Exploring the Potential of Mathematical Self-Purification Models Used for Evaluating Water Quality in Rivers

Abstract

The quality of water in rivers and their self-purification capacity are critical for maintaining healthy aquatic ecosystems. This study aims to analyze and compare various mathematical models of self-purification, assessing their applicability in restoring water quality and proposing recommendations for their improved use. A systematic review of the scientific literature was conducted following PRISMA 2020 guidelines to ensure a rigorous approach. Research questions were framed using the PICO model, which includes Population, Intervention, Comparison, and Outcomes. Relevant studies published between 2015 and 2024 regarding mathematical models of river self-purification were selected. Inclusion and exclusion criteria were applied, and a critical analysis of findings was performed, highlighting methodologies and results. The results indicate that the effectiveness of self-purification models varies significantly depending on environmental and geographic characteristics. A need for more specific models and the integration of local variables was identified as a research gap that requires attention in future studies. Furthermore, recommendations were made to enhance model calibration and validation, as well as to incorporate innovative approaches for optimizing water quality management in rivers. These mathematical models are essential tools for managing river water quality, promoting public health, and contributing to the achievement of Sustainable Development Goal 6 (SDG 6).

1. Introduction

Water quality in rivers is fundamental to preserving biodiversity and the health of ecosystems [1,2]. Rivers act as links between terrestrial and marine ecosystems, actively participating in the global biogeochemical cycle [3]. Industrial development and urbanization have increased discharges of pollutants such as heavy metals, pesticides, pathogens, and nutrients, resulting in a negative alteration of water quality [4,5,6]. The quality of water, in turn, determines its capacity to dissolve and transport pollutants [7,8]. This phenomenon is particularly critical in countries experiencing rapid economic and urban growth, where runoff from agricultural, industrial, and residential areas has intensified non-point source pollution in rivers and connected bodies of water [9,10]. This affects not only aquatic ecosystems but also groundwater sources [11], posing significant risks to human and environmental health [12].

Direct discharge of wastewater, the expansion of activities such as agriculture and livestock farming, and other anthropogenic activities have increased pollutant levels in rivers, limiting the use of these resources and causing eutrophication and the loss of ecosystem services [13,14]. Additionally, extreme climate events exacerbated by human intervention threaten the availability of clean water [15,16], prompting many countries to implement fluvial restoration measures that enhance self-purification processes of these bodies of water [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17].

Increased pollution of water bodies worldwide threatens water quality and aquatic biodiversity [18]. According to the World Health Organization (WHO), more than 80% of urban effluents are discharged into rivers without proper treatment, which significantly increases pollutant loads and affects life in these ecosystems [19,20].

Sustainable Development Goal 6 (SDG 6) which aims to ensure access to clean water and sanitation, has become a central focus of international efforts [21]. The 2023 United Nations (UN) Water Conference emphasized the importance of global cooperation to improve water quality, promoting the SDG 6 Global Acceleration Framework, which focuses on partnerships for efficient water management [22]. Organisms such as the European Environment Agency (EEA) and the United Nations Environment Programme (UNEP) promote the use of self-purification models as essential tools to monitor water quality and anticipate changes in vulnerable fluvial systems [23]. As the United Nations (UN) and the WHO’s Global Antimicrobial Resistance and Use Surveillance System (GLASS) Report 2022 state, these models enable the evaluation of the recovery capacity of rivers under high pollution loads [24].

The self-purification of rivers is an essential natural process to preserve water quality through organic matter degradation and nutrient recycling [25]. This process, involving physical, chemical, and biological transformations [26], partially mitigates the impact of pollutants [14]. Self-purification is an essential ecosystem service that helps maintain water quality in rivers and other water bodies [27]. Nonetheless, factors such as increasing water demand and inadequate waste management [20], including the lack of sanitation and direct discharge of wastewater into water bodies, affect the self-purification capacity of rivers [28]. This self-purification capacity is becoming ever more limited, especially in small and medium-sized rivers in rural areas, where sanitation infrastructure is usually deficient [29].

Rivers assimilate pollutants thanks to their self-purification capacity, which is closely related to the availability of dissolved oxygen (DO) [30]. However, this process also depends on several other variables, such as temperature, flow rate, organic load and biochemical oxygen demand (BOD) [31]. All of these variables significantly influence the effectiveness of self-purification [32,33]. Temperature, as an environmental factor, especially affects the self-purification of rivers with lower flow rates as they are more vulnerable to pollution [29]. On the other hand, BOD is considered a key measure for evaluating self-purification capacity [34].

River systems have a self-purification capacity that can be surpassed when pollution levels exceed their resilience threshold [35]. This impacts communities and aquatic fauna that depend on these resources and compromises the ecosystem services that rivers provide [17,36]. In this context, mathematical modeling of the self-purification processes emerges as a key tool to understand and predict a river’s capacity to deal with pollutant loads [37,38], highlighting the need to develop mathematical models that anticipate and mitigate the deterioration of water quality [23]. These models enable the evaluation, identification, prediction, and quantification of the impact of factors that contribute to pollution [39], which allows for the design of control scenarios, minimization, and cost analysis [28,40]. In addition, they provide a fundamental scientific basis for quality control and decision-making in environmental management. Models like MIKE, CE-QUAL-W2, EFDC, and WASP, which are based on physicochemical principles, simulate the evolution of water quality, although they require precise parameter adjustments, and they may have limitations in high complexity environments. Other models, like those based on neural networks, take advantage of the great volume of historical data to learn complex relationships providing useful predictions, although sensible to the quality and consistency of input data [41].

The goal of this study is to carry out a systematic literature review of the main mathematical models for self-purification that are used for the evaluation of water quality in rivers. This review focuses on an analysis of the application of different models of river self-purification in response to the presence of pollutants that negatively affect water quality. Through this analysis, the aim is to provide a theoretical basis for the different models for such implementation.

Incorporating the PICO and PRISMA (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) methodologies in the systematic review not only improves the review’s effectiveness by providing clarity for research questions, standardization in the processes, rigor in the selection of relevant studies, validity and reproducibility of results, but also provides a solid basis for future research [42,43]. The PRISMA 2020 methodology and the PICO model were adopted to define the systematic review search, guaranteeing transparency and precision in the selection of scientific information. PRISMA 2020 ensures a rigorous process in the identification, selection, and evaluation of relevant studies, providing a structured framework that increases the reproducibility and validity of results [44]. Additionally, the PICO model serves as an effective strategy to formulate clear and specific research questions focusing on Population, Intervention, Comparison, and Outcomes. This facilitates the comparison between different mathematical models for self-purification and their effectiveness for the evaluation of water quality in rivers. The combination of the proposed methodologies ensures an exhaustive analysis focused on the objectives and provides a solid basis for future research and practical applications in water management [45].

2. Materials and Methods

2.1. Research Process

For this study, the methodology of systematic review of scientific literature was used, which involves a critical summary of available data concerning a specific topic [46]. For this work, the PRISMA 2020 research framework (Preferred Reporting Items for Systematic Reviews and Meta-Analyses), published in 2021, was followed. PRISMA 2020, an updated version of PRISMA 2009, aims to support authors in optimizing the submission of systematic reviews. It offers essential data to interpret and correctly apply the findings from this type of analyses, and it establishes a series of recommendations for structuring the manuscripts of these reviews [43,47]

Focusing on PRISMA methodology, the following stages were considered: topic delineation, formulation of focus questions, formulation of keywords, database selection for the application of keywords, information filters to ensure relevance (inclusion-exclusion criteria), analysis of results, and report writing [48,49].

2.2. Protocol and Focus Questions

The PICO model was the protocol followed to help structure the research questions. This model evaluates the study Population, Intervention, Comparison, and Outcomes. It is a robust model that initially emerged with the aim of developing structured questions containing the necessary elements to further build the most efficient bibliographic search strategy possible [46,50].

In this context, the population/problem refers to the question: “What is the problem to be addressed?”, thought of as a dependent variable. The intervention relates to the question: “What action or change would affect the population/problem?”, which represents the independent variable. The comparison implies the analysis of “What is the alternative to the intervention?” or “Does there exist a different intervention?”, which is seen as a “control group”. Finally, the outcome focuses on identifying “What are the relevant results?” [44]. The implementation of the PICO strategy was carried out, and its results are shown in

Table 1. PICO methodology in the systematic review of river self-purification models.

Population/Problem	Pollution of rivers that affect water quality
Intervention	Application of models to analyze the self-purification of rivers
Comparison	Analysis of different river self-purification models
Outcomes	Determination of the most effective self-purification models, and their applicability in the restoration of water quality in rivers

. This table details each component of the research question and its connection to the present study.

Based on the PICO strategy, the following questions were formulated:

Q1. What are the most used mathematical models to evaluate river self-purification, and what are their theoretical foundations?

Q2. What are the most relevant variables used in self-purification models, and how do they influence the prediction of water quality?

Q3. What are the strengths and weaknesses of the self-purification models in polluted rivers?

2.3. Systematic Search According to PRISMA

For the initial selection of scientific articles in the Scopus database, the period between 2015 and 2024 was defined, and keywords such as “Self-purification”, “River”, and “Model” were used, resulting in a total of 480 articles. To further filter the search, the keyword “water quality” was applied in addition to the previous keywords, obtaining 193 articles. For an even stricter selection, the publication year was limited, the type of document was restricted to articles, and English was chosen as the search language. The search string used was TITLE-ABS-KEY (“Self-purification” AND “River” AND “Model” AND “water quality”) AND PUBYEAR > 2014 AND PUBYEAR < 2025 AND (LIMIT-TO (LANGUAGE, “English”)). In total, 67 documents were identified as relevant for this study.

