One-dimensional Advection-dispersion Equation Research Articles

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.

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.

Master equation model for solute transport in river basins: part I channel fluvial scale

Normal and anomalous diffusion are ubiquitous in many physical complex systems. Here we define a system of diffusion equations generalized in time and space, using the conservation principles of mass and momentum at channel scale by a master equation. A numerical model for describing the steady one-dimensional advection-dispersion equation for solute transport in streams and channels imposed with point-loading is presented. We find the numerical model parameter as the solution of this system by estimating the transition probability that characterizes the physical phenomenon in the diffusion regime. The results presented (Part I) refer to the channel scale and represent the first part of a research project that has been extended to the basin scale.

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.

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.

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 Field-Based Evaluation of the Reliability of Empirical Formulae for Quantifying the Longitudinal Dispersion Coefficient in Small Channels

The majority of formulae for predicting in-channel mixing by longitudinal dispersion are based on empirical evidence from medium to large rivers, whereas small creeks and streams are under-represented despite their hydrological and ecological importance. In this study, twenty-six formulae for predicting the longitudinal dispersion coefficient (Kx) were evaluated for their applicability to small channels using field measurements and hydraulic modeling. Predicted values for Kx, following guidelines recommended in the original publications for the formulae, were compared to measured dispersion coefficients obtained from sodium-chloride plumes injected into two small channels (a concrete-lined, trapezoidal channel and a small, natural stream) based on fitting the Hayami solution to the one-dimensional advection-dispersion equation (ADE) to each plume. The predicted Kx coefficients from the formulae were also utilized to create model-simulated plumes, which were compared to those measured well downstream of the point of injection. The findings demonstrate that the predictive accuracy of the twenty-six formulae was extremely variable; none were able to predict the dispersion process in the small channels with better than ± 50% accuracy. These results show that “universal” formulae are plagued with a large degree of uncertainty and should be used with caution when applied to small channels, although more robust predictions are possible with some formulae if site-specific data are available for calibration.

Surrogate prediction of the breakthrough curve of solute transport in rivers using its reach length dependence

Techniques for predicting the contaminant cloud propagation along a stream are necessary for swift action against contaminant spill accidents in fluvial systems. Due to their low computational cost, one-dimensional solute transport models have conventionally been employed, in which the complex channel characteristics are considered using model parameters. However, the determination of such parameters relies predominantly on optimization techniques based on pre-measured tracer data, which are usually unavailable for unexpected accidents. The present paper suggests an alternative method for predicting a breakthrough curve (BTC) variation along an unmeasured stream reach where no flow information is provided. In this study, we investigated the relationship between directly-measured flow properties and BTC characteristics based on field tracer experiments. Using statistical features of the tracer BTCs, we devised a regressive prediction method for estimating the BTC features as a function of travel distance, and validated the method by comparison with simulations using both a one-dimensional advection–dispersion equation (ADE) and transient storage model (TSM), whose parameters were calibrated at upstream reaches. The proposed regressive predictions were relatively accurate than those from parameter-calibrated models, and this advantage was more apparent for long-distance predictions for the unmeasured river reach.

The effects of pH, ionic strength, and natural organics on the transport properties of carbon nanotubes in saturated porous medium

Carbon Nanotubes (CNTs) are widely utilized, and their inadvertent release into the soil and ground water systems may have detrimental environmental implications. Therefore, it is critical to comprehend the effect of several environmental conditions on the fate and transport of CNTs in porous media. In previous studies, a narrow range of pH, ionic strength, and collector grain size have only been considered while evaluating the transport and retention of functionalized multiwalled carbon nanotubes (f-MWCNTs). None have highlighted the effect of environmentally relevant concentrations of natural organic matter and particle concentrations on the mobility of f-MWCNTs in porous medium. The current study aims to fill the knowledge gap by evaluating the fate and transport of f-MWCNTs at different, environmentally relevant physicochemical conditions of pH (5, 7, and 9), ionic strengths (10, 50, and 100 mM), humic acid as representative of natural organic matter (0.1, 1, and 10 mg/L) and particle concentrations (10, 25, and 50 mg/L) through saturated porous medium (quartz sand). It was noted that a wide range of pH and humic acid imparted an overall negative charge on the particles, enhancing their mobility (C/C0 > 0.8). However, transport of the f-MWCNTs was hindered in the presence of high ionic strength and particle concentrations (C/C0 < 0.1). The transport profile was mathematically simulated using a one-dimensional advection-dispersion equation (ADE) and colloidal filtration theory (CFT). Additionally, interaction energy profiles based on the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory were estimated to gain a better understanding of the mechanism of f-MWCNT deposition. In light of the potential environmental consequences, the impact of real-world water systems (tap, river, lake, and ground water) on the transport behaviour of f-MWCNTs was also investigated. The study considerably advances our understanding of the various physicochemical and environmental factors that can critically impact the transport of f-MWCNT in porous media, which could prove beneficial for environmental applications and risk management.

