Advection-dispersion Equation Research Articles

A note on the solution to one-dimensional advection-dispersion-reaction problems

The series representation of the reactant concentration in one-dimensional advection-dispersion-reaction problems within a container of finite length is derived in a compact and computationally less complex form than other representations found in the literature. The initial concentration of the reactant is assumed to be uniform (C 0), while the boundary conditions are assumed to be a constant reactant flux ( U C e ) at the inlet and a zero concentration gradient at the outlet of the container, where U is the average fluid velocity. The concentration is expressed in terms of the introduced constants a and b for the steady-state part, and In , Jn and Kn for the transient part of the response. The derived solution to the advection-dispersion-reaction problem can be readily generalized to include a uniform source/sink term (zero-order production rate σ) by making the replacements of C 0 and C e with C 0 − σ / k and C e − σ / k , where k is the reaction rate constant, and by adding the particular solution σ / k . A simple form of the solution to the advection-dispersion equation with a source term is also derived; its transient part is expressed in terms of the integrals In and Kn only. The formulation of the analysis is cast in such a way that the same eigenfunctions and the same eigenvalue condition apply to all three considered cases, advection-dispersion-reaction with and without a source-term, and advection-dispersion with a source term, independently of the values of k and σ.

Modeling the transport of microplastics along river networks

The excessive use of plastics in modern life has led to a significant increase in production and a corresponding rise in plastic waste generation. The slow degradation of plastics results in the introduction and accumulation of microplastics (MP) in the environment, posing environmental and health risks. River networks, acting as conduits between terrestrial and marine environments, play a crucial role in controlling the transport of MP. Predicting the complex processes of MP pathways in these environments is an ongoing challenge. To address this issue, we propose a model that integrates the advection-dispersion equation with anthropogenic MP loads and hydraulic river network characteristics. The validity of the model was assessed using literature data from three river networks worldwide. Model results show a good agreement between predictions and field observations (R2=0.72). Consequently, predicted MP data was used to perform a potential pollution assessment through the pollution load index, revealing in most cases higher MP contamination in headwaters stream and a dilution effect along the river network. The structure of the proposed model allows its further implementation to account for other transport mechanisms, interactions with other emerging contaminants (i.e., pharmaceuticals), and connections with other riverine environments, making it a valuable tool for understanding and mitigating MP pollution.

Solution of Two-Dimensional Solute Transport Model for Heterogeneous Porous Medium Using Fractional Reduced Differential Transform Method

This study contains a two-dimensional mathematical model of solute transport in a river with temporally and spatially dependent flow, explicitly focusing on pulse-type input point sources with a fractional approach. This model is analyzed by assuming an initial concentration function as a declining exponential function in both the longitudinal and transverse directions. The governing equation is a time-fractional two-dimensional advection–dispersion equation with a variable form of dispersion coefficients, velocities, decay constant of the first order, production rate coefficient for the solute at the zero-order level, and retardation factor. The solution of the present problem is obtained by the fractional reduced differential transform method (FRDTM). The analysis of the initial retardation factor has been carried out via plots. Also, the influence of initial longitudinal and transverse dispersion coefficients and velocities has been examined by graphical analysis. The impact of fractional parameters on pollution levels is also analyzed numerically and graphically. The study of convergence for the FRDTM technique has been conducted to assess its efficacy and accuracy.

Study on fuzzy fractional European option pricing model with Mittag-Leffler kernel

