Contaminant Transport Equation Research Articles

Forward and inverse river contaminant transport modeling using group preserving scheme

Environmental concerns have necessitated the development of computational models predicting pollutant dispersion in natural water systems. Due to the ill-posed nature of the inverse contaminant transport equation, solving this equation using all stable and convergent inverse methods is impossible. Factors such as river geometry, unsteady and non-uniform flow, and tidal influences add to the complexity of the inverse problem. These factors have prompted the evaluation of Group Preserving Scheme for environmental applications. The inverse solution method derives a general equation for solving ordinary differential equations by addressing a dynamical system at negative time steps, ensuring convergence. Three test cases have been presented to evaluate Backward Group Preserving Scheme (BGPS). These have included validation using observational data from Missouri River, inverse simulation in a tidal river, and sensitivity analysis of parameters such as pollutant patterns, advection, dispersion, and decay coefficients. The dataset includes calibrated data from Missouri River, which demonstrated high accuracy, with Mean Relative Error (MRE) ranging from 2.8% to 5.0% for the inverse model. Under tidal conditions, accuracy decreases over time but remains robust, with a Nash–Sutcliffe efficiency of 0.77–0.96 and an MRE of 0.9%–5.9%. Sensitivity analysis revealed optimal model performance for Péclet numbers greater than 500. The model performed best with gradual, wide-peaked pollutant patterns and moderate decay rates (Damköhler number between 5 and 10). BGPS proves effective for transport simulation and concentration history reconstruction in complex rivers, including those with tidal influences, offering a robust tool for water contamination analysis in various flow conditions.

Estimation of contamination release histories using meshless radial point collocation method and multiverse optimization

Abstract Contaminants in groundwater may enter through various sources which are required to be identified for informed decision-making regarding remediation. In early stages of contamination of an aquifer, the sources of contamination are generally unknown. To estimate the unknown release histories at potential sources, the governing equations of contaminant transport are solved backwards in time which renders the inverse problem as ill-posed. Furthermore, due to measurement or operational errors, observed breakthrough curves can be noisy which can induce uncertainty. Simulation-optimization (SO) techniques are generally used for solving the inverse model. Nevertheless, swarm population-based optimization algorithms may get trapped in local optima in the nonlinear search space related to the problem. In this study, SO model based on meshless radial point collocation method (RPCM) and multiverse optimizer (MVO) is proposed. MVO implements gradual transformation from global exploration to local exploitation phase which helps in better estimation of unknowns. The performance of RPCM-MVO-SO model is compared with two other SO models developed by combining RPCM with grey wolf optimizer and particle swarm optimization in two case studies. The proposed model performs better compared to the other two SO models considered which suggests the efficacy of the model for release history estimation in groundwater.

Numerical Simulation of Effects of Velocity and Diffusion Coefficient on Concentration of Contaminants in the Fluid Flow

The study developed and implemented Implicit and explicit schemes for solving convection –diffusion equation in one dimension on concentration of contaminant in a fluid flow. The stability of the scheme was analyzed and the accuracy of the solution of the contaminant transport equation was validated by exact available solution. Graphical presentation of the solution for varying velocity and diffusion coefficient was given. The explicit method (EM) involved one unknown on lift hand side (LHS) while implicit method (IM) involved several unknowns on LHS. The study analyzed the effect of velocity and diffusion coefficient on concentration of contaminant in a fluid flow. The developed schemes were solved numerically using MATLAB was to generate the result and in analysis of results. Results showed that concentration of contaminants increased inversely with velocity and directly to diffusion coefficient. Therefore, for proper treatment of water for example, it is necessary to reduce the flow velocities to reduce the trend of contaminants. As Velocity increases the concentration of contaminant decreases and as diffusion coefficient increases the concentration of contaminant increases.

