The simplicity of the water balance equation contrasts with the difficulty of determining the value of each of the variables involved, especially in basins with few or no records. At the same time, the determination of water availability at large temporal and spatial scales does not identify the regions with the highest pressure on it, so it is of interest for the development of water management and use plans to know the spatial variation precipitation (the principal source of water), evapotranspiration (the primary precipitation loss), as well as water demand. This study applies the water balance equation to determine the spatially distributed water availability. Methods for calculating evapotranspiration (Turc and Thornthwaite) and runoff (curve number, runoff coefficient, and Thornthwaite monthly calibration) are compared. The results show differences between the mountainous and rainy areas of the basin. The low area near the sea is flatter, has less rainfall, and concentrates most of the agricultural activity. The latter are areas with water scarcity, although, on average, the entire basin has water availability.

  • Annual excess precipitation using the runoff coefficient method yields satisfactory results.

  • The Turc method is more dependent on precipitation while the Thornthwaite method is more dependent on temperature.

  • The decrease in precipitation and/or increase in temperature due to climate change, together with the increase in evapotranspiration, will have a significant impact on water availability.

  • The availability of water in the lower part of the basin is highly dependent on the conservation of the forested ecosystem in the upper part of the basin and on summer precipitation.

  • The spatial characterization of the climatological variables involved in the water balance in a basin allows the identification of areas with a water deficit or surplus.

The water distribution on Earth changes continuously over time (Chow et al. 1988), making it challenging to determine the water availability of a region due to the wide variety of factors involved (topography, geology, vegetation, and anthropogenic impacts). The availability of water depends mainly on the spatiotemporal distribution of precipitation since it gives rise to the formation of different water bodies (Xu & Singh 1998), and surface bodies such as rivers and lakes are the primary sources of water due to their accessibility (Van Beek et al. 2011). The spatiotemporal variation of precipitation differs from that of demand, so the lack of water is accentuated in the year's dry season and in areas where it does not rain (Postel et al. 1996). This heterogeneity means that studies to determine water availability are limited, given that there may be water availability for a particular region and a certain period but not in other areas with lower rainfall, where the demand is higher or it does not rain (Oki et al. 2001; Van Beek et al. 2011).

In Mexico, the rains are largely due to tropical storms, whose trajectories, together with the continental relief, favor their concentration in the south of the country and on the windward side of the mountain ranges close to the oceans (Pacific and Atlantic), together with convective storms and to a lesser extent the cold fronts coming from the north of the continent, give rise to two well-defined seasons, one of rain from June to October and another of the dry season (Jáuregui 1970; Arreguín et al. 2011). Of the total volume of precipitation, 72.5% is evapotranspirated, 6.3% infiltrates, and 21.2% runs off superficially. Of the latter, approximately 75.7% is granted to agriculture, 14.7% to public urban supply, and 4.9% to industry; therefore, in the regions with the greatest agricultural activity and the largest population (center and north), water sources are under greater pressure, and their quality is also affected (Oswald & Sánchez 2011).

Traditionally, surface runoff is a volume of water available for exploitation, which, in the case of Mexico, is stipulated in the Official Mexican Standard (NOM) NOM-011-CONAGUA-2015 (SEMARNAT 2015) (CONAGUA is Mexico's National Water Commission, and SEMARNAT is Mexico's Secretariat of Environment and Natural Resources) but does not consider the volume of water that ecosystems require for their development, the waste of water, its contamination and the lack of adequate control mechanisms in the extraction of the resource, since they accentuate the problem of water availability (Jiménez et al. 1998; Silva et al. 2013). For this reason, the knowledge and quantification of the phases of the hydrological cycle in the past, present, and especially in the future will have the potential to develop water management plans for better economic and social development since climate change has impacted hydrological processes (IPCC 2021).

Thus, to determine water availability, applying the water balance equation, which presents different variants depending on the phases of the hydrological cycle, spatiotemporal scale, and anthropogenic activities in the basin is necessary. In this sense, precipitation is the primary input to the system. However, the outputs correspond to evapotranspiration, infiltration, and surface runoff; the volume of water in lakes, reservoirs, etc., is considered a change in storage (Chow et al. 1988). Precipitation and surface runoff are generally the only variants measured; the others must be estimated using different formulations or remote sensing methods (Bouaziz et al. 2020). The latter case's precision is still low (Zhao et al. 2013). Du et al. (2016) found that, for annual time scales, the water stored in the soil and groundwater can be considered negligible in the analysis. However, they are also determinants for calculating the monthly runoff and evapotranspiration values.

Evapotranspiration is the most important of the output variables of the system, as it constitutes the variable of the greatest magnitude and is fundamental in the process of the hydrological cycle and the evaluation of water resources, which links the water and energy balance of the terrestrial surface (Zhao et al. 2013). The calculation of evapotranspiration can be carried out by a large number of methods, classified according to the spatial scale (one or several underlying surfaces isolated or grouped), temporal (daily, monthly, and annual), and by its origin (energy, temperature, and mass transfer) and the number of input variables (Xu & Singh 2001). In this way, two groups of methods stand out: on the one hand, those that separately estimate the evaporation of water and soil and transpiration due to vegetation; on the other hand, those that first calculate potential evapotranspiration and subsequently adjust it to an actual value based on soil moisture (Xu & Singh 1998; Zhao et al. 2013). The former is more used in physically based hydrological models, due to the inherent difficulty and a number of input variables to estimate, while the latter is used in conceptual models, as they do not require a large amount of information, which is difficult to access, and they are easy to apply and adapt (Xu & Li 2003; Li et al. 2016; Wang et al. 2018).

