Successive identification of biodegradation rates for multiple sequentially reactive contaminants in groundwater.

At the field scale, the biodegradation rate is usually estimated from analytical solutions to single species transport with first-order reactions, using measured data as input. Because many contaminants, e.g., chlorinated solvents, are degraded in a sequential pattern, with degradation products further reacting to produce new species, it is of great interest to quantify the transformation rate of every reaction. The conventional inverse solutions for identifying the transformation rates are limited to single species problems. In the present study, we propose a successive optimization approach to identify the biodegradation rate for each species by using a previously developed analytical solution to multi-species first-order reactive transport using data obtained at the field scale. By specifying a link between analytical solutions to sequentially reactive transport problems and optimization methods and assuming constant transport parameters (velocity, dispersivities, and retardation factors), the first-order transformation rates are optimized successively from parent species to its daughter species.     

Operator-splitting procedures for reactive transport and comparison of mass balance errors

Operator-splitting (OS) techniques are very attractive for numerical modelling of reactive transport, but they induce some errors. Considering reactive mass transport with reversible and irreversible reactions governed by a first-order rate law, we develop analytical solutions of the mass balance for the following operator-splitting schemes: standard sequential non-iterative (SNI), Strang-splitting SNI, standard sequential iterative (SI), extrapolating SI, and symmetric SI approaches. From these analytical solutions, the operator-splitting methods are compared with respect to mass balance errors and convergence rates independently of the techniques used for solving each operator. Dimensionless times, NOS, are defined. They control mass balance errors and convergence rates. The following order in terms of decreasing efficiency is proposed: symmetric SI, Strang-splitting SNI, standard SNI, extrapolating SI and standard SI schemes. The symmetric SI scheme does not induce any operator-splitting errors, the Strang-splitting SNI appears to be O(N2OS) accurate, and the other schemes are first-order accurate.     

A reaction-based paradigm to model reactive chemical transport in groundwater with general kinetic and equilibrium reactions

This paper presents a reaction-based water quality transport model in subsurface flow systems. Transport of chemical species with a variety of chemical and physical processes is mathematically described by M partial differential equations (PDEs). Decomposition via Gauss-Jordan column reduction of the reaction network transforms M species reactive transport equations into two sets of equations: a set of thermodynamic equilibrium equations representing N(E) equilibrium reactions and a set of reactive transport equations of M-N(E) kinetic-variables involving no equilibrium reactions (a kinetic-variable is a linear combination of species). The elimination of equilibrium reactions from reactive transport equations allows robust and efficient numerical integration. The model solves the PDEs of kinetic-variables rather than individual chemical species, which reduces the number of reactive transport equations and simplifies the reaction terms in the equations. A variety of numerical methods are investigated for solving the coupled transport and reaction equations. Simulation comparisons with exact solutions were performed to verify numerical accuracy and assess the effectiveness of various numerical strategies to deal with different application circumstances. Two validation examples involving simulations of uranium transport in soil columns are presented to evaluate the ability of the model to simulate reactive transport with complex reaction networks involving both kinetic and equilibrium reactions.     

Intercomparison of reactive transport models applied to UO2 oxidative dissolution and uranium migration

Oxidative dissolution of uranium dioxide (UO(2)) and the subsequent migration of uranium in a subsurface environment and an underground waste disposal have been simulated with reactive transport models. In these systems, hydrogeological and chemical processes are closely entangled and their interdependency has been analyzed in detail, notably with respect to redox reactions, kinetics of mineralogical evolution and hydrodynamic migration of species of interest. Different codes, where among CASTEM, CHEMTRAP and HYTEC, have been used as an intercomparison and verification exercise. Although the agreement between codes is satisfactory, it is shown that the discretization method of the transport equation (i.e. finite elements (FE) versus mixed-hybrid FE and finite differences) and the sequential coupling scheme may lead to systematic discrepancies.     

Intercomparison of reactive transport models applied to UO2 oxidative dissolution and uranium migration

Oxidative dissolution of uranium dioxide (UO₂) and the subsequent migration of uranium in a subsurface environment and an underground waste disposal have been simulated with reactive transport models. In these systems, hydrogeological and chemical processes are closely entangled and their interdependency has been analyzed in detail, notably with respect to redox reactions, kinetics of mineralogical evolution and hydrodynamic migration of species of interest.Different codes, where among CASTEM, CHEMTRAP and HYTEC, have been used as an intercomparison and verification exercise. Although the agreement between codes is satisfactory, it is shown that the discretization method of the transport equation (i.e. finite elements (FE) versus mixed-hybrid FE and finite differences) and the sequential coupling scheme may lead to systematic discrepancies.     

