Advection-dispersion Model Research Articles

A Method for Calibrating the Transient Storage Model from the Early and Late-Time Behavior of Breakthrough Curves

Solute transport in rivers is controlled by mixing processes that occur over a wide spectrum of spatial and temporal scales. Deviations from the classic advection–dispersion model observed in tracer test studies are known to be generated by the temporary trapping of solutes in storage zones where velocities and mixing rates are relatively small. In this work, the relation between the early and late-time behavior of solute breakthrough curves (BTCs) and the key parameters of the Transient Storage Model (TSM) is analyzed using non-asymptotic approximations of the model equations. Two main slopes are identified corresponding to the rising and decreasing limbs of the BTCs which are linked by specific relationships to transport and storage parameters. The validity of the proposed approximations is demonstrated with both synthetic and experimental data. Consistent with the TSM assumptions, the range of validity of the proposed approximations represents the limit of separability between surface dispersion and transient storage and can be expressed as a function of a nondimensional parameter. The results of this work can help environmental scientists identify solute transport and transient storage parameters and support the design of enhanced field tracer experiments.

Solutions of the mobile–immobile advection–dispersion model based on the fractional operators using the Crank–Nicholson difference scheme

In this manuscript, based on the Atangana–Baleanu Caputo (ABC) fractional derivative and the Caputo fractional derivative for the mobile–immobile advection–dispersion model are considered. For the proposed model, the Crank–Nicholson difference method (C-NDM) scheme is created. Utilizing the Von-Neumann analysis technique, the stability estimate for this difference scheme is demonstrated. The numerical results were obtained for the proposed model using the (C-NDM). The numerical results were produced using the MATLAB program. Then, the obtained approximate solutions for the proposed model through these two fractional derivative operators using the (C-NDM) were compared with the exact solution. The obtained results demonstrate the viability and utility of the proposed technique, and these two fractional derivatives yield a more comparable result. This study differs from previous studies by comparing this model which is based on the Caputo and the Atangana–Baleanu Caputo fractional derivatives using the Crank–Nicholson difference technique.

Experimental investigations on spreading and displacement of fluid plumes around an injection well in a contaminated aquifer

ｄe concentration of the injected fluid decays smoothly around the location of the corresponding sharp interface, which is beyond the theoretical range. This concentration spread should be considered in redefining outer mixing boundaries around injection well between the injected and ambient contaminated solutions. In this study, a radial transport experiment was performed (naphthol green B was first injected as a contaminant plume followed by potassium permanganate as the remediation agent) to mimic the interface mixing in more realistic conditions. The radial advection–dispersion model (e.g., dominated by interfacial driving, R (C/C0 = 0.5), combined with dispersion mixing) was used to obtain the position of the “boundary”, rf, at the mixing fringe of the injected fluid. The results show that the injected oxidant front ran after the contaminated plume at a migration velocity of more than 1.29 times, increasing the mechanical dispersion difference between potassium permanganate and naphthol green B. The increase in the injection rate and ambient groundwater flow velocity improved the mixing of oxidant and contaminated plume (the area difference between the contaminated plume and the oxidant decreased) but also accelerated the contaminated plume coverage. Changing the injection mode (such as linear injection) was an effective method, which enhanced the mixing of the oxidant-contamination plume and weakly expanded the contaminated plume (The swept area, 254 cm2, was the smallest among the three injection modes). The relationship between injection and mixing needs to be further strengthened.

A CONVERGENCE ANALYSIS OF THE MOBILE–IMMOBILE ADVECTION–DISPERSION MODEL OF TEMPORAL FRACTIONAL ORDER ARISING IN WATERSHED CATCHMENTS AND RIVERS

An efficient high-order computational procedure is going to be created in this paper to determine the solution to the mobile–immobile advection–dispersion model (MIAD) of temporal fractional order [Formula: see text], which can be employed to model the solute forwarding in watershed catchments and floods. To do it, the temporal-first derivative of MIAD is discretized by using the finite-difference technique’s first-order precision and the linear interpolation’s temporal-fractional derivative. On either side, the space derivative is simulated using a collocation approach based on the Legendre basis to generate the full-discrete method. The order of MIAD-convergence for the implicit numerical structure is explained. Additionally, a basic conceptual discussion of the temporal-discretized stability mechanism is included in this paper. Finally, two models are provided to show the reliability and excellence of the organized approach.