In this way, the Penman–Monteith method (Penman 1948), based on mass transfer, the Turc method (Turc 1961; Pike 1964), based on radiation, and the Thornthwaite method (Thornthwaite 1948), based on temperature, stand out. These and other similar methods are associated with a certain degree of uncertainty because they were calibrated for specific basins, climates, and sizes (Li et al. 2016). In principle, the Penman–Monteith method due to its physical basis should be more precise, but in areas with limited information and due to a large number of input variables, it loses precision, so more straightforward methods such as those based on energy (Turc method) and temperature (Thornthwaite's method) perform better (Singh & Xu 1997; Oudin et al. 2005). To overcome certain limitations in Thornthwaite's method, Thornthwaite & Mather (1957) established a monthly iteration method to determine the actual value of annual evapotranspiration. To date, several studies have been carried out in different parts of the world, calibrating the original equation locally and simplifying the obtaining of the hydrological variables (Farzanpour et al. 2019; Quej et al. 2019; Trajkovic et al. 2019).

On the other hand, Budyko's theoretical framework (Budyko 1948) establishes that the actual value of evapotranspiration is limited by the available water and the energy of the Sun to evaporate the water. If the precipitation value is insufficient to satisfy the potential evapotranspiration, then the actual evapotranspiration equals the precipitation. In this way, the value of actual evapotranspiration is between the precipitation and potential evapotranspiration values, considering the water retained in the soil as the source, i.e., considering the impact of vegetation cover and soil type. In the water balance, human activities can be considered from the demand for water or the volume of reservoirs (Chow et al. 1988), as well as through land and vegetation cover (Li et al. 2016).

Infiltration is a slightly more complex process, and although there are methodologies based on the hydraulic characteristics of the soil, the application in conceptual models is not straightforward due to the lack of information and the amount of input variables (Horton 1941; De Smedt et al. 2000), so in practice it is calculated indirectly from runoff and precipitation through semi-empirical formulations based on events, such as the curve number, runoff coefficient, and methods described in the Mexican law (NOM-011-CONAGUA-2015; SEMARNAT 2015), which allows knowing the runoff and applying the water balance equation, the infiltration is estimated. The first two methods were developed for isolated storm events (Chow et al. 1988). However, the last one is used for annual balances and considers evapotranspiration (SEMARNAT 2015).

The water balance equation is traditionally applied at the basin level (aggregate model), which facilitates the estimation of the corresponding variables. However, they represent an average value over the basin, and thus, precision is lost in the results (Rivadeneira-Tassara et al. 2022). With the development of new Information and Communication Technologies and geographic information systems (GIS), many hydrological models and among them, the water balance equation can be applied at the level of sub-basins (semi-distributed model) or cells, distributed model (Lim et al. 2006). The use of distributed models brings specific challenges related to the quantity and quality of the input data, their application, and the validation of the results (Pao-Shan et al. 2001). On the one hand, not having such data limits the application of these models, especially in areas where the concentration of hydrometric and climatological stations (CS) is scarce, the record is incomplete, or there is no information for the most recent years. The above requires the application of different methods to determine the reliability and reconstruction of the data (Coria et al. 2016).

In general, we will have more effective and reliable hydrological models by guaranteeing the availability, quantity, and quality of precipitation information (Koutsouris et al. 2016). However, the validation of results is complicated since they only sometimes have the real values of the different variables measured spatially, which will always have some uncertainty (Zhang et al. 2008). Therefore, hydrological models are validated from measured data only at a particular point, generally at the exit of the study area, as if it were an aggregate model (Du et al. 2016).

In this way, in this work, a methodology for applying the spatially distributed water balance equation in the Piaxtla River basin, located in northwest Mexico, is proposed, and calibrated with runoff data. With this, the objective is to analyze some simple methods for calculating evapotranspiration and infiltration that are easy to apply without losing precision due to the availability of information, the variables involved, the economic activities, and the availability and needs of water in the said basin. The methodology is calibrated and validated with runoff information at two hydrometric stations (HS) in the main channel of the basin, as well as with official publications on water availability for the basin. To that end, the study area and the available information are defined, then the proposed methodology is described, and finally, the results are validated, and the corresponding analysis is carried out.

Study area

The Piaxtla River basin is in northwest Mexico, on the continental side of the Gulf of California and south of the Mexican states of Sinaloa and Durango. The basin has a contribution area of 6,984 km2 and a central channel with a length of approximately 220 km and discharges at the height of the town of San Dimas, next to the Meseta de Cacaxtla protected natural area in the Gulf of California. The basin partially covers the municipalities of San Dimas, Tamazula, Canatlán, and Durango in the Mexican state of Durango and San Ignacio, Elota, and Mazatlán in Sinaloa (Figure 1).
Figure 1

Location of the Piaxtla River basin.

Figure 1

Location of the Piaxtla River basin.

Close modal

In the study area, there are a significant number of CS with influence on the Piaxtla River basin (19); however, due to the lack of information, only eight were used for this work, two in the Mexican state of Durango (10031 Huahuapan and 10042 Las Truchas) and six in the state of Sinaloa (25001 Acatitán, 25021 Dimas, 25024 El Limón CFE, 25045 Ixpalino, 25,084 San Ignacio, and 25,118 San Ignacio CFE). In addition, there are two HS (10065 Ixpalino – on the same site as the CS 25045 Ixpalino – and 10111 Piaxtla – in the town of San Ignacio, near the CS 25084 San Ignacio) (Figure 1). The HS gives rise to the exit point of two homonymous sub-basins, which will be used for calibration and comparison purposes. The watershed divides (and other spatial calculations too) were determined with the support of QGIS (QGIS 2023), based on the digital terrain model (DTM) from the Mexican Elevation Continuum (CEM 3.0) with a resolution of 15 m (INEGI 2024) from Mexico's National Institute of Statistics, Geography, and Informatics (INEGI) (Figure 1).

