Particular application of a mathematical transport model incorporating sub-surface reactive pollutants

In the first part of the paper, a 2D transport equation was used, which takes into consideration only the non-linear adsorption term described by means of non-linear Freundlich isotherm. The equation of this type was the basis of numerical calculations of concentration for chosen indicators (nitrates and biochemical oxygen demand). In the second part, the presented equation took into consideration not only the nonlinear adsorption term, but also the term of biodegradation (biological denitrification), described by the first-order microbial reaction, most often applied in practice. Numerical calculations in the first and second parts of the paper were carried out using the same numerical scheme and the author’s program PCCS-2, presented in Aniszewski (2009) and compared afterwards with the measured concentration values. The calculation results made it possible to define the reduction degree of concentration reduction for the chosen chemicals as a result of reversible sorption and biodegradation processes.     

Biodegradation and transport of benzene,toluene, and xylenes in a simulated aquifer: comparison of modelled and experimental results

Both laboratory experiments and numerical modelling were conducted to study the biodegradation and transport of benzene–toluene–xylenes (BTX) in a simulated semi‐confined aquifer. The factors incorporated into the numerical model include advection, hydrodynamic dispersion, adsorption, and biodegradation. The various physico‐chemical parameters required by the numerical model were measured experimentally. In the experimental portion of the study, BTX compounds were introduced into the aquifer sand. After the contaminants had been transported through the system, BTX concentrations were measured at 12 equally spaced wells. Subsequently, microorganisms obtained from the activated sludge of a sewage treatment plant and cultured in BTX mixtures were introduced into the aquifer through the 12 sampling wells. The distribution data for BTX adsorption by the aquifer sand form a nonlinear isotherm. The degree of adsorption by the sand varies, depending on the composition of the solute. The degradation time, measured from the time since the bacteria were added to the aquifer until a specific contaminant was no longer detectable, was 35–42 h for BTX. The dissolved oxygen, after degradation by BTX compounds and bacteria, was consumed by about 40–60% in the entire simulated aquifer; thus the aerobic conditions were maintained. This study provides insights for the biodegradation and transport of BTX in aquifers by numerical modelling and laboratory experiments. Experimental and numerical comparisons indicate that the results by Monod degradation kinetics are more accurate than those by the first‐order degradation kinetics. Copyright © 2002 John Wiley & Sons, Ltd.     

Modeling of Contaminant Movement Near Pumping Wells: Saturated-Unsaturated Flow with Particle Tracking

A transient axisymmetric saturated-unsaturated numerical flow model was coupled with a particle tracking model to investigate the movement of contaminants when a shallow unconfined aquifer is pumped at a constant rate. The particle tracking model keeps track of locations and masses of solutes in the aquifer, and the time of capture by the well. At the end of each time-step the flow model solves the Richard's equation for the hydraulic head distribution from which elemental velocities are calculated. Solutes are then displaced for a period equivalent to the time-step using both the magnitude and direction of the elemental velocities. Numerical experiments were performed to investigate effluent concentrations in wells with screens of different length and in different positions relative to zones of stratified contamination. At early times of pumping the effluent concentrations were similar to the concentrations adjacent to the well screen, but at late times, the concentrations approached the vertically averaged concentration in the aquifer. Time to attain the vertically averaged concentration was determined by the well geometry, initial location of the contaminant plume in relation to the well screen, and hydraulic properties of the aquifer. The results are consistent with the hydraulics of flow to a pumping well and of particular importance, they demonstrate that short-term pump tests could give erroneous design concentrations for pump-and-treat systems. The model provides a means of quantifying arrival times and mixing ratios. It could therefore provide a useful means of designing production wells in aquifers with stratified contamination and more efficient recovery systems for aquifer remediation.     

Positive solution of two-dimensional solute transport in heterogeneous aquifers

The transport of contaminants in aquifers is usually represented by a convection-dispersion equation. There are several well-known problems of oscillation and artificial dispersion that affect the numerical solution of this equation. For example, several studies have shown that standard treatment of the cross-dispersion terms always leads to a negative concentration. It is also well known that the numerical solution of the convective term is affected by spurious oscillations or substantial numerical dispersion. These difficulties are especially significant for solute transport in nonuniform flow in heterogeneous aquifers. For the case of coupled reactive-transport models, even small negative concentration values can become amplified through nonlinear reaction source/sink terms and thus result in physically erroneous and unstable results. This paper includes a brief discussion about how nonpositive concentrations arise from numerical solution of the convection and cross-dispersion terms. We demonstrate the effectiveness of directional splitting with one-dimensional flux limiters for the convection term. Also, a new numerical scheme for the dispersion term that preserves positivity is presented. The results of the proposed convection scheme and the solution given by the new method to compute dispersion are compared with standard numerical methods as used in MT3DMS.     

