Advection-dispersion Equation Research Articles

A Review on Storage Process Models for Improving Water Quality Modeling in Rivers

Water quality is intricately linked to the global water crisis since the availability of safe, clean water is essential for sustaining life and ensuring the well-being of communities worldwide. Pollutants such as industrial chemicals, agricultural runoff, and untreated sewage frequently enter rivers via surface runoff or direct discharges. This study provides an overview of the key mechanisms governing contaminant transport in rivers, with special attention to storage and hyporheic processes. The storage process conceptualizes a ubiquitous reactive boundary between the main channel (mobile zone) and its surrounding slower-flow areas (immobile zone). Research from the last five decades demonstrates the crucial role of storage and hyporheic zones in influencing solute residence time, nutrient cycling, and pollutant degradation. A review of solute transport models highlights significant advancements, including models like the transient storage model (TSM) and multirate mass transport (MRMT) model, which effectively capture complex storage zone dynamics and residence time distributions. However, more widely used models like the classical advection–dispersion equation (ADE) cannot hyporheic exchange, limiting their application in environments with significant storage contributions. Despite these advancements, challenges remain in accurately quantifying the relative contributions of storage zones to solute transport and degradation, especially in smaller streams dominated by hyporheic exchange. Future research should integrate detailed field observations with advanced numerical models to address these gaps and improve water quality predictions across diverse river systems.

Modeling the Process of Critical Exhaustion of Oxygen in a River Caused by Exponential Source of Pollution

The analysis and improvement of the river's change dynamics are critical to the global investigation of water quality issues. This study investigates the mathematical model representation of one dimension of a pair of advection-dispersion equations. These equations pertain to the concentrations of both pollutants and dissolved oxygen. Certain terms, which demonstrate an increase in the decay rate of pollution through exponential sources and assume a decreasing river's cross-sectional measurement, lead to the diminished oxygen levels observed in this study. The crucial points are demonstrated using both analytical and numerical analysis of the steady-state model, which allows for examining multiple situations about the elimination of oxygen and aquatic survival in a river with severe pollution.

Legendre Polynomials and Techniques for Collocation in the Computation of Variable-Order Fractional Advection-Dispersion Equations

The paper discusses a numerical approach to solving complicated partial differential equations, with a particular emphasis on fractional advection-dispersion equations of space-time variable order. With the use of fractional derivative matrices, Legendre polynomials, and numerical examples and comparisons, it surpasses current methods by utilizing spectral collocation techniques. It resolves equations involving spatial and time variables that are variable-order fractional advection–dispersion (VOFADE). Legendre polynomials serve as basis functions in this method, whereas Legendre operational matrices are employed for fractional derivatives. The technique is more computationally efficient since it reduces fractional advection–dispersion equations to systems of algebraic equations. Numerical examples and a comparison with current approaches illustrate the method’s superior performance in solving complicated partial differential equations, especially in the context of transport processes.

Transport of microplastics treated with dielectric barrier discharge (DBD) plasma in saturated porous media

The performance of discharge plasma in treating organic pollutants and micro-organisms in water is impressive. When discharge plasma is used to treat polluted water containing organic pollutants and microorganisms, the presence of a certain amount of microplastics (MPs) in the water is unavoidable due to the complexity of the components contained in the water and the prevalence of MPs. MPs, as one of the pollutants that are difficult to be degraded by discharge plasma, undergo physical and chemical changes that increase their risk in the environment after treatment. Therefore, it is necessary to understand the fate of MPs after being treated with discharge plasma. In this study, the surface morphology of plastics before and after discharge plasma treatment was observed by scanning electron microscopy (SEM). The plastics after discharge plasma treatment were characterized by Fourier transform infrared (FT-IR) and X-ray photoelectron spectroscopy (XPS) to determine the changes in oxygen-containing functional groups on the surface. The recovery of microplastics (MPs) in saturated porous media under different physicochemical and plasma oxidation conditions was investigated by column experiments. It has been shown that MPs exhibit increased recovery under conditions of increased flow rate and pH. A decrease in recovery was observed at elevated ionic strength and co-existing cation valence. High voltages and low air flow rates increase the oxidation of MPs by increasing the thermal effects of the dielectric barrier discharge (DBD) plasma system, the amount of reactive oxygen species (ROS) and the intensity of ultraviolet ray (UV) irradiation. The mobility of MPs is enhanced by a combination of these factors. The advection–dispersion equation (ADE) fits the transport data of MPs well. The interaction energy between quartz sand and MPs was calculated using the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory. This study provides a new perspective on the potential risks of discharge plasma in water treatment.