The population in the basin is small (32,115 inhabitants) and is distributed mainly in rural localities, and urban land use does not represent more than 5 km2 (0.07% of the total surface of the basin). In contrast, the area dedicated to agriculture is more extensive, covering almost 11%, and three types are distinguished: temporary, irrigated, and induced pasture-livestock farming. In the basin, the pine-oak forest and sub-deciduous lowland forest extend significantly, mainly on luvisols (24.1%), leptosols (23.6%), and phaozems (15.8%) (WRB-IUSS 2015), in contrast to the urban areas that can barely be distinguished (Figure 2). The type of soil in the basin corresponds mainly to fine sands with silt – hydrologic soil type B – (57.8%) and very fine sands with clay (38%; C), which gives rise to soils with medium and high permeability, respectively. The hydrological soil (Figure 2, right) will also be used to calculate the curve number (Chow et al. 1988; USDA-NRCS 2010).
Figure 2

Land use, vegetation cover (left panel; INEGI 2024), and soil type (right panel; INEGI 2024).

Figure 2

Land use, vegetation cover (left panel; INEGI 2024), and soil type (right panel; INEGI 2024).

Close modal

The Piaxtla River basin only has a small dam (h = 4 m depth) called ‘Derivadora Piaxtla de Arriba’, at an equal elevation to the town of the same name (CONAGUA 2024), so it is considered that it does not generate a change in storage. In addition, there is no information on importing and exporting water to and from other basins. For analysis purposes, the CONAGUA divides the Piaxtla River basin into two sub-basins, the 1,016 Piaxtla River 1 (from its source to HS 10111) and the 1017 Piaxtla River 2 (from HS 10111 to the mouth to the sea).

Climatological information

The climatological (precipitation and temperature) and hydrometric (flow) information were obtained from the Climatological Statistical Information site (CONAGUA 2024) and the National Database of Surface Waters (BANDAS in Spanish) (CONAGUA 2024). As is typical, it was necessary to complete the daily values in some stations, from some days (25045 Ixpalino) to some months (10031 Huahuapan and 10042 Las Truchas). First, statistical linear regression was used (Montgomery et al. 2021), and second, the inverse distance method (Moral 2010; Aragón-Hernández et al. 2019). Thus, the daily values of precipitation (accumulated in 24 h), temperature (average of maximum and minimum value), and mean flow in each station were obtained in the period 1961–2017 for the former, and 1951–2010 for the latter. From this information, the flow (volume) and accumulated precipitation for each year were obtained. In the case of temperature, the average for each year was determined. Finally, the mean annual value of each time series was calculated.

Since the water balance of the study area will be calculated in a distributed manner (at the cell level with resolution equal to the DTM, 15 m), the value of the measured climatological variables (P, T) should have the same format, but these were measured punctually (Figure 1), so, the ordinary kriging interpolation method was used (Moral 2010; Aragón-Hernández et al. 2019), which has proven to provide adequate results (Figure 3).
Figure 3

Average annual accumulated precipitation (left panel) and average annual temperature (right panel) in the Piaxtla River basin.

Figure 3

Average annual accumulated precipitation (left panel) and average annual temperature (right panel) in the Piaxtla River basin.

Close modal

The highest values of precipitation were obtained in the highest areas of the mountains (1,099 mm) and decreased, as did the topography, to 649 mm in the lowest part (Figure 3, left panel). The average annual accumulated precipitation of the Piaxtla sub-basin is 956.1 mm, 933.4 mm for the Ixpalino sub-basin, and 906.3 mm for the entire basin. In the case of the average annual temperature, there is an inverse behavior to precipitation, with values of up to 25.9 °C in lower elevation areas and close to the coast and 11.6 °C in the highest areas of the mountains (Figure 3, right panel). The average annual temperature of the Piaxtla sub-basin is 14.5 °C, 15.6 °C for the Ixpalino sub-basin, and 16.6 °C for the entire count.

Hydric balance

The water balance equation of the natural processes that occur in a basin for monthly or annual periods, based on Budyko's theoretical framework (Budyko 1948) is (Du et al. 2016):
(1)
where P is the total precipitation, ETR is the actual evapotranspiration, Pe is the direct runoff (excess precipitation), ΔS is the water content change stored in the soil root zone, D is the water demand of the population and the economic activities carried out in the basin (agricultural, urban, and industrial uses), R is the returns of water to the environment, and ΔG is the change in groundwater storage.
The terms ΔS + ΔG, in which infiltration is also included, are grouped in the term ΔS due to the complexity of obtaining them separately; the final equation that defines the balance (availability) of water in the basin is:
(2)

The following sections describe the procedure to determine the value of each of the variables in each cell in which the basin is discretized based on the information and conditions available.

Evapotranspiration

Evapotranspiration (actual, ETR) is calculated from the simplified Turc method, which depends only on precipitation and temperature (Turc 1961; Pike 1964). In addition, for comparative purposes, potential evapotranspiration (ETP) is also calculated using the Thornthwaite method (Thornthwaite 1948). To obtain the ETR value, ETP must be iterated monthly as (Thornthwaite & Mather 1957):
(3)
where m is month 1–12, Pm is the precipitation in month m, ETRm is the actual evapotranspiration in month m, and ΔSTm is the water change stored in the soil in month m, calculated as:
(4)
The water stored in the soil in month m (STm) depends on land use, vegetation cover (USV in Spanish), and soil type. STm changes throughout the year, decreasing in the warm months of the year when Pm < ETPm and increasing in the humid months when Pm > ETPm. Furthermore, the value of STm cannot exceed the soil holding capacity (SHC). The SHC can be estimated from the proposed values (Dunne & Willmott 1996). STm changes as a function of Pm and ETPm, as (Thornthwaite & Mather 1957):
(5)
Thus, the calculation of the ETP begins in the last month of the year's wet season, where it is assumed that the soil has reached its maximum storage capacity. This is valid if the annual value of ETP is less than that corresponding to the accumulated precipitation (Paccu). Otherwise, the initial value of ST is unknown and must be proposed. In those cases, in which the annual precipitation is not enough to recharge the water stored in the soil, the initial value can be proposed by Thornthwaite & Mather (1957):
(6)
where RP is the potential water recharge (Equation (7)) and PP is the potential water loss in the year (Equation (8)), which are written as:
(7)
(8)

