Evaluation of longitudinal and transverse dispersivities/distance ratios for tracer test in a radially convergent flow field with scale-dependent dispersion

This study presents a novel mathematical model for analysis of non-axisymmetrical solute transport in a radially convergent flow field with scale-dependent dispersion. A two-dimensional, scale-dependent advection–dispersion equation in cylindrical coordinates is derived based on assuming that the longitudinal and transverse dispersivities increase linearly with the distance of the solute transported from its injected source. The Laplace transform finite difference technique is applied to solve the two-dimensional, scale-dependent advection–dispersion equation with variable-dependent coefficients. Concentration contours for different times, breakthrough curves of average concentration over concentric circles with a fixed radial distance, and breakthrough curves of concentration at a fixed observation point obtained using the scale-dependent dispersivity model are compared with those from the constant dispersivity model. The salient features of scale-dependent dispersion are illustrated during the non-axisymmetrical transport from the injection well into extraction well in a convergent flow field. Numerical tests show that the scale-dependent dispersivity model predicts smaller spreading than the constant-dispersivity model near the source. The results also show that the constant dispersivity model can produce breakthrough curves of averaged concentration over concentric circles with the same shape as those from the proposed scale-dependent dispersivity model at observation point near the extraction well. Far from the extracting well, the two models predict concentration contours with significantly different shapes. The breakthrough curves at observation point near the injection well from constant dispersivity model always produce lesser overall transverse dispersion than those from scale-dependent dispersivity model. Erroneous dimensionless transverse/longitudinal dispersivity ratio may result from parametric techniques which assume a constant dispersivity if the dispersion process is characterized by a distance-dependent dispersivity relationship. A curve-fitting method with an example is proposed to evaluate longitudinal and transverse scale-proportional factors of a field with scale-dependent dispersion.     

Modeling Transport in Transient Ground-Water Flow: An Unacknowledged Approximation

Abstract. During unsteady or transient ground-water flow, the fluid mass per unit volume of aquifer changes as the potentiometric head changes, and solute transport is affected by this change in fluid storage. Three widely applied numerical models of two-dimensional transport partially account for the effects of transient flow by removing terms corresponding to the fluid continuity equation from the transport equation, resulting in a simpler governing equation. However, fluid-storage terms remaining in the transport equation that change during transient flow are, in certain cases, held constant in time in these models. For the case of increasing heads, this approximation, which is unacknowledged in these models'documentation, leads to transport velocities that are too high, and increased concentration at fluid and solute sources. If heads are dropping in time, computed transport velocities are too low. Using parameters that somewhat exaggerate the effects of this approximation, an example numerical simulation indicates solute travel time error of about 14 percent but only minor errors due to incorrect dilution volume. For horizontal flow and transport models that assume fluid density is constant, the product of porosity and aquifer thickness changes in time: initial porosity times initial thickness plus the change in head times the storage coefficient. This formula reduces to the saturated thickness in unconfined aquifers if porosity is assumed to be constant and equal to specific yield. The computational cost of this more accurate representation is insignificant and is easily incorporated in numerical models of solute transport.     

Analysis of solute transport in a divergent flow tracer test with scale‐dependent dispersion

It has been known for many years that dispersivities increase with solute displacement distance in a subsurface. The increase of dispersivities with solute travel distance results from significant variation in hydraulic properties of porous media and was identified in the literature as scale‐dependent dispersion. In this study, Laplace‐transformed analytical solutions to advection‐dispersion equations in cylindrical coordinates are derived for interpreting a divergent flow tracer test with a constant dispersivity and with a linear scale‐dependent dispersivity. Breakthrough curves obtained using the scale‐dependent dispersivity model are compared to breakthrough curves obtained from the constant dispersivity model to illustrate the salient features of scale‐dependent dispersion in a divergent flow tracer test. The analytical results reveal that the breakthrough curves at the specific location for the constant dispersivity model can produce the same shape as those from the scale‐dependent dispersivity model. This correspondence in curve shape between these two models occurs when the local dispersivity at an observation well in the scale‐dependent dispersivity model is 1·3 times greater than the constant dispersivity in the constant dispersivity model. To confirm this finding, a set of previously reported data is interpreted using both the scale‐dependent dispersivity model and the constant dispersivity model to distinguish the differences in scale dependence of estimated dispersivity from these two models. The analytical result reveals that previously reported dispersivity/distance ratios from the constant dispersivity model should be revised by multiplying these values by a factor of 1·3 for the scale‐dependent dispersion model if the dispersion process is more accurately characterized by scale‐dependent dispersion. Copyright © 2007 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.     