Solute transport in heterogeneous model porous media under different flow rates: Experimental and modelling study

The present paper focuses on solute transport behaviour in a Model Heterogeneous Porous Medium (MHPM) under different flow rates. We report tracer experiments under stationary hydraulic conditions, with 7 different stationary flow rates spanning two orders of magnitude. Several replicates are carried out on several MHPMs, allowing for a sound statistical assessment of experimental imprecision. The experimental BreakThrough Curves (BTCs) exhibit a dual transport mode in agreement with previously-reported field scale experiments. This dual transport mode is shown to be flow-rate independent under a suitable variable change, with the BTCs superposing in the limit of experimental uncertainty. The experiment is modelled using a classical Multi-Region Advection-Dispersion (MRAD) model with only two mobile regions. We present a flow rate independent reformulation of the MRAD model that allows both water and solute continuity to be preserved during the calibration process. Assuming a linear dependence of the dispersion and exchange coefficients on the flow rate is shown to yield satisfactory model behaviour. This confirms the linearity of the dispersion coefficient with respect to the flow rate, often suggested in the literature, over a wide range of flow conditions.

Numerical simulation of time variable fractional order mobile–immobile advection–dispersion model based on an efficient hybrid numerical method with stability and convergence analysis

In this paper, we propose a hybrid numerical method based on the weighted finite difference and the quintic Hermite collocation methods. The proposed method is used for solving the variable-order time fractional mobile–immobile advection–dispersion(VOMIM-AD) model, such that the discretization is done by applying collocation method with Hermite splines in the spatial direction and weighted finite difference method in the temporal direction. Provided examples confirm the studied stability and convergence properties of the proposed method. The obtained results from the graphical illustration and numerical simulations, in comparison with other methods in the literature, demonstrate that the reported method is very robust and accurate.

On the Numerical Approximation of Mobile-Immobile Advection-Dispersion Model of Fractional Order Arising from Solute Transport in Porous Media

The fractional mobile/immobile solute transport model has applications in a wide range of phenomena such as ocean acoustic propagation and heat diffusion. The local radial basis functions (RBFs) method have been applied to many physical and engineering problems because of its simplicity in implementation and its superiority in solving different real-world problems easily. In this article, we propose an efficient local RBFs method coupled with Laplace transform (LT) for approximating the solution of fractional mobile/immobile solute transport model in the sense of Caputo derivative. In our method, first, we employ the LT which reduces the problem to an equivalent time-independent problem. The solution of the transformed problem is then approximated via the local RBF method based on multiquadric kernels. Afterward, the desired solution is represented as a contour integral in the left half complex along a smooth curve. The contour integral is then approximated via the midpoint rule. The main advantage of the LT-RBFs method is the avoiding of time discretization technique due which overcomes the time instability issues, second is its local nature which overcomes the ill-conditioning of the differentiation matrices and the sensitivity of the shape parameter, since the local RBFs method only considers the discretization points in each local domain around the collocation point. Due to this, sparse and well-conditioned differentiation matrices are produced, and third is the low computational cost. The convergence and stability of the numerical scheme are discussed. Some test problems are performed in one and two dimensions to validate our numerical scheme. To check the efficiency, accuracy, and efficacy of the scheme the 2D problems are solved in complex domains. The numerical results confirm the stability and efficiency of the method.