The method is applied until completing a year, ending in the same month in which it began; if the proposed initial value of STm for month m (m = 1) and year i, is equal to the calculated value of STm for the same month m (m = m + 11), but for the following year (i = i + 1), the process ends; otherwise, it is repeated. The above holds without problems for humid areas where precipitation is always sufficient to satisfy the ETP (Thornthwaite & Mather 1957), and the method converges in the first iteration. In those basins with more than one dry season and one wet season, in addition to Paccu < ETP, the method takes time to converge. To avoid excessive calculations, given that the above will be applied in each cell, that is, each cell has its own annual condition of the Paccu–ETP relationship, a term threshold was established when the global relative error (ER) is less than 5% (Montgomery et al. 2021). The application of the method is considered correct if the annual value of ΔS is zero.

Runoff

Runoff in a basin can be determined differently, depending on whether it is measured or not. In the first case, it is determined directly from the record of the HS; in the second, it is necessary to use event-based methods such as the runoff coefficient and curve number (Chow et al. 1988), which allow calculating direct runoff and infiltration but need to be calibrated with measured data. Another approach is the Thornthwaite physically based method based on ETR and soil moisture (Thornthwaite & Mather 1957).

Thornthwaite method

Thornthwaite & Mather (1957) established that the excess precipitation of a basin can be obtained from the ETR. The concept is similar to the case of the curve number method, where runoff only occurs if the infiltration capacity has been reached, that is:
(9)

where Pd is the available precipitation, which corresponds to the precipitation with the potential to infiltrate and runoff, and that is available for exploitation.

Water demands

Water demands in a basin can be determined based on land use (agricultural and industrial) and population size (urban). The demand for agricultural water or consumptive use can be determined from the ET, the industrial demand from public records available, and the demand for urban use based on the surface area number of inhabitants, and climate (AWWA 2014). For water balance purposes, returns can be determined as a percentage of demands based on water use (SEMARNAT 2015).

Change in storage

The change in storage ΔS can be interpreted as a surplus (ΔS > 0) or deficit of water (ΔS < 0), also known as water availability, for the analysis cell once water demands are included.

In the case of agricultural demands, which were proposed to be equal to ETR, this term is neglected in the water balance equation so as not to be considered double (Equation (10)).
(10)

Calibration and validation

To calibrate and validate part of the described methodology, some calculated results, such as excess precipitation, are compared with the runoff measured in the 2 HS within the basin.

Evapotranspiration

With the precipitation and temperature values in each cell of the study area (Figure 3) and using the Thornthwaite method (Thornthwaite 1948), first, the ETP and then the ETR (Equations (3)–(8)) were calculated; the last one, based on the values of the SHC (Dunne & Willmott 1996) and the ETP (Figure 4, left panel). The monthly iterative process began in October (end of the rainy season); the convergence values (iteration 6) correspond to the actual evapotranspiration. Minimum values of 571 mm mainly in agricultural or grassland areas and maximum values of 847 mm in the transition zone between the mountainous and coastal regions are obtained. The average value of the Piaxtla sub-basin is 685.7 mm, 698.2 mm for the Ixpalino sub-basin, and 697.9 mm for the entire basin. Also, the ETR by the Turc method (Turc 1961) was calculated. The average value for the Piaxtla sub-basin is 626 mm, 639 mm for the Ixpalino sub-basin, and 642 mm for the entire basin (Figure 4, right panel). With both ETR methods, available precipitation was calculated (Equation (9)).
Figure 4

ETR calculated with the Thornthwaite method (left panel) and the Turc method (right panel) in the Piaxtla River basin.

Figure 4

ETR calculated with the Thornthwaite method (left panel) and the Turc method (right panel) in the Piaxtla River basin.

Close modal

Runoff

From the records of the average daily flows at the HS (Piaxtla and Ixpalino), the normal hydrograph was calculated. The average annual runoff is 31.5 and 47.7 hm3, respectively, with a similar flow variation concerning time between both seasons.

From the normal hydrograph, the base runoff was determined graphically (Chow et al. 1988), resulting in 7.8 and 10.8 hm3, respectively, which implies that direct runoff (precipitation in excess) is 23.7 hm3 (139.5 mm) and 36.9 hm3 (187.9 mm), respectively. With this information and the P (Figure 3, left panel), a mean Ce value of 0.146 and 0.201 and a mean NC value of 48.3 and 66.9, respectively, were calculated (Chow et al. 1988), the latter only depending on the available precipitation (Equation (9)).

On the other hand, the runoff coefficient, curve number, and Thornthwaite methods were used to calculate excess precipitation (Pe, direct runoff) in each cell. For the first, a runoff coefficient (Ce) was assigned based on values proposed in the literature (Chow et al. 1988), depending on land use and vegetation cover (Figure 2, left panel) and soil type (Figure 2, right panel). An average Ce of 0.152 was obtained for the Piaxtla sub-basin and 0.151 for the Ixpalino sub-basin; with errors of 4.1 and −24.9%, respectively, compared with those obtained with measured values. Subsequently, a calibration process was carried out, resulting in a coefficient of 0.155 (error of 6.2%) for the first, 0.163 (error of −18.9%) for the second, and 0.177 for the entire basin.

In the case of the curve number method, based on the use of land and vegetation cover (Figure 2, left panel), the slope of the terrain calculated from DTM, and the hydrologic soil (Figure 2, right panel), the curve number (CN) to each cell was assigned (Chow et al. 1988). After the calibration process, the resulting CN varies from 30 in some central regions to 98 in urban areas; an average of 62.0 (error of 33.7%) for the Piaxtla sub-basin, 63.2 (error of −7.3%) for Ixpalino, and 64.2 for entire basin were obtained.

