Flow distribution and trends in the Das Velhas River Basin

In the management of water resources, it is necessary to balance the demands of multiple uses of water and water availability, while enabling use in an environmentally sustainable way. Probability distributions of flow rates are essential tools for assessing water availability. The objectives of this work were to analyze the best probability distribution that conforms to the annual minimum daily average discharge for periods of seven consecutive days (Q7) for 14 stream gauging stations in the Das Velhas River Basin and to identify possible trends in Q7 time series and in bi monthly and annual sets of daily discharges in three key stream gauging stations. The quality of fit was verified by the Anderson-Darling test (A-D). The selection of the models that presented the best fit was done according to the Bayesian Information Criterion (BIC). The Mann-Kendall test was used to verify trends in time series of discharge. In general, better measures of quality of fit were obtained for the probability distributions Gumbel and Rayleigh. Negative trends in discharge distributions were verified in the three stations. For the Várzea da Palma station, the closest to the river mouth, negative and significant trends were found for the Q7 data and daily average discharge for every bimester except the first.


INTRODUCTION
Concerns about sustainability of water resources arise frequently in discussions related to environmental and socioeconomic issues at global, regional and local levels. The exploitation of natural resources, without regard for sustainability, negatively impacts relationships between water availability and demand, limiting regional strategies of social and economic development covering multiple uses, among them public water supply and irrigation.
The growing importance given to rational water use has been motivating evolutions in systems of planning, administration and use of the water resources, in the search for an effective answer to social demands. To adequately manage water potential, it is fundamental to know the hydrological characterization of the basin, considering temporal and spatial variability. Knowledge of flow distribution, identifying the probability of maximum and minimum discharges events, is a fundamental aspect of this characterization.
In the management of water resources, it is necessary to continually balance water demands and availability, in an environmentally sustainable way. Mello et al. (2010) emphasizes the importance of the study of the probability distribution of hydrological variables, since from this knowledge it is possible to determine the appropriate reference values of discharge to estimate regional water availability.
Each state in Brazil has adopted specific criteria for the establishment of minimum reference flows that will inform studies on the granting of water-use rights. For situations in which water catchment will occur in rivers, the hydrological regime or discharge probability distributions should be considered in the analysis. For perennial rivers, the grant request is usually analyzed based on a percentage of the minimum discharge for a period of seven consecutive days and return period of ten years (Q7,10) or minimum flows associated with the 95% (Q95) or 90% (Q90) of probability.
From time series of discharges, it is possible to evaluate water availability and significant changes in the hydrological regime in the long run. Best-fit analysis to identify probability distribution can be applied to these series . Santos (2010) argues that the randomness of hydrological processes make it difficult to predict. However, a quality of fit analysis of flow time series can be carried out for probabilistic or stochastic models. From this analysis, it is possible to infer (at a certain level of confidence) extreme flow values that are useful to the management of water resources.
On the other hand, trend analysis of discharges gives a basis to understand the impact of climate change and variations in the executed demands of multiple water uses, supporting the development and improvement of hydrological models applicable to water-resource management and scenario forecasting (Joseph et al., 2013).
The trend study of the components of the hydrological cycle is fundamental in the management of water resources. Through this analysis, government management councils can seek adjustments and courses of actions to ensure sustainability in the supply of water resources in harmony with the demand dynamics (Vilanova, 2014b;Kibria et al., 2016).
