Advection-dispersion Equation Research Articles

A semi-analytical solution for one-dimensional pollutant transport equation in different types of river networks

This study proposes a semi-analytical solution to a one-dimensional advection-dispersion equation considering reaction and source terms to predict and describe pollutant transport in various river networks using the Laplace transform approach. Moreover, the proposed semi-analytical solution is expanded for variable coefficients of the transport equation by increasing the number of middle nodes and associated intervals. The presented method solves the mass transport equation by considering advection and dispersion phenomena for one branch of the river network. The final mass balance equation is thus obtained in different types of river networks for each node. Regarding the connection matrix, the matrices of the characteristics of flow, pollutant, and network geometry are first determined as the model inputs. Then, by considering the diffusion and final mass conservation (Laplace transform of mass balance equation) equations for each node in the arbitrary river network, the mass balance and diffusion matrices are calculated in terms of the Laplace variable (s). Thus, a complex system of nonlinear algebraic equations is produced in terms of s, which can be solved using inverse Laplace algorithms to determine the concentration value at every node. The present study introduces four practical scenarios of pollutant transport in river networks in order to evaluate the proposed semi-analytical solution; the first is a branch-type river network with constant coefficients. The second case is a loop-type river network with variable coefficients. The third case simultaneously considers a point pollutant source and a distributed pollutant source in a branch-type river network. The fourth case is a branch-type river network under the dispersion-dominant phenomenon with a point source. Results show that the proposed semi-analytical solution is adequately capable of modeling pollutant transport and indicating its critical features in complex states of various river networks. Moreover, exploring the literature shows that no analytical solution has been introduced for river networks before undertaking the present study.

Effects of biofilms on the retention and transport of PFOA in saturated porous media.

Understanding the fate and transport of perfluorooctanoic acid (PFOA) in soil and groundwater is essential to reliable assessments of its risks. This study investigated the impacts of Gram-positive Bacillus subtilis (BS), Gram-negative Pseudomonas aeruginosa (PA) and wild microbiota (WM) biofilm on the transport of PFOA in saturated sand columns at two ionic strengths (i.e., 1.0 and 20.0mM NaCl). The retention of PFOA in biofilm-coated sand columns was higher than that in uncoated sand columns, due to biofilm-induced reinforced hydrophobic interactions and surface roughness, and decreased zeta potential. However, the retention effects varied among biofilm bacterial species with PFOA retardation factors in PA, WM and BS columns of 1.29-1.38, 1.21-1.29 and 1.11-1.15, respectively. Notably, PA biofilm had the most pronounced effect on PFOA retention. While increasing ionic strength promoted the retention of PFOA in BS biofilm-coated sand, it had no significant impact on PFOA transport in PA and WM biofilm-coated sand. This could be attributed to the differences in biofilm composition, deviating the ionic strengths effects on electrostatic double layer compression. The advection dispersion equation coupled with two-site kinetic retention model well described the transport of PFOA in all saturated columns. Our findings reveal that biofilm plays important roles in PFOA transport in porous media, instructive for risk assessment and remediation of PFOA contamination.

A fast Alikhanov algorithm with general nonuniform time steps for a two‐dimensional distributed‐order time–space fractional advection–dispersion equation

AbstractIn this paper, we propose a fast Alikhanov algorithm with nonuniform time steps for a two dimensional distributed‐order time–space fractional advection–dispersion equation. First, an efficient fast Alikhanov algorithm on the general nonuniform time steps for the evaluation of Caputo fractional derivative is presented to sharply reduce the computational work and storage, and are applied to the distributed‐order time fractional derivative or multi‐term time fractional derivative under the nonsmooth regularity assumptions. And a generalized discrete fractional Grönwall inequality is extended to multi‐term fractional derivative or distributed‐order fractional derivative for analyzing theoretically our algorithm. Then the stability and convergence of time semi‐discrete scheme are investigated. Furthermore, we derive the corresponding fully discrete scheme by finite element method and discuss its convergence. At last, the given numerical examples adequately confirm the correctness of theoretical analysis and compare the computing effectiveness between the fast algorithm and the direct method.

