Scientific article

Study area

The geographical location of the spate irrigation system area is from 19° 50’ 00’’ to 19° 56’ 00’’ NL and from 101° 40’ 00” to 101° 31’ 00” WL (Figure 2), which is within the Angulo River sub-basin, which belongs to the Lerma-Chapala basin. This area is integrated into the 12 Lerma-Santiago Hydrological Region, with an area of 2 062 km² (from 19° 35’ to 20° 15’ NL and from 101° 20’ to 102° 00’ WL) covering 15 municipalities: Angamacutiro, Chucándiro, Coeneo, Erongarícuaro, Huaniqueo, Jiménez, Morelia, Morelos, Nahuatzen, Panindícuaro, Penjamillo, Purépero, Puruándiro, Quiroga, and Zacapu (Figure 3).

Edaphology. The type of soil in the study area is mainly clay (49.50 %) and silty (47.85 %), so it can be said that not only one dominates in terms of occupied area (Table 1).

Table 1. Area occupied by soil type in study area.

Land use and vegetation. Figure 4 shows that almost half the land in the study area is for agricultural-livestock-forestry use (44.68 %), while hydrophilic halophyte vegetation occupies the smallest area (0.01 %) (Table 2).

Table 2. Land use and vegetation area in the study area.

Tributary area of each channel

Determination of the areas of influence of each channel, within the study area, was made by means of the topography provided by the CEM, and was defined based on the characteristic slopes of each area by means of the HEC-GeoHMS program. This was done because they are tributary areas or surfaces that provide runoff to the channel (Figure 5).

Runoff curve number

The runoff curve number (CN) method for estimating runoff depth is based on CN-tabulated (dimensionless) values, which were developed by the Soil Conservation Service (SCS) of the United States, where the curve number chosen will depend, in practical terms, on the soil texture (hydrologic soil group) and land use, the density of plant cover (hydrologic condition) and the possible existence of soil conservation practices (Pérez-Nieto, Ibáñez-Castillo, Arellano-Monterrosas, Fernández-Reynoso, & Chávez-Morales, 2015). In Mexico, this method is used due to its simplicity and lack of instrumentation in the basins (Alonso-Sánchez, Ibáñez-Castillo, Arteaga-Ramírez, & Vázquez-Peña, 2014).

In this study, the CN value was obtained from data tabulated by Mockus et al. (2009), based on soil type and land use (Table 3), in combination with the edaphological map (Domínguez-Mora et al., 2008). The spatial distribution, according to the NC obtained, is shown in Figure 6.

Table 3. Runoff curve number.

Subsequently, the CNs obtained were adjusted, since the SCS defines three soil moisture conditions: I-dry (wilting point), II-medium moisture and III-completely wet (field capacity). The CNs for soil conditions I and III were calculated with the equations proposed by Neitsch, Arnold, Kiniry, and Williams (2011).

Weather stations

The weather data used in the model were from 1983 to 2012 (30 years), which were obtained from Sistema Meteorológico Nacional (National Meteorological System) stations operated by Comisión Nacional del Agua (National Water Commission), located both inside and outside the study area (Figure 7).

Precipitation

The behavior of the average monthly precipitation recorded at the weather stations closest to the study area is shown in Figure 8.

Missing precipitatin data were calculated by the inverse square distance method (Equations 1 and 2) (Gallegos-Cedillo, Arteaga-Ramírez, Vázquez-Peña, & Juárez-Méndez, 2015).

P x = ∑ p i * w i ∑ w i (1)

w i = 1 D i 2 (2)

Where P _x is the missing precipitation data (mm), P _i is the precipitation recorded at the surrounding auxiliary stations (minimum two) at the date of the missing data (mm) and D _i is the distance between each surrounding station and the incomplete station (km).

Calculation of maximum expected rainfall

To calculate the maximum rains in different return periods, namely 10, 20, 50 and 100 years, we used the methodology proposed by the SCS, which consists of obtaining and grouping the maximum rains presented in a certain recording period (30 years in this case), calculating statistical parameters of the theoretical distribution and adjusting the rains.

Frequency analysis

The objective of hydrological information frequency analysis is to relate the magnitude of events, especially extreme ones, to their occurrence through the use of probability distributions, for which it is assumed that the hydrological information analyzed is independent of space and time (Chow, Maidment, & Mays, 1988). In this case, frequency analysis of extreme rains was carried out by the by the method of moments, for which the original rainfall values were fitted to a theoretical distribution: normal, log normal, pearson III (gamma), log pearson and Gumbel.

Kolmogorov-Smirnov goodness-of-fit test

To verify whether the data fittted the proposed methodology, the Kolmogorov-Smirnov goodness-of-fit test was applied (Equation 3), which consists of comparing the observed probability distribution function [F ₀ (X _m )] (Equation 4) with the theoretical one [F(X _m )]:

D = m a x F 0 X m - F ( X m ) (3)

F 0 X m = 1 - m n + 1 (4)

where m is the order number of the given value, X _m is a list from lowest to highest and n is the total number of data.

To use this test, the proposed null hypothesis (H₀) is that the probability distribution function is D (α, β).

The maximum observed difference (D) is compared with a Kolmogorov-Smirnov table statistic; if D is lower, the theoretical distribution is accepted. The value of this statistic depends on the length of the record and the level of significance. If several distributions are accepted, the one with the smallest observed difference is chosen.