Additional filters were then applied to discard articles that were not related to the research questions, resulting in a final total of 30 articles. These selected articles allowed the analyzation of the most used mathematical models to evaluate river self-purification, highlighting their theoretical foundations and methodologies. The two most significant variables that influence the prediction of water quality were also identified: BOD and nutrients. In addition, the advantages and limitations of the models applied to polluted rivers, considering their predictive capacity and their application in different contexts, were evaluated. These topics contributed to a better understanding of the self-purification process and its management in aquatic ecosystems. Figure 1 shows the search guidelines based on the PRISMA methodology in a detailed manner, as well as the exclusion criteria applied to certain documents.

The following inclusion criteria were considered: keywords have to be found in the title and/or abstract of each article; studies have to focus specifically on rivers; studies have to focus on the mathematical modeling of self-purification for the evaluation of water quality; and studies have to have been published between 2015 and 2024.

Studies related to other water bodies, groundwater, lacustrine and marine environments were excluded. Studies related to the purification of wastewater, repeated studies between databases, and studies that do not contribute relevant information to answer the research questions were also excluded.

The inclusion criteria focused on the presence of the keywords “self-purification,” “river,” “model,” and “water quality” in the title or abstract, which allowed for thematic filtering of the results. Of the 480 articles initially obtained, 193 were considered relevant, and after applying additional filters, the number was reduced to 67. The exclusion criteria ruled out studies focused on other bodies of water (lakes, reservoirs, marine ecosystems, or groundwater), research without mathematical modeling, superficial analyses (such as quality indices without modeling), as well as duplicate publications or publications without relevant contributions. This delimitation responds to the hydrodynamic differences between ecosystems: in lotic systems, self-purification models depend on advection-dispersion processes, flow dynamics, and atmospheric reaeration, while in lentic systems, thermal stratification and residence times intervene, and in marine environments, salinity gradients. Following the PRISMA guidelines, which guarantee transparency and standardization in systematic reviews, 30 articles were selected. This sample constitutes the most relevant and consistent evidence for building a solid theoretical basis for modeling river self-purification.

3. Results and Discussion

Before addressing each of the research questions, it is crucial to present a synthesis of the key findings obtained from the selected articles. The analyzed articles provide detailed data about the use of several mathematical models for self-purification, their application in different software environments, and the parameters needed to evaluate water quality in river systems. Also, these studies highlight the environmental variables and the characteristics of rivers that influence the behavior and efficiency of such models. In this way, it is possible to identify patterns, limitations, and specific applications of the models, providing a solid basis to answer the research questions regarding their frequent use, influence factors, and strengths and weaknesses in the modeling of self-purification of polluted rivers.

3.1. Research Question 1: What Are the Most Used Mathematical Models to Evaluate River Self-Purification, and What Are Their Theoretical Foundations?

3.1.1. Temporal Trends in the Publication of Studies on Self-Purification Models Used to Assess Water Quality in Rivers

Figure 2 illustrates a notable evolution in the number of documents published on river self-purification models from 2015 to 2024. The graph reveals a considerable increase in research starting in 2020, with significant growth in the publication of studies and fluctuating behavior in previous years. In the earlier years (2015–2017), the number of published studies was quite low, with sporadic peaks that did not exceed six annual publications in Scopus. From 2018 onwards, a level of increase is observed, but it is in 2020 when the number begins to increase significantly, reaching a maximum of 14 publications in 2021 and 2022. This growth trend continues until 2024, although a slight stabilization occurs towards the end of the period. During the initial phase (2015–2017), the paucity of publications could reflect limited interest or a lack of focus on mathematical models and simulations related to river self-purification. During this period, studies likely focused more on general environmental characterization and the use of water quality indices to assess river health, rather than on specific modeling of the self-purification process. Although the number of publications began to grow in 2018, the increase was moderate (2018–2019), suggesting that the field of study is beginning to gain traction in the scientific community, although it still lacks widespread interest. This phenomenon could be related to a growing concern about water pollution and the need to investigate natural river purification processes, driven by greater awareness of environmental issues. Starting in 2020, there was a significant increase in the number of publications, peaking between 2021 and 2022. This change can be attributed to several factors, such as increased investment in environmental research due to the intensification of problems related to water pollution worldwide, as well as a growing concern about water quality and the health of aquatic ecosystems, driven by industrialization, population growth, and urbanization. In recent years, there has been a growing interest in sustainable and natural solutions, such as self-purification models, which describe how water bodies, such as rivers, can spontaneously cleanse themselves through biological, chemical, and physical processes. Furthermore, the COVID-19 pandemic may have had an indirect influence, as during that time there was a renewed focus on environmental issues due to the visible effects of reduced human activity on ecosystems. In recent years (2023–2024), the number of publications has remained relatively high, although it has not increased as sharply as in previous years. This trend could reflect a certain level of maturity in research on self-purification models, with a more consolidated and specialized approach to studies. The scientific community appears to have reached a threshold of knowledge and is now focusing more on optimizing existing models or their practical application in real-life situations.

3.1.2. Geographic Distribution of Studies on River Self-Purification Models

Figure 3 presents the most prominent countries in terms of publications. Countries with the highest number of studies in this field published in Scopus are shown in larger font; countries with the lowest number of studies are represented in smaller font. Among the most prominent countries are China, the United States, Brazil, India, Russia, and Indonesia, while countries such as Bosnia and Herzegovina, Kazakhstan, and Ecuador have a smaller representation. Countries with the greatest concerns about water pollution, such as China, India, and the United States, tend to lead in the number of published research studies. This is due to the urgent need for sustainable solutions for the natural purification of their rivers, which are affected by industrial, urban, and agricultural growth. China clearly stands out as the country with the highest number of publications in this field. This may be due to several factors. China has experienced massive economic growth in recent decades, which has increased water pollution in rivers and other bodies of water. Furthermore, the Chinese government has invested significantly in environmental research, focusing on ecological restoration and improving water quality through natural processes such as self-purification [51]. The United States is also among the countries with the largest number of publications. This country has a strong tradition of scientific research and a growing focus on environmental sustainability. Attention to water quality in rivers and lakes has been driven by federal and state policies requiring the treatment and monitoring of polluted water bodies. India and Brazil are also among the most prominent countries, reflecting their growing interest in water quality and pollution challenges. In India, rapid urban and industrial growth has exacerbated water quality problems, especially in rivers like the Ganges, which is the subject of numerous self-purification studies. Brazil, with its vast river network and the importance of the Amazon, has prioritized research into sustainable methods for purifying its rivers [52].

The spatial representation of the reviewed literature was generated using a visualization technique based on georeferenced nodes (or geospatial word clouds). In Figure 3, countries with the largest volume of research in water quality modeling (e.g., China, the United States, and India) are shown in larger font sizes, while countries with fewer studies are represented in smaller font sizes. This size coding provides an immediate visualization of the relative frequency of publications by country and facilitates the identification of the main research centers in the field.

These results highlight a major limitation in the global applicability of river self-cleaning models, as their use is predominantly concentrated in basins in countries such as China, Brazil, India, and the United States. This concentration not only reflects inequalities in funding and scientific infrastructure but also introduces considerable methodological bias. Complex models, such as QUAL2K and WASP, are primarily designed and calibrated to operate in contexts with high-resolution data and extensive monitoring capabilities, limiting their effectiveness in regions with scarce or low-quality information [53,54], thereby restricting their practical applicability in underrepresented hydroclimatic systems characterized by data scarcity (a predominant reality in many regions of Africa, the Middle East, or certain parts of Eastern Europe and Latin America) [55,56]. The need to develop and validate models in these contexts is urgent. Recent studies in underrepresented regions have demonstrated the feasibility of simple, tailored models. For example, the successful application of QUAL2K in the Ndarugu River, Kenya (Africa) [57] and in basins in the Middle East (Iran and Iraq) [58,59], as well as the use of the WASP model in the Ravi River, Pakistan [60], are examples of crucial efforts to validate the robustness of these tools under local data constraints. Therefore, the need to urgently develop and test these models in underrepresented hydroclimatic contexts, such as intermittent, seasonal rivers or arid and semi-arid regions, will be emphasized in order to overcome the systemic risk of applying parameters calibrated for one system to a radically different one without due local recalibration, which would severely compromise global water management [61].

3.1.3. Analysis of Keyword Trends in Research on River Self-Purification Models

Figure 4 shows a co-occurrence analysis of keywords extracted from studies on self-purification models in rivers, revealing three main conceptual clusters. The most prominent cluster focuses on “water quality,” accompanied by terms such as “rivers,” “river listening,” “purification,” and “quality control,” reflecting that much of the research is geared toward evaluating the impact of self-purification on pollution and the restoration of river ecosystems. A second cluster, focused on environmental management and monitoring, includes keywords such as “wastewater,” “water management,” and “environmental monitoring,” indicating an interest in the applied use of models in decision-making and public policy. The third cluster highlights chemical and geographic variables, with terms such as “nitrogen,” “ammonia,” “self-purification,” and “China,” suggesting both the relevance of specific compounds in modeling processes and the high concentration of studies in that country. The interrelationship between these terms demonstrates that self-purification models are approached from a multidisciplinary perspective, combining physical-chemical, ecological, and territorial aspects, which is essential for their effective application in the comprehensive management of water quality in rivers.

3.1.4. Most Commonly Used Mathematical Models for Evaluating Self-Purification in Rivers: Frequencies, Characteristics and Limitations

Figure 5 shows the most used models in river self-purification studies according to the systematic review. Of the 30 studies reviewed, 5 models stand out as the most widely used to analyze river self-purification. QUAL2K and QUAL-UFMG are the most frequent models, each model appearing in 6 different articles. The model classically denominated as Streeter-Phelps is in third place and was used in 3 different studies. Finally, WASP and MIKE models were employed in 2 studies each.