Multicomponent simulation of wastewater-derived nitrogen and carbon in shallow unconfined aquifers. I. Model formulation and performance

One of the most common methods to dispose of domestic wastewater involves the release of septic effluent from drains located in the unsaturated zone. Nitrogen from such systems is currently of concern because of nitrate contamination of drinking water supplies and eutrophication of coastal waters. The objectives of this study are to develop and assess the performance of a mechanistic flow and reactive transport model which couples the most relevant physical, geochemical and biochemical processes involved in wastewater plume evolution in sandy aquifers. The numerical model solves for variably saturated groundwater flow and reactive transport of multiple carbon- and nitrogen-containing species in a three-dimensional porous medium. The reactive transport equations are solved using the Strang splitting method which is shown to be accurate for Monod and first- and second-order kinetic reactions, and two to four times more efficient than sequential iterative splitting. The reaction system is formulated as a fully kinetic chemistry problem, which allows for the use of several special-purpose ordinary differential equation (ODE) solvers. For reaction systems containing both fast and slow kinetic reactions, such as the combined nitrogen-carbon system, it is found that a specialized stiff explicit solver fails to obtain a solution. An implicit solver is more robust and its computational performance is improved by scaling of the fastest reaction rates. The model is used to simulate wastewater migration in a 1-m-long unsaturated column and the results show significant oxidation of dissolved organic carbon (DOC), the generation of nitrate by nitrification, and a slight decrease in pH.     

Simultaneous transport of parent and daughter compounds in aquifers with linear first-order sorption kinetics

In soils, daughter compounds may be generated from a parent compound by microbial metabolism, chemical reactions, radioactive decay, or other mechanisms. These daughter compounds are also acted upon by soil physical, chemical and biological processes. A system often referred to as a cascade or chain of compounds system results. While a great deal of attention has been given to this problem with the linear equilibrium assumption applied uniformly to all transport and reacting compounds, little attention has been given to the simultaneous transport and fate of a parent-daughter chain with a first-order rate assumed for the adsorption-desorption kinetics of each compound and with the soil partitioned into three sorption classes.A general one-dimensional cascade or chain model for the simultaneous transport of parent and daughter compounds in sorbing, homogeneous, water table aquifers is presented. The model is based on an advective-dispersive mass accounting formulation for both compounds and includes: (a) first-order rate of conversion of parent to daughter; (2) first-order rates of loss of either parent or daughter or both due to metabolism, chemical reaction and/or irreversible processes; (3) partitioning of the aquifer material into three sorption classes, namely mildly sorbing, strongly sorbing and organic matter; (4) linear first-order kinetic rules for adsorption and desorption operating on each of the sorbing soil fractions for each compound; (5) constantly emitting sources of rectangular shape of parent compound; and (6) mass accounting boundary conditions; and a tailorable initial distribution on [0, ∞). Mathematical analysis yields a coupled, linear system of equations including two transport and fate equations, initial and boundary data, and six kinetic rules, namely three each for parent and daughter compound. A numerical scheme for solving the system of equations was developed using readily available procedures since analytical solutions could not be found. Solutions for scenarios based on leaking underground sources are presented.     

One-dimensional model for biogeochemical interactions and permeability reduction in soils during leachate permeation

This paper uses the findings from a column study to develop a reactive model for exploring the interactions occurring in leachate-contaminated soils. The changes occurring in the concentrations of acetic acid, sulphate, suspended and attached biomass, Fe(II), Mn(II), calcium, carbonate ions, and pH in the column are assessed. The mathematical model considers geochemical equilibrium, kinetic biodegradation, precipitation-dissolution reactions, bacterial and substrate transport, and permeability reduction arising from bacterial growth and gas production. A two-step sequential operator splitting method is used to solve the coupled transport and biogeochemical reaction equations. The model gives satisfactory fits to experimental data and the simulations show that the transport of metals in soil is controlled by multiple competing biotic and abiotic reactions. These findings suggest that bioaccumulation and gas formation, compared to chemical precipitation, have a larger influence on hydraulic conductivity reduction.     

