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.     

VS2DRTI: Simulating Heat and Reactive Solute Transport in Variably Saturated Porous Media

Variably saturated groundwater flow, heat transport, and solute transport are important processes in environmental phenomena, such as the natural evolution of water chemistry of aquifers and streams, the storage of radioactive waste in a geologic repository, the contamination of water resources from acid‐rock drainage, and the geologic sequestration of carbon dioxide. Up to now, our ability to simulate these processes simultaneously with fully coupled reactive transport models has been limited to complex and often difficult‐to‐use models. To address the need for a simple and easy‐to‐use model, the VS2DRTI software package has been developed for simulating water flow, heat transport, and reactive solute transport through variably saturated porous media. The underlying numerical model, VS2DRT, was created by coupling the flow and transport capabilities of the VS2DT and VS2DH models with the equilibrium and kinetic reaction capabilities of PhreeqcRM. Flow capabilities include two‐dimensional, constant‐density, variably saturated flow; transport capabilities include both heat and multicomponent solute transport; and the reaction capabilities are a complete implementation of geochemical reactions of PHREEQC. The graphical user interface includes a preprocessor for building simulations and a postprocessor for visual display of simulation results. To demonstrate the simulation of multiple processes, the model is applied to a hypothetical example of injection of heated waste water to an aquifer with temperature‐dependent cation exchange. VS2DRTI is freely available public domain software.     

Analytical solutions to the transient,unsaturated transport of water and contaminants through horizontal porous media

We present a range of analytical solutions to the combined transient water and solute transport for horizontal flow. We adopt the concept of a scale and time dependent dispersivity used for contaminant transport in aquifers and apply it to transient, unsaturated horizontal flow to develop similarity solutions for both constant solute concentration and solute flux boundary conditions. Through the use of a specific form of the water profile as used by Brutsaert [Water Resour Res 1968:4;785], the solute profiles can be reduced to a simple quadrature. We also derive a solution for the instantaneous injection of water and solute into a horizontal media for an arbitrary dispersivity. It is found that the solute concentration remains constant in both space and time as the water redistributes, suggesting that the solute does not disperse relative to the water.     

Finite-element analysis of the transport of water and solutes in title-drained soils

The finite-element method based on a Galerkin technique was used to formulate the problem of simulating the two-dimensional (cross-sectional) transient movement of water and solute in saturated or partially saturated nonuniform porous media. The numerical model utilizes linear triangular elements. Nonreactive, as well as reactive solutes whose behaviour can be described by a distribution coefficient or first-order reaction term were considered. The flow portion of the model was tested by comparison of the model results with experimental and finite-difference results for transient flow in an unsaturated sand column and the solute transport portion of the model was tested by comparison with analytical solution results. The model was applied to a hypothetical case involving movement of water and solutes in tile-drained soils. The simulation results showed the development of distinct solute leaching patterns in the soil as drainage proceeded. Although applied to a tile drainage problem in this study, the model should be equally useful in the study of a wide range of two-dimensional water and solute migration problems.     

Multi-block lattice Boltzmann simulations of solute transport in shallow water flows

We report a two-dimensional multi-block lattice Boltzmann model for solute transport in shallow water flows, which is developed based on the advection–diffusion equation for mass transport and the shallow water equations for the flows. A weighting factor is included in the centered scheme for improved accuracy. The model is firstly verified by simulating three benchmark tests: wind-driven circulation in a dish-shaped lake, jet-forced flow in a circular basin, and flow formed by two parallel streams containing different uniform concentrations at the same constant velocity; and then it is applied to a practical wind-induced flow, Baiyangdian Lake, which is characterized by irregular geometries and complex bathymetries. The numerical results have shown that the model is able to produce accurate and detailed results for both water flows and solute transport, which is attractive, especially for flows in narrow zones of practical terrains and certain areas with largely varying pollutant concentrations.     

