Two-dimensional Solute Transport Research Articles

A reduced proper orthogonal decomposition (POD) element free Galerkin (POD-EFG) method to simulate two-dimensional solute transport problems and error estimate

One meshless method based on the variational weak form is the element free Galerkin method. The element free Galerkin (EFG) method is similar to the finite element method but the test and trial functions for the EFG method are chosen from moving least squares (MLS) approximations. The shape functions of MLS approximation don't have the δ-Kronecker property thus the essential boundary conditions (Dirichlet boundary conditions) can not be applied, directly. On the other hand, there are some shape functions that have the mentioned property. One of these functions is the radial point interpolation method (RPIM). In the current paper, we employ the shape functions of RPIM as the test and trial functions. We apply the EFG method based on the RPIM (EFG-RPIM) for solving two-dimensional solute transport problems. To reduce the used CPU time, the proper orthogonal decomposition (POD) approach has been combined with the EFG-RPIM technique. Also, the unconditional stability and convergence of POD-EFG-RPIM method are proved by the energy method. Finally, some numerical results have been reported to show the efficiency and computational order of the new method.

Explicit finite-difference solution of two-dimensional solute transport with periodic flow in homogenous porous media

AbstractThe two-dimensional advection-diffusion equation with variable coefficients is solved by the explicit finitedifference method for the transport of solutes through a homogenous two-dimensional domain that is finite and porous. Retardation by adsorption, periodic seepage velocity, and a dispersion coefficient proportional to this velocity are permitted. The transport is from a pulse-type point source (that ceases after a period of activity). Included are the firstorder decay and zero-order production parameters proportional to the seepage velocity, and periodic boundary conditions at the origin and at the end of the domain. Results agree well with analytical solutions that were reported in the literature for special cases. It is shown that the solute concentration profile is influenced strongly by periodic velocity fluctuations. Solutions for a variety of combinations of unsteadiness of the coefficients in the advection-diffusion equation are obtainable as particular cases of the one demonstrated here. This further attests to the effectiveness of the explicit finite difference method for solving two-dimensional advection-diffusion equation with variable coefficients in finite media, which is especially important when arbitrary initial and boundary conditions are required.

Assessing the performance of a permeable reactive barrier–aquifer system using a dual-domain solute transport model

Transport behavior through a permeable reactive barrier (PRB)-aquifer system is complicated because of the different physical and chemical properties of the PRB and the aquifer. Dual-domain solute transport models are efficient tools for better understanding the various processes and mechanisms of reactive solute transport through a PRB–aquifer system. This study develops a dual-domain analytical model to assess the physical and chemical processes of two-dimensional reactive solute transport through a PRB–aquifer system. The dispersion processes of a dual–domain system on the solute transport are investigated. The results show that the dispersion parameters in a dual-domain system synchronously govern the dynamic shape of the contaminant plume. The low longitudinal and transverse dispersion coefficients of a dual-domain system may restrict the spreading of the plume and elevate the plume’s concentration level. The derived analytical solution is applied to explore how the different reactive transport processes affect the performance of a PRB-aquifer system. The results show that the first-order decay rate constant of the PRB has a critical effect on the performance of the PRB-aquifer system, whereas the effects of the physical dispersion properties on PRB performance are less significant.

An alternative smooth particle hydrodynamics formulation to simulate chemotaxis in porous media

Chemotaxis, the microorganisms autonomous motility along or against the concentration gradients of a chemical species, is an important, yet often neglected factor controlling the transport of bacteria through saturated porous media. For example, chemotactic bacteria could enhance bioremediation by directing their own motion to residual contaminants trapped in low hydraulic conductive zones of contaminated aquifers. The aim of the present work is to develop an accurate numerical scheme to model chemotaxis in saturated porous media and other advective dominating flow systems. We propose to model chemotaxis by using a new class of meshless Lagrangian particle methods we recently developed for applications in fluid mechanics. The method is based on the Smooth Particle Hydrodynamics (SPH) formulation of (Ben Moussa et al., Int Ser Numer Math, 13(1):29–62, 2006), combined with a new Weighted Essentially Non-Oscillatory (WENO) reconstruction technique on moving point clouds in multiple space dimensions. The purpose of this new numerical scheme is to fully exploit the advantages of SPH among traditional mesh-based and mesh-free schemes and to overcome drawbacks related to the use of standard SPH for modeling chemotaxis in porous media. First, we test the new scheme against analytical reference solutions. Then, under the assumption of complete mixing at the Darcy scale, we perform two-dimensional conservative solute transport simulations under steady-state flow conditions, to show the capability of the proposed new scheme to model chemotaxis.