Reactive solute transport in macroscopically homogeneous porous media: analytical solutions for the temporal moments

In this work, we investigate one-dimensional solute transport affected by rate-limited sorption, first-order mass transfer, and first-order transformation. Analytical expressions are obtained for the temporal moments of the solute in the solution phase. The effect of various rate coefficients on the temporal moments is examined. It was found that, in the presence of transformation reactions, the mean arrival time, and the spread and skewness of the breakthrough curves, are not monotonic functions of the rate coefficients. These solutions will be useful as a preliminary analysis tool for ascertaining the relative importance of various processes under given conditions. They may also be used to analyze the accuracy of various numerical techniques used for simulation of reactive transport.     

A numerical model for transient-hysteretic flow and solute transport in unsaturated porous media

A two-dimensional flow and transport model was developed for simulating transient water flow and nonreactive solute transport in heterogeneous, unsaturated porous media containing air and water. The model is composed of a unique combination of robust and accurate numerical algorithms for solving the Richards', Darcy flux, and advection-dispersion equations. The mixed form of Richards' equation is solved using a finite-element formulation and a modified Picard iteration scheme. Mass lumping is employed to improve solution convergence and stability behavior. The flow algorithm accounts for hysteresis in the pressure head-water content relationship. Darcy fluxes are approximated with a Galerkin and Petrov-Galerkin finite-element method developed for random heterogeneous porous media. The transport equation is solved using an Eulerian-Lagrangian method. A multi-step, fourth-order Runge-Kutta, reverse particle tracking technique and a quadratic-linear interpolation scheme are shown to be superior for determining the advective concentration. A Galerkin finite-element method is used for approximating the dispersive flux. The unsaturated flow and transport model was applied to a variety of rigorous problems and was found to produce accurate, mass-conserving solutions when compared to analytical solutions and published numerical results.     

Transport of a decay chain in homogenous porous media: analytical solutions

With the aid of integral transforms, analytical solutions for the transport of a decay chain in homogenous porous media are derived. Unidirectional steady-state flow and radial steady-state flow in single and multiple porosity media are considered. At least in Laplace domain, all solutions can be written in closed analytical formulae. Partly, the solutions can also be inverted analytically. If not, analytical calculation of the steady-state concentration distributions, evaluation of temporal moments and numerical inversion are still possible. Formulae for several simple boundary conditions are given and visualized in this paper. The derived novel solutions are widely applicable and are very useful for the validation of numerical transport codes.     

Semi-analytical solution of two-dimensional sharp interface LNAPL transport models

In this paper, we present semi-analytical solutions for two-dimensional equations governing transport of Light Non-Aqueous Phase Liquids (LNAPL) in unconfined aquifers. The proposed model is based on sharp interface displacement and steady groundwater flow assumptions, where both the water–LNAPL interface and the LNAPL–air interface are represented as sharp interfaces. In the case of steady groundwater flow, these equations can be reduced to a two-dimensional nonlinear solute transport equation, with the LNAPL thickness in the free product lens being the primary unknown variable. The linearized form of this solute transport equation falls into the category of two-dimensional transport equation with time-dependent dispersion coefficients. This equation can be solved analytically for an infinite domain region. In this paper, the general form of the analytical solution for the transport equation, as well as the solutions for some specific cases are presented. To demonstrate the utility of the proposed solution, numerical results obtained for two example problems are discussed and presented comparatively with a finite-element solution and other more restrictive solutions available in the literature. Although the solutions discussed in this paper have some simplifying assumptions, such as sharp-interfaces between fluid phases, steady groundwater flow and homogeneous aquifer properties, the semi-analytical solutions presented in this study may be used effectively as bench mark solutions in evaluating LNAPL migration in the subsurface. These solutions are simple and cost effective to implement and may be used in the calibration of other more complex numerical solutions that can be found in the literature.     