Evaluating the arsenic attenuation of soil amended with calcium–magnesium composites of different particle sizes

An attenuation layer composed of ground mixed with stabilising agents can prevent the contamination of the surrounding area when using soils and rocks with geogenic contaminants in embankments. The optimum particle size of the stabilising agent must be selected based on the requirements of the construction site because the mechanical and chemical properties of the attenuation layer are site-specific. However, the relationship between the particle size of the stabilising agent and the attenuation performance of soil–agent mixtures has yet to be fully clarified. This study employs batch sorption tests to evaluate the attenuation of arsenic by a soil mixed with a calcium–magnesium composite with different particle sizes, ranging from powder particles (<0.075 mm in size) to granular particles with diameters between 2.0 and 9.5 mm. Amended soil more effectively attenuates the contaminant than the original soil. In one experiment, a stabilising agent of granular particles (between 2.0 and 9.5 mm) for the amendment increased the soil’s partition coefficient Kd from 14.5 to 22.2 cm3/g, which is more than a 50% improvement in the attenuation. Using a stabilising agent with a smaller particle size for the amendment has a greater impact. Kd increases linearly as the particle size of the stabilising agent decreases down to 0.075 mm. Using the Kd from laboratory tests, simulations with a one-dimensional advection–dispersion equation demonstrate the durability of the attenuation layer. Both the powder and the granular particles show promise as attenuation layer materials.

Release estimation of pollutants in river by the variational analysis approach

After accidental release of pollutants in rivers, the release estimation of the pollutant is necessary for the dealing of this accident. With the increase of people's attention to river pollution, the monitoring data of river pollution will become more and more abundant and diversified. At this point, data assimilation approaches will be more advantageous. In this paper, a new model called variational analysis inverse model (VAIM) based on the variational data assimilation is proposed to solve the pollutant release estimation problem. In the framework of the variational analysis, the conjugate gradient method and one-dimensional imprecise line search Wolfe-Powell conditions are combined to solve this problem. The implicit finite difference scheme is adopted to solve the one-dimensional advection-dispersion equation. Some synthetic cases are conducted to evaluate the robustness of the proposed model under the effect of the observational error and the effect of initial estimates for release rates in the iteration. Results show that VAIM successfully recovers the accurate release. There are only slight differences among estimated releases by VAIM under different initial estimates. Field tracer experiments are used to evaluate the practicability of VAIM. Results show that the relative error between the estimated release and the real release in the field tracer experiment is only 1.2%. In conclusion, VAIM is a release estimation method with high accuracy and will play an important role in river pollution management.

Analytical Modeling of Solute Transport in a Two-Zoned Porous Medium Flow

The objective of this study is to model the transport of groundwater contamination in one-layered and two-zoned porous medium flows by an analytical approach. The one-dimensional advection–dispersion equation (ADE) has usually been used to describe the problems of pollutant transport in a water environment. This study presents some exact solutions to the one-dimensional ADE to examine the variation of solute concentration with and without the biodegradable effect in an unconfined aquifer of a finite domain by the generalized integral transform method (GITM). The modeling results for the concentration of groundwater contamination show that the pollutant concentration is more sensitive to the Peclect number than to the retardation factor and the first-order decaying coefficient in uniform groundwater flow. In composite soil zones, decaying and diffusion factors have significant effects on the contamination concentration around the interface, especially over a long-term period. The transport flux between the two regions is determined by the concentration gradient at the interface of the two regions. The contaminated concentration decreases as the retardation factor, Peclect number, and the first-order decaying coefficient increase for every location at a fixed duration. Moreover, the contaminated concentration is more sensitive to the Peclect number than to the retardation factor and the first-order decaying coefficient in uniform groundwater flow.

