Advantages of the analytic element method for the solution of groundwater management problems

In the simulation‐optimization approach, a coupled optimization and groundwater flow/transport model is used to solve groundwater management problems. The efficiency of the numerical method, which is used to simulate the groundwater flow, is one the major reason to obtain the best solution for a management problem. This study was carried out to examine the advantages of the analytic element method (AEM) in the simulation‐optimization approach, for the solution of groundwater management problems. For this study, the AEM and finite difference method (FDM) based flow models were developed and coupled with the particle swarm optimization (PSO)‐based optimization model. Furthermore, the AEM‐PSO and FDM‐PSO models developed were applied in hypothetical as well as real field conditions to address groundwater management problems and the results were compared. For the real field situation, the models developed were applied to the Dore River basin in France to minimize the installation and operational cost of new pumping wells taking the location and discharge of the pumping wells as decision variables. The constraints of the problem were identified with the help of stakeholders and water authority officials. The AEM flow model was developed to facilitate the management model particularly when at each iteration, the optimization model calls for a simulation model to calculate the values of groundwater heads. The results show that, at some points, the AEM‐PSO model is efficient in identifying the optimal location of wells and consequently results in optimal costs, sometimes difficult when using the FDM. Copyright © 2011 John Wiley & Sons, Ltd.     

Analytical model for fully three‐dimensional radial dispersion in a finite‐thickness aquifer

Analytical solutions for contaminant transport in a non‐uniform flow filed are very difficult and relatively rare in subsurface hydrology. The difficulty is because of the fact that velocity vector in the non‐uniform flow field is space‐dependent rather than constant. In this study, an analytical model is presented for describing the three‐dimensional contaminant transport from an area source in a radial flow field which is a simplest case of the non‐uniform flow. The development of the analytical model is achieved by coupling the power series technique, the Laplace transform and the two finite Fourier cosine transform. The developed analytical model is examined by comparing with the Laplace transform finite difference (LTFD) solution. Excellent agreements between the developed analytical model and the numerical model certificate the accuracy of the developed model. The developed model can evaluate solution for Peclet number up to 100. Moreover, the mathematical behaviours of the developed solution are also studied. More specifically, a hypothetical convergent flow tracer test is considered as an illustrative example to demonstrate the three‐dimensional concentration distribution in a radial flow field. The developed model can serve as benchmark to check the more comprehensive three‐dimensional numerical solutions describing non‐uniform flow contaminant transport. Copyright © 2009 John Wiley & Sons, Ltd.     

A parallel mesh-free contaminant transport model based on the Analytic Element and Streamline Methods

This paper introduces a new method for simulating large-scale subsurface contaminant transport that combines an Analytic Element Method (AEM) groundwater flow solution with a split-operator Streamline Method for modeling reactive transport. The key feature of the method is the manner in which the vertically integrated AEM flow solution is used to construct three-dimensional particle tracks that define the geometry of the Streamline Method. The inherently parallel nature of the algorithm supports the development of reactive transport models for spatial domains much larger than current grid-based methods. The applicability of the new approach is verified for cases with negligible transverse dispersion through comparisons to analytic solutions and existing numerical solutions, and parallel performance is demonstrated through a realistic test problem based on the regional-scale transport of agricultural contaminants from spatially distributed sources.     

Finite difference modeling of contaminant transport using analytic element flow solutions

The two-dimensional implementation of the analytic element method (AEM) is commonly used to simulate steady-state saturated groundwater flow phenomena at regional and local scales. However, unlike alternative groundwater flow simulation methods, AEM results are not ordinarily used as the basis for simulation of reactive solute transport. The use of AEM-simulated flow fields is impeded by the discrepancy between a continuous representation of flow and a typically discrete representation of transport, and requires translation of the flow solution to a discrete analog. This paper presents a variety of methods for analytically calculating conservative discrete water fluxes and integrated components of the dispersion tensor across cell interfaces. An Eulerian finite difference method based on these AEM-derived parameters is implemented for use in simulation of 2D (vertically averaged) solute transport. This implementation is first benchmarked against existing methods that use standard finite difference flow solutions, then used to investigate the effects of an inaccurate discrete water balance. It is shown that improper translation of AEM fluxes leads to significant water balance errors and inaccurate simulation of contaminant transport.     

