Reactive Transport Equations Research Articles

Double conditioning of reactive scalar transport equations in turbulent nonpremixed flames

Double conditioning of reactive scalar transport equations on mixture fraction and sensible enthalpy is presented as a promising extension to conventional conditional moment closure (CMC) methods. Simple first order, singly conditioned CMC cannot accurately predict combustion phenomena such as partially premixed flames and flames with local extinction and reignition. CMC is applied to turbulent flames with significant temperature fluctuations that bring to light the deficiencies of the conventional CMC approach. Fluctuations of temperature and species around their conditional means cannot be neglected when these quantities are conditioned on mixture fraction only. Comparison with direct numerical simulation (DNS) results demonstrates, however, the great potential for doubly conditioned CMC. Doubly conditioned CMC gives excellent agreement with DNS at all times and it captures phenomena like local extinction and the onset of reignition accurately.

Generalized solution to multi-dimensional multi-species transport equations coupled with a first-order reaction network involving distinct retardation factors

This paper presents a general method for solving coupled multi-dimensional, multi-species reactive transport equations. The new method can be used for solving multi-species transport problems involving first-order kinetic interactions and distinct retardation factors. The solution process employs Laplace transformation and linear transformation steps to uncouple the governing set of coupled partial differential equations. The uncoupled equations are solved using an elementary solution. The details of the solution algorithm are illustrated by deriving an explicit analytical solution to a two-species reactive transport problem. In addition, three one-dimensional problems and two three-dimensional problems are solved to illustrate the use of the method. The proposed solution scheme is a robust procedure for solving a variety of multi-dimensional, multi-species transport problems that are coupled with a first-order reaction network.

Natural attenuation of trichloroethene and its degradation products at a lake-shore site

Subsurface contamination by trichloroethene (TCE) was detected at a Michigan National Priorities List (NPL) site in 1982. The TCE plume resulted from the disposal of spent solvent and other chemicals at an industrial facility located in the eastern shore of Lake Michigan. TCE degradation products of three dichloroethene (DCE) isomers, vinyl chloride (VC) and ethene were present. The plume was depleted of oxygen and methanogenic at certain depths. Transects of the plume were sampled by slotted auger borings the year after the TCE plume was first discovered. Water samples were also taken from lake sediments to a depth of 12 m about 100 m offshore. Later samples were taken along the shoreline of the lake with a hand-driven probe. Later in 1998 water was taken from sediments about 3-m from the shoreline. The average concentration of each chemical and net apparent base coefficient between appropriate pairs of transects between the lower site and lakeshore were calculated. Loss rates were then calculated from an analytical solution of the two-dimensional advective–dispersive–reactive transport equation. Net apparent rate coefficients and a set of coupled reaction rate equations were used to extract the apparent loss coefficients. This study showed the field evidence for natural attenuation of TCE.

Dynamical features of reaction-diffusion fronts in fractals.

The speed of front propagation in fractals is studied by using (i) the reduction of the reaction-transport equation into a Hamilton-Jacobi equation and (ii) the local-equilibrium approach. Different equations proposed for describing transport in fractal media, together with logistic reaction kinetics, are considered. Finally, we analyze the main features of wave fronts resulting from this dynamic process, i.e., why they are accelerated and what is the exact form of this acceleration.

Simulation and Control of Chlorine Levels in Water Distribution Networks

Chlorine concentration is an important parameter used to assess the quality of water supplied by a distribution network. While several water quality simulators are readily available (e.g., EPANET software), water quality optimization decision support tools are currently lacking. A new control-oriented calculation model for chlorine concentrations in distribution networks is presented for general water distribution networks operating under periodic demand conditions. The model is based upon the analysis of the characteristic curves of the concentration transport-reaction equation over each pipe in a network. Numerical results are comparable with EPANET results, but unlike other water quality algorithms, the current method requires neither discretizations along the pipes, nor the choice of initial conditions. Moreover, it is a first step toward a method for controlling concentrations, yielding linear relationships between the concentrations at both ends of each pipe by the explicit inclusion of system dynamics. These relationships can be used as constraints in a linear programming optimization model that minimizes deviations from permissible concentration levels.