The identification of flow trends also contributes to the understanding of global climate Rev. Ambient. Água vol. 14 n. 3, e2289 -Taubaté 2019 variability and is essential for the development of hydrological models, hydrological forecasting and water-resource management. Climatic variability and disorderly use of the soil affect several components of the hydrological cycle, impacting the pattern of time series (Salvadori, 2013;Damázio and Costa, 2014;Tan and Gan, 2015;Agevap, 2016). Statistical tests such as Pettitt and Mann-Kendall can be applied to trend analysis in the time series of meteorological and hydrological variables (Barua et al., 2013;Uliana et al., 2015). The Mann-Kendall test (MK) is recommended by the World Meteorological Organization (WMO) and widely applied (Blain, 2010;Caloiero et al., 2011;Lima et al., 2011;Suhaila et al., 2010;Tao et al., 2011). The Mann-Kendall test has been used in several studies of hydrological trends, mainly in flow-trend analysis. Its use is justified by the fact that it presents greater robustness regarding deviations of normality and non-stationarity of the data of the time series when compared to parametric tests. In addition, this test allows the detection and approximate location of the starting point of a certain trend (Costa et al., 2015;Salviano et al., 2016). Dale et al. (2015) applied the Mann-Kendall seasonal trend analysis method to test mean daily streamflow in gauging stations from 1948 through 2010 in the Cimarron-Skeleton Watershed (North Central Oklahoma), with a drainage area of 8275 km 2 . They verified the importance of adjusting irrigation to mitigate the impact of increasing climate variability on streamflow. Soares et al. (2018) performed the Mann-Kendal test to analyze climatic indicators of desertification in the Pajeú River Basin in the State of Pernambuco, Brazil, identifying trends for rainfall and dryness indexes.
In the State of Minas Gerais, Brazil, the Das Velhas River Basin and some others the criterion adopted was a maximum limit of 30% of Q7,10 to be granted for consumptive water use in the hydrographic basin portion bounded by each section considered under natural conditions. In Das Velhas River Basin, there are no studies related to probability distributions and flow trends. Segmentation of these analyses on a bimonthly basis makes it possible to identify critical periods in terms of management. The Das Velhas River is the largest tributary of the São Francisco River. In its basin is the region with the highest population density in the hydrographic unit of the São Francisco River, having significant economic importance for the State of Minas Gerais. The largest water catchment in the entire basin occurs at the upper reaches of the Das Velhas River channel, responsible for supplying about 74% of the urban demand in the Belo Horizonte and Metropolitan Region. In Strategic Territorial Units (STUs) of the high Das Velhas River region, the sectors that most draw water are industry and mining, except for the STU Nascentes, which has significant demand for irrigation. Irrigation is also the main sector responsible for the withdrawal of water in the STUs of the regions Middle Low and Lower of the Das Velhas River Basin.
The present work contributes to the updating of base studies for the management of water resources in the Das Velhas River Basin, from the perspective of expansion of irrigated agriculture, especially, among multiple uses. This study therefore analyzed the quality of fit of probability distribution models to data of annual minimum of average daily discharge for a period of seven consecutive days (Q7) for 14 stream-gauging stations in Das Velhas River Basin and identified possible trends in Q7 time series and in bi monthly and annual sets of daily discharges in the stream gauging stations at Ponte Raul Soares, Santo Hipólito and Várzea da Palma.

Study Area
The study area was the Das Velhas River Basin, located between the latitudes 17°15' S and 20°25' S and longitudes 43°25' W and 44°50' W, in the central region of the state of Minas Gerais, comprising an area of 27,850 km 2 , which corresponds to 5% of the state area. The basin has a significant drainage density that feeds the main water course, which is 806.84 km long.
Considering data of the current master plan of the basin (Comitê da Bacia Hidrográfica do Rio das Velhas, 2015), the Q7,10 is 48.31 m 3 s -1 . The lower Das Velhas River stretch is the one with the highest water availability in relation to Q7,10 (31.54 m 3 s -1 ), followed by stretches mediumhigh (25.94 m 3 s -1 ), middle-lower (17.46 m 3 s -1 ) and high (13.58 m 3 s -1 ). The main segments responsible for effective water consumption are urban supply and irrigation, which account for 50.48% and 36.44% of total consumption, respectively. The water consumption by irrigation prevails especially in the STUs of the regions Middle and Low of the Das Velhas River Basin. In the STUs of the High region of the Das Velhas River Basin, the consumption of water for mining, industry and urban supply prevails.