This research paper presents an innovative approach for modeling and analyzing complex systems with uncertain data. Our strategy leverages fuzzy calculus and time-fractional differential equations to achieve this goal. Specifically, we propose the utilization of the fuzzy Atangana-Baleanu time-fractional derivative, which incorporates non-singular kernels for fuzzy functions. This derivative type is particularly suitable for qualitative analysis of fractional differential equations in fuzzy space. We establish the existence and uniqueness of solutions for fuzzy linear time-fractional problems based on this differentiability concept. Additionally, we introduce a numerical solution method, namely the fuzzy homotopy perturbation transform method (FHPTM), to solve these problems. To demonstrate the effectiveness and practical applicability of our approach, we provide concrete examples such as the fuzzy time-fractional Advection-Dispersion equation, the fuzzy time-fractional Diffusion equation, and the fuzzy time-fractional Black-Scholes European option pricing problem. These examples not only illustrate the solution steps involved but also showcase the potential of our method in addressing real-world problems. The outcomes of our research underscore the significance of considering fuzzy calculus and time-fractional differential equations when modeling and analyzing intricate systems with uncertain data.

Performance expectation of coal waste in permeable reactive barrier for removal of cadmium considering contamination level and pore water velocity

The effectiveness and longevity of permeable reactive barriers (PRBs) depend on the performance of the reactive materials and the subsurface environment. The relationship of the groundwater velocity on performance of coal waste for the heavy metal removal was reported in our previous study. In this study, we investigated the performance and longevity of coal waste as a PRB material for the removal of Cd considering subsurface environmental conditions such as contamination level and groundwater velocity. The artificial groundwater contaminated by Cd were prepared with various concentrations ranging from 10 to 100 mg L−1. Lab-scale column experiments were conducted using coal waste filled columns by injecting the artificial groundwater. The breakthrough curves were analyzed advection dispersion equation coupled with equilibrium sorption model to determine the retardation factor. The Cd breakthrough curves exhibited different retardation with respect to the contamination levels. The Cd transport was more retarded as the contamination level lowered. The relationship between the retardation factor and the contamination levels could be explained with empirical equations based on non-linear sorption isotherms. By adopting the velocity dependency of sorbent performance in our previous study, transport of Cd within coal waste was simulated under various subsurface environmental conditions to construct the longevity function. The function could be used for the longevity prediction of coal waste as a PRB material considering groundwater velocity and contamination level in subsurface environment.

Effective removal of arsenic from contaminated groundwater using an iron-based metal-organic framework

This study investigates the potential of an iron-based metal–organic framework (MOF) material, Fe-benzene-1,3,5-tricarboxylate (Fe-BTC), for arsenic removal from water. We conducted batch and column experiments to evaluate the effects of varying mass dosages of MOF and pH ranges on arsenic adsorption. Our batch experiments revealed that increasing the mass of Fe-BTC MOF led to higher adsorption capacity. Furthermore, within the range of pH values analyzed, Fe-BTC demonstrated stable arsenic adsorption capacity, suggesting that pH conditions did not significantly affect its performance. In the column experiments, we used granitic material amended with compost and compared arsenic concentration breakthrough curves with and without the presence of MOF to assess its efficiency in arsenic removal. Without MOF, we observed rapid arsenic arrival followed by a slow increase in concentration, indicating anomalous transport dynamics induced by the compost. The addition of MOF resulted in prolonged arsenic arrival and a stabilized lower concentration, indicating effective arsenic adsorption. Fe-BTC MOF exhibited a six-fold increase in arsenic adsorption compared to its absence when added to the soil material. We employed a fractional order advection–dispersion equation model to characterize and predict the physical and chemical dynamics in the column experiments. The transport model accurately matched the arsenic breakthrough curves in the column experiments by capturing the non-Fickian transport and sorption chemical dynamics. The model indicated that compost had an insignificant impact on arsenic adsorption due to flow heterogeneity, while higher dosages of MOF resulted in increased arsenic adsorption and non-Fickian dynamics.