Advancing the knowledge of solute transport in the presence of bound water in mixed porous media based on low-field nuclear magnetic resonance

Bound water (BW) has significant effects on the physical and chemical properties of soils and other geologic formations. However, little is known about its effect on the non-Fickian transport of solute in porous media. This paper investigated the effect of BW on the non-Fickian solute transport characteristics in mixed porous media. Different porous media of quartz sand with variable contents of powdered minerals were prepared. The low-field nuclear magnetic resonance (LF-NMR) was used to quantitatively characterise BW and its spatial distribution. Then solute transport experiments were conducted in different prepared porous media and the measured breakthrough curves (BTCs) were analysed using the advection–dispersion equation (ADE), mobile-immobile (MIM) and continuous time random walk (CTRW) models. The results revealed that all measured BTCs exhibited the characteristics of early arrival and late-time tailing. However, stronger non-Fickian characteristics were observed as BW increased in content, indicating that BW influenced the non-Fickian transport of the solute. The optimised immobile water fraction of the MIM model consistently increased with increasing estimated BW fraction. Moreover, the coefficient of mass transfer rate (α) of MIM decreased as BW increased, meaning that the solute exchange between the mobile and immobile fluids became slower and consequently yielded strongly tailed BTCs. From the CTRW model, the solute transport velocity (vCTRW) increased with increasing BW, leading to a reduction in the breakthrough time of the solute. An increase in BW yielded a decreasing power law exponent (βCTRW) of the CTRW model, implying an enhancement of the non-Fickian transport characteristics. Furthermore, the porous media with BW<16.4% showed moderately non-Fickian solute transport whereas those with BW≥16.4% exhibited highly non-Fickian solute transport.

Paper-based evaporation concentrators: Comparison of linear and radial geometries.

Paper-based evaporation concentrators with linear and radial geometries are compared. A new method of finding approximate analytic solutions of the advection-dispersion equation is proposed, based on the behavior of concentrators with infinite sources. Analytic approximations are compared with numerical solutions, and the advantage of radial concentration is highlighted: linear concentration rates scale with the square root of the Péclet number Pe while radial rates scale with Pe itself, leading to faster radial concentration beyond a critical value. Experiments are performed with Brilliant Blue FCF dye, using optical transmission and the Beer-Lambert law for quantitation. Dye concentrations are chosen for operation in the linear absorbance regime. Radial concentration is demonstrated under ambient conditions on filter paper disks with 60 mm diameter evaporation areas fed from a perimeter source, in a reverse of the well-known "coffee stain" experiment. Airflow enhanced concentration in strips and wedges is compared directly, using laser-patterned chromatography paper. The advantage of radial concentration is confirmed (and enhanced by diversion of concentrate to the corners of strips) and concentration factors greater than (the dynamic range of measurement) are obtained in ∼2 h using 30 mm long columns.

Non-asymptotic Disturbances Estimation for Time Fractional Advection-Dispersion Equation

In this paper, a method for disturbance estimation for a class of non-homogeneous time fractional advection dispersion equation with general boundary conditions, is proposed. The disturbances are considered to affect the boundary conditions or the measurements or both. The proposed method is based on modulating functions. Numerical simulations are presented to illustrate the performance of the approach.

An efficient two-level factored method for advection-dispersion problem with spatio-temporal coefficients and source terms

&lt;abstract&gt;&lt;p&gt;A two-level factored implicit scheme is considered for solving a two-dimensional unsteady advection-dispersion equation with spatio-temporal coefficients and source terms subjected to suitable initial and boundary conditions. The approach reduces multi-dimensional problems into pieces of one-dimensional subproblems and then solves tridiagonal systems of linear equations. The computational cost of the algorithm becomes cheaper and makes the method more attractive. Furthermore, the two-level approach is unconditionally stable, temporal second-order accurate and spatial fourth-order convergent. The developed numerical scheme is faster and more efficient than a broad range of methods widely studied in the literature for the considered initial-boundary value problem. The stability of the proposed procedure is analyzed in the $ L^{\infty}(t_{0}, T_{f}; L^{2}) $-norm whereas the convergence rate of the algorithm is numerically analyzed using the $ L^{2}(t_{0}, T_{f}; L^{2}) $-norm. Numerical examples are provided to verify the theoretical result.&lt;/p&gt;&lt;/abstract&gt;

Numerical Solution for Advection-Dispersion Equation with Uniform and Varying Boundary Conditions

In the present study, one-dimensional advection-dispersion equation with variable coefficients is solved numerically with help of PDEPE in a finite porous domain. The pollutant is entering from the left end of the domain along the direction of the flow. Two different types of groundwater velocities have been considered, one rapidly decreasing with position and time and the other one being of sinusoidal nature over position and time. The dispersion coefficient is taken proportional to the groundwater velocity. Transport is included the first order decay and zero-order production parameters being proportional to the exponentially decreasing function with position and time and also being of sinusoidal nature over position and time. The nature of pollutant and porous medium are considered chemically non-reactive. Initially, porous domain is considered not to be solute free. Numerical solutions are obtained for uniform and varying type point sources. In heterogeneous porous media, variations in the parameters of solute transport such as: seepage velocities, dispersion coefficients etc. can be easily deal through numerical models. The effects of various physical parameters on solute concentration profiles are illustrated graphically.

