Deterministic and probabilistic modeling of microbiological quality using the QUAL-UFMG: a water resource management tool applied on the slope waters of the Grande River, Brazil

Brazil has a population of approximately 213.3 million inhabitants (IBGE 2022) and is a country with dimensions similar to those of a continent, which results in very different regional conditions in some aspects. The greatest water availability is found in the North region, which has the lowest population density. On the other hand, the Southeast concentrates the largest portion of the population and does not have the same water availability, which leads to greater demand, a greater amount of discharges and less dilution capacity of water bodies (Ana 2009). It is known that the disposal of untreated or insufficiently treated effluents in water sources can cause a series of damages to the aquatic environment (Ashouri & Rafei 2018; Bui et al. 2019; Liu et al. 2022), such as, for instance, contamination by pathogenic organisms, which is a problem, especially in developing countries (Pujol-Vila et al. 2016; Andersson et al. 2018; Madani et al. 2021).

The presence of etiological agents of waterborne diseases affects some of the main uses intended for it, such as drinking water supply, irrigation and bathing (Von Sperling 2014a; Alegbeleye et al. 2016; Madani et al. 2021), which may lead to an increase in the number of ill people and/or imply the technical and financial impossibility of water resource use. As it is costly and economically unfeasible to quantify the presence of all possible causes of diseases, monitoring is done using groups of organisms that are indicators of fecal contamination, the group of thermotolerant coliforms, along with detection methodologies for quantification of the bacteria Escherichia coli (E. coli), are the most used in the assessment of health hazards (Batista & Harari 2016; Silva et al. 2021).

According to the thematic diagnosis of water and sewage services (SNIS 2021), only 63.2% of the sewage generated in Brazil is collected and the percentage of treatment is even lower, at 50.8%, with unequal service among large urban centers (with higher rates) and smaller towns (with low population coverage). Even in locations that have wastewater treatment plants (WWTPs), it is considered that there is low removal of pathogenic organisms, since the occurrence of tertiary treatment with disinfection is very infrequent (Ferreira et al. 2021; SNIS 2021).

Aiming to guarantee surface waters quality compatible with their uses, according to their framework class, Federal Law No. 14.026 (Brasil 2020) established the new Legal Framework for Sanitation in Brazil, determining, among other things, that public basic sanitation services must guarantee 90% of the population with sewage collection and treatment by 31 December 2033. It should be noted, however, that this legal provision does not establish criteria regarding the minimum efficiency of removal of the various contaminants existing in the sewage, especially the fecal contamination indicator organisms.

It is of crucial importance to know the behavior of disease-transmitting agents in a water body, from the moment it is discharged to the places where it is used, in order to infer forms of control or to estimate points where the water collection can be installed for different uses. It is known that most of these agents have in the human intestinal tract the optimal conditions for their growth and reproduction. Once subjected to the adverse conditions prevailing in the water body, they tend to decrease in number, characterizing the so-called decay (Von Sperling 2014b; Madani et al. 2021). Thus, depending on the degree of contamination of a watercourse, the reduction in the count of these pathogenic organisms may be greater or dependent on a long stretch of recovery.

Water quality models are important tools for the effective management of water resources, assisting in decision-making by providing water quality simulations for a variety of management actions (Koo et al. 2020; Bello & Haniffah 2021; Kaufman et al. 2021; White et al. 2021). Furthermore, these models reduce the need for water collection and analysis, reducing costs in terms of resources and time (Von Sperling 2014b; Quijano et al. 2017; Bui et al. 2019). It is noteworthy that the monitoring of water quality in Brazil is quite limited, with a small number of sampling points, monitored variables and short historical series (ANA 2009). As a guide for assessing water quality using these models, it is necessary to have pertinent legislation and established parameters, such as the framework classes of water bodies.

The framework aims to ensure that the water quality is compatible with the most demanding uses for which it is intended and to reduce the costs of combating pollution, through permanent preventive actions (Machado et al. 2019; Costa 2021). With specific regard to fresh water, there are five classes, fundamentally established in accordance with the predominant uses of water resources (Brasil 2005; minas Gerais 2008): the special class, which cannot receive sewage discharges and serves to preserve the natural balance of aquatic communities; and Classes 1–4, the first of which has more restrictive parameters of quality and destination for more noble uses, with an increase in the permissiveness of variable values in the following classes.