Upscaling solute transport in rough single-fractured media with matrix diffusion using a time fractional advection-dispersion equation

Complex structures in fractures and the rock matrix, which are difficult to map exhaustively at the local scale, can dominate contaminant transport in fractured aquifers. An upscaled model is therefore needed to effectively characterize solute transport in a heterogeneous fracture-matrix system on a large spatiotemporal scale. Most existing upscaling methods, however, do not specifically quantify the influence of matrix diffusion and fracture surface roughness on solute transport, and their upscaling parameters are usually difficult to obtain. To fill this knowledge gap, first, a time fractional advection–dispersion equation (t-FADE) model, which can accurately describe solute transport in straight fractured media, is proposed. Next, the influence of local roughness and tortuosity of fractures on transport is comprehensively considered by introducing trend lines. The t-FADE is used to characterize solute transport in each segment of rough fractures, and an upscaled model describing solute transport in rough fractures is established by averaging the governing equations of each segment. Finally, a time-dependent attenuation function is introduced into the convolutional function of the upscaled model to overcome the error caused by overlapping diffusion regions. Model comparison and analysis reveal that the rough wall of fractures strengthens matrix retention capacity, leading to a delayed peak of the breakthrough curve (BTC) with a lower peak concentration, which can be quantitatively characterized by the ratio of the actual fracture length to its longitudinal length; however, the fracture structure does not affect the solute BTC’s trailing concentration at the constant fracture mean flow velocity without considering mechanical dispersion, so the relatively simple straight fracture model can be used to describe the trailing stage of transport in rough-walled fracture media.

An Analytical Approach to Contaminant Transport with Spatially and Temporally Dependent Dispersion in a Heterogeneous Porous Medium

This study presents an analytical solution to the one-dimensional advection-dispersion equation (ADE) for a semi-infinite heterogeneous aquifer system with space and time-dependent groundwater velocity and dispersion coefficient. The dispersion coefficient is assumed to be proportional to the groundwater flow velocity. In addition, retardation factor, first-order decay and zero-order production terms are also considered. Contaminants and porous media are assumed to be chemically inert. Initially, it is assumed that some uniformly distributed solutes are already present in the aquifer domain. The input point source is considered uniformly continuous and increasing nature in a semi-infinite porous medium. The solutions are obtained analytically using the Laplace Integral Transform Technique (LITT). The nature of the concentration profile of the resulting solution for different parameters in different time domains is illustrated graphically.

A New Compact Numerical Scheme for Solving Time Fractional Mobile-Immobile Advection-Dispersion Model

This work is focused on the derivation and analysis of a novel numerical technique for solving time fractional mobile-immobile advection-dispersion equation which models many complex systems in engineering and science. The scheme is derived using the effective combination of Euler and Caputo numerical techniques for approximating the integer and fractional time derivatives respectively, and a fourth order exponential compact scheme for spatial derivatives. The Fourier analysis technique is used to prove that the proposed numerical scheme is unconditionally stable and perform convergence analysis. To assess the viability and accuracy of the proposed scheme, some numerical examples are demonstrated with constant as well as variable order time fractional derivatives for this model.

In-depth understanding of transport behavior of sulfided nano zerovalent iron/reduced graphene oxide@guar gum slurry: Stability and mobility

Nanoscale zero-valent iron (NZVI), which has the advantages of small particle size, large specific surface area, and high reactivity, is often injected into contaminated aquifers in the form of slurry. However, the prone to passivation and agglomeration as well as poor stability and mobility of NZVI limit the further application of this technology in fields. Therefore, sulfided NZVI loaded on reduced graphene oxide (S-NZVI/rGO) and guar gum (GG) with shear-thinning properties as stabilizers were used to synthesize S-NZVI/rGO@GG slurries. SEM, TEM, and FT-IR confirmed that the dispersion and anti-passivation of NZVI were optimized in the coupled system. The stability and mobility of the slurry were improved by increasing the GG concentration, enhancing the pH, and decreasing the ionic strength and the presence of Ca2+ ions, respectively. A modified advection-dispersion equation (ADE) was used to simulate the transport experiments considering the strain and physicochemical deposition/release. Meanwhile, colloidal filtration theory (CFT) demonstrated that Brownian motion plays a dominant role in the migration of S-NZVI/rGO@GG slurry, and the maximum migration distance can be increased by appropriately increasing the injection rate. Extended-Derjaguin-Landau-Verwey-Overbeek (XDLVO) theory showed that the excellent stability and migration of S-NZVI/rGO@GG slurry mainly came from the GG spatial forces. This study has important implications for the field injection of S-NZVI/rGO@GG slurry. According to the injection parameters, the injection range of S-NZVI/rGO@GG slurry is effectively controlled, which lays the foundation for the promotion of application in actual fields.

