Analytical solutions for multiple species reactive transport in multiple dimensions

Many numerical computer codes used to simulate multi-species reactive transport and biodegradation have been developed in recent years. Such numerical codes must be validated by comparison of the numerical solutions with an analytical solution. In this paper, a method for deriving analytical solutions of the partial differential equations describing multiple species multi-dimensional transport with first-order sequential reactions is presented. Although others have developed specific solutions of multi-species transport equations, here a more general analytical approach, capable of describing any number of reactive species in multiple dimensions is derived. A substitution method is used to transform the multi-species reactive transport problem to one that can be solved using previously published single-species solutions for various initial and boundary conditions. One- and three-dimensional examples are presented to illustrate the steps involved in extending single-species solutions to a four-species system with sequential first-order reactions.     

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.     

Assessment of intrinsic bioremediation of a coal-tar-affected aquifer using Two-dimensional reactive transport and Biogeochemical mass balance approaches.

Expedited site characterization and groundwater monitoring using direct-push technology and conventional monitoring wells were conducted at a former manufactured gas plant site. Biogeochemical data and heterotrophic plate counts support the presence of microbially mediated remediation. By superimposing solutions of a two-dimensional reactive transport analytical model, first-order degradation rate coefficients ((day-1) ) of various compounds for the dissolved-phase plume were estimated (i.e., benzene [0.0084], naphthalene [0.0058], and acenaphthene [0.0011]). The total mass transformed by aerobic respiration, nitrate reduction, and sulfate reduction around the free-phase coal-tar dense-nonaqueous-phase-liquid region and in the plume was estimated to be approximately 4.5 kg/y using a biogeochemical mass-balance approach. The total mass transformed using the degradation rate coefficients was estimated to be approximately 3.6 kg/y. Results showed that a simple two-dimensional analytical model and a biochemical mass balance with geochemical data from expedited site characterization can be useful for rapid estimation of mass-transformation rates.     

Truncation errors in finite difference models for solute transport equation with first-order reaction

The truncation errors associated with finite difference solutions of the advection-dispersion equation with first-order reaction are formulated from a Taylor analysis. The error expressions are based on a general form of the corresponding difference equation and a temporally and spatially weighted parametric approach is used for differentiating among the various finite difference schemes. The numerical truncation errors are defined using Peclet and Courant numbers and a new Sink/Source dimensionless number. It is shown that all of the finite difference schemes suffer from truncation errors. In particular it is shown that the Crank–Nicolson approximation scheme does not have second order accuracy for this case. The effects of these truncation errors on the solution of an advection–dispersion equation with a first order reaction term are demonstrated by comparison with an analytical solution. The results show that these errors are not negligible and that correcting the finite difference scheme for them results in a more accurate solution.     

Operator-splitting errors in coupled reactive transport codes for transient variably saturated flow and contaminant transport in layered soil profiles

One possible way of integrating subsurface flow and transport processes with (bio)geochemical reactions is to couple by means of an operator-splitting approach two completely separate codes, one for variably-saturated flow and solute transport and one for equilibrium and kinetic biogeochemical reactions. This paper evaluates the accuracy of the operator-splitting approach for multicomponent systems for typical soil environmental problems involving transient atmospheric boundary conditions (precipitation, evapotranspiration) and layered soil profiles. The recently developed HP1 code was used to solve the coupled transport and chemical equations. For steady-state flow conditions, the accuracy was found to be mainly a function of the adopted spatial discretization and to a lesser extent of the temporal discretization. For transient flow situations, the accuracy depended in a complex manner on grid discretization, time stepping and the main flow conditions (infiltration versus evaporation). Whereas a finer grid size reduced the numerical errors during steady-state flow or the main infiltration periods, the errors sometimes slightly increased (generally less than 50%) when a finer grid size was used during periods with a high evapotranspiration demand (leading to high pressure head gradients near the soil surface). This indicates that operator-splitting errors are most significant during periods with high evaporative boundary conditions. The operator-splitting errors could be decreased by constraining the time step using the performance index (the product of the grid Peclet and Courant numbers) during infiltration, or the maximum time step during evapotranspiration. Several test problems were used to provide guidance for optimal spatial and temporal discretization.     

Estimating first-order reaction rate coefficient for transport with nonequilibrium linear mass transfer in heterogeneous media

A travel-time based approach is developed for estimating first-order reaction rate coefficients for transport with nonequilibrium linear mass transfer in heterogeneous media. Tracer transport in the mobile domain is characterized by a travel-time distribution, and mass transfer rates are described by a convolution product of concentrations in the mobile domain and a memory function rather than predefining the mass transfer model. A constant first-order reaction is assumed to occur only in the mobile domain. Analytical solutions in Laplace domain can be derived for both conservative and reactive breakthrough curves (BTCs). Temporal-moment analyses are presented by using the first and second moments of conservative and reactive BTCs and the mass consumption of the reactant for an inverse Gaussian travel-time distribution. In terms of moment matching, there is no need for one to specify the mass transfer model. With the same capacity ratio and the mean retention time, all mass transfer models will lead to the same moment-derived reaction rate coefficients. In addition, the consideration of mass transfer generally yields larger estimations of the reaction rate coefficient than models ignoring mass transfer. Furthermore, the capacity ratio and the mean retention time have opposite influences on the estimation of the reaction rate coefficient: the first-order reaction rate coefficient is positively linearly proportional to the capacity ratio, but negatively linearly proportional to the mean retention time.     

