The present study is focused on the development of the analytical solution of a one-dimensional advection-dispersion equation (ADE) by using Laplace transform in a semi-infinite multilayer heterogeneous porous medium. Seepage velocity, hydrodynamics dispersion, and retardation factor are considered to decrease exponentially with space. The model is developed with the basic principle of general dispersion theory, where the dispersion parameter is considered to be directly proportional to the flow velocity. The solution is formulated under the assumption that the aquifer is free of pollutants initially. At the input location of the initial boundary of the uppermost layer (UML), a point source is considered, while the concentration at the intermediate inlet layer is directly obtained from the outlet concentration of the preceding layer. The zero gradient condition describes the outlet boundary. The semi-infinite aquifer is divided into discrete sub-layers in a like manner. To demonstrate the validity of the model, the developed analytical solutions are additionally verified by corresponding numerical solutions. Also, the impact of spatially dependent parameters on pollutant migration are graphically depicted. Such a study will be advantageous for the groundwater hydrologist to predict and analyze the flow and transport behavior of the groundwater in various multilayer aquifer systems.

Predicting hydraulic parameters in porous media, such as Darcy and non-Darcy coefficients, is critical for understanding flow dynamics and designing hydraulic structures. This study develops a stochastic modelling framework using bivariate copula models to estimate these parameters, leveraging easily measurable quantities like packing diameter etc. A synthetic dataset, derived from experimental distributions reported in the literature, was utilized to overcome challenges associated with direct data collection. The analysis revealed that hydraulic radius is strongly correlated with Darcy (Kendall's tau =-0.74) and non-Darcy coefficients (Kendall's tau =-0.76), and an inverse relationship was observed between hydraulic radius and Darcy coefficient. For example, the probability of the Darcy coefficient not exceeding 2 s/m, 20 s/m, and 100 s/m was 10 %, 50 %, and 80 %, respectively, for a hydraulic radius of 0.005 m. Furthermore, the strong correlation between Darcy and non-Darcy coefficients (Kendall's Tau = 0.67) enables the use of the former as a predictor for the latter. For a Darcy coefficient of 50 s/m, the probability of the non-Darcy coefficient not exceeding 300 s2/m2 and 750 s2/m2 was approximately 50 % and 75 %, respectively. This approach provides designers and engineers with a probabilistic framework for selecting hydraulic parameter values with varying degrees of confidence. It offers practical applications in the design of hydraulic structures, such as rockfill dams, and in estimating discharge through fractures, allowing for more reliable assessments of head loss and stability. The findings underscore the potential of copula models in enhancing the predictive accuracy and practicality of hydraulic analyses in porous media.

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.

Groundwater quality is one of the major concerns. Quality of the groundwater directly impacts human health, growth of plants and vegetables. Due to the severe impacts of inadequate water quality, it is imperative to find a swift and economical solution. Water quality prediction may help us to manage water resources properly. The present study has been carried out considering thirty-seven water sample data points form the Pindrawan tank command area of Raipur district, Chhattisgarh, India. A total of nineteen physicochemical parameters were measured, out of which seventeen parameters were used to compute the weight-based groundwater quality index (WQI). In this present work, the primary goal is to identify the most effective parameters for WQI prediction. Out of the seventeen parameters tested, the Mann—Whitney—Wilcoxon (MWW) statistical test has revealed that five parameters Fe, Cr, Na, Ca, and Mg hold a strong statistical significance in distinguishing between drinkable and non-drinkable water. Out of these five parameters, Cr is the only parameter that maintains a different range of values for drinkable water and non-drinkable water. To validate the efficiency of these statistically significant parameters, machine learning techniques like Artificial Neural Networks (ANN) and Logistic Regression (LR) were used. The experimental results clearly demonstrate that out of all the seventeen parameters tested, utilizing only Cr yields remarkably high classification accuracy. ‘Cr’ achieved an accuracy of 91.67% using artificial neural networks. This is much higher than the accuracy of 66.67% obtained using a parameter set with all seventeen parameters. The proposed methodology achieved good accuracy when classifying water samples into drinkable and non-drinkable water using only one parameter, ‘Cr’.

Saltwater intrusion in a coastal region is one of the major issues facing the agricultural, domestic, and industrial sectors. Groundwater withdrawal for an increasing population and agricultural and industrial demands are the main reasons for saltwater intrusion into coastal aquifers. Saltwater intrusion into freshwater aquifers directly affects a population's health, economy, and social development. This study explores an innovative approach to modeling saltwater intrusion into freshwater using a prey-predator model. Freshwater is analogized as the prey and saltwater as the predator, providing a unique perspective on understanding this phenomenon. The present model was formulated using a balance of freshwater and saltwater and is applicable to coastal aquifers. An amount of freshwater withdrawal such that the volume of saltwater does not produce adverse impacts is predicted graphically. Based on a certain simulation, the current model suggests that if the withdrawal rate drops to 50%, freshwater is projected to surpass saltwater dominance for about 90 years. The model has been validated through a benchmark saltwater intrusion problem.