Effects of rate-limited mass transfer on water sampling with partially penetrating wells

Nonequilibrium concentration type curves are numerically developed and sensitivity analyses are performed to examine the relationships between effluent concentrations in partially penetrating monitoring/extraction wells, the vertical plume shape, and the mass transfer characteristics of the aquifer. The governing two-dimensional, axisymmetric nonequilibrium solute transport equation is solved in three stages using an operator-splitting approach. In the first two stages, the advection and dispersion terms are solved with the Eulerian-Lagrangian method, based on the backward method of characteristics for advection and the standard implicit Galerkin finite element method for dispersion. In the third step, the first-order, immobile-mobile domain mass transfer term is computed analytically for both two-site and lognormally distributed, multirate models. Effluent concentration variations with time and contour plots of the pore water concentration distribution in the aquifer are compared for a wide range of field- and laboratory-measured mass transfer rates, various plume shapes, and relevant physical/chemical parameter values, including pumping rate, vertical anisotropy ratio, retardation factor, and porosity. The simulation results show that rate-limited mass transfer can have a significant impact on sample and aquifer pore water concentrations during three-dimensional transport to a partially penetrating well. An alternative dimensionless form of the nonequilibrium solute transport equation is derived to illustrate the key parameter groupings that quantify rate-limited sorption effects and show the relative importance of individual parameters. A hypothetical field application example demonstrates the fitting of dimensional type curves to discrete-interval sampling data in order to evaluate the mass transfer characteristics of an aquifer and shows how type curve superposition can be used to model complex plume shapes.     

Predictive optimal control for seismic analysis of non-linear and hysteretic structures

In this paper, an effective active predictive control algorithm is developed for the vibration control of non-linear hysteretic structural systems subjected to earthquake excitation. The non-linear characteristics of the structural behaviour and the effects of time delay in both the measurements and control action are included throughout the entire analysis (design and validation). This is very important since, in current design practice, structures are assumed to behave non-linearly, and time delays induced by sensors and actuator devices are not avoidable. The proposed algorithm focuses on the instantaneous optimal control approach for the development of a control methodology where the non-linearities are brought into the analysis through a non-linear state vector and a non-linear open-loop term. An autoregressive (AR) model is used to predict the earthquake excitation to be considered in the prediction of the structural response. A performance index that is quadratic in the control force and in the predicted non-linear states, with two additional energy related terms, and that is subjected to a non-linear constraint equation, is minimized at every time step. The effectiveness of the proposed closed-open loop non-linear instantaneous optimal prediction control (CONIOPC) strategy is presented by the results of numerical simulations. Since non-linearity and time-delay effects are incorporated in the mathematical model throughout the derivation of the control methodology, good performance and stability of the controlled structural system are guaranteed. Copyright © 1999 John Wiley & Sons, Ltd.     

Transport of three herbicides in ground water at Twin Lake test site,Chalk River,Ontario, Canada

A field tracer test performed under natural flow conditions at the Twin Lake test site, Chalk River Laboratories of the Atomic Energy of Canada Ltd. in Chalk River, Ontario, Canada, using tritium and three herbicides (Chlortoluron, Terbuthylazine, and Pendimethalin) was interpreted using the dispersion equation with a combined reaction model. The reaction model couples an instantaneous equilibrium reaction governed by a linear adsorption isotherm with a reversible or irreversible kinetic reaction of the first order, and decay. An improved interpretation method consists of a simultaneous fitting of theoretical concentration and mass-recovery curves to the experimental data, which leads to a more reliable determining of reaction models and improves the accuracy of fitting. Tritium served as the reference tracer to determine the flow velocity, dispersivity, and the recovery of the herbicides. Chlortoluron was slightly delayed by equilibrium exchange with strongly reduced concentration due to an irreversible kinetic reaction and/or decay. Terbuthilazine was slightly delayed by equilibrium exchange, with strongly reduced concentration due to a reversible kinetic reaction with some influence of decay. A strong equilibrium reaction and a strong reversible kinetic reaction without degradation governed the transport of Pendimethalin, reducing considerably its concentration. The results obtained show that simulations based only on Kd and decay constant, especially if these parameters are found in the laboratory, may considerably differ from those performed with reaction parameters determined in properly performed field tests. The dominant reaction types, and the values of parameters found in the study, supply useful information on the transport of the investigated herbicides in sandy aquifers under natural flow conditions.     