Numerical solution of non-linear advection-dispersion equation in a finite porous domain

This study presents a numerical simulation of the one-dimensional concentration-dependent convection-dispersion equation in a finite heterogeneous porous formation. The solution of the convection-diffusion equation with variable coefficients is obtained with the help of MATLAB pdepe solver. The groundwater flow velocity depends on the pollutant concentration, and the dispersion coefficient is proportional to the groundwater flow velocity. The effects of the zero-order production term and the first-order decay are also considered. The aquifer is assumed to be heterogeneous and finite, with sources concentrated in the flow direction. It is assumed that the porous media and the pollutant are chemically non-reactive. Initially porous domain is considered not solute free. The model assumes a uniform continuous input point source and a variable input point source released from the left end of the aquifer domain. The obtained results graphically describe the importance of the dispersion coefficient and other relevant parameters for solute transport in porous media. The developed numerical solution is verified with an analytical solution, and it is found that they are in good agreement.

Integrated numerical modeling to quantify transport and fate of microplastics in the hyporheic zone.

Despite the significance of rivers and streams as pathways for microplastics (MP) entering the marine environment, limited research has been conducted on the behavior of MP within fluvial systems. Specifically, there is a lack of understanding regarding the infiltration and transport dynamics of MP across the streambed interface and within the hyporheic sediments. In this study, transport and retention of MP are investigated using a new numerical modeling approach. The model is built as a digital twin of accompanying flume experiments, which are used to validate the simulation results. The model accurately represents particle transport in turbulent water flow and within the hyporheic zone (HZ). Simulations for transport and infiltration of 1µm MP particles into a sandy streambed demonstrate that the advection-dispersion equation can be used to adequately represent particle transport for pore-scale sized MP within the HZ. To assess the applicability of the modeling framework for larger MP, the experiment was repeated using 10µm particles. The larger particles exhibited delayed infiltration and transport behavior, and while the model successfully represented the spatial extent of particle transport through the HZ, it was unable to fully replicate hyporheic transit times. This study is the first to combine explicit validation against experimental data, encompassing qualitative observations of MP concentration patterns and quantification of fluxes. By that, it significantly contributes to our understanding of MP transport processes in fluvial systems. The study also highlights the advantages and limitations of employing a fully integrated modeling approach to investigate the transport and retention behavior of MP in rivers and streams.

A Simplified 3D Model for Remediation of Pollution in a River by Releasing Clean Water Using the Advection-Dispersion Equation

A Simplified 3D Model for Remediation of Pollution in a River by Releasing Clean Water Using the Advection-Dispersion Equation

A Matrix Transform Technique for Distributed-Order Time-Fractional Advection–Dispersion Problems

Invoking the matrix transfer technique, we propose a novel numerical scheme to solve the time-fractional advection–dispersion equation (ADE) with distributed-order Riesz-space fractional derivatives (FDs). The method adopts the midpoint rule to reformulate the distributed-order Riesz-space FDs by means of a second-order linear combination of Riesz-space FDs. Then, a central difference approximation is used side by side with the matrix transform technique for approximating the Riesz-space FDs. Based on this, the distributed-order time-fractional ADE is transformed into a time-fractional ordinary differential equation in the Caputo sense, which has an equivalent Volterra integral form. The Simpson method is used to discretize the weakly singular kernel of the resulting Volterra integral equation. Stability, convergence, and error analysis are presented. Finally, simulations are performed to substantiate the theoretical findings.