The onset of double-diffusion convection in an inclined porous medium with a concentration-based internal heat source is investigated by performing linear and nonlinear stability analysis. The Brinkman model is employed to model the momentum equation. Effects of different parameters, such as the thermal Rayleigh number ( Ra-T) and solutal Rayleigh number ( Ra-s), the angle of inclination ( phi), the Lewis number (Le), the Darcy number (Da), and the concentration-based internal heat source (Q), are shown. A normal mode technique has been employed on the disturbance equations to get the generalized eigenvalues problem, which is solved by the Chebyshev-tau method coupled with the QZ algorithm in MATLAB. It was observed that increasing the solutal Rayleigh number stabilizes the system due to the higher concentration at the lower boundary than the upper boundary. It has also been observed that decreasing the Darcy number has a destabilizing effect, which means that decreasing permeability advances the onset of double-diffusion convection. Furthermore, it was observed that an increase in the concentration-based internal heat source destabilizes the system. Our numerical results show that for Ra-s>0 and phi>0 degrees, for all Q values, the subcritical instability only exists for transverse rolls.

In this article, an incisive analysis had been carried out to investigate the impact of air pollution on groundwater through a water reservoir. The primary objective of the present work was to develop a mathematical model that shows the direct impact of air pollution on groundwater contamination. The advection-dispersion equation (ADE) was considered to model the contaminant transport through the different media. Due to its inherent complexities, contaminant transport through the different domains was often difficult to model. To overcome it, the simple solution was to consider different velocities and dispersion for different domains. It was considered that the air zone was directly connected with the groundwater through a large water body. Initially, water body and aquifer had been considered contaminant free along with the flux type boundary condition at the semi-infinite part of the boundary. The model had been solved analytically using Laplace transformations. The current study demonstrates that depending upon the length of the medium the impact of contaminant would vary significantly. The air pollution source height had a direct impact on the contaminant concentration level in the aquifer aquitard system. The current study also illustrates that around 25% of air pollutants were induced in the groundwater.

Contaminant transport in a soil formation is described by advection dispersion equation. In this study, a horizontal and transversal contaminant transport along transient groundwater flow under non-linear sorption is solved numerically to examine the contaminant distribution profiles in finite soil media. The horizontal and transversal pore-water seepage velocities are defined as varying exponentially with time. Dispersion is considered directly proportional to the first power of the pore-water velocity. Axial input sources varying exponentially with time are assigned along the two-dimensional coordinate axes. For approximating the governing solute transport equation into algebraic equations, Crank-Nicolson (CN) and alternating direction implicit (ADI) methods are used. Both the numerical solutions are illustrated graphically with the help of computer software for various hydrological input data. In a special case, the numerical solutions are also compared with an approximate solution obtained by PDEtool. The comparison is performed with the help of contour plots. The CN method gives more accurate result than ADI method for the present model problem.

In this study, a two-dimensional contaminant transport model with time-varying axial input sources subject to non-linear sorption, decay, and production is numerically solved to find the concentration distribution profile in a heterogeneous, finite soil medium. The axial input sources are assigned on the coordinate axes of the soil medium, with background sources varying sinusoidally with space. The groundwater velocities are considered space-dependent in the longitudinal and transversal directions. Various forms of axial input sources are considered to study their transport patterns in the medium. The alternating direction implicit (ADI) and Crank-Nicolson (CN) methods are applied to approximate the two-dimensional governing equation, and the obtained algebraic system of equations in each case is further solved by MATLAB scripts. Both approximate solutions are illustrated graphically for various hydrological input data. The influence of various hydrogeological input parameters, such as the medium’s porosity, density, sorption conditions, dispersion coefficients, etc., on the contaminant distribution is analyzed. Further, the influence of constant and varying velocity parameters on groundwater contaminant transport is studied. The stability of the proposed model is tested using the Peclet and Courant numbers. Substantial similarity is observed when the approximate solution obtained using the CN method is compared with the finite element method in a special case. The proposed approximate solution is compared with the existing numerical solutions, and an overall agreement of 98–99% is observed between them. Finally, the stability analysis reveals that the model is stable and robust.