Coupling water flow and solute transport into a physically-based surface-subsurface hydrological model

A distributed-parameter physically-based solute transport model using a novel approach to describe surface-subsurface interactions is coupled to an existing flow model. In the integrated model the same surface routing and mass transport equations are used for both hillslope and channel processes, but with different parametrizations for these two cases. For the subsurface an advanced time-splitting procedure is used to solve the advection-dispersion equation for transport and a standard finite element scheme is used to solve Richards equation for flow. The surface-subsurface interactions are resolved using a mass balance-based surface boundary condition switching algorithm that partitions water and solute into actual fluxes across the land surface and changes in water and mass storage. The time stepping strategy allows the different time scales that characterize surface and subsurface water and solute dynamics to be efficiently and accurately captured. The model features and performance are demonstrated in a series of numerical experiments of hillslope drainage and runoff generation.     

Experimental and analytical studies of solute transport during run‐off over vegetated surfaces

Solute transport in overland flow is considered as one of the main contributors to water pollution. Although many models of pollutant transport mechanism from soil to run‐off water have been proposed, the characteristics of solute transport accompanying the water run‐off over vegetated surface have not been well studied. In this study, a series of laboratory experiments were conducted to study the solute transport over vegetated surfaces. Based on the experimental results, an idea of the “stationary water layer” in run‐off was proposed. Applying the complete mixing theory in the stationary water layer, an analytical solute transport model was developed with the assumption that the upper run‐off completely mixes with the underlying water in the stationary water layer for each site. The results show that the predictions made by the present model are in good agreement with the measured experimental data. For the vegetated surfaces, the depth of stationary water layer is related to the rainfall intensity, bed slope, and vegetation density. The analytical solution shows that the maximum solute transport occurs at the time of concentration. This study advances our understanding of the mechanisms of solute transport over vegetated areas.     

Solute transport and water content measurements in clay soils using time domain reflectometry

Abstract

Clayey and saline soils have been shown to be problematic for time domain reflectometry (TDR) measurements. This study presents some of these problems and discusses solutions to them. Thirteen solute transport experiments were carried out in three undisturbed soil columns of swelling clay soil from Tunisia, labelled S1, S2, and S3 respectively. The columns were collected at three different physiographical regions within a catchment. Water fluxes ranged from 1.2 to 7.2 cm day^?1. The large solute transport heterogeneity and large tailing indicated that preferential flow was most pronounced in S1. The preferential flow took place in voids between structural elements and in wormholes. In S3, preferential flow was also evident, but not to the same extent as in S1. In S2, the solute transport was more uniform with little preferential flow. The heterogeneity of the solute transport increased with the water flux in S1 and to a smaller extent in S3, whereas it remained constant in S2. In a previous dye experiment in the field, preferential flow in cracks was observed at those sites where S1 and S3 were collected. In the column experiments, preferential flow in these cracks was less due to the higher initial water content compared to the dye experiments, indicating that the desiccation cracks were closed by the swelling clay.     

A novel platform to evaluate the dampening of water and solute transport in an experimental channel under unsteady flow conditions

This paper presents a novel platform to study the dampening of water and solute transport in an experimental channel under unsteady flow conditions, where literature data are scarce. We address the question about what could be the smallest size of experimental platform that is useful for research, project studies, and teaching activities and that allows to do rational experiments characterized by small space occupation, short experimental duration, high measurement precision, high quality and reproducible experimental curves, low water and energy consumption, and the possibility to test a large variety of hydrograph scenarios. Whereas large scale hydraulic laboratories have focused their studies on sediment transport, our platform deals with solute transport. The objectives of our study are (a) building a platform that allows to do rational experiments, (b) enriching the lack of experimental data concerning water and solute transport under unsteady state conditions, and (c) studying the dampening of water and solute transport. We studied solute transport in a channel with lateral gain and lateral loss under different experimental configurations, and we show how the same lateral loss flow event can lead to different lateral loss mass repartitions under different configurations. In order to characterize water and solute dampening between the input and the output of the channel, we calculate dampening ratios based on peak coordinates of time flow curves and time mass curves and that express the decrease of peak amplitude and the increase of peak occurrence time between the input and output curves. Finally, we use a solute transport model coupling the diffusive wave equation for water transfer and the advection–diffusion equation for solute transport in order to simulate the experimental data. The simulations are quite good with a Nash–Sutcliffe efficiency NSE > 0.98 for water transfer and 0.84 < NSE < 0.97 for solute transport. This platform could serve hydrological modellers because it offers a variety of measured parameters (flow, water height, and solute concentration), at a fine time step under unsteady flow conditions.     