On the use and error of approximation in the Domenico (1987) solution

A mathematical solution for solute transport in a three-dimensional porous medium with a patch source under steady-state, uniform ground water flow conditions was developed by Domenico (1987). The solution derivation strategy used an approximate approach to solve the boundary value problem, resulting in a nonexact solution. Variations of the Domenico (1987) solution are incorporated into the software programs BIOSCREEN and BIOCHLOR, which are frequently used to evaluate subsurface contaminant transport problems. This article mathematically elucidates the error in the approximation and presents simulations that compare different versions of the Domenico (1987) solution to an exact analytical solution to demonstrate the potential error inherent in the approximate expressions. Results suggest that the accuracy of the approximate solutions is highly variable and dependent on the selection of input parameters. For solute transport in a medium-grained sand aquifer, the Domenico (1987) solution underpredicts solute concentrations along the centerline of the plume by as much as 80% depending on the case of interest. Increasing the dispersivity, time, or dimensionality of the system leads to increased error. Because more accurate exact analytical solutions exist, we suggest that the Domenico (1987) solution, and its predecessor and successor approximate solutions, need not be employed as the basis for screening tools at contaminated sites.     

Effect of anisotropy on solute transport in degraded fen peat soils

Peat soils are heterogeneous, anisotropic porous media. Compared to mineral soils, there is still limited understanding of physical and solute transport properties of fen peat soils. In this study, we aimed to explore the effect of soil anisotropy on solute transport in degraded fen peat. Undisturbed soil cores, taken in vertical and horizontal direction, were collected from one drained and one restored fen peatland both in a comparable state of soil degradation. Saturated hydraulic conductivity (K _s) and chemical properties of peat were determined for all soil cores. Miscible displacement experiments were conducted under saturated steady state conditions using potassium bromide as a conservative tracer. The results showed that (1) the K _s in vertical direction (K _sv) was significantly higher than that in horizontal direction (K_sh), indicating that K _s of degraded fen peat behaves anisotropically; (2) pronounced preferential flow occurred in vertical direction with a higher immobile water fraction and a higher pore water velocity; (3) the 5% arrival time (a proxy for the strength of preferential flow) was affected by soil anisotropy as well as study site. A strong correlation was found between 5% arrival time and dispersivity, K _s and mobile water fraction; (4) phosphate release was observed from drained peat only. The impact of soil heterogeneity on phosphate leaching was more pronounced than soil anisotropy. The soil core with the strongest preferential flow released the highest amount of phosphate. We conclude that soil anisotropy is crucial in peatland hydrology but additional research is required to fully understand anisotropy effects on solute transport.     

Two-dimensional power series solution for non-axisymmetrical transport in a radially convergent tracer test with scale-dependent dispersion

It has been known for many years that dispersivity increases with solute travel distance in a subsurface environment. The increase of dispersivity with solute travel distance results from the significant variation of hydraulic properties of heterogeneous media and was identified in the literature as scale-dependent dispersion. This study presents an analytical solution for describing two-dimensional non-axisymmetrical solute transport in a radially convergent flow tracer test with scale-dependent dispersion. The power series technique coupling with the Laplace and finite Fourier cosine transform has been applied to yield the analytical solution to the two-dimensional, scale-dependent advection–dispersion equation in cylindrical coordinates with variable-dependent coefficients. Comparison between the breakthrough curves of the power series solution and the numerical solutions shows excellent agreement at different observation points and for various ranges of scale-related transport parameters of interest. The developed power series solution facilitates fast prediction of the breakthrough curves at any observation point.     

Analysis of solute transport with a hyperbolic scale-dependent dispersion model

An empirical hyperbolic scale-dependent dispersion model, which predicts a linear growth of dispersivity close to the origin and the attainment of an asymptotic dispersivity at large distances, is presented for deterministic modelling of field-scale solute transport and the analysis of solute transport experiments. A simple relationship is derived between local dispersivity, which is used in numerical simulations of solute transport, and effective dispersivity, which is estimated from the analysis of tracer breakthrough curves. The scale-dependent dispersion model is used to interpret a field tracer experiment by nonlinear least-squares inversion of a numerical solution for unsaturated transport. Simultaneous inversion of concentration-time data from several sampling locations indicates a linear growth of the dispersion process over the scale of the experiment. These findings are consistent with the results of an earlier analysis based on the use of a constant dispersion coefficient model at each of the sampling depths.     