River quality management: Integrating uncertainty, failure probability, and assimilation capacity

Managing river water quality is challenging due to uncertainties in hydraulic and hydrologic parameters. This study integrates the symmetric exponential function (SEF) approach for solving the advection-dispersion equation with the Monte Carlo method in MATLAB. This combination allows us to explore the river's assimilation capacity and the failure probability (Pf) of maintaining desired water quality standards. Here, Pf represents the likelihood of pollutant concentrations exceeding acceptable limits under varying river conditions. A key contribution of this study is the introduction of a novel equation, developed using the Genetic Programming (GP) soft computing tool, to calculate assimilation capacity considering the failure probability of water quality provision. This equation provides a valuable tool for risk assessment in water resource management by quantifying pollutant assimilation dynamics. Its robustness is validated through high Coefficient of Determination (R2) and Overall Index (OI) values near 1, along with low Root Mean Square Error (RMSE) and Mean Absolute Error (MAE). The study identifies critical river characteristics, such as flow velocity and pollutant load, significantly influencing the reliability index (β). By outlining how adjustments in these parameters can achieve a target reliability index (β=4.526), our study offers a practical approach to safeguarding river ecosystems. For example, increasing flow velocity by 76 % can shift the river from a safe state (Pf=3×10−5) to a hazardous state (Pf=1), while a 44 % decrease in velocity allows for 57 % more pollutant assimilation. These findings highlight the importance of flow control as a cost-effective strategy for mitigating high pollutant concentrations and ensuring sustainable water quality management. By integrating numerical approaches with reliability sampling methods and soft computing techniques, this study enhances understanding of river system dynamics and supports informed decision-making for protecting water resources.

Electrical conductivity fluctuations as a tracer to determine time-dependent transport characteristics in hyporheic sediments

Assessing solute transport in riverbed sediments is important for quantifying the effective reactivity of hyporheic sediments and the magnitude of exchange flows between rivers and their river beds. A typical approach of estimating transport in riverbed sediments is by measuring natural tracers such as fluctuations of temperature or electrical conductivity (EC) and fitting models to them that assume time-independent travel time distributions, implying steady-state flow. Here, we use a transport parameterization that is based on the advection–dispersion equation (ADE) with coefficients that continuously vary in time. The ADE is solved numerically and its solution is fitted to measured EC time series using Bayesian parameter inference. A continuous function of model parameters is constructed by smoothly interpolating between point values with different temporal resolution, and Tikhonov regularization is used to avoid spurious parameter fluctuations. The approach is tested using EC time series synchronously measured in river water and hyporheic porewater of two urban rivers in Germany and one urban river in South Australia. For all datasets the goodness of fit was improved by introducing a time-dependent EC offset. Estimated porewater velocities were highly transient in three out of the four datasets with values increasing by up to a factor of six over the course of a day. Flux transients were likely related to both variations of hydraulic gradients along and spatial shifting of flow paths. Comparison to stationary, non-parametric deconvolution indicated that transient flow may induce multimodality in stationary travel time distributions. Given the high temporal dynamics of transport in the streambed sediments of the three investigated urban rivers, we envision that the presented model is also a valuable tool to improve the assessment of reactive transport in riverbed sediments.

Existence and Uniqueness of Solution Represented as Fractional Power Series for the Fractional Advection–Dispersion Equation

The fractional advection–dispersion equation is used in groundwater hydrology for modeling the movements of contaminants/solute particles along with flowing groundwater at the seepage velocity in porous media. This model is used for the prediction of the transport of nonreactive dissolved contaminants in groundwater. This paper establishes the existence and the uniqueness of solutions represented as fractional bi-variate power series of some initial-value problems and boundary-value problems for the fractional advection–dispersion equation. Moreover, a method to approximate the solutions using fractional polynomials in two variables and to evaluate the errors in a suitable rectangle is designed. Illustrative examples showing the applicability of the theoretical results are presented.