Experimental and numerical study of bed roughness effect on longitudinal dispersion

Abstract The effects of bed roughness on the longitudinal dispersion coefficient (DL) were experimentally and numerically investigated in the present study. The tracer experiments were first carried out in a circular flume with a diameter of 1.6 m over both smooth and rough beds (coarse sand) with four sizes (ks = d65) of 1.04, 2.09, 3.01, and 4.24 mm. In addition, the one-dimensional advection–dispersion equation was numerically solved. The longitudinal dispersion coefficient was calculated by comparing the numerical and experimental breakthrough curves. The results showed that by increasing the bed roughness height (from zero to 4.24 mm), the longitudinal dispersion coefficient increased by 34%. In addition, the longitudinal dispersivity (λ = DL/V) increased with increasing relative roughness (ks/h), so that the range of longitudinal dispersivities in smooth bed experiments were 0.037–0.049 m and for rough bed (ks = 4.24 mm) were 0.07–0.084 m. In other words, with increasing the bed roughness height from zero (smooth bed) to 4.24 mm, the longitudinal dispersivities increased from 0.037 to 0.077 m, indicating an increase of about 108%. Furthermore, a relationship was developed using non-dimensional longitudinal dispersion (DL/(Vh)) as a function of relative roughness (ks/h). It can be concluded that taking into consideration bed roughness as the driving force of shear dispersion would improve predictive equations of the longitudinal dispersion in the rivers. As the bottom of all natural rivers has roughness elements with different sizes, the results of this study will definitely be useful in estimating the longitudinal dispersion coefficient in natural rivers and quantifying the effect of roughness in the longitudinal dispersion coefficient equations.

Sorption-desorption column tests to evaluate the attenuation layer using soil amended with a stabilising agent

Sorption-desorption column tests using acrylic columns (ϕ 5 cm × h 10 cm) were employed to evaluate the sorption performance of an attenuation layer against geogenic contamination. The attenuation layer material was silica sand amended with 1, 5, or 10% of a stabilising agent. The main component of the agent was magnesium oxide. The sorption behaviour of the materials was determined by a fluoride solution (C0 = 80 mg/L F-), while the desorption behaviour was determined by distilled water. Breakthroughs (C/C0 > 0.05) occurred after approximately 1, 20, and 50 PVF for stabilising agent contents of 1, 5, and 10%, respectively. The one-dimensional advection-dispersion equation modelled the breakthrough curves obtained from the tests. The predictions gave unrealistic estimates, especially for the breakthrough point where C/C0 = 0.05. For the 1% agent content, approximately 20% of the sorbed mass, Ss, was desorbed, but the percentage of desorbed mass, Sd, was much smaller for the higher agent contents. The difference between the sorbed and desorbed masses was defined as the immobilised fraction, Ss - Sd. For the 5% agent content, Ss - Sd = 4.0 mg/g. The results suggest that when silica sand is amended with magnesium oxide as an agent, the mixture can immobilise the fluoride in the attenuation layer.

An Analytical Solution of Spatially Dependent One-Dimensional Advection-Dispersion Equation for a Varying Pulse Type Input Source

In present study, solution of advection-dispersion equation is obtained to determine concentration distribution of solute introduced from a varying pulse type point source in one-dimensional heterogeneous semi-infinite porous medium. Heterogeneity of the medium gives rise to space dependent groundwater velocity, dispersion coefficient and retardation factor. Groundwater velocity is some exponent ξ to a linear function of space. The dispersion and retardation factor are also exponents of same linear function with exponents (ξ+1) and (ξ-1), respectively, where ξ takes the value 0 or 1. At one end of the domain, a time dependent varying nature source, which involves step-size increasing function of time, acts along the flow up to a certain time the n eliminated while concentration gradient is considered zero at the other end of the domain. Initially, medium is uniformly polluted. Firstly, the advection-dispersion equation is reduced into constant coefficients by using certain transformations and then Laplace Integral Transformation Technique is utilized to get the solution. The obtained result is illustrated with numerical examples to study the effect of various parameters.

Comparative application and optimization of different single-borehole dilution test techniques

Single-borehole dilution tests (SBDTs) are a method for characterizing groundwater monitoring wells and boreholes, and are based on the injection of a tracer into the saturated zone and the observation of concentration over depth and time. SBDTs are applicable in all aquifer types, but especially interesting in heterogeneous karst or fractured aquifers. Uniform injections aim at a homogeneous tracer concentration throughout the entire saturated length and provide information about inflow and outflow horizons. Also, in the absence of vertical flow, horizontal filtration velocities can be calculated. The most common method for uniform injections uses a hosepipe to inject the tracer. This report introduces a simplified method that uses a permeable injection bag (PIB) to achieve a close-to-uniform tracer distribution within the saturated zone. To evaluate the new method and to identify advantages and disadvantages, several tests have been carried out, in the laboratory and in multiple groundwater monitoring wells in the field. Reproducibility of the PIB method was assessed through repeated tests, on the basis of the temporal development of salt amount and calculated apparent filtration velocities. Apparent filtration velocities were calculated using linear regression as well as by inverting the one-dimensional (1D) advection-dispersion equation using CXTFIT. The results show that uniform-injection SBDTs with the PIB method produce valuable and reproducible outcomes and contribute to the understanding of groundwater monitoring wells and the respective aquifer. Also, compared to the hosepipe method, the new injection method requires less equipment and less effort, and is especially useful for deep boreholes.