The Pe in each cell was then calculated with the runoff coefficient method using P and with the curve number method with Pd (Chow et al. 1988); Pd was calculated with the ETR of Turc and Thornthwaite methods. The results reflect the spatial distribution, both the runoff coefficient and the curve number. With the last method, non-representative values close to zero are obtained.

With the runoff coefficient method, the highest values of Pe are observed in urban areas (Figure 5, left panel), with an average value for the entire basin of 137.8 mm (145.9 and 141.8 mm in the Piaxtla and Ixpalino sub-basins, respectively). With the curve number method (ETR with the Turc method), Pe is reduced to a minimum of 0 mm in the west and a maximum of 460 mm in the upper zone of the basin with an average of 155.7 for the entire basin and 199.8 and 175 mm in the Piaxtla and Ixpalino sub-basins, respectively (Figure 5, center panel).
Figure 5

Excess precipitation: runoff coefficient method (left panel), curve number method (center panel), and Thornthwaite method (right panel).

Figure 5

Excess precipitation: runoff coefficient method (left panel), curve number method (center panel), and Thornthwaite method (right panel).

Close modal

Finally, with the Thornthwaite method (Equation (9)), the value of Pe was calculated (Figure 5, right panel). The highest values of up to 487 mm are observed in the mountainous and high areas of the basin, which decreases to 0 mm in the lowest area. The average value for the entire basin is 208.4, 270.4, and 235.2 mm for the Piaxtla and Ixpalino sub-basin, respectively. Contrary to previous methods, here, a greater influence of both ETR and P is observed than soil type and vegetation cover.

Water demands and returns

The water demand in agricultural areas was considered equal to ETR; the water demand for urban use was calculated based on the number of inhabitants (31,320), surface area, location of the population, and climate (INEGI 2024). Therefore, considering a supply of 338 L/habitant/day (AWWA 2014), the water demand is 3.86 hm3. The demand for industrial water was not considered due to the lack of information regarding its location. In this way, with the ETR value calculated with the Turc method, a water demand of 10.5 mm was obtained for the Piaxtla sub-basin, 22.8 mm for the Ixpalino sub-basin and 43.1 mm for the entire basin (Figure 6, left).
Figure 6

Water demands calculated from urban and agricultural areas. ETR with the Turc method (left panel) and concessioned (right panel).

Figure 6

Water demands calculated from urban and agricultural areas. ETR with the Turc method (left panel) and concessioned (right panel).

Close modal

In addition, a water demand sheet was obtained from the volume granted by the municipality (urban, agricultural, and industrial), according to the Public Registry of Water Demand of Mexico, REPDA (CONAGUA 2024). Such volume was divided by the area of each municipality, obtaining a layer of 4.1 and 4.5 mm for the Piaxtla and Ixpalino sub-basins, respectively, and 6 mm for the entire basin (Figure 6, right panel). In both cases, the returns to the environment established by NOM-011-CONAGUA-2015 were proposed as 30% for agricultural use and 40% for urban use (SEMARNAT 2015).

Water balance

Finally, the water balance equation (Equation (2)) was applied to determine the change in storage. The above gave rise to different combinations due to the calculation methods of the ETR, the D, and the Pe, grouped according to the latter's calculation method, that is, the runoff coefficient method (Figure 7), the number method curve (Figure 8), and the Thornthwaite method. In the latter, the results with granted and calculated water demands are very similar.
Figure 7

Change in storage: Pe is calculated with the runoff coefficient method, ETR with the Turc (upper panel), and Thornthwaite (lower panel) methods, and D granted (left panel) and calculated (right panel).

Figure 7

Change in storage: Pe is calculated with the runoff coefficient method, ETR with the Turc (upper panel), and Thornthwaite (lower panel) methods, and D granted (left panel) and calculated (right panel).

Close modal
Figure 8

Change in storage: Pe is calculated with the curve number method, ETR with the Turc (top panel), and Thornthwaite (bottom panel) methods, and D granted (left panel) and calculated (right panel).

Figure 8

Change in storage: Pe is calculated with the curve number method, ETR with the Turc (top panel), and Thornthwaite (bottom panel) methods, and D granted (left panel) and calculated (right panel).

Close modal

The water balance calculated with the Pe with the runoff coefficient method, the ETR with the Turc method, and the concessioned demands showed the highest average value of ΔS for the Piaxtla and Ixpalino sub-basins (177.2 and 139.9 mm, respectively); when considering Thornthwaite's ETR, the values are reduced to 115 and 79, respectively. The results with the Pe with the curve number method were similar and contrasted with the Pe of the monthly Thornthwaite calibration, where ΔS is close to zero.

Finally, the different methods and combinations used determined the average values of the basins and sub-basins of each of the variables involved in the water balance (Table 1).