Coupling advection–dispersion equation and dynamic model for phytoremediation of 134Cs in soil

Coupling advection–dispersion equation and dynamic model for phytoremediation of 134Cs in soil

Fractional weak adversarial networks for the stationary fractional advection dispersion equations

Fractional weak adversarial networks for the stationary fractional advection dispersion equations

New results on fractional advection–dispersion equations

In this paper, a class of fractional Sturm–Liouville advection–dispersion equations with instantaneous and noninstantaneous impulses is considered, in particular, the nonlinearities discussed here include Caputo fractional derivatives. Since the nonlinear terms contain fractional derivatives, this problem does not directly have variational structure, we need to combine critical point theory and an iterative method to deal with such problems. Finally, the existence of at least one nontrivial solution is proved by the mountain pass theorem and the iterative method. At the same time, an example is given to illustrate the main result.

Using index and physically-based models to evaluate the intrinsic groundwater vulnerability to non-point source pollutants in an agricultural area in Sardinia (Italy)

This research aims at studying the intrinsic vulnerability of groundwater to diffuse environmental pollutants in the Muravera coastal agricultural area of Sardinia, Italy. The area faces contamination risks arising from agricultural practices, especially the use of fertilizers, pesticides, and various chemicals that can seep into the groundwater. The study examined the interplay among hydrological elements, including soil characteristics, groundwater depth, climate conditions, land use, and aquifer properties. To do that, the outcomes of FLOWS 1D physically-based agrohydrological model were analyzed in parallel with those of the overlay-and-index model SINTACS, in a sort of reciprocal benchmarking. By using FLOWS, water movement and solute transport in the unsaturated zone were simulated by, respectively, solving the Richard Equation (RE) and the Advection-Dispersion equation (ADE). As such, this model allowed to account for the role of soil hydraulic and hydro-dispersive properties variability in determining the travel times of a conservative solute through the soil profile to the groundwater. For FLOWS simulations, a complete dataset was used as input, including soil horizons, soil physical and hydraulic properties of 36 soil profiles, average annual depth to groundwater table at each soil profile (ranging from 1 to 50 meters), and climatic temporal series data on rainfall and evapotranspiration. Detailed analyses of travel times for the movement of 25, 50, 75, and 100% of the solute mass to reach groundwater were conducted, revealing that the depth to groundwater predominantly influences vulnerability. This result was coherent with SINTACS vulnerability map due to the large impact of the depth to groundwater on SINTACS analysis.

Modeling of Microplastics Migration in Soil and Groundwater: Insights into Dispersion and Particle Property Effects.

Migration of microplastics (MPs) in soil-groundwater systems plays a pivotal role in determining its concentration in aquifers and future threats to the terrestrial environment, including human health. However, existing models employing an advection-dispersion equation are insufficient to incorporate the holistic mechanism of MP migration. Therefore, to bridge the gap associated with MP migration in soil-groundwater systems, a dispersion-drag force coupled model incorporating a drag force on MPs along with dispersion is developed and validated through existing laboratory and field-scale experiments. The inclusion of the MP dispersion notably increased the global maximum particle velocity (vmaxp) of MPs, resulting in a higher concentration of MPs in the aquifer, which is also established by sensitivity analysis of MP dispersion. Additionally, increasing irrigation flux and irrigation areas significantly accelerates MP migration downward from soil to deep saturated aquifers. Intriguingly, vmaxp of MPs exhibited a nonlinear relationship with MPs' sizes smaller than 20 μm reaching the highest value (=1.64 × 10-5 m/s) at a particle size of 8 μm, while a decreasing trend was identified for particle sizes ranging from 20 to 100 μm because of the hindered effect by porous media and the weaker effect of the drag force. Moreover, distinct behaviors were observed among different plastic types, with poly(vinyl chloride), characterized by the highest density, displaying the lowest vmaxp and minimal flux entering groundwater. Furthermore, the presence of a heterogeneous structure with lower hydraulic conductivity facilitated MP dispersion and promoted their migration in saturated aquifers. The findings shed light on effective strategies to mitigate the impact of MPs in aquifers, contributing valuable insights to the broader scientific fraternity.

Recursive analytical solution for nonequilibrium multispecies transport of decaying contaminant simultaneously coupled in both the dissolved and sorbed phases

Multispecies transport analytical models that solve advection-dispersion equations (ADEs) are efficient tools for evaluating the transport of decaying contaminants and their sequential products. This study develops a novel semi-analytical model to simulate the multispecies transport of decaying contaminants, considering nonequilibrium sorption and decay in both dissolved and sorbed phases. First-order reversible kinetic sorption equations with decay processes are coupled to ADEs. Recursive analytical solutions, using the Laplace transform and generalized integral transform, are developed to address the mathematical complexity of the governing equations. The model's simulation results show excellent agreement with both numerical models and existing analytical solutions. Applied to a four-member radionuclide decay chain, the model reveals that including decay in the sorbed phase results in a lower concentration of the first member and avoids underestimating the radioactivity concentrations of daughter elements. These differences in dissolved radioactivity concentrations between models with and without sorbed phase decay may impact health risk assessments for radioactive waste disposal. Finally, this study provides a more sophisticated mathematical tool for analyzing multispecies transport in real field conditions where nonequilibrium sorption processes predominantly occur.

