The kinematic wave model

The kinematic wave model considers that in the equation of the amount of Barré de Saint-Venant movement, the inertia and pressure terms are insignificant with regard to the friction and gravity terms (^{Fuentes et al., 2012}). With these assumptions, the model is:

where A=A(x,t) is the hydraulic area (L²), Q=Q(x,t) is the flow rate (L³ T^-1), W is the infiltrated volume per unit of length of the furrow in the unit of time (L³ T^-1), t is the time (T), S₀ is the slope of the depths of the furrow (LL^-1) and S_f is the slope of the energy line (LL^-1)..

The equation (1), which considers the flow rate as a function of the hydraulic area, is (^{Litrico, 2001}):

The coefficient C_k (Q) is obtained from a flow resistance law with S_f =S₀ . The laws of power resistance, such as that proposed by Chezý and Manning (^{Sotelo, 1977}), allow obtaining the generic relationship Q=αA ^β , from which C_k=αβ(Q/α) ^1-1/β is deduced.

Given that the hydraulic radius intervenes in the resistance laws, it is necessary to consider the geometrical shape of the furrow. Since it is considered that S_f =S₀ , the friction and gravitational forces are equal and there is no significant acceleration of the flow; therefore, the model is applied to runoffs with small flow rates and soft slopes as in gravity irrigation. ^{Woolhiser (1975)}, based on simulations in canals with different geometric and operative characteristics, established that the kinematic wave model represents adequately the flow dynamics insofar as the following inequality is fulfilled:

where K is a kinematic number, L₀ the furrow length, F02=Q02T0/gA03 is the Froude number calculated at the position of the flow entrance, Q₀ is the flow rate at the canal’s entry, T₀ the width of the free surface in the same position, A₀ is the area at the canal’s entrance, and h_n the normal depth.

The parameters of the furrow geometry are obtained from the consideration that the power functions described adequately the depth-area and area-flow rate relationships, which are represented as follows (^{González-Camacho et al., 2006}):

where R_h is the hydraulic radius (L) and a, b, c, and d are parameters of the furrow geometry that are obtained by linear regression.

The parameters c and d are obtained with the Manning equation, for uniform flow:

where n is the Manning rugosity coefficient (L^-1/3 T).

If equation (6) is introduced into equation (7), and the terms are redefined, the equation obtained provides the flow rate in function of the furrow geometry:

The complete solution requires the understanding of the progress of the water infiltrated in the whole time; this is calculated with the Green and Ampt equation.

The Green and Ampt equation

The Green and Ampt equation is established under the following hypotheses: 1) the content of initial moisture (θ_i ) is constant throughout the length of the soil column, that is θ_i =θ₀ in the whole profile; 2) during the infiltration process two moisturing zones are formed, one completely saturated (θ= θ_s), 0 ≤ z < z _f (t) and another dry one with the initial moisture content (θ_i = θ₀), z_f (t) < z, where z_f (t) is the position of the moisturing front in the saturated zone K=K_s , where K_s is the hydraulic conductivity at saturation; 3) in the saturated zone the distribution of the pressures is hydrostatic ψ=hsup-hf+hsupzzf, where ψ is the water pressure, h_sup is the water depth on the surface and h_f the suction on the moisturing front; and 4) the Darcy flow is independent of z in the saturated zone and is equal to the infiltration flow that takes place on the surface. Thus, the resulting Green and Ampt equation is:

where Δθ = θ_s - θ₀ is the capacity for storage and I is the infiltrated volume accumulated per surface unit of the soil or water infiltrated.

The infiltrated volume per unit of furrow length in the time unit (W) of the equation (3) is obtained with the methodology proposed by ^{González-Camacho et al. (2006)}.

