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Article

A Study on the Coexistence of Anthropogenic and Natural Sources in a Three-Dimensional Aquifer

by
Animesh Samanta
1,
Ayan Chatterjee
2,* and
Mritunjay Kumar Singh
1
1
Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India
2
School of Engineering and Allied Sciences, The Neotia University, Diamond Harbour 743368, India
*
Author to whom correspondence should be addressed.
Water 2024, 16(1), 177; https://doi.org/10.3390/w16010177
Submission received: 26 November 2023 / Revised: 26 December 2023 / Accepted: 31 December 2023 / Published: 3 January 2024
(This article belongs to the Section Soil and Water)
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Abstract

:
A study using mathematical modeling has been conducted to analyze how both man-made and natural sources of contaminants affect various layers of an aquifer-aquitard system. The xy-, yz-, and zx-plane have been used to depict the locations where the natural sources of contaminant occur on the xz- and yz-plane, and where the man-made sources occur, on the xy-plane. It is assumed that the sources occurring in different planes are constant, while the velocity of groundwater flow has been considered only along the x-axis. A three-dimensional advection dispersion equation (ADE) has been used to accurately model the flow of groundwater and contaminants through a porous medium. Three distinct sources exert their influence on three separate planes throughout the entire duration of this study, thus making it possible to model these sources using initial conditions. This study presents a profile of contaminant concentration in space and time when constant sources are located on different planes. Some physical assumptions have been considered to make the model relatable to real-world phenomena. Often, finding stability conditions for numerical solutions becomes difficult, so an unconditionally stable solution is more appreciable. The homotopy analysis method (HAM), a method known for its unconditional stability, has been used to solve a three-dimensional mathematical model (ADE) along with its initial conditions. Man-made sources show more impact than equal-strength natural sources in the aquifer-aquitard system.