A spreadsheet program for two-well tracer test data analysis

Two-well tracer tests are often conducted to investigate subsurface solute transport in the field. Analyzing breakthrough curves in extraction and monitoring wells using numerical methods is nontrivial due to highly nonuniform flow conditions. We extended approximate analytical solutions for the advection-dispersion equation for an injection-extraction well doublet in a homogeneous confined aquifer under steady-state flow conditions for equal injection and extraction rates with no transverse dispersion and negligible ambient flow, and implemented the solutions in Microsoft Excel using Visual Basic for Application (VBA). Functions were implemented to calculate concentrations in extraction and monitoring wells at any location due to a step or pulse injection. Type curves for a step injection were compared with those calculated by numerically integrating the solution for a pulse injection. The results from the two approaches are similar when the dispersivity is small. As the dispersivity increases, the latter was found to be more accurate but requires more computing time. The code was verified by comparing the results with published-type curves and applied to analyze data from the literature. The method can be used as a first approximation for two-well tracer test design and data analysis, and to check accuracy of numerical solutions. The code and example files are publicly available.     

Kinematic wave solutions for pollutant transport over an infiltrating plane with finite‐period mixing and mixing zone

Kinematic wave solutions are derived for transport of a conservative non‐point‐source pollutant during a rainfall‐runoff event over an infiltrating plane for two cases: (i) finite‐period mixing and (ii) soil‐mixing zone. Rainfall is assumed to be steady, uniform and finite in duration, and it is assumed to have zero concentration of pollutants. Infiltration is assumed constant in time and space. Prior to the start of rainfall, the pollutant is distributed uniformly over the plane. In the first case, when rainfall occurs, the mixing of pollutant in the runoff water occurs in a finite period of time. In the second case, the chemical concentration is assumed to be a linearly decreasing function of rainfall intensity and overland flow. The solute concentration and discharge are found to depend on the flow characteristics as well as the solute concentration parameters. The characteristics of solute concentration and discharge graphs seem to be similar to those reported in the literature and observed in laboratory experiments. Copyright © 2002 John Wiley & Sons, Ltd.     

Influence of transient flow on contaminant biodegradation

The rate of biodegradation in contaminated aquifers depends to a large extent on dispersive mixing processes that are now generally accepted to result from spatial variations in the velocity field. It has been shown, however, that transient flow fields can also contribute to dispersive mixing. The influence of transient flow on biodegrading contaminants is particularly important since it can enhance mixing with electron acceptors, further promoting the reactive process. Using numerical simulations, the effect of transient flow on the behavior of a biodegradable contaminant is evaluated here both with respect to the development of apparently large horizontal transverse dispersion and also with respect to enhanced mixing between the substrate (electron donor) and electron acceptor. The numerical model BIO3D, which solves for advective-dispersive transport coupled with Monod-type biodegradation of substrates in the presence of an electron acceptor, was used for the simulations. The model was applied in a two-dimensional plan view mode considering a single substrate. Transient flow fields were found to yield larger apparent transverse dispersion because the longitudinal dispersivity also acts transverse to the mean flow direction. In the reactive case, the transient flow field increases substrate-oxygen mixing, which in turn enhances the overall rate of biodegradation. The results suggest that in the case of moderate changes of flow directions, a steady-state flow field can be justified, thereby avoiding the higher computational costs of a fully transient simulation. The use of a higher transverse horizontal dispersivity in a steady flow field can, under these conditions, adequately forecast plume development.     

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.     

