Reactive Solute Transport Research Articles

A small perturbations solution for nonequilibrium chemical transport through soils with relatively high desorption rate

An analytical solution is presented for reactive solute transport or nonuniform flow when the reaction rate or the diffusive transfer rate to stagnant water is small compared to the flow velocity or large compared to the contact time. For these cases many transport models assume local chemical or physical equilibrium for conceptual and mathematical simplification. In this paper the commonly used nonequilibrium formulation for one‐dimensional, steady‐flow transport was used as the starting point. Dimensionless variables were defined and substituted into the mass balance equations forming dimensionless equations and boundary and initial conditions. There are two characteristic timescales associated with the two rate‐limited processes: The fast characteristic timescale is related to the chemical desorption and the slow timescale is related to the convective‐dispersive transport. By scaling the first‐order rate equation that accounts for linear sorption kinetics (or the dissolved chemical exchange between mobile and stagnant porosities in structured soils), a singular perturbations problem was obtained in which a small parameter, ϵ, multiplies the time derivative. This small parameter is the ratio between the slow and fast timescales. Inner, т=O(ϵ), and outer,т=O(1), solutions were developed, and the method of matched asymptotic expansion was used to form a uniform solution over the entire time domain. The leading‐order approximation, obtained for ϵ → 0 and valid for т=O(1), yields equations that are usually obtained by the local equilibrium assumption (LEA). A initial layer, т=O (ϵ), near t=0 is formed within which the initial concentrations equilibrate. The condition under which the local equilibrium assumption is valid, ϵ=ν2/krD ≪ 1, which depends on the system parameters such as flow velocity, dispersion coefficient, and the variables. This condition was also obtained in other studies by solving the problem in either kinetic or LEA form and comparing the solutions to determine under what conditions the local equilibrium approximation is close to the kinetic solution. As ϵ decreases, the LEA is valid, and the initial layer, in which the initial concentrations reach local equilibrium, is shorter.

Abstract: Reactive Solute Transport Modeling of the Acid Mine Drainage Contamination in the Snow Fork of the Monday Creek Watershed

Abstract: Reactive Solute Transport Modeling of the Acid Mine Drainage Contamination in the Snow Fork of the Monday Creek Watershed

Reactive Solute Transport with a Variable Selectivity Coefficient in an Undisturbed Soil Column

AbstractThe spatial distribution of major ion concentrations limits the pre‐dictability of solute transport processes in field soils. Therefore, it is important to analyze solute transport with chemical reactions based on results obtained from field soils and numerical simulation. A simulation model with cation‐exchange reactions was developed and applied to solute‐transport analysis of an undisturbed field soil. Chemical reaction terms in the convective‐dispersive equation were estimated by the Levenberg‐Marquardt nonlinear least‐squares regression technique to satisfy physical and chemical processes simultaneously. The reliability of the model was tested with liquid‐phase and solid‐phase concentrations of measured spatial distributions of Ca2+, Mg2+, Na+, and K+ after continuous infiltration of KCl solution into an undisturbed soil column. The experimental results revealed that the selectivity coefficients for Ca‐Na and Ca‐Mg exchange could be kept constant, while those for Ca‐K exchange increased with the equivalent fraction of K+ in the solid phase. The effects of the exchange selectivity coefficient on reactive solute transport are discussed based on the simulation results. When a constant selectivity coefficient was used, the model failed to predict the spatial distributions of cation concentrations in the solid phase. Thus, model predictions can be improved by use of variable instead of constant selectivity coefficients.

Nonideal transport of reactive solutes in heterogeneous porous media 2. Quantitative analysis of the Borden natural-gradient field experiment

