Two-dimensional Solute Transport Research Articles

A novel four phase slug single-well push–pull test with regional flux: forward modeling and parameter estimation

The single-well push–pull (SWPP) test has been widely used for several applications, including determination of transport parameters such as regional groundwater flux velocity, effective porosity and dispersivity. A complete SWPP test usually includes four phases of injection, chase, rest, and extraction, while previous models mostly ignore the chase phase for the sake of simplification. Actually, the chase phase is important to minimize the impact of well skins and wellbore storage on the shape of the breakthrough curves (BTCs), thus is necessary and should be included in the model. To this end, two simplified methods are proposed in this study based on two different time tp and ta during a four-phase SWPP test to estimate the parameters related to transport, where tp and ta correspond to the time of the peak concentration during the injection phase and the time when the center of tracer mass returns to the wellbore during the extraction phase, respectively. Specifically, the tp method can be used to estimate the regional groundwater flow velocity and dispersivity (transverse and longitudinal), while the ta method is used to estimate the regional groundwater flow velocity and the effective porosity. Finite-element numerical modeling of the SWPP test based on the coupling of two-dimensional transient groundwater flow and solute transport is employed to test the applicability of the proposed methods. Results indicate that both the tp and ta methods proposed in this study perform satisfactorily in estimation of regional groundwater flow velocity, effective porosity and dispersivity, and these two methods require fewer observations than traditional curve fitting methods and do not require any optimization algorithms. We have also compared the proposed two methods with several previous methods, and conclude that these two new methods generally perform better than those of previous studies in terms of parameter estimation reliability.

A Novel Cellular Automata Framework for Modeling Depth-Averaged Solute Transport during Pluvial and Fluvial Floods

Currently, for modeling two-dimensional (2D) solute transport during pluvial and fluvial floods, the finite volume (FV) models are widely used because of their strong ability to handle steep concentration and velocity gradients from the flow advection term. However, heavy computational requirements are subsequently introduced which limit the numerical efficiency. To further increase numerical efficiency but keep the required accuracy, this study proposes a novel Solute Transport Modeling based on Cellular Automata framework (STMCA) to simulate solute transport due to the flow advection, turbulent diffusion, and material decay mechanisms in several sets of explicit algebraic equations. Four studied cases involving steep gradients of solute concentration and velocities in steady/unsteady violent flow conditions are used to compare the accuracy of the STMCA approach with a Godunov-type FV solute transport approach with a total variation diminishing (TVD) scheme. Then, the performances of the two approaches on water quality modeling are assessed through the E. Coli transport modeling during pluvial/fluvial floods on a real-scale terrain. The proposed STMCA approach is found to achieve almost the same accuracy as the FV approach. As to the numerical efficiency, the STMCA approach is faster than the FV approach by 289.6–328.6%. Hence, the proposed STMCA approach is proven to be an effective tool for simulating solute transport.

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.

Development of Simple Formula for Transverse Dispersion Coefficient in Meandering Rivers

This study aims to develop a straightforward and practical formula to estimate transverse dispersion coefficients in meandering natural rivers, a critical factor for predicting solute transport. We present a novel expression for the transverse dispersion coefficient based on dispersion and hydraulic data sets obtained from tracer experiments conducted in natural rivers. A distinctive feature of the formula is its reliance on one dimensionless hydraulic parameter, u¯u*hRc. To assess the effectiveness and accuracy of our proposed formula, we compare it with previously established equations commonly employed in the field. Furthermore, we apply the formula to natural river bends situated in the Nakdong River of Korea. This equation serves to estimate the initial value of the dispersion coefficient in two-dimensional solute transport modeling. As a result, the calibrated value of the dimensionless transverse dispersion coefficient is 0.97, which is only a 16% difference between the initial value of 0.81 as obtained from the formula. The formula presented in this study, simplifying and utilizing the dimensionless hydraulic parameter, offers a promising approach to estimating transverse dispersion in natural meandering rivers in cases where tracer and secondary flow data are unavailable. Additionally, the formula can be refined with more recent dispersion data, leading to a clearer, more straightforward, and validated formulation that captures the intricate interplay between topography and transverse dispersion.

Random Walk Particle Tracking to Model Dispersion in Steady Laminar and Turbulent Pipe Flow.

To accurately model a two-dimensional solute transport in drinking water pipes and determine the effective dispersion coefficients for one-dimensional water quality models of water distribution systems, a random walk particle tracking approach was developed to analyze the advection and dispersion processes in circular pipes. The approach considers a solute particle's two-dimensional random movement caused by molecular or turbulent diffusion and associated velocity profile, and can simulate any mixing time and accurately model the longitudinal distribution of the solute concentration. For long mixing times, the simulation results agreed with a previous analytically derived solution. For turbulent flow conditions, simulations showed that the longitudinal dispersion of the solute is very sensitive to the utilized cross-sectional velocity profiles. This approach is easy to implement programmatically and unconditionally stable. It can predict the mixing characteristics of a pipe under various initial and boundary conditions.

