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Hi, Can you help me out with the hydro dynamic part of the model? |
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Introduction
Water is a natural resource essential for all life on earth. Protecting water and the ecosystems that depend on it is therefore of utmost importance for human health and sustaining human activities. Widespread contamination of water resources, including groundwater and surface water, observed in recent years at national, African and global levels makes water quality situation worrying.
Water pollution [1] is nowadays a major global challenge that humanity is facing, adversely affecting not only human health but also environmental balance and economic development. It refers to the release of unwanted substances, such as chemicals or microorganisms, into water bodies, making them unsafe for human use and related activities and disrupting ecosystem functions [2]. Water pollution, whether surface water pollution [3] or groundwater pollution [4], is both natural and anthropogenic in origin and poses existential threats to human health and the environment [5]. However, it turns out that the most common sources of water pollution are anthropogenic, such as sewage discharge, agricultural runoff, urban runoff, and industrial activities. Water pollutants, such as pathogens, organic matter, nutrients, sediments, thermal substances and radioactive materials, originate from point sources or non-point (or dispersed) sources. Point source pollutants are dis- charged into water bodies from discrete identifiable conveyances, such as factory facilities, sewage treatment plants or storm drains. In contrast, non-point sources of pollution release a large amounts pollutants into water bodies from a broad unconfined area. They are more diffuse and primarily caused by runoff.
Contaminated water and poor sanitation are associated with the spread of several bacterial, viral and parasitic diseases such as cholera, diarrhoea, dysentery, hepatitis A, encephalitis, typhoid, polio, amoebiasis and giardiasis [6]. It also fuels malnutrition, and in particular, childhoot stunting [7]. In 2019, about 1.4 million premature deaths worldwide stemmed from water pollution [8], making it the second leading cause of premature death in the world after air pollution. It has been reported that approximately 829,000 people worldwide die every year from diseases directly attributable to unsafe drinking water, poor sanitation and hygiene practices [9]. This number includes about 300,000 children under five years of age, representing 5.3% of all deaths in this age group. Developing countries are the most affected by water pollution since up to 80% of illnesses in this part of the world are attributed to contaminated water and other water-related issues [10]. African continent, in particular, bears the brunt of the severe consequences of this kind of pollution, overwhelming healthcare infrastructure and hampering efforts to improve human capital and economic growth [11]. As a result, it has become essential to track down and effectively survey the slightest discharge into surface water and groundwater.
Over the decades, major decisions have been taken to address the health and environmental hazards caused by water pollution by drawing up and applying legislation and regulations [12] governing water use, the implementation of pollution control strategies and the development advanced technical tools for monitoring groundwater and surface water quality [13]. These techniques include traditional and modern methods, in order to detect pollutants and better identify their sources and to sup- port authorities in making decisions for preserving water resources. Conventional approaches [14], including analysis in laboratory, portable field instruments and tests and visual field inspections [15], are far too time-consuming and costly due to the high maintenance of laboratory facilities and the involvement of chemical materials, and are particularly inefficient for many practical applications. Modern methods, which include advanced and technologically innovative approaches such as re- mote sensing [16], earth observation satellite imagery [17], Cyber-Physical Systems (CPS) [18], Internet of Things (IoT) based smart sensors [19] and other sophisticated optical techniques [20], provide reliable, continuous and near-real-time data for accurate and high spatial and temporal resolution water quality monitoring. Despite the fact that these smart solutions are more cost-effective, reliable and easier to maintain, they are challenging to implement in developing countries [21], particularly in Sub-Saharan Africa, due to the paucity of financial resources to acquire them on the one hand, and the lack of technical capabilities to implement them effectively on the other. This clearly demonstrates that these approaches are not concretely viable for these countries and therefore, it is imperative to investigate more suitable alternative methods.
Relevant alternatives to these conventional approaches include mathematical modeling and numerical simulation [22]. They are fundamental and efficient tools widely used for understanding and predicting the dynamics of pollutants in aquatic environments [23] and evaluating their possible effects on the ecosystems. They deal with the transport and diffusion of pollutants in water bodies and address chemical processes. Rapid growth in computational power and recent advances in robust numerical methods [24] have contributed significantly to the remarkable success of these tools.