Rainfall-runoff relationship

According to Aparicio-Mijares (2001), precipitation records are more abundant than runoff records, and do not change due to anthropogenic modifications in the basin; therefore, it is recommended to use methods that determine runoff from rainfall data and basin characteristics. The CN method, for the calculation of precipitated depth to runoff depth, was developed in 1950 by the U.S. Department of Agriculture. The main parameters involved in the process of converting rainfall to runoff are the area of the basin, the amount of rainfall, the slope of the basin, the time of concentration, the vegetation, and the spatial and temporal distribution of the rain.

For the above, Equations 5 and 6 are used, which estimate the direct runoff from a storm (Chow et al., 1988):

P e = ( P - 0.2 S ) 2 ( P + 0.8 S ) (5)

for P > 0.2 S, otherwise Pe = 0

S = 25400 N C - 2540 (6)

where Pe is the direct runoff depth (mm), P is the precipitation (mm), S is the maximum potential retention (mm).

The most commonly used methodological tool to model the rainfall-runoff process is based on the unit hydrograph theory (Juárez-Méndez, Ibáñez-Castillo, Pérez-Nieto, & Arellano-Monterrosas, 2009), which assumes that the discharge at any time is proportional to the volume of runoff evenly distributed in the basin (Snider et al., 2007), as it considers the rainfall depth, the area, shape, slope and vegetation of the basin, although not explicitly (Aparicio-Mijares, 2001). The fundamental principles of base flow time constant, overlap and proportionality make the unit hydrograph an extremely flexible tool for developing synthetic hydrographs.

Chow et al. (1988) mention that the dimensionless SCS hydrograph is a synthetic triangular unit hydrograph transformed into a curvilinear shape, in which the flow rate is expressed by the ratio of the flow rate (q) with respect to the peak flow rate (q _p , m³·s^-1·mm^-1) and the occurrence time of the peak in the unit hydrograph (T _p , h). If a unit hydrograph of 1 mm is considered, it can be demonstrated that:

q p = 0.208 A T p (7)

where A is the drainage area (km²) and T _p is egual to 1.67 h d according to the SCS.

Additionally, Aparicio-Mijares (2001), in SCS basins, indicates that in the unit hydrographs the delay time (t _r ) is egual to 0.6 T _c , where T _c is the time of concentration of the basin. The T _p can be expressed in terms of the t _r and the duration of the effective rainfall d _e (Equation 8):

T p = d e 2 + t r (8)

On the other hand, runoff can be estimated in two ways: 1) probabilistic prediction and 2) "real time" forecast, so it is necessary to adequately characterize the runoff and the phenomena related to it (the rain and the basin). In addition, based on the laws of physics and probability theory, a series of models needs to be developed, which should be manageable in practice, and lead to correct forecasts and predictions (Domínguez-Mora, 1990).

Once the runoff comes down from the slopes, it begins its route in the channels that are already defined. The channels cushion the speed and temporarily store part of the flow. Therefore, at the end of the channel, an outlet hydrograph different from the initial one is generated.

There are different methods for calculating flood routing in channels; one of the simplest is the method described by Muskingum (Equation 9) (Aparicio-Mijares, 2001):

O i + 1 = C 1 I i + C 2 I i + 1 + C 3 O i (9)

where I _i and O _i are the ordinates of the inlet and outlet hydrographs (m³·s^-1), respectively; the coefficients C₁, C₂ are C₃ are calculated as follows:

C 1 = K x + ∆ t / 2 K 1 - x + ∆ t / 2 ;   C 2 = - K x + ∆ t / 2 K 1 - x + ∆ t / 2 ; C 3 = K 1 - x - ∆ t / 2 K 1 - x + ∆ t / 2 (10)

where C ₁ + C ₂ + C ₃ = 1, K and x are the Muskingum parameters, the first in time units and the second dimensionless.

The spate irrigation system as a storage area

The spate irrigation system formed by "boxes" works similar to a storage dam, but without discharging immediately. It is considered that in times of major floods, runoff can be divided between a maximum volume stored by the boxes and the continuation of its course in the channel.

HEC-HMS model

According to McCuen (1998), hydrological models are simplified representations of real hydrological systems that allow studying the functionality and response of a basin to several input conditions, to better understand hydrological events. The Hydrologic Engineering Center-Hydrologic Modeling System (HEC-HMS) is a model of hydrologic parameters aggregated by sub-basin, with the option of a model distributed in space; it subdivides a basin into sub-basins, taking as the initial criterion the tributaries of the main current (Miranda-Aragón, Ibáñez-Castillo, Valdéz-Lazalde, & Hernández-de la Rosa, 2009). In the present work, this model was used, since it simulates the precipitation and runoff processes of the dendritic basin systems. In addition, it is applied in diverse geographical areas to solve a large number of problems; this includes the water supply to the watershed and flood hydrology. In Mexico, runoff from urban or natural basins is studied per extreme weather event (Vargas-Castañeda, Ibáñez-Castillo, & Arteaga-Ramírez, 2015).

Hydrographs, or some of their elements (such as peak flow), are used in the planning and design of water control structures, as well as to show existing hydrological effects or proposed changes in land use (Snider et al., 2007). Hydrographs produced by the HEC-HMS are used directly or together with other software to study water availability, urban drainage, flow forecasting, future urbanization impact, reservoir spillway design, flood damage reduction, floodplain regulation, and systems operation (U.S. Army Corps of Engineers, 2016). On the contrary, hydrological studies with traditional techniques for surface runoff do not present the study needs for small structures.