Simple transport models for karst systems

Tracer tests are useful field methods with a wide range of applications in karst aquifers. The measured tracer breakthrough curve at a karst spring allows the estimation of the important transport properties of the karst system. The breakthrough curve is typically characterized by a long tailing and a low tracer recovery that is not captured by the classical one-dimensional advection dispersion model. Partitioning models are often used and require a large number of parameters. In the present work, simplified transport models are proposed to simulate the typical karst breakthrough curve. The model setup consists of a karst conduit that is interacting laterally with the surrounding matrix. The solute transport in both the conduit and matrix are governed by the one-dimensional advection-dispersion equation and the resulting mathematical model consists of a coupled system of differential equations. The Laplace transform method is used to derive the transport models for various flow and boundary conditions. The models are in terms of four physically meaningful parameters that represent the advection and dispersion properties of the flow in the conduit and matrix. Algebraic relationships to estimate the model parameters are also derived in terms of the key features of the breakthrough curves. The main advantage of the proposed models are their simple formulation and ability to capture the long tailing of the breakthrough curve in terms of just a few parameters. Application of the models to two real karst aquifers demonstrate their effectiveness in simulating the observed breakthrough curves and estimating the key transport parameters.

Analytical and Numerical Solutions of Pollution Concentration with Uniformly and Exponentially Increasing Forms of Sources

The study of pollution movement is an important basis for solving water quality problems, which is of vital importance in almost every country. This research proposes the motion of flowing pollution by using a mathematical model in one-dimensional advection-dispersion equation which includes terms of decay and enlargement process. We are assuming an added pollutant sources along the river in two cases: uniformly and exponentially increasing terms. The unsteady state analytical solutions are obtained by using the Laplace transformation, and the finite difference technique is utilized for numerical solutions. Solutions are compared by relative error values. The result appears acceptable between the analytical and numerical solutions. Varying the value of the rate of pollutant addition along the river (q) and the arbitrary constant of exponential pollution source term (λ) is displayed to explain the behavior of the incremental concentration. It is shown that the concentration increases asqandλincrease, and the exponentially increasing pollution source is a suitable model for the behavior of incremental pollution along the river. The results are presented and discussed graphically. This work can be applied to other physical situations described by advection-dispersion phenomena which are affected by the increase of those source concentrations.

Fundamental Study on Transport Model for Radionuclides under Unsaturated Condition around Near-Surface Underground

ABSTRACTThe low-level nuclear wastes such as decontamination waste from Fukushima are disposed in near-surface underground, where the intermittent recharge of rain and groundwater causes spatial distribution of water content. Therefore, pores of soils are not filled with water, that is, an unsaturated zone will be formed. In such a condition, since the water flow path are detoured by clogged gas in pores of soil in the unsaturated zone, the migration path of radionuclide would be different from the saturated zone. So far, the one-dimensional advection-dispersion equation (ADE) model has been widely used in order to explain experimental results under an unsaturated condition. However, the detouring of local flow-paths remarkably affects the mass transfer. The one-dimensional ADE evaluates such a detouring effect by using Peclet number and retardation coefficient as fitting parameters. In other words, the one-dimensional ADE model is difficult to explain mass transfer under an unsaturated condition. Therefore, the purpose of this study is explaining such complicated transport of radionuclides using a multi-path model based on phenomena in underground. The proposed multi-path model considering both water saturation and permeability distributions showed good agreement with the experimental data under an unsaturated condition.

Groundwater contamination in mega cities with finite sources

Groundwater contamination due to multiple sources occurring in mega cities was modelled. One constant source contamination was considered at the source boundary, whereas other sources may join in between at various locations at different times. Initially, the aquifer was contamination-free in mega cities and was subsequently contaminated by means of different sources in due course of time. One-dimensional ADE (Advection Dispersion Equation) for modelling groundwater contamination was used and solved analytically in the semi-infinite aquifer domain for a finite number of point sources. A numerical solution was also obtained for two sources to compare analytical solutions. Results were examined for different velocity profiles to show the maximum contaminant concentration level with distance. This may be helpful to model the maximum possible number of point sources of contamination (i.e., it represents approximately what happens in the field situation). Some remedial measures may be taken to overcome these kinds of contamination problems in mega cities by treating the sources so that recharge of the aquifer is not affected.