Calculation of Sustainability Indicators for Water Objects based on the Example of Water Use in the Arctic Basin of the Yenisei River

Abstract

In the article, the fractal approach to the description of nonequilibrium water ecosystems in their exploitation conditions is considered. The article demonstrates clearly that, in these conditions, the management of the development of water objects should be a precautionary provision, which should furthermore be reduced to the management of potential anthropogenic risks arising from a violation of the coevolution of anthropogenic and natural processes equivalent to the disruption in sustainability of the ecosystem in its exploitation conditions. The fractal approach opens up new opportunities for the assessment of water management based on the calculation of the fractal sustainability indicator (anthropogenic transformation) for water ecosystems that is directly related to the maximum permissible environmental load (MPEL).

1. Introduction

The Yenisei, like other great Siberian rivers, directly affects the state of water (including coastal) ecosystems of the Arctic Ocean [1].

At the mouth of the Yenisei River, complex deltaic regions and beaches prevail. These areas of active mixing of river and sea waters represent natural geochemical barriers. These barriers partially prevent pollutants that could be washed up into the Arctic Ocean from entering the environment. At the same time, Russian scientists say that the main sources of pollution are household waste waters washed by large rivers into the waters of the Arctic seas. It follows that the control of water use in the Yenisei River Basin is an extremely urgent problem for the Blue Economy [2].

Comprehensive study of the dynamics of the development and preservation of water ecosystems is important for the practical implementation of environmental management [3].

The methodological papers note that the complexity of natural objects and diversity of factors affecting them dictate the need to develop a complex indicator that will provide understanding of the relationship between natural objects and anthropogenic stressors. This requires a paradigm shift in the research—one which facilitates describing the ecological footprint of a water ecosystem as a result of the self-organization of factor loads, of both exogenous and endogenous origin, interacting with one other.

An extensive amount of work on the application of the theory of fractals in this direction is presented in these papers [4,5,6,7,8].

Successful solutions to various ecological problems should be determined based on the methods of comprehensive ecosystem analysis that facilitate undertaking a comprehensive process analysis at different levels of the ecosystem organization and determine ways to control these processes.

It is worth noting that today there is no clear methodology for both performing an integrated analysis and selecting sustainability indicators (anthropogenic transformation) [9].

The main reasons for this are [10]:

• The difficulties in assessing the ecosystem response to the combination of external anthropogenic factors of a different nature.
• The multifactored and diverse nature of the load on the ecosystem forming a scenario for its future development.
• The presence of natural mechanisms of self-organization in the ecosystem that partially or fully compensate for the negative effects of anthropogenic stressors.
• The time delay between the action of a stressor and the ecosystem response.

The complexity of ecosystems and variety of acting anthropogenic factors that can affect their functioning highlight the need to develop a comprehensive ecosystem state indicator that will provide a comprehensive understanding of their reaction to multiple anthropogenic stressors [11].

2. Materials and Methods

In view of the fact that the anthropogenic transformation of the ecosystem under the influence of external load leads to its structural and functional transformation, the main requirements for the selection of indicators should reflect the characteristics necessary for managing the exploitation of the ecosystems for various purposes (

Table 1. Requirements for the Selection of Indicators of Anthropogenic Transformation of Water Ecosystems [10].

Indicator Properties	Description
Concreteness	The indicators should reflect the main ecosystem features and have a clear and unambiguous interpretation of the calculated value.
Community	The indicators should be based on the standard parameters of the ecosystem time series that arise from the ecologic monitoring to identify the trends and direction of the anthropogenic processes. The indicators should allow the use of long-term observations and comparative analysis.
Cost-effectiveness	The cost of information provided by the indicator should be higher than the cost of its reception.
Availability	The indicators for the target audience should be simple and available.
Targeted Linkage to Management	The indicators should reflect the change in the quantitative characteristics of the ecosystem and have a targeted linkage to management and decision-making; in particular, identify the most dangerous components of pollution that accumulate in the ecosystem and contribute most to ecosystem degradation.
Sensitivity	The indicators should be sensitive to small perturbations in the ecosystem and predict catastrophic changes in its key characteristics.
MPEL Indication	The indicators should provide advance warning of achieving the MPEL at which environmental risks reach their maximum and the ecosystem loses its capacity for reproduction.
Reproducibility	The values of the indicators should be repeatable and reproducible for different sets of spatial-temporal, empirical data.