A Solution of a Dispersion Problem for the Concentration of Pollutants in River Water

In this study, one-dimensional advection dispersion equation is solved for the concentration of water pollutant. To solve the differential equation, the Laplace’s transform technique has been used assuming no added pollutants on the way of water along the river. From the study, it is found that for various time period and speed, the concentration of the water pollutants decreases continuously, as the distance of the river increases from the point source.

Deterministic versus stochastic modelling of unsaturated flow in a sandy field soil based on dual tracer breakthrough data

The 216 km2 Neuenhagen Millcreeck catchment can be characterized as a drought-sensitive landscape in NE Germany. It is therefore of fundamental human interest to understand how water that fell as precipitation moves through the unsaturated soils and recharges groundwater. Additionally, a better knowledge of nutrient transport from soil to groundwater is important, especially in landscapes with light sandy soils. For a better understanding of these processes a dual tracer field experiment with bromide (Br–) and deuterium (D2O) was carried out some years ago. The aim of the present study is to use the results of this experiment to model tracer transport in the unsaturated zone via two different concepts, the classical deterministic advection-dispersion equation and a new stochastic approach. The advantage of the stochastic modelling method proposed here for field-scale tracer application is to produce reliable information about expected total solute fluxes from the unsaturated zone to groundwater and about mean transit times. Moreover, this allows one to evaluate the mass of solute in the soil profile and to determine the range of water velocity fluctuations. Field experiments should be concentrated on estimation of fluctuation of water flow velocity to make stochastic models more accurate. To summarize, this work contributes to new modelling methods for the simulation of water and solute transport in unsaturated sandy soils which are heavily affected by droughts and irregular hydrological processes in the subsurface

Interwell simulation model for the advection dispersion equation

We propose an interwell simulation model for the advection dispersion equation (ISADE) to predict the concentration of contaminant observed at the pumping well. This method comprises of two main steps. Initially, the model divides the aquifer or reservoir into a series of 1D injector-producer pairs and uses the historical contaminant observation data to estimate five major unknowns in each of these control volumes: the interwell connectivity, the pore volume, the volumetric flowrate at the grid face, the dispersion coefficient, and the number of grid cells in each control volume. Finally, once the history matching process is complete, the estimated variables are used to predict the concentration of contaminant observed at the wells. One main advantage of modeling contaminant production using the ISADE model is that it is computationally cheaper than the full physics model as it uses a series of one-dimensional connections between each producer-injector pair with all pairs connected to each other by interconnectivity factors. The advection diffusion equations is then solved for each of these pairs independently. Three examples were presented to test the effectiveness of the ISADE model. Each of the three examples involves flow of fluid in a synthetic heterogeneous aquifer with different well configurations and varying concentrations of the contaminant released into the aquifers at some injection locations. In Examples 1 and 3, non-reactive contaminants were introduced into the respective aquifers through injection wells, while in Example 2, nonreactive contaminant was introduced into the aquifer from a from a node on one of its boundary. All the examples studied show that the results obtained from ISADE closely match those from the full-scale simulation model. Furthermore, the model can easily handle changes in input parameters such as the concentration of contaminants released in the aquifer, and the injection and production rates.

Integrated modeling for simulating sediment production and transport in agricultural landscapes

A new modeling platform was developed to simulate spatially distributed sediment production and transport in agricultural landscapes. The sediment production at the grid scale was computed using the stand-alone Water Erosion Prediction Project-Hillslope Erosion code, while an advection-dispersion equation represented sediment transport. The model's performance was tested in the Water Resources and Erosion watersheds at the Oklahoma and Central Plains Agricultural Research Center, El Reno, OK, USA. Results showed that this modeling approach could capture the complex behavior of sediment under different management practices in the watersheds. The modeling framework provides mechanistic processes to simulate the fate and transport of sediment across the watershed at spatio-temporal scales that are useful in assessing the environmental impacts of management schemes. Therefore, this study is expected to help advance the current approaches to estimating soil erosion by bridging scale differences to capture the large-scale effects of small-scale soil erosion processes.