According to information on the portal HidroWeb (2013), in this area there are 54 stream gauging stations, 14 are the responsibility of ANA, 30 of IGAM, 5 of CEMIG, 4 of CODEVASF and 1 of DNOS. The focus of the work was on the 14 stations that are the responsibility of the National Water Agency (ANA), whose dispositions in the basin are presented in Figure 1, for which discharge data are available. The time series of discharge data was obtained from the HidroWeb system of the ANA. The stream gauging stations considered in this study are shown in Table 1. Data of the stream gauging stations and characteristics of the data set are presented in Table 1 and Table 2, respectively.

Calculation of the minimum flow Q7
The moving average of seven consecutive days of mean daily discharge was calculated. For each station, the annual lowest values of the moving averages for seven consecutive days (Q7) were classified in ascending order forming a series for which the statistical analyses were carried out.
For non-complete consecutive seven-day sets of discharge values, it would not be possible to calculate the moving average without filling-in fail procedures. It was chosen in this work to not fill-in failures, discarding incomplete data sets, in order to not insert new sources of uncertainties in data series. The number of failures in seven-day datasets was counted for each station (Table 2).
Rev. Ambient. Água vol. 14 n. 3, e2289 -Taubaté 2019 Anderson-Darling test (Razali and Wah, 2011) at 5% significance level was applied to assess the quality of fit between empirical and theoretical distributions, i.e., for model selection. P-values and critical values were obtained by using the technique of parametric bootstrapping. The possibility of applying the technique of parametric bootstrapping due to the computational procedures programmed in the software used was considered as an additional criterion for selecting models. After that, the Bayesian Information Criterion (BIC) (Emiliano et al., 2009;Detzel et al., 2014;Pinto et al., 2015) was applied to rank the selected models. The software @Risk version 7.0.1 (Palisade, 2016) was employed to run the tests.

Trend analysis
The mean daily discharge (volumetric flow rate) time series were analyzed in three stream gauging stations: Ponte Raul Soares, Várzea da Palma and Santo Hipólito. The data were grouped in the following categories: discharge Q7 (minimum average daily discharge in a period of seven consecutive days, for each year); general (daily discharge over the study period); daily discharges grouped in bimesters, that is: 1 st bimester (January and February); 2 nd bimester (March and April); 3 rd bimester (May and June); 4 th bimester (July and August); 5 th bimester (September and October); and 6 th bimester (November and December).
The Mann-Kendall test was applied at a 5% significance level. The analyses were performed using the software R version 3.3.1 (R Core Team, 2016), with the "Kendall" package (McLeod, 2011).
Rainfall trend was also investigated by using the Mann-Kendall test for the three INMET

Model fit and selection
Probability distributions that were fit to the data of annual minimum discharge for a period of seven days (Q7) according to the Anderson-Darling test are presented in Table 3.
The probability distributions of Gumbel and Rayleigh were highlighted, since they fit to 8 and 5 stations of 14, respectively, according to the Anderson-Darling test and the possibility to apply the technique of parametric bootstrapping by the @Risk software. Quality of fit was observed for the distributions of Inverse Gaussian, Log Logistic, Log Normal, Pearson 5 and Weibull for two stations. Adjustment to Q7 data was not verified for the Erlang, Gamma and Triangular distributions. No distribution was suitable to describe the data for the stations Água Limpa Farm, Água Limpa Fazenda, Honório Bicalho Montante, Pinhões and Ponte Raul Soares (p-value <0.05).
After checking the quality of fit of the models to the observed data, the ranking of these distributions was done through the BIC selection criterion, as presented in Table 4. The Gumbel distribution was highlighted, for which the best performance was verified for 4 of the studied stations, followed by the Rayleigh distribution.   In a similar study, Silvino et al. (2007) used the Normal, Log-Normal, Exponential Range and Weibull probability distributions to estimate the minimum discharges of the Paraguay River. The distributions for which the data of the time series were best fitted were Weibull and Normal. Almeida et al. (2014) verified the quality of fit of the distributions Normal, Log Normal to 2 parameters, Weibull, Gumbel and Log Gumbel to data of time series of Q7 in the Miranda River Watershed. Among the studied distributions, Log Gumbel was the one that provided the best fit to the time series, proving to be the most reliable in the determination of minimum flows in the reference sub-basin. On the other hand, the Normal Log distribution was the worst distribution regarding the adjustment to the series data.