).

The questions of choice and formalization of sustainability indicators (anthropogenic transformation) are closely linked with the problem of assessing the nominal load on a ecosystem. According to experts, the nominal anthropogenic load should not be higher than that to which an ecosystem can adapt. Therefore, a permissible level of impact should be defined by comparison to a reaction that occurs in the ecosystem, with its natural variations (undisturbed state).

At the moment, the most common approaches in the sphere of regulation search for compromise between ecological and economical approaches; an approach which is consistent with the concept of sustainable development for ensuring the acceptable level of environmental risks regarding anthropogenic transformation of water ecosystems.

In this regard, it is suggested that the maximum permissible environmental load (MPEL) is used as an aggregated indicator of impact on water ecosystems. This indicator most fully reflects the opportunities of the ecosystem connected with the maximum load level that it sustains for a long time without any signs of violation of its structural and functional characteristics [11].

The main factors limiting the description of ecosystem morphology in its exploitation conditions (water use) and complicating the formalization of sustainability indicators (anthropogenic transformation) are:

• The nonlinearity of the ecosystem that causes the possibility of its self-organization and natural search for new states being in equilibrium with the external environment under the influence of changes in external factor loads (stressors).
• The existence of metastable states in the ecosystems where there may exist limit cycles of matter and energy exchange between the ecosystem and the external environment. It is confirmed that many ecosystems consume their resources and show functional resistance on the verge of chaos due to the balance of speeds of processes, but even an insignificant imbalance in parameters and speeds can lead to the emergence and development of catastrophic processes [4].

In theory, the above-mentioned problems arise from the problem of chaos-order ratio in complex systems as a mathematical assessment of the relationship between the components (ecosystem components) and the whole (maintaining the functional integrity).

The economic development of water resources is always accompanied by an anthropogenic transformation of the ecosystem. The character and volume of these transformations determine the ecological situation of the water ecosystem: the greater the anthropogenic transformation, the worse the controllability of economic activities; and vice versa, the insignificance of anthropogenic transformation points to a maximum efficiency of anthropogenic processes.

Environmental safety is largely determined by the effectiveness of applied technologies making it possible to keep the anthropogenic cycle closed even under the maximum load. But even the presence of advanced technology requires the creation of reliable environmental indicators ensuring control over the situation leading to degradation of the anthropogenic cycle; such indicators will let us know that the growth of pollution has begun, and the ecosystem cannot cope with it with its own resources. In particular, we are talking about indication of the maximum permissible environmental load (MPEL) under which environmental risks are maximized and the ecosystem loses its ability to recover.

The advantages of using a fractal as a mathematical tool for the description of water object ecosystems are due to the following fractal properties:

• Scale invariance (self-similarity) as a fractal property allows analyzing the ratio of hierarchically organized resource components to their functional whole in dynamics. In this case, self-organization is a mechanism that combines the constituent components into a single system whole.
• As a self-similar structure, the fractal makes it possible to quantify the nature of coevolution (proportionality) of two opposites—anthropogenic and natural components—throughout the studied time interval as a change of the ecosystem morphology. It is indicative that the ecosystem degeneration connected with the “environment-object” relation antagonism is directly related to the loss of its fractality at the integer values of D = 1 and D = 2 for which the coevolution of components is completely lost. Thus, violation of the coevolution is stronger, the higher the value of D, and it is accompanied by an increase in the difference between the current and optimal parameters of the ecosystem; for the optimum, the fractal indicator is D = 1.5 (Hurst constant H = 0.5) (Figure 1).

3.

Evaluation of the fractality level of the processes allows us to estimate a degree of the anthropogenic cycle closure provided by the self-organization of anthropogenic and natural components of the ecosystem; and the cycle closure means its sustainability in the external environment.

The fractal as a scale-invariant (self-similar) set best simulates the process of self-organization of anthropogenic processes and its violations related to the change of factor loads. In this case, the process fractality boundaries determine those critical states in which the self-organization is not supported, and the ecosystem cannot exist as a single whole [12].