Field experiments constitute an integral component of research on transport and fate of contaminants in the subsurface. One of the most well known of the few field experiments performed with reactive solutes is the natural-gradient experiment conducted at the Borden site during 1982 to 1984. A major finding of the experiment was that the transport of the reactive, organic compounds was nonideal. First, the velocities of the centers of mass of the plumes decreased with time, which was reflected in a temporal increase in effective retardation. Second, the longitudinal spreading observed for the organic solutes was about three times larger than that of the nonreactive tracers for an equivalent travel distance. Third, the breakthrough curves measured at selected monitoring points exhibited greater asymmetry compared to the nonreactive tracers. The cause(s) of the nonideal transport observed for the organic solutes has remained unexplained, despite a number of attempts. We have used a multi-scale, multi-factor mathematical model to successfully predict the displacement and spreading behavior of the tetrachloroethene and tetrachloromethane plumes. Based on our analyses, we conclude that a near-field trend of increasing sorption capacity was a primary cause of the deceleration of the centers of mass of the organic-solute plumes. The coupled effects of nonlinear sorption and enhanced spreading caused by spatially variable hydraulic conductivity and spatially variable sorption also influenced plume displacement. In addition, it is possible that the combination of spatially variable hydraulic conductivity and sorption contributed directly to plume deceleration. However, a magnitude of sorption variability larger than has been measured to date is required for this contribution to be significant. The combined spatial variability of hydraulic conductivity and sorption, and a potential negative cross correlation between them, appears to have been the major cause of the enhanced longitudinal spreading observed for the organic-solute plumes in comparison to the nonreactive-solute plumes. However, nonlinear sorption, the spatial trend of increasing sorption capacity, and rate-limited sorption/mass transfer also influenced spreading behavior. In total, it is evident that the transport of the organic compounds during the Borden natural-gradient field experiment was influenced by several interacting factors and coupled processes, and that accurate prediction of the observed behavior requires the use of a mathematical model that accounts for this complexity.

Nonequilibrium Transport of Reactive Solutes through Layered Soil Profiles with Depth-Dependent Adsorption

The pesticide atrazine was used as a reactive tracer in a series of miscible displacement experiments that were conducted with soil columns containing the same amount of sludge or manure, but distributed either as a layer or uniformly in depth. Travel time probability density function (pdf) of Cl- and atrazine measured from both types of columns was analyzed based upon a transfer function model that assumes a two-site nonequilibrium advection−dispersion equation (ADE). Sorption sites contributed by both organic materials were primarily rate-limited. Travel time of atrazine, estimated by temporal moment analysis on the measured travel time pdf, was consistently increased in organic material-amended columns for both distributions. When flux concentration of atrazine was fitted to the nonequilibrium ADE, either a higher overall distribution coefficient (Kd) or a higher fraction of instantaneous adsorption sites (f) or both were found for layered columns, indicating an enhanced accessibility to sorption sites of organic amendments probably due to less shielding by soil minerals.

Finite Element Approximation of The Transport of Reactive Solutes in Porous Media. Part II: Error Estimates for Equilibrium Adsorption Processes

In this paper we analyze a fully practical piecewise linear finite element approximation involving numerical integration, backward Euler time discretization, and possibly regularization and relaxation of the following degenerate parabolic equation arising in a model of reactive solute transport in porous media: find $u(x,t)$ such that $$ \partial_t u + \partial_t [ \p (u)] - \Delta u = f \quad {\rm in} \quad \Omega \times (0,T\,], $$ \vspace*{-12pt} $$ u = 0 \quad {\rm on} \quad \partial \Omega \times (0,T\,] \qquad u(\cdot ,0) = g(\cdot ) \quad {\rm in} \; \Omega $$ for known data $ \Omega \subset {\bf R}^d, \; 1 \leq d \leq 3 $, f, g, and a monotonically increasing $ \p \in {\rm C}^0({\bf R}) \cap {\rm C}^1(- \infty , 0] \cup (0,\infty)$ satisfying $ \p (0) = 0$, which is only locally Hölder continuous with exponent $ p \in (0,1)$ at the origin; e.g., $ \p (s) \equiv [s]_+^p $. This lack of Lipschitz continuity at the origin limits the regularity of the unique solution u and leads to difficulties in the finite element error analysis.

Analysis of split operator methods for nonlinear and multispecies groundwater chemical transport models

Coupled solute transport and reaction models are computationally demanding when multispecies, multidimensional simulations are considered. Split operator methods provide approximate solutions to the reactive solute transport problem that are both relatively efficient to compute and to construct. The transport and reaction operators are split into two separate computational steps. Split operator methods are introduced in the context of single species sorption to the soil, with an emphasis on the splitting errors that are induced. For standard two-step methods, the splitting error is proportional to Δt , the temporal step size of the numerical scheme. The alternating split operator scheme, in which the order of the operations is switched at succeeding time steps, apparently does not remove the splitting error for nonlinear reactions, whereas it is removed for linear cases. The truncation error is extended to the case of two reacting species.