1. Introduction

Freshwater is one of the most essential requirements of mankind as well as living organisms. Groundwater is one of the main sources of fresh water. Commonly, groundwater is used for drinking, household work, agriculture, and industrial purposes. The gradual decline of the water table is primarily caused by the surging demand for crops and industry-based resources, driven by a growing population. As a result, groundwater is constantly being depleted on a large scale. Alongside the declining water table, groundwater contamination stands out as a significant global issue. In 25 countries around the world, recent studies show that 80% of the population is at risk of dying from infectious diseases caused by contaminated groundwater [1,2]. Groundwater contamination may happen in different scenarios, but the sources of groundwater contamination are either point sources or non-point sources. In this study, sources of groundwater contamination are classified in two categories: anthropogenic and natural sources. Anthropogenic activities that affect the groundwater quality are agricultural practices, urbanization, and industrial applications. However, other natural influences impacting groundwater quality include climate change, natural disasters, geological factors, etc. [2]. Groundwater contamination from natural sources in rural and urban areas was found to be quite similar, but groundwater contamination from anthropogenic sources in rural and urban areas was found to be different [3]. Droughts, floods, water-rock interactions, saltwater intrusion, leaching organic matter, evapotranspiration, and atmospheric deposition are the most important natural causes of groundwater contamination, although pharmaceuticals, agricultural activities, industrial waste, mining activities, over-pumping, and urban development are the major anthropogenic causes of groundwater contamination [2,3,4,5]. Industrial effluents (such as pesticides, fertilizers, hydrocarbons, phenols, plasticizers, detergents, oils, etc.) are the major man-made sources of groundwater contamination. The contamination of groundwater caused by human and geological activities has a detrimental effect on human health [6,7,8,9]. As such, it is very important to control groundwater contamination that occurs due to anthropogenic as well as geogenic sources.
Many researchers explore groundwater contamination that is due to anthropogenic as well as geogenic sources. These studies are given in tabular form below.
Authors and yearNature of source
Kreitler and Browning [10], Colten [11], Pimentel et al. [12], Datta et al. [13], Chakraborti et al. [14], Gilliom et al. [15], Gilliom [16], Kaplay and Patode [17], Mondal et al. [18], Ghanem et al. [19], Ewea [20], Lamers et al. [21], Al-Wabel et al. [22], Sasakova et al. [23], Rehman et al. [24], Li et al. [25], Li et al. [26]Anthropogenic source
Ullah et al. [27], Payus et al. [28], Huang et al. [29]Natural sources
Singh et al. [30], Devic et al. [31], Coyte et al. [32], Karunanidhi et al. [33], Perraki et al. [34], Shukla and Saxena [35], Makaya and Maphosa [36]Both anthropogenic and natural sources (No mathematical modelling)
The majority of sources of groundwater contamination are non-point sources in nature, e.g., line and plane sources. Many researchers have considered groundwater contamination using point sources as a simplifying assumption while solving one-, two-, and three-dimensional problems. They used different techniques/approaches to formulate the problem mathematically. Statistical and mass balance modeling were mostly used to analyze and predict the contaminant concentration in an aquifer. Yeh and Yeh [37] utilized a one-dimensional ADE equation to scrutinize the movement of solute caused by a point source and boundary source. In their study, Guo et al. [38] employed a comprehensive two-dimensional mathematical model to illustrate the mechanism of solute transport resulting from point source mixing on the free water surface, while also taking into account bed adsorption within a channel. Barilari et al. [39] developed a hazard index to find the most effective point source and control the point source.
Most of the literature reviews have indicated that anthropogenic and natural sources have been treated as separate entities, despite the fact that they both operate concurrently in an aquifer system. The main goal of the paper is to develop a model that incorporates both anthropogenic and natural sources using only initial conditions, representing a novel approach in groundwater hydrology modeling. The authors of this recent work have developed a sophisticated mathematical model to accurately capture the complex interplay between anthropogenic and natural sources within a system. Human activities introduce anthropogenic sources into the environment. Most of the anthropogenic sources consists inorganic materials and it is harmful for human health. These sources of contamination primarily come from the upper plane of the aquifer. In contrast, natural sources of contamination can occur at various depths within the aquifer. To understand the impact of different types of sources in an aquifer, both types of sources have been considered with the same strength. Groundwater transport problems in an aquifer due to point sources were modeled using one-, two-, and three-dimensional ADE. Additionally, we considered a three-dimensional homogenous aquifer for our problem. As such, a three-dimensional advection-dispersion equation has been used to model the system mathematically. All sources are simultaneously active, and the contaminant’s velocity is evaluated along the positive x-axis. This elevates the problem and mathematical modeling to a higher level of nobility. In most boundary value problems, the Heaviside function is commonly used to include the boundary values. However, in this study, the authors have developed a new function that seamlessly incorporates more than one initial condition. It is, again, a novel approach for solving the modeled problem with more than one initial condition using HAM. The homotopy analysis method (HAM) has been used to solve the problem analytically. The impact of different sources corresponding to space and time has been studied to find out the most impactful source. We have examined the homogeneous aquifer as a solution to this problem, but it may also be applicable to a heterogeneous porous medium through incorporating time- and space-dependent dispersion and velocity.

2. Mathematical Model and Solution Methodology

Mathematical modeling of physical real-life problems often requires some simplified assumptions. While preserving the essential nature of the issue, we have made simplified assumptions that enable our model to be both applicable to real-world problems and easily comprehensible. The following assumptions have been taken into consideration to describe our model.

2.1. Assumptions

Domain of the model: The aquifer is vertically semi-infinite in the z-direction and also horizontally semi-infinite in both the x- and y-direction.
Flow direction: Flow is unidirectional, where fluid flow along the x-direction has been considered.
Source nature: Two types of sources have been considered; namely, anthropogenic and geogenic sources. The source of constant strength originates at the surface of the different planes of the aquifer for geogenic sources, but for anthropogenic sources, contaminant enters into the system through the upper plane (Figure 1). The source’s strength is at its peak at the exact point of occurrence and gradually diminishes as the distance from that point increases.
Initial and boundary conditions: At any point in time, the aquifer is contaminated by one or more than one contaminant. Contaminants from various sources at different levels affect different planes. This is an initial condition for the sources. The governing equation directly incorporates another source. Thus, initially, contaminant concentration is considered throughout the planes near the origin and along the axes of the aquifer.