The minimum efficiencies and maximum concentrations defined in the legislation must be observed when discharging wastewater into receiving bodies, as well as respecting the non-alteration of the watercourse class in which the stretch was classified, according to the intended uses (Classes 1–4). However, due to the inefficiency of sanitation services in many locations, the river is often in a condition inferior to what it should be, making it difficult to meet the legal requirements for the disposal of treated wastewater. This is the case in many hydrographic basins, such as the Rio Grande Basin and its planning units, such as GD2 (water slopes of the Rio Grande), with a high degree of contamination (in certain stretches), as observed in studies such as Amâncio et al. (2018). Thus, actions must be taken to improve the quality of the watercourse, with an increase in sewage treatment rates, in addition to reviewing the frameworks in place.

The general objective of this work was to implement a mathematical model of surface water quality for thermotolerant coliforms, of simple and free use for the watershed in question, with deterministic and probabilistic results, aiming at:

• (a)

  Checking the current basin water quality panorama, based on the few monitoring points, against the quality required for its main uses;

• (b)

  Inferring the future situation of the basin, in case new investments in sewage collection and treatment are not adopted;

• (c)

  Checking how the basin water quality will be if the obligations foreseen in the sanitation legal framework are met, indicating whether the quality will be compatible with the preponderant uses foreseen. Otherwise, propose the necessary interventions;

• (d)

  Identifying stretches where the proposed framework is not being met and assessing the necessary conditions for compliance;

• (e)

  Assessing the adequacy of the model used for the management of Brazilian watersheds.

Brazil has eight large hydrographic basins and the São Francisco River Basin, together with the Paraná River Basin, covers a large part of the most populated region of the country (ANA 2009). One of the main subbasins of the Paraná River is the Grande River Basin, which is subdivided into 24 Water Resources Planning and Management Units – UPGRH (IGAM 2013). For the present work, one of these planning units was selected, named GD2 (waters slopes of Rio Grande), which comprises the areas drained by Mortes, Jacaré and Cervo Rivers.

Table 1 shows a summary of the sewage systems in the basin, in which it is observed that only 13 municipalities have some form of sewage treatment. Of the total urban population of the basin (541,576 inhabitants), 95% are served with sewage collection, however, only 21.7% of the collected sewage is treated (without disinfection) before being discharged into the receiving bodies, a situation that is worrying. Of the total discharges generated, 67.7% correspond to the four largest municipalities in the basin (Barbacena, São João Del-Rei, Lavras and Oliveira).

In the evaluation, the watercourses that receive discharges from urban areas or have water quality monitoring points were modeled and are presented in Figure 1, as well as the allocation of municipal seats with their respective points of liquid discharge.

The pollutant loads considered in the discharges are those related to sanitary sewage discharges from the urban areas of 30 municipalities located within the basin. The considered flows are shown in Table 1, and the discharges are indicated in Figure 1. As there is no systematic qualitative monitoring of sanitary sewage in the municipalities, for the deterministic modeling, the value of 10⁶ MPN (100 mL)⁻¹ was considered, which is consistent with reports in the literature (Arceivala 1981; Metcalf & Eddy 1991; Von Sperling & Von Sperling 2013; Von Sperling 2014a) and with the calibration process.

The model chosen was the QUAL-UFMG, which is widely used in Brazil, due to its versatility and ease of use, allowing greater applicability and good reliability (Fan et al. 2012; Srikrishnan & Keller 2021). It consists of an Excel platform developed by von Sperling (2014b), based on simplifications of the QUAL2E model (Brown & Barnwell 1987) and adaptation of its equations to electronic spreadsheets (De Oliveira Filho & Lima Neto 2018; Lima et al. 2018). For technical simplification, the algae component of its modeling and the longitudinal dispersion are excluded from the model, since advection is the main transport phenomenon in rivers. According to von Sperling (2014b), for rivers (small and large), the inclusion of the phenomenon of dispersion is not important, at least on a large time scale (stationary conditions), as in the case of the present study. However, for verification purposes, in item 2.5, its influence is verified.

The calculations are made considering numerical integration by the Euler method (Teodoro et al. 2013). In QUAL-UFMG, it is possible to evaluate the dynamics of some variables; however, the study will be restricted only to thermotolerant coliforms.

In Brazilian legislation (BRASIL 2005; Minas Gerais 2008), there is no discharge standard for organisms that indicate fecal contamination. However, the respective discharge, even if treated, cannot give the receiving body the adverse characteristics of the class in which it was framed.