Predicting the wash behaviour of filter cakes with inhomogeneous cake geometry on a macroscopic scale

Displacement washing of isotropic filter cakes with inhomogeneous cake geometry is investigated theoretically. Fluid flow is described by Darcy’s law, while the advection–dispersion equation is used to describe solute transport within the cake. An additional mass transfer term is taken into account to describe mass transfer from non-permeated to permeated domains of the filter cake. The equations are solved numerically in three spatial dimensions. Model predictions are compared to experimental data from preceding publications of the authors and show good agreement. An explorative case study demonstrates the negative impact of typical inhomogeneous cake geometries on displacement washing.

High order numerical treatment of the Riesz space fractional advection-dispersion equation via matrix transform method

In this paper, a mixed high order finite difference scheme-Padé approximation method is applied to obtain numerical solution of the Riesz space fractional advection-dispersion equation(RSFADE). This method is based on the high order finite difference scheme that derived from fractional centered difference and Padé approximation method for space and time integration, respectively. The stability analysis of the proposed method is discussed via theoretical matrix analysis. Numerical experiments are presented to confirm the theoretical results of the proposed method.

Dynamic leaching assessment of recycled polyurethane-coated tire rubber for sustainable engineering applications

Assessing the material ramifications of waste tire rubber stands as a pivotal endeavor for its practical utility. Dynamic leaching analysis emerges as indispensable in delineating its potentialities within sustainable engineering domains. Waste materials stemming from tire recycling industries, notably polyurethane-coated rubber (PUcR), may exhibit promising viability owing to attenuated leaching, thus fostering the circular economy paradigm. Towards this end, upflow column percolation leaching tests were conducted on recycled rubber-soil mixture (RSM) alongside PU-coated rubber-soil mixture (PUcRSM). Analysis through Inductively Coupled Plasma-Atomic Emission Spectroscopy (ICP-AES) unveiled the leaching dynamics of assorted metals in RSM and PUcRSM, notably showcasing markedly diminished cumulative zinc content (by 99.4%) in PUcRSM, coupled with mitigated electrical conductivity, redox potential, and eluate pH. Leveraging HYDRUS-1D for analytical modeling predicated on the advection-dispersion equation further underscored these insights. These findings suggest that waste PUcR presents itself as a viable option, distinguished by its diminished leachability, suitable for a wide array of applications. These applications include vibration damping in infrastructure, tunnel linings, asphalt roads, rubberized concrete, and lightweight backfill materials for bridges and retaining walls, among others. This initiative will contribute to recycling a substantial quantity of over 1 billion used tires produced annually, ultimately supporting the circular economy ethos through assured and adaptable repurposing of waste from tire rubber recycling industries.

Application of Oversampling Techniques for Enhanced Transverse Dispersion Coefficient Estimation Performance Using Machine Learning Regression

The advection–dispersion equation has been widely used to analyze the intermediate field mixing of pollutants in natural streams. The dispersion coefficient, manipulating the dispersion term of the advection–dispersion equation, is a crucial parameter in predicting the transport distance and contaminated area in the water body. In this study, the transverse dispersion coefficient was estimated using machine learning regression methods applied to oversampled datasets. Previous research datasets used for this estimation were biased toward width-to-depth ratio (W/H) values ≤ 50, potentially leading to inaccuracies in estimating the transverse dispersion coefficient for datasets with W/H > 50. To address this issue, four oversampling techniques were employed to augment the dataset with W/H > 50, thereby mitigating the dataset’s imbalance. The estimation results obtained from data resampling with nonlinear regression method demonstrated improved prediction accuracy compared to the pre-oversampling results. Notably, the combination of adaptive synthetic sampling (ADASYN) and eXtreme Gradient Boosting regression (XGBoost) exhibited improved accuracy compared to other combinations of oversampling techniques and nonlinear regression methods. Through the combined ADASYN–XGBoost approach, it is possible to enhance the transverse dispersion coefficient estimation performance using only two variables, W/H and bed friction effects (U/U*), without adding channel sinuosity; this represents the effects of secondary currents.