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.     

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.     

Assessing impacts of partial mass depletion in DNAPL source zones: II. Coupling source strength functions to plume evolution

Analytical solutions, describing the time-dependent DNAPL source-zone mass and contaminant discharge rate, derived previously in Part I [Falta, R.W., Rao, P.S., Basu, N., this issue. Assessing the impacts of partial mass depletion in DNAPL source zones: I. Analytical modeling of source strength functions and plume response. J. Contam. Hydrol.] are used as a flux-boundary condition in a semi-analytical contaminant transport model. These analytical solutions assume a power relationship between the flow-averaged source concentration, and the source DNAPL mass; the empirical exponent (gamma) is a function of the flow field heterogeneity, DNAPL architecture, and the correlation between them. The DNAPL source strength terms can account for partial source remediation, either at time zero, or at some later time after the DNAPL release. The transport model considers advection, retardation, three-dimensional dispersion, and sequential first-order decay/production of several species. A separate solution is used to compute the time-dependent mass of each contaminant in the plume. A series of examples using different values of gamma shows how the benefits of partial DNAPL source remediation can vary with site conditions. In general, when gamma>1, relatively large short-term reductions in the plume concentrations and mass occur, but the source longevity is not strongly affected. Conversely, when gamma<1, the short-term reductions in the plume concentrations and mass are smaller, but the source longevity can be greatly reduced. In either case, the source remediation effort is much more effective if it is undertaken at an early time, before much contaminant mass has entered the plume. If the remediation effort is significantly delayed, the leading parts of the plume are not affected by the source remediation, and additional control or remediation of the plume itself is required.     

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.     

Modeling radium and radon transport through soil and vegetation

A one-dimensional flow and transport model was developed to describe the movement of two fluid phases, gas and water, within a porous medium and the transport of 226Ra and 222Rn within and between these two phases. Included in this model is the vegetative uptake of water and aqueous 226Ra and 222Rn that can be extracted from the soil via the transpiration stream. The mathematical model is formulated through a set of phase balance equations and a set of species balance equations. Mass exchange, sink terms and the dependence of physical properties upon phase composition couple the two sets of equations. Numerical solution of each set, with iteration between the sets, is carried out leading to a set-iterative compositional model. The Petrov-Galerkin finite element approach is used to allow for upstream weighting if required for a given simulation. Mass lumping improves solution convergence and stability behavior. The resulting numerical model was applied to four problems and was found to produce accurate, mass conservative solutions when compared to published experimental and numerical results and theoretical column experiments. Preliminary results suggest that the model can be used as an investigative tool to determine the feasibility of phytoremediating radium and radon-contaminated soil.     

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.     

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.     

Modelling of diffusion-limited retardation of contaminants in hydraulically and lithologically nonuniform media

A new reactive transport modelling approach and examples of its application are presented, dealing with the impact of sorption/desorption kinetics on the spreading of solutes, e.g. organic contaminants, in groundwater. Slow sorption/desorption is known from the literature to be strongly responsible for the retardation of organic contaminants. The modelling concept applied in this paper quantifies sorption/desorption kinetics by an intra-particle diffusion approach. According to this idea, solute uptake by or release from the aquifer material is modelled at small scale by a "slow" diffusion process where the diffusion coefficient is reduced as compared to the aqueous diffusion coefficient due to (i) the size and shape of intra-particle pores and (ii) retarded transport of solutes within intra-particle pores governed by a nonlinear sorption isotherm. This process-based concept has the advantage of requiring only measurable model parameters, thus avoiding fitting parameters like first-order rate coefficients.In addition, the approach presented here allows for modelling of slow sorption/desorption in lithologically nonuniform media. Therefore, it accounts for well-known experimental findings indicating that sorptive properties depend on (i) the grain size distribution of the aquifer material and (ii) the lithological composition (e.g. percentage of quartz, sandstone, limestone, etc.) of each grain size fraction. The small-scale physico-chemical model describing sorption/desorption is coupled to a large-scale model of groundwater flow and solute transport. Consequently, hydraulic heterogeneities may also be considered by the overall model. This coupling is regarded as an essential prerequisite for simulating field-scale scenarios which will be addressed by a forthcoming publication.This paper focuses on mathematical model formulation, implementation of the numerical code and lab-scale model applications highlighting the sorption and desorption behavior of an organic contaminant (Phenanthrene) with regard to three lithocomponents exhibiting different sorptive properties. In particular, it is shown that breakthrough curves (BTCs) for lithologically nonuniform media cannot be obtained via simple arithmetic averaging of breakthrough curves for lithologically uniform media. In addition, as no analytical solutions are available for model validation purposes, simulation results are compared to measurements from lab-scale column experiments. The model results indicate that the new code can be regarded as a valuable tool for predicting long-term contaminant uptake or release, which may last for several hundreds of years for some lithocomponents. In particular, breakthrough curves simulated by pure forward modelling reproduce experimental data much better than a calibrated standard first-order kinetics reactive transport model, thus indicating that the new approach is of high quality and may be advantageously used for supporting the design of remediation strategies at contaminated sites where some lithocomponents and/or grain size classes may provide a long-term pollutant source.     