Theory-guided full convolutional neural network: An efficient surrogate model for inverse problems in subsurface contaminant transport

Identification of the location and strength of a contaminant source in an aquifer is a challenging but crucial task. Efficient surrogate models can be constructed to replace traditional time-consuming simulators while solving this inverse problem. In recent years, with the rapid development of machine learning (ML) algorithms, the artificial neural network (ANN) has been proven to be an efficient way for surrogate modeling. However, it may be difficult for ANN-based algorithms to learn the convection-dispersion equation and predict the contaminant concentration field due to their point-to-point learning scheme. Because of their strong localized features, the concentration fields can be seen as images. In contrast, the convolutional neural network (CNN) can extract spatial information better due to its convolutional structure. Herein, a theory-guided full convolutional neural network (TgFCNN) model is proposed to solve inverse problems in subsurface contaminant transport. TgFCNN can construct robust and reliable surrogate models with limited training realizations, and be further utilized for inverse modeling tasks. The loss function of TgFCNN comprises the residual of governing equations of contaminant transport, as well as data mismatch. Moreover, the iterative ensemble smoother (IES) method is employed to update the parameters while solving the inverse problems. The proposed TgFCNN model is evaluated in four scenarios. The results demonstrate that the TgFCNN model possesses strong generalization and extrapolation abilities, and satisfactory accuracy when estimating unknown contaminant source parameters, as well as the permeability field. The time consumption of the TgFCNN surrogate model for inverse tasks is also greatly reduced compared to using traditional simulators directly.

A fourth-order block-centered compact difference scheme for nonlinear contaminant transport equations with adsorption

Nonlinear contaminant transports through porous media are important in many scientific and engineering applications. In this paper, we develop and analyze fourth-order block-centered compact difference scheme (BCCDS) for the nonlinear contaminant transport equations with adsorption process in porous media. Based on block-centered mesh, a fourth order compact difference scheme of solution and its flux is derived. We prove the mass conservation of the proposed scheme and its unconditional stability. We analyze the convergence and obtain the fourth-order error estimate under the smooth regularity of exact solution. Numerical experiments are presented to show the performance of the schemes.

Impact of Pumping Rate on Contaminant Transport in Groundwater—A Numerical Study

Public supply wells are commonly considered one of the most significant sources of freshwater on Earth. Therefore, potential well water contamination can conceivably be regarded as a crucial issue that is closely correlated with both environmental protection and water demand. In the present study, a three-dimensional numerical model is developed to simulate unsteady and spatially varying groundwater flow, along with contaminant migration. Besides, the proposed model is capable of investigating well water quality by the change of the wells’ pumping rates. The developed model uses a finite-volume time splitting numerical technique to solve governing groundwater flow and soluble contaminant transport equations. Comparison of the numerical simulation results with analytical solutions, as well as experimental and field data, clearly demonstrates the satisfactory performance of the present model. The fundamental aim of the study is to evaluate the effect of pumping rate and its variations on pollution migration through saturated porous media. To meet this purpose, contaminant concentrations and contaminants’ travel time were studied under different pump flow rate conditions. The modeling results revealed that choosing an optimum range for the pumping rate increases contaminant travel time and reduces aquifer vulnerability.

Markov chain Monte Carlo with neural network surrogates: application to contaminant source identification

Subsurface remediation often involves reconstruction of contaminant release history from sparse observations of solute concentration. Markov Chain Monte Carlo (MCMC), the most accurate and general method for this task, is rarely used in practice because of its high computational cost associated with multiple solves of contaminant transport equations. We propose an adaptive MCMC method, in which a transport model is replaced with a fast and accurate surrogate model in the form of a deep convolutional neural network (CNN). The CNN-based surrogate is trained on a small number of the transport model runs based on the prior knowledge of the unknown release history. Thus reduced computational cost allows one to reduce the sampling error associated with construction of the approximate likelihood function. As all MCMC strategies for source identification, our method has an added advantage of quantifying predictive uncertainty and accounting for measurement errors. Our numerical experiments demonstrate the accuracy comparable to that of MCMC with the forward transport model, which is obtained at a fraction of the computational cost of the latter.

Optimal design of in-situ bioremediation system using the meshless element-free Galerkin method and particle swarm optimization

An optimal design of in-situ bioremediation system for contaminated aquifer sites mainly depends on the injection/extraction well locations and their pumping rates. Previous studies have used discretization approaches such as finite difference method (FDM) or finite element method (FEM) to solve the equations of groundwater flow and contaminant transport (GFCT) that characterize the bioremediation processes. In these numerical models, well locations can only lie at the grid nodes and therefore, the locations other than the grid nodes remain unexplored for the optimal in-situ bioremediation. To explore such locations, in this study a meshless simulation model called BIOEFGM is developed using the element-free Galerkin method (EFGM) and coupled with the particle swarm optimization (PSO). BIOEFGM model provides flexibility in adding injection/extraction wells anywhere in the computational domain during the entire optimization procedure. However, FDM/FEM requires meshing and re-meshing of the computational domain for each set of wells and it increases the computational efforts significantly for solving the optimization problem. In this paper, the proposed BIOEFGM-PSO simulation-optimization (S/O) model is first applied to a well-known bioremediation problem and then to a field type large aquifer problem. The simulation results of BIOEFGM are verified with those from the RT3D simulations. The estimated bioremediation cost from the BIOEFGM-PSO model for the first problem is found to be lesser in comparison to the same calculated with different S/O models in the previous studies. The results of this problem also show that optimized in-situ bioremediation system designed by proposed S/O model takes less remediation time and also favor effective biodegradation of contaminants. For both the problems, BIOEFGM-PSO identified optimal well locations at positions other than that of discretized nodes and it leads to efficient bioremediation. It indicates the effectiveness of the BIOEFGM-PSO model and therefore, can be applied for designing the better optimized in-situ bioremediation systems for field problems.