When exposed to environmental conditions that differ from those prevailing within the intestinal tract of warm-blooded animals, ideal for their development and reproduction, coliforms and other enteric organisms present natural mortality. Among the various factors that contribute to decay, we can mention mainly: sunlight, temperature, sedimentation, pH, lack of nutrients and predation (Almeida 1979; Arceivala 1981; Thomann & Mueller 1987).

The mortality of microorganisms in different water bodies is generally associated with different K_b values, depending on the nature of the organism and the conditions in the aquatic environment (Arceivala 1981). Due to the dependence of several factors, the K_b values obtained in different studies in freshwater vary over a wide range. Typical values, however, are in the range of K_b = 0.5 a 1.5 day⁻¹ (base e, 20 °C), with 1.0 day⁻¹ being commonly adopted (Arceivala 1981; Thomann & Mueller 1987).

The flow rate considered for the dilution of wastewater into receiving bodies was the minimum, average of seven consecutive days, for a recurrence time of 10 years (Q_7,10), according to the legislation of the State of Minas Gerais (Minas Gerais 2008).

In this way, the flow rate of the receiving bodies was updated for each segment considered, according to its drainage area. Table 2 shows the total Q_7.10 for each of the 10 subbasins.

Regarding quality data, data from 11 surface water quality monitoring stations (Figure 1) operated by the Instituto Mineiro de Gestão das Águas (Minas Gerais Water Management Institute) – IGAM were used, with quarterly analysis and historical series since 1997 for the variables thermotolerant coliforms and temperature (IGAM 2022), and six stations operated by the National Water Agency – ANA, with data only on temperature (ANA 2022), with average values for the last 5 years during the dry season (April–October), shown in Table 3.

The data presented in Table 3 were used in the calibration of the model, for Q_7.10.

None of the quality stations with data on thermotolerant coliforms are located in upstream stretches of watercourses, in places with little or no human intervention. Thus, in defining the head values for the modeled watercourses, it was decided to assume the upper limit of Class 1 (Minas Gerais 2008), that is, 200 MPN (100 mL)⁻¹. Therefore, this was the value considered for the river, immediately before the first discharge of sanitary sewage, in all modeled watercourses.

The performance of a model is considered adequate and good if the CD value exceeds 0.75 and it is considered acceptable if the CD value is between 0.36 and 0.75 (Collischonn 2001). According to Thomann & Mueller (1987), the adequacy of the model will be as good as the CD value approaches 1. Calibration was performed using the Excel Solver tool, seeking to maximize CD, varying the model coefficients within the ranges reported in the literature.

The values adopted for K_b, for the various modeled watercourses after calibration, are presented in Table 4.

CD = 0.989 (with a relative error of 1.75%) was obtained, which shows an adequate performance.

If the expression is true, it means that the concentration profile will not be affected by more than 10% if dispersion is ignored.

Values were calculated for the right portion of Equation (6) for all subbasins under study. The lowest value found (worst situation) was 1.5 × 10⁶ m²·h⁻¹, in the Peixe River subbasin (v = 0.31 m·s⁻¹ and K_b = 1.0 day⁻¹).

According to Arceivala (1981), typical L_D values for larger water courses and rivers are between 10³ and 10⁵ m²·h⁻¹. Therefore, as the value found (in the worst situation) is above the D_L value, it is assumed that the expression is true and that the longitudinal dispersion can be neglected.

In mathematical models of water quality, there are uncertainties in the determination of input data that can significantly alter the expected results, which can lead to wrong decisions (Costa & Teixeira 2011; White et al. 2021). Thus, it is necessary to identify the input variables that directly affect the uncertainty of the output results of the simulation model (De Menezes et al. 2016; Koo et al. 2020; Soares & Calijuri 2021). There are many sources of inaccuracy in the modeling process, for example, those related to the estimation of coefficients, input data and system structure (Lindenschmidt & Fleischbein 2007; Khorashadi Zadeh et al. 2022). According to von Sperling (2014b), the uncertainty analysis allows, in addition to expressing the results in probabilistic terms, the sensitivity analysis of the model's response to the input data, allowing inferences about the importance of a given parameter or variable.

In the present work, the Monte Carlo simulation technique was used (Beck 1987), which has been widely applied in the uncertainty analysis of mathematical models in general (Brum et al. 2022). It is based on running a large number of simulations (which can reach thousands). In each simulation of the model, a different set of values of the input data about which there is uncertainty is selected. Each value is randomly generated, according to its distribution and within a pre-specified range. The results obtained are later statistically analyzed, generating results in terms of probability (Kroese et al. 2014).