Groundwater pollution is a one of the major problems of our environment caused by various sources such as industries, pesticides, fertilizers, and mining activities. Convection–dispersion equation (CDE) is employed to model the transport of groundwater contamination mathematically, but it can be challenging due to complex geometries and hydrogeological characteristics. Moreover, for several realistic scenarios, the input contaminant source might be located at an intermediate location of the domain, leading to dispersion in forward and backward directions from the source point. Few previous works in this context have been approached analytically and limited to the one-dimensional (1D) assumption of the medium. In this study, a pollutant dispersion with an intermediate time-dependent point source is modelled mathematically. The forward–backward pollutant distribution in a two-dimensional (2D) semi-infinite transient groundwater flow field is investigated. The concentration gradients are taken as zero across the final boundaries of the domain. The effect of off-diagonal dispersion is included in the model equation. The impact of various hydrological input parameters, such as dispersion, porosity, distribution coefficient, decay parameter, etc., on the pollutant transport is examined graphically. The proposed transport model problem is solved numerically and analytically using the Crank–Nicolson (CN) method, and Laplace transform technique (LTT), respectively. The accuracy of the proposed numerical method is evaluated by comparing it with the analytical solution using graphical and statistical measures. The obtained numerical solution of the 2D model problem shows a good agreement with the analytical solution. This may interest researchers working in surface water and vadose zone hydrology areas.

This study adopts generalized dispersion theory in one-dimensional advection–dispersion equation (ADE), where time-dependent dispersion and velocity are considered. The generalized dispersion theory allows mechanical dispersion to be directly proportional to seepage velocity with power n, where n is any real number. Homotopy analysis method (HAM) that uses a simple algorithm is adopted to handle the non-linearity that occurred in the ADE under the generalized dispersion. A point source is introduced to the entry boundary and a line source is introduced to the entire model domain. Three time-dependent point sources in the form of (i) exponentially decreasing function, (ii) linear function and (iii) sinusoidal function, at the entry boundary are considered. Two-line sources are considered in the form of (i) linear space-dependent function and (ii) nonlinear space-time-dependent function. Using the HAM, semi-analytical solutions for any power n are derived and semi-analytical solutions for n = 1 and n = 1.5 are discussed in particular. Comparison with the analytical solution is discussed and found good agreement for 6th order of solution obtained by HAM.

The prediction of pollutant migration and its concentration variation in the subsurface hydrology is vitally requisite for the assessment and restorative treatment of polluted groundwater system. Pollutant migration for the multispecies reactive system cannot be reliably investigated by classical form of convection-dispersion equation (CDE), due to the presence of more than one reactive species. This paper establishes a time-fractional model for multispecies reactive system under the first order sequential reaction network to understand the anomalous or non-Fickian migration phenomenon for reactive pollutants. At present, most of the fractional models are presented for the classical CDE to investigate migration phenomenon for single species system, not for the multispecies reactive system due to the complexity of the modelled framework. The impact of fractional derivative model is analysed for variable dependent migration parameters and constant parameters, both for the multispecies reactive migration phenomenon. The fractional derivative is expressed in the Caputo sense and to handle the non-linearity of problem, Homotopy perturbation method (HPM) is adopted. The advantage of this method, to get the solutions, is that the HPM is independent of small parameters required for the deformation process as used in other existing perturbation techniques, which make it much more convenient to use for non-linear systems. The impact of the fractional derivative index and other migration parameters is graphically depicted for the reactive species and significant influence of fractional term is observed. The derived solutions are then validated by using the corresponding solutions obtained by other existing well-established methods to ensure the convergence of the HPM solutions. As there do not exist any solutions for such fractional models for multispecies reactive system, this study may be advantageous to convey better understanding for the anomalous or non-Fickian migration for reactive pollutants and their remediation planning in the groundwater resources.

In this study, a one-dimensional non-linear advection–dispersion equation subject to spatial–temporal dependent advection and dispersion coefficients is solved for a heterogeneous groundwater system. The non-linearity of the governing equation is based on the Freundlich and Langmuir sorption isotherms. The groundwater flow is considered to vary exponentially with time. Also, a generalized theory of the dispersion coefficient is used for extensive study of the model problem. The approximate solutions of the model problem are obtained in a semi-infinite and finite heterogeneous media by employing the Crank–Nicolson scheme. The exact solutions are obtained in both domains by the Laplace transform technique subject to linear sorption isotherm and non-transient flow conditions. Further, various graphical solutions are obtained using MATLAB scripts to examine the contaminant transport behaviour. For quantitative evaluation of the proposed model, a root mean square (RMS) error is computed. Overall, the results show that RMS error of the approximate solutions with respect to the exact solutions is within acceptable limits (less than 5%) for different combinations of discretization parameters. The robustness of the proposed model suggests its better suitability for modelling groundwater transport phenomena under the consideration of a non-linear sorption isotherm.