Geocomputação aplicada no problema de transporte acoplado ao fluxo subterrâneo em aquífero livre

A geocomputação, sendo a aplicação de métodos e técnicas computacionais para apoiar a resolução de problemas georreferenciados, auxilia a compreensão de fenômenos naturais relacionados aos recursos hídricos. Algumas ferramentas de geoprocessamento proporcionam mapas digitais que delimitam a bacia hidrográfica em estudo e ainda, a distribuição potenciométrica em conhecidas coordenadas no domínio computacional podem ser usadas na condição inicial e de fronteira do modelo hidrogeológico. Neste sentido, o artigo visa a implementação de técnicas da geocomputação na solução do problema do transporte de contaminante em aquífero livre. A metodologia proposta inicia na geração de uma malha não estruturada; gera a função para a condição de fronteira; aplica um método de iterações de Picard com estimativa inicial calculada numericamente; simula o campo de velocidades de águas subterrâneas em situações de poços de bombeamento e áreas de recarga e, finalmente, acopla a solução estacionária e não-linear do fluxo livre, obtida pelo método de elementos finitos, nos parâmetros advectivo e dispersivo da equação transiente do transporte de contaminantes. Resultados gráficos ilustram a simulação do comportamento hidráulico e as isolinhas de concentração em subsuperfície georreferenciada da bacia hidrográfica do Reservatório do Lobo no município de Itirapina-SP.

Machine learning subsurface flow equations from data

Governing equations of physical problems are traditionally derived from conservation laws or physical principles. However, some complex problems still exist for which these first-principle derivations cannot be implemented. As data acquisition and storage ability have increased, data-driven methods have attracted great attention. In recent years, several works have addressed how to learn dynamical systems and partial differential equations using data-driven methods. Along this line, in this work, we investigate how to discover subsurface flow equations from data via a machine learning technique, the least absolute shrinkage and selection operator (LASSO). The learning of single-phase groundwater flow equation and contaminant transport equation are demonstrated. Considering that the parameters of subsurface formation are usually heterogeneous, we propose a procedure for learning partial differential equations with heterogeneous model parameters for the first time. Derivative calculation from discrete data is required for implementing equation learning, and we discuss how to calculate derivatives from noisy data. For a series of cases, the proposed data-driven method demonstrates satisfactory results for learning subsurface flow equations.

Alternating direction implicit scheme as Finite-Difference method to solve coupled groundwater flow and contaminant transport model in the coastal aquifer

The simulation of coupled groundwater flow-contaminant transport equation can be conducted numerically. The Finite-Difference Method (FDM) through Alternating Direction Implicit (ADI) Scheme will be applied in this research. The ADI scheme is chosen because the FDM in two-dimensional form. The ADI scheme recognized to solve two-dimensional PDE (Partial Differential Equation) due the Crank-Nicholson has matrix dimension problem and other problem from another scheme of FDM. This research will use ADI scheme to transform the groundwater flow and contaminant transport equation become the discrete form. The model algorithm translated into VBA (Visual Basic Algorithm) code and the physical parameter determined hypothetically. The result of model simulation describes that the ADI scheme can be applied to solve coupled groundwater flow-contaminant transport equation. The result can be stated as stable because it reached the steady condition after 60 days of simulation.