Analytic solutions to the two-dimensional solute advection-dispersion equation coupled with heat diffusion equation in a vertical aquifer section

Analytical and semi-analytical solutions to solute transport equations are not only used to predict the fate and transport of contaminants in soil or groundwater systems, but also to provide reference standards for validation of numerical models. However, when the solute transport equation is coupled with heat, the corresponding analytical solutions are rarely reported, perhaps due to the complexity of the coupled models. The main purpose of this paper is to present an analytical solution to the solute transport equation coupled with heat for the case of modeling solute transport in a vertical section of a homogenous infinite aquifer with steady uniform groundwater flow under constant-flux solute source. In earlier work (Shan and Javandel, 1997), analytical solutions to the solute transport equation in both finite and infinite aquifer thickness were presented, influence of heat (Soret effect) was generally not considered. Heat can, however, have a significant impact on movement of concentration front. In the present work, we first couple the solute equation with a generalized classic heat diffusion equation. To do this, we assume that the heat is not influenced by the solute (Dufour effect) such that only a one-way coupled model is considered. The repeated integral transformation method (RITM) is utilized to obtain the analytical solutions to the proposed coupled model. As a limiting case, a comparison with analytical solution (Shan and Javandel, 1997), equation (21) and other literature findings is done verify the new proposed solutions and validate our results. The influences of the thermopheresis effect and other model parameters are pictured graphically by the use of MATLAB R2015a. In comparison with the earlier work, the present results show that movement of contaminant in the medium is affected by the presence of heat, in particular along the direction of flow, consistent with the literature. The analysis of longitudinal and transverse concentration profiles suggest that the proposed solutions are applicable for monitoring of groundwater contaminants and risk assessment. Also, the solutions should be useful for comparison with numerical models.

Chebyshev spectral method for the variable-order fractional mobile–immobile advection–dispersion equation arising from solute transport in heterogeneous media

Chebyshev spectral method for the variable-order fractional mobile–immobile advection–dispersion equation arising from solute transport in heterogeneous media

Simplified dispersion analysis based on dye tests at a small stream

Abstract The modelling of solid transport in open channels requires good knowledge about parameters related to basic processes such as hydrodynamic dispersion, advection and decay rates. Such parameters are usually determined by dye tests. Numerous tracer studies have been performed on laboratory flumes and natural rivers. However, on-site sampling is often difficult, expensive and needs special apparatus. The main aim of the study was to justify simplified method based on the monitoring of the dye cloud shape in order to determine both longitudinal and transversal dispersion coefficients. In this study, four dye tests were carried out on a small local stream (the Lipkovsky) using Rhodamine WT fluorescein dye as a tracer. The tests were carried out in such a manner that both longitudinal and horizontal transversal dispersion data were obtained. For this purpose, the visually determined extent of the dye cloud was interpreted via the analytical solution of the advection-dispersion equation. The results obtained by this simplified approach indicated that the longitudinal dispersion coefficient Dx = 0.051–0.057 m2/s and the coefficient of horizontal transversal dispersion Dy = 0.00024–0.00027 m2/s. The method was justified by corresponding root mean square error (RMSE) counting RMSE = 0.65–1.02 m for the dye cloud centre, RMSE = 1.87–2.46 m for the head and tail of the cloud and RMSE = 0.025–0.11 m for the cloud width, the Nash-Sutcliffe efficiency coefficients ranged from 0.9 to 0.998. The comparison of these values with empirical formulae and other tracer studies indicated significant overestimation of the mentioned values of Dx , which can be attributed to the uniform velocity distribution along the width of Lipkovsky Stream. Much better agreement was achieved for Dy .