Advancing understanding of in-river phosphorus dynamics using an advection–dispersion model (ADModel-P)

The objectives of the present research are (1) to predict phosphorus compounds transport along river stretches at high spatio-temporal resolution by developing an original approach based on advection–dispersion modelling (ADModel-P); (2) to advance the understanding of in-stream phosphorus transformation processes, and (3) to explore their relation to controlling factors (water temperature, seasonality and water flow). For a case study of the River Swale (UK) modelling results, in agreement with results based on experimental data, show that resuspension is the largest contributor to the variability of organic phosphorus, while adsorption–desorption are the largest contributors to the variability of soluble reactive phosphorus. Additionally, simulations reveal that conversion of inorganic to organic forms is important. In-channel sinks appear more important than sources for Soluble Reactive Phosphorus (SRP) during 80% of the time, while there is no clear evidence that Organic Phosphorus (OP) sinks or sources are dominant except the beginning of spring (around 20% of the total time). The findings are valuable because they advance knowledge regarding: (1) which in-stream processes are important; (2) values associated to transformation rates for mineralization, sedimentation, resuspension, uptake, adsorption – desorption; (3) at which times rivers are net sources or sinks and which source/sink processes might be dominant. ADModel-P is a robust model which has the benefits of simple field data requirements (compared to more complex models) and less assumptions (e.g. compared to simple models assuming perfect mixing in reaches) but without the drawbacks of lack of process representation to enable confidence in predictions to change. There is extensive scope for transferability to other rivers: rate constants can be estimated from easily attainable information on water temperature, seasonality and water flow.

Diffusion and Sorption Studies of Cs, Sr and Co in Intact Crystalline Rock

Three cationic tracers, Sr2+, Co2+ and Cs+ were tested with a modified electromigration device by applying 2V, 3V and 4V voltage gradients over an intact Grimsel granodiorite rock sample. An ideal plug-flow model and an advection-dispersion model were applied to analyze the breakthrough curves. Matrix characterization by C-14-PMMA autoradiography and scanning electron microscopy showed that in the centimeter scale of Grimsel granodiorite rock, the interconnected matrix porosity forms a well-connected network for diffusion. Micrometer-scale fissures are transecting biotite and chlorite minerals, indicating sorption of the studied cations. The ideal plug-flow model indicated that the effective diffusion coefficients (De values) for Sr2+, Co2+ and Cs+ tracer ions within the Grimsel granodiorite rock were 3.20 × 10−13 m2/s, 1.23 × 10−13 m2/s and 2.25 × 10−12 m2/s, respectively. De values were also derived from the advection-dispersion model, from which 2.86 × 10−13 m2/s, 1.35 × 10−13 m2/s and 2.26 × 10−12 m2/s were calculated. The diffusion speed for the tracers was in the sequence of Cs+ > Sr2+ > Co2+ that is in the same sequence as their diffusion in diluted water. The distribution coefficients (Kd values) calculated from the models covered the range of two magnitudes (from 10−7 m3/kg to 10−5 m3/kg). The result indicated that the sorption process of the studied elements did not reach equilibrium during the electromigration process, mainly due to the too much acceleration of the migration speed by the voltage gradients applied over the rock sample.

Time fractional advection-dispersion model to study transportation of particles with time-memory for unsteady nonequilibrium suspension in open-channel turbulent flows

In this study, the effects of time-memory on the mixing and nonequilibrium transportation of particles in an unsteady turbulent flow are investigated. The memory effect of particles is captured through a time-fractional advection-dispersion equation rather than a traditional advection-dispersion equation. The time-fractional derivative is considered in Caputo sense which includes a power-law memory kernel that captures the power-law jumps of particles. The time-fractional model is solved using the Chebyshev collocation method. To make the solution procedure more robust three different kinds of Chebyshev polynomials are considered. The time-fractional derivative is approximated using the finite difference method at small time intervals and numerical solutions are obtained in terms of Chebyshev polynomials. The model solutions are compared with existing experimental data of traditional conditions and satisfactory results are obtained. Apart from this, the effects of time-memory are analyzed for bottom concentration and transient concentration distribution of particles. The results show that for uniform initial conditions, bottom concentration increases with time as the order of fractional derivative decreases. In the case of transient concentration, the value of concentration initially decreases when T < 1 and thereafter increases throughout the flow depth. The effects of time-memory are also analyzed under steady flow conditions. Results show that under steady conditions, transient concentration is more sensitive for linear, parabolic, and parabolic-constant models of sediment diffusivity rather than the constant model.