The Numerical Simulation of the Freshwater-Seawater Mixing in the Single Phase Coastal Aquifer

The interaction between freshwater and seawater usually occurs in the coastal aquifer. To simply explain this condition, describing the interface between them is conducted. This approach is a multiphase condition. Unfortunately, this condition is on immiscible condition, whereas the miscible condition should be obtained. The miscible condition of freshwater and seawater along coastal aquifer occurs on the single phase. This condition neglects the concept of the interface between them. The purpose of this research is to simulate the mixing of freshwater and seawater in the coastal aquifer by using the numerical method. Groundwater flow equation is governed involving the effect of contaminant concentration. On the other hand, the conservative groundwater contaminant transport equation is used. The discretization process uses the Finite-Difference Method. Coupling two equations will obtain the spreading of contaminant concentration along the model domain. The result shows that the freshwater-seawater interface does not appear and the change of concentration can be reviewed in every section and every point of the computational domain.

In-situ Observation and Nitrate-N Load Assessment in Madhubani District, Bihar, India

Abstract Fertilizers may leach through the vadose zone and eventually reach groundwater in agriculturally intensive areas. Thus, the main focus of this study was to investigate Nitrate-N load and vulnerability of groundwater resources using in-situ observed hydrogeological data. Soil water flow and contaminant transport equation was numerically simulated using HYDRUS 1D for constant head and atmospheric top boundary conditions. Sub-surface materials were distributed based on the lithologs of the target area. Observed water table locations were considered as bottom boundary condition to respective numerical domain. The time taken by Nitrate-N to reach groundwater table was considered to estimate vulnerability index. The results show that Nitrate-N load is higher in constant head boundary conditions than atmospheric boundary conditions. The eastern part of the study area shows high vulnerability than northern part followed by western part. In-situ observed nitrate concentrations were well matched with simulated results. The high vulnerability in eastern and northern part is due to alluvial sandy lithologs and very shallow groundwater table. These findings are in line with the observed low water table depths, less runoff, and higher hydraulic conductivity of the vadose zone material in these area. In western part, forest cover dominated land use causes low pollution vulnerabilities to groundwater resources. This study may help to frame agricultural and soil-water conservation practices with more sustainable remedial techniques.

Simulation of saltwater intrusion into coastal aquifer of Nagapattinam in the lower cauvery basin using SEAWAT

The groundwater that represents 30.1% of the global freshwater supplies is at risk of being contaminated by saltwater intrusion. Saltwater intrusion is the induced flow of seawater into freshwater aquifers. Saltwater intrusion can occur due to natural processes as well as over-extraction of groundwater from coastal aquifers. Growing urban populations in the coastal regions and thus a higher demand for freshwater, emphasise the importance of proper management of these aquifers and prevention of saltwater intrusion by over-extraction. Numerical modelling is a useful tool in helping hydrologists to understand and predict how saltwater intrusion occurs in coastal aquifers. There are various numerical models that can predict groundwater flow and contaminant transport such as SUTRA, SEAWAT, FEFLOW and MODFLOW. These numerical models are based on the governing equations of groundwater flow and contaminant transport. In this paper, the extension of saltwater intrusion into the coastal aquifers of the Nagapattinam coastal region, Tamil Nadu has been investigated by modelling the region using SEAWAT engine in Visual MODFLOW.

The Finite-Difference Model of Fully Saturated Groundwater Contaminant Transport

Groundwater quality is one of water resource problem. This problem is driven by contaminant transport phenomena and can be described as a mathematical model. Contaminant transports equation usually is composed by advection and dispersion flux. In the porous medium or aquifer, contaminant meet tortuosity effect, therefore hydrodynamic dispersion must be considered as the development of dispersion flux. This paper explains the mathematical model of groundwater contaminant on fully saturated condition. It starts from governing equation of contaminant transport. Advection flux is based on groundwater velocity. In steady state condition, groundwater velocity can be determined as certain value. In another hand, in the transient condition, groundwater velocity must be determined based on the solution of groundwater flow model. Dispersion flux is calculated through first Fick’s law and this component is distinguished into two parts follow mechanical dispersion and molecular diffusion. Mechanical dispersion affected by groundwater velocity and dispersivity. Contaminant transport equation is solved numerically using Finite Difference Method (FDM). This final model is validated theoretically and then this model is simulated into transient condition. The result of the simulation is described and explained graphically. Based on this research, the result of FDM model has similar physical behavior to FEM model from CTRAN example.