Multidimensional analytical models for isotope ratios in groundwater pollutant plumes of organic contaminants undergoing different biodegradation kinetics

This work presents analytical models which are able to predict contours of concentrations and isotope ratios of organic pollutants in homogeneous aquifers. Four analytical solutions of the advective–dispersive transport equation for reactive transport from the literature differing in assumptions regarding biodegradation kinetics were used. Stable isotope ratios are computed after modelling the individual reactive transport of isotopic species in the aquifer, which respond differently to fractionation by biodegradation or sorption. The main finding of this study is that the isotope ratios in the plumes are very sensitive to the assumptions underlying the biodegradation kinetics in the models. When biodegradation occurs throughout the core of the plume as first-order reaction, the transversal gradients in isotope ratios are smooth. When biodegradation occurs in a bi-molecular reaction with an electron acceptor (modelled by double-Monod kinetics), steep transversal isotope gradients are predicted. When the reaction rates approach instantaneous reaction along the plume fringes, isotope shifts in the core of the plume disappear. A model incorporating plume and fringe degradation produces the most plausible predictions of isotope ratios in this study. It is shown furthermore that isotope fractionation by sorption causes an even different pattern of isotope ratios, with positive shifts restricted to near the forerunning front of an expanding plume. The models developed in this work can serve for the validation of numerical models and may be incorporated in natural attenuation support systems such as e.g. BIOSCREEN.     

Biogeochemical evolution of a landfill leachate plume, Norman, Oklahoma

Leachate from municipal landfills can create groundwater contaminant plumes that may last for decades to centuries. The fate of reactive contaminants in leachate-affected aquifers depends on the sustainability of biogeochemical processes affecting contaminant transport. Temporal variations in the configuration of redox zones downgradient from the Norman Landfill were studied for more than a decade. The leachate plume contained elevated concentrations of nonvolatile dissolved organic carbon (NVDOC) (up to 300 mg/L), methane (16 mg/L), ammonium (650 mg/L as N), iron (23 mg/L), chloride (1030 mg/L), and bicarbonate (4270 mg/L). Chemical and isotopic investigations along a 2D plume transect revealed consumption of solid and aqueous electron acceptors in the aquifer, depleting the natural attenuation capacity. Despite the relative recalcitrance of NVDOC to biodegradation, the center of the plume was depleted in sulfate, which reduces the long-term oxidation capacity of the leachate-affected aquifer. Ammonium and methane were attenuated in the aquifer relative to chloride by different processes: ammonium transport was retarded mainly by physical interaction with aquifer solids, whereas the methane plume was truncated largely by oxidation. Studies near plume boundaries revealed temporal variability in constituent concentrations related in part to hydrologic changes at various time scales. The upper boundary of the plume was a particularly active location where redox reactions responded to recharge events and seasonal water-table fluctuations. Accurately describing the biogeochemical processes that affect the transport of contaminants in this landfill-leachate-affected aquifer required understanding the aquifer's geologic and hydrodynamic framework.     

A new analytical solution of tidal water table fluctuations in a coastal unconfined aquifer

This paper presents a new perturbation solution of the non-linear Boussinesq equation for one-dimensional tidal groundwater flow in a coastal unconfined aquifer. Built upon the work of Parlange et al. [Parlange, J.-Y., Stagnitti, F., Starr, J.L., Braddock, R.D., 1984. Free-surface flow in porous media and periodic solution of the shallow-flow approximation, J. Hydrol., 70, 251–263], the solution adopts a new perturbation parameter that is by definition less than unit, and thus is applicable to a wider range of physical conditions within the constraint of the Boussinesq approximation. This approach avoids a secular term in the third-order perturbation equation of Parlange et al. (1984), enabling the derivation of the third- and higher-order solutions. In comparison with a numerical (“exact”) solution, the new perturbation solution is shown to be slightly more accurate than that of Parlange et al. (1984) with the second-order approximation. The obtained third-order solution exhibits considerable improvement in accuracy. In relatively simple analytical forms, the present perturbation solution will help to understand better the non-linear characteristics of tidal water table fluctuations in as modeled by the non-linear Boussinesq equation coastal unconfined aquifers.     