In this case, the set of acting factors (both internal and external) can be partitioned into two subsets: the external factors of anthropogenic pollution in water ecosystems reinforcing the chaos, and the internal system-forming factors shaping the order. Here the ratio of these factors determines how pollution affects the anthropogenic transformation of the ecosystem whose maximum deviation from the optimum determines its viability.

To quantify the ecological safety of anthropogenic impacts, it is advisable to develop a classification of environmental risks according to degree of disturbance of the environmental properties of the ecosystem; and to introduce fractal indicators of early warning of the MPEL on the basis of a criterion for violation of the anthropogenic process coevolution. Violation of the anthropogenic process coevolution means that the value of the anthropogenic transformation of the ecosystem exceeds the ecological capacity of the environment [13].

The targeted linkage to exploitation management is provided as follows: the calculation of fractal values of the anthropogenic transformation of the ecosystem is performed on the basis of the order parameters, i.e., the limited number of variables determining the ecosystem self-organization [14].

• Violation of the anthropogenic process coevolution occurs when the anthropogenic transformation value reaches the limits of sustainability of the ecosystem. The limit proximity control should be performed on the basis of the indicator of early warning of the MPEL, the achievement of which shows that the economic exploitation of the ecosystem should be stopped [15].
• The ecosystem development management during economic exploitation is seen as a proactive management of potential risks arising in cases of violation of the anthropogenic processes. The control is reduced to artificial compensation of negative anthropogenic impacts, which increases the natural process sustainability and extends the environmental capacity of the ecosystem [16].
• Violation of the anthropogenic process coevolution is due, as a rule, to the excessive levels of anthropogenic exploitation of the water ecosystem. Reflecting its natural response to external influences, the ecosystem thereby changes its state either gradually or abruptly. In such a mode, the ecosystem state will exponentially tend to an state of unsustainability [17].

In the reports by the authors of the present article that verify the application of multifractal methods for the assessment of the sustainability of water objects, we can highlight the following:

• The fractal analysis of the Moscow River Basin in the region of the Strogino Floodplain [18].
• The fractal assessment of the effectiveness of algoremediation of water objects using green microalgae [16,18].
• The implementation of the Federal Target Program, “Development of Methods for Assessing a Recreational Potential of Water Objects and Regulation of Recreational Activities”.

To assess critical (limiting) water ecosystem development factors, we will introduce the continuous function of fractal “temperature” of the processes in the interval of D ∈ (1; 2) as follows:

$$T_f=\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(2-D\right)$}\right.-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.\right)$$

(1)

where D is a fractal indicator of the water object; n is a dimension of the space in which an object is embedded (in our case, n = 2).

The fractal “temperature” is linear depending on the environmental entropy; S_f is an indicator of irreversible energy dissipation by the ecosystem when interacting with the external environment.

$$S_f = \propto · T_f^D$$

(2)

where α is a correction coefficient defined by the correlation of acting factors; D is a fractal indicator of the ecosystem.

From Formulas (1) and (2) it follows that the minimum energy dissipation corresponds to a state of complete resource-saving balance with the environment; H = 0.5 is a fractal Hurst indicator serving as an indicator of conditional optimum—a state to which complex systems functioning in non-equilibrium conditions tend. Really, the values of H = 0.5: D = 1.5 are markers of random processes which have no correlation (that is, trend) to what is determined from the relation:

K = 2^2H − 1

(3)

where K is a measure of correlation of acting factors; H is a fractal Hurst indicator.

The appearance of the trend contributing to the deviation of the ecosystem from the optimum is determined by the interval of 0.5 < H < 1, and the higher the Hurst constant value, the more strongly the chaotic nature of the ecosystem is expressed. Therefore, H is interpreted as a ratio of the trend strength to the noise level (chaotic nature) in the ecosystem.

In other words, the more pronounced the external ecosystem determinism (economic exploitation intensity), the more chaotic its behavior is.

Moreover, on the basis of introduced ratios (1–3), there were introduced the fractal indicators D = 1.2, D = 1.7, which are the limiting ecosystem sustainability indicators, within the limits of which the ecosystem self-organization (adaptability to external anthropogenic disturbance) is provided [13] (

Table 2. Fractal Indicators of the Limiting Factors and their Features.