An assessment of using the predictor–corrector technique to solve reactive transport equations

AbstractWe examined, through comparison among the full‐coupling (FC), operator‐splitting (OS), and predictor–corrector (PC) techniques, the effectiveness of using the PC technique to solve depth‐averaged reactive transport equations in the shallow water domain. Our investigation has led to three major conclusions. Firstly, both the OS and PC techniques can efficiently solve reactive transport equations because the advection–diffusion transport equations are solved outside the non‐linear iteration loop and the reaction equations are solved node by node. However, these two techniques may risk sacrificing computational accuracy. Secondly, the OS or PC technique incorporated with the Lagrangian–Eulerian (LE) approach can handle boundary sources more precisely than alternatively with the conventional Eulerian (CE) approach. Thirdly, with the LE approach incorporated, the numerical results from the three techniques agreed highly with one another except when diffusion became significant. In this case, the PC technique's result still matched well with the FC technique's result, but differences between the OS and FC techniques' results arose as diffusion increased. Based on this study, we recommend to apply as a first step the PC technique to solving reactive transport equations with respect to both computational efficiency and accuracy. Copyright © 2002 John Wiley & Sons, Ltd.

Developing high-resolution chronologies in glacimarine sediments: examples from southeastern Alaska

Abstract Glacial systems release sediment to the marine environment over a range of time periods, including short-term seasonal, tidal, and diurnal scales. Often, the sedimentary record providing the highest temporal resolutions of short-term processes is found in relatively inaccessible areas such as proximal to glacier termini or below floating ice sheets. To assess the importance of short-term glacial and oceanographic processes in creating glacimarine strata, it is necessary to evaluate strata production over similar time scales. Time-series coring, sediment traps, or bathymetric profiles may be difficult to perform in these harsh settings. Coupling the observations of sedimentary structures seen in core X-ray radiographs with sound chronologies allows for the evaluation of short-term glacimarine sedimentation. For seasonal time-scales, the highly particle-reactive radioisotope 234 Th (24 day half-life) can be used to measure the rates of strata production under both steady and non-steady sediment deposition. To create appropriate age-depth relationships, two approaches are used: a rigorous and exacting mathematical model developed from a steady-state transport-reaction equation; and the less exacting CIC (constant initial concentration) and CRS (constant rate of supply or constant flux) point transformations. These methods are used for developing chronologies of cores collected in 1995 in Icy Bay, Alaska at 4, 12, and 32 km from the tidewater terminus of the Guyot Glacier. Examples are given of the boundary conditions that must be satisfied to use each approach in developing age-depth relationships. For the two cores collected closest to the terminus, age-depth relationships can be generated using all three approaches, whereas the presence of bioturbation in the most ice-distal sample severely complicates matters. The high (∼0.5 cm d −1 ) but non-steady sedimentation rates at the ice-proximal station create a high-resolution sedimentary record. The controls on sedimentation at this location are evaluated by examining sedimentary structures (alternating light and dark laminae and beds) seen in digitized X-ray radiographs. The CRS method is used to convert depth to time in X-ray radiographs. The down-core pixel intensities of the grey-scale X-ray radiograph positives are used to create a time series of sedimentation reflected in proxy data (the bulk density of the sediments). By using Blackman-Tukey spectral analyses and relatively new wavelet techniques to evaluate periodic processes, sedimentation at this location can be related to seasonal variability in meltwater production, fortnightly tidal influences on particle settling rates, and episodic precipitation-induced sediment deposition.