Analytical representation of the optimal flow rate

According to ^{Fuentes et al. (2012)}, the formula to calculate the optimal flow rate per unit of width is a function of the length of the border or furrow, the net irrigation depth, and the parameters characteristic of the infiltration represented by the capillary forces, sorptivity, and gravitational forces, that is, the hydraulic conductivity at saturation (^{Rendón et al., 2012}):

where K_sL=q_m is the minimum optimal unitary discharge needed for the water to arrive at the end of the border or furrow, S is the sorptivity of the medium expressed by S₂ = 2K_sh_f (θ_s-θ₀) and ln is the net irrigation depth. The optimal flow rate per furrow is calculated as Q₀ = bq₀ , where b is the width of the furrow.

Numerical solution

The solution of equation 3 is obtained through several methods, among which those of characteristics and finite differences stand out. The method of finite differences is used primarily because of its adaptability to the change of the initial and frontier conditions. In our study the method of finite differences with the Eulerian approach was used, because there are not numerical instabilities. The model of the kinematic wave was solved with the algorithm by ^{González-Camacho et al. (2006)} with coefficients a = 1.58, b = 1.87, c = 0.23, d = 2.68 and Manning coefficient value n = 0.05 for the plots in the first irrigation event and n = 0.03 for assistance irrigation events. The entry data to the simulation model were measured in the field stemming from irrigation tests and direct measurements of the moisture contents, and the infiltration parameters (K_s y h_f) were estimated through an inverse method using the Levenverg-Marquardt optimization algorithm (^{Moré, 1978}).

The study zone

The irrigation district 085 La Begoña, Guanajuato, Mexico, is in the central-eastern zone of the state of Guanajuato and includes the municipalities of Celaya and Comonfort (20º 38’ and 21º 07’ N, 100º 45’ and 100º 53’ W). The extension of the district is 12 389.5 ha, offers services to 3 288 users and is divided into four irrigation modules: Neutla, Comonfort, Margen Izquierda and Margen Derecha. In these, 197 irrigation tests were performed on a surface of 749.37 ha.

Data collection

The characteristics and properties measured in the plots were: length, slope, texture, apparent density, initial moisture contents and at saturation. The first two were obtained with a total station, the initial moisture contents with calibrated TDR 300^®, the texture was obtained in the laboratory with the Bouyucos method, the apparent density (ρ_a) with the known volume cylinder method, and the moisture content at saturation (θ_s) was assimilated to the total soil porosity (ϕ), which was obtained from the apparent density, and the solids density (ρ_a) was considered to be 2.65 g cm^-3, that is, θ_s = ϕ =1-ρ_a/ρ_s .

Irrigation tests analyses

To carry out the irrigation tests, the soil was characterized and the initial moisture content was measured, the lengths with pennants were fixed every 15 m along the furrows to measure the phases of advancement and recession. The times at which the water reached the streamers were measured once the irrigation began. During the irrigation test the flow rate was measured continually at the entrance of the sprinkling can and the furrows, with a Doppler Ultrasonic Flow Meter (FluxSense^®). All the irrigation tests were done in open furrow.

When the water almost reached the end of the furrows, the flow rate stopped at the entrance and the recession phase was measured. In the irrigation tests there was no interference with the irrigators, since the current irrigation and sheets to be applied were expected to be evaluated with these tests.

With the data of advance, recession and characteristics of the soils where the irrigation tests were performed, the kinematic wave model was calibrated, with the Levenberg-Marquardt (^{Moré, 1978}) algorithm, to obtain the infiltration parameters of the Green and Ampt (K_s y h_f) equation representing the advance, storage and recession phases from each of the tests.

Irrigation design

The parameters of the calibration process (K_s y h_f), the net irrigation depth that is expected to be applied in the plot, the content of initial moisture, the entry flow rate, and the length of the terrain were used in equation 10 to calculate the optimal analytical unitary flow rate (q₀). The plot flow rate was divided by the optimal flow rate (Q₀) to obtain the number of furrows from irrigation length and the result approached the next whole number.