The Niger River [25] is the principal river in West Africa, with a main course length of about 4200 km, and the third-longest river in Africa after the Nile and the Congo River. It rises on the mountain range of Fouta Djalon (Upper Niger), a highland region of west-central Guinea with an average altitude of about 1100 m, and flows northeastward forming in central Mali, during the wet season, a vast floodplain (Inner Delta) which absorbs a large part of its hydraulic potential [26]. On the fringes of the Sahara, downstream of the Inner Delta, the Niger turns and flows due south-east (Middle Niger) into Nigeria. Then, the Niger continues its course (Lower Niger) to the Atlantic Ocean where it finally runs into the Gulf of Guinea after its confluence with the Benue River, its largest tributary. The Niger River Basin [27], the fourth largest in Africa after the Congo, Nile and Lake Chad, covers an area of about 2.2 million km2, including 1.5 million km2 of an active hydrological basin, is shared by nine West and Central African countries: Benin, Burkina Faso, Cameroon, Chad, Côte d’Ivoire, Guinea, Mali, Niger and Nigeria. This portion of land drained by the Niger is an extraordinary asset for the nine countries that share it, especially for the three riparian countries, which account for more than 80% of its area: Mali, Niger and Nigeria. Indeed, the Niger Basin abounds with immense development potential [28], covering irrigation for food production, including agricultural, fisheries, pastoral, rice and market gardening activities, hydroelectric power generation, transport, industrial and mining development and other related activities such as trade, investment in communications and increased mobility of people and goods [29]. The resources of this basin are therefore vital for sustainable and integrated regional development, and in particular for the riparian population, estimated at over 130 million people [30] with an average annual growth rate of around 3%. However, despite these important socio-economic opportunities in the Niger Basin, it is seriously threatened by major phenomena such the continuing decline in its flows, unsustainable use of its resources, desertification, silting of the riverbed and upstream pollution due to various activities such as industry, mining, agriculture and urban enterprises, resulting from the combined effects of decades of drought, climate change and strong demographic growth [27]. The assaults of these various anthropogenic polluting activities [31] on the Niger Basin over many years not only weaken ecosystems and biodiversity [32], but also undermine its capacity to meet the expectations of the riparian population, expand their impoverishment and seriously imperil their health [33].
In Mali, the Niger River [27] follows a curve of 1700 km and flows through several major cities of the country, including Bamako, Koulikoro, Ségou, Mopti, Timbuktu and Gao. It is one of the few permanent sources of water in the heart of the arid and semi-arid Sahel [34] and is the main source of drinking water [35] for the city of Bamako, the political and economic capital of the country. The portion of the Niger Basin in Mali, the largest share, makes up around 30% of the total basin area [27] and is home approximately 75% of the malian population [36], many of whom live in Bamako on the banks of the Niger. It represents a vital artery for Mali, making an invaluable contribution to the socio-economic development since a large majority of the population depends directly or indirectly on its resources. Despite its immense potential for sustainable development of Mali, the Niger River is severely strained along its course through the country, particularly in the major urban centers, including Bamako, by several destructive factors such as overuse of river banks and shallows and pollution generated by urban wastewater, which is highly dense in microplastic and includes domestic wastewater, runoff, water produced by the industrial units and water from certain informal economic activities such as gold panning, artisanal dyeing and sand and gravel extraction [37]. As the self-purification capacity [38] of Niger River has been severely impaired by the nature, and extent of the pollution, it is essential to identify, track down and effectively monitor the multiple sources of pollutant discharge into this river, in order to gain in-depth knowledge of their dynamics and thus support and accommodate the authorities decision-making process for preserving the river and its resources [39].
For this purpose, we propose a unified framework for the mathematical modeling and numerical simulation of the pollution of the Niger River in Bamako in order to contribute to a better understanding of the dispersion pattern of water pollutants. In terms of specific contribution to the current literature, this paper provides the scientific community interested in water pollution issues with a non-trivial application based on new robust algorithms and advanced numerical tools. The model we are interested in consists of a coupling of the shallow-water equations, which govern unsteady water flows in rivers, with an advection-diffusion equation for pollutant transport in water flows.
Mathematical model
Mathematical modeling and computer simulation of pollutant transport in natural water bodies, including rivers, streams, lakes, floods, dams and coastal areas, are fundamental and important tools for the prediction and risk assessment of water pollution involving in a wide variety of applications in environmental science, hydraulic engineering and industry. These powerful numerical tools provide an accurate and reliable estimation of spatial and temporal distribution of pollutant, essential for an effective prediction of water pollution and the establishment of better control strategies and risk management. The transport of pollutants in natural water systems is a complex process [40] that depends on both the flow characteristics of the fluid and the physical and chemical properties of the pollutants. Thus, this process can be described by a mathematical model with two different but dependent variables, namely a hydrodynamic variable representing the fluid flow dynamics and a scalar transport variable defining the pollutant concentration.