Correction of residence time distributions and hydraulic indexes affected by tracer release duration in surface flow constructed wetlands

In tracer experiments of surface flow constructed wetlands, residence time distributions (RTDs) are generally originated from the normalized concentration data of recovered tracers based on ideal instantaneous tracer release mode. However, this tracer release mode is commonly approximated by finite-continuous tracer release over a relatively short duration. As a result, the accuracy of RTDs and hydraulic indexes cannot be guaranteed if the RTD data and index values were still calculated based on the instantaneous release. To solve this problem, a method for correcting the original RTD data and the index values affected by tracer release duration was proposed. The correction of hydraulic indexes can be performed using analytical formulas, whereas the correction of the RTDs requires an RTD model (such as the gamma distribution and one-dimensional advection-dispersion model) or a smoothing algorithm (such as the simple moving average algorithm) to avoid the increasing effect of random measurement errors caused by the direct use of the RTD correction formulas. The correction method was successfully verified by a specially designed tracer test, and its practical application effect was exemplified. A Monte Carlo test also proved that the RTD correction owed certain robustness to tracer release instability, while the correction of some hydraulic indexes that reflect flow dispersion and comprehensive hydraulic efficiency had inferior robustness. Aside from this, the influence of tracer release duration on RTDs and hydraulic indexes without any correction was analyzed. It was found that the release duration caused the shape of the original RTDs to be more skewed in the positive direction of the x-axis and changed from “thin and high” to “fat and short”, while the duration of the release caused the positive relative errors of most indexes, which became more serious with an increase in the duration or worsening hydraulic performance. Based on these influence rules, 0.01 was defined as the threshold of the normalized release duration which needs to correct RTDs and hydraulic indexes, considering that the relative errors between the uncorrected and corrected indexes were less than 5% when the normalized duration was less than this threshold.

Variable-order Mittag-Leffler fractional operator and application to mobile-immobile advection-dispersion model

The main target of our research is to achieve the solution to a variable-order fractional mobile-immobile advection-dispersion equation within the Atangana-Baleanu (AB) fractional operator. This equation is the best tool for modeling dynamical systems. The presented scheme is based on the collocation method and Lagrange polynomials. The given approach uses derivative operational matrices of integer and variable orders. The operational matrix of variable-order derivative in the AB sense is computed based on Lagrange polynomials in the presented study. The key theory of this approach is to convert the corresponding problem to a system of algebraic equations. The solution to the problem under study can be computed by utilizing the mentioned system and the collocation points. The aim of this paper is to show that this technique is easily used to solve fractional partial differential equations with excellent accuracy in results. To demonstrate the accuracy and efficiency of the suggested method, some numerical examples are provided.

Transport of silica encapsulated DNA microparticles in controlled instantaneous injection open channel experiments

Surface water tracing is a widely used technique to investigate in-stream mass transport including contaminant migration. Recently, a microparticle tracer was developed with unique synthetic DNA encapsulated in an environmentally-friendly silica coating (Si-DNA microparticle). Previous tracing applications of such tracers reported detection and quantification, but a massive loss of tracer mass. However, the transport behavior of these DNA-tagged microparticle tracers has not been rigorously quantified and compared with that of solute tracers. Therefore, we compared the transport behavior of Si-DNA microparticles to the behavior of solute NaCl in 6 different, environmentally representative water types using breakthrough curves (BTCs), obtained from laboratory open channel injection experiments, whereby no Si-DNA microparticle tracer mass was lost. Hereafter, we modelled the BTCs using a 1-D advection-dispersion model with one transient storage zone (OTIS) by calibrating the hydrodynamic dispersion coefficient D and a storage zone exchange rate coefficient. We concluded that the transport behavior of Si-DNA microparticles resembled that of NaCl in surface-water relevant conditions, evidenced by BTCs with a similar range of D; however, the Si-DNA microparticle had a more erratic BTC than its solute counterpart, whereby the scatter increased as a function of water quality complexity. The overall larger confidence interval of DSi-DNA was attributed to the discrete nature of colloidal particles with a certain particle size distribution and possibly minor shear-induced aggregations. This research established a solid methodological foundation for field application of Si-DNA microparticles in surface water tracing, providing insight in transport behavior of equivalent sized and mass particles in rivers.

