Advection Dispersion Reaction Equation Research Articles

A Semi-Analytical Model of Contaminant Transport in Barrier Systems with Arbitrary Numbers of Layers

In regions with sanitary landfills, unsuitable liner designs can result in significant soil and groundwater contamination, leading to substantial environmental remediation costs. Addressing this challenge, we propose a semi-analytical model for solute transport that uses the advection–dispersion–reaction equation in a multi-layered liner system. A distinctive feature of our model is its ability to account for infiltration velocity, arbitrary numbers of layers, thin layers such as geomembranes, and mass flow. We validated our model against existing published models and applied it to a case study of a real sanitary landfill in the capital of Brazil. Through parametric analyses, we simulated contaminant transport across various layers, including the geomembrane (GM), geosynthetic clay liner (GCL), soil liner (SL), and compacted clay liner (CCL). The analyses showed the importance of choosing the most appropriate construction system based on the location and availability of materials. Considering toluene contamination, a GM molecular diffusion coefficient (DGM) greater than 10−13 m2 s−1 exhibited similar efficiency when compared with CCL (60 cm thick). In addition, the results showed that the liner system may have the same efficiency in changing SL (60 cm thick) for a GCL (1 cm thick).

Behavior characteristics of bimolecular reactive transport in heterogeneous porous media

In order to study the mechanism of bimolecular reactive solute transport in heterogeneous porous media, the chemical reaction (CuSO4 + Na2EDTA2−→CuEDTA2) was carried out by laboratory experiments and numerical simulation in heterogeneous porous media. Three different kinds of heterogeneous porous media (Sd2 = 1.72, 1.67 and 0.80 mm2) and flow rates (1.5, 2.5 and 5.0 mL/s) were considered. The increase of flow rate would promote the mixing between reactants, resulting in a greater peak value and a slighter “tailing” of product concentration, while the increase of medium heterogeneity would result in a more significant “tailing”. It was found that the concentration breakthrough curves of reactant CuSO4 had a peak in the early stage of the transport, and the peak value increased with the increase of flow rate and medium heterogeneity. The concentration peak of CuSO4 was caused by the delayed mixing and reaction of reactants. The IM-ADRE (The advection–dispersion–reaction equation considering incomplete mixing) model could well simulate the experimental results. The simulation error of IM-ADRE model for the concentration peak of product was less than 6.15%, and the fitting accuracy for “tailing” increased with the increase of flow. The dispersion coefficient increased logarithmically with the increase of flow, and was negatively correlated to the heterogeneity of the medium. In addition, the dispersion coefficient of CuSO4 simulated by IM-ADRE model was one order of magnitude larger than that simulated by ADE model, indicating that the reaction promoted dispersion.

Modeling the Dynamics of Mixture Toxicity and Effects of Organic Micropollutants in a Small River under Unsteady Flow Conditions.

The presence of anthropogenic organic micropollutants in rivers poses a long-term threat to surface water quality. To describe and quantify the in-stream fate of single micropollutants, the advection-dispersion-reaction (ADR) equation has been used previously. Understanding the dynamics of the mixture effects and cytotoxicity that are cumulatively caused by micropollutant mixtures along their flow path in rivers requires a new concept. Thus, we extended the ADR equation from single micropollutants to defined mixtures and then to the measured mixture effects of micropollutants extracted from the same river water samples. Effects (single and mixture) are expressed as effect units and toxic units, the inverse of effect concentrations and inhibitory concentrations, respectively, quantified using a panel of in vitro bioassays. We performed a Lagrangian sampling campaign under unsteady flow, collecting river water that was impacted by a wastewater treatment plant (WWTP) effluent. To reduce the computational time, the solution of the ADR equation was expressed by a convolution-based reactive transport approach, which was used to simulate the dynamics of the effects. The dissipation dynamics of the individual micropollutants were reproduced by the deterministic model following first-order kinetics. The dynamics of experimental mixture effects without known compositions were captured by the model ensemble obtained through Bayesian calibration. The highly fluctuating WWTP effluent discharge dominated the temporal patterns of the effect fluxes in the river. Minor inputs likely from surface runoff and pesticide diffusion might contribute to the general effect and cytotoxicity pattern but could not be confirmed by the model-based analysis of the available effect and chemical data.