Deficit of factors for D = (1.2 ± 0.1) Particularity: Deterministic dynamics of the ecosystem, the behavior of which is regulated by uncompensated, one-way influencing factors with which the ecosystem does not cope on its own.

Ecological optimum for D = 1.5 ± 0.1 Particularity: Deterministic dynamics of the ecosystem, the behavior of which is regulated by uncompensated one-way influencing factors with which the ecosystem does not cope on its own.

Redundancy of factors for Dk = (1.7 ± 0.1) Particularity: Chaotic dynamics of the ecosystem accompanied by sharp jumps in its system characteristics. The ecosystem stagnation and the beginning of uncontrolled destruction under the influence of random factors.

).

When the fractal indicator of the water ecosystem deviates from the optimum for D = 1.5 in the direction of increasing the systemic complicity, its reproductivity logically decreases; and the ecological environment becomes more “viscous”, which is accompanied by the partial resource degradation and inhibition of the processes of exchange with the environment. For D = 1.7, the compensatory mechanisms of the water ecosystem are completely lost and it stagnates. Such a state is achieved when the economic exploitation level of the water ecosystem is equivalent to the possibilities of its natural production, which indicates the imbalance.

Based on the above, the fractal dynamics-based equation of anthropogenic processes can be represented by the following ratios:

$$D_e={{\displaystyle \sum }}_{j=1}^2{a}_{ij}{F}_j(D^{+};D^{-});1.2\le D_e\le 1.7$$

(4)

$${{\rm I}}_e=\left|1.5-D_e\right|$$

(5)

where D_e is a fractal indicator of the water ecosystem; I_e is a fractal sustainability index determined by the deviation of water ecosystem development from the optimum F_j (D⁺; D⁻) are fractal indicators of competing anthropogenic processes; a_ij are weighting coefficients of the acting factors.

Equations (4) and (5) describe the self-organization of anthropogenic processes within the ecosystem technology intensity as an alternation of competing phases of anthropogenic disturbances—the adaptation of the water ecosystem.

These equations can be interpreted physically as follows: the viability domain of the water ecosystem is defined by its deviation from the ecological optimum F_j(D = 1.5): the greater the deviation, the greater the specific factors oppress its viability. The maximum F_j(D = 1.7) and minimum F_j(D = 1.2) permissible values are those limiting values, beyond which the existence of the water ecosystem is no longer possible [14], (Figure 2).

It is clear that the existence of the water ecosystem is impossible if the limit cycle of reproduction is violated, which is associated with the inability to compensate for anthropogenic disturbances (

Table 3. Classification of Anthropogenic Processes by Sustainability Violation Risks.

Process Classes	Ecosystem Dynamics Type	Environmental Risk
Deterministic	Random	Low Re → 0
Self-organized	Fractal	Acceptable 0 < Re < 1
Bistable	Unsustainable	High Re → 1
Chaotic	Catastrophic	Extremely high Re = 1

).

The Maximum Permissible Environmental Load (MPEL) at which the limit stable cycle of the ecosystem is formed can be defined by the following ratio:

MPEL: [D_e → (1.2V1.7); I_e → 1]

(6)

As a result, the solutions obtained from Equations (4) and (5) identify a specific type of water ecosystem dynamics corresponding to the established metabolism. The most productive metabolism corresponds to the solution D = 1.5; (I_e; R_e) → 0, against which all environmentally friendly solutions are defined, right up to the sustainability limit, in which it stops self-sustaining [15].

Thus, the increase of water ecosystem complexity described by Equations (4) and (5) leads to its self-organization as restoration of disturbed equilibrium with the environment right up to the excessive complexity (MPEL) in which the ecosystem cannot independently cope with the external load. Since the functioning of real water ecosystems happens in states far from equilibrium, the control of the MPEL as a bistability indicator becomes paramount in the quality control structure for anthropogenically transformed objects.