Semianalytical solutions for contaminant transport under variable velocity field in a coastal aquifer

Existing closed-form solutions of contaminant transport problems are limited by the mathematically convenient assumption of uniform flow. These solutions cannot be used to investigate contaminant transport in coastal aquifers where seawater intrusion induces a variable velocity field. An adaptation of the Fourier-Galerkin method is introduced to obtain semi-analytical solutions for contaminant transport in a confined coastal aquifer in which the saltwater wedge is in equilibrium with a freshwater discharge flow. Two scenarios dealing with contaminant leakage from the aquifer top surface and contaminant migration from a source at the landward boundary are considered. Robust implementation of the Fourier-Galerkin method is developed to efficiently solve the coupled flow, salt and contaminant transport equations. Various illustrative examples are generated and the semi-analytical solutions are compared against an in-house numerical code. The Fourier series are used to evaluate relevant metrics characterizing contaminant transport such as the discharge flux to the sea, amount of contaminant persisting in the groundwater and solute flux from the source. These metrics represent quantitative data for numerical code validation and are relevant to understand the effect of seawater intrusion on contaminant transport. It is observed that, for the surface contamination scenario, seawater intrusion limits the spread of the contaminant but intensifies the contaminant discharge to the sea. For the landward contamination scenario, moderate seawater intrusion affects only the spatial distribution of the contaminant plume while extreme seawater intrusion can increase the contaminant discharge to the sea. The developed semi-analytical solution presents an efficient tool for the verification of numerical models. It provides a clear interpretation of the contaminant transport processes in coastal aquifers subject to seawater intrusion. For practical usage in further studies, the full open source semi-analytical codes are made available at the website https://lhyges.unistra.fr/FAHS-Marwan.

Analytical solution of the one-dimensional contaminant transport equation in groundwater with time-varying boundary conditions

In this paper, an analytical solution for the one-dimensional advection–diffusion equation for studying the contaminant transport in groundwater is presented. The solution is obtained for spatially varying diffusivity and velocity terms along with time-varying boundary conditions. The differential equation considered in the paper is in the form of Legendre Linear Differential Equation which is reduced to a linear differential equation having constant coefficients by a suitable transformation. The final solution for the differential equation in the transformed domain is obtained by the method of Eigenfunction expansions. The solution is tested by considering a test problem.

Solution of fully-coupled shallow water equations and contaminant transport using a primitive-variable Riemann method

A Riemann-solver scheme, using primitive variables rather than conserved variables, is configured and tuned for the solution of the fully-coupled two-dimensional shallow water and contaminant transport equations. This scheme is based on the unstructured finite volume discretization using primitive-variable Roe-flux approximation with an entropy fix. The primitive-variable flux associated with the exact source-term balancing is well-behaved and well-balanced for both still-water and dry regions with arbitrary bed topography. Second-order accuracy is used in space and time. The present study uses a nonlinear implicit scheme based on Newton-iterative algorithm for the time integration. In order to show the accuracy of the scheme, numerical results are verified by different test cases for contaminant advection and diffusion. A scenario of contaminant transport in a complex geometry with wet and dry elements is also simulated to demonstrate that the present work can be implemented on practical applications involving flooding and contaminant transport.

Analytical simulation of two dimensional advection dispersion equation of contaminant transport

The study was designed to investigate the analytical simulation of two dimensional advection dispersion equation of contaminant transport. The steady state flow condition of the contaminant transport where inorganic contaminants in aqueous waste solutions are disposed of at the land surface where it would migrate through the verdoze zone to underground water is considered. We solved the two dimensional advection dispersion equation analytically which is solute transport model without sorption or degradation using change of variable method. We critically reviewed two dimensional equations depicting the transport of contaminant in groundwater and investigate with the help of graphical representation the effect of Peclet number on the concentration of contaminant and established real life interpretation of contaminant transport. Two cases were considered, when Peclet number is less than one and when Peclet number is greater than one. The result obtained revealed that the contaminant concentration increases along x direction and decreases along y direction for both values of peclet number greater than one and less than one. The study has contributed to knowledge through the method utilized to achieve the model analytical solution and the Physical interpretation of Peclet number. From the analysis, we recommend for further studies on the contaminant transport which also depends on the available data, that the extension of advection –dispersion model to three dimensions and comparison of travel time of contaminant transport solution to Kinetic or multi-component mode.Keywords: Contaminant, Seepage Velocity, Aquifer, Advection-dispersion Equation, change of variable methods

Industry Effluent Disposal into Rivers: Coupled Multiobjective-Analytical Optimization Model

AbstractIn this work, a linear system theory approach produces an analytical solution of contaminant mass transport equations for its subsequent use in multiobjective optimization for optimal waste...