Modelling of the groundwater flow and of tracer movement in the porous and fissured media: Chalk Aquifer (Northern part of Paris Basin, France)

A groundwater flow model has been developed in order to study the chalk aquifer of Paris Basin, based on most of the geological and hydrological available data. The numerical processes are intended to modelling the groundwater flow in the Senonian (Late Cretaceous) formations and to visualize the tracer movement in groundwater resources in the experimental site of LaSalle Beauvais (northern part Paris Basin). Both objectives were achieved as follows: (i) the comprehension of the spatial distribution of the hydraulic conductivity in the chalk aquifer taking into account the characteristics of the hydrogeological system and (ii) the use of the analytical solution for describing one-dimensional to two-dimensional solute transport in a unidirectional steady-state flow tracer with scale-dependent dispersion. Advection and diffusion mechanisms are taken into account. Comparison between the breakthrough curves of the analytical and the numerical solutions provided an excellent agreement for various ranges of scale-related transport parameters of interest. The developed power series solution facilitates fast prediction of the breakthrough curves at each observation point. Thus, the derived new solutions are widely applicable and are very useful for the validation of numerical transport. The numerical approach is carried out by MT3DMS, a Modular 3-D Multi-Species Transport Model for Simulation of Advection, Dispersion, and Chemical Reactions of Contaminants in Groundwater Systems, and based on total variation-diminishing method using the ULTIMATE algorithm. The estimation of the infected surface could constitute an approach in water management and allows to prevent the risks of pollution and to manage the groundwater resource from a durable development perspective. Copyright © 2015 John Wiley & Sons, Ltd.

Quantitative and Qualitative Assessment of Seawater Intrusion in Wadi Ham under Different Pumping Scenarios

AbstractArid coastal regions are more vulnerable to groundwater deterioration problems. Intensive groundwater abstraction from the coastal aquifer of Wadi Ham, United Arab Emirates, caused a severe saltwater intrusion problem. Given the deterioration of groundwater quality, domestic water supply from well fields have been terminated and replaced by desalinated seawater. In addition, many farms in southeast Fujairah city have been abandoned. This paper develops a two-dimensional finite element groundwater flow and solute transport model to simulate the spatial and temporal variations of the salinity distribution in the coastal aquifer of Wadi Ham, taking into account the transition zone between freshwater and seawater bodies. All simulations were conducted in the horizontal view under transient conditions. The available historical records of the water table levels were used to calibrate and validate the developed model. Emphasis was placed on the response of the transition zone to different pumping scenari...

Impact of thin aquitards on two-dimensional solute transport in an aquifer

The influence of aquitards on solute transport in an aquifer is an important and often overlooked process for subsurface contaminant transport. In particular, slow advection (leakage) into an aquitard is often neglected in previous analytical treatment of solute transport, making such analytical solutions unsuitable for benchmarking numerical simulations of transport when aquitard leakage exists. In this study, a semi-analytical solution to the two-dimensional conservative solute transport in an aquifer bounded by thin aquitards is derived in the Laplace domain. The governing equation in the aquifer (not aquitard) incorporates terms accounting for advection, longitudinal dispersion, and transverse vertical dispersion. Both one-dimensional vertical advection and molecular diffusion are considered for aquitard transport. The solutions are derived under conditions of steady-state flow and the first- and third-type transport boundary conditions in the aquifer along with assuming the continuity of concentration and vertical mass flux at aquifer and aquitard interfaces. The solutions in the real time domain are obtained by numerically inverting the solutions in the Laplace domain using the Stehfest (1970) algorithm. The semi-analytical solutions are compared with those from Zhan et al. (2009b), which considered aquitard leakage in infinitively thick aquitards. The concentration profiles, breakthrough curves and distribution profiles in the aquifer are different from those of Zhan et al. (2009b) at small ratios of the aquitard/aquifer thickness; whereas, the results of both are consistent for thick bounding aquitards. This study reveals that the residence time distribution (RTD) in the main aquifer is related to the aquitard/aquifer thickness ratios, Peclet numbers and porosities of adjacent aquitards. The results also suggest that MT3DMS (a commonly applied transport code) cannot successfully simulate solute transport at the aquifer-aquitard interfaces. The presented solutions improve available solutions for transport processes in an aquifer bounded by thin aquitards with leakage. The developed solutions can be directly extended to cases when the vertical hydrodynamic dispersion of the aquitards is considered by simply replacing the effective molecular diffusion coefficients of the aquitard by the vertical hydrodynamic dispersion coefficients of the aquitards.