To achieve for the irrigation design generated to be applied constantly in the plots evaluated, it was necessary to work in the awareness of the irrigators, since the incorrect practice known as “letting the water sleep” is rooted. This practice consists in opening many furrows to have time to monitor several plots with simultaneous irrigation and thus receive a higher payment, or opening during the night more furrows, and on the next day for the water that went to the drainage not to be considered.

Analysis of texture and moisture content at saturation

The plots analyzed are classified into eight textural classes, the clay loam texture predominated (38.18 %) and the sandy clay loam was less predominant (2.60 %) (Table 1 and Figure 1).

Figure 1 Characterization of the plots according to their texture. 

The moisture content at saturation was variable between plots, even in the same textural class, which is why the calculation of the optimal irrigation flow rate with an average value can give different results. Thus, it is convenient to obtain the experimental measure (Figure 2).

Figure 2 Moisture content at saturation per textural class. 

Evaluation of irrigation tests

In the study plots, irrigation was observed with lengths from six to 119 furrows and the variation depended on the entry flow rate to the plot, which ranged between 4 and 142 L s^-1. The tests in the same textural class showed different results (Table 2). The surface of irrigation test was sown with maize (Zea Mayz) (29.42 %), sorghum (Sorghum vulgare) (16.75 %), alfalfa (Medicago sativa) (16.45 %), bean (Phaseolus vulgaris) (15.12 %), jicama (Pachyrhizus erosus) (8.79 %), barley (Hordeum vulgare) (5.27 %), onion (Allium cepa) (4.35 %), and wheat (Triticum aestivum) (3.84 %).

The average efficiency of application was 53 %; however, in some cases it was lower than 20 %. In the soils with higher sand content, the irrigation time was higher than in those where clays predominated (Figure 3). The atypical points in the graph correspond to plots with irrigation lengths over 130 m. The water depths applied were higher, about 15 cm, to those recommended, since the average was 30.31 cm, although in some cases water depths measured up to 138 cm (Figure 4).

Figure 3 Conventional irrigation time and with design per hectare. 

Figure 4 Irrigation water depths applied in a conventional way and using the analytical formula. 

Irrigation design with the optimal flow rate formula

In all the textural classes there was a considerable reduction in the number of furrows per length (Figure 5). In the most critical case, it went from 105 to 39 furrows per length, since in that plot to irrigate 1 ha the entry flow rate was 38.85 L s^-1 for 57.3 h. After the recommendation, the time was reduced to 12.97 h, which was equivalent to using in the traditional case a volume of 7 972.02 m³ and with the recommendation, 1 814.4 m³; the savings was 6 157.62 m³ ha^-1 in each irrigation event.

Figure 5 Furrows per length of conventional irrigation and with the analytical formula for the plot flow rates provided by the modules. 

After applying the optimal flow rate to the plot, the irrigaiton times decreased in average 11.76 h ha^-1 per irrigation event, although in some cases, the reduction was 61.69 h (Figures 3 and 4). The irrigation depths decreased in average 19.63 cm, and in some cases the water depth that ceased to be applied was 124.68 cm. According to the design formula, for the plots with length of more than 150 m, the optimal flow rates were higher than 4 L s^-1 per furrow. These flow rates are inviable in the field because they erode the furrows, which do not have the capacity for leading them. Because of this, in these cases it was recommended to shorten the length of irrigation to around 100 m.

Volume saved

In average, 2 000 m³ ha^-1 ceased to be applied. However, in some groups of sandy loam and sandy clay loam textures the saving was higher (Figure 6).

Figure 6 Volume of water saved per hectare of irrigation. 

The volume that was applied in the 749.37 ha, per irrigation event, was 2.1846 million m³. With the irrigation design, the volume (1.1339 million m³) decreased 48 %. The saving of 1.0507 million m³ per irrigation, in the three average irrigation events that producers apply per cycle, is equivalent to 3.1513 million m³ in the 749.37 ha. The average irrigation efficiency achieved in the zone was higher than 85 %, which is equivalent to slightly over 30 % of the efficiency with which the users were irrigating (53 %).