Table 2 highlights these five models describing their theoretical foundations, applications, and limitations so that there is a particular approach to improve understanding of the models. The QUAL-UFMG model, derived from QUAL2E, is noteworthy for its application in tropical zones and its accessibility through Microsoft Excel. However, its effectiveness is compromised in non-tropical climates, thus limiting its applications. In addition, the calibration of parameters such as BOD₅ has shown deficiencies, suggesting that the results may be unreliable if the real conditions of the effluents are not known [62,63]. This highlights the need for precise and contextualized data for effective modeling, a common aspect in many models.

The QUAL2K model represents a significant evolution by including additional variables such as algae growth and denitrification. Although the simulation of water quality is improved, challenges are faced regarding the collection of detailed data, which can limit its precision [18]. This model requires a lot of information and therefore can be less accessible for practical applications, especially in contexts where data is scarce.

The Streeter-Phelps model is a classic approach that allows the identification of natural recovery zones and critical pollution points. Nevertheless, it depends on homogeneous and stationary conditions which may result in an underestimation of the concentration of dissolved oxygen in more complex scenarios, such as those with diffuse sources of pollution [64,65]. This limitation underscores the need for models that are able to adapt to seasonal variations and the hydrodynamic complexity of rivers.

The WASP model, developed by USEPA, allows for simulating the interaction of multiple parameters and their impact on water quality. Although it is versatile and useful for the analysis of polluting impacts, the WASP model has limitations regarding the representation of extreme events, which can be critical in environmental management contexts where climate changes are forecasted [66]. The dependency on precise data and the simplification of assumptions also affects the reliability of its predictions [67].

Meanwhile the MIKE model is distinguished for its capacity to simulate fluxes in complex river systems and for evaluating the impact of discharges on water quality. Nonetheless, its precision highly depends on the quality and quantity of input data, and its unidimensional nature may not adequately capture the tridimensional complexity of certain river systems [68,69]. This suggests that even though MIKE is an advanced model, its application may be limited in scenarios where spatial variability is significant [70].

The mathematical models for the evaluation of river self-purification are valuable tools for water quality management, but they have to be carefully selected and applied, taking into consideration their theoretical foundations and limitations. The availability and precision of data, as well as the adaptability to different environmental contexts, are critical factors that determine the effectiveness of these models in practice. The continual evolution and refinement of these models will be essential to address emerging challenges in water resources management.

The SWAT (Soil and Water Assessment Tool) model is widely used in the scientific literature due to its ability to simulate hydrological, sedimentary and water quality processes at the basin scale [60]. However, its distributed approach and its design oriented to assess long-term impacts of land use and agricultural practices on the production and transport of water, sediment and non-point source (NPS) pollutants make it more suitable for basin-scale studies than for detailed simulations of water quality in river reaches [11]. Therefore, its exclusion from direct performance comparison with models such as QUAL2K or WASP, which are specifically designed to simulate in situ self-purification processes in river reaches, is justified [53].

Table 2. Mathematical models used for the evaluation of river self-purification: theoretical foundations, applications, and limitations.

Authors	Model	N° Articles	Theoretical Foundation	Application	Limitations
[28,62,63,71,72,73]	QUAL-UFMG	6	It is an adaptation of the QUAL2E model from USEPA developed by the Federal University of Minas Gerais (Portuguese: Universidade Federal de Minas Gerais, UFMG) in Brazil.   The calculation is done by processing coupled differential equations in Microsoft Excel. It uses the equations from the QUAL family with the corresponding adaptations.   Main parameters - Biochemical oxygen demand (BOD) - Dissolved oxygen (DO) - Total nitrogen and its fractions - Total phosphorous and its fractions - Thermotolerant coliforms	Given that the model is developed in Brazil, its application mainly focuses on tropical zones (similar to the design conditions) and in studies that require simple tools of easy access based on Excel to evaluate the impact of discharges.	It is limited to zones of tropical conditions. It has less effective precision for other climate zones [71].   According to the work of Da Luz et al. [63], for the DO calibration a good data fit was shown, but this was not the case for BOD5. This adjustment was presented due to the fact that real conditions in which effluents are discharged are unknown.   The discharge of pollutant loads in a water body has to be analyzed with much caution because misinterpreted data and incorrect results compromise predominant applications [63].
[18,65,74,75,76,77]	QUAL2K	6	The QUAL2K model is an evolution of the QUAL2E model, designed to improve water quality simulations through the inclusion of variables such as algae growth, denitrification, and an adjustment in the calculation of dissolved oxygen.   Main parameters - DO - BOD5 - pH - Temperature - Electrical conductivity - Organic nitrogen - Ammoniacal nitrogen - Inorganic nitrogen - Phosphorous - Sedimentable solids - Phytoplankton and Periphyton	The application of this model is used in simulations of dissolved oxygen and biochemical oxygen demand.	If the model uses variables such as nitrogen or phosphorous, it can extend to problems with these pollutants. Also, it presents significant challenges due to the quantity of detailed data that is required, which is why the information is often very limited [18].   Many times the behavior of variables in the section studied may tend to underestimate in this case the concentration of DO [18] The adjustment to the QUAL2K model may be significantly lower [77].
[64,65,78]	Streeter-Phelps	3	The Streeter-Phelps model is the main historical basis model from which many other models for the evaluation of water quality in rivers and other water bodies are derived through the explication of the interaction between BOD and DO. It is specially designed in response to wastewater discharges.   Main parameters - DO - BOD5	It is a model used for water quality simulation in rivers. It is a classic approach that does not include advanced technological tools.   The model allows the identification of natural recovery zones and critical pollution points, considering the organic matter decomposition rate, and the process of reoxygenation.	The authors mention that for this model homogeneous, stationary, and constant flux conditions have to be assumed [64,65,78]   According to the work of Díaz et al. [64], the precision of the model is lowered by the presence of diffuse sources of pollution, and in cases of high hydrodynamic complexity.   According to the work of Pazmiño-Rodríguez et al. [65], neither multiple pollutants nor seasonal variability are considered.
[58,66]	WASP	2	The WASP model is a tool developed in 1980 by USEPA. It is designed to simulate various parameters and their interaction for the evaluation of water quality in such a way that the effect of pollutant loads over water bodies can be predicted. It basically uses a set of differential equations to describe transport, dispersion, and reaction of pollutants.   Main parameters - DO - BOD5 - NH4-N - Organic nitrogen - NO3-N - Organic phosphorous - PO4-P - Phytoplankton - Coliform bacteria - Silica	The model is used to simulate water quality under different pollution scenarios, allowing the analysis of impacts of either anthropogenic or natural pollutant loads.   Speaking of environmental management, the model helps to design optimal strategies for control measures of pollutant sources such as optimal systems and for future decisions based on the information provided on water quality.	According to the work of Żelazny et al. [60], there are limitations for precise representation of extreme events, such as runoff caused by extreme rainfall.   According to the work of Ma et al. [67], there is a precision dependency on input data, and on the simplification of assumptions in the behavior of the interception system.
[69,79]	MIKE	2	The MIKE model was developed in 1972 by the Danish Hydraulic Institute. It is a deterministic model that allows the simulation of fluxes in a non-permanent state within river systems, adapting both to simple and complex configurations.   This model can be used to evaluate the impact of discharges on water quality in rivers, besides working as a hydraulic model for the analysis of floods.   It integrates specialized modules, among which those of rainfall-runoff, advection-dispersion, and hydrodynamics are included.   Main parameters Physicochemical - BOD5 - DO - NO3-N - NH4-N - Heavy metals - Coliform bacteria - Hydrodynamic - Velocity - Flow rate - Water level - Water quality - Sediment transport	It is an advanced model that allows hydrodynamic and water quality simulation in water bodies, evaluating diverse pollutant dispersion scenarios, such as the optimal location of pumping stations.   The model analyzes self-purification processes, identifies critical pollution zones, and allows the design of environmental management strategies for improving water quality.	According to the work of S. Han et al. [79], the precision of the model depends on the quantity of input data. Also, the accuracy of the predictions may be affected by the complexity of the fluvial system that may require simplifications.   According to the work of H. Wang et al. [69], the model is limited by the accuracy and availability of input data. The unidimensionality of the MIKE 11 model may not fully capture the complexity of fluvial systems with significant tridimensional variations.

Table 2. Mathematical models used for the evaluation of river self-purification: theoretical foundations, applications, and limitations.