Using PHREEQC to simulate solute transport in fractured bedrock

The geochemical computer model PHREEQC can simulate solute transport in fractured bedrock aquifers that can be conceptualized as dual-porosity flow systems subject to one-dimensional advective-dispersive transport in the bedrock fractures and diffusive transport in the bedrock matrix. This article demonstrates how the physical characteristics of such flow systems can be parameterized for use in PHREEQC, it provides a method for minimizing numerical dispersion in PHREEQC simulations, and it compares PHREEQC simulations with results of an analytical solution. The simulations assumed a dual-porosity conceptual model involving advective-reactive-dispersive transport in the mobile zone (bedrock fracture) and diffusive-reactive transport in the immobile zone (bedrock matrix). The results from the PHREEQC dual-porosity transport model that uses a finite-difference approach showed excellent agreement compared with an analytical solution.     

Salinity sources of Kefar Uria wells in the Judea group aquifer of Israel. Part 2—quantitative identification model

The Kefar Uria group of wells have experienced an increase of salinity of the pumped water in the last two decades. The source of salinity is not known. Geohydrological and geochemical considerations of Part 1 (Avissar et al., in press) suggest two possible mechanisms and sources. The first source is infiltrating at the top of the aquifer, in contact with the overlaying formation and is close to the wells. The second source is deep and more distant and is attributed to salt leaching. In the present study, a methodology is developed in order to select the most probable source by quantitative modeling of flow and transport. The problem is more difficult than similar ones investigated in the literature, because of the complex three-dimensional flow pattern. However, simplified models can be used for the limited scope of source identification. The aquifer is modeled as a semi-infinite confined one under steady state conditions. The saline water input area and the 13 wells are represented by source distributions and an analytical solution is obtained for the flow field. A best fit between computed and measured well pressure heads leads to reasonable value of the identified permeability. Advective solute transport from salinity sources to wells is modeled numerically by particle tracking and leads to partition of solute flux among wells. Comparison of calculated wells salinity and the measured ones leads to the conclusive selection of one of the sources as the most probable one.     

Characterizing heterogeneous soil water flow and solute transport using information measures

Heterogeneous water flow and solute transport in soils are an important phenomenon and difficult to be characterized. The objectives of this study were to investigate the heterogeneity of solute transport related to heterogeneous soil water flow using dye infiltration experiments, and to characterize heterogeneous water flow and solute transport in soils using the information theory. Field experiments of dye infiltration were performed in four plots. Various information measures were applied to characterize information content and complexity of water flow and solute transport in soils. Information contents and complexities of the maximum and apparent infiltration depths, and the mean and standard deviation of concentrations in the vertical direction of the plots were calculated. More heterogeneous processes of soil water flow and transport result in higher information/complexity values. The probability distributions of mean concentration were similar to those of the corresponding apparent infiltration depths for the plots, indicating that heterogeneity of dye concentrations was closely related to that of soil water flow. However, the range of information entropy and complexity of the water flow sequences was much narrower than that of the sequences of the concentrations. The results suggested that the transport processes were more heterogeneous than the water flow processes. Compared with the probability distributions of flow parameters, the information measures appeared to be a more versatile tool to describe flow and transport heterogeneities in soils.     

Multi-functional heat pulse probe measurements of coupled vadose zone flow and transport

Simultaneous measurement of coupled water, heat, and solute transport in unsaturated porous media is made possible with the multi-functional heat pulse probe (MFHPP). The probe combines a heat pulse technique for estimating soil heat properties, water flux, and water content with a Wenner array measurement of bulk soil electrical conductivity (EC_bulk). To evaluate the MFHPP, we conducted controlled steady-state flow experiments in a sand column for a wide range of water saturations, flow velocities, and solute concentrations. Flow and transport processes were monitored continuously using the MFHPP. Experimental data were analyzed by inverse modeling of simultaneous water, heat, and solute transport using an adapted HYDRUS-2D model. Various optimization scenarios yielded simultaneous estimation of thermal, solute, and hydraulic parameters and variables, including thermal conductivity, volumetric water content, water flux, and thermal and solute dispersivities. We conclude that the MFHPP holds great promise as an excellent instrument for the continuous monitoring and characterization of the vadose zone.     