Microbial transport in soils and groundwater: A numerical model

To serve as a tool in the long term evaluation of the risk of accumulation of microbial contaminants (bacteria and viruses) entering soil and groundwater, a mathematical model is developed to predict the spatial and temporal distribution of pollutant concentration. The governing equation for bacterial transport is coupled with a transport equation for the bacterial nutrient present in the seeping wastewater. The deposition and declogging mechanisms are incorporated into the model as a rate process for bacteria and as an equilibrium partitioning for viruses. While the decay is assumed to be a first order reaction and the growth of bacteria is assumed to follow the Monod equation, the model equations exhibit nonlinearity and coupling. A simplified set of equations is solved analytically to test the numerical results. Coupled numerical solutions in one and two dimensions are obtained by the Galerkin method at spatial and temporal locations of interest. Cases studied included a soil column and a horizontal two-dimensional field coupled with the one dimensional solution. For these examples, the bacteria are removed almost totally within the top 7 cm of soil with minimal risk of clogging.     

Semi-analytical solutions for solute transport in fractured porous media using a strip source of finite width

Transient and steady-state analytical solutions are derived to investigate solute transport in a fractured porous medium consisting of evenly spaced, parallel discrete fractures. The solutions incorporate a finite width strip source, longitudinal and transverse dispersion in the fractures, source decay, aqueous phase decay, one-dimensional diffusion into the matrix, sorption to fracture walls, and sorption within the matrix. The solutions are derived using Laplace and Fourier transforms, and inverted by interchanging the order of integration and utilizing a numerical Laplace inversion algorithm. The solutions are verified for simplified cases by comparison to solutions derived by Batu [Batu V. A generalized two-dimensional analytical solution for hydrodynamic dispersion in bounded media with the first-type condition at the source. Wat Resour Res 1989;25(6):1125] and Sudicky and Frind [Sudicky EA, Frind EO. Contaminant transport in fractured porous media: analytical solutions for a system of parallel fractures. Wat Resour Res 1982;18(6):1634]. The application of the solutions to a fractured sandstone demonstrates that narrower source widths and larger values of transverse dispersivity both lead to lower downstream concentrations in the fractures and shorter steady-state plumes. The incorporation of aqueous phase decay and source concentration decay both lead to lower concentrations and shorter plumes, with even moderate amounts of decay significantly shortening the persistence of contamination.     

Multicomponent solute transport with sorption and soluble complexation

A numerical algorithm for predicting the migration of multiple subsurface pollutants has been developed accounting for dispersion, convection, soluble complexation and solid phase accumulations (sorption). The basis of the model is a finite element solution of the mass transport equation. The essence of the algorithm is the treatment of the sorption terms as implicit functions of the total soluble concentrations facilitating a general and modularized treatment of solution phase and solid phase chemistry. Examples are presented illustrating the effects of soluble complexation and competitive sorption on the transport of multicomponent solutions.     

Irreversible sorption of benzene in sandy aquifer materials

Hydrocarbon compounds in aquifers are generally known to show a retardation effect due to sorption onto the surfaces of solid particles. In this study, we investigated the effect of sorption on the transport of benzene in sandy aquifer materials by conducting batch and column tests for both sandy aquifer materials and sandy materials to which had been added 0·5% powdered activated carbon. The batch test was conducted by equilibrating dry materials with benzene solutions of various initial concentrations, and by analysing the concentrations of benzene in the initial and equilibrated solutions using high‐performance liquid chromatography (HPLC). The column test was performed to monitor the concentrations of effluent versus time, known as a breakthrough curve (BTC). We injected KCl and benzene solutions as tracers into the inlet boundary as two different types of square pulse and step, and monitored the effluent concentrations at the exit boundary under a steady‐state condition using an electrical conductivity meter and HPLC. Simulation of benzene transport was performed using the convective–dispersive equation model with the distribution coefficients obtained from the batch test and the transport parameters of the conservative solute KCl from the column test. The observed BTCs of KCl and benzene for pulse injection showed that the arrival times of the peaks of both tracers coincided well, but the relative peak concentration of benzene was much lower than that of KCl. Comparison of the simulated and observed BTCs showed a great discrepancy for all cases of injection mode and material texture, indicating the absence of retardation effect. These results reveal that the predominant process affecting the benzene transport in the sandy aquifer materials is an irreversible sorption rather than retardation. This tentative conclusion was verified by simulation of benzene transport using an irreversible sorption parameter that led to a good agreement between the simulated and observed BTCs. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