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.     

Modelling of physical and reactive processes during biodegradation of a hydrocarbon plume under transient groundwater flow conditions

Numerical experiments of non-reactive and reactive transport were carried out to quantify the influence of a seasonally varying, transient flow field on transport and natural attenuation at a hydrocarbon-contaminated field site. Different numerical schemes for solving advective transport were compared to assess their capability to model low transversal dispersivities in transient flow fields. For the field site, it is shown that vertical plume spreading is largely inhibited, particularly if sorption is taken into account. For the reactive simulations, a biodegradation reaction module for the geochemical transport model PHT3D was developed. Results of the reactive transport simulations show that under the site-specific conditions the temporal variations in groundwater flow do, to a modest extent, affect average biodegradation rates and average total (dissolved) contaminant mass in the aquifer. The model simulations demonstrate that the seasonal variability in groundwater flow only results in significantly enhanced biodegradation rates when a differential sorption of electron donor (toluene) and electron acceptor (sulfate) is assumed.     

One-dimensional simulation of solute transfer in saturated-unsaturated porous media using the discontinuous finite elements method

A one-dimensional transport model for simulating water flow and solute transport in homogeneous-heterogeneous, saturated-unsaturated porous media is presented. The model is composed of a combination of accurate numerical algorithms for solving the nonlinear Richard's and advection-dispersion equations (ADE). The mixed form of Richard's equation is solved using a standard finite element method (FEM) with primary variable switching. The transport equation is solved using operator splitting, with the discontinuous finite element method (DFE) for discretization of the advective term. A slope limiting procedure for DFE avoids numerical instabilities but creates very limited numerical dispersion for high Peclet numbers. An implicit finite differences scheme (FD) is used for the dispersive term. The unsaturated flow and transport model (Wamos-T) is applied to a variety of rigorous problems including transient flow, heterogeneous medium and abrupt variations of velocity in magnitude and direction due to time-varying boundary conditions. It produces accurate and mass-conservative solutions for a very large range of grid Peclet numbers. The Wamos-T model is a good and robust alternative for the simulation of mass transport in unsaturated domain.     

Biodegradation of hydrocarbons vapors: Comparison of laboratory studies and field investigations in the vadose zone at the emplaced fuel source experiment, Airbase Vaerløse, Denmark

The natural attenuation of volatile organic compounds (VOCs) in the unsaturated zone can only be predicted when information about microbial biodegradation rates and kinetics are known. This study aimed at determining first-order rate coefficients for the aerobic biodegradation of 13 volatile petroleum hydrocarbons which were artificially emplaced as a liquid mixture during a field experiment in an unsaturated sandy soil. Apparent first-order biodegradation rate coefficients were estimated by comparing the spatial evolution of the resulting vapor plumes to an analytical reactive transport model. Two independent reactive numerical model approaches have been used to simulate the diffusive migration of VOC vapors and to estimate degradation rate coefficients. Supplementary laboratory column and microcosm experiments were performed with the sandy soil at room temperature under aerobic conditions. First-order kinetics adequately matched the lab column profiles for most of the compounds. Consistent compound-specific apparent first-order rate coefficients were obtained by the three models and the lab column experiment, except for benzene. Laboratory microcosm experiments lacked of sensitivity for slowly degrading compounds and underestimated degradation rates by up to a factor of 5. Addition of NH3 vapor was shown to increase the degradation rates for some VOCs in the laboratory microcosms. All field models suggested a significantly higher degradation rate for benzene than the rates measured in the lab, suggesting that the field microbial community was superior in developing benzene degrading activity.     

One-dimensional simulation of solute transfer in saturated–unsaturated porous media using the discontinuous finite elements method

A one-dimensional transport model for simulating water flow and solute transport in homogeneous–heterogeneous, saturated–unsaturated porous media is presented. The model is composed of a combination of accurate numerical algorithms for solving the nonlinear Richard's and advection–dispersion equations (ADE). The mixed form of Richard's equation is solved using a standard finite element method (FEM) with primary variable switching. The transport equation is solved using operator splitting, with the discontinuous finite element method (DFE) for discretization of the advective term. A slope limiting procedure for DFE avoids numerical instabilities but creates very limited numerical dispersion for high Peclet numbers. An implicit finite differences scheme (FD) is used for the dispersive term.The unsaturated flow and transport model (Wamos-T) is applied to a variety of rigorous problems including transient flow, heterogeneous medium and abrupt variations of velocity in magnitude and direction due to time-varying boundary conditions. It produces accurate and mass-conservative solutions for a very large range of grid Peclet numbers. The Wamos-T model is a good and robust alternative for the simulation of mass transport in unsaturated domain.     

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.