A reduced-order finite volume element formulation based on POD method and numerical simulation for two-dimensional solute transport problems

Proper orthogonal decomposition (POD) method has been successfully used in the reduced-order modeling of complex systems. In this paper, we extend the applications of POD method, i.e., combine the classical finite volume element (FVE) method with the POD method to obtain a reduced-order FVE formulation with lower dimensions and sufficiently high accuracy for two-dimensional solute transport problems, which have real life practical applications. We then provide error estimates between the reduced-order POD FVE solutions and classical FVE solutions and we provide implementation of an extrapolation algorithm for solving the reduced-order FVE formulation. Thus, we provide the theoretical basis for practical applications. A numerical example is then used to ascertain that the results of numerical computation are consistent with the theoretical derivations. Moreover, it is shown that the reduced-order FVE formulation based on POD method is both feasible and efficient for solving two-dimensional solute transport problems.

Nitrate leaching from a potato field using adaptive network-based fuzzy inference system

The conventional methods of application of nitrogen fertilizers might be responsible for the increased nitrate concentration in groundwater of areas dominated by irrigated agriculture. Appropriate water and nutrient management strategies are required to minimize groundwater pollution and to maximize nutrient use efficiency and production. Design and operation of a drip fertigation system requires understanding of nutrient leaching behavior in cases of shallow rooted crops such as potatoes which cannot extract nutrient from a lower soil depth. This study deals with neuro-fuzzy modeling of nitrate (NO3−) leaching from a potato field under a drip fertigation system. In the first part of the study, a two-dimensional solute transport model was used to simulate nitrate leaching from a sandy soil with varying emitter discharge rates and fertilizer doses. The results from the modeling were used to train and validate an adaptive network-based fuzzy inference system (ANFIS) in order to estimate nitrate leaching. Two performance functions, namely mean absolute percentage error (MAPE) and correlation coefficient (R), were used to evaluate the adequacy of the ANFIS. Results showed that ANFIS can accurately simulate HYDRUS-2D behavior regarding nitrate leaching under the circumstances of the present study.

Two-Dimensional Surface Flow and Solute Transport Model for Basin Irrigation with Conventional Fertilization

The authors propose a two-dimensional surface flow and solute transport model of basin irrigation with conventional fertilization. Specifically, based on the mean velocities and water depth obtained from the complete hydrodynamic model, the nonmean water flow velocities along the basin surface plane and vertical direction was constructed by use of the linear-log law in turbulence and the continuous equation of incompressible hydrodynamics. Based on these constructed velocities, the advection-diffusion equation, which can describe nonuniform solute concentration along the basin surface plane and vertical direction, was employed to describe the two-dimensional basin surface solute transport. Based on the experimental data, the proposed model was validated. The results show that the curves from the observed and simulated results are in sufficient agreement, and the proposed model can successfully simulate the basin surface water flow and solute transport for basin irrigation with conventional fertilization. The analyzed results, based on the Markov stochastic process, show that the proposed model can simulate practical effects by approximately 71.09–87.93%. The other stochastic effects, approximately 12.07–28.91%, would be simulated when the model contains other stochastic factors, such as the spatial variation of the soil physical parameters, fertilizer uniformity applied on the basin surface, and so on.