Authors	Model	N° Articles	Theoretical Foundation	Application	Limitations
[28,62,63,71,72,73]	QUAL-UFMG	6	It is an adaptation of the QUAL2E model from USEPA developed by the Federal University of Minas Gerais (Portuguese: Universidade Federal de Minas Gerais, UFMG) in Brazil.   The calculation is done by processing coupled differential equations in Microsoft Excel. It uses the equations from the QUAL family with the corresponding adaptations.   Main parameters - Biochemical oxygen demand (BOD) - Dissolved oxygen (DO) - Total nitrogen and its fractions - Total phosphorous and its fractions - Thermotolerant coliforms	Given that the model is developed in Brazil, its application mainly focuses on tropical zones (similar to the design conditions) and in studies that require simple tools of easy access based on Excel to evaluate the impact of discharges.	It is limited to zones of tropical conditions. It has less effective precision for other climate zones [71].   According to the work of Da Luz et al. [63], for the DO calibration a good data fit was shown, but this was not the case for BOD5. This adjustment was presented due to the fact that real conditions in which effluents are discharged are unknown.   The discharge of pollutant loads in a water body has to be analyzed with much caution because misinterpreted data and incorrect results compromise predominant applications [63].
[18,65,74,75,76,77]	QUAL2K	6	The QUAL2K model is an evolution of the QUAL2E model, designed to improve water quality simulations through the inclusion of variables such as algae growth, denitrification, and an adjustment in the calculation of dissolved oxygen.   Main parameters - DO - BOD5 - pH - Temperature - Electrical conductivity - Organic nitrogen - Ammoniacal nitrogen - Inorganic nitrogen - Phosphorous - Sedimentable solids - Phytoplankton and Periphyton	The application of this model is used in simulations of dissolved oxygen and biochemical oxygen demand.	If the model uses variables such as nitrogen or phosphorous, it can extend to problems with these pollutants. Also, it presents significant challenges due to the quantity of detailed data that is required, which is why the information is often very limited [18].   Many times the behavior of variables in the section studied may tend to underestimate in this case the concentration of DO [18] The adjustment to the QUAL2K model may be significantly lower [77].
[64,65,78]	Streeter-Phelps	3	The Streeter-Phelps model is the main historical basis model from which many other models for the evaluation of water quality in rivers and other water bodies are derived through the explication of the interaction between BOD and DO. It is specially designed in response to wastewater discharges.   Main parameters - DO - BOD5	It is a model used for water quality simulation in rivers. It is a classic approach that does not include advanced technological tools.   The model allows the identification of natural recovery zones and critical pollution points, considering the organic matter decomposition rate, and the process of reoxygenation.	The authors mention that for this model homogeneous, stationary, and constant flux conditions have to be assumed [64,65,78]   According to the work of Díaz et al. [64], the precision of the model is lowered by the presence of diffuse sources of pollution, and in cases of high hydrodynamic complexity.   According to the work of Pazmiño-Rodríguez et al. [65], neither multiple pollutants nor seasonal variability are considered.
[58,66]	WASP	2	The WASP model is a tool developed in 1980 by USEPA. It is designed to simulate various parameters and their interaction for the evaluation of water quality in such a way that the effect of pollutant loads over water bodies can be predicted. It basically uses a set of differential equations to describe transport, dispersion, and reaction of pollutants.   Main parameters - DO - BOD5 - NH4-N - Organic nitrogen - NO3-N - Organic phosphorous - PO4-P - Phytoplankton - Coliform bacteria - Silica	The model is used to simulate water quality under different pollution scenarios, allowing the analysis of impacts of either anthropogenic or natural pollutant loads.   Speaking of environmental management, the model helps to design optimal strategies for control measures of pollutant sources such as optimal systems and for future decisions based on the information provided on water quality.	According to the work of Żelazny et al. [60], there are limitations for precise representation of extreme events, such as runoff caused by extreme rainfall.   According to the work of Ma et al. [67], there is a precision dependency on input data, and on the simplification of assumptions in the behavior of the interception system.
[69,79]	MIKE	2	The MIKE model was developed in 1972 by the Danish Hydraulic Institute. It is a deterministic model that allows the simulation of fluxes in a non-permanent state within river systems, adapting both to simple and complex configurations.   This model can be used to evaluate the impact of discharges on water quality in rivers, besides working as a hydraulic model for the analysis of floods.   It integrates specialized modules, among which those of rainfall-runoff, advection-dispersion, and hydrodynamics are included.   Main parameters Physicochemical - BOD5 - DO - NO3-N - NH4-N - Heavy metals - Coliform bacteria - Hydrodynamic - Velocity - Flow rate - Water level - Water quality - Sediment transport	It is an advanced model that allows hydrodynamic and water quality simulation in water bodies, evaluating diverse pollutant dispersion scenarios, such as the optimal location of pumping stations.   The model analyzes self-purification processes, identifies critical pollution zones, and allows the design of environmental management strategies for improving water quality.	According to the work of S. Han et al. [79], the precision of the model depends on the quantity of input data. Also, the accuracy of the predictions may be affected by the complexity of the fluvial system that may require simplifications.   According to the work of H. Wang et al. [69], the model is limited by the accuracy and availability of input data. The unidimensionality of the MIKE 11 model may not fully capture the complexity of fluvial systems with significant tridimensional variations.

3.2. Research Question 2: What Are the Most Relevant Variables Used in Self-Purification Models, and How Do They Influence the Prediction of Water Quality?

To answer this question, the five most common and relevant mathematical self-purification models identified in the reviewed studies have been highlighted. In particular, each model employs specific equations to simulate the dynamics of pollutants, allowing the adjustment of the simulations to essential environmental factors such as flow rate, flow velocity, and the physicochemical parameters of water.

Having a variety of equations and input data provides flexibility to adapt to different river types and pollution scenarios. Therefore, to answer this research question,

Table 3. Most relevant variables used in river self-purification models and their evaluation methods.

Model	Variables	Category	Evaluation Method	Calibration	Model Robustness	Reference
QUAL-UFMG	BOD5, DO, Total nitrogen and its fractions, Total phosphorous and its fractions, Thermotolerant coliforms	Biochemistry, Physical chemistry, Chemistry, Microbiology	Coupled differential equations from the QUAL equations family, Microsoft Excel spreadsheet simulation.	The associated coefficients to the DO and BOD5 variables, as well as the organic matter decomposition coefficient are manually calibrated in this model. The calibration method consists of varying the coefficients until obtaining the minimum sum of squares through a process that integrates a database of observed and modeled data, making the determination of the coefficient more precise. Coefficient values described in the literature are equally considered as in Pani et al. [28] or in Da Luz et al. [63].	High, as long as it is applied in tropical zones and according to the calculation of the coefficients for their appropriate calibration.	[28,63,71,73]
QUAL2K	pH, Temperature, Sedimentable solids, BOD5, DO, Electrical conductivity, Nitrogen (organic, ammoniacal, nitric), Phosphorous (organic, inorganic)	Physics, Biochemistry, Physical chemistry, Chemistry	Mass balance using differential equations; specific parameter analysis.	For this model, calibration is performed on the constants associated with the main variables (DO, BOD) as well as those associated with chemical and physical variables. The calibration methodology is described as a Montecarlo simulation using the GLUE method based on databases of the considered variables. Calibrations from literature are also taken into account, like Gutiérrez [73]	High, although it depends on the modeling processes and the calibration performed.	[18,65,74,76,77]
Streeter-Phelps	DO, BOD5	Physical chemistry, Biochemistry	Simulates the dynamics of DO and BOD based on natural self-purification processes, including degradation and reoxygenation.	The deoxygenation (k1) and reoxygenation (k2) constants are used. These are obtained from equation calculations proposed by different authors. In De Menezes et al. [78] the equations proposed by von Sperling are used, in Pazmiño-Rodríguez et al. [65] the equations from Owen and Gibbs are used, and in Díaz et al. [64] a statistical approach is taken.	Average since it does not integrate other variables for modeling.	[64,65,78]
WASP	DO, BOD5, Nitrogen, Phosphorous, Suspended solids, Pathogens	Physical chemistry, Chemistry, Physics, Microbiology	The dynamic analysis allows the modeling of multiple pollutants and their interactions in water bodies.	In this model, a series of coefficients are considered according to the variables used. It is a more complex model, so only the coefficients for physical variables were calculated, while for the other variables, data were considered from the existing literature.	Average	[67,68]
MIKE	Velocity, Flow rate, Water level, Water quality, Sediment transport     BOD5, DO, NO3-N, NH4-N, Heavy metals, Coliform bacteria	Physics,           Biochemistry, Physical chemistry, Chemistry, Microbiology	1D, 2D, and 3D detailed simulations based on advanced numerical methods.	More general coefficients are considered for each variable, they can be summarized in the diffuse convection coefficient and the matter attenuation coefficient. These are usually extracted from existing literature.   For hydraulic variables, the most relevant coefficient is the Manning coefficient.	High, due to the integration of multiple modules that allow a more complete modeling.	[69,79]

is analyzed in detail, where the variables considered in each model are presented, noting that the variables present in all models correspond to biochemical oxygen demand (BOD) and dissolved oxygen (DO), which stand out as key parameters when analyzing self-purification in water bodies. In some models, other variables such as total and organic nitrogen, and nitrogen fractions, as well as total and organic phosphorous are identified, which gives the models additional potential by considering nutrients found in water.

Models such as the QUAL-UFMG and QUAL2K focus on biochemical and physicochemical variables, including BOD₅ and DO. These variables are essential as they reflect the capacity of the aquatic ecosystem to decompose organic matter and maintain adequate conditions for aquatic life. The manual calibration of coefficients in QUAL-UFMG which is based on the minimization of the sum of squares, allows a precise adaptation to local conditions that increases the robustness of the model for tropical contexts [71]. However, this dependency on local calibrations can limit the applications of the model in other regions where historical data is scarce or unavailable.

The Streeter-Phelps model, although simpler, focuses only on BOD₅ and DO, which can be a significant limitation as it does not consider other relevant pollutants such as nutrients or pathogens. The usefulness of this model could be restricted to situations where these factors are not critical [64]. By integrating multiple contaminants, the WASP model provides a more holistic vision of the quality of water. Nevertheless, the complexity of this model can make the calibration and interpretation of results more challenging [66].

The MIKE model stands out due to its capacity to simulate in 1D, 2D, and 3D, allowing a more detailed analysis of the interactions between physical and chemical variables, as well as sediment transport. This complexity is advantageous since it provides a more complete model for watershed management, but it also requires a greater effort for the calibration and validation of the coefficients [79].

Regarding robustness, models such as QUAL-UFMG and MIKE are considered highly reliable due to their capacity to integrate multiple variables and processes. However, their effectiveness depends largely on the quality of input data and the calibration process, which can represent a challenge in contexts where data is limited. In contrast, simpler models like Streeter-Phelps may be easier to implement, but their lack of consideration of additional variables can affect the precision of water quality predictions.

The selection of the appropriate model for evaluating river self-purification should be based on a balance between the model’s complexity, data availability, and the specific objectives of the study. The most relevant variables like BOD₅, DO, and nutrients are crucial in predicting water quality, and their inclusion in the models must be carefully considered to ensure accurate results that are applicable to water resources management.

3.3. Research Question 3: What Are the Strengths and Weaknesses of the Self-Purification Models in Polluted Rivers?

When analyzing the mathematical models for river self-purification, it was determined that each model has its own set of strengths and weaknesses that affect their effectiveness for the evaluation of water quality in polluted rivers (

Table 4. Comparative evaluation of mathematical models for river self-purification.