Backward probabilistic model of groundwater contamination in non-uniform and transient flow

Backward location and travel time probabilities, which provide information about the former location of contamination in an aquifer, can be used to identify unknown contamination sources. Backward location probability describes the possible upgradient positions of contamination at a known time in the past, and backward travel time probability describes the time required for contamination to travel from a known upgradient location to an observation point. These probabilities are related to adjoint states of resident concentration, and their governing equation is the adjoint of a forward contaminant transport model. Using adjoint theory to obtain the appropriate governing equation, we extend the backward probability model for conservative solutes to more general non-uniform and transient flow fields. In particular, we address three important extensions, spatially-varying porosity, transient flow and temporally-varying porosity, and internal distributed sources and sinks of solute and water. For the first time we learn that forward and backward location and travel time probabilities are not necessarily equivalent to adjoint states, but are related to them. The extensions are illustrated using a vertically-integrated groundwater model, creating transient flow by a step change in pumping and using areal recharge as an internal distributed source. Both the movement and spread of probabilities are affected. With internal sources of water, there are two interpretations of backward probability, depending on whether or not the source of water is also a source of solute. The results demonstrate how the backward probability model can be applied to other, perhaps more important, non-uniform and transient flow conditions, with time- and space-varying water storage, such as time-varying pumping or unsaturated (or saturated–unsaturated) flow and transport with spatially- and temporally-varying moisture content.     

Estimation of water saturation dependence of dispersion in unsaturated porous media: experiments and modelling analysis

In the dispersion theory, a linear relationship has been verified between the coefficient of hydrodynamic dispersion and water velocity, both in saturated and in unsaturated porous media. But for unsaturated soils the variability of flow directions and microscopic velocities can be larger than in saturated soils because of the lower degree of water saturation. This leads to an increased dispersion. Therefore, relationships between water content and relative water velocity fluctuations and water content together with the coefficient of dispersivity in unsaturated porous media respectively have been investigated systematically by displacement experiments in glass beads and coarse-textured sandy soil columns. The breakthrough curves (BTCs) of chloride showed that an increase of solute mixing with a decrease of water content was caused by an increase of flow velocity fluctuations for different pathways. In order to explain the observed tailing effect in unsaturated flow, two mathematical models were used to fit theoretically derived nonlinear functions of water content dependent dispersivities for both porous media. The close agreement between the observed and computed results suggests that the theoretical model of hydrodynamic dispersion can be extended to transport in unsaturated porous media, providing that BTCs of the effluent water are used to estimate representative dispersivity parameters of soils.     

Can Contaminant Transport Models Predict Breakthrough?

A solute breakthrough curve measured during a two-well tracer test was successfully predicted in 1986 using specialized contaminant transport models. Water was injected into a confined, unconsolidated sand aquifer and pumped out 125 feet (38.3 m) away at the same steady rate. The injected water was spiked with bromide for over three days; the outflow concentration was monitored for a month. Based on previous tests, the horizontal hydraulic conductivity of the thick aquifer varied by a factor of seven among 12 layers. Assuming stratified flow with small dispersivities, two research groups accurately predicted breakthrough with three-dimensional (12-layer) models using curvilinear elements following the arc-shaped flowlines in this test.
Can contaminant transport models commonly used in industry, that use rectangular blocks, also reproduce this breakthrough curve? The two-well test was simulated with four MODFLOW-based models, MT3D (FD and HMOC options), MODFLOWT, MOC3D, and MODFLOW-SURFACT.
Using the same 12 layers and small dispersivity used in the successful 1986 simulations, these models fit almost as accurately as the models using curvilinear blocks. Subtle variations in the curves illustrate differences among the codes. Sensitivities of the results to number and size of grid blocks, number of layers, boundary conditions, and values of dispersivity and porosity are briefly presented. The fit between calculated and measured breakthrough curves degenerated as the number of layers and/or grid blocks decreased, reflecting a loss of model predictive power as the level of characterization lessened. Therefore, the breakthrough curve for most field sites can be predicted only qualitatively due to limited characterization of the hydrogeology and contaminant source strength.     

Effects of compound-specific dilution on transient transport and solute breakthrough: A pore-scale analysis

This pore-scale modeling study in saturated porous media shows that compound-specific effects are important not only at steady-state and for the lateral displacement of solutes with different diffusivities but also for transient transport and solute breakthrough. We performed flow and transport simulations in two-dimensional pore-scale domains with different arrangement of the solid grains leading to distinct characteristics of flow variability and connectivity, representing mildly and highly heterogeneous porous media, respectively. The results obtained for a range of average velocities representative of groundwater flow (0.1–10 m/day), show significant effects of aqueous diffusion on solute breakthrough curves. However, the magnitude of such effects can be masked by the flux-averaging approach used to measure solute breakthrough and can hinder the correct interpretation of the true dilution of different solutes. We propose, as a metric of mixing, a transient flux-related dilution index that allows quantifying the evolution of solute dilution at a given position along the main flow direction. For the different solute transport scenarios we obtained dilution breakthrough curves that complement and add important information to traditional solute breakthrough curves. Such dilution breakthrough curves allow capturing the compound-specific mixing of the different solutes and provide useful insights on the interplay between advective and diffusive processes, mass transfer limitations, and incomplete mixing in the heterogeneous pore-scale domains. The quantification of dilution for conservative solutes is in good agreement with the outcomes of mixing-controlled reactive transport simulations, in which the mass and concentration breakthrough curves of the product of an instantaneous transformation of two initially segregated reactants were used as measures of reactive mixing.     