A reduced finite element formulation based on POD method for two-dimensional solute transport problems

A proper orthogonal decomposition (POD) method is applied to a usual finite element (FE) formulation for two-dimensional solute transport problems with real practical applied background such that it is reduced into a reduced FE formulation with lower dimensions and high enough accuracy. The error estimates between the reduced POD FE solutions and the usual FE solutions are provided. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that this validates the feasibility and efficiency of POD FE method.

Exact analytical solutions for two-dimensional advection–dispersion equation in cylindrical coordinates subject to third-type inlet boundary condition

Exact analytical solutions for two-dimensional advection–dispersion equation (ADE) in cylindrical coordinates subject to the third-type inlet boundary condition are presented in this study. The finite Hankel transform technique in combination with the Laplace transform method is adopted to solve the two-dimensional ADE in cylindrical coordinates. Solutions are derived for both continuous input and instantaneous slug input. The developed analytical solutions are compared with the solutions for first-type inlet boundary condition to illustrate the influence of the inlet condition on the two-dimensional solute transport in a porous medium system with a radial geometry. Results show significant discrepancies between the breakthrough curves obtained from analytical solutions for the first-type and third-type inlet boundary conditions for large longitudinal dispersion coefficients. The developed solutions conserve the solute mass and are efficient tools for simultaneous determination of the longitudinal and transverse dispersion coefficients from a laboratory-scale radial column experiment or an in situ infiltration test with a tracer.

Numerical simulation based on POD for two-dimensional solute transport problems

A proper orthogonal decomposition (POD) method is applied to a usual finite element scheme for two-dimensional solute transport problems such that it is reduced into a reduced finite element formulation with lower dimensions and high enough accuracy. Numerical examples show that the results of numerical computations are consistent with accurate solutions. Moreover, this validates the feasibility and efficiency of POD method.

Modeling uncertainty analysis in flow and solute transport model using Adaptive Neuro Fuzzy Inference System and particle swarm optimization

Groundwater flow and contaminant transport simulations require the determination of various hydro geological parameters such as transmissivity, aquifer thickness, seepage velocity, dispersibility etc. Due to the complex behavior of Groundwater flow and contaminant transport simulations, reliable measurement of the parameters involved is often not possible while performing groundwater system simulations. Hence a methodology is developed in this study wherein a neural-fuzzy model based on an Adaptive Neuro Fuzzy Inference System (ANFIS) is integrated with the particle swarm optimization to estimate the uncertainties in output parameters due to imprecision in input parameters. The Particle Swarm Optimization (PSO) technique is used to find a global optimal solution to a groundwater flow and contaminant transport problem. This is achieved by incorporating the Adaptive Neuro Fuzzy Inference System (ANFIS) to evaluate the objective function within the PSO framework. Later, the ANFIS-PSO algorithm is applied to four problems taking a) a single imprecise parameter for radial flow to a well, b) two imprecise parameters for one dimensional solute transport in steady uniform flow, c) three imprecise parameters for a two-dimensional heterogeneous steady flow problem and finally d) four imprecise parameters for the problem of two-dimensional solute transport. The results show that with the ANFIS-PSO algorithm, the computational burden is reduced considerably when compared to the commonly used vertex method.

Analytical Solution for Two-Dimensional Solute Transport in Finite Aquifer with Time-Dependent Source Concentration

Using the Hankel Transform Technique, an analytical solution is derived for two-dimensional solute transport in a homogeneous isotropic aquifer. The aquifer is subjected to time-dependent point source contamination. The solution is derived under two conditions: (1) the flow velocity in the aquifer is a sinusoidally varying function and (2) the flow velocity is an exponentially decreasing function. Initially the aquifer is assumed solute free. The analytical solution is illustrated using an example.