Model	Advantages	Disadvantages
QUAL -UFMG	Model mostly used in Brazil for its adaptation to tropical conditions [71].	Limited for tropical conditions, so its accuracy will be less effective in other climates [71].
	DO and BOD5 parameters have the greatest presence in most of the literature and studies reviewed. These 2 parameters are considered key evaluation parameters [28,62,63,71,72,73]	According to the work of Da Luz et al. [63], for the DO calibration a good data fit was shown, but this was not the case for BOD5. This adjustment was presented due to the fact that real conditions in which effluents are discharged are unknown.
	The QUAL-UFMG excel platform provided by this model is an easy-to-use and highly productive tool for modeling water quality. It is a bidimensional model [28,53,54,62,63,64].	The discharge of pollutant loads in a water body has to be analyzed with much caution because misinterpreted data and incorrect results compromise predominant applications [63].
	The model reproduces physical, chemical, and biological processes, considering point and diffuse inputs of pollutant loads, as well as point extractions, both consumptive and non-consumptive [62,72].
QUAL2K	The QUAL2K model complies with the study of the behavior of water quality of surface sources. Its simulation determines the behavior of the pollutant quality in a certain time for a specific section [18,65,74,75,76,77]	If the model uses variables such as nitrogen or phosphorous, it can extend to problems with these pollutants. Also, it presents significant challenges due to the quantity of detailed data that is required, which is why the information is often very limited [18].
	QUAL2K can estimate and simulate an unknown location of the pollution source, it provides a dynamic and accurate estimation of the following factors: TEM, EC, pH, DO, and BOD5 [76].	Many times the behavior of variables in the section studied may tend to underestimate in this case the concentration of DO [18].
	The QUAL2K model is based on differential equations for unidimensional systems, and a stationary flux. It is an efficient method to simulate water quality and hydrologic conditions in rivers, as well as in systems with diffuse pollutant loads [77].	The adjustment to the QUAL2K model may be significantly lower [77].
	This model can solve the Streeter–Phelps equation. It makes an analytic expansion for the DO and BOD5 relation. The QUAL2K model is a good approximation for estimating the BOD5 load capacity [18].
Streeter-Phelps	It facilitates the development of other advanced numerical models like QUAL-I, II, E, and QUAL-2K. This model describes how oxygen demand decreases in a river over distance due to BOD5 degradation [78].	It tends to overestimate the DO concentration and the efficiency of atmospheric reaeration in certain sections [65].
	The Streeter-Phelps classic model is a reference point in sanitary and environmental engineering, being the precursor of water quality models [64].	The mass balance of the model is only performed at a specific point, unlike more advanced models that do it throughout the entire section [65].
	It is a simple and useful model to evaluate water quality in rivers, even with limited data. It achieves a good fit in dissolved oxygen simulations with a high coefficient of determination and Nash-Sutcliffe index [65].	It may present an error in the prediction of DO of up to 40% in some cases [65].
WASP	The WASP model can also be used for operational purposes. With advective and dispersive transport between discrete physical compartments it can be applied in 1D, 2D, or 3D for any type of water [18].	This model does not determine the extent of the intermediate zone, because the short section of the river covered by the selected model is selected as an analysis example [18].
	The WASP model can handle multiple types of pollutants in one single run. It has two kinetic models: advanced toxic transformation and advanced eutrophication [18].	The WASP model does not have adequate treatment facilities [18].
	Using the model, it is possible to determine the hazards that can be generated by the facilities already present in the watersheds [18].
MIKE	This model is responsible for analyzing complex problems with a certain precision [79].	The specialization level has to be high for its configuration and operation, limiting the use of modeling [69].
	Using the water quality module, the MIKE software allows the user to create different templates of pollutant composition, which helps to achieve greater modeling accuracy [79].	It is limited by license costs and software access, restricting its use [69].
	It is widely used for hydrodynamic analysis, water quality modeling, pollutant migration, and quality improvement [79].

). The QUAL-UFMG model, even though popular in Brazil, is intrinsically limited by its focus on tropical conditions. This geographical specification may compromise the transfer of the model to other regions with different climates, which raises a dilemma about its universality [71]. Furthermore, its dependency on parameters such as BOD₅ and DO, although relevant, can lead to an excessive simplification of the complexity of aquatic ecosystems. The calibration of the model, especially regarding BOD₅, is affected by the lack of precise data about the real discharge conditions which can produce misleading results [63].

The QUAL2K model, on the other hand, offers a dynamic estimation of the behavior of water quality, but its need for detailed data makes its use more challenging. Where pollutant information is scarce, the effectiveness of the model is compromised [18]. In addition, the underestimation of the DO concentration can have profound implications for water resources management, as it can lead to inadequate decisions about remediation and water use [18,80].

As for the Streeter-Phelps model, its simplicity can represent both an advantage and a disadvantage. It is accessible and easy to understand, but its limited focus on one single mass balance point can be insufficient in sections of the river that have significant variation in water quality [65]. The overestimation of the concentration of dissolved oxygen is another critical issue, as it can generate a sensation of false security in water quality management [65,80].

The WASP model is promising due to its capacity to handle multiple pollutants. However, its limitation in determining the extent of the intermediate zone can be a significant obstacle [66,81]. This suggests that even though the model can offer a general vision, it may not be sufficiently detailed to address specific problems in polluted rivers. The lack of adequate treatment facilities in the model also indicates that its applicability in real management scenarios may be limited [66].

The MIKE model presents itself as a powerful tool to address complex problems, but its high specialization level and elevated costs limit its accessibility [69]. This raises questions about equity in the access to advanced modeling tools, especially in developing countries where resources are scarce. Despite the potential of MIKE, the entrance barrier can result in an underutilization of its capabilities in areas that need it most [79].

Even though the models for river self-purification provide valuable tools for the evaluation of water quality, their inherent limitations require critical analysis. The effectiveness of each model greatly depends on the context in which it is applied and on the quality of available data. This underlines the need for a holistic approach that combines modeling with the collection of precise and relevant data, as well as the consideration of specific characteristics of each aquatic ecosystem.

There are crucial limitations in the reviewed models (oversimplification, input dependence, and lack of climate adaptability: QUAL-UFMG, QUAL2K, Streeter-Phelps, WASP, MIKE) [61]. Nevertheless, it is imperative to illustrate how these theoretical shortcomings translate into quantifiable prediction errors and operational biases in actual water quality management [82]. Oversimplification in classical steady-state models, particularly Streeter-Phelps, leads to significant management bias and notable accuracy failures [4,76]: by assuming ideal plug flow conditions and single point sources, the real dynamics of complex rivers with high hydrodynamic variability or diffuse loadings (non-point sources or NPS) are ignored [53,83]. This methodological omission directly influences the overestimation of dissolved oxygen (DO) and a critical underestimation of actual DO deficits [84,85]. When these models are used to estimate pollutant carrying capacity (PLC), errors in the representation of key processes, including self-purification mechanisms, can lead to unreliable predictions [86]. Furthermore, the extreme dependence on detailed data required by complex simulation models such as QUAL2K and WASP results in practical inoperability in catchments with scarce information or limited monitoring capabilities, a predominant reality in many developing countries where financial resources and institutional capacity restrict the collection of high-resolution data [87,88].

This bias is mitigated by the development of simplified, management-oriented models, such as the Water Quality Systems Assessment Model (WQSAM), developed in South Africa, which can simulate representative frequency distributions of nutrients and salinity in catchments with limited data [55,89]. Furthermore, the lack of climate adaptability in self-cleaning models becomes particularly evident under future hydrological stress scenarios [90,91]. The WASP7 model was used to dynamically simulate the water quality of the Upper Qu’Appelle River under future climate change scenarios, specifically for the periods 2050–2055 and 2080–2085. Results showed that, with projected increases in air temperature, nutrient concentrations, including nitrate (NO₃⁻ N), tend to decrease during open water conditions, although this trend varies depending on the season and the climate scenario applied. In particular, monthly mean NO₃⁻ N concentrations were observed to deviate by up to approximately 9.79% from reference conditions, highlighting the potential impact of climate change on nutrient dynamics in regulated rivers [92]. An error of this magnitude, if not integrated through complex model couplings, would cause catastrophic failures in water quality planning in river-reservoir systems, confirming that model limitations are, in essence, quantifiable sources of environmental and public health risk.

The classical Streeter-Phelps (S-P) model was specifically designed for a single point source discharge, assuming uniform mixing across the river cross section [84]. The strict reliance on these assumptions turns the methodological rigidity of S-P into a direct source of systematic prediction error [83]. Contemporary river environments, especially those influenced by urbanization and agriculture, are characterized by high hydrodynamic complexity and the prevalence of non-point source (NPS) pollution loadings, such as agricultural or urban streamflow [83,93]. These factors directly violate the steady and homogeneous flow assumption of S-P, since NPS loadings are dynamic and dispersed, making the model unsuitable for representing the temporal and spatial variability of pollutant loading [84,93]. The limitation of the Streeter-Phelps model to incorporate multiple pollutants, seasonal variability or diffuse pollution sources has been extensively pointed out in the recent literature [84,94]. The inability of the model to handle this complexity results in a significant reduction in its predictive accuracy [84]. Studies indicate that under conditions of high hydrodynamic complexity or the presence of diffuse sources, the predictive capacity of the S-P is compromised [94]. The most critical consequence of this simplification is observed in the DO simulation results [95]. The S-P model tends to overestimate DO concentration and the efficiency of the atmospheric reaeration process in complex or impacted river sections [96]. The Streeter-Phelps model assumes ideal mixing conditions and does not consider additional processes such as benthic oxygen demand, photosynthesis, or additional sources and sinks of DO, which can result in inaccurate predictions of dissolved oxygen concentration in rivers with complex characteristics [97].