2.2. Governing Equation

The advection-dispersion equation provides a model for studying the transport of contaminants through porous media in various dimensions [40,41,42,43]. A three-dimensional advection-dispersion equation with a source or production term is used to model the problem mathematically [44,45,46].
𝜕 C 𝜕 t = 𝜕 𝜕 x ( D x 𝜕 C 𝜕 x − U x C ) + 𝜕 𝜕 y ( D y 𝜕 C 𝜕 y − U y C ) + 𝜕 𝜕 z ( D z 𝜕 C 𝜕 z − U z C ) + λ F C i
with the initial conditions given below:
C ( x , y , 0 , 0 ) = f 1 ( x , y ) x ≥ 0 , y ≥ 0 , z = 0 , t = 0
C ( 0 , y , z , 0 ) = f 2 ( y , z ) y ≥ 0 , z ≥ 0 , x = 0 , t = 0
C ( x , 0 , z , 0 ) = f 3 ( x , z ) x ≥ 0 , z ≥ 0 , y = 0 , t = 0
where C ( x , y , z , t ) [ M L − 3 ] is the solute concentration at the point ( x , y , z ) at time t ; D x , D y , and D z are the components of dispersion coefficient along the co-ordinate x-,y-, and z-axis, respectively; U x , U y , and U z  are the components of pore water velocity along the co-ordinate x-,y-, and z-axis, respectively; λ [ T − 1 ] is the first order growth or decay term; and F is a function that represents the source nature as well as the plane of the source lies; F is considered in such a way that its source strength is highest at the point of occurrence, gradually diminishing as the distance from that point increases. C i [ M L − 3 ] is constant source strength.

2.3. Solution

HAM is the most commonly used method for solving initial and boundary value problems. In this present study, we solved this problem using HAM [42,47,48]. To our knowledge, no other works have tried to solve a differential equation with multiple initial conditions. The study focuses on a new trial solution approach for solving partial differential equations. This approach is specifically designed to handle equations with multiple initial conditions.

2.3.1. CASE 1: Source Lies on the x y -Plane

When the source lies on the x y -plane at a point ( x i , y i , 0 ) , F should be a function of x and y . We calculate F in such a way that validates our assumption. We consider F as
F ( x , y ) = 1 1 + ( x − x i ) 2 + ( y − y i ) 2
Hence, it becomes evident that the function F reaches its peak at point ( x i , y i , 0 ) and its value gradually declines as we move further away from this point.
We chose the initial approximation with the help of the initial conditions.
C 0 ( x , y , z , t ) = e x p { − ( p x y + q y z + r x z ) } e x p ( − m t ) = f ( x , y , z ) e x p ( − m t )
where p , q , r , and m are constant.
Using the HAM method, we can obtain a higher-order approximation by solving the governing equation with an initial approximation.
The parameter h is known as the control parameter for convergence.
A finite number of approximations are necessary to achieve an accurate result. After conducting our calculations, we observed that the solution converges after the third approximation. We have concluded that it is satisfactory to take into account only the third approximation, as there is no need for any additional approximations in our solution.
Finally, we get the solution for the x y -plane, given below.
C ( x , y , z , t ) = C 0 ( x , y , z , t ) + C 1 ( x , y , z , t ) + C 2 ( x , y , z , t ) + C 3 ( x , y , z , t )
where C 0 ( x , y , z , t ) , C 1 ( x , y , z , t ) , C 2 ( x , y , z , t ) , and C 3 ( x , y , z , t ) are given using the Equations (6), (A1), (A2), and (A3), respectively.

2.3.2. CASE 2: Source lies on the y z -Plane

Similarly, when the source lies on the y z -plane at a point ( 0 , y j , z j ) , F should be a function of y and z . We calculate F in such a way that validate our assumption. We consider F as
F ( y , z ) = 1 1 + ( y − y j ) 2 + ( z − z j ) 2
Therefore, it is clear that the functional value of F reaches its maximum at the point ( 0 , y j , z j ) and the value decreases as the distance increases from the point ( 0 , y j , z j ) .
Similarly, we get all approximations for the y z -plane
There is no need for any additional approximations in our solution. Finally, we get the solution for the y z -plane considering up to third-order approximation.
C ( x , y , z , t ) = C 0 ( x , y , z , t ) + C 1 ( x , y , z , t ) + C 2 ( x , y , z , t ) + C 3 ( x , y , z , t )
where C 0 ( x , y , z , t ) , C 1 ( x , y , z , t ) , C 2 ( x , y , z , t ) , and C 3 ( x , y , z , t ) are given using the Equations (A4), (A5), (A6), and (A7) in Appendix A, respectively.

2.3.3. CASE 3: Source Lies on the x z -Plane

Since the flow is along the x direction, the solution for the x z -plane is similar to the solution of the x y -plane, so we can easily obtain the solution for the x z -plane by replacing z in the place of y in the solution obtained for the x y -plane. The function F will be in the form
F ( x , z ) = 1 1 + ( x − x k ) 2 + ( z − z k ) 2
The solution we are considering is when the source lies on the xy-plane (upper plane of the aquifer), which is the solution for anthropogenic sources. On the other hand, the solutions where the source lies on the xz- and yz-plane are considered as solutions for natural sources in order to proceed further.

