**1. Introduction**

In Mexico there are 6.5 million hectares under irrigation. Of this, 3.3 million are distributed in 86 irrigation districts and 3.2 million are distributed in just over 40 thousand irrigation units located mainly in the center and north of the country. Gravity surface irrigation (border or furrows) is the most used method in these areas [1].

According to recent statistics and studies that have been registered by the National Water Commission (CONAGUA, according to its Spanish acronym), application efficiencies in Mexico are less than 50% [1–3], 5% below the global average [4,5], and consequently the water productivity (Kg/m3) is very low compared to pressurized irrigation systems [1,6]. Despite the fact that support has been given

to modernize the traditional sprinkler and drip irrigation systems, there are still various factors that cause farmers not to opt for these technologies; for example, a lack of investment, little government support, payment of trained personnel to operate the equipment, social factors, risks in the acquisition of equipment that they do not know, among others [4,6–8].

The low application efficiencies obtained by gravity irrigation systems are mainly due to water losses associated with deep infiltration, coleus, and flooding in some parts of the soil as a result of a poor flow design at the entrance of the furrow [2,3], as well as unevenness in the plots and little knowledge about the water depth that must be supplied to the crops so that they can develop fully. However, in a sector where the demand for water occupies a high percentage of the available water, the efficient use of irrigation water and the increase in productivity are crucial factors that must be addressed as soon as possible to sustainably manage the water and adaptation to climate change [5,9–11].

Faced with this problem, during the years 2015–2019 CONAGUA maintained a modernization program in gravity irrigation systems (RIGRAT for its acronym in Spanish) in an area of 200,000 ha in some irrigation districts of Mexico. The objective of the program was to increase the efficiency of water application in the crops through two main axes: a timely design for optimal flow at the entrance of the furrows and a reduction in irrigation time in each plot.

A correct design of the flow rate that must be applied to each border or furrow requires knowledge of the characteristics of the plots (length, moisture content, and apparent density), the established crop, phenological stage, and the irrigation depth to be applied, as well as the mean parameters of the infiltration equation being used: Richards or Green and Ampt [12].

Several simulation models can be found in the literature to model gravity irrigation, which are completely empirical [13–15] to those that use the complete Barré de Saint-Venant and Richards equations to model runoff and infiltration, respectively [16–18]. The use of these models (numerical or analytical) helps to better understand the behavior of water during the processes of advancing, storage, and recession [12,17,19]. However, the complexity with which they were developed or the limitations they have make their use impractical and only used for research purposes, leaving aside their practical application.

In recent studies, it has been reported that it is possible to have high application efficiencies in gravity irrigation systems, by applying the optimal flow rate at the entrance of the border or furrow and maintaining a high uniformity coefficient in the plot, called the Christiansen Uniformity Coefficient [2,3,12].

This methodology was applied by Chávez and Fuentes [2,3] for seven crops planted in 1010 ha: *Zea mays* L., *Sorghum vulgare* Pers., *Medicago sativa* L., *Phaseolus vulgaris* L., *Pachyrhizus erosus* L., *Hordeum vulgare* L., *Triticum aestivum* L. and *Allium cepa* L. They found that with the application of the optimal flow rate of the calculated irrigation, the irrigation depths decreased on average 19 cm. The irrigation times decreased on average 11.76 h ha−<sup>1</sup> per irrigation event and, in addition, the average volume saved was 2000 m<sup>3</sup> ha−<sup>1</sup> per irrigation event, which increased the average efficiency from 51 to 86%. However, to have an efficient design, knowledge of the plots and the water requirements of the crops established in the different phenological stages are necessary [12].

The objective of this work is to show that it is possible to increase the application efficiency and productivity of water in gravity irrigation systems in Irrigation District 023, San Juan del Río, Querétaro, Mexico. For the simulation, the kinematic wave model is used, and an analytical formula is used to calculate the optimal flow rate that takes into account the characteristics of the plot (length, moisture content, density, and texture), the parameters of the equation of infiltration (Ks and Hf), and the net irrigation depth to be applied to the crops in their different phenological stages.

#### **2. Materials and Methods**