In the current work, the water flows in Niger river is governed by the two-dimensional shallow- water equations [41], often known as Saint-Venant equations. This hydrodynamic model, fairly well known in the literature, is derived from the integration of the three-dimensional incompressible Navier–Stokes equations over the fluid depth under the assumptions of hydrostatic pressure distribu- tion and the uniform velocity distribution along the vertical direction. This system of equations [42] is a suited model widely used for describing the flows in environments where the fluid depth is much smaller than the horizontal scale of motion, which is absolutely the case of the context of this study. On the other hand, for modeling the dynamics of pollutants in water flows, we consider a two-dimensional advection-diffusion equation [43]. Thus, the global model we consider for fully de- scribing the transport of pollutants by shallow-water flows, regarding concurrently flow and transport phenomena, results in the coupling [44] of the shallow-water equations with the advection-diffusion equation. The resulting system of partial differential equations (PDEs) is written in the so-called conservative form as follows:
where
The first equation of the system above represents the mass conservation of the water. It is also referred as continuity equation. The second equation refers to the momentum conservation, while the third equation represents the transport of pollutants by the water flows. The pollutant is supposed to be passive (non-reactive) and we assume that it does not induce any feedback to the water flow.
The bottom shear stress$\boldsymbol{\tau}_b$ is computed using the Manning law as follows:
where$\rho_w$ is the water density, $n_b$ is the Manning roughness coefficient and $\left\Vert\mathrm{\mathbf{q}}\right\Vert$ is the Euclidean norm of the flow discharge. The surface stress $\boldsymbol{\tau}_s$ is expressed as a quadratic function of the wind velocity $\mathrm{\mathbf{w}}$ as follows:
where$c_s$ is the friction coefficient of the wind. It is usually defined by:
where$\rho_a$ is the air density. The additional body forces $\mathrm{\mathbf{f}}$ include atmospheric$f_c$ is defined by:
pressure gradient and tidal potential forces. They will not be considered in this work.
The Coriolis parameter
where$\omega=7.2921\mbox{E}-5~\mbox{rads}^{-1}$ is the angular speed of Earth rotation and $\varphi$ is the geographic latitude of the site.
The diffusion tensor$\mathrm{\mathbf{D}}$ is a $2\times 2$ matrix defined by:
where$D_{xx}$ , $D_{xy}$ , $D_{yx}$ and $D_{yy}$ correspond to the diffusion coefficients of the pollutant in each combination of $x$ and $y$ directions. The diagonal elements $D_{xx}$ and $D_{yy}$ represent the diffusion coefficients along the principal directions and the off-diagonal terms $D_{xy}$ and $D_{yx}$ reflect the correlation between diffusion coefficients along the respective pair of principal directions. In general, the components of the diffusion tensor $\mathrm{\mathbf{D}}$ , accounting for molecular diffusion and turbulent mixing, depend on the flow characteristics and the nature of pollutant considered, including water depth, flow velocity, bottom roughness, wind and vertical turbulence. Since turbulent mixing is the dominant component of dispersion in river, it is feasible to neglect molecular diffusion, which is small scale process that occurs much more slowly than turbulent mixing. Thus, in longitudinal (or streamwise) and transverse reference system, the diffusion tensor can be expressed in the form of a diagonal matrix as follows:
where$D_L$ and $D_T$ are longitudinal (along the flow) and transverse (perpendicular to the flow) dispersion coefficients, respectively. These coefficients are estimated by:
where$u^{\ast}$ represents the shear velocity and $\alpha_L$ and $\alpha_T$ are dimensionless constants quantifying the magnitude of longitudinal and transverse dispersion, respectively. Typical values of these constants for river flows are $\alpha_L=5.93$ and $\alpha_T=0.15$ .
In order to compute the components of the diffusion tensor$\mathrm{\mathbf{D}}$ in the Cartesian coordinate system, the following expressions hold:
where$\theta=\arctan(q_y/q_x)$ represents the angle between the flow direction and the x-axis.
The fields and physical parameters of the model are listed in Table below:
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