3. Results and Discussion

Different scenarios for different conditions have been predicted when anthropogenic and natural sources occur in different planes. For homogeneous aquifers, the dispersion and velocity parameters are considered as D x = 0.07 , D y = 0.07 , D z = 0.07 , and U x = 0.03  [30]. Other parameters are considered as λ = 1 , p = 1 , q = 1 , r = 1 , and m = 1 . The figures are depicted using MATLAB software (R2013a).
The convergence control parameter is an essential parameter to find for any semi-analytical solution. The convergence of the solution using HAM is entirely dependent on the convergence control parameter h . Therefore, we must first analyze the convergence of the present model’s solution. From this graph (Figure 2), we can predict the range of h where the solution will become stable. The contaminant concentration vs. h curve has been plotted via keeping concentration fixed at point  ( 1.5 , 1.5 , 1.5 )  and fixed time as t = 1 . By analyzing Figure 2, we can identify a specific position where the concentration aligns parallel to the h-axis. The values of h at this position can be utilized as the convergence control parameter.
The contaminant concentration is parallel to the h-axis within the range of ( − 1.2 , − 0.8 ) , ensuring the solution achieves rapid convergence. For convergence, we have the flexibility to select any values within this region. However, we will focus on a value of the convergence control parameter, h = − 1 , for our subsequent proceedings. This particular value lies conveniently in the middle of the domain and will significantly reduce the calculation required.
A profile was plotted to represent the distribution of contaminant concentration against the x and y axes. This profile depicts the impact of an anthropogenic source located at point ( 1 , 1 , 0 ) on the x y plane. The measurements were taken at a fixed distance of z = 1 , a fixed time of t = 3 , and fixed source strength of C i = 2 (Figure 3). The concentration of contaminants initially starts at 14 units and then gradually increases along the y -axis. However, it remains relatively unchanged along the x -axis. As we move along the diagonal, the concentration gradually decreases, approaching zero asymptotically.
Figure 4 displays the distribution profile of the concentration of contaminants against the y - and z -axis. This profile is illustrated when a natural source is positioned on the y z -plane at a specific point, ( 0 , 1 , 1 ) , with a constant value of x = 1 , fixed time of t = 3 , and fixed source strength of C i = 2 . The concentration of contaminants starts at 22 units and gradually increases along the y-axis, while remaining relatively stable along the z-axis. Similarly, as we move along the diagonal, the concentration gradually decreases, approaching zero asymptotically.
Figure 5 shows the distribution profile of contaminant concentration against the x- and y-axis. The illustration showcases the influence of a human-induced source positioned at a particular point, ( 1 , 1 , 0 ) , on the plane. It maintains a consistent value of z = 1.5 , a fixed source intensity of C i = 2 , and different time values of t. The concentration profiles begin at distinct values of 10 units, 24 units, and 40 units for specific time intervals of 2.5 units, 3 units, and 3.5 units, respectively (Figure 5). However, as the distance increases, these concentrations gradually decrease and approach zero asymptotically. It is clear from the figure (Figure 5) that contaminant concentration increases with time. If the anthropogenic source of contamination is continuously active, then the concentration of contaminants within the aquifer will steadily increase over time at a specific location.
The graph in Figure 6 shows the distribution of contaminant concentration along the y- and z-axis. This is for a fixed value of x = 1.5 and fixed source strength of C i = 2 , while the natural source is located at point ( 0 , 1 , 1 ) on the plane. The concentration profiles begin at distinct values of 18 units, 50 units, and 95 units for specific time intervals of 2.5 units, 3 units, and 3.5 units, respectively (Figure 6). However, as the distance increases, these concentrations gradually decrease and approach zero asymptotically. It is also clear from the figure (Figure 6) that contaminant concentration increases with time. As such, it is obvious that if the natural source of contamination is continuously active, then the concentration of contaminants within the aquifer will steadily increase over time at a specific location.
We can observe in Figure 7 the comparison of the concentration distribution of contaminants along the x-axis. This comparison is made between an anthropogenic source located on the x y -plane and a natural source situated on the y z -plane. For the x y -plane, the anthropogenic source occurs at the point ( 1 , 1 , 0 ) and for the y z -plane the natural source occurs at the point ( 0 , 1 , 1 ) .The concentration of contaminants, which initially stands at 4.35 units, reaches its maximum value at x = 1 due to the occurrence of the source at ( 1 , 1 , 0 ) , where the anthropogenic source is located on the x y -plane. When the natural source is located at point ( 0 , 1 , 1 ) on the y z -plane, the concentration of contaminants initially at 4.8 units, and then decreases as the value of x increases.
In Figure 8, we compare the contaminant concentration distribution against the y -axis when an anthropogenic sources lies on the x y -plane and a natural source lies on the y z -plane, wherein the anthropogenic source occurs at the point ( 1 , 1 , 0 ) and the natural source which occurs at the point ( 0 , 1 , 1 ) . Figure 8 illustrates two comparable distributions of contaminant concentrations for both the anthropogenic and natural sources. Contaminants from natural sources start with lower concentrations than those from human activities, but they gradually increase and eventually surpass them. However, after reaching a maximum at y = 1, they start to decline and ultimately finish below the contaminant concentration of the anthropogenic source at y = 3.
If we compare Figure 7 and Figure 8, the impact of the velocity is quite clear from the figures. The velocity of the groundwater has been considered along the x-axis. The velocity in the x-axis direction has a significant impact on dispersion. As a result, the initial decrease in water contamination is followed by a gradual increase as we continue to measure, resulting in a distinct pattern. However, as we move even farther, the concentration of contaminants decreases once more.
Figure 9 and Figure 10 are plotted to show the contaminant concentration distribution against time for an anthropogenic source and natural source, respectively. In Figure 9 and Figure 10, the contaminant concentration profiles follow the exact same pattern for three distinct diagonal points as time progresses. In all scenarios, the concentration of the contaminant consistently and steadily increases linearly over time.
It is evident from the comprehensive discussion of the various graphs that both anthropogenic and natural sources significantly contribute to groundwater contamination. It is an indisputable fact that anthropogenic sources solely result from human activity. Therefore, in order to conform to this well-established knowledge, any human-made source within the system should be situated on the xy- plane.
In this present study, anthropogenic sources show more impact, as the source lies on the xy-plane and the velocity of the groundwater is along the x-axis. As such, the major part of the contaminant flows in the direction of the velocity, which is quite a natural phenomenon. Anthropogenic and natural sources show similar results concerning time for a fixed point. The contaminant concentration increases with time slowly but steadily, following a linear pattern.