Simulation of thick biofilm growth at the microscale

The use of thick biofilms in porous media to form biobarriers is a new way of controlling pollution in aquifers. Biofilm growth at the pore scale needs to be understood in order to determine how to best construct biobarriers to block the flow of pollutants into sources of water. The physical, chemical and biological processes involved are very complicated. It is necessary to solve the Navier–Stokes equations for the flow, transport equations for the movement of nutrients and pollutants and growth and transport-reaction equations for the biofilm. Since the growth of the biofilm changes the physical characteristics of the medium, all the processes are interrelated. In this paper we concentrate on the biofilm growth equations and in how to deal with the coupling of the equations.

Finite element modeling of fluid–rock interaction problems in pore-fluid saturated hydrothermal/sedimentary basins

In order to use the finite element method for solving fluid–rock interaction problems in pore-fluid saturated hydrothermal/sedimentary basins effectively and efficiently, we have presented, in this paper, the new concept and numerical algorithms to deal with the fundamental issues associated with the fluid–rock interaction problems. These fundamental issues are often overlooked by some purely numerical modelers. (1) Since the fluid–rock interaction problem involves heterogeneous chemical reactions between reactive aqueous chemical species in the pore-fluid and solid minerals in the rock masses, it is necessary to develop the new concept of the generalized concentration of a solid mineral, so that two types of reactive mass transport equations, namely, the conventional mass transport equation for the aqueous chemical species in the pore-fluid and the degenerated mass transport equation for the solid minerals in the rock mass, can be solved simultaneously in computation. (2) Since the reaction area between the pore-fluid and mineral surfaces is basically a function of the generalized concentration of the solid mineral, there is a definite need to appropriately consider the dependence of the dissolution rate of a dissolving mineral on its generalized concentration in the numerical analysis. (3) Considering the direct consequence of the porosity evolution with time in the transient analysis of fluid–rock interaction problems, we have proposed the term splitting algorithm and the concept of the equivalent source/sink terms in mass transport equations so that the problem of variable mesh Peclet number and Courant number has been successfully converted into the problem of constant mesh Peclet and Courant numbers. The numerical results from an application example have demonstrated the usefulness of the proposed concepts and the robustness of the proposed numerical algorithms in dealing with fluid–rock interaction problems in pore-fluid saturated hydrothermal/sedimentary basins.

Gel Placement in Porous Media: Constant Injection Rate

In this paper we analyse advective transport of polymers, crosslinkers and gel, taking into account non-equilibrium gelation, gel adsorption and crosslinker precipitation. In absence of diffusion/dispersion the resulting model consists of hyperbolic transport-reaction equations. These equations are studied in several steps using mainly analytical techniques. For simple cases, we obtain explicit travelling wave solutions, whereas for more complicated cases we rely on analytical techniques to analyse the problem qualitatively. Finally, a numerical solution for the full system of equations is obtained. The results developed in this study can be used to validate numerical solutions obtained from commercial simulators.

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.

A numerical model simulating reactive transport in shallow water domains: model development and demonstrative applications

We present a two-dimensional depth-averaged numerical model simulating reactive transport in shallow water domains subject to given flow conditions. A new approach is used to account for more conceivable kinetically-controlled chemical reactions than have been included in previous models. In this model, a chemical may dissolve in the water column or in the interstitial water of the bed sediments, react with other dissolved chemicals to form new chemicals also in the dissolved phase, or be sorbed onto sediments to become particulate chemicals. To save computer time, we employ a predictor–corrector strategy to deal with the reactive transport equations of mobile substances where transport equations are computed in the predictor step with reaction rates estimated at the previous time step, and rate correction is achieved node by node in the corrector step to obtain the concentration distributions at the present time step. For immobile substances, we solve the corresponding ordinary differential equations together with the corrector equations of mobile substances in the corrector step by using the Newton–Raphson method. Two examples are demonstrated to show the capability of our model.