Table 1

Water balance variables

VariableMethodPiaxtlaIxpalinoComplete
(mm/year)(mm/year)(mm/year)
Temperature (T, °C) Ordinary kriging 14.5 15.6 16.6 
Precipitation (POrdinary kriging 956.1 933.4 906.3 
Evapotranspiration (ET) 1: Turc 626.0 639.0 642.0 
Thornthwaite (potential) 941.8 897.0 861.4 
2: Thornthwaite (actual) 685.7 698.2 697.9 
Available precipitation (Pd1: Turc 330.0 294.0 264.0 
2: Thornthwaite 270.1 235.2 208.4 
Runoff (PeMeasured 139.5 187.9 233.0a 
10: Ce (P145.9 141.8 137.8 
Ce (Pd150.7 45.0 40.3 
NC (P707.5 698.4 686.2 
21: NC (Pd1199.8 175.0 155.7 
22: NC (Pd2132.6 114.5 101.6 
Thornthwaite 270.4 235.2 208.4 
Demands (D1: Agricultural (ETR1) and urban uses 10.5 22.8 43.1 
2: Agricultural (ETR2) and urban uses 10.7 24.7 43.7 
3: Concessions 4.1 4.5 6.0 
ΔSb Pe10 + ETR1 + D1 174.7 138.5 107.9 
Pe10+ ETR2+D2 115.0 79.0 51.6 
Pe21+ ETR1+D1 130.6 119.8 109.2 
Pe22+ ETR2+D2 138.0 121.0 107.0 
Thornthwaite + D2 0.2 0.1 0.1 
Pe10+ ETR1+D3 177.2 139.9 107.1 
Pe10+ ETR2+D3 117.2 80.3 50.8 
Pe21+ ETR1+D3 126.6 115.5 103.4 
Pe22+ ETR2+D3 133.8 116.2 101.1 
 Thornthwaite + D3 −2.8 −3.1 −4.1 
VariableMethodPiaxtlaIxpalinoComplete
(mm/year)(mm/year)(mm/year)
Temperature (T, °C) Ordinary kriging 14.5 15.6 16.6 
Precipitation (POrdinary kriging 956.1 933.4 906.3 
Evapotranspiration (ET) 1: Turc 626.0 639.0 642.0 
Thornthwaite (potential) 941.8 897.0 861.4 
2: Thornthwaite (actual) 685.7 698.2 697.9 
Available precipitation (Pd1: Turc 330.0 294.0 264.0 
2: Thornthwaite 270.1 235.2 208.4 
Runoff (PeMeasured 139.5 187.9 233.0a 
10: Ce (P145.9 141.8 137.8 
Ce (Pd150.7 45.0 40.3 
NC (P707.5 698.4 686.2 
21: NC (Pd1199.8 175.0 155.7 
22: NC (Pd2132.6 114.5 101.6 
Thornthwaite 270.4 235.2 208.4 
Demands (D1: Agricultural (ETR1) and urban uses 10.5 22.8 43.1 
2: Agricultural (ETR2) and urban uses 10.7 24.7 43.7 
3: Concessions 4.1 4.5 6.0 
ΔSb Pe10 + ETR1 + D1 174.7 138.5 107.9 
Pe10+ ETR2+D2 115.0 79.0 51.6 
Pe21+ ETR1+D1 130.6 119.8 109.2 
Pe22+ ETR2+D2 138.0 121.0 107.0 
Thornthwaite + D2 0.2 0.1 0.1 
Pe10+ ETR1+D3 177.2 139.9 107.1 
Pe10+ ETR2+D3 117.2 80.3 50.8 
Pe21+ ETR1+D3 126.6 115.5 103.4 
Pe22+ ETR2+D3 133.8 116.2 101.1 
 Thornthwaite + D3 −2.8 −3.1 −4.1 

aEstimated by the proportion of areas.

bIncludes the corresponding returns.

Analysis of results

First, the differences between the watersheds defined by CONAGUA and those calculated for this work stand out, and thus the contribution areas (Figure 1). For the entire basin (Piaxtla River basin), CONAGUA considers additional sub-basins that do not contribute to the main channel (they discharge to the Gulf of California, between the mouth of the Piaxtla River and the Elota and Quelite rivers), resulting in an area of 7,441 km2, in contrast to the 6,984 km2 calculated. Also, the CONAGUA watershed divide (area of 4,854 km2) and the calculated one (5,329 km2) of the Piaxtla 1 sub-basin (from the watershed divide to Piaxtla HS) do not coincide, especially on the left bank of the channel. In such a way, differences could be expected between the availability of water from CONAGUA, and the results obtained.

Climatological information

The highest P values are in the mountainous area, before the plateau, and the lowest in the coastal area; the T behaves inversely (Figure 3). The ETR calculated with the Turc method presents the highest values in the areas of the highest P; however, with the Thornthwaite method, it is observed where the T is highest (Figures 3 and 4). However, the values of the mean ETR are similar. With the last method, it is 10% higher in the three sub-basins. The above clarifies the role of precipitation and temperature in the actual value of evapotranspiration, so any trend in such variables, such as those expected due to climate change (IPCC 2021), will impact the water balance.

The analysis carried out indicates that the ETR represents between 71 and 77% of the total precipitation of the entire basin; that is similar to the national average of Mexico, which corresponds to 72% (Arreguín et al. 2011), runoff (excess precipitation) between 11 and 28% and ΔS between 6 and 11%. For the case of the study, the ΔS corresponds to the volume of water that infiltrates, given the absence of lakes and large dams to store water, therefore, also in the order of the national average of 6% (Arreguín et al. 2011). Finally, the water demands represent a minimum volume, between 0.75 and 3% (Figure 9).
Figure 9

ETP, ETR, and Pe compared with P.

Figure 9

ETP, ETR, and Pe compared with P.

Close modal

Figure 9 shows that the ETP, Pe, and ΔS decrease depending on the size of the basin, in contrast to the ETR and demands. Furthermore, the proportion of the variables concerning total precipitation indicates that the ETR is the largest, followed by Pe, ΔS, and finally, the water demands. From this, it follows that the ΔS, due to its difficulty in estimation, can be considered negligible just as Du et al. (2016) found. The little water impact of human activities in the basin is also confirmed, which is consistent with the distribution of vegetation cover and land use.

Runoff

The runoff measured at both HS has a similar behavior and a specific correlation with the contribution area; that is, the volume measured at HS Piaxtla (located upstream) represents 66% of HS Ixpalino and 86% concerning the contribution area.

The calculated excess precipitation (runoff) tends to be greatest in urban areas, followed by agricultural and grassland areas. Total precipitation and available precipitation were used in two methods (runoff coefficient and curve number). With the curve number method, the runoff values are generally lower. In the lowest areas, runoff is minimal, even with values of zero (Figure 5). The latter also occurs with the Thornthwaite method, in which the effect of the ETR is notable, so the spatial variation differs from the other methods (Figure 5).