Hydraulic diffusivity and macrodispersivity calculations embedded in a geographic information system

Abstract

A numerical technique is presented whereby aquifer hydraulic diffusivities (D) and macrodispersivities (α) are calculated by linear equations rewritten from flow and solute transport differential equations. The approach requires a GIS to calculate spatial and temporal hydraulic head (h) and solute concentration gradients. The model is tested in Portugal, in a semi-confined aquifer periodically monitored for h and chloride/sulphate concentrations. Average D (0.46 m²/s) and α (1975 m) compare favourably with literature results. The relationship between α and scale (L) is also investigated. In this context, two aquifer groups could be identified: the first group is heterogeneous at the “macroscopic” scale (solute travelled distances <1 km), but homogeneous at the “megascopic” scale. The overall scale dependency in this case is given by an equation of logarithmic type. The second group is heterogeneous at the macroscopic and megascopic scales, with a scale dependency of linear type.

Citation Pacheco, F.A.L., 2013. Hydraulic diffusivity and macrodispersivity calculations embedded in a geographic information system. Hydrological Sciences Journal, 58 (4), 930–944.     

Ground-Water Quality at a Creosote Waste Site

An abandoned creosote facility in Conroe, Texas, has become a field site for the National Center for Ground Water Research (NCGWR) at Rice University. Ground-water contamination in the shallow aquifer beneath the site was characterized by sampling soils and water quality at 14 monitoring wells and 35 boreholes. Results from six sampling trips over two years for inorganic and organic chemical concentrations in the ground water show wells around the site were contaminated to levels above 800 μg/l for naphthalene, 400 μg/1 for methyl naphthalene, and 150 μg/1 for dibenzofuran. Conservative constituents, traced by chloride concentrations up to 75 mg/l, have migrated 300 ft (90 m) downgradient of the site. Organic contaminants have been adsorbed and microbially degraded in their migration from the waste source as evidenced by their attenuated concentrations. Detailed field pump tests have been performed to evaluate hydraulic conductivity at several of the shallow wells. The U.S. Geological Survey (USGS) Solute Transport Model (Konikow and Bredehoeft, 1978) has been used to predict chloride plume patterns and evaluate parameters which govern transport processes at the Conroe waste site.     

A quasi three-dimensional model for flow and transport in unsaturated and saturated zones: 1. Implementation of the quasi two-dimensional case

A quasi three-dimensional (QUASI 3-D) model is presented for simulating the subsurface water flow and solute transport in the unsaturated and in the saturated zones of soil. The model is based on the assumptions of vertical flow in the unsaturated zone and essentially horizontal groundwater flow. The 1-D Richards equation for the unsaturated zone is coupled at the phreatic surface with the 2-D flow equation for the saturated zone. The latter was obtained by averaging 3-D flow equation in the saturated zone over the aquifer thickness. Unlike the Boussinesq equation for a leaky-phreatic aquifer, the developed model does not contain a storage term with specific yield and a source term for natural replenishment. Instead it includes a water flux term at the phreatic surface through which the Richards equation is linked with the groundwater flow equation. The vertical water flux in the saturated zone is evaluated on the basis of the fluid mass balance equation while the horizontal fluxes, in that equation, are prescribed by Darcy law. A 3-D transport equation is used to simulate the solute migration. A numerical algorithm to solve the problem for the general quasi 3-D case was developed. The developed methodology was exemplified for the quasi 2-D cross-sectional case (QUASI2D). Simulations for three synthetic problems demonstrate good agreement between the results obtained by QUASI2D and two fully 2-D flow and transport codes (SUTRA and 2DSOIL). Yet, simulations with the QUASI2D code were several times faster than those by the SUTRA and the 2DSOIL codes.     

利用积分方程法计算双侧向测井仪的井眼校正曲线

对电测井积分方程中的各积分项进行了理论推导，利用任意闭合曲面对曲面外一点所张立体角为零的特点将电势积分项由对柱体侧面的积分转化为对柱体底面的积分，从而使在纵向采用线性插值条件下给出了电势积分项的解析解．对电流积分项进行了化简，使之由曲面积分化为一维数值积分．利用所得结果计算了双侧向测井仪的井眼校正曲线．