Transport of nanoscale zero-valent iron in saturated porous media: Effects of grain size, surface metal oxides, and sulfidation

Knowledge of the fate and transport of nanoscale zero-valent iron (nZVI) in saturated porous media is crucial to the development of in situ remediation technologies. This work systematically compared the retention and transport of carboxymethyl cellulose (CMC) modified nZVI (CMC-nZVI) and sulfidated nZVI (CMC-S-nZVI) particles in saturated columns packed with quartz sand of various grain sizes and different surface metal oxide coatings. Grain size reduction had an inhibitory effect on the transport of CMC-S-nZVI and CMC-nZVI due to increasing immobile zone deposition and straining in the columns. Metal oxide coatings had minor effect on the transport of CMC-S-nZVI and CMC-nZVI because the sand surface was coated by the free CMC in the suspensions, reducing the electrostatic attraction between the nZVI and surface metal oxides. CMC-S-nZVI displayed greater breakthrough (C/C0 = 0.82–0.90) and higher mass recovery (84.9%–89.3%) than CMC-nZVI (C/C0 = 0.70–0.80 and mass recovery = 70.9%–79.6%, respectively) under the same experimental conditions. A mathematical model based on the advection-dispersion equation simulated the experimental data of nZVI breakthrough curves very well. Findings of this study suggest sulfidation could enhance the transport of CMC-nZVI in saturated porous media with grain and surface heterogeneities, promoting its application in situ remediation.

Hybridizable discontinuous Galerkin methods for space-time fractional advection-dispersion equations

This work focuses on the numerical solution of the initial and boundary value problems for space-time fractional advection-diffusion equations. The well-posedness of the weak solutions is shown by Lax-Milgram lemma. Two fully discrete methods are established. The main idea is based on a hybridizable discontinuous Galerkin approach in spatial direction and two finite difference schemes in temporal direction: L1 formula, the weighted and shifted Grünwald-Letnikov formula. The stability and convergence analyses of the proposed methods are derived in detail. Several numerical experiments are provided to illustrate the theoretical results.

One-dimensional numerical model for unsteady solute transport in open channels under multi-point loading

A numerical method for solving the unsteady-one-dimensional advection-dispersion equation for solute transport in streams and channels imposed with multiple point-loading is presented. The numerical technique proposes to solve the equation, which also includes a first-order decay term and a loading term, using techniques similar to those employed by the finite volume method (FVM). The channel length is segmented into cells and the solution aims at evaluating the concentrations at the cell interfaces or the internal nodes, at prescribed time steps for the steady condition of the flow. The results are computed commencing from a specified time level with known spatial concentration within the domain. It is assumed that the boundary nodes at the two ends of the channel, one each at the upstream and the downstream, are specified with solute concentration values or its spatial gradient at all times. Unlike the standard within-the-cell spatial approximations of a variable assumed in the finite volume or finite element schemes, the proposed method models the spatial variation of the solute concentration within a cell by an analytical expression in terms of the concentrations at the two boundary nodes of the cell and the influx rate of point-loading, if any, located within the cell. Using such a closed-form function based upon the true nature of the problem not only helps in capturing the spatial variation of concentration within a cell more precisely but also allows a better approximation of its spatial gradients at the cell interfaces, thus helping to achieve a more accurate computation of the inter-cell dispersive fluxes. The performance of the proposed algorithm is evaluated against analytical benchmarks, a standard finite difference scheme, and one public-domain computational package. Data sets reported from field tests are used for demonstrating the potential of the technique in simulating real situations.

Competitive control of multiscale aquifer heterogeneity on solute transport in an alluvial aquifer

Natural aquifers can contain multiscale medium heterogeneity, whose joint impact on solute transport, however, remains obscure. To fill this knowledge gap, this study selected an alluvial aquifer as the representative medium, where the spatial distribution of multiple hydrofacies (with a typically nonuniform hydraulic conductivity (K) within each facies) could represent both regional-scale (large-scale) and subhydrofacies-scale (local-scale) heterogeneities. To obtain first-hand data, a field tracer test was conducted in an alluvial aquifer located at the Tongzhou site in Beijing, China, where the breakthrough curves (BTCs) obtained at various aquifer depths indicated a significant influence of both regional- and subhydrofacies-scale heterogeneities on solute transport. Further numerical analysis revealed that the impact of regional-scale heterogeneity on solute transport increased with increasing discrepancy in K between adjacent hydrofacies. In comparison, subhydrofacies-scale heterogeneity embedded within each hydrofacies played an important role in controlling solute transport when the mean Ks value of adjacent hydrofacies became more similar. Bimodal transport at early stages was mainly impacted by multiscale preferential flow paths, while solute retention at later times was induced by immobile zones dominating late-time transport. Anomalous transport induced by multiscale heterogeneity could be captured by the dual heterogeneous domain model (DHDM), while the dual advection–dispersion equation (DADE) or fractal mobile-immobile (fMIM) model could only provide reasonable predictions at the early or later stage, respectively. This study sheds light on the control mechanism of multiscale heterogeneity on solute transport.