4. Enhancing Model Application across Various Anthropogenic and Geogenic Scenarios

This present model may be useful to model different anthropogenic and geogenic scenarios. In this present section, we discuss some other real-life applications of the present model.

4.1. Impact on Air Quality

Air pollution is caused by both human activities and natural events. Volcanic eruptions are a natural cause of air pollution, while emissions from chimneys are examples of pollution caused by human activities. The origins of pollution can be found across various areas, which our current model can effectively address by considering different velocities and dispersion patterns.

4.2. Implications of Surface Water Pollution

Air flow plays a major role in contaminating the surface water bodies. Airborne substances can have a significant impact on surface water bodies, directly affecting the upper layers. These substances, known as geogenic sources of contamination, are carried through the air and can cause a variety of effects, whereas human activities again contaminate the water body from different planes and behave like anthropogenic sources. This situation can also be modelled using the present mathematical modelling.
Two direct applications of the anthropogenic and geogenic sources are described here. We can think of various real-life situations where contamination may occur due to both activities. For example, we can consider the sea and lakes, but our current model may not be able to accurately simulate them because of turbulence. In that case, we would need to couple Navier-Stokes and continuity equations with advection dispersion equations.

5. Conclusions

From the present study, the authors can conclude the following:
  • Anthropogenic and natural sources act at the same time in three different planes. The impact of contaminant concentration varies depending on the source location, and it is clear that the overall impact is significant, as shown in this study. However, high contaminant concentration levels decrease and tend to zero after a certain distance.
  • Contaminant concentration is increased with respect to time for a fixed point in the aquifer. The linear increment at the contaminant concentration profile happens due to three different sources acting at the same time.
  • The impact of contaminant concentration, resulting from both anthropogenic and natural sources, acting across different planes, is bound to be more significant than that caused by a single source alone. Depending upon the number of sources, the contaminant concentration level increases.
  • Human activities have a greater effect than natural sources, which is influenced by where the source is located and how fast it is moving along the x-axis. In contrast, natural sources show the same impact with respect to space and time.
  • The impact of the h-value can be understood from the h-curve. The h-value plays a crucial role in solving the current problem, making it imperative to consider the h-value in order to achieve a reliable solution.
  • Velocity is considered in the x directions, which creates a stable decrease in the contaminant concentration profile with respect to space, although the strength of the source is high.
  • A new function has been developed by the authors, providing seamless integration of multiple initial conditions. This allows for the effective resolution of problems solely based on initial conditions using HAM. The ADE incorporates a function that displays the strength of the source. Present work may be extended to sources with different types of strength.

Author Contributions

Conceptualization, A.S. and A.C.; methodology, A.S. and A.C.; software, A.S. and A.C.; formal analysis, A.S. and A.C.; writing—original draft preparation A.S.; writing—review and editing, A.C. and M.K.S.; visualization, A.S. and A.C.; supervision, M.K.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