Solution of the nonlinear transport equation using modified Picard iteration

The transport and fate of reactive chemicals in groundwater is governed by equations which are often difficult to solve due to the nonlinear relationship between the solute concentrations for the liquid and solid phases. The nonlinearity may cause mass balance errors during the numerical simulation in addition to numerical errors for linear transport system. We have generalized the modified Picard iteration algorithm of Celia et al.⁵ for unsaturated flow to solve the nonlinear transport equation. Written in a ‘mixed-form’ formulation, the total solute concentration is expanded in a Taylor series with respect to the solution concentration to linearize the transport equation, which is then solved with a conventional finite element method. Numerical results of this mixed-form algorithm are compared with those obtained with the concentration-based scheme using conventional Picard iteration. In general, the new solver resulted in negligible mass balance errors (< ∥10⁻⁸∥%) and required less computational time than the conventional iteration scheme for the test examples, including transport involving highly nonlinear adsorption under steady-state as well as transient flow conditions. In contrast, mass balance errors resulting from the conventional Picard iteration method were higher than 10% for some highly nonlinear problems. Application of the modified Picard iteration scheme to solve the nonlinear transport equation may greatly reduce the mass balance errors and increase computational efficiency.     

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.     

A coupled model for simulating surface water and groundwater interactions in coastal wetlands

Coastal wetlands are characterized by strong, dynamic interactions between surface water and groundwater. This paper presents a coupled model that simulates interacting surface water and groundwater flow and solute transport processes in these wetlands. The coupled model is based on two existing (sub) models for surface water and groundwater, respectively: ELCIRC (a three‐dimensional (3‐D) finite‐volume/finite‐difference model for simulating shallow water flow and solute transport in rivers, estuaries and coastal seas) and SUTRA (a 3‐D finite‐element/finite‐difference model for simulating variably saturated, variable‐density fluid flow and solute transport in porous media). Both submodels, using compatible unstructured meshes, are coupled spatially at the common interface between the surface water and groundwater bodies. The surface water level and solute concentrations computed by the ELCIRC model are used to determine the boundary conditions of the SUTRA‐based groundwater model at the interface. In turn, the groundwater model provides water and solute fluxes as inputs for the continuity equations of surface water flow and solute transport to account for the mass exchange across the interface. Additionally, flux from the seepage face was routed instantaneously to the nearest surface water cell according to the local sediment surface slope. With an external coupling approach, these two submodels run in parallel using time steps of different sizes. The time step (Δt_g) for the groundwater model is set to be larger than that (Δt_s) used by the surface water model for computational efficiency: Δt_g = M × Δt_s where M is an integer greater than 1. Data exchange takes place between the two submodels through a common database at synchronized times (e.g. end of each Δt_g). The coupled model was validated against two previously reported experiments on surface water and groundwater interactions in coastal lagoons. The results suggest that the model represents well the interacting surface water and groundwater flow and solute transport processes in the lagoons. Copyright © 2011 John Wiley & Sons, Ltd.     

Using time‐domain reflectometry to characterize shallow solute transport in an oak woodland hillslope in northern California,USA

The natural heterogeneity of water and solute movement in hillslope soils makes it difficult to accurately characterize the transport of surface‐applied pollutants without first gathering spatially distributed hydrological data. This study examined the application of time‐domain reflectometry (TDR) to measure solute transport in hillslopes. Three different plot designs were used to examine the transport of a conservative tracer in the first 50 cm of a moderately sloping soil. In the first plot, which was designed to examine spatial variability in vertical transport in a 1·2 m² plot, a single probe per meter was found to adequately characterize vertical solute travel times. In addition, a dye and excavation study in this plot revealed lateral preferential flow in small macropores and a transport pattern where solute is focused vertically into preferential flow pathways. The bypass flow delivers solute deeper in the soil, where lateral flow occurs. The second plot, designed to capture both vertical and lateral flow, provided additional evidence confirming the flow patterns identified in the excavation of the first plot. The third plot was designed to examine lateral flow and once again preferential flow of the tracer was observed. In one instance rapid solute transport in this plot was estimated to occur in as little as 3% of the available pore space. Finally, it was demonstrated that the soil anisotropy, although partially responsible for lateral subsurface transport, may also homogenize the transport response across the hillslope by decreasing vertical solute spreading. Copyright © 2002 John Wiley & Sons, Ltd.     