Nickel removal by biogenic selenium nanoparticles (SeNPs): Characterization and simulation of non-symmetric breakthrough curves in continuous-flow packed-bed columns

Nanoparticles offer an alternative mean for heavy metals removal from aquatic environment. In this work we study the sequestering of nickel by biogenic selenium nanoparticles (SeNPs) both in native and in immobilized form in alginate matrix (alginate-SeNPs). The apparent sorption like process is attributed to the coordination chemistry of Ni with the functional groups of the extracellular polymeric substances (EPS) covering the SeNPs. The kinetics and the equilibrium of the process are presented, while the morphology and the properties of the adsorption beads are studied by SEM, FTIR and rheological analysis. Langmuir isotherm describes the adsorption equilibrium with maximum loading Q0 29.411 mg Ni/g SeNPs and 0.651 mg Ni/g alginate-SeNPs. The adsorption of nickel is fast and equilibrium is attained within one hour. Continuous flow experiments in packed beds reveal early elution patterns and non-symmetric sigmoidal profiles due to the reduced adsorption capacity of the immobilized SeNPs, the short depth of the beds and the apparent non-uniform flow pattern. The Bohart-Adams equation fits adequately the main sigmoidal part of the breakthrough curves, however fails to predict the early breakthrough data. By using the advection–dispersion-reaction equation (ADR) under two novel aspects: (a) the rapid equilibrium model with the introduction of an efficiency coefficient accounting for all the kinetic and sorption limitations and (b) the split flow model accounting for any non-uniform flow conditions in the column, the model accuracy is significantly improved. Repeated sorption–desorption experiments show that alginate-SeNPs retain 59% of the original adsorption capacity even after eight cycles.

Colloid transport through a variable-aperture fracture under unfavorable attachment conditions: Characterization with a continuous time random walk model

Although the effects of fracture aperture variability on colloid advection and dispersion are well understood, how aperture variability affects colloid deposition under unfavorable geochemical conditions remains underexplored. To address this knowledge gap, we simulated 6200 cases of colloid transport in variable-aperture fractures with different magnitudes of fracture aperture variability and correlation lengths. Flows were laminar (Reynolds number < 1) with dispersion-dominated transport (Peclet number < 1) in the simulations. The numerically obtained breakthrough curves (BTCs) reflected non-Fickian transport behavior with power-law tailing where the power-law exponent increased with aperture variability rather than correlation length. Solutions to the advection-dispersion-reaction equation (ADRE) and a continuous time random walk (CTRW) model were fit to the BTCs. The CTRW model faithfully reproduced BTCs that decayed according to a power law, while the ADRE failed to do so. The ensemble mean of the attachment rate (equivalent to the reaction rate in the ADRE) in the CTRW model was correlated to a measurable property (i.e., aperture variability) using a power-law function. This newly established function facilitates accurate prediction of non-Fickian colloid transport using the CTRW model where dynamic attachment and detachment of colloids are considered under unfavorable deposition conditions.

Modeling contact oxygenation and adhesive wear extension in axisymmetric flat circular fretting interfaces

In this study, adhesive wear extension in axisymmetric fretting contacts is simulated by applying the Contact Oxygenation Concept (COC) which illustrates the influence of fretting loadings on oxygen transport within the interface. Di-oxygen flow is modeled using an Advection-Dispersion-Reaction (ADR) approach which was formerly proposed assuming third body as a porous medium crossed by atmospheric gases. It was shown that ADR model is not only predictive in square contacts but also in circular flat interfaces where it is capable of simulating the partition of adhesion and abrasion for a broad range of contact pressures and sliding frequencies. Moreover, a simplified analytical solution is proposed for ADR equation which is proven to be highly correlated with the numerical and experimental results.