For each set of random data generated, the model was run and output data were produced, which were stored in new spreadsheets. A thousand simulations were carried out, with the help of an extension, implemented at QUAL-UFMG, in (visual basic for applications (VBA)), as proposed by Brum et al. (2022), generating results that can be analyzed probabilistically.

Table 5 presents the input data that suffered variation, with the respective percentages considered and justifications/references.

Along with uncertainty analysis, sensitivity analysis is of crucial importance in the use of water quality models, especially when applied to water resource management. This consists of evaluating the influence of each input variable on the model's output data (Soares & Calijuri 2021). In this way, it is possible to verify which variables are more important and deserve greater investment in obtaining more accurate data (Saltelli et al. 2021; Khorashadi Zadeh et al. 2022).

The results generated in the 1,000 Monte Carlo simulations were used, applying the regionalized sensitivity analysis, as initially proposed by Spear (1980). Initially, the raw input data were separated into two samples. The first aggregated the values of the coefficients (and other input data) whose rounds generated the 50% highest values of maximum thermotolerant coliforms and the other gathered the other simulations. Then, the samples were compared to verify if they would be significantly different, using the ‘t’ test for dependent samples, with a significance level of α = 0.05. The test was performed for all model input data that showed variation. Those data that obtained a value of p ≤ 0.05 (hypothesis test) were considered important for the process, that is, the model is sensitive to them.

The calculation process was also automated with the help of the extension in VBA proposed by Brum et al. (2022).

After its calibration, the model was used to make deterministic and probabilistic simulations for the four scenarios proposed below, always considering the Q_7,10 of the receiving body. In the first scenario (C-01), the current populations, current flows and levels of sewage collection and treatment were considered, as shown in Table 1 (SNIS 2021). In the future scenario (C-02), the urban populations of the municipalities were projected for 2033 (Table 6) and the current rates of sewage collection and treatment were maintained (Table 1). In the population projection calculations, data from the last two Brazilian demographic censuses, 2000 and 2020, were considered (IBGE 2022) and a geometric progression was applied to obtain the population in 2033.

The third scenario (C-03) consisted of considering the urban populations projected for the year 2033 and a value of 90% for sewage collection and treatment, as determined by the Sanitation Legal Framework (Brasil 2020).

The standard for effluent discharge in Brazil does not include organisms that indicate fecal contamination (Brasil 2005; Minas Gerais 2008), only establishing that the discharge of effluents cannot give the receiving body adverse characteristics according to its framework class. As there is no disinfection in the treatment systems already implemented and the environmental legislation does not explicitly require its implementation, it was decided to consider, in the first three scenarios, the removal of only 90%, compatible with natural sewage treatment systems (Von Sperling 2014a; Mailler et al. 2021). Finally, the fourth scenario (C-04) refers to the deterministic model to find the necessary efficiencies for removing thermotolerant coliforms, so that the proposed framework for the basin (CBH-GD2 2018), illustrated in Figure 1, would be reached, considering the population of 2033. The limit values for the watercourse to be in each class are presented in Table 7.

In Brazil, especially in the state of Minas Gerais, licensing environmental bodies have as a rule not to allow new discharges of effluents (even treated ones) in watercourses where the current condition is out of the framework class. Thus, the present work aims to identify the stretches of rivers where this occurs for the four scenarios studied, where the implementation of enterprises generating liquid effluents, which should be discharged there, would not be authorized.

In this approach, the input data of the model were fixed, being the average values for the characteristics of sanitary sewages and rivers and the values found in the calibration process for the model coefficients, as described in items 2.1–2.8.

For Scenarios 1 and 2, we can verify a predominance of the condition of the watercourses in Classes 3 and 4, with Class 4 being observed almost exclusively downstream of the sanitary sewage discharge points. This fact brings a special health concern since it makes the use of water for nobler uses, such as primary contact recreation and irrigation of vegetables that are consumed raw, not recommended (Brasil 2005; Minas Gerais 2008). This is a direct consequence of the release of municipal sanitary discharges without treatment or with insufficient treatment, making the water unfit for most uses, from a sanitary point of view, for long stretches.