Ensemble‐averaged equations for reactive transport in porous media under unsteady flow conditions

We present a method for deriving the ensemble‐averaged reactive solute transport equation for unsteady, non‐divergence‐free flow field conditions. Our approach uses a cumulant expansion, Lie group theory, and time‐ordered exponentials to develop the ensemble‐averaged transport equation. The cumulant expansion is in powers of a αтc, where α measures the magnitude of the perturbations of the transport and reaction operators and тc is the correlation time of these perturbations. Because the cumulant expansion avoids secular terms (terms in powers of time), the problem can be closed by rationally truncating the expansion. The truncated terms can be shown to be of lower order than those terms that are kept, provided that a particular constraint (in terms of the Kubo number) is met. The use of Lie group theory allows one to automatically combine the Eulerian and Lagrangian approaches. A particular time‐ordered exponential that arises in the analysis can be interpreted as a translation operator that possesses a well‐defined algebra. These translation operators appear in the second‐order (covariance) terms of the cumulant expansion, and their effect is to shift one of the terms of the covariance functions relative to the other along the trajectory formed by the ensemble‐averaged velocity field. This approach does not require neglecting the local dispersion tensor and has the advantage that no integral transformations are conducted; therefore all results are expressed in terms of real space variables.

Non-standard Numerical Methods Applied to Subsurface Biobarrier Formation Models in Porous Media

Biofilm forming microbes have complex effects on the flow properties of natural porous media. Subsurface biofilms have the potential for the formation of biobarriers to inhibit contaminant migration in groundwater. Another example of beneficial microbial effects is the biotransformation of organic contaminants to less harmful forms, thereby providing an in situ method for treatment of contaminated groundwater supplies. Mathematical models that describe contaminant transport with biodegradation involve a set of coupled convection–dispersion equations with non-linear reactions. The reactive solute transport equation is one for which numerical solution procedures continue to exhibit significant limitations for certain problems of groundwater hydrology interest. Accurate numerical simulations are crucial to the development of contaminant remediation strategies. A new numerical method is developed for simulation of reactive bacterial transport in porous media. The non-standard numerical approach is based on the ideas of the ‘exact’ time-stepping scheme. It leads to solutions free from the numerical instabilities that arise from incorrect modeling of derivatives and reaction terms. Applications to different biofilm models are examined and numerical results are presented to demonstrate the performance of the proposed new method.

Development and demonstrative application of a 3-D numerical model of subsurface flow, heat transfer, and reactive chemical transport: 3DHYDROGEOCHEM

This paper presents the development of a 3-D coupled model of subsurface flow, heat transfer, and reactive chemical transport where the local equilibrium assumption is applied. Strong coupling is used for steady-state simulations, while weak coupling may be employed for transient simulations to save computation time. Four types of boundary conditions and a complete suite of geochemical reactions are taken into account. The flow equation is discretized by the Galerkin finite element method. Both the reactive chemical transport and heat transfer equations are solved by either the hybrid Lagrangian–Eulerian or the conventional Eulerian finite element method. Two approaches are considered to compute the transport of aqueous components. The Picard method is used to handle non-linearity in both reactive chemical transport and unsaturated subsurface flow, while the Newton–Raphson method is employed to calculate chemical equilibrium. Two realistic, albeit hypothetical example problems are used to demonstrate the capability of the model.

Reactive transport modeling of plug-flow reactor experiments: quartz and tuff dissolution at 240°C