Numerical validation of a Eulerian hydrochemical code using a 1D multisolute mass transport system involving heterogeneous kinetically controlled reactions

It is demonstrated that at steady state, the 1D thermo-kinetic hydrochemical Eulerian mass balance equations in pure advective mode are indeed identical to the governing mass balance equations of a single reaction path (or geochemical) code in open system mode. Thus, both calculated reaction paths should be theoretically identical whatever the chemical complexity of the water-rock system (i.e., multicomponent, multireaction zones kinetically and equilibrium-controlled). We propose to use this property to numerically test the thermo-kinetic hydrochemical Eulerian codes and we employ it to verify the algorithm of the 1D finite difference code KIRMAT. Compared to the other methods to perform such numerical tests (i.e., comparisons with analytical, semi-analytical solutions, between two Eulerian hydrochemical codes), the advantage of this new method is the absence of constraints on the chemical complexity of the modelled water-rock systems. Moreover, the same thermo-kinetic databases and geochemical functions can be easily and mechanically used in both calculations, when the numerical reference comes from the Eulerian code with no transport terms (u and D = 0) and modify to be consistent with the definition of the open system mode in geochemical modelling. The ability of KIRMAT to treat multicomponent pure advective transport, subjected to several kinetically equilibrium-controlled dissolution and precipitation reactions, and to track their boundaries has been successfully verified with the property of interest. The required numerical validation of the reference calculations is bypassed in developing the Eulerian code from an already checked single reaction path code. A forward time-upstream weighting scheme (a mixing cell scheme) is used in this study. An appropriate choice of grid spacing allows to calculate within the grid size uncertainty the correct mineral reaction zone boundaries, despite the presence of numerical dispersion. Its correction enables us to improve the convergence and to extend the numerical test to mixed advective-dispersive mass transport. However, the skewness factor involves numerical oscillations that prevent to compute different grid spacing. The use of a different chemically controlled time step constraint in both calculations induces some inconsistencies into the validation tests. This numerical validation method may be applied as well as to check a thermo-kinetic hydrochemical finite element based code, from a 1D heterogeneous systems, and 2D-3D systems provided that they are designed so as to be 1D equivalent. A one-step algorithm and the use of a numerical reference coming from the Eulerian code to be tested ensure the potential success (accuracy) of the numerical validation method.     

Multidimensional, multicomponent, subsurface reactive transport in nonuniform velocity fields: code verification using an advective reactive streamtube approach

High performance computing has made possible the development of high resolution, multidimensional, multicomponent reactive transport models that can be used to analyze complex geochemical environments. However, as increasingly complex processes are included in these models, the accuracy of the numerical formulation coupling the nonlinear processes becomes difficult to verify. Analytical solutions are not available for realistically complex problems and benchmark solutions are not generally available for specific problems. We present an advective reactive streamtube (ARS) transport technique that efficiently provides accurate solutions of nonlinear multicomponent reactive transport in nonuniform multidimensional velocity fields. These solutions can be compared with results from Eulerian-based advection-dispersion-reaction models to evaluate the accuracy of the numerical formulation used. The ARS technique includes mixed equilibrium and kinetic complexation and precipitation-dissolution reactions subject to the following assumptions: (1) transport is purely advective (i.e., no explicit diffusion or dispersion), and (2) chemistry is described by a canonical system of reactions that evolves with time and is unaffected by position in space. Results from the ARS technique are compared with results from the massively parallel, multicomponent reactive transport model MCTRACKER on a test problem involving irreversible oxidation of organic carbon and reaction of the oxidation products with two immobile mineral phases, gypsum and calcite, and fifteen aqueous complexes. Truncation error, operator splitting error, and the nonlinear transformation of these errors in the high-resolution reactive transport model are identified for this problem.     

Models of multi-media partitioning of multi-species chemicals: The fugacity/aquivalence approach

The fugacity or aquivalence approach to environmental modelling is extended to treat chemicals which may be present as several inter-converting species in a multi-media environment. Species-specific and total Z values are defined expressing the capacity of each phase for each species of chemical. Similarly, species-specific and total D values are defined to quantify transport and transformation rates. Fugacity or aquivalence is used as the criterion for partitioning behaviour or physical equilibrium. By defining species proportions on an aquivalence or fugacity fraction basis, the equations for multiple species can be consolidated into a single “pseudo single-component” mass balance equation. Steady-state multi-species, multi-compartment mass balance models can be assembled for such systems and solved in various ways to give mass balances and distributions for all species amounts, concentrations, fluxes and species conversion rates.

Examples are presented for a hypothetical substance present as three species in a system under equilibrium and non-equilibrium, steady-state conditions.     

An analytically solved kinetic model for pesticide degradation in single compartment systems

An analytical kinetic model was developed to simulate the degradation of pesticides in systems such as soil or water. Based on a single compartment system, a set of simultaneous first-order differential equations was analytically solved by the eigenvalue and eigenvector method. The developed model is capable of simulating the concentrations of parent compound and any net of degradation products connected by irreversible reactions.     