A different situation only occurs in the lower reaches of some watercourses, such as the Elvas, Pirapetinga, Peixe and Cervo Rivers, where the large drainage area and the distance from the discharge point favor improvement of water quality. This fact shows the importance of the dilution ratio, which comes with the increase in the flow of the watercourse and the phenomenon of bacterial decay in water quality, which should always be important factors to be considered in the management of water resources.

A significant improvement can be observed in Scenario 3, with compliance with the sanitation legal framework (Brasil 2020), especially for the main bed of the Mortes River. However, this improvement is not enough to meet the proposed framework for the basin and consequently, the current and future uses of water (CBH-GD2 2018). This fact proves that even the universalization of sewage collection and treatment, if tertiary treatment with disinfection is not foreseen, is not enough, in the vast majority of cases, to safeguard the quality of the receiving bodies, from the point of view of indicator organisms of fecal contamination. Secondary treatment may only be sufficient in very specific situations, where we have a low discharge flow and a high receiving body flow, providing high dilution ratios. This can be proven, for example, by the launch of the urban headquarters of Ibituruna in the main bed of the Mortes River.

The main bed of the Mortes River is framed (CBH-GD2 2018) between km 51 (confluence with the Caieiros stream) and km 181 (confluence with the Peixe River), in Class 3, with the remainder being Class 2.

For Scenario 3, which aims to meet the Sanitation Legal Framework (Brasil 2020), we can see significant improvement for the main bed of the Mortes River. Non-compliance with the classification only occurs only in two short stretches: just downstream of Bandeirinha Stream (with discharges from Antônio Carlos – km 17) and then downstream of Água Limpa Stream (with discharges from São João Del-Rei – km 123). Unfortunately, this situation is not repeated in the other watercourses modeled in the basin, where the actions proposed in Scenario 3 will not be sufficient to meet the framework.

Table 9 presents the sections with a situation that does not comply with the proposed framework, according to the studied scenarios.

We can see that, in Amparo River, compliance with the framework (Class 2) will only occur in Scenario 4. In Scenarios 1 and 2, almost the entire length of the watercourse would be in Class 4. In Scenario 3, we would have Class 4 until close to km 10 and Class 3 up to its mouth on JAR.

As stated above, the efficiency of 90% in the removal of thermotolerant coliforms is not enough to comply with the framework in the vast majority of municipalities. According to the results of the deterministic modeling, for C-04 (compliance with compliance), the efficiencies shown in Table 10 would be necessary.

Only the municipalities of Barroso, Conceição da Barra de Minas, Ibituruna, Ijaci, Ribeirão Vermelho, Santana do Jacaré and Tiradentes would not require disinfection to comply with the framework and the efficiency of the removal of thermotolerant coliforms, of the order of 90%, in a secondary treatment would be enough. Some municipalities, such as Ibertioga and Ressaquinha need efficiencies of around 99.99%. The urban center of Lavras, on the other hand, would need an efficiency of 99.999% to meet the Class 2 framework of the Vermelho Stream. This is mainly due to the placement of sewage discharges within the basin, in regions with a greater or lesser drainage area and, consequently, a greater or lesser dilution ratio.

The framework is made according to the predominant uses found or foreseen in the basin, reflecting the necessary and, mainly, desired water quality. However, it must be done judiciously, taking into account technical and mainly economic criteria related to the treatment of sanitary sewage, under penalty of imposing heavy burdens on municipalities (mainly smaller ones) and sanitation companies. An example is the small municipality of Ressaquinha, which would need 99.99% efficiency in coliform removal, in addition to universal sewage collection and treatment, to comply with the framework. It should be evaluated on a case-by-case basis, bearing in mind the real and necessary uses of each stretch, as well as the existing releases of evictions and the economic capacities of each municipality. If necessary, the uses can be reviewed or even intermediate goals created. This is of fundamental importance in Brazil and other developing countries, where investments in sanitation are very limited.

In this item, the results of the probabilistic modeling will be presented. Longitudinal profiles were generated for all simulations performed, creating uncertainty band graphs (Brum et al. 2022). Due to Excel's technical limitations, for these graphs, the profiles were presented only for the first 250 simulations (out of 1000 Monte Carlo simulations).

The Brazilian environmental legislation (Brasil 2005; Minas Gerais 2008), in general, does not deal with the discharge standard and compliance with the framework in a probabilistic way. Analyses are usually made in absolute terms, such as attendance and non-attendance. Also, no bibliographical references were found to guide acceptable values for the probability of meeting the framework. In this way, it is up to the water resources manager, based on their reality and the risks they are willing to assume, to define the probability of meeting the framework in which they will work. Here is an interesting knowledge gap to be filled in future studies.