Coordinate mapping of analytical contaminant transport solutions to non-uniform flow fields

Existing analytical solutions to 2D and 3D contaminant transport problems are limited by the mathematically convenient assumption of uniform flow. An approximate method is developed herein for coordinate mapping of 2D (vertically-averaged) transport solutions to non-uniform steady-state irrotational and divergence-free flow fields in single-layer aquifers. The method enables existing analytical transport solutions to be applied to aquifer systems with wells, non-uniform saturated thickness, surface water features, and (to a limited degree) heterogeneous hydraulic conductivity and recharge. This mass-conservative coordinate mapping approach is inexact in its approximation of the dispersion process but is still sufficiently accurate for many simple flow systems. The degree of model error is directly proportional to the variation of velocity magnitude within the domain. These mapped analytical solutions are compared to numerical simulation results and the coordinate mapping errors are investigated. The methods described herein may be used in the traditional capacity of analytical transport models, i.e., screening and preliminary site assessment, without sacrificing accuracy by assuming locally uniform flow conditions or applying an ad-hoc coordinate transformation. The solutions benefit from the traditional advantages of analytical methods, particularly the removal of artifacts due to spatial and temporal discretization: no time-stepping or numerical discretization is required.     

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 © 2016 John Wiley & Sons, Ltd.     

One-dimensional uniform and time varying solute dispersion along transient groundwater flow in a semi-infinite aquifer

An analytical solution for the space-time variation of contaminant concentration in one-dimensional transient groundwater flow in a homogenous semi-infinite aquifer, subjected to time-dependent source contamination, is derived. The uniform and time varying dispersion along transient groundwater flow is investigated under two conditions. First, the flow velocity distribution in the aquifer is considered as a sinusoidally varying function, and second, the flow velocity distribution is treated as an exponentially increasing function of time. It is assumed that initially the aquifer is not solute free, so the initial background concentration is considered as an exponentially decreasing function of the space variable which is tending to zero at infinity. It is assumed that dispersion is directly proportional to the square of the velocity, noting that experimental observations indicate that dispersion is directly proportional to the velocity with a power ranging from 1 to 2. The analytical solution is illustrated using an example and may help benchmark numerical codes and solutions.     

Laplace-transform analytic element solution of transient flow in porous media

A Laplace-transform analytic element method (LT-AEM) is described for the solution of transient flow problems in porous media. Following Laplace transformation of the original flow problem, the analytic element method (AEM) is used to solve the resultant time-independent modified Helmholtz equation, and the solution is inverted numerically back into the time domain. The solution is entirely general, retaining the mathematical elegance and computational efficiency of the AEM while being amenable to parallel computation. It is especially well suited for problems in which a solution is required at a limited number of points in space–time, and for problems involving materials with sharply contrasting hydraulic properties. We illustrate the LT-AEM on transient flow through a uniform confined aquifer with a circular inclusion of contrasting hydraulic conductivity and specific storage. Our results compare well with published analytical solutions in the special case of radial flow.     

Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: A comparative study

This work deals with a comparison of different numerical schemes for the simulation of contaminant transport in heterogeneous porous media. The numerical methods under consideration are Galerkin finite element (GFE), finite volume (FV), and mixed hybrid finite element (MHFE). Concerning the GFE we use linear and quadratic finite elements with and without upwind stabilization. Besides the classical MHFE a new and an upwind scheme are tested. We consider higher order finite volume schemes as well as two time discretization methods: backward Euler (BE) and the second order backward differentiation formula BDF (2). It is well known that numerical (or artificial) diffusion may cause large errors. Moreover, when the Péclet number is large, a numerical code without some stabilising techniques produces oscillating solutions. Upwind schemes increase the stability but show more numerical diffusion. In this paper we quantify the numerical diffusion for the different discretization schemes and its dependency on the Péclet number. We consider an academic example and a realistic simulation of solute transport in heterogeneous aquifer. In the latter case, the stochastic estimates used as reference were obtained with global random walk (GRW) simulations, free of numerical diffusion. The results presented can be used by researchers to test their numerical schemes and stabilization techniques for simulation of contaminant transport in groundwater.     