In a study on minimum flows, Victorino et al. (2014) used the Gumbel distributions for minimums, Fréchet, Log 2p, Log 3p, Gamma 2p, Gamma 3p and Weibull. The Log 2P, Log 3P and Gama 2P distributions were considered statistically adequate for the series of minimum annual discharges of the Grande River in the region of Barreiras (Bahia, Brazil), and Log 2P presented a more accurate adjustment than the others. Silva et al. (2006) applied the Log Normal 3P, Weibull and Gumbel probability distributions to time series of annual minimum daily and Q7 discharges from the region of the high Grande River. The best fit was observed for the distribution Log Normal 3P.
Descriptive statistics measures for the empirical distributions of Q7 of the stream gauging stations are presented in Table 5 ). Values of kurtosis close to the normal distribution, that is, near 3 (mesokurtic), were found for most stations. However, high values of kurtosis, indicating a higher concentration of discharge values near the center of the distribution, were verified for the Ponte do Licínio and Ponte Raul Soares Jusante stations. Probability density functions for the empirical (input) and theoretical distributions for the stations of Santo Hipólito and Várzea da Palma are presented in Figure 2. Higher asymmetry and kurtosis indexes were observed in Várzea da Palma, characteristics that influenced the fitting of a different set of distributions in relation to that obtained for Santo Hipólito. The morphology of the upstream basin, relief, soil types, transmissivity and dimensions of aquifers, as well as rainfall patterns (intensity, duration and frequency) are some factors that can influence the characteristics of the distribution of flow values in a stream gauging such as asymmetry and kurtosis. These characteristics will determine the potential of theoretical probability models to fit to distributions of empirical data.

Trend Analysis
The results of the trend analysis for the Ponte Raul Soares, Santo Hipólito and Várzea da Palma stations are presented in Tables 6 to 8. For the stream-gauging station Ponte Raul Soares (Table 6), negative trends were observed in all data sets, except for the 1 st and 6 th bimester (S positive). However, the negative trend was significant for the Q7 and 4 th bimester. Negative (decreasing) trends in all data sets (S negative) were detected for the Santo Hipólito station (Table 7). Based on the Mann-Kendall test, there were significant trends (p-value <0.05) for Q7, 4 th and 5 th bimesters. For the Várzea da Palma station (Table 8), negative (decreasing) trends for all data sets (S negative) were pointed out. Only for the 1 st bimester this trend was not significant (p-value> 0.05). Table 6. Trend analysis of flow rate in the stream gauging station Ponte Raul Soares (S statistic, S variance (VAR (S)), denominator (D), Kendall correlation coefficient (τ)).   The lowest mean daily flows occurred in the fourth bimester in 65.0%, 61.6% and 61.3% of the years with all bimonthly sets in the stations of Raul Soares, Santo Hipólito and Várzea da Palma, respectively. The fifth bimester is second in relevance in the concentration of low flows. Significant negative trends for the Q7 and fourth bimester were indicated for the three stream-gauging stations. The time series of flow and nonparametric lowess (locally weighted scatterplot smoothing) regression (Lindsey and Sheather, 2010) for these two sets are shown in Figure 3. More pronounced decreases are observed in the first-and final thirds of the series. Changes in hydrological series can occur due to different causes, such as variations in rainfall amounts and water withdrawals to meet multiple uses. The lowest values of S are in the 4 th and 5 th bimesters, showing trends of decreasing flows. During these periods droughts occur, which contribute to a significant increase in irrigation demand. In Q7 data, a strong downward trend was also observed for the three stations, which is a worrisome factor regarding the management of water resources in the basin. According to the Water Resources Master Plan of the Das Velhas River Basin of 2015, for the strategic territorial units (STUs) Rio Bicudo, Ribeirão Picão, Ribeirão da Mata, Ribeirão Jequitibá, Rio Itabirito, Carste and high stretch of Das Velhas River, channel water withdrawal has already exceeded the criterion used (30% of Q7,10), preventing new grants (Comitê da Bacia Hidrográfica do Rio das Velhas, 2015).