Identification of multiple unknown point sources occurring in the 2D transport equation: application to groundwater pollution source identification

Accessing quality water is of crucial importance to both society and the environment. Deterioration in water quality through groundwater pollution presents a substantial risk to human health, plant and animal life, and detrimental effects on the local economy. To ensure groundwater quality, there is need to identify locations of unknown groundwater pollution sources. In this paper, the locations of groundwater contaminant sources have been identified using inverse problem technique. The work in this paper concerns the inverse source problem in the Advection Dispersion Reaction Equation (ADRE) with an emphasis on groundwater pollution source identification. Mathematically, inverse source problem involves the reconstruction of the source function in the ADRE from the boundary and interior measurements. An inverse source problem technique for identifying the unknown groundwater pollution source utilizing only the boundary and interior measurements is developed. The finite volume discretization method is employed on the adjoint ADRE to provide the data. The data from the finite volume method results into Volterra integral equation which after discretizing transforms into an ill-posed inverse problem. Tikhonov regularization method is used to achieve stability on the ill-posed problem. The results indicates that our proposed inverse problem is accurate with the data.

Exploring Compatibility of Sherwood-Gilland NAPL Dissolution Models with Micro-Scale Physics Using an Alternative Volume Averaging Approach

The dynamics of NAPL dissolution into saturated porous media are typically modeled by the inclusion of a reaction term in the advection-dispersion-reaction equation (ADRE) with the reaction rate defined by a Sherwood-Gilland empirical model. This stipulates, among other things, that the dissolution rate is proportional to a power of the NAPL volume fraction, and also to the difference between the local average aqueous concentration of the NAPL species and its thermodynamic saturation concentration. Solute source models of these sorts are ad hoc and empirically calibrated but have come to see widespread use in contaminant hydrogeology. In parallel, a number of authors have employed the method of volume averaging to derive upscaled transport equations describing the same dissolution and transport phenomena. However, these solutions typically yield forms of equations that are seemingly incompatible with Sherwood-Gilland source models. In this paper, we revisit the compatibility of the two approaches using a radically simplified alternative volume averaging analysis. We begin from a classic micro-scale formulation of the NAPL dissolution problem but develop some new simplification approaches (including a physics-preserving transformation of the domain and a new geometric lemma) which allow us to avoid solving traditional closure boundary value problems. We arrive at a general, volume-averaged governing equation that does not reduce to the ADRE with a Sherwood-Gilland source but find that the two approaches do align under straightforward advection-dominated conditions.

Reactive contaminant transport simulation using the analytic element method, random walk particle tracking and kernel density estimator

In this study, we propose a simulation model to solve the two-dimensional groundwater flow and advection-dispersion-reaction equation (ADRE) by coupling analytic element method (AEM), random walk particle tracking (RWPT) and kernel density estimator (KDE). In the AEM-RWPT-KDE model, AEM solves the groundwater flow equation, RWPT solves the advection-dispersion-reaction equation and KDE enhances the accuracy and computational efficiency of RWPT. AEM generates a continuous velocity distribution which is suitable for RWPT. An analytic expression is modified to simulate radioactive decay for discrete transport time steps and is embedded in the RWPT model. Linear adsorption is also incorporated in the RWPT model by tracking particles with a retarded velocity. Unlike Eulerian transport models such as finite element or finite difference methods, RWPT is completely free from numerical dispersion. The comparison of results illustrates the superiority of the RWPT model over accurate Eulerian-Lagrangian models with reference to the analytical solution. Further, the AEM-RWPT-KDE model is used to simulate the transport processes of Radium-228 and Trichloroethene (TCE) in two hypothetical aquifer cases. Both case studies reflect the practical applicability of the proposed methodology.

A parsimonious transport model of emerging contaminants at the river network scale

Abstract. Waters released from wastewater treatment plants (WWTPs) represent a relevant source of pharmaceuticals and personal care products to the aquatic environment, since many of them are not effectively removed by the treatment systems. The consumption of these products increased in the last decades and concerns have consequently risen about their possible adverse effects on the freshwater ecosystem. In this study, we present a simple, yet effective, analytical model of transport of contaminants released in surface waters by WWTPs. Transport of dissolved species is modeled by solving the advection dispersion reaction equation (ADRE) along the river network by using a Lagrangian approach. We applied this model to concentration data of five pharmaceuticals, diclofenac, ketoprofen, clarithromycin, sulfamethoxazole, and irbesartan, collected during two field campaigns, conducted in February and July 2015 in the Adige River, northeastern Italy. The model showed a good agreement with measurements and the successive application at the monthly timescale highlighted significant variations of the load due to the interplay between streamflow seasonality and variation of the anthropogenic pressure, chiefly due to the variability of touristic fluxes. Since the data required by the model are widely available, our model is suitable for large-scale applications.