To quantitatively evaluate the different methods, the mean runoff and relative error of the basin and sub-basins were determined (Table 2). The best option corresponds to the calibrated and proposed runoff coefficient methods, both using P, followed by the curve number method with Pd; in the rest of the methods, the errors are greater than 50%.

Table 2

Measured and calculated mean runoff and relative errors in the basin and sub-basins

MethodSourcesPiaxtlaERaIxpalinoERaERMbComplete
(mm)(%)(mm)(%)(%)(mm)
Measured  139.5 – 187.9   233.0c 
Ce proposed P 148.6 6.5 151.8 −19.2 12.4 157.1 
Pd1 50.7 −63.7 45.0 −76.1 69.4 40.3 
Ce calibrated P 145.9 4.6 141.8 −24.5 13.8 137.8 
Pd1 43.6 −68.7 38.9 −79.3 73.6 35.2 
NC P 707.5 407.2 698.4 271.7 344.5 686.2 
Pd1 199.8 43.2 175.0 −6.9 26.4 155.7 
Pd2 132.6 −4.9 114.5 −39.1 20.7 101.6 
Thornthwaite P 270.4 93.8 235.2 25.2 62.1 208.4 
MethodSourcesPiaxtlaERaIxpalinoERaERMbComplete
(mm)(%)(mm)(%)(%)(mm)
Measured  139.5 – 187.9   233.0c 
Ce proposed P 148.6 6.5 151.8 −19.2 12.4 157.1 
Pd1 50.7 −63.7 45.0 −76.1 69.4 40.3 
Ce calibrated P 145.9 4.6 141.8 −24.5 13.8 137.8 
Pd1 43.6 −68.7 38.9 −79.3 73.6 35.2 
NC P 707.5 407.2 698.4 271.7 344.5 686.2 
Pd1 199.8 43.2 175.0 −6.9 26.4 155.7 
Pd2 132.6 −4.9 114.5 −39.1 20.7 101.6 
Thornthwaite P 270.4 93.8 235.2 25.2 62.1 208.4 

aRelative error (ER).

bRelative error as the average (ERM) of both hydrometric stations.

cEstimated value based on contribution areas.

The calculated runoff decreases as the contribution area increases, which does not coincide with the measured runoff. Furthermore, in the Piaxtla sub-basin, runoff is overestimated and underestimated in the Ixpalino sub-basin and the entire basin (except for the curve number method using P; Table 2). The above is, partly, because in the contribution surface downstream of the Piaxtla HS, the water demand is greater than the rest of the basin and does not correspond to land use and, consequently, with the allocation of the Ce and the NC.

In the application of the different alternatives, the runoff coefficient method with Pd with very low results, the curve number method with P with very high values (it is necessary to consider the ETR), and the Thornthwaite method with results high in the sub-basins, but similar to the estimated mean value (with area ratio) for the entire basin stand out; in the latter case, P is a function of ETR and SHC.

Finally, given the adequate results, the runoff coefficient method indirectly considers the ETR (SEMARNAT 2015); therefore, the calibration process was proportional based on the demands, land use, and vegetation cover, and this process can still be improved.

Hydric balance

The ΔS encompasses storage and infiltration, but in the Piaxtla River basin, there are no surface water bodies (lakes, lagoons, and dams); therefore, the ΔS can be related more to potential infiltration than storage.

The calculation of such variables shows the areas where there is availability (ΔS > 0) and water deficit (ΔS < 0). Its spatial variation is influenced by the method used to calculate excess precipitation (runoff coefficient, curve number, and Thornthwaite), giving rise to three groups of results. The spatial variation of the results between the first two methods is similar (Figures 7 and 8).

The runoff coefficient method results show water availability in the middle and upper part of the basin and water deficit in the lower part of it (Figure 7). With the curve number method, there is water availability in the entire basin, with low values in the lower part and a water deficit in very localized areas (Figure 8). These differences are because with the first method, for any value of P, there is runoff; however, with the second method, runoff is generated from a particular value of NC, given that the moisture deficit of the water must first be satisfied. With the Thornthwaite method, the annual ΔS is close to zero, as established by theory. This is a way to validate that the calculations are correct, but it does not offer information on the availability or deficit of water and does not represent reality. The availability or deficit of water with this method can be known by analyzing the monthly results. In the first two methods, the ΔS is influenced by land use and vegetation cover, and in the third, by the ETR.

With all combinations, the average ΔS shows that the Piaxtla sub-basin has the highest water availability, followed by the Ixpalino sub-basin and, finally, the entire basin. This indicates that water availability is directly proportional to the basin area and the elevation of the terrain, in part due to a similar behavior of P and a behavior inversely proportional to the ETR and water demand. The behavior of ETR, D, and R in the basin's lower part results from vegetation cover and land use, which generate negative ΔS values. On the other hand, regardless of the method and combination of variables, the impact of human activities in the basin, considered from demands and returns, is not very significant.

In particular, ΔS calculated by combining Pe with the curve number method and ETR with the Turc or Thornthwaite method is very similar and corresponds to intermediate values. The highest values correspond to Pe with the runoff coefficient, ETR with Turc, and the calculated demands; the values less than Pe with the runoff coefficient, ETR with Thornthwaite, and the concessioned demands are also present.

Cells with ΔS > 0 represent infiltration zones, which recharge the aquifer. In this work, an average recharge of 644.4 hm3 (minimum of 354.79 and maximum of 762.65 hm3) was determined, 8.9% higher than that calculated by CONAGUA of 591.2 hm3 (CONAGUA 2020); the CONAGUA value was estimated based on results from the lower area of the basin.