There is a perspective of advancing irrigated agriculture in regions such as the lower Rio das Velhas. This advance imposes pressure on the demand for water resources with significant effects on water availability, which was partially detected at the Várzea da Palma station. The milder relief in this region favors the use of irrigation methods as a central pivot, among others, as can be observed in Figure 4. The presence of irrigated agriculture is also observed in the region of the county of Sete Lagoas (STU Ribeirão Jequitibá) in the region of the Middle Velhas -High Stretch, as shown in Figure 5. This figure also shows the great urban conglomerate of the capital of the state of Minas Gerais (Belo Horizonte) in the region of High Velhas. Results from Mann-Kendall test for rainfall time series for annual totals and for bimesters indicated significant negative trends only for the fifth bimester for Sete Lagoas meteorological station (p-value = 0.0497). Significant positive trends were detected for the second bimester in the meteorological stations of Belo Horizonte (p-value = 0.0034) and Sete Lagoas (p-value = 0.0118). However, some remarks should be considered. The time series of rainfall began in the early 1960s and the series of flow rate began in the late 1930s. In addition, there are only three INMET old conventional stations in the basin. These aspects impose a certain level of uncertainty in the understanding of the effects of rainfall variability on the flow distributions in the studied area.
In their study on streamflow trend, Marengo and Alves (2005) concluded that the discharges of the Paraíba do Sul River, observed in stream gauging stations of the states São Paulo and Rio de Janeiro, has a negative trend for the last 50 years. The study pointed to a possible impact of anthropic influence (energy generation, irrigation and demand associated to population growth) as possible causes for these trends and not to a climate change of the rainfall distribution in the basin. Nevertheless, Santana et al. (2014) evaluated time series of rainfall and discharge in the Paraguay River Basin in the stations Barra do Bugres, Cáceres and Descalvado during the period of 1980-2005. According to the analyses, the discharge is basically a function of precipitation. Rosin et al. (2015) studied five stream-gauging stations located in the interior of the Rio das Mortes Basin, in the State of Mato Grosso, Brazil. The authors verified a strong connection in relation to the type of use and withdraw of water resources for the station whose analyzes showed a negative trend in all series. Uliana et al. (2015) studied trends in time series of rainfall and flow data for the city of Alegre, in the state of Espirito Santo, Brazil, and found that flow rate in October had increased by 34.2% since 1963. They also detected a tendency of increase in rainfall for the months of August and September, since the years of 1967 and 1964, respectively, and concluded that this trend, among other possible factors, may have influenced the increase of the average flow rate for the month of October.
In a similar work, Vilanova (2014a) evaluated the existence of trends in time series of discharge and rainfall in the high Mogi-Guaçu region from 1973 to 2006. A negative trend was detected in both time series. The author suggests that the identified trends may be the result of climate change, initially affecting the region's rainfall and possibly generating a similar response at the discharges.

CONCLUSIONS
Among the probability distributions studied to describe discharge data of Q7, the best fits, in general, were verified for Gumbel and Rayleigh distributions, based on Anderson Darling and Criterion of Bayesian Information tests. However, the quality of fit by different distributions was differentiated for each stream gauging station.
Significant negative trends for the three stream gauging stations were detected for time series of Q7 and mean daily flow in the 4 th bimester. There were significant negative trends for the Santo Hipólito station also for the 5 th bimester. For the Várzea da Palma station, located near the mouth of the Das Velhas River, only for the first bimester significant negative trend in the flow series were not detected. Corresponding negative trends were not observed for the rainfall series from the studied meteorological stations.