The study was conducted using data from the analysis of hydrochemical indicators of pollution from the surface water basin of the Yenisei River from 2019–2021. The above indicators were provided by the Yenisei Basin Water Management Board, which is a part of the Federal Water Resources Agency of the Russian Federation.

The empirical pollution data were analyzed at Station 1 (6 km downstream from Krasnoyarsk), Station 2, located directly in the Arctic region (1.6 km downstream from Igarka), and Station 3, also located in the Arctic region (1.6 km downstream from Dudinka).

The aims of the study were to calculate the fractal indicators of water ecosystem sustainability in the stations under consideration; and, on the basis of the values obtained, to conduct zoning of the Yenisei River basins on the environmental water use risks directly influencing the quality of water resources.

Thus, for Station 1 (6 km downstream from Krasnoyarsk), the measurement of the fractality of surface water pollution time series is as follows (

Table 4. Fractal Indicators of the Yenisei River Ecosystem at Station 1 (6 km downstream from Krasnoyarsk) from 2019–2021.

Analyzed Indicators	Fractal Indicators D over the Period of 2019–2021
Suspended solids, mg/dm3	1.52
pH	1.41
Oxygen, mg/dm3	1.43
Chlorides, mg/dm3	1.48
Sulfates (SO4), mg/dm3	1.47
BOD5, mg/dm3	1.55
Nitrates (NO3), mg/dm3	1.49
Total iron, mg/dm3	1.42
Manganese, mg/dm3	1.39
Oil products, mg/dm3	1.49

). The fractal dimension was calculated based on the monthly data obtained at the water monitoring station located 6 km downstream the Yenisei River from Krasnoyarsk.

3. Results

The results obtained from the study allow assessment of the ecological state of a water body in conditions of its economic and recreational use as a result of self-organization of technogenical processes. A violation of self-organization indicates the degradation of hydrobionts and, therefore, the need for external management for their natural regeneration.

With regard to the sustainable water management, the methods and models presented open up new opportunities for specialists:

• Regulation of maximum permissible environmental loads (MPEL) on water objects at which hydrobionts lose the ability to self-repair. This allows setting the maximum permissible environmental loads on water objects at which they move towards a condition of instability.
• Identification of particularly dangerous anthropogenic impacts on the hydrobiont structure that contribute to its degradation and the diminishing quality of water resources. This opens up a possibility for management of potential environmental risks. A good example is growth and spread of toxic blue-green algae and algal blooms.
• Water quality management at the river basin level in conditions of economic and recreational use through the regulation of anthropogenic load levels. This is especially true for large water objects with different shapes and regimes of economic and recreational use.
• Control of the effectiveness of the environmental protection measures carried out to restore water objects destabilized by anthropogenic impacts using bioremeditation (algoremediation) technologies.

In order to convert the calculation data into the factor load coordinate system, we used the “Main component method” function of the SPSS program. This allowed us to carry out a morphological analysis of the water ecosystem by constructing the analyzed data projections in the binary system of variables; where F1 is a component of the positive correlations between the variables showing the ecosystem pollution growth by increasing the intensity of water use; and F2 is a component of the negative correlations between the variables, showing the opportunity to structure the ecosystem by reproducing biological resources.

Solutions (4, 5) directly describing the ecosystem self-organization process are reflected in the structure of the factor load matrix constructed using the SPSS program,

Table 5. Factor Load Matrix of the Yenisei River Ecosystem at Station 1 (6 km downstream from Krasnoyarsk).

Analyzed Indicators	Weighting Coefficients of Factor Loads
	F1	F2
Suspended solids, mg/dm3	0.327	−0.135
pH	−0.156	0.023
Oxygen, mg/dm3	−0.155	−0.135
Chlorides, mg/dm3	−0.029	0.004
Sulfates (SO4), mg/dm3	0.117	0.332
BOD5, mg/dm3	0.355	−0.073
Nitrates (NO3), mg/dm3	0.287	−0.160
Total iron, mg/dm3	0.333	0.001
Manganese, mg/dm3	−0.091	0.457
Oil products, mg/dm3	0.425	0.402

, and its projections in the binary phase space, Figure 2

Accordingly, the center of the phase diagram demonstrates the states of balance, the remoteness from which depresses equilibrium and maximizes the amount of environmental entropy. The peripheral areas of the phase diagram correspond to the states of anthropogenic imbalance of the ecosystem we have identified as reaching the MPEL [16] (Figure 3).