Extension of reactive transport modeling to predict the coupled thermal, hydrological, and chemical evolution of complex geological systems is predicated on successful application of the approach to simulate well-constrained physical experiments. In this study, steady-state effluent concentrations and dissolution/precipitation features associated with crushed quartz and tuff dissolution at 240°C have been determined experimentally using a plug-flow reactor (PFR) and scanning electron microscopy (SEM) techniques, then modeled with the reactive transport simulator GIMRT ( Steefel and Yabusaki, 1996) using a linear rate law from transition state theory (TST) . For quartz dissolution, interdependence of the specific surface area ( A m ) and reaction rate constant ( k m ) predicted from the modeling agrees closely with that obtained from an analytical solution to the reaction–transport equation without diffusion/dispersion, verifying the advection-dominant nature of the PFR experiments. Independently-determined A qtz and k qtz from the literature are shown to be internally consistent with respect to the model and analytical interdependence, implying appropriateness of the linear TST rate law and adequacy of BET-determined A m for use in modeling PFR experiments. Applications of this integrated approach for monomineralic dissolution include assessment of internal consistency among independent A m and k m data, estimation of k m from BET-determined A m , and rapid evaluation of alternative rate laws. For tuff dissolution, accurate simulation of the experimental steady-state effluent concentrations (to within 3% for Na, Al and K; to within 15% for Si and Ca) and dearth of alteration phases (<1 vol.% in the model) defines minimum supersaturation thresholds for nucleation of primary minerals (to suppress their secondary precipitation) and the (non-unique) relative k m of hydrous aluminosilicates. For such multicomponent, multiphase experiments, reactive transport simulators provide the only practical means of estimating and constraining these kinetic parameters. Despite the approximations and uncertainties inherent in present-day models, close agreement between the experimental and simulation results reported herein provides an important measure of confidence for extending the modeling approach to address increasingly complex systems for which development of experimental analogs is impractical or impossible.

Reactive Transport Modelling of Microbially-Driven Redox Chemistry: Application to Subsurface Environments

The complexity of multi-component, multi-reaction subsurface environments requires sophisticated models to realistically simulate the biogeochemical dynamics of these systems. Traditional geochemical modelling of groundwater composition has been based primarily on thermodynamic equilibrium calculations. While equilibrium-based models are sometimes able to reproduce field and laboratory observations pertaining to relatively fast inorganic reactions, they are not adequate to represent the irreversible microbial degradation of organic matter, secondary redox reactions, or mineral precipitation and dissolution reactions (Hunter et al., 1998). Hence, kinetic models are needed for describing the reactions that drive the redox plus acid-base chemistry of groundwaters. The biogeochemical dynamics of subsurface environments are characterized by complex reaction networks. Frequently, the product of one reaction acts as a reactant in one or more other reactions. Furthermore, parallel reactions often compete for the same chemical compounds. The competition may involve microbially-mediated reaction pathways, inorganic reactions, or both. Central to the redox chemistry of groundwaters is the microbiallymediated decomposition of organic compounds which results in the consumption of dissolved and solid oxidants (02, , Mn(III, IV), Fe(III), ) and the production of reduced inorganic compounds (e.g. HzS, Fe z+, ). Reaction fronts may develop when the reduced species are transported to more oxidized areas and are subsequently reoxidized. A fully kinetic, multi-component, reactive transport model is presented here. The model equations describe transport by flow and dispersion, and homogeneous plus heterogeneous react ions, including both inorganic and microbially-mediated reactions. In general, all the chemical species are involved in kinetically-controlled reactions, while only a subset of the species also participate in reactions that are sufficiently fast to remain in local equilibrium. Because the rate expressions incorporate the concentrations of multiple species, the reactive transport equations of the independent species are directly coupled to one another via the reaction rate terms. All of the reactions included in the model, with the exception of homogeneous eomplexation reactions and sorption processes, are kineticallycontrolled. When groundwater transporting dissolved organic compounds and reduced inorganic species reaches an aerobic zone, some of the oxidants will be consumed by secondary redox reactions (i.e. redox reactions not involving the oxidation of organic carbon), thereby reoxidizing the reduced inorganic species and leaving only a fraction of the oxidants available for use in the microbial respiratory pathways (i.e. aerobic respiration, denitrification, manganese reduction, iron reduction and sulphate reduction). To demonstrate this, a numerical simulation was done in which leachate from a landfill infiltrates into a previously pristine, aerobic aquifer. In the simulation, reduced, organic-rich leachate is transported away from the source area, causing the aquifer to become anoxic within the boundaries of the plume. In the plume, the dissolved organic matter from the leachate is degraded via microbiallymediated pathways, resulting in the complete consumption of all available electron acceptors. At the downstream edge of the plume, the highly reduced porewaters come into contact with oxidized porewaters from the surrounding aquifer and a reaction front develops. There is significant competition between primary (i.e. microbial biodegradation) and secondary reaction pathways for the electron acceptors present at this interface. For example, the calculations reported in Table 1 predict that 90% of the O2 and all of the supplied by the groundwater are diverted to the oxidation of reduced inorganic species carried by the contaminant plume. Similarly, large

Models of mobility-shift assay of complexes between dimerizing protein and DNA.