A novel wavelets operational matrix method for the time variable-order fractional mobile–immobile advection–dispersion model

A novel computational technique for the solution of the variable-order fractional mobile–immobile advection–dispersion equation has been presented in this paper. Firstly, operational integration matrices and variable-order fractional derivatives were deduced using Boubaker wavelets to implement this proposed technique. Utilizing Boubaker wavelets basis for functions approximations and the operational matrices of integration and variable-order fractional derivative along with collocation points, the variable-order fractional mobile–immobile advection–dispersion equation is reduced into the system of algebraic equations. In addition, to determine the convergence analysis and error estimate of the proposed numerical technique, some useful theorems are discussed. Finally, to analyze the computational efficiency and applicability of the proposed numerical method, several numerical examples are presented.

Determining Adsorption Parameters of Potentially Contaminant-Releasing Materials Using Batch Tests with Differing Liquid-Solid Ratios.

Adsorption parameters such as the distribution coefficient are required to predict the release behavior of contaminants using advection-dispersion models. However, for potentially contaminant-releasing materials (PCMs) such as dredged sludge and coal ash, these parameters cannot be obtained by conventional adsorption tests. This study developed a method to determine adsorption parameters for PCMs from a set of batch tests conducted in parallel as a function of the liquid-solid ratio (LS-parallel test). This LS-parallel test was performed on sandy soil derived from marine sediment using liquid-solid ratios from 1 to 300 L/kg. The water-contact time was also changed from 10 min to 28 d to elucidate the kinetics or equilibrium of contaminants released from the sample. Adsorption parameters were successfully obtained if the substance was under adsorption control. A column percolation test was performed to confirm the effectiveness of the obtained parameters. Good agreements were observed for SO42− and B, but discrepancies remained for other substances such as F− and As suggesting that improvements are necessary in both the LS-parallel test procedure and the advection-dispersion model.

Long-term behavior of PFAS in contaminated agricultural soils in Germany

Per- and polyfluoroalkyl substances (PFAS) contaminated compost materials have been applied over the last few decades to agricultural fields in Germany, resulting in large-scale diffuse PFAS plumes. The leaching behavior of PFAS from the first two identified contaminated agricultural sites in Germany were investigated, one at Brilon-Scharfenberg, North Rhine-Westphalia (BS-NRW), and the other at Rastatt/Mannheim, Baden-Württemberg. The specific objectives of this study were to assess the longevity of the PFAS agricultural sources and compare standardized column percolation tests to long-term leaching of PFAS from contaminated sites. The advection-dispersion model (ADM) was used to compare the leaching behavior of perfluorooctanoic acid (PFOA) and perfluorooctanesulfonic acid (PFOS) from standardized column percolation tests and long-term field leaching data from the BS-NRW site. Column leaching tests conducted with PFOS and PFOA contaminated soil simulated the initial rapid decline but did not predict the long-term behavior (tailing) observed at the field site over 12 years. Trend analyses of the PFAS field data from the BS-NRW showed that concentrations had stabilized and that individual PFAS exhibited distinct seasonal fluctuations; the latter is likely due to the ongoing transformation of precursors and a seasonal influence on production rates of mobile PFAS. Mass balances conducted at both sites indicate that complete removal of these compounds will likely take years to decades to occur, which is expected from the results of the column leaching tests.