Composite analytical solutions for a soil vapour extraction system

Soil vapour extraction (SVE) is a common remediation technique for cleaning up unsaturated soils contaminated by volatile organic compounds (VOCs). Analytical solutions, which result from simple mathematical models, can allow the fast approximation of the time‐dependent effluent concentration and the gaining of insight into the processes that take place during soil remediation. Deriving the analytical solutions to advection–dispersion equations that simultaneously take into account the mechanical dispersion and molecular diffusion is very difficult because of the variable dependence of governing equations' coefficients. In this study, we first present two simplified analytical solutions that only consider mechanical dispersion or molecular diffusion. The two developed analytical solutions are compared with the numerical solution that simultaneously considers both mechanical dispersion and molecular diffusion to examine the applicability of the two simplified analytical solutions and distinguishes the individual contribution of the mechanical dispersion and molecular diffusion to total VOCs transport in an SVE system. Results show that dispersion plays an important role during SVE decontamination and neither the diffusion‐dominated solution nor the dispersion‐dominated solution can agree well with the numerical solution when both mechanical dispersion and molecular diffusion have significant contributions to the total VOCs transport flux. A composite analytical solution that linearly couples the diffusion‐ and dispersion‐dominated analytical solutions, which is proposed herein to eliminate the discrepancy between the analytical solutions and the numerical solution. Results indicate that the proposed composite analytical solution agrees well with the numerical solution and is an effective tool for quickly and accurately evaluating the time‐dependent effluent concentration for parameters of the different ranges of interest in an SVE remedial system. Copyright © 2006 John Wiley & Sons, Ltd.     

Analytical Solutions for Water Flow and Solute Transport in the Unsaturated Zone

Analytical solutions for the water flow and solute transport equations in the unsaturated zone are presented. We use the Broadbridge and White nonlinear model to solve the Richards’ equation for vertical flow under a constant infiltration rate. Then we extend the water flow solution and develop an exact parametric solution for the advection-dispersion equation. The method of characteristics is adopted to determine the location of a solute front in the unsaturated zone. The dispersion component is incorporated into the final solution using a singular perturbation method. The formulation of the analytical solutions is simple, and a complete solution is generated without resorting to computationally demanding numerical schemes. Indeed, the simple analytical solutions can be used as tools to verify the accuracy of numerical models of water flow and solute transport. Comparison with a finite-element numerical solution indicates that a good match for the predicted water content is achieved when the mesh grid is one-fourth the capillary length scale of the porous medium. However, when numerically solving the solute transport equation at this level of discretization, numerical dispersion and spatial oscillations were significant.     

A double-porosity model for a fractured aquifer with non-Darcian flow in fractures

Abstract

Non-Darcian flow in a finite fractured confined aquifer is studied. A stream bounds the aquifer at one side and an impervious stratum at the other. The aquifer consists of fractures capable of transmitting water rapidly, and porous blocks which mainly store water. Unsteady flow in the aquifer due to a sudden rise in the stream level is analysed by the double-porosity conceptual model. Governing equations for the flow in fractures and blocks are developed using the continuity equation. The fluid velocity in fractures is often too high for the linear Darcian flow so that the governing equation for fracture flow is modified by Forcheimer's equation, which incorporates a nonlinear term. Governing equations are coupled by an interaction term that controls the quasi-steady-state fracture—block interflow. Governing equations are solved numerically by the Crank-Nicolson implicit scheme. The numerical results are compared to the analytical results for the same problem which assumes Darcian flow in both fractures and blocks. Numerical and analytical solutions give the same results when the Reynolds number is less than 0.1. The effect of nonlinearity on the flow appears when the Reynolds number is greater than 0.1. The higher the rate of flow from the stream to the aquifer, the higher the degree of nonlinearity. The effect of aquifer parameters on the flow is also investigated. The proposed model and its numerical solution provide a useful application of nonlinear flow models to fractured aquifers. It is possible to extend the model to different types of aquifer, as well as boundary conditions at the stream side. Time-dependent flow rates in the analysis of recession hydrographs could also be evaluated by this model.     