The theory of mass transport coupled to macromolecular interactions under chemical kinetic control forms the basis of four different models of the electrophoretic mobility-shift assay of complexes formed between dimerizing proteins and DNA. The theory of mass action was applied to the set of simultaneous dimerization (either simple or ligand-induced) and DNA-binding reactions in order to fix the initial equilibrium composition of mixtures to be assayed. Theoretical mobility-shift patterns were obtained for a range of protein concentrations at constant DNA concentration by numerical solution of the set of simultaneous transport-reaction equations appropriate for each model. In those cases in which dimerization in solution is modeled (including heterodimerization), analysis of the peaks in the patterns provides apparent binding constants, which, when extrapolated to infinite dilution of protein, yield acceptable estimates of equilibrium constants. Those for binding of dimer are products of two or three equilibrium constants, from which the equilibrium binding constant can be extracted, provided that dimerization and, where required, ligand-binding constants are determined by independent physicochemical methods. Dimerization of protein when bound to DNA is distinctive in that extrapolation to infinite dilution of protein is not required.

Eulerian–Lagrangian localized adjoint methods for reactive transport with biodegradation

AbstractThe microbial degradation of organic contaminants in the subsurface holds significant potential as a mechanism for in‐situ remediation strategies. The mathematical models that describe contaminant transport with biodegradation involve a set of advective–diffusive–reactive transport equations. These equations are coupled through the nonlinear reaction terms, which may involve reactions with all of the species and are themselves coupled to growth equations for the subsurface bacterial populations. In this article, we develop Eulerian–Lagrangian localized adjoint methods (ELLAM) to solve these transport equations. ELLAM are formulated to systematically adapt to the changing features of governing partial differential equations. The relative importance of retardation, advection, diffusion, and reaction is directly incorporated into the numerical method by judicious choice of the test functions that appear in the weak form of the governing equation. Different ELLAM schemes for linear variable–coefficient advective–diffusive–reactive transport equations are developed based on different operator splittings. Specific linearization techniques are discussed and are combined with the ELLAM schemes to solve the nonlinear, multispecies transport equations. © 1995 John Wiley &amp; Sons, Inc.

Theory of the mobility-shift assay of nonspecific protein-DNA complexes governed by conditional probabilities: the HU:DNA complex.

Complexes of an 88 bp DNA and the HU protein were studied by both experimental and theoretical electrophoretic mobility-shift analyses. Experimental analysis defined the stoichiometry of binding and estimated an apparent intrinsic dissociation constant (Kd = 1 to 3 x 10(-7) M) for the HU:DNA complexes. The theory of conditional probabilities was applied to the binding of HU to DNA in order to fix the initial equilibrium composition of mixtures to be assayed theoretically by the mobility-shift procedure. Electrophoretic mobility-shift patterns were obtained by numerical solution of a set of simultaneous transport-reaction equations, in which the chemical kinetic term is formulated in terms of dissociation of the different DNA:HU complexes in gel cages. The computed patterns simulated the experimental patterns describing the titration of a fixed concentration of an 88 bp DNA fragment with dimeric HU. These insightful results provide guidelines for interpretation of the electrophoretic behavior of systems in which a ligand binds nonspecifically to DNA. In particular, the narrow unresolved zone observed both experimentally and theoretically beyond 50-60% saturation is a reaction zone characteristic of noncooperative ligand-binding governed by conditional probabilities. The discrepancy between the theoretically assigned and experimental values of the intrinsic binding constant is attributed to an HU-induced change in the conformation of DNA.