  • Case-1: Source lies on the xy-plane
The values of C 1 ,   C 2   a n d   C 3
C 1 ( x , y , z , t ) = h m R ( x , y , z ) f ( x , y , z ) ( e x p ( − m t ) − 1 ) − h λ F ( x , y ) C i t
C 2 ( x , y , z , t ) = C 1 ( x , y , z , t ) + h 2 m R ( x , y , z ) f ( x , y , z ) ( e x p ( − m t ) − 1 ) − h 2 λ F ( x , y ) C i t + h 2 m T ( x , y , z ) f ( x , y , z ) { t + 1 m ( e x p ( − m t ) − 1 ) }
+ [ D x F x ″ ( x , y ) + D y F y ″ ( x , y ) ] h 2 λ C i t 2 2 − U x F x ′ ( x , y ) h 2 λ C i t 2 2 C 3 ( x , y , z , t ) = C 2 ( x , y , z , t ) + h 2 m R ( x , y , z ) f ( x , y , z ) ( e x p ( − m t ) − 1 ) − h 2 λ F ( x , y ) C i t + h 3 m R ( x , y , z ) f ( x , y , z ) ( e x p ( − m t ) − 1 ) − h 3 λ F ( x , y ) C i t + h 3 m T ( x , y , z ) f ( x , y , z ) { t + 1 m ( e x p ( − m t ) − 1 ) } + [ D x F x ″ ( x , y ) + D y F y ″ ( x , y ) ] h 3 λ C i t 2 2 − U x F x ′ ( x , y ) h 3 λ C i t 2 2 − D x [ h 2 m S 1 ( x , y , z ) f ( x , y , z ) { 1 m ( 1 − e x p ( − m t ) ) − t } ( 1 + h ) − h 2 λ F x ″ ( x , y ) C i t 2 2 ( 1 + h ) − U x h 3 λ F x ‴ ( x , y ) C i t 3 6 + h 3 m W 1 ( x , y , z ) f ( x , y , z ) { t 2 2 + 1 m ( 1 m ( e x p ( − m t ) − 1 ) + t ) } + [ D x F x i v ( x , y ) + D y ( F y ″ ) x ″ ( x , y ) ] h 3 λ C i t 3 6 ] − D y [ h 2 m S 2 ( x , y , z ) f ( x , y , z ) { 1 m ( 1 − e x p ( − m t ) ) − t } ( 1 + h ) − h 2 λ F x ″ ( x , y ) C i t 2 2 ( 1 + h ) − U x h 3 λ ( F x ′ ) y ″ x , y ) C i t 3 6 + h 3 m W 2 ( x , y , z ) f ( x , y , z ) { t 2 2 + 1 m ( 1 m ( e x p ( − m t ) − 1 ) + t ) } + [ D x F x i v ( F x ″ ) y ″ ( x , y ) + D y F y i v ( x , y ) ] h 3 λ C i t 3 6 ] − D z [ h 2 m S 3 ( x , y , z ) f ( x , y , z ) { 1 m ( 1 − e x p ( − m t ) ) − t } ( 1 + h ) + h 3 m W 3 ( x , y , z ) f ( x , y , z ) { t 2 2 + 1 m ( 1 m ( e x p ( − m t ) − 1 ) + t ) } ] + U x [ h 2 m S 0 ( x , y , z ) f ( x , y , z ) { 1 m ( 1 − e x p ( − m t ) ) − t } ( 1 + h ) − h 2 λ F x ″ ( x , y ) C i t 2 2 − h 3 λ F x ′ ( x , y ) C i t 2 2 − U x h 3 λ F ″ x , y ) C i t 3 6 + h 3 m W 0 ( x , y , z ) f ( x , y , z ) { t 2 2 + 1 m ( 1 m ( e x p ( − m t ) − 1 ) + t ) } + [ D x F x ‴ ( x , y ) + D y ( F y ″ ) x ′ ( x , y ) ] h 3 λ C i t 3 6 ]
  • Case-2: Source lies on the yz-plane
The values of C 0 ,   C 1 ,   C 2   a n d   C 3
C 0 ( x , y , z , t ) = f ( x , y , z ) exp ( − mt )
C 1 ( x , y , z , t ) = h m R ( x , y , z ) f ( x , y , z ) ( exp ( − mt ) − 1 ) − h λ F ( y , z ) C i t
C 2 ( x , y , z , t ) = C 1 ( x , y , z , t ) + h 2 m R ( x , y , z ) f ( x , y , z ) ( e x p ( − mt ) − 1 ) − h 2 λ F ( y , z ) C i t + h 2 m T ( x , y , z ) f ( x , y , z ) { t + 1 m ( e x p ( − mt ) − 1 ) } + [ D y F y ″ ( y , z ) + D z F z ″ ( y , z ) ] h 2 λ C i t 2 2