In the specific case of the thermotolerant coliforms variable, the legislation (Minas Gerais 2008) determines that the class limit should not be exceeded by at least 80% of at least six annual samples, with a bimonthly frequency. Thus, in the absence of other references, it can be assumed that the probability of meeting the framework, for this parameter in question, will be accepted if it is at least 80%.

We can verify that up to km 17, where it receives Bandeirinha Stream with discharges from Antônio Carlos, the probability (P) of Class 2 compliance is 100%, for the four studied scenarios. After this first significant discharge of sanitary sewage, meeting the framework is compromised, for the first three scenarios, but mainly for the first two, where the probability of meeting the framework, right after the confluence, is nil. For the third scenario, this probability is around 50%.

The probability of complying with the framework varies downstream, as phenomena related to bacterial decay occur and new discharges of sanitary sewage happen. A little downstream, just after the confluence with Caieiros Stream, bringing sewage from Barbacena, the compliance probabilities are approximately 60% for C-01 and C-02 and 100% for C-03.

In general, the probabilities of compliance with the framework for Scenarios 1 and 2 are practically nil, with some short stretches where there is some probability of compliance.

Regarding C-03, we can verify that it will not be enough for the entire length of the Mortes River. Right at the confluence with the Bandeirinha Stream (km 17), the probability of compliance is only 32.8%. At km 64, where it receives sewage from Barroso, the probability of compliance is only 44.5%. At the confluence with the Água Limpa Stream, carrying sewage from São João Del-Rei, the probability of compliance is only 6.7%. Thus, treatment efficiencies for the Municipalities of Antônio Carlos, Barroso and São João Del-Rei should be greater. Right after the confluence with the Peixe River, the probability is 76.1%.

In Scenario 4, we have a 100% probability of compliance for almost the entire length of Rio Mortes. The only exception is just downstream from the urban center of Barroso, where the probability of compliance is 57.1%, but which returns to 100% just 9 km downstream.

For the Amparo River, which receives the discharges from the urban center of Santo Antônio do Amparo, the probability of meeting the framework (Class 2) is nil, in the first two scenarios, in all its extension. In Scenario 3, there is some probability of compliance starting at km 15.6, reaching 50% at km 28.9 and a maximum of 70.2% at its mouth, at km 32.

As for Scenario 4, we have 89.1% just downstream of the discharge point, reaching 100% 2.65 km downstream.

Table 11 presents a summary of the results of the probabilistic model, for all modeled watercourses, according to each scenario.

We can see that the probability of complying with the framework, for C-01 and C-02, except for the Barba- de-Lobo River, is zero for all watercourses, which shows the poor quality of the water from a sanitary point of view. As phenomena related to bacterial decay occur, the situation improves downstream. Pinhão, Quilombo, Elvas, Sujo, Pirapetinga and Algodão watercourses have a 50% probability of compliance a few kilometers downstream. Only Quilombo, Elvas and Sujo watercourses, due to their large extensions, manage to reach a 100% probability of compliance. Barba-de-Lobo River has excellent sanitary quality, even receiving discharges from Conceição da Barra de Minas, which treats only 71.8% of its sewage, without disinfection, due to its great dilution capacity, which does not occur in the rest of the watercourses. The Q_7,10 at the point of discharge is 2,360 L s⁻¹, for a current sewage discharge of 1.8 L s⁻¹, which gives a dilution ratio of 1,311 times. In Scenario 2, we can see a worsening of the situation, with the increase in distances needed for the purification of watercourses to occur.

In Scenario 3, where 90% of the sewage is collected and treated, however, without disinfection, we see that right downstream from the discharge points, the situation of non-compliance with the framework does not change, with zero probability of compliance equal to the first two scenarios. The improvement in water quality can only be seen downstream, reducing the necessary distance to 50 and 100% probability of complying with the framework. However, even in this scenario, some watercourses do not have sufficient distance.