The present study deals with the generalized one-dimensional (1-D) advection fractal dispersion (AFD) equation. The numerical simulation is carried out to simulate the plume's fate and transport phenomenon in a finite porous medium. The present model is developed for chemical substitute migration in soil with spatial varying advection and its corresponding fractal dispersion. Moreover, it is adequate for deliberating spatial varying sorption and first-order biological degradation due to reaction processes in the soil. As in generalized dispersion theory, the dispersion term is expressed to be directly proportional to seepage velocity by some power alpha, where alpha varies from 1 to 2. The present numerical study is capable of dealing with various types of advection and its corresponding fractal dispersion. Initially, we have assumed that the aquifer was pollutant-free. At the inlet of the domain, some pollutant source has been considered, and at the outlet boundary, it is assumed that the pollutant concentration disappears to zero. The pollutant concentration distribution has been observed in different types of porous medium as well as for various hydrological parameters. The current generalized AFD model is based on the implicit finite difference technique, which can illustrate chemical migration in the soil to the groundwater, and further, the MATLAB version R2013a(8.1.0.604) Pdepe tool technique validates it. A special case study is also discussed regarding the migration of Iron (Fe) and Zinc (Zn) in the soil. Afterwards, the numerical model is used to illustrate the breakthrough curves (BTCs) for the various transport parameters, and the accuracy of the model problem is also discussed using root mean square error (RMSE). (C) 2022 American Society of Civil Engineers.

Multispecies solute dispersion with sequential first-order decay often occurs in contaminated geological formation with radioactive wastes and their biodegradation products. As a result, analytical result is an essential and effective means with a variability of applications, such as providing approximate analysis of contamination dispersion hazard scenarios, executing sensitivity analysis to exploring how various decay parameters affect the processes of contaminant transport. The main objective is to analyze solute dispersion for two-species solute migration model subject to uniform source concentration. The effect of linear isotherm, porosities, densities, and first-order decay terms is also analyzed for concentration distribution. For the prediction of solute dispersion, use solute included domain as a linear combination of the initial source concentration with zero-order production at initially. An exact solution is derived for unsteady-state two-species transport model with uniform dispersion coefficient for semi-infinite saturated geological formation. The concentration distribution pattern is depicted for the first and second species for a finite length of the domain in the presence and absence of species for different values of densities and variable porosities. The results showed that the contaminant concentration remained the same during the early period of injection for the first species, whereas for the second species, it attained its maximum level of concentration. Also, we found that beyond a certain distance, it attains the minimum level of concentration for both the first and second species. Results were compared with the existing research works and were shown noticeable good agreement between them.

A glacier's mass balance and dynamics are regulated by changes in ice velocity. As a result, estimating glacier flow velocity is a crucial part of temporal glacier monitoring of its health, response to climate change parameters, and its effect on increasing the sea level rise. In this study, we estimated the Gangotri glacier surface velocities from 2014 to 2021. We used remote sensing-based techniques to estimate the Gangotri surface velocity since it provides such measurements regularly for a vast geographical area. Sub-pixel correlation of Landsat 8 imagery was used by using the COSI- Corr (co-registration of optically sensed images and correlation) tool to determine surface velocities over the Gangotri glacier. Our derived velocities values match the ground truth velocities values comparatively well. Gangotri surface velocities vary over the various regions of the glacier, and from year to year. Our study indicated that the middle region of the ablation zone and the accumulation zone had higher velocities across all the years, while the boundary regions of the glacier show lower speeds. The average velocities range varies from ∼13 m/year in the accumulation zone to ∼22m/year. For the ablation zone, the average velocities range from ∼ 11m/year in the ablation zone to ∼18m/year. The average surface velocity showed a high decrement of 26 % from 2014 to 2021. The general surface velocities in Gangotri vary from ∼8 m/y to∼ 61 m/y± 1.9 from 2014 to 2021.

This study proposes two-dimensional (2D) pollutant migration in a semi-infinite geological formation with spatial varying transport parameters. Because the groundwater flow is bidirectional, the impact of off-diagonal dispersion also was taken into account. A decay parameter was considered in the aqueous phase as well as in the solid phase. We assumed that the groundwater reservoir was not plume-free, because some scale-varying pollutant exists there very initially, decaying with space. A change in the source at the inlet boundary in the presence of off-diagonal dispersion (ODD) alters the strength of pollutant concentration. The existing solutions can be reduced into other existent solutions for various geological formations. The Laplace transform technique (LTT) was applied to obtain the pollutant concentration profile in the 2D anisotropic heterogeneous porous medium. The Crank-Nicolson finite-difference (CNFD) technique was adopted for a numerical simulation. We demonstrated the validity of the analytical result. The numerical result and some previously available data from literature were compared and good agreement was found.