The need for models that address the complexity of nonpoint sources has driven the development of tools with different levels of complexity [98]. Watershed-scale NPS pollution models range from rapid screening tools that require little data and provide reasonable average values (e.g., GWLF) [99] to complex models that are data-intensive and require rigorous calibration to simulate particular processes (e.g., SWAT) [100]. The lesson from this comparison is that model choice represents a critical trade-off between desired accuracy and data availability [18]. While a simple model like GWLF may be suitable for rapid assessments or for producing better average values of observed data, it is not appropriate for extreme event management or for projects that demand high accuracy [18]. Extreme simplification becomes an impediment when the management objective is to understand and mitigate pollution spikes during runoff events, moments in which phenomena are inherently dynamic and non-stationary.

Advanced models, such as QUAL2K and WASP, represent significant evolutions over Streeter-Phelps by incorporating a greater number of physicochemical variables, dynamic processes (algal growth, denitrification), and the ability to model multiple pollutants in 1D, 2D, or 3D. However, this sophistication comes at a direct operational cost: the need for comprehensive and accurate input data [101]. QUAL2K, for example, requires detailed data on watershed hydrology, water quality parameters, and statistics on point and non-point sources (emissions from factories, sewage pumping stations) [54]. Similarly, the accuracy of the WASP model depends critically on the quality and quantity of input data, including the linkage with hydrodynamic and sediment transport models [53]. This high data demand creates a complexity paradox in water management. Globally, deteriorating water quality has made models essential tools, but the reality in many regions, particularly in developing countries, is that a lack of observed data, limited financial resources, and institutional instability severely restrict the applicability of these complex models [87]. Where information is scarce or limited, the predictive effectiveness of QUAL2K or WASP is compromised, hampering their practical use in decision-making [54]. Water quality planning must look to the future, considering the impacts of climate change, which include rising water temperatures, altered reception patterns, and more frequent extreme hydrological events (droughts and floods) [102]. Water quality models based on deterministic, historically calibrated physicochemical principles, such as QUAL2K and WASP, prove to be fragile in the face of these changes [103]. Their greatest vulnerability lies in their focus on steady-state conditions. For example, QUAL2K, although advanced, operates primarily under this assumption, complicating the integration of dynamic data associated with climate change. WASP, although dynamic, has shown limitations in accurately representing extreme events, such as flooding caused by intense rainfall. The rigidity of the historical calibration means that the kinetic coefficients used for the base scenario (e.g., year 2010) are inadequate for a future where temperature and flow deviate excessively from the design conditions, resulting in significant predictive errors [19,53].

To demonstrate the impact of warming on nutrient kinetics, it is essential to resort to simulations of future scenarios. A study that employed the WASP7 dynamic model to assess the impacts of global warming on water quality in the upper Qu’Appelle River (Canada), projecting conditions for the periods 2050–2055 and 2080–2085, provided a clean quantification of the predictive error associated with climate change [92]. The analysis focused on how increasing water temperature affects nutrient kinematics. The results showed significant deviations in the concentration of nitrogen species compared to the 2010 baseline scenario. A maximum deviation of 6.71% will be observed for ammonium (NH₄-N) and, critically, up to 9.79% for nitrate (NO₃ -N) in the most severe future scenarios. These deviations were most pronounced during the open water phase (late summer), the period most vulnerable to eutrophication. An error of almost 10% in nitrate prediction over a long-term planning horizon is a critical flaw for eutrophication management. This finding demonstrates that the climatic fragility of deterministic models is not an abstract limitation, but a quantifiable source of risk that, if ignored, guarantees the failure of long-term planning for pollution mitigation and treatment infrastructure [104].

Traditional models, such as Streeter-Phelps, rely on oversimplification by explicitly excluding vital processes such as sediment oxygen demand (SOD), photosynthesis, and respiration, resulting in underestimation or overestimation of dissolved oxygen (DO) in critical reaches [105]. This limitation extends to more advanced models such as QUAL-UFMG and QUAL2K, where the representation of biological dynamics, although present, often relies on simplified kinetic coefficients and manual calibration, compromising accuracy by not reflecting actual discharge conditions or seasonal variability of biota [106,107]. The omission of microbial biodiversity is particularly critical, as river biofilms, composed of complex algal and bacterial communities, act as the primary kinetic driver for pollutant degradation, driving key processes such as nitrification and denitrification essential for the removal of nitrogen and phosphorus from water. The efficiency of this microbial degradation is highly dynamic and dependent on environmental conditions and biofilm structure, a complexity not captured by constant decay coefficients [108]. In parallel, the scant consideration of sediment fluxes in this model overlooks that the benthic layer acts as an active biogeochemical interface and a fundamental reservoir of particulate organic matter. The sediment exerts a significant Sediment Oxygen Demand (SOD), which constitutes a crucial consumption of DO in the water column and acts as a source or sink of nutrients, an indispensable factor for predicting assimilative capacity and eutrophication in impacted rivers [109]. To realistically represent these processes, a dynamic interaction between the hydrodynamic, water quality and benthic modules is necessary. This capability, unique to models such as WASP and MIKE (with specialized modules such as ICM/HEM3D), allows the simulation of both bacterial conversion of benthic organic matter and oxygen consumption under different flow regimes [110]. However, as detailed in Table 4 of the manuscript, the robust implementation of WASP and MIKE is inherently limited by the massive amount of detailed field data and the high specialization required for their calibration and operation, restricting their practical applicability in data-poor settings. This limitation highlights the knowledge gap identified in the review and underlines the need to integrate approaches that model complex biological interactions and the influence of aquatic biota to provide more accurate and holistic predictions for water quality management [23].

3.4. Final Considerations Between Models

After analyzing the different models, it is concluded that the accuracy of the simulations highly depends on the employed coefficients in each equation, as these are intrinsically related to the physical and chemical characteristics of the water systems, such as their shape, size, and configuration. Adequate calibration of these coefficients is fundamental as a variation in them can significantly alter the simulation results.

It is also crucial to consider factors such as sedimentation, organic matter processing, and oxygen consumption, as these processes do not only influence water quality and affect the dynamic of aquatic ecosystems, but they also determine a river’s ability to purify itself. Sedimentation, for example, can modify nutrient availability and the habitat for aquatic organisms, whereas organic matter decomposition is a main indicator of an ecosystem’s health.

The specific parameters of each river have a considerable impact in the accuracy of the models, often exceeding the influence of general environmental variables. This occurs as these parameters directly respond to the particularities of each river system, allowing models to achieve more accurate simulations representative of reality. Hence, it is paramount that these models are adaptive and that they consider the particularities of each water body by integrating local and specific data.

To achieve effective modeling of river self-purification and proper water quality management, it is imperative to take on a holistic approach that contemplates both the specific coefficients and the fundamental ecological processes. The combination of these elements will not only improve the accuracy of simulations, but also contribute to the conservation and sustainability of aquatic ecosystems.

3.5. Recent Developments and Prospects for Self-Purification Models

Recent research on modeling self-purification in rivers has gone beyond traditional approaches, incorporating new techniques and more sophisticated methodologies [111,112,113]. Mathematical models of self-purification have evolved significantly in recent years, driven by advances in computational capacity, the availability of real-time data, and the development of hybrid methodologies that combine physical-chemical approaches with artificial intelligence techniques [13,85].

Among the most widely used models are equilibrium and dynamic models, such as QUAL2K, QUAL2E, Streeter-Phelps, and simulation models based on water quality models such as WASP and MIKE [19,114]. These models are based on principles of mass conservation, transport, and chemical reaction, allowing simulation of processes such as BOD degradation, oxidation of biochemical oxygen demand, and algal blooms. Recent developments focus on the integration of machine learning approaches and big data analytics to address traditional limitations, such as the dependence on precise parameters and the difficulty in modeling processes with high spatial and temporal variability [111,112]. For example, neural networks and evolutionary algorithms have been used to optimize parameters and improve predictive capacity under conditions of limited data availability or high uncertainty [115,116]. Furthermore, deep neural networks (deep learning) have been implemented to model self-purification in river systems, with the aim of predicting the temporal and spatial distribution of pollutants along rivers [115]. The integration of physical models with neural networks has improved the accuracy of predictions, allowing for more detailed simulations of the variability of self-purification processes under different flow, temperature, and pollution conditions [116].

In recent years, there has been a growing emphasis on multiscale models that combine macroscopic (watershed) information with microscopic processes such as the biological activity of bacteria or microorganisms [117,118]. These models allow the integration of remote sensing data, real-time monitoring, and hydrodynamic models to accurately assess the self-purification capacity of rivers with different types of pollution [119,120]. Furthermore, models that integrate ecological interactions, such as the effects of aquatic species on water quality and self-purification processes, are increasingly relevant. These models can include simulations of microbial biomass, biodiversity, and nutrient cycling in the river ecosystem, allowing us to understand how the biodiversity of microorganisms and aquatic organisms impacts the river’s ability to purify its waters [121,122]. A critical aspect of these advances is the need to improve the transferability and scalability of models developed in local contexts to different geographic regions, taking into account specific ecological, climatic, and anthropogenic variables [123]. Incorporating climate variables, land use, and hydrometeorological data into hybrid models is essential for obtaining more accurate and adaptable predictions, especially under climate change scenarios [124,125]. Looking ahead, prospects point to the integration of self-healing models with real-time monitoring systems using remote sensors and artificial intelligence platforms. This will enable adaptive and proactive management for the protection of water resources. In this context, the use of digital technologies, sensors, and information systems will be essential to monitor, manage, and optimize the use of water resources, as well as to respond to extreme events and promote holistic and resilient management [126]. Digital twins are powerful tools that can significantly improve the understanding and management of self-healing processes in rivers and other water bodies.

Furthermore, the standardization of calibration and validation methodologies, along with the development of open collaborative platforms, will facilitate knowledge transfer and the incorporation of technological innovations, promoting more robust, flexible, and universal models.

