation 6th of 1996 conditions (Qin ˆ 1.4 l/s, Cv(Qin) ˆ 8%, tco ˆ 300 min, MET ˆ 57 mm) on P1 Sub-plot 2. The same procedure such as that applied in the previous example of model application is used to derive the statistical values of the infiltration parameters. In order to propose an easy comparison (e.g. a graphical one) between field observations and simulations results, simulation is conducted on a furrow number equivalent to that monitored although, as evoked above, this is not really acceptable from a theoretical point of view. In this operation, generally average value is conserved, whereas, SD differs significantly from that derived from simulation conducted on 500 draws (from around 0.5% to 1.5% regarding Cv(TL)). The result of the application is presented in Fig. 4 where simulation of mean advance is compared to observed mean advance. Advance time simulation (TL) from the faster to the slower furrow is compared with observed advance time in Fig. 5. Observed and simulated advance time are 222 (Cv ˆ 10%) and 217 min (Cv ˆ 11.5%) for this validation test, respectively. In spite of a slight over-estimation of the variability (as explained above the number of observations is limited), the performance of the model can be considered as acceptable even for a limited

Fig. 5. Validation of the stochastic simulation of the advance process at the plot scale on P1 (Sub-plot 2) for 6th irrigation 1996.

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number of draws (the Cv value equals 10.5% using 500 draws) this presents the advantage of allowing concrete comparison with field observation. Average minimal water application depth is 59 mm with Cv ˆ 9% and averaged efficiency equals 80% with Cv ˆ 6%. Using neutron probe measurements a WAD of 53 mm was obtained at x ˆ 25 m and 50 mm at x ˆ 75 m for P3, whereas, WAD equals 55 mm at x ˆ 25 m on P1. Since soil moisture profile before and after irrigation are comparable to those presented in Fig. 2, a part of WAD is assumed to be lost by drainage on downstream parts of the plots. 3. Discussion and conclusion The simulation of WAD variability and its impact through the irrigation season is obviously of interest in a sustainable agricultural context. An analytical model adapted to cracking soils (Mailhol and Gonzalez, 1993) simulates advance-infiltration phenomenon in FDF or CEF conditions (Mailhol et al., 1997) for a given furrow with Qin as inflow value. A possibility of accounting for heterogeneous infiltration conditions, induced by the CEF practice in the study context, was added here to that basic model. As shown in Mailhol and Gonzalez (1993) and Mailhol et al. (1997) WAD is correctly predicted when model calibration is successfully conducted on the advance trajectory. This was verified in the field context for each irrigation test using runoff monitoring of the three FDFs. In our field context, Nash criterion is generally greater than 0.995% for 95% of the whole of furrows where the infiltration parameters B (Bup and Bdn) and Cs are calibrated. Due to difficulties in using mechanistic approaches in furrow irrigation modelling, particularly in non-darcian flux conditions, empirical relationships are proposed to update the infiltration parameter values and their variance in local soil conditions. Coupling these relations with the Monte-Carlo stochastic simulation method in the advance infiltration model, it is, therefore, possible to simulate advance and water application depth variability at the plot scale with an acceptable level of reliability. Sensitivity tests conducted on r2 (Bup and Bdn correlation) attest that a 20% r2 variation induces a 7% variation in Cv(TL) variation, whereas, a 20% SD(B) variation gives a 13% Cv(TL) variation. Model simulations aim at determining the optimal irrigation practices as illustrated by the following example where a minimal water application depth (MWAD) of 50 mm is required on a 230 m length plot. Irrigation decision making should occur when MET reaches 65 mm. So, an interesting solution regarding irrigation performance is provided with Qin ˆ 1 l/s Cv(Qin) ˆ 8% (the lowest value obtained in our field context) while tco is calculated so that MWAD attains at least 50 mm. In the case of CEF practice irrigation performances are: AE ˆ 78%, Cv(AE) ˆ 8%; Cv(MWAD) ˆ 12%; TL ˆ 287 min, Cv(TL) ˆ 15% with tco ˆ 380 min. In case of FDF (where Bup ˆ Bdn) we have: AE ˆ 70%, Cv(AE) ˆ 11%; Ro ˆ 23%, Cv(Ro) ˆ 39%; TL ˆ 332 min, Cv(TL) ˆ 14%; Cv(MWAD) ˆ 9% with tco ˆ 420 min, where AE ˆ application efficiency, Ro ˆ runoff, MWAD ˆ minimal WAD. Using Qin ˆ 0.75 l/s, averaged application efficiency would be sensitively improved for CEF

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practice in contrast with FDF. But for the two practices the variability of TL, MWAD and Ro would be significantly greater. The proposed simulation model would contribute to the identification of the best irrigation management strategy for improving yields, water saving and reduction of ground water pollution in heavy clay soil context. 4. Nomenclature The following symbols are used in this paper: AE B Bup Bdn CEF EUWR Cv(Qin) Cs ET0 FDF I L MAWS MET MSWC MSV PAWD R2 Ro r2 SWD SWC SD(B) TL tco T WAD WBE Z

application efficiency in % parameter of the cumulative infiltration equation in l/m B value in the upstream part of the plots B value in the downstream part of the plots closed-end furrow easily usable water reserve variation coefficient of discharge inlet stabilized infiltration rate of the cumulative infiltration in l/m/min daily reference evapotranspiration free draining furrow cumulative infiltration (l/m) furrow length maximal available water stored Maximal evapotraspiration maximal soil water capacity mean stored volume in the micro basins of the furrows (under no-clean till furrow conditions) Irrig(t ÿ 1). Efficiency (80%) determination coefficient of a correlation between two variables runoff losses in % determination coefficient between Bup and Bdn soil water depletion soil water content standard deviation of parameter B advance time (time required by the water to reach the end of the furrow at x ˆ L) cutoff time statistical variable (representing Bdn in the Monte-Carlo simulation method) water application depth water balance equation (i.e. Lewis and Milne equation) statistical variable (representing Bup in the Monte-Carlo simulation method) having a realization zi ˆ z ‡ z Vi, where Vi is generated by the RND function

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