C 3 ( x , y , z , t ) = C 2 ( x , y , z , t ) + h 2 m R ( x , y , z ) f ( x , y , z ) ( e x p ( − mt ) − 1 ) − h 2 λ F ( y , z ) C i t + h 3 m R ( x , y , z ) f ( x , y , z ) ( e x p ( − mt ) − 1 ) − h 3 λ F ( y , z ) C i t + h 3 m T ( x , y , z ) f ( x , y , z ) { t + 1 m ( e x p ( − mt ) − 1 ) } + [ D y F y ″ ( y , z ) + D z F z ″ ( y , z ) ] h 3 λ C i t 2 2 − D x [ h 2 m S 1 ( x , y , z ) f ( x , y , z ) { 1 m ( 1 − e x p ( − m t ) ) − t } ( 1 + h ) + h 3 m W 1 ( x , y , z ) f ( x , y , z ) { t 2 2 + 1 m ( 1 m ( e x p ( − m t ) − 1 ) + t ) } ] − D y [ h 2 m S 2 ( x , y , z ) f ( x , y , z ) { 1 m ( 1 − e x p ( − m t ) ) − t } ( 1 + h ) − h 2 λ F y ″ ( y , z ) C i t 2 2 ( 1 + h ) + h 3 m W 2 ( x , y , z ) f ( x , y , z ) { t 2 2 + 1 m ( 1 m ( e x p ( − m t ) − 1 ) + t ) } + [ D y F y i v ( y , z ) + D z ( F z ″ ) y ″ ( y , z ) ] h 3 λ C i t 3 6 ] − D z [ h 2 m S 3 ( x , y , z ) f ( x , y , z ) { 1 m ( 1 − e x p ( − m t ) ) − t } ( 1 + h ) − h 2 λ F z ″ ( y , z ) C i t 2 2 ( 1 + h ) + h 3 m W 3 ( x , y , z ) f ( x , y , z ) { t 2 2 + 1 m ( 1 m ( e x p ( − m t ) − 1 ) + t ) } + [ D y ( F y ″ ) z ″ ( y , z ) + D z F z i v ( y , z ) ] h 3 λ C i t 3 6 ] + U x [ h 2 m S 0 ( x , y , z ) f ( x , y , z ) { 1 m ( 1 − e x p ( − m t ) ) − t } ( 1 + h ) + h 3 m W 0 ( x , y , z ) f ( x , y , z ) { t 2 2 + 1 m ( 1 m ( e x p ( − m t ) − 1 ) + t ) } ]
  • Case-3 Source lies on the xz-plane
Functions, variables, and parameters used in this study are given below:
f ( x , y , z ) = exp { − ( pxy + qyz + rxz ) }
R ( x , y , z ) = [ m + D x ( py + rz ) 2 + D y ( px + qz ) 2 + D z ( qy + rx ) 2 + U x ( py + rz ) 2 ]
T ( x , y , z ) = D x S 1 ( x , y , z ) + D y S 2 ( x , y , z ) + D z S 3 ( x , y , z ) − U x S 0 ( x , y , z )
S 0 ( x , y , z ) = R x ′ ( x , y , z ) − ( py + rz ) R ( x , y , z )
S 1 ( x , y , z ) = R x ″ ( x , y , z ) − 2 ( py + rz ) R x ′ ( x , y , z ) + ( py + rz ) 2 R ( x , y , z )
S 2 ( x , y , z ) = R y ″ ( x , y , z ) − 2 ( px + qz ) R y ′ ( x , y , z ) + ( px + qz ) 2 R ( x , y , z )
S 3 ( x , y , z ) = R z ″ ( x , y , z ) − 2 ( qy + rx ) R z ′ ( x , y , z ) + ( qy + rx ) 2 R ( x , y , z )
W 0 ( x , y , z ) = T x ′ ( x , y , z ) − ( py + rz ) T ( x , y , z )
W 1 ( x , y , z ) = T x ″ ( x , y , z ) − 2 ( py + rz ) T x ′ ( x , y , z ) + ( py + rz ) 2 T ( x , y , z )
W 2 ( x , y , z ) = T y ″ ( x , y , z ) − 2 ( px + qz ) T y ′ ( x , y , z ) + ( px + qz ) 2 T ( x , y , z )
W 3 ( x , y , z ) = T z ″ ( x , y , z ) − 2 ( qy + rx ) T z ′ ( x , y , z ) + ( qy + rx ) 2 T ( x , y , z )