Upscaling Solute Transport in Rough Single-Fractured Media with Matrix Diffusion Using Time Fractional Advection-Dispersion Equation

Upscaling Solute Transport in Rough Single-Fractured Media with Matrix Diffusion Using Time Fractional Advection-Dispersion Equation

A Water and Heat Coupling Model for GW-SW Interaction Considering Nonuniform Heat Transfer Effects of Soil.

An accurate groundwater-surface water (GW-SW) and heat coupling model (WHCM) aids in exploration of water migration and heat transport laws in the sedimentary layers of riverbeds and riparian zones, which is critical for understanding the transport patterns of contaminants in sediments. In this paper, a novel WHCM for GW-SW interaction is proposed by incorporating an effective soil thermal conductive model (ESTCM) into the advection-dispersion equation to address the shortcomings of current simulations that fail to account for the nonuniform heat transfer of soil. The model is developed in COMSOL, and the numerical implementation method is illustrated in detail. Furthermore, six recommended ESTCMs are examined in the WHCM to validate and compare the model simulation effects based on field test observations in the Walker River, USA. The results demonstrate that the proposed model performs better than the model that does not account for the nonuniform heat transfer effects of the soil in simulating GW-SW water and heat exchange. Meanwhile, the proposed model's simulation effects based on the Y2019 model perform the best. The orthogonal test revealed that the hydraulic conductivity ks and porosity n are the two most sensitive parameters influencing the model output results of groundwater seepage velocity and temperature, respectively. The efforts of this work can offer technical assistance and a guide for enhancing the simulation effect of river GW-SW water and heat exchange.

A COMPOSITE COLLOCATION METHOD BASED ON THE FRACTIONAL CHELYSHKOV WAVELETS FOR DISTRIBUTED-ORDER FRACTIONAL MOBILE-IMMOBILE ADVECTION-DISPERSION EQUATION

In this study, an accurate and efficient composite collocation method based on the fractional order Chelyshkov wavelets is proposed for obtaining approximate solution of distributed-order fractional mobile-immobile advection-dispersion equation with initial and boundary conditions. Operational matrices based on the fractional Chelyshkov wavelets are constructed. The proposed method reduce the solution to a system of algebraic equations, which is solved by Newton’s iterative method. Provided examples confirm the accuracy and applicability of the proposed method in line with the studied convergence analysis and error estimation. The obtained results of demonstrated numerical schemes illustrate that this approach is very accurate and efficient.

Numerical investigation of solute transport into deformable marine sediments driven by ocean waves

Most existing studies do not consider the effect that the deformability of submarine sediments may have on solute transport under the action of ocean waves. In this study a two-dimensional numerical model based on Biot’s poro-elastic consolidation theory and an advection–dispersion equation is established to investigate the transport of non-reactive solute into a deformable porous seabed under the influence of ocean waves. The numerical model is then validated against experimental and analytical results in order to demonstrate its accuracy and robustness. The simulation results indicate that ignoring the deformability of seabed soil and associated seepage force due to ocean waves may lead to a significant underestimation of the rate of solute transport in shallow layers of marine sediment, particularly fine sand or silt. The effects of parameters such as the shear modulus, non-dimensional mechanical dispersion coefficient (related to particle size and permeability), dimensionless sediment thickness, wave steepness and the relative water depth have on solute transport into deformable sediments were investigated. The results indicate that the ratio of the longitudinal coefficient of mechanical dispersion between deformable and non-deformable sediments that can represent the enhancement of solute migration due to soil deformation may reach up to 85. This becomes more pronounced as the shear modulus and permeability of sediments decrease and the depth of water increases when the sediment thickness is less than 0.8 times the wavelength. The rate of solute transport in deformable sediments reaches its maximum when the sediment thickness is about 0.2 times the wavelength. In fact, as the relative water depth increases from 0.1 to 0.3, the enhancement of solute migration due to wave-induced soil deformation increases by about 60%-80%. In comparison to linear waves, shallow water waves, e.g., the first-order cnoidal waves, can increase the transport of solute more due to its strong non-linear behaviour, particularly in deformable seabed.