Simulation of field‐scale water flow and bromide transport in a cracked clay soil

Single domain models may seriously underestimate leaching of nutrients and pesticides to groundwater in clay soils with shrinkage cracks. Various two‐domain models have been developed, either empirical or physically based, which take into account the effects of cracks on water flow and solute transport. This paper presents a model concept that uses the clay shrinkage characteristics to derive crack volume and crack depth under transient field conditions. The concept has been developed to simulate field average behaviour of a field with cracks, rather than flow and transport at a small plot. Water flow and solute transport are described with basic physics, which allow process and scenario analysis. The model concept is part of the more general agrohydrological model SWAP, and is applied to a field experiment on a cracked clay soil, at which water flow and bromide transport were measured during 572 days. A single domain model was not able to mimic the field‐average water flow and solute transport. Incorporation of the crack concept considerably improved the simulation of water content and bromide leaching to the groundwater. Still deviations existed between the measured and simulated bromide concentration profiles. The model did not reproduce the observed bromide retardation in the top layer and the high bromide dispersion resulting from water infiltration at various soil depths. A sensitivity analysis showed that the amounts of bromide leached were especially sensitive to the saturated hydraulic conductivity of the top layer, the solute transfer from the soil matrix to crack water flow and the mean residence time of rapid drainage. The shrinkage characteristic and the soil hydraulic properties of the clay matrix showed a low sensitivity. Copyright © 2000 John Wiley & Sons, Ltd.     

Solute Migration from the Aquifer Matrix into a Solution Conduit and the Reverse

A solution conduit has a permeable wall allowing for water exchange and solute transfer between the conduit and its surrounding aquifer matrix. In this paper, we use Laplace Transform to solve a one‐dimensional equation constructed using the Euler approach to describe advective transport of solute in a conduit, a production‐value problem. Both nonuniform cross‐section of the conduit and nonuniform seepage at the conduit wall are considered in the solution. Physical analysis using the Lagrangian approach and a lumping method is performed to verify the solution. Two‐way transfer between conduit water and matrix water is also investigated by using the solution for the production‐value problem as a first‐order approximation. The approximate solution agrees well with the exact solution if dimensionless travel time in the conduit is an order of magnitude smaller than unity. Our analytical solution is based on the assumption that the spatial and/or temporal heterogeneity in the wall solute flux is the dominant factor in the spreading of spring‐breakthrough curves, and conduit dispersion is only a secondary mechanism. Such an approach can lead to the better understanding of water exchange and solute transfer between conduits and aquifer matrix. Highlights:

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Modeling interaction of fluid and salt in an aquifer/lagoon system

To simulate the dynamic interaction between a saline lagoon and a ground water system, a numerical model for two-dimensional, variable-density, saturated-unsaturated, and coupled flow and solute transport (saltwater intrusion by finite elements and characteristics [SIFEC]) was modified to allow the volume of water and mass of salt in the lagoon to vary with each time step. The modified SIFEC allows the stage of a lagoon to vary in accordance with a functional relation between the stage and water volume of the lagoon, and also allows the salt concentration of the lagoon to vary in accordance with the salt budget of the lagoon including chemical precipitation and dissolution of salt. The updated stage and salt concentration of the lagoon are in turn used as transient boundary conditions for the coupled flow and solute transport model. The utility of the modified model was demonstrated by applying it to the eastern Mediterranean coastal region of Turkey for assessing impacts of climate change on the subsurface environment under scenarios of sea level rise, increased evaporation, and decreased precipitation.