The integration of Artificial Intelligence (AI) and Machine Learning (ML) in water quality modeling is the main recent trend, offering high technical feasibility due to its ability to accurately model nonlinear relationships and high dimensionality of water parameters, outperforming traditional mechanistic approaches in predictive accuracy for short-term forecasts and early detection of anomalies in real-time (via IoT sensors) [127,128]. However, this perspective must be approached with a critical analysis of its inherent challenges. The first major challenge is the need for Big Data, unlike process-based models, ML algorithms are inherently “data-hungry”, requiring large volumes of continuous and high-quality historical data for robust training, which often proves unfeasible in regions with limited monitoring or incomplete time series [129]. The second challenge is the risk of overfitting, where complex models such as Deep Neural Networks (DNNs) can memorize the noise in the training set and catastrophically fail to generalize their predictions to extreme or unseen scenarios (such as floods or droughts), critical situations for water management [130]. The third and perhaps biggest obstacle is the poor interpretability of black-box models. The inability of AI to extract causal relationships or underlying physical coefficients (fundamental to ecology and engineering) generates a lack of trust and hampers scientific diagnosis or the justification of regulatory decisions to stakeholders [82]. To address this criticism, the most promising solution is the development of hybrid models, which combine the predictive power of ML with the physical consistency of process-based models (or PIML, Physics-Informed Machine Learning), where physical equations act as penalty constraints. This balanced approach ensures that, despite using a “black box” for prediction, the final result is scientifically plausible and physically consistent, achieving the transparency and reliability necessary for a responsible implementation of AI in water resources management [129].

Recent Advances and Perspectives for Self-Purification Models

The advancement of geospatial technologies, particularly Geographic Information Systems (GIS) and Remote Sensing (RS) through satellite approaches, has opened a new frontier in water quality modelling by allowing the incorporation of critical spatial and temporal variability that one-dimensional models, such as QUAL2K and WASP, tend to simplify [131]. GIS is fundamental for watershed analysis, facilitating the precise delineation of micro-basins, the generation of Digital Elevation Models (DEMs) and the detailed mapping of land use and land cover, inputs that significantly improve the estimation of hydrodynamic parameters and diffuse pollutant loads [132]. For its part, TD, using sensors such as Landsat or MODIS, enables synoptic and near real-time monitoring of key water quality indicators such as surface temperature, suspended solids, turbidity and chlorophyll-a, at scales that would be unfeasible through traditional field monitoring [133]. However, the integration of these tools faces several challenges: there is a significant disparity in scales between mathematical models, which often require high-precision point data, and the spatial resolution of satellites [131]. Processing remote sensing data is complex and demands rigorous atmospheric corrections for effective calibration, especially in turbid inland waters [134]. Furthermore, software integration between hydrodynamic models and GIS platforms is often laborious, requiring the development of hybrid models or coupled platforms. Overcoming these obstacles is essential to move towards spatially clean modeling and optimize the management of diffuse pollution.

3.6. Applicability of the Findings Identified in the Review

To determine a river’s capacity for self-purification it is necessary to carry out a rigorous evaluation of water quality. The QUAL2K model is a viable and detailed tool for this objective, allowing the simulation of the behavior of key self-purification parameters, such as the evaluation of organic matter oxidation processes, nitrification rates, and decay of pathogens. An example of this applicability is found in Chile, where specific kinetics measure the rate of oxygen gain or loss in water [74].

The Streeter-Phelps model is useful for analyzing mechanisms that affect dissolved oxygen levels in a surface water channel receiving wastewater discharges. An interesting example of its applicability is found in Quevedo, Ecuador, where the Streeter-Phelps mathematical model and the QUAL2K program were used to obtain the variability of dissolved oxygen concentrations and biochemical oxygen demand [135]. Additionally, the impact of wastewater discharges on these parameters was simulated, providing a comprehensive assessment of water quality and its self-purification capacity in the presence of pollutants.

Thus, the combination of both models becomes a robust tool for the management and evaluation of water quality in polluted rivers, as it produces a more solid comprehension of the self-purification process and the effects of pollution.

The QUAL-UFMG model is a hydrologic and water quality simulation tool developed by the Federal University of Minas Gerais in Brazil. It is based on the QUAL2K model with adaptations to characteristics of Brazilian rivers, and other rivers from similar regions. This model is useful for tropical climate rivers or for rivers with similar climate conditions to Brazil, where the QUAL2K model could not simulate [19].

As for the MIKE and WASP models, both have robust and advanced software for the calculation of hydrodynamic variables which make them especially effective in the simulation of transport phenomena and water flux. Although both models include water quality modules, their application in this field could be complemented by using them together with other water quality specialized models like QUAL2K or the denominated classic Streeter-Phelps model. These specific models allow a more detailed adjustment of self-purification parameters [53].

3.7. Strengths, Limitations, and Knowledge Gaps

This analysis highlights several strengths that add robustness and utility to the revision carried out in this study. The first is its systematic methodology, based on the PRISMA protocol, which guarantees a rigorous and representative selection of the studies reviewed. This allows an exhaustive and reliable analysis of water self-purification models. Another strong point is the detailed comparison between models, including their advantages, disadvantages, applicability, and technical specifications, which provides a useful integral vision for researchers, professionals, and water resources managers.

Nonetheless, this work has some limitations as well, the main one being that although the most commonly used models were selected, emergent or recent approaches that can provide novel perspectives in self-purification modeling were not. For example, the use of geographic information systems (GIS) like the ArcGIS or HEC software, as well as modeling through the use of satellite images were not included [122]. Moreover, the review focused on studies of specific databases, which could limit the geographical and environmental diversity of the scenarios evaluated. Yet another limitation is the variability in the coefficients and input data parameters, which often depend on specific local conditions and can be difficult to calibrate without accurate field data. This implies that the results of the models can vary significantly depending on the context and require additional adjustments to maximize their accuracy.

These strengths and limitations suggest that even though the review provides a solid theoretical foundation for the selection and application of models, a broader exploration of recent models and the collection of field data could enhance adaptability and accuracy in future research.

Unfortunately, knowledge gaps in the field of river self-purification are significant and affect the effectiveness of the existing models. Although the fundamental mass balance principles of highly utilized models like QUAL2K and WASP allow for global application across varied climates, a key challenge remains the reliable transferability of parameters and kinetic rate coefficients derived from specific regional studies. Both WASP and QUAL2K originated in North American contexts (e.g., US EPA applications for the Great Lakes and major estuaries), and their successful application spans temperate, subtropical, and tropical zones worldwide. However, models specifically adapted for regional conditions, such as QUAL-UFMG (detailed in Table 2), are explicitly limited to their intended geographical scope (e.g., tropical environments). For the widely used global models (WASP and QUAL2K), the principal gap lies in the rigorous calibration and validation of kinetic coefficients—which are highly sensitive to local temperature, solar radiation, and biogeochemical processes—to accurately reflect the substantial variability between temperate and tropical river systems. This translates into a lack of understanding about how climatic and geographic variations can influence the self-purification processes, which can lead to inaccurate results in water resources management in various areas [107,128].

Furthermore, there is a lack of studies that integrate environmental variables and local characteristics in an exhaustive manner. Many models tend to simplify the complexity of aquatic ecosystems, ignoring important factors like biodiversity, sediment dynamics, and interaction between different pollutants [14,122]. This can result in an over or under estimation of a river’s self-purification capacity, affecting management and conservation decisions.

Another critical aspect is the need for further investigation regarding the calibration of models and validation of results in different geographic and ecologic contexts [129,130]. Many studies are based on limited data or experimental conditions that do not reflect the reality of aquatic ecosystems. This underscores the importance of conducting broader and more detailed field studies that can provide robust data for model validation.

Finally, the inclusion of emergent approaches, such as the use of geographic information systems (GIS) and modeling through the use of satellite images, could offer new perspectives for the analysis of self-purification [126,131]. However, these approaches have not yet been effectively integrated into most existing studies, representing a remarkable opportunity to advance knowledge and management of water quality in rivers.

3.7.1. Fundamental Limitations: The Trade-Off Between Complexity and Utility

Despite their heuristic value and effectiveness as management tools, one-dimensional self-healing models such as QUAL2K and WASP are inherently limited by their need to oversimplify the ecological complexity of river ecosystems. This simplification is not a failure, but a deliberate result of the trade-off between model complexity and practical usefulness (usability) [132]. The need for parsimony guides the design of these models, forcing the representation of three-dimensional, dynamic processes (such as water-sediment interactions or vertical/lateral mixing) in one spatial dimension and under steady-state or quasi-steady-state conditions, respectively [133]. This reduction is necessary to keep input data requirements and computational cost low, making the model accessible to regulatory agencies with limited budgets or scarce monitoring data. Highly complex and dimensional models (such as 3D hydrodynamic models), although offering greater accuracy in the physical and biological re-presentation of the river, demand a quantity and quality of field data and expert calibration that often exceed the management capacity of many users, reducing their applicability and usability [134]. Therefore, the “over-simplification” of ecosystems is justified in the context of efficient management decision-making, where a fast and reasonably accurate response is often preferable to a very precise but unattainable or unusable response within the timeframe of the environmental policy [132].

3.7.2. Uncertainty Analysis, Sensitivity, and Hybrid Models in Water Quality Prediction

A critical aspect derived from the need for robust data and the parametric complexity of models such as QUAL-UFMG, QUAL2K, and MIKE is the imperative to adopt rigorous methodologies that transcend traditional manual calibration. The predictive efficacy of self-healing models is limited by parametric and structural uncertainty, making the application of post-modeling techniques essential. In this context, uncertainty analysis (UA) is essential to quantify the range of plausible outcomes arising from variations in input data, model parameters, and structural assumptions. Tools such as the GLUE (Generalized Likelihood Uncertainty Estimation) algorithm or the MCMC (Markov Chain Monte Carlo) method allow the generation of confidence bands around model predictions, improving transparency in decision-making. For example, recent studies have applied the MCMC-based SCEM-UA algorithm to assess uncertainties in river water temperature models, demonstrating its effectiveness in estimating relative errors of up to 15% [135]. Additionally, sensitivity testing (ST) allows for the identification of parameters that exert the greatest influence on critical output variables (e.g., BOD and DO). Methods such as Fourier Sensitivity Analysis (FAST) or variance-based methods (such as Sobol) are essential for prioritizing sampling and calibration efforts for the most influential coefficients, such as deoxygenation and re-aeration rates. For example, recent research has applied the Sobol method to analyze the sensitivity of water levels to different operational objectives, highlighting its usefulness in identifying key parameters [136].