Time-dependent dispersion coefficients for the evolution of displacement fronts in heterogeneous porous media

We present an approach for quantifying displacement fronts in heterogeneous porous media based on the concept of time-dependent apparent dispersion coefficients. The concept of constant asymptotic macrodispersion generally overestimates the area swept by a displacement front and leads to unrealistic upstream dispersion. We show that the large-scale front spreading can be captured by a one-dimensional advection–dispersion equation that is parameterized by a suitably chosen temporally evolving dispersion coefficient. For purely advective front spreading, we derive an analytical expression based on a predictive continuous time random walk approach, which applies to highly heterogeneous porous media. This analysis elucidates the variability of solute travel times as the key longitudinal spreading mechanism. It shows that the evolution of dispersion can be captured as the sum of exponentials that decay on two dominant time scales. In a particle-based picture, these scales mark the short time at which transported particles start exploring the flow variability and the large time at which the slowest particles start decorrelating their transport velocity. Based on these insights, we propose a heuristic formula that accounts for the impact of local-scale dispersion as an additional decorrelation mechanism. The heuristic expression for the longitudinal dispersion coefficient captures solute spreading for a broad range of Péclet numbers and heterogeneity variances. The proposed approach is tested against direct numerical simulations. It provides a robust and fast method for quantifying the evolution of displacement fronts in heterogeneous porous media with possible applications, for example, in groundwater contamination modelling, underground gas storage, and geothermal energy production.

Study on advection–dispersion behavior for simulation of 3H, 99Tc, and 90Sr transport in crushed sandstone of column experiments

Abstract In order to establish a universal and safety-compliant post-closure safety assessment technique, it is necessary to develop appropriate models to explain the migration behavior of radioactive materials within the rock system. Advection-dispersion experiments (ADE) have proven successful in designing transport models through a calibration/validation process of breakthrough curves (BTCs). In the present investigation, we employed a dynamic column device to examine the transport of Tritium (3H), Technetium-99 (99Tc), and Strontium-90 (90Sr) in crushed sandstone. Non-reactive transport experiments utilizing conservative tracers enabled us to determine the transport parameters, including retardation factors (R) and dispersivity (α). Our study focused on investigating the surface complexation model (SCM) using an additive approach coupled with the advection-dispersion equation to simulate the reactive transport behavior of 90Sr. The results affirm the robustness of our chosen thermodynamic database and modeling approach, emphasizing the criticality of accurately modeling and predicting reactive transport behavior in porous media.

Architecture dependent transport behavior of iron (0) entrapped biodegradable polymeric particles for groundwater remediation

Polylactic acid based spherical particles with three architectural variations (Isotropic (P1), Semi porous (P2), and Janus (P3)) were employed to encapsulate zero valent iron nanoparticles (ZVINPs), and their performance was extensively evaluated in our previous studies. However, little was known about their transportability through saturated porous media of varying grain size kept under varying ionic strength. In this particular study, we aimed to investigate the architectural effect of polymeric particles (P1–P3) on their mobility through the sand column of varying grain size in presence of mono, di, and tri-valent ions of varying concentrations (25–200 mM (millimoles)). As per column breakthrough experiments (BTCs) using various types of sands, amphiphilic Janus type (P3) particles exhibited the maximum transportability among all the tested particles, irrespective of the nature of the sand. Owing to the narrower travel path, sands with lower porosity (31%) delayed the plateau by shifting it to a higher pore volume with a minimum retention of iron (C/Co: 0.94 for P3) in the column. The impact of mono (Na+, K+), di (Ca2+, Mg2+), and trivalent (Al3+) ions on their transportability was progressively increased from P3 to P1, especially at higher ionic concentrations (200 mM), with P3 being the most mobile particles (C/Co:0.54 for Al3+). Among all the ions, Al3+ exhibited maximum hindrance to their mobility through the sand column. This could be due to their strong charge screening effect coupled with cation bridging complex formation with moving particles. Experimental results obtained from BTCs were found to be well-fitted with a theoretical model based on advection-dispersion equation, showing minimum retention for P3 particles. Overall, it can be inferred that encapsulation of ZVINPs inside Janus particles (P3) with a right balance of amphiphilicity and highly negative surface charge would be required to achieve considerable transportability through sand aquifers to target contaminants in polluted groundwater existing under harsh conditions (high ionic concentrations).