The result obtained in station 1 is a clear sign of the well-balanced dynamics of water use, and as a result, the high ecosystem sustainability in the 2019–2021 period. The impacts of potential environmental risk factors—among which the most influential are (with the maximum factor load values) oil products, organic, and suspended matter—are random and practically do not affect the restoration of ecosystem hydrobionts,

Table 6. Indicators of Sustainability of the Yenisei River Ecosystem at Station 1 (6 km downstream from Krasnoyarsk) as of 2021.

Potential Risk Factors	Fractal Indicator, De	Sustainability Index	Environmental Risk, Re
Suspended solids (0.32);	1.49	0.01	Low
Oil products (0.42)
BOD5 (0.35);

. This result is also correlated with the standard calculations of surface water pollution in 2020 [17].

Similarly, for Stations 2 and 3, located directly in the Arctic region, the following results are achieved,

Table 7. Indicators of Sustainability of the Yenisei River Ecosystem at Station 2 (1.6 km downstream from Igarka) as of 2021.

Potential Risk Factors	Fractal Indicator, De	Sustainability Index	Environmental Risk, Re
Suspended solids (0.55);	1.41	0.09	Acceptable
Oil products (0.32)
BOD5 (0.67);

and

Table 8. Indicators of Sustainability of the Yenisei River Ecosystem at Station 3 (1.6 km downstream from Dudinka) as of 2021.

Potential Risk Factors	Fractal Indicator, De	Sustainability Index	Environmental Risk, Re
Suspended solids (0.8);	1.4	0.1	Acceptable
Oil products (0.3)
BOD5 (0.5);

.

4. Discussion

At stations 2 and 3, the dynamics of analyzed emission shows that the overall environmental situation is close to a slightly violated self-sustaining mode in which the partial degradation of hydrobionts is not significant, and the ecosystem can recover on its own after the weakening of the load on oil products, organic and suspended matter. However, the sustainability index value of the technology intensity at Station 1 is significantly higher than at Stations 2 and 3. This may be due to the difference in water content of the water area under consideration and less expressed processes of water eutrophication when there is a change of zoning of the river water area. To improve the quality of water use, the reduction of the economic activity level directly near the river water area at Stations 2 and 3 may be recommended.

Thus, by comparing the results obtained we can conclude that it is possible to zone the Yenisei River basins according to environmental risks of water use, Figure 4.

5. Conclusions

In general, in all stations studied in the Yenisei River Basin (from 6 km downstream from Krasnoyarsk to 1.6 km downstream from Dudinka), water use is balanced with the ecosystem state, as indicated by the potential environmental risk factors, among which the most influential are oil products, organics, and suspended matter. Their impact is random at Station 1 and athropogenically deterministic, but controlled at Stations 2 and 3. Water governance in these stations consists in reducing the influence of the above potential risk factors, after which the ecosystem will cope with the partial degradation of hydrobionts and recover on its own. The technology intensities of Stations 2 and 3 have approximately the same sustainability indicator values; however, the sustainability indicator value of Station 1 differs sharply, which may be due to the difference in water content of the considered water area and less expressed processes of water eutrophication due to the change of zoning of the river water area.

The study provided quantitative parameters for the sustainability indicator directly related to the maximum permissible environmental load (MPEL) of the water ecosystem. Physically, the indicator reflects the ecosystem self-organization result—the competitive natural and anthropogenic components determining their coevolution.

It was found that violation of the coevolution of anthropogenic processes is directly related to the deviation of the water ecosystem development from the optimum to which the condition De → 1.5 corresponds. The preservation of the coevolution is provided under the condition that the water ecosystem fractality is maintained when the growth of environmental entropy does not change abruptly. This indicates the acceptable level of economic exploitation of the water object and the preservation of its qualitative characteristics.

The crisis situation of unsustainability, in which the coevolution of anthropogenic processes is violated, corresponds to the condition De → (1.2V1.7), where the water ecosystem turns from an open system into a closed one, loses its relationship with the external environment, and finally degrades.