Finally, hybrid approaches represent an innovative methodological alternative for improving accuracy and reducing uncertainty. These approaches combine the mechanistic and physical basis of traditional models (such as QUAL2K or WASP) with the pattern recognition capabilities of machine learning, such as Artificial Neural Networks (ANN). This integration of physical models with neural networks has proven effective in modeling complex nonlinear processes and in optimizing effluent quality prediction, overcoming the limitations of purely physical models in long-term prediction or in extreme conditions [137].

3.7.3. Practical Implications and Recommendations for Water Management

The synthesis of the strengths, limitations, and knowledge gaps identified herein has direct implications for sustainable water resources management. Implication for Model Selection (Strength): Managers should prioritize the selection of models such as QUAL2K or WASP not only for their mechanistic capability, but also for the availability of input data. The robustness demonstrated in the simulation of key parameters (OD and BOD) justifies their use in the assessment of mitigation scenarios, such as the determination of the Total Maximum Daily Load [138]. Implication for Data Collection (Limitation): Given the high parametric sensitivity and the need to reduce uncertainty, the main implication for practice is the reorientation of field monitoring efforts. Regulatory and management bodies should invest in continuous measurement systems (real-time sensors) and focus on obtaining long and high-resolution time series of flow, temperature and pollutant discharges, especially during critical low-water periods. This will allow a shift from static calibration exercises to dynamic and operational modeling [139]. Implication for Policy Development (Knowledge Gap): The identified gap in the modeling of specific biogeochemical processes (such as microbial assimilation or sedimentation of emerging pollutants) implies that water quality policies should not rely solely on DO and BOD prediction. It is crucial to integrate hybrid modeling (mechanistic + Machine Learning) to develop early warning systems and manage risks derived from non-traditional pollutants, moving towards an Integrated Water Resources Management (IWRM) that covers water quantity and quality in a coordinated manner [140].

3.8. Comparative Analysis and Methodological Limitations of the Compared Models

Classical models such as Streeter-Phelps, although widely used for their conceptual simplicity, have limitations in that they only represent deoxygenation and reaeration processes, omitting essential components such as nutrient cycling, sediment-water exchange and microbial dynamics that are determinants in complex fluvial systems [23,141]. This methodological simplification results in biased predictions when applied to rivers with multiple sources of pollution or active biogeochemical processes, as demonstrated by Nugraha et al. [141], who found that applying the Streeter-Phelps model to polluted rivers can lead to erroneous conclusions about resilience due to the exclusion of other sinks and sources of oxygen. In contrast, intermediate complexity models such as QUAL2K and QUAL2Kw incorporate modular representations of multiple constituents (BOD, DO, nutrients) with more realistic kinetic schemes, allowing for better characterization of longitudinal water quality gradients and management scenarios [129,142]. However, these models are highly sensitive to oxidation, nitrification, and denitrification coefficients, requiring intensive local calibration to obtain reliable predictions, as reported by Babamiri et al. [129] in their application of QUAL2KW in mountainous rivers. Structural differences are accentuated when compared to coupled hydrodynamic models such as MIKE11, which specifically address unsteady flow, heat transfer, and reaeration processes, providing better control over the variables determining water quality differences in systems with complex hydraulics [143,144]. The literature shows that omitting sediment, phytoplankton, or microbial process modules from medium-complexity models tends to result in higher organic matter and lower dissolved oxygen concentrations compared to models that include these processes, highlighting significant conceptual gaps [143,145,146]. Nguyen and Willems [143] demonstrated that model structural uncertainty substantially influences water quality assessment, finding that models that include sediment and biological process modules reduce prediction bias. The contextual applicability of these models varies considerably depending on specific hydrological and geographic factors. In data-limited catchments, hybrid approaches that integrate catchment hydrological models (such as SWAT) with one-dimensional water quality models have been shown to produce low-bias and robust calibration results, as evidenced by the work of Bui et al. [142] in the Cau River Basin in Vietnam. This hybrid strategy overcomes the limitations of direct spatial data and represents an important methodological solution for regions with sparse hydrological information. The most critical structural limitations include the inadequate representation of kinetic processes in simplified models, the omission of hydrodynamics and biogeochemical processes in models of intermediate complexity, and the widespread parametric uncertainty in reaction coefficients that have wide ranges in the literature [130,147]. Salla et al. [130] emphasize the critical importance of calibration for mathematical modeling of self-cleaning, showing that the use of default values from the literature without local calibration produces inconsistent predictions of DO, nutrients, and metals. The selection of the appropriate model should be based primarily on the specific research question and data availability. For rapid assessments of assimilative capacity in single reaches, parsimonious DO-BOD approaches provide useful capacity bounds, but their results should be interpreted recognizing the omission of multiple processes [141,148]. For longitudinal multi-constituent management scenarios, QUAL2K or WASP are more appropriate when lateral loads are known, whereas fully coupled hydrodynamic tools are necessary when unsteady hydraulics or temperature are controlling factors [101,129,130,142,143]. This methodological and contextual differentiation represents a critical advance in understanding the capabilities and limitations of each modeling approach, providing a solid basis for informed selection of modeling tools according to the specific characteristics of the river system and research objectives, but their results should be interpreted recognizing the omission of multiple processes [141,145]. For longitudinal multi-constituent management scenarios, QUAL2K or WASP are more appropriate when lateral loads are known, whereas fully coupled hydrodynamic tools are necessary when unsteady hydraulics or temperature are controlling factors [101,129,130,142,143]. This methodological and contextual differentiation represents a critical advance in understanding the capabilities and limitations of each modeling approach, providing a solid basis for informed selection of modeling tools according to the specific characteristics of the river system and research objectives, but their results must be interpreted recognizing the omission of multiple processes [141,145]. For longitudinal multi-constituent management scenarios, QUAL2K or WASP are more appropriate when lateral loads are known, whereas fully coupled hydrodynamic tools are necessary when unsteady hydraulics or temperature are controlling factors [101,129,130,142,143]. This methodological and contextual differentiation represents a critical advance in understanding the capabilities and limitations of each modeling approach, providing a solid basis for informed selection of modeling tools according to the specific characteristics of the river system and research objectives.

3.9. Future Direction of Next-Generation Research and Modeling

To overcome the inherent limitations of one-dimensional mechanistic models (such as QUAL2K and WASP), and given the increasing availability of real-time monitoring data, the next generation of water quality modeling research is focusing on two key paradigms: coupled models and the use of machine learning (ML) techniques.

The most significant criticism of traditional models is their inability to capture the interdependence between hydrology (water quantity) and ecology/biogeochemistry (water quality and ecosystem health). Coupled ecohydrological models represent a necessary evolution, as they simulate the bidirectional dynamics between physical and biological processes in a watershed [149]. These models allow water resource managers to assess ecological flow and biodiversity in an integrated manner with pollution parameters, which is critical in the context of climate change. Examples include the coupling of basin hydrological models (such as WEAP or SWAT) with water quality modules (QUAL2K), enabling a more holistic and spatially distributed simulation of pollutant transport and its impact on river health [150].

Machine learning (ML) and deep learning (DL) offer a robust alternative for rapid prediction and classification of water quality, especially in scenarios with large volumes of data (Big Data) and complex nonlinear relationships that mechanistic models struggle to calibrate [151]. Techniques such as artificial neural networks (ANN), random forests, and neuro-fuzzy inference systems (ANFIS) have demonstrated high predictive accuracy for parameters such as DO, BOD, and fecal coliforms in rivers, serving as complementary tools for environmental monitoring. The future trend points to hybrid models, where ML is used to optimize the calibration of complex parameters in mechanistic models (e.g., re-aeration coefficients) or to perform real-time early warning forecasts from IoT sensor data, overcoming the bottleneck of purely physics-based modeling [152].

4. Conclusions

This systematic review, based on the PRISMA methodology, identified five prominent river self-purification models commonly used in water quality management: QUAL-UFMG, QUAL2K, Streeter-Phelps, WASP, and MIKE. However, the comparative review demonstrates that the choice of any of these models should be based on a critical analysis of the specific application conditions, as their practical utility depends on the appropriate adaptation of their parameters and input conditions to the local characteristics of the water system. In this sense, mechanistic models such as QUAL2K or WASP are more appropriate when seeking a thorough understanding of ecosystem kinetic processes, such as re-aeration or sedimentation, or in contexts with limited but high-quality data focused on key physicochemical variables. In contrast, AI-based models should be applied exclusively when extensive, high-frequency time series are available, and when high real-time predictive accuracy is required, for example, in pollution early warning systems. Since some models require specific coefficients and detailed data to optimize their accuracy, the importance of selecting the model based on the type of pollutant and the specific conditions of the river is highlighted. To avoid overfitting in machine learning models, the need to move toward the standardization and expansion of IoT sensor networks is emphasized, ensuring the continuity, traceability, and reliability of time series related to climate and water quality variables. In the face of the challenges posed by climate change, we propose overcoming traditional static approaches and explicitly incorporating climate uncertainty through hybrid models capable of simulating extreme events such as droughts or floods, which is essential for projecting the future vulnerability of water bodies. Consequently, it is recommended that agencies responsible for water management invest in modeling systems that not only represent the current state but also facilitate forward-looking, resilient, and evidence-based planning.