In Scenario 4, where disinfection was considered, with the efficiencies shown in Table 10, we have a 100% probability of meeting the framework, just downstream of the discharge point, in the vast majority of modeled watercourses. Some watercourses have a probability of compliance greater than 80%, which reaches 100% a little downstream, as is the case of Caieiros, Ressaquinha, Amparo, Tanque Grande and Pirapetinga Streams. Even in this scenario, some watercourses showed a low probability of compliance, right downstream of the joining point of Loures (sewage from Alfredo Vasconcelos), Maracanã (sewage from Oliveira) and Algodão (sewage from São Bento do Abate), indicating the need to increase the efficiencies proposed in Table 10 or the implementation of treated sewage outfalls, aiming to carry out the discharge further downstream, where the dilution ratio would be higher.

An interesting situation occurs with the Quilombo Stream, which receives discharges from the urban center of Resende Costa. In Scenario 3, the results for water quality are worse than in Scenarios 1 and 2. We see that, with 90% of sewage collection and treatment, the necessary distance for a 50% probability of meeting the framework is 14.8 km, against 7.7 and 9.2 km in the first two scenarios. This is due to the fact that, in the first two scenarios, there is only 39.4% of sewage collection and the rest is disposed of in sump/sinkhole systems, not being discharged directly into the stream. In Scenario 3, 90% of the collection is considered.

In general, the probabilistic results came to confirm the deterministic model, also showing the precarious current quality of the water, in terms of thermotolerant coliforms, as well as the ineffectiveness of the New Legal Sanitation Framework (Brasil 2020) in meeting the framework of the courses of water in the basin, which will only be achieved with tertiary treatment, with disinfection, in the vast majority of municipalities.

It was found that the current condition of the basin (C-01), as well as Scenarios 2 and 3, present water quality, in terms of thermotolerant coliforms, in disagreement with the proposed framework (CBH-GD2 2018), in the vast majority of modeled stretches. These stretches are problematic for the environmental licensing of enterprises that generate liquid effluents and the State Environmental Agency does not authorize new discharges until compliance with the framework is established.

This is a very delicate issue, as the legislation (Brasil 2005; Minas Gerais 2008) establishes that effluent releases cannot give the receiving body characteristics that are adverse to the class in which it was classified. However, there is no indication of how to proceed when the class is already violated upstream of the release point. The understanding, for the vast majority of analysts of the Environmental Agency, is that under these conditions, new releases of effluents cannot be authorized. This fact brings several inconveniences to undertakings that generate liquid effluents located close to watercourses in this situation.

Sensitivity analysis was performed for all model simulations, showing slightly different results for each watercourse studied. As an example, Table 12 presents the results for River Elvas.

Sensitivity analysis was performed for all watercourses studied and in general, the model was sensitive to the variables.

• - Flow and thermotolerant coliforms in raw sewage;

• - River velocity and temperature;

• - Bacterial decay rate – K_b.

Due to the high concentration of thermotolerant coliforms in raw sewage, it was already expected that the model would be very sensitive to this variable, as well as to the discharge flow (De Paoli & Von Sperling 2013; Von Sperling 2014b). The flow velocity of the receiving body also plays an important role, as it is conditioned by the distance at which the necessary decomposition of microorganisms will occur. The temperature, together with the bacterial decay rate – K_b, conditions the velocity with which the decay kinetics will occur (Batista & Harari 2016; Silva et al. 2021).

Based on the sensitivity analysis, it is concluded that systematic monitoring of the basin is extremely important, mainly for the quantitative and qualitative characterization of raw sewage. Research is also important in order to reduce the uncertainty in the definition of the coefficient K_b.

Based on the results obtained, it can be concluded that:

• The joint analysis of the deterministic and probabilistic model adopted proved to be an applicable tool for the management of watersheds, especially in Brazil and other developing countries, where the universalization of the collection and treatment of sanitary sewage is not yet a reality, due to its relative simplicity of application.

• The current situation of the basin, in terms of thermotolerant coliforms, is worrying, since almost all of the modeled watercourses, in most of their extensions, are out of their framework class. This has very serious health implications since the current and future uses of the basin's waters are compromised, due to the risk of waterborne diseases. The projection for the year 2033, without forecasting interventions in the basin, further aggravates the scenario, due to population growth.

• The actions proposed in the sanitation legal framework (Brasil 2020), which, despite providing for the obligation of 90% of sewage collection and treatment for all municipalities, are not sufficient to ensure compliance with the framework. This study showed that compliance with the framework, for the variable thermotolerant coliforms, will only be reached with the implementation of tertiary treatment, with disinfection, in the overwhelming majority of municipalities, which is not explicitly provided for in the legislation.

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

The authors declare there is no conflict.