On the separate treatment of mixing and spreading by the reactive-particle-tracking algorithm: An example of accurate upscaling of reactive Poiseuille flow

The Eulerian advection-dispersion-reaction equation (ADRE) suffers the well-known scale-effect of reduced apparent reaction rates between chemically dissimilar fluids at larger scales (or dimensional averaging). The dispersion tensor in the ADRE must equally and simultaneously account for both solute mixing and spreading. Recent reactive-particle-tracking (RPT) algorithms can, by separate mechanisms, simulate 1) smaller-scale mixing by inter-particle mass transfer, and 2) mass spreading by traditional random walks. To test the supposition that the RPT can accurately track these separate mechanisms, we upscale reactive transport in Hagen-Poiseuille flow between two plates. The simple upscaled 1-D RPT model with one velocity value, an upscaled Taylor macro-dispersivity, and the local molecular diffusion coefficient matches the results obtained from a detailed 2-D model with fully described velocity and diffusion. Both models use the same thermodynamic reaction rate, because the rate is not forced to absorb the loss of information upon upscaling. Analytic and semi-analytic upscaling is also performed using volume averaging and ensemble streamtube techniques. Volume averaging does not perform as well as the RPT, while the streamtube approach (using an effective dispersion coefficient along with macro-dispersion) performs almost exactly the same as RPT.

A non-iterative method for identifying multiple unknown time-dependent sources compactly supported occurring in a 2D parabolic equation

We address the non-linear inverse source problem of identifying multiple unknown time-dependent sources spatially supported in some subdomains of a two-dimensional bounded domain subject to an evolution advection–dispersion–reaction equation. Provided to be available within the monitored domain interfaces for recording the state and its flux crossing each suspected zone where a source could occur, we establish an identifiability theorem that yields uniqueness of the unknown elements defining all occurring sources. We develop a non-iterative detection–identification method that goes throughout the monitored domain to detect in each suspected zone whether there exists or not an occurring source. Once a source is detected, the developed method determines lower and upper bounds of the total amount discharged by the occurring source and localizes the geometric centre of its unknown spatial support. Then, given two desired reference geometries for example, squares/discs centred at the already localized geometric centre, the method determines the biggest domain defined by a first reference geometry included in the sought source spatial support as well as the smallest domain defined by a second reference geometry containing this unknown support. In addition, the developed method gives an approximation of the surface area of this latest and identifies its time-dependent intensity function. Some numerical experiments on a variant of the surface water Biological Oxygen Demand pollution model are presented.

Detection-Identification of multiple unknown time-dependent point sources in a 2D transport equation: application to accidental pollution

We address the nonlinear inverse source problem of identifying multiple unknown time-dependent point sources occurring in a two-dimensional evolution advection–dispersion–reaction equation. Provided to be available within the monitored domain interfaces for recording the generated state and its flux crossing each suspected zone where a source could occur, we establish a constructive identifiability theorem based on an introduced dispersion-current function that yields uniqueness of the unknown elements defining all occurring sources. Then, the established theorem leads to develop a detection-identification method that goes throughout the monitored domain to detect in each suspected zone whether there exists or not an occurring source. Once a source is detected, the developed method determines lower and upper bounds of the mean value discharged by its unknown time-dependent intensity function. Thereafter, the method localizes the sought position of the detected source as the unique solution of an equation satisfied by the introduced dispersion-current function and identifies its unknown intensity function from solving an associated deconvolution problem. Ultimately, the unknown number of occurring sources is deduced as the sum of all detected-identified active sources. Some numerical experiments on a variant of the surface water BOD pollution model are presented.