The annual water availability in the Piaxtla sub-basin is 987.82 and 1,407.17 hm3 until its mouth to the sea (CONAGUA 2020). Such values were obtained only from the estimated runoff at the corresponding hydrometric station, less the different uses of water recorded in the REPDA. On the other hand, with the results of this work, the available volume in the Piaxtla sub-basin is 944.30 hm3, 4.4% lower (Pe with the runoff coefficient method, ETR with Turc method, and granted water demands) and 762.65 hm3 (Pe with the curve number method, ETR with Tur, and calculated water demands) in the entire basin. As mentioned above, the CONAGUA volume for the entire basin must be overestimated, given that the watershed it considers includes other basins that flow into the sea between the Piaxtla, Elota, and Quelite rivers and provide 6.5% more area. Although there is an available volume of water at the annual and basin levels, a spatial analysis such as the one presented makes it possible to identify the areas of the basin where there is availability or deficit of water due to the impact of economic activities.

Finally, in applying the methodology, the interaction of the water flow between cells was not considered; that is, the result of each cell is independent of the adjacent cells, which does not adequately represent reality. For example, the runoff generated in urban areas runs downstream and can evaporate, infiltrate, be used in economic activity, or reach the main channel; however, in areas or cells where the P is not high enough to satisfy the ETP, no runoff is generated, it could receive water from natural or anthropogenic sources from upstream and thereby satisfy its requirements and even generate runoff.

Spatial variability factors

The spatial distribution of climatological variables depends on different factors, one of which is topography. In this way, the precipitation increases proportionally to the terrain’s elevation; however, the temperature has an inverse behavior.

As a consequence, the ETR is generally higher in high areas and also depends on land use in coastal areas, but the ETR is also a function of the calculation method used; with the Turc method, it is more dependent on precipitation and in the Thornthwaite method on temperature; the mean values are similar.

Also, depending on land use, excess precipitation is lower in low areas (especially urban, agricultural, and grassland areas), even with values of zero, due to low precipitation values, high temperature, and ETR; and greater in temperate and high zones.

Finally, water availability is heterogeneous, but it is lower in the lower part of the basin, due to the combination of lower precipitation and higher temperature, ETR, and water demands.

Limits of application

The calculations were carried out annually, but some methodologies were used that were not entirely suitable for this purpose, since the runoff coefficient and curve number method were proposed to estimate direct runoff (excess precipitation) for isolated storm events. Even so, the runoff coefficient method is the one that provides the best results.

On the other hand, the Turc method for calculating the ETR was initially proposed for humid areas but is now widely used. Furthermore, the application of either method depends on an adequate characterization of soil moisture; that is, the Turc method of soil retention capacity and the Thornthwaite method of water stored in the soil.

In any case, an adequate calibration of the methods involved with measured data is most recommended.

In this work, the Piaxtla River basin was characterized spatially (15 × 15 m cells) and hydrologically with information on precipitation, temperature, land use and vegetation cover, soil type, terrain slope, water demands, and returns to the environment, with information from Mexican official sites (CONAGUA and INEGI). With this information, evapotranspiration was calculated with the Thornthwaite and Turc methods, runoff using the curve number and runoff coefficient methods, and water demands based on the available information and the volume granted according to the REPDA. The calculated runoff was calibrated and validated with the runoff measured at the two available HS. Finally, the value of the change in storage and infiltration was determined by combining the previous methods.

In the exercise carried out, it was found that by the monthly iterative method proposed by Thornthwaite & Mather (1957), values similar to the simplified equation of annual Turc method data are obtained. On the other hand, the use of annual accumulated precipitation in the curve number method overestimates runoff significantly; however, when considering evapotranspiration and available precipitation, more consistent runoff values are obtained, in contrast, for the runoff coefficient method, including evapotranspiration through available precipitation decreases runoff values significantly, proving that it is conceptually incorrect. The calculation of runoff with the Thornthwaite method yields the highest values. In the annual hydrological cycle, the percentage of infiltration and water storage concerning the other variables is low. Therefore, they can be considered negligible.

In the calculation of runoff, given that methods developed for isolated events were used (runoff coefficient and curve number), in the first, the annual precipitation was used, and in the second, available precipitation; both are associated with some uncertainty due to the assignment of the runoff coefficient and curve number respectively, which depend on soil characteristics. An accurate estimate of these will allow a better estimate of water availability. The results show differences, especially spatial ones since the average value is similar at both the basin and sub-basin levels.

The application of GIS tools was very useful in analyzing water availability spatially and at the cell level, adding a new perspective to the analysis (distributed analysis). This leads to the identification of areas with potential water deficits and surpluses without considering the interaction between neighboring cells (water flow). Thus, it can be analyzed whether the water demand can be satisfied by excess precipitation and from areas with a significant surplus without compromising natural runoff or the demand of the different ecosystems.

Under climate change scenarios, the methodology presented here would allow for identifying areas that could compromise access to water since evapotranspiration depends on temperature and precipitation, variables significantly affected by the phenomenon. In this sense, the lower area of the Piaxtla River basin could be affected since water availability, without considering river runoff, is very limited, even in the rainy season. Thus, correct water management in the basin would imply not overexploiting the resource in the highest areas since downstream activities would be affected, and the aquifer's recharge would be compromised since it already has a deficit.

The authors of this work are grateful for the support of the Dirección General de Asuntos del Personal Académico DGAPA of the Universidad Nacional Autónoma de México UNAM (General Directorate of Academic Personnel Affairs DGAPA of the National Autonomous University of Mexico UNAM), through the Program ‘Apoyo a Proyectos de Investigación e Innovación Tecnológica PAPIIT – IG100421’ (Support for Research and Technological Innovation Projects PAPIIT IG100421) ‘Análisis de las interacciones entre aguas continentales y marinas en el Golfo de California bajo el enfoque de la fuente al mar como base para su gestión sustentable’ (‘Analysis of the interactions between continental and marine waters in the Gulf of California under the source-to-sea approach as a basis for its sustainable management’).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.

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