Finite Element Approximation of the Transport of Reactive Solutes in Porous Media. Part 1: Error Estimates for Nonequilibrium Adsorption Processes

In this paper we analyze a fully practical piecewise linear finite element approximation involving numerical integration, backward Euler time discretization, and possibly regularization of the following degenerate parabolic system arising in a model of reactive solute transport in porous media: find $\{u(x,t),v(x,t)\}$ such that \begin{eqnarray*} &\partial_t u + \partial_t v - \Delta u =f \quad\mbox{in } \Omega \times(0,T] \qquad u=0\quad\mbox{on } \partial\Omega\times(0,T]& &\partial_tv=k(\varphi(u)-v) \quad\mbox{in }\Omega\times(0,T]& &u(\cdot\,,0)=g_1(\cdot)\quad v(\cdot\,,0) =g_2(\cdot)\quad\mbox{in } \Omega\subset \Real^d,\quad 1\le d\le 3& \end{eqnarray*} for given data $k\in \Real^+$, $f$, $g_1$, $g_2$ and a monotonically increasing $\varphi\in C^0(\Real)\cap C^1(-\infty,0]\cup(0,\infty)$ satisfying $\varphi(0)=0$, which is only locally Holder continuous with exponent $p\in(0,1)$ at the origin, e.g., $\varphi(s)\equiv[s]_+^p$. This lack of Lipschitz continuity at the origin limits the regularity of the unique solution $\{u,v\}$ and leads to difficulties in the finite element error analysis.

On the transport of reactive solutes in partially saturated heterogeneous anisotropic porous formations

First-order analysis, based on a stochastic continuum presentation of the Eulerian retarded velocity in heterogeneous, partially saturated porous formations and a general Lagrangian description of the transport, was used to investigate the effect of both the anisotropy of the formation and the texture of the soil materials constituting the formation, on the spreading of reactive (sorptive) solutes under unsaturated flow. The physically plausible situation in which the retardation factor is negatively correlated with log-unsaturated conductivity was considered in this study. Results of the analyses show that for a given water saturation and gravity-dominated unsaturated flow, both the physical and chemical heterogeneities of the formation enhance the spreading of the reactive solute in the longitudinal direction. This applies especially in formations with typical flow barriers which are small in a direction perpendicular to the mean flow, relative to their size in the direction parallel to the mean flow (large aspect ratio, e), and in formations of fine-textured soil materials with significant capillary forces (large macroscopic capillary length scale, λ). The relative impact of the chemical heterogeneity of the formation on the spreading of the reactive solute, however, is expected to increase in formations of coarse-textured soil materials (small λ) and in formations which exhibit a significant stratification (small e). This result stems from the fact that when the mean flow is vertical, small λ and small e both diminish the impact of the heterogeneity in the physical properties of the formation on the variability of the flow in the longitudinal direction.

Large Time Profiles in Reactive Solute Transport

In this paper we consider the large time structure of reactive solute plumes in two dimensional, macroscopically homogeneous, flow domains. The reactions between the dissolved chemicals and the porous matrix are equilibrium adsorption reactions, given by an isotherm of Freundlich type. We also incorporate the effect of partial and full decay. We use the method of asymptotic balancing to obtain, to leading order, the large time behaviour of the solute concentration and the relevant moments (mass, centre of mass,variance). The method of balancing is based on certain conjectures about the form of the temporal decay and partial spreading of the solute. These conjectures are verified numerically.

A numerical study of fronts in random media using a reactive solute transport model

We simulate the random front solutions of a nonlinear solute transport equation with spatial random coefficients modeling inhomogeneous sorption sites in porous media. The nonlinear sorption function is chosen to be Langmuir type, and the random coefficients are two independent stationary processes with fast decay of correlations. The model equation is in conservation form, and the random fronts are similar to random viscous shocks. We find that the average front speed is given by an ensemble averaged explicit Rankine–Hugoniot relation, and the front position fluctuates about its mean. Our numerical calculations show that the standard deviation is of the order O( $$\sqrt t $$ ) for large time, and the front fluctuation scaled by $$\sqrt t $$ converges to a Gaussian random variable wih mean zero. We come up with a formal theory of front fluctuation, yielding an explicit expression of the root t normalized front standard deviation in terms of the random media statistics. The theory agrees remarkably with the numerically discovered empirical formula.