Numerical Simulation and Experimental Study of Bimolecular Reactive Transport in Porous Media

The traditional advection–dispersion–reaction equation (ADRE) often meets difficulty in simulating diffusion-controlled reactive transport, because the grid-based ADRE with a laboratory-measured reaction rate overpredicts pore-scale mixing and the product concentration. In this study, we chose bimolecular reactive transport (A $$\,+\,$$ B $$\,\rightarrow \,$$ AB) in porous media as an example and developed a fully implicit Galerkin finite-element method with Picard’s linearization scheme for solving the ADRE considering incomplete mixing at pore scale (IM-ADRE) based on a continuum approximation by Sanchez-Vila et al. (Water Resour Res 46:W12510, 2010). Sensitivity analysis showed that the IM-ADRE model was most sensitive to the parameter “m,” which was the power index of time used to control the decline rate of the time-dependent kinetic reaction term considering effects of incomplete mixing at pore scale. We used the IM-ADRE model to interpret the column experiment reported by Raje and Kapoor (Environ Sci Technol 34(7):1234–1239, 2000), which studied bimolecular reactive transport of aniline (AN) +1,2-naphthoquinone-4-sulfonic acid (NQS) $$\rightarrow $$ 1,2-naphthoquinone-4-aminobenzene (NQAB) in porous media. Previous studies found that the discrepancy between the simulated and observed peak product concentrations was as large as 55 % by the traditional ADRE, while the discrepancy was reduced to 5 % by the IM-ADRE model. We further conducted new reactive transport column experiments with NQS and AN to systematically understand the nature of bimolecular reactive transport in porous media and further check the validity of the IM-ADRE model. Our experiments differed from previous ones in two aspects. First, we used two different realistic porous media with six different flow velocities while previous experiments mostly involved artificial media such as glass beads under two similar flow velocities. Second, our experiment used a column (with the length of 100 cm) relatively longer than that (18 cm) in Raje and Kapoor (2000), and thus, the boundary effect of the column was minimized, and the time dependence of the effective reaction term could be further checked rigorously. Our study reveals that the IM-ADRE is a feasible tool in quantifying reactive transport at a travel distance up to 100 cm, further validating the effective, time-dependent rate coefficient proposed empirically by Sanchez-Vila et al. (2010).

Identification of time active limit with lower and upper bounds of total amount loaded by unknown sources in 2D transport equations

We address the nonlinear inverse source problem, which consists in identifying the time limit from which some unknown sources in a two-dimensional advection–dispersion-reaction equation become inactive. Under some reasonable assumptions and without requiring any a priori information about the form of the involved unknown sources, we prove the identifiability of the desired time active limit. We establish an identification method based on some boundary null controllability results that uses the records of the associated state on the outflow boundary and of its flux on the inflow boundary of the monitored domain to determine the time active limit. The application of this method to the most encountered forms of surface water pollution sources leads us to determine also lower and upper bounds that provide a significant approximation of the total amount loaded by the involved unknown sources without requiring any a priori information either on the number of active sources or on the form of their time-dependent intensity functions. Some numerical experiments on a variant of the surface water biological oxygen demand pollution model are presented.

Predictions of dynamic changes in reaction rates as a consequence of incomplete mixing using pore scale reactive transport modeling on images of porous media

We present a pore scale model capable of simulating fluid/fluid reactive transport on images of porous media from first principles. We use a streamline-based particle tracking method for simulating flow and transport, while for reaction to occur, both reactants must be within a diffusive distance of each other during a time-step. We assign a probability of reaction (Pr), as a function of the reaction rate constant (kr) and the diffusion length. Firstly, we validate our model for reaction against analytical solutions for the bimolecular reaction (A+B→C) in a free fluid. Then, we simulate transport and reaction in a beadpack to validate the model through predicting the fluid/fluid reaction experimental results provided by Gramling et al. (2002). Our model accurately predicts the experimental data, as it takes into account the degree of incomplete mixing present at the sub-pore (image voxel) level, in contrast to advection–dispersion–reaction equation (ADRE) model that over-predicts pore scale mixing. Finally, we show how our model can predict dynamic changes in the reaction rate accurately accounting for the local geometry, topology and flow field at the pore scale. We demonstrate the substantial difference between the predicted early-time reaction rate in comparison to the ADRE model.

Dilution Effect of the Apparatus on Contaminant Transport Parameters Assessed by Column Testing

AbstractThis technical note describes a method aimed at correcting the dilution effect of the apparatus on the solute transport parameters determined by column testing. The method requires quantification of the dead volume (i.e., not drainable during sample collection) of the permeation apparatus and makes use of a closed-form solution to the advection-dispersion-reaction equation (ADRE) equation used to infer transport parameters from column test results. Simulations of the dilution effect were performed and actual experimental data were analyzed, with reference to testing equipment frequently used in column studies (flexible-wall permeameter and bladder accumulators) and to porous materials commonly used in pollutant containment [geosynthetic clay liners (GCLs) and compacted clay liners (CCLs)]. Results showed that the dilution effect of the testing apparatus can significantly affect the determined transport parameters, in particular by increasing the retardation factor and reducing the diffusion coeffi...