Influence of rate-limited sorption on the cleanup of layered soils by vapor extraction

The conditions under which rate-limited sorption is important for cleanup of layered soils by vapor extraction are investigated. The investigation includes two steps: (a) the cleanup time is estimated for a number of scenario cases by means of a numerical model and (b) the numerical results are approximated using analytical solutions derived for simplified models. In this way, equations are derived, which give insight into the influence of different parameters characterizing the properties of the soil, the geometry of the formation, the mass transfer mechanisms in it, and the distribution of the contaminant mass in the different phases (gas phase, water phase and solid phase). The numerical model used is based on the advection-dispersion differential equations for Darcian isothermal airflow, local equilibrium contaminant mass transfer between gas phase and soil water and first-order kinetics for mass transfer between soil water and solid phase. The numerical results are approximated combining an analytical solution to estimate cleanup time in layered formations for local equilibrium sorption, which has been presented in a previous work (J. Contam. Hydrol., 36 (1999) 105). with an analytical solution based on the well-mixed reservoir model under consideration of rate-limited sorption. The analytical approximation of the cleanup time is in reasonable agreement with the numerical results and allows its estimation with small computational effort.     

Contaminant transport in groundwater in the presence of colloids and bacteria: Model development and verification

Colloids and bacteria (microorganisms) naturally exist in groundwater aquifers and can significantly impact contaminant migration rates. A conceptual model is first developed to account for the different physiochemical and biological processes, reaction kinetics, and different transport mechanisms of the combined system (contaminant–colloids–bacteria). All three constituents are assumed to be reactive with the reactions taking place between each constituent and the porous medium and also among the different constituents. A general linear kinetic reaction model is assumed for all reactive processes considered. The mathematical model is represented by fourteen coupled partial differential equations describing mass balance and reaction processes. Two of these equations describe colloid movement and reactions with the porous medium, four equations describe bacterial movement and reactions with colloids and the porous medium, and the remaining eight equations describe contaminant movement and its reactions with bacteria, colloids, and the porous medium. The mass balance equations are numerically solved for two-dimensional groundwater systems using a third-order, total variance-diminishing scheme (TVD) for the advection terms. Due to the complex coupling of the equations, they are solved iteratively each time step until a convergence criterion is met. The model is tested against experimental data and the results are favorable.     

Analytical model of solute transport by unsteady unsaturated gravitational infiltration

Penetration of reactive solute into a soil during a cycle of water infiltration and redistribution is investigated by deriving analytical closed form solutions for fluid flux, moisture content and contaminant concentration. The solution is developed for gravitational flow and advective transport and is applied to two scenarios of solute applications encountered in the applications: a finite pulse of solute dissolved in irrigation water and an instantaneous pulse broadcasted onto the soil surface. Through comparison to simulations of Richards' flow, capillary suction is shown to have contrasting effects on the upper and lower boundaries of the fluid pulse, speeding penetration of the wetting front and reducing the rate of drying. This leads to agreement between the analytical and numerical solutions for typical field and experimental conditions. The analytical solution is further incorporated into a stochastic column model of flow and transport to compute mean solute concentration in a heterogeneous field. An unusual phenomenon of plume contraction is observed at long times of solute propagation during the drying stage. The mean concentration profiles match those of the Monte-Carlo simulations for capillary length scales typical of sandy soils.     

Effect of non-zero divergence wind fields on atmospheric transport calculations

The use of wind fields that do not satisfy the equation of continuity exactly can introduce significant errors in the calculation of the atmospheric transport of chemical species. Results are presented showing that non-zero divergence winds, used as inputs to a transport equation of conservation form, can introduce local fictitious production or destruction rates into the numerical calculations. A divergence-corrected form of the equation, which is mathematically equivalent but not numerically equivalent to an advection form of the equation, can eliminate these undesirable effects as well as the advection form equation can. The divergence-corrected form of the equation leads exactly to the conservation form when massconsistent winds are used. Also, it has the desirable property that the numerical form can preserve, to a large extent, integral conservation relations of the original mass balance equation even when Δ ·V ≠ 0. The effects of using non-zero divergence winds appear as first-order chemical reactions. These terms are compared quantitatively with the rates of several tropospheric chemical reactions.