Experimental Investigation and Modeling of Uranium (VI) Transport Under Variable Chemical Conditions

The transport of adsorbing and complexing metal ions in porous media was investigated with a series of batch and column experiments and with reactive solute transport modeling. Pulses of solutions containing U(VI) were pumped through columns filled with quartz grains, and the breakthrough of U(VI) was studied as a function of variable solution composition (pH, total U(VI) concentration, total fluoride concentration, and pH‐buffering capacity). Decreasing pH and the formation of nonadsorbing aqueous complexes with fluoride increased U(VI) mobility. A transport simulation with surface complexation model (SCM) parameters estimated from batch experiments was able to predict U(VI) retardation in the column experiments within 30%. SCM parameters were also estimated directly from transport data, using the results of three column experiments collected at different pH and U(VI) pulse concentrations. SCM formulations of varying complexity (multiple surface types and reaction stoichiometries) were tested to examine the trade‐off between model simplicity and goodness of fit to breakthrough. A two‐site model (weak‐ and strong‐binding sites) with three surface complexation reactions fit these transport data well. With this reaction set the model was able to predict (1) the effects of fluoride complexation on U(VI) retardation at two different pH values and (2) the effects of temporal variability of pH on U(VI) transport caused by low pH buffering. The results illustrate the utility of the SCM approach in modeling the transport of adsorbing inorganic solutes under variable chemical conditions.

Nonideal transport of reactive solutes in heterogeneous porous media: 1. Numerical model development and moments analysis

The transport of reactive solutes at the field scale is characteristically nonideal, and this nonideality is often caused by more than one factor. In these cases, mathematical models that explicitly account for multiple factors are necessary for proper simulation of transport and for accurate analysis of causative factors. The purpose of this paper is to present a mathematical model for simulating the transport of reactive solute in heterogeneous porous media. We have taken a multi-scale, multi-factor approach that explicitly accounts for such factors as hydraulic-conductivity variability, structured porous media, rate-limited diffusive mass transfer, and nonlinear, rate-limited, spatially variable sorption. The influence of these factors on the displacement and spreading of solute plumes and on mass flux is illustrated with a series of two-dimensional (vertical) simulations. It is shown that rate-limited sorption/mass transfer and nonlinear sorption can significantly influence the first, second, and third spatial moments, whereas hydraulic-conductivity variability significantly influences only the second spatial moment. Plume skewness is especially sensitive to the specific factor controlling nonideal transport. For example, both rate-limited sorption/mass transfer and nonlinear sorption can create negatively skewed plumes during early stages of transport. However, the plume influenced by rate-limited sorption/mass transfer tends toward symmetry as global residence time increases. Conversely, the plume influenced by nonlinear sorption tends toward a constant degree of asymmetry as the spreading forces balance the concentration-dependent retardation behavior associated with nonlinear sorption. Furthermore, the results illustrate that unique behavior can result from the coupling of multiple processes. For example, when influenced by coupled heterogeneity and nonlinear sorption, the shape of a plume may change from positive to negative skewness during the course of transport.

Coupled transport and dispersion of multicomponent reactive solutes in rectilinear flows

Simultaneous transport of two-component solution in flows in tubes and channels, coupled via a matrix wall reaction coefficient, is considered. Expressions for long-time effective axial matrix solute properties are derived in the particular case of a channel formed by two parallel plates. These matrix coefficients include effective axial solute reactivity, velocity and diffusivity. These coefficients are systematically investigated both analytically and numerically for a two-component species, having different molecular diffusivities and moving in a plane Poiseuille flow between two parallel walls, on which surfaces they undergo coupled surface reactions. It is shown that for small coupling reactivity coefficient the species initially behave like uncoupled constituents, the dispersive transport of which can be studied via single-species solution scheme. At large times the species transport becomes coupled and all constituents are characterized by the same nonmatrix transport properties. These properties constitute leading modes of the corresponding matrix coefficients, governing the species transport for earlier times. Physically, in this situation the effective properties of both constituents are controlled by the (microscale) molecular diffusivity of the slowest component.

Reactive Solute Transport in an Acidic Stream: Experimental pH Increase and Simulation of Controls on pH, Aluminum, and Iron

Solute transport simulations quantitatively constrained hydrologic and geochemical hypotheses about field observations of a pH modification in an acid mine drainage stream. Carbonate chemistry, the formation of solid phases, and buffering interactions with the stream bed were important factors in explaining the behavior of pH, aluminum, and iron. The precipitation of microcrystalline gibbsite accounted for the behavior of aluminum; precipitation of Fe(OH)3 explained the general pattern of iron solubility. The dynamic experi ment revealed limitations on assumptions that reac tions were controlled only by equilibrium chemistry. Temporal variation in relative rates of photoreduction and oxidation influenced iron behavior. Kinetic limita tions on ferrous iron oxidation and hydrous oxide precipitation and the effects of these limitations on field filtration were evident. Kinetic restraints also characterized interaction between the water column and the stream bed, including sorption and desorption of protons f...