An efficient numerical method for the solution of 2D variable order time fractional mobile–immobile advection–dispersion model

In the literature, there are a few efficient numerical methods with stability and convergence analysis for the solution of variable order fractional partial differential equations in higher dimensions. In this work, we propose an efficient numerical method for the solution of two‐dimensional variable order time fractional mobile–immobile advection–dispersion model with spatially variable convection coefficient. At first, we discretize the time derivatives with a second‐order scheme. Then, we apply the Legendre spectral element method in spatial directions and obtain the fully discrete scheme. We prove that the semi‐discrete scheme is unconditionally stable and present an error estimate for the fully discrete scheme. The numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.

Numerical treatment of the space fractional advection–dispersion model arising in groundwater hydrology

This paper studies a new computational method for the approximate solution of the space fractional advection–dispersion equation in sense of Caputo derivatives. In the first method, a time discretization is accomplished via the compact finite difference, while the fourth kind shifted Chebyshev polynomials are used to discretize the spatial derivative. The unconditional stability and convergence order of the method are studied via the energy method. Three examples are given for illustrating the effectiveness and accuracy of the new scheme when compared with existing numerical methods reported in the literature.

Coupled 1D Hydrodynamic-Water Quality Model of Kobadak-Sibsa River System for Salinity

Salinity intrusion creates severe problems in agriculture and drinking water sources in the floodplain of tidal rivers due to low upstream flow and climate changes. This study focuses on 1-D advection-dispersion modeling of the Kobadak-Sibsa river by HEC-RAS. This work has been carried out in simulation of temporal distribution of salinity concentration in different locations which help to identify which location exceeds the chloride limit of drinking water of 1000 ppm and chloride limit in agricultural water limit of 1500 ppm. The hydrodynamic model was coupled with 1D salinity modeling. The calibration was done for the month June 2016. Then the validation was done using the data of July 2015. It has been found a good agreement between simulated and observed water level for Manning’s roughness coefficient as 0.021. Once the hydrodynamic model was calibrated and validated, the 1D advection-dispersion model was performed and calibrated and validated for the year 2016 and 2015 for different dispersion coefficients (D) for different reaches as a tuning parameter. It has found that the higher the dispersion co-efficient the higher the salinity concentrations that the dispersion co-efficient varied from 25 m2/s to 9000 m2/s which showed good agreement between simulated and observed salinity data. From December salinity gradually increases and reaches its peak in April or May. Maximum salinity concentration has been determined in different locations by simulating this model. This hydrodynamic and water quality coupled modeling can be helpful for lesson learning to prevent salinity intrusion in the Kobadak river.

Fractional and fractal advection-dispersion model

<p style='text-indent:20px;'>A fractal advection-dispersion equation and a fractional space-time advection-dispersion equation have been developed to improve the simulation of groundwater transport in fractured aquifers. The space-time fractional advection-dispersion simulation is limited due to complex algorithms and the computational power required; conversely, the fractal advection-dispersion equation can be solved simply, yet only considers the fractal derivative in space. These limitations lead to combining these methods, creating a fractional and fractal advection-dispersion equation to provide an efficient non-local, in both space and time, modeling tool. The fractional and fractal model has two parameters, fractional order (<inline-formula><tex-math id="M1">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>) and fractal dimension (<inline-formula><tex-math id="M2">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>), where simulations are valid for specific combinations. The range of valid combinations reduces with decreasing fractional order and fractal dimension, and a final recommendation of <inline-formula><tex-math id="M3">\begin{document}$ \; 0.7 \leq \alpha, \beta \leq 1 $\end{document}</tex-math></inline-formula> is made. The fractional and fractal model provides a flexible tool to model anomalous diffusion, where the fractional order controls the breakthrough curve peak, and the fractal dimension controls the position of the peak and tailing effect. These two controls potentially provide tools to improve the representation of anomalous breakthrough curves that cannot be described by the classical model.