Numerical investigation of solute migration and release from sediments driven by wave-induced accumulation of pore water pressure

Sediments with poor drainage conditions are prone to build-up of excess pore water pressure when exposed to dynamic wave loading. This results in accumulated seepage flow that may have a significant impact on migration of solutes in marine sediments. In this study, the complex interplay between solute transport and wave-induced seepage process is investigated through a pioneering approach that combines and synchronously solves the two-dimensional advection-dispersion equation governing nonreactive solute transport with the governing equations for wave-induced oscillatory and residual pore pressures. The simulated results have demonstrated that the accumulated seepage flow significantly contributes to the advective and dispersive fluxes of solute, where the advection leads to movement of the solute layer and the hydrodynamic dispersion primarily enlarges the thickness of solute layer. The distribution of residual pore water pressure along depth exhibits two patterns that are primarily influenced by the thickness of the seabed. When the seabed is thinner than 0.3 times the wavelength, the residual pore pressure increases monotonically with depth (Pattern I). In this situation, the solute always migrates upwards to the seabed surface. Otherwise, Pattern II depicts the distribution of residual pore pressure, where the peak value of residual pore pressure attains at a certain depth and may advance downwards with wave action time. In this case, the direction of solute migration in regions shallower than a certain depth (e.g., 0.6 times the thickness of sediment) may alternate over time but the solute can eventually be transported upwards to seabed surface. The parametric study shows that when the seabed thickness is 0.2–0.3 times the wavelength, accumulated seepage flow has less effect on solute migration, whereas decreasing relative density and shear modulus, as well as increasing soil permeability can enhance the effect of accumulated pore water pressure on solute migration.

Numerical method for fractional Advection–Dispersion equation using shifted Vieta–Lucas polynomials

In the pursuit of creating more precise and flexible mathematical models for complex physical phenomena, this study constructs a unique fractional model for the Advection–Dispersion equation. The Advection–Dispersion equation is a fundamental mathematical tool for analyzing fluid dynamics and mass transfer processes, but traditional integer-order models often fail to accurately capture anomalous transport behaviors. To overcome this limitation, we employ a non-singular non-integer operator based on the Heydari–Hosseininia notion. This operator allows us to transform the classical advection–dispersion equation into a more robust and flexible fractional model, which can better represent a variety of transport phenomena across different scales and mediums.The numerical approximation of the proposed fractional Advection–Dispersion equation model is achieved using a set of specific polynomials known as shifted Vieta–Lucas polynomials. Aided by their unique properties, the shifted Vieta–Lucas polynomials enable us to transform the original differential system into a more computationally tractable algebraic system via a non-integer derivative matrix. Furthermore, we undertake a detailed analysis of convergence and truncation error associated with the shifted Vieta–Lucas polynomials, underpinning the validity and stability of our numerical method. To illustrate the efficiency and robustness of the derived method, we provide several example problems which are solved using our method. The corresponding solutions are extensively presented with the aid of figures and tables, demonstrating the method’s remarkable performance in solving the fractional Advection–Dispersion equation model. The proposed method shows promising potential for further application and development in solving other fractional differential equations and related scientific problems.

Solvent-pumped evaporation concentration on paper in linear and radial geometries.

Solvent-pumped evaporation-driven concentration of an initial distribution of solutes on a porous substrate is considered in one and two dimensions. Approximate analytic solutions to the isotropic advection-dispersion equations are first found for a Gaussian kernel and an infinite domain, following the smoothed particle approximation. Analytic solutions for more general initial distributions are then found as sums of Gaussians, and comparison is made with numerical solutions. In each case, initial distributions are advected toward the stagnation point and concentrated. Two-dimensional analysis is then extended to describe anisotropy in permeability and diffusion, and hydrodynamic dispersion. Radial-flow experiments are performed using filter papers and water-soluble dyes. Diffusion coefficients, temperature and humidity profiles, and the evolution of spot distributions are measured. The results confirm minor anisotropy in permeability and diffusion, limited hydrodynamic dispersion, and largely uniform evaporation. Péclet numbers over 2500 are demonstrated. Evaporation-driven concentration provides a mechanism for solute transport over long timescales. Potential applications lie in the design of paper spray microanalytical devices operating by solvent pumping rather than capillary flow.