A Solution for a Water Quality Model in a Uniform Stream Channel using New Iterative Method

During the last decade, political awareness of river water quality issues has increased significantly. New environmental policies have a requirement for improved methods for the investigation and evaluation of river water quality. Moreover, the derivation and assessment of management practices is also necessary. An efficient Perturbation Iteration Algorithm for solving the water quality assessment model has been developed. The employed equation of this uniform flow model is one-dimensional Advection-DispersionReaction equation that has variable coefficients. This water quality model requires the calculation of the substance dispersion considering the water velocity in the channel. Numerical values are obtained by using the Runge-Kutta-Fehlberg fourth-fifth order method for comparison. It was discovered that Perturbation Iteration Algorithm solution gels well with the numerical solution. Two examples are included to demonstrate the efficiency, accuracy, and simplicity of the proposed method.

Improvement of numerical simulation of bimolecular reactive transport using time-dependent parameters

The overestimation of the contaminant concentration is a main issue in simulating the reactive transport using the common advection-dispersion-reaction equation (ADRE). To solve this problem, a new modeling method is developed. The parameters of the model are identified based on experimental data. The results of the model are compared with previous results in the literature, and the sensitivity of the model is analyzed by examining the model's response to changes of the model parameters. Main conclusions are as follows: (1) The numerical modeling approach is feasible with an improved simulating accuracy. The predicted values are in agreement with the experimental measurements. The relative errors of the peak concentration between the simulated and experimental values are less than 2.5%. The errors are greatly reduced as compared with the results (about 67.8% as the maximum) based on a traditional ADRE in the literature. (2) There are three parameters (m, β0 and D) which can be calibrated on the basis of experimental data in the model. The reactive product concentrations are mainly influenced by the parameters involved in the reactive ratio such as m and β0. The hydrodynamic dispersion coefficient D has almost no influence on the reactive product transport. (3) Our model does not provide better fitting curves for the “early arrival” and the “long tail”. The “early arrival” and the “long tail” are associated with low values of the product AB. Under these conditions, the reaction rate is close to 0, and the model of the ADRE reduces to the advection-dispersion equation (ADE). Further mechanism study is needed in the future.

Modeling improved ISCO treatment of low permeable zones via viscosity modification: Assessment of system variables

A major challenge to successfully using in situ chemical oxidation (ISCO) for groundwater treatment is achieving uniform contact between the oxidant and contaminants in a heterogeneous aquifer. Viscosity modification technology, where a water-soluble polymer is mixed with remedial fluids, has been introduced in recent years to improve oxidant coverage of the target zone (i.e., sweep efficiency) and thus, treatment efficacy. In this work, we developed a numerical model to simulate the remedial fluid coverage from an ISCO injection with viscosity modification. Specifically, solution mixtures of xanthan and NaMnO4 were injected into a two-dimensional (2D) transport flow box that contained heterogeneous layers. Xanthan solutions were simulated as shear-thinning non-Newtonian fluids, where viscosity is a function of shear rate, polymer and NaMnO4 concentrations. Reactive transport of the polymer, NaMnO4, TCE, and reaction products were simultaneously modeled using advection dispersion reaction (ADR) equations coupled with the simulated flow field. The numerical model was validated using experimental data from the 2D cell experiments. Sensitivity analysis was conducted to investigate the relative contributions of system variables, such as polymer and permanganate concentrations, flow rate, permeability contrast, and different geometry settings. Results showed that higher concentration of permanganate and slower flow rate of the shear-thinning non-Newtonian fluids improved the oxidants ability to enter low permeable zones and react with the TCE. Higher permeability contrast decreased the velocity of the xanthan–MnO4− mixture inside the low permeable zone (LPZ), which increased TCE oxidation and product recovery. Changing the architecture of the LPZ from one zone to two smaller zones separated by a transmissive zone increased the overall product recovery. Thus, viscosity modification can improve both the sweeping efficiencies and TCE removal.