Probabilistic modeling of aquifer heterogeneity using reliability methods

A probabilistic model of groundwater contaminant transport is presented. The model is based on coupling first- and second-order reliability methods (FORM and SORM) with a two-dimensional finite element solution of groundwater transport equations. Uncertainty in aquifer media is considered by modeling hydraulic conductivity as a spatial random field with a prescribed correlation structure. FORM and SORM provide the probability that a contaminant exceeds a target level at a well, termed the probability of failure. Sensitivity of the probability of failure to basic variabilities in grid block conductivity is also obtained, at no additional computational effort. The effect of the choice of the predetermined target level at the observation well is provided, along with its effect on the relevant sensitivity information. Considerable saving in computational time was achieved by superimposing a coarse random variables mesh on a finer numerical mesh. The presence of regions of lower conductivity on the probabilistic event is analyzed, and the regions in which conductivity most affects the results are identified.     

Discussion about the validity of sharp‐interface models to deal with seawater intrusion in coastal aquifers

Analytical models have been exhaustively used to study simple seawater intrusion problems and the sustainable management of groundwater resources in coastal aquifers because of its simplicity, easy implementation, and low computational cost. Most of these models are based on the sharp‐interface approximation and the Ghyben–Herzberg relation, and their governing equations are expressed in terms of a single potential theory to calculate critical pumping rates in a coastal pumping scenario. The Ghyben–Herzberg approach neglects mixing of fresh water and seawater and implicitly assumes that salt water remains static. Therefore, the results of the analytical solutions may be inaccurate and unacceptable for some real‐complex case studies. This paper provides insight into the validity of sharp‐interface models to deal with seawater intrusion in coastal aquifers, i.e. when they can be applied to obtain accurate enough results. For that purpose, this work compares sharp‐interface solutions, based on the Ghyben–Herzberg approach, with numerical three‐dimensional variable‐density flow simulations for a set of heterogeneous groundwater flow and mass transport parameters, and different scenarios of spatially distributed recharge values and spatial wells placement. The numerical experiment has been carried out in a 3D unconfined synthetic aquifer using the finite difference numerical code SEAWAT for solving the coupled partial differential equations of flow and density‐dependent transport. This paper finds under which situations the sharp‐interface solution gives good predictions in terms of seawater penetration, transition zone width and critical pumping rates. Additionally, the simulation runs indicate to which parameters and scenarios the results are more sensitive. Copyright © 2013 John Wiley & Sons, Ltd.     

Analytical solutions for benchmarking cold regions subsurface water flow and energy transport models: One-dimensional soil thaw with conduction and advection

Numerous cold regions water flow and energy transport models have emerged in recent years. Dissimilarities often exist in their mathematical formulations and/or numerical solution techniques, but few analytical solutions exist for benchmarking flow and energy transport models that include pore water phase change. This paper presents a detailed derivation of the Lunardini solution, an approximate analytical solution for predicting soil thawing subject to conduction, advection, and phase change. Fifteen thawing scenarios are examined by considering differences in porosity, surface temperature, Darcy velocity, and initial temperature. The accuracy of the Lunardini solution is shown to be proportional to the Stefan number. The analytical solution results obtained for soil thawing scenarios with water flow and advection are compared to those obtained from the finite element model SUTRA. Three problems, two involving the Lunardini solution and one involving the classic Neumann solution, are recommended as standard benchmarks for future model development and testing.     

Mixing cell method for solving the solute transport equation with spatially variable coefficients

The advection–dispersion equation with spatially variable coefficients does not have an exact analytical solution and is therefore solved numerically. However, solutions obtained with several of the traditional finite difference or finite element techniques typically exhibit spurious oscillation or numerical dispersion when advection is dominant. The mixing cell and semi-analytical solution methods proposed in this study avoid such oscillation or numerical dispersion when advection dominates. Both the mixing cell and semi-analytical solution methods calculate the spatial step size by equating numerical dispersion to physical dispersion. Because of the spatial variability of the coefficients the spatial step size varies in space. When the time step size Δt→0, the mixing cell method reduces to the semi-analytical solution method. The results of application to two cases show that the mixing cell and semi-analytical solution methods are better than a finite difference method used in the study. © 1998 John Wiley & Sons, Ltd.     