Analytical power series solutions to the two-dimensional advection–dispersion equation with distance-dependent dispersivities

As is frequently cited, dispersivity increases with solute travel distance in the subsurface. This behaviour has been attributed to the inherent spatial variation of the pore water velocity in geological porous media. Analytically solving the advection–dispersion equation with distance-dependent dispersivity is extremely difficult because the governing equation coefficients are dependent upon the distance variable. This study presents an analytical technique to solve a two-dimensional (2D) advection–dispersion equation with linear distance-dependent longitudinal and transverse dispersivities for describing solute transport in a uniform flow field. The analytical approach is developed by applying the extended power series method coupled with the Laplace and finite Fourier cosine transforms. The developed solution is then compared to the corresponding numerical solution to assess its accuracy and robustness. The results demonstrate that the breakthrough curves at different spatial locations obtained from the power series solution show good agreement with those obtained from the numerical solution. However, owing to the limited numerical operation for large values of the power series functions, the developed analytical solution can only be numerically evaluated when the values of longitudinal dispersivity/distance ratio e_L exceed 0·075. Moreover, breakthrough curves obtained from the distance-dependent solution are compared with those from the constant dispersivity solution to investigate the relationship between the transport parameters. Our numerical experiments demonstrate that a previously derived relationship is invalid for large e_L values. The analytical power series solution derived in this study is efficient and can be a useful tool for future studies in the field of 2D and distance-dependent dispersive transport. Copyright © 2008 John Wiley & Sons, Ltd.     

Direct simulation of solute recycling in irrigated areas

Solute recycling from irrigation can be described as the process that occurs when the salt load that is extracted from irrigation wells and distributed on the fields is returned to the groundwater below irrigated surfaces by deep percolation. Unless the salt load leaves the system by means of drains or surface runoff, transfer to the groundwater will take place, sooner or later. This can lead to solute accumulation and thus to groundwater degradation, particularly in areas where extraction rates exceed infiltration rates (semi-arid and arid regions). Thus, considerable errors can occur in a predictive solute mass budget if the recycling process is not accounted for in the calculation. A method is proposed which allows direct simulation of solute recycling. The transient solute response at an extraction well is shown to be a superposition of solute mass flux contributions from n recycling cycles and is described as a function of the travel time distribution between a recycling point and a well. This leads to an expression for a transient ‘recycling source’ term in the advection–dispersion equation, which generates the effect of solute recycling. At long times, the ‘recycling source’ is a function of the local capture probability of the irrigation well and the solute mass flux captured by the well from the boundaries. The predicted concentration distribution at steady state reflects the maximum spatial concentration distribution in response to solute recycling and can thus be considered as the solute recycling potential or vulnerability of the entire domain for a given hydraulic setting and exploitation scheme. Simulation of the solute recycling potential is computationally undemanding and can therefore, for instance, be used for optimisation purposes. Also, the proposed method allows transient simulation of solute recycling with any standard flow and transport code.     

Pollutant’s Horizontal Dispersion Along and Against Sinusoidally Varying Velocity from a Pulse Type Point Source

An analytical solution of a two-dimensional advection diffusion equation with time dependent coefficients is obtained by using Laplace Integral Transformation Technique. The horizontal medium of solute transport is considered of semi-infinite extent along both the longitudinal and lateral directions. The input concentration is assumed at an intermediate position of the domain. It helps to evaluate concentration level along the flow as well as against the flow through one model only. The source of the input concentration is considered to be of pulse type. In the presence of the source, it is assumed to be decreasing very slowly with time, and just after the elimination of the source it is assumed to be zero. The dispersion coefficient and the advection parameter are considered directly proportional to each other. The analytical solution may be used to predict the solute concentration level with position and time in an open medium as well as in a porous medium. The effect of heterogeneity on the solute transport may also be predicted.     

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.