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Water 16 00177 g001
Figure 1. Geometry of the problem.
Figure 1. Geometry of the problem.
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Figure 2. Contaminant concentration distribution versus h graph.
Figure 2. Contaminant concentration distribution versus h graph.
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Figure 3. Contaminant concentration distribution profile against the x - and y -axis for an anthropogenic source that lies on the x y -plane.
Figure 3. Contaminant concentration distribution profile against the x - and y -axis for an anthropogenic source that lies on the x y -plane.
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Figure 4. Contaminant concentration distribution profile against the y - and z -axis for a natural source that lies on the y z -plane.
Figure 4. Contaminant concentration distribution profile against the y - and z -axis for a natural source that lies on the y z -plane.
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Figure 5. Contaminant concentration distribution profile against the x - and y -axis for different time periods for an anthropogenic source that lies on the x y -plane.
Figure 5. Contaminant concentration distribution profile against the x - and y -axis for different time periods for an anthropogenic source that lies on the x y -plane.
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Figure 6. Contaminant concentration distribution profile against the y - and z -axis for different time periods for a natural source that lies on the y z -plane.
Figure 6. Contaminant concentration distribution profile against the y - and z -axis for different time periods for a natural source that lies on the y z -plane.
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Figure 7. Contaminant concentration distribution profile against the x -axis when an anthropogenic source lies on the x y -plane and a natural source lies on the y z -plane.
Figure 7. Contaminant concentration distribution profile against the x -axis when an anthropogenic source lies on the x y -plane and a natural source lies on the y z -plane.
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Figure 8. Contaminant concentration distribution profile against the y -axis when an anthropogenic source lies on the x y -plane and a natural source on the y z -plane.
Figure 8. Contaminant concentration distribution profile against the y -axis when an anthropogenic source lies on the x y -plane and a natural source on the y z -plane.
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Figure 9. Contaminant concentration distribution profile against time for an anthropogenic source.
Figure 9. Contaminant concentration distribution profile against time for an anthropogenic source.
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Figure 10. Contaminant concentration distribution profile against time for a natural source.
Figure 10. Contaminant concentration distribution profile against time for a natural source.
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Samanta, A.; Chatterjee, A.; Singh, M.K. A Study on the Coexistence of Anthropogenic and Natural Sources in a Three-Dimensional Aquifer. Water 2024, 16, 177. https://doi.org/10.3390/w16010177

AMA Style

Samanta A, Chatterjee A, Singh MK. A Study on the Coexistence of Anthropogenic and Natural Sources in a Three-Dimensional Aquifer. Water. 2024; 16(1):177. https://doi.org/10.3390/w16010177

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Samanta, Animesh, Ayan Chatterjee, and Mritunjay Kumar Singh. 2024. "A Study on the Coexistence of Anthropogenic and Natural Sources in a Three-Dimensional Aquifer" Water 16, no. 1: 177. https://doi.org/10.3390/w16010177

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