Saltwater intrusion modeling in heterogeneous confined aquifers using two-relaxation-time lattice Boltzmann method

This study develops a lattice Boltzmann method (LBM) with a two-relaxation-time collision operator (LTRT) to solve saltwater intrusion problems. A directional-speed-of-sound (DSS) technique is introduced to take into account the hydraulic conductivity heterogeneity and discontinuity, as well as the velocity-dependent dispersion coefficient. The forcing terms in the LTRT model are customized in order to recover the density-dependent groundwater flow and mass transport equations. Using the LTRT with the squared DSS achieves at least second-order accuracy. The LTRT results are verified with Henry’s analytical solution as well as compared with several numerical examples and modified Henry problems that consider heterogeneous hydraulic conductivity and velocity-dependent dispersion. The numerical results show good agreement with the Henry analytical solution and with the numerical solutions obtained by other numerical methods.     

Mathematical modeling and practical verification of groundwater and contaminant transport in a chosen natural aquifer

The paper addresses the 2D mathematical equation of conservative contaminant transport in an aquifer for chosen contaminants. The contaminants (chlorides and sulfates) are subject to instantaneous reversible part of sorption process. The term of instantaneous reversible sorption in the presented equation has been described by the non-linear Freundlich adsorption isotherm, widely applied in practice in relation to static processes (for local equilibrium). The numerical solution (using the finite difference method) has been based on the previously calculated values of longitudinal and transverse dispersion coefficients and the non-linear adsorption parameters for the chosen contaminants. Based on this model, the values of chloride and sulfate concentration isolines have been calculated and compared with the measured maximal concentrations in the chosen natural aquifer (installed piezometers). Additionally, the values of chloride concentrations have been calculated taking into account the influence of radioactive decay term, using the numerical value of the firstorder decay rate constant for an adopted theoretical radionuclide.     

Upstream Dispersion in Solute Transport Models: A Simple Evaluation and Reduction Methodology

This article outlines analytical solutions to quantify the length scale associated with “upstream dispersion,” the artificial movement of solutes in the opposite direction to groundwater flow, in solute transport models. Upstream dispersion is an unwanted artifact in common applications of the advection-dispersion equation (ADE) in problems involving groundwater flow in the direction of increasing solute concentrations. Simple formulae for estimating the one-dimensional distance of upstream dispersion are provided. These show that under idealized conditions (i.e., steady-state flow and transport, and a homogeneous aquifer), upstream dispersion may be a function of only longitudinal dispersivity. The scale of upstream dispersion in a selection of previously presented situations is approximated to highlight the utility of the presented formulae and the relevance of this ADE anomaly in common transport problems. Additionally, the analytical solution is applied in a hypothetical scenario to guide the modification of dispersion parameters to minimize upstream dispersion.     

Modeling Variably Saturated Multispecies Reactive Groundwater Solute Transport with MODFLOW‐UZF and RT3D

A numerical model was developed that is capable of simulating multispecies reactive solute transport in variably saturated porous media. This model consists of a modified version of the reactive transport model RT3D (Reactive Transport in 3 Dimensions) that is linked to the Unsaturated‐Zone Flow (UZF1) package and MODFLOW. Referred to as UZF‐RT3D, the model is tested against published analytical benchmarks as well as other published contaminant transport models, including HYDRUS‐1D, VS2DT, and SUTRA, and the coupled flow and transport modeling system of CATHY and TRAN3D. Comparisons in one‐dimensional, two‐dimensional, and three‐dimensional variably saturated systems are explored. While several test cases are included to verify the correct implementation of variably saturated transport in UZF‐RT3D, other cases are included to demonstrate the usefulness of the code in terms of model run‐time and handling the reaction kinetics of multiple interacting species in variably saturated subsurface systems. As UZF1 relies on a kinematic‐wave approximation for unsaturated flow that neglects the diffusive terms in Richards equation, UZF‐RT3D can be used for large‐scale aquifer systems for which the UZF1 formulation is reasonable, that is, capillary‐pressure gradients can be neglected and soil parameters can be treated as homogeneous. Decreased model run‐time and the ability to include site‐specific chemical species and chemical reactions make UZF‐RT3D an attractive model for efficient simulation of multispecies reactive transport in variably saturated large‐scale subsurface systems.