Reactive solute transport in acidic streams

Spatial and temporal profiles of pH and concentrations of toxic metals in streams affected by acid mine drainage are the result of the interplay of physical and biogeochemical processes. This paper describes a reactive solute transport model that provides a physically and thermodynamically quantitative interpretation of these profiles. The model combines a transport module that includes advection-dispersion and transient storage with a geochemical speciation module based on MINTEQA2. input to the model includes stream hydrologic properties derived from tracer-dilution experiments, headwater and lateral inflow concentrations analyzed in field samples, and a thermodynamic database. Simulations reproduced the general features of steady-state patterns of observed pli and concentrations of aluminum and sulfate in St. Kevin Gulch, an acid mine drainage stream near Leadville, Colorado. These patterns were altered temporarily by injection of sodium carbonate into the stream. A transient simulation reproduced the observed effects of the base injection.

Correction to “Development of the Fokker‐Planck Equation and its solutions for modeling transport of conservative and reactive solutes in physically heterogeneous media”

Correction to “Development of the Fokker‐Planck Equation and its solutions for modeling transport of conservative and reactive solutes in physically heterogeneous media”

The modeling of reactive solute transport with sorption to mobile and immobile sorbents: 1. Experimental evidence and model development

Modeling carrier‐influenced transport needs to take into account the reactivity of the carrier itself. This paper presents a mathematical model of reactive solute transport with sorption to mobile and immobile sorbents. The mobile sorbent is also considered to be reactive. To justify the assumptions and generality of our modeling approach, experimental findings are reviewed and analyzed. A transformation of the model in terms of total concentrations of solute and mobile sorbents is presented which simplifies the mathematical formulations. Breakthrough data on dissolved organic carbon are presented to exemplify the need to take into account the reactivity of the mobile sorbent. Data on hexachlorobiphenyl and cadmium are presented to demonstrate carrier‐introduced increased mobility, whereas data on anthracene and pyrene are presented to demonstrate carrier‐introduced reduced mobility. The experimental conditions leading to the different findings are pointed out. The sorption processes considered in the model are both equilibrium and nonequilibrium processes, allowing for different sorption sites and nonlinear isotherms and rate functions. Effective isotherms, which describe the sorption to the immobile sorbent in the presence of a mobile sorbent and rate functions, are introduced and their properties are discussed.

Application of a model accounting for kinetic sorption and degradation to in situ microcosm observations on the fate of aromatic hydrocarbons in an aerobic aquifer

The fate of seven aromatic hydrocarbons was investigated under aerobic conditions by in situ microcosm (, one‐dimensional in situ column) in a sandy aquifer low in organic carbon. The results were simulated by an one‐dimensional reactive solute transport model accounting for kinetically controlled sorption and degradation in order to obtain degradation parameters. The kinetic sorption process affected the fate of the aromatic hydrocarbons and must be accounted for in simulation of the degradation process. The kinetic sorption constants were determined either by laboratory batch sorption experiments or from the fate curves of a biological deactivated ISM. The fate curves of biological active ISMs were adequately described both by the first‐order degradation model, including a lag phase, and by the Monod model. The first‐order degradation model was preferred to the Monod model, because the Monod model approaches the first‐order degradation model at the low concentrations studied (initial concentrations of the organic compounds were 150 μg L−) and because the basic assumptions of the Monod model may be violated in a system where several aromatic hydrocarbons are present at the same time. The results obtained by the first‐order degradation model showed lag phase ranging from 2–20 days. First‐order degradation rate constants for aromatic hydrocarbons determined in situ ranged between 0.05 and 0.8 day−1. Sensitivity analysis revealed the importance of independent determination of sorption constants in order to obtain reliable first‐order degradation rate constants.

The modeling of reactive solute transport with sorption to mobile and immobile sorbents: 2. Model discussion and numerical simulation

We analyze mathematically and numerically a model derived in part 1 of this paper [Knabner et al., this issue]. The model deals with carrier‐influenced transport and besides advective and dispersive transport and equilibrium and nonequilibrium sorption takes into account both carrier facilitation and cosorption. Guidelines are derived, whether the overall mobility is enhanced or reduced and various other properties of the model elucidated, in particular for varying carrier concentrations. We indicate how to modify existing numerical codes for the usual adsorption model and discuss simulations for experimental data sets.