Analytical solution of two-dimensional solute transport in an aquifer–aquitard system

This study deals with two-dimensional solute transport in an aquifer–aquitard system by maintaining rigorous mass conservation at the aquifer–aquitard interface. Advection, longitudinal dispersion, and transverse vertical dispersion are considered in the aquifer. Vertical advection and diffusion are considered in the aquitards. The first-type and the third-type boundary conditions are considered in the aquifer. This study differs from the commonly used averaged approximation (AA) method that treats the mass flux between the aquifer and aquitard as an averaged volumetric source/sink term in the governing equation of transport in the aquifer. Analytical solutions of concentrations in the aquitards and aquifer and mass transported between the aquifer and upper or lower aquitard are obtained in the Laplace domain, and are subsequently inverted numerically to yield results in the real time domain (the Zhan method). The breakthrough curves (BTCs) and distribution profiles in the aquifer obtained in this study are drastically different from those obtained using the AA method. Comparison of the numerical simulation using the model MT3DMS and the Zhan method indicates that the numerical result differs from that of the Zhan method for an asymmetric case when aquitard advections are at the same direction. The AA method overestimates the mass transported into the upper aquitard when an upward advection exists in the upper aquitard. The mass transported between the aquifer and the aquitard is sensitive to the aquitard Peclet number, but less sensitive to the aquitard diffusion coefficient.

Convergence analysis of a vertex-centered finite volume scheme for a copper heap leaching model

In this paper a two-dimensional solute transport model is considered to simulate the leaching of copper ore tailing using sulfuric acid as the leaching agent. The mathematical model consists in a system of differential equations: two diffusion–convection-reaction equations with Neumann boundary conditions, and one ordinary differential equation. The numerical scheme consists in a combination of finite volume and finite element methods. A Godunov scheme is used for the convection term and an P1-FEM for the diffusion term. The convergence analysis is based on standard compactness results in L2. Some numerical examples illustrate the effectiveness of the scheme. Copyright © 2009 John Wiley & Sons, Ltd.

Comparison of laboratory-scale solute transport visualization experiments with numerical simulation using cross-bedded sandstone

Using a slab of Massillon Sandstone, laboratory-scale solute tracer experiments were carried out to test numerical simulations using the Advection–Dispersion Equation (ADE). While studies of a similar nature exist, our work differs in that we combine: (1) experimentation in naturally complex geologic media, (2) X-ray absorption imaging to visualize and quantify two-dimensional solute transport, (3) high resolution transport property characterization, with (4) numerical simulation. The simulations use permeability, porosity, and solute concentration measured to sub-centimeter resolution. While bulk breakthrough curve characteristics were adequately matched, large discrepancies exist between the experimental and simulated solute concentration fields. Investigation of potential experimental errors suggests that the failure to fit solute concentration fields may lie in loss of intricate connectivity within the cross-bedded sandstone occurring at scales finer than our property characterization measurements (i.e., sub-centimeter).

Density-driven flow and solute transport problems. A 2-D numerical model based on the network simulation method

The network simulation method, based on the formal equivalence between physical systems and electrical networks, solves numerical problems of relatively mathematical complexity in a versatile, efficient and computationally fast way. In this paper, the method is applied for the first time to the design of a general purpose model for simulating two-dimensional transient density-driven flow and solute transport through porous media, a mathematical model made up by coupled, nonlinear differential equations. Using the Boussinesq approximation and the stream function formulation, the model is used to solve two typical problems related with groundwater flows. Isochlor concentration and stream function curves are presented and successfully compared with those of other authors. Simulation is carried out using the digital computer program Pspice with relatively low computing times.

Characterizing soil properties by the meshless local Petrov-Galerkin method

This study presents a preliminary development of a direct back analysis procedure by the meshless local Petrov-Galerkin (MLPG) method and its application to characterize soil properties using in-situ test results. As compared to a direct back analysis based on the finite element method (FEM), it is intended to show that doing a direct back analysis by the MLPG method may reduce the computation costs or treat with the non-homogeneity of characterized soil properties more easily. Taking a two-dimensional (2D) solute transport problem as an illustration, an MLPG1 model of such a problem is derived to predict the solute concentration with trial coefficients of dispersion. To account for the non-homogeneity of these coefficients, variation of them is described by the moving least-squares interpolation. The Akaike information criterion is then introduced to find those coefficients of dispersion with which errors between predicted and measured data are minimized. A benchmark problem is studied to test the precision of numerical results provided by the proposed MLPG1 model. Another example illustrates the experiences of doing an MLPG-based direct back analysis. Comparison of MLPG-based and FEM-based direct back analyses is taken.