An efficient analytical approach with novel integral transform to study the two-dimensional solute transport problem

The q-homotopy analysis method (q-HAM) in combine with the novel integral transform known as Elzaki transform (ET) leads to an efficient analytical technique called, the q-homotopy analysis Elzaki transform method (q-HAETM). In the present study, the two- dimensional advection–dispersion (AD) problem is investigated using an analytical technique q-HAETM. These equations are mainly used to describe the fate of pollutants in aquifers. The analytical solutions to the AD equations are more interesting since they serve as benchmarks against which numerical solutions can be compared. The novelty of the work is to discuss the two-dimensional (2D) solute transport problem in the fractional sense. The reliability and the efficiency of the considered algorithm are demonstrated by employing the 2D fractional solute transport problem. The solute concentration profile is shown in terms of surface plots. The comparison of the exact solution and the approximate solution is done by the 2D plots. The numerical approximate error solutions are presented for different fractional orders. q-HAETM offers us to modulate the range of convergence of the series solution using ℏ, called auxiliary parameter or convergence control parameter. By performing appropriate numerical simulations in comparison with other existing techniques, the effectiveness and reliability of the considered technique are validated. The obtained findings show that the proposed method is very gratifying and examines the complex challenges that arise in science and innovation.

Modifying Elder’s Longitudinal Dispersion Coefficient for Two-Dimensional Solute Mixing Analysis in Open-Channel Bends

Elder’s equation for the longitudinal dispersion coefficient in two-dimensional solute transport analysis cannot be applied to curved channels because the vertical distribution of the longitudinal velocity does not obey the logarithmic law in the bends of an open channel. In this study, a two-dimensional longitudinal dispersion coefficient based on an equation that can appropriately describe the vertical distribution of flow velocity in open-channel bends is derived theoretically. The proposed equations for the vertical velocity distribution and dispersion coefficient are compared and verified with values measured from two different types of open channels, i.e., a laboratory channel and a natural-like channel. The increase in the longitudinal dispersion coefficient based on the difference in the vertical distribution of the flow velocity is evaluated quantitatively. In terms of the longitudinal dispersion coefficient, no significant difference is observed between the observed dispersion coefficient based on the concentration data and the coefficient value calculated using the equation proposed in this study. The dispersion equation proposed in this study can be easily applied to assign the value of the longitudinal dispersion coefficient for the two-dimensional mixing modelling in bends using basic hydraulic factors.

Evaluation of longitudinal / transverse dispersion coefficients and prediction of concentration at river confluence in two-dimensional solute transport analysis

Evaluation of longitudinal / transverse dispersion coefficients and prediction of concentration at river confluence in two-dimensional solute transport analysis

Analytical solution of two-dimensional conservative solute transport in a heterogeneous porous medium for varying input point source

In the present study, an analytical solution is obtained for two-dimensional advection–dispersion equation with variable coefficients in a semi-infinite heterogeneous porous medium. Dispersion coefficient and groundwater velocity are assumed to vary temporally as well as spatially while the retardation factor varies spatially only. The first-order decay and zero-order production terms are also considered into account. The variations in parameters along both longitudinal and transverse directions are of exponentially decreasing nature. Initially the geological formation is considered not solute free. The nature of pollutant is considered of conservative and is assumed to be originating from time-dependent varying point source. New time and space variables are introduced to convert the variable coefficients of the advection–dispersion equation into constant coefficients. Analytical solution of the proposed model is obtained by employing Laplace integral transformation technique (LITT). The proposed analytical solution is illustrated graphically to study the effects of various parameters on the solute transport for temporally exponentially decreasing and sinusoidal nature velocities.

Laboratory observations for two-dimensional solute transport in an aquifer-aquitard system.

Low-permeability media such as clay appear in nearly all hydrogeological systems. To date, although significant efforts have been put forward by hydrologists, transport mechanism is still not well understood in such media, especially in an aquifer-aquitard system. In this study, two-dimensional experiments of groundwater flow and solute transport were conducted in a clay-sand two-layer system to investigate the characteristics of flow and transport in such a system. Sodium chloride (NaCl) (a conservative tracer) from a tank was injected after passing by the pre-inlet reservoir where the mixing effect and flow transiency were analyzed. A new numerical model considering the mixing effect and flow transiency was developed to interpret the experimental data based on the finite-element COMSOL Multiphysics platform. Transport parameters were assessed by best fitting the observed breakthrough curves (BTCs). Several important results were obtained. Firstly, aquitard advection was found to be non-negligible and should be considered in a proper mathematical model for describing the transport process. Secondly, advective velocities were temporally variable and showed decreasing trends in the sand and clay layers, mainly due to the impacts of physical and biological clogging. Thirdly, the mixing effect in the pre-inlet reservoir led to a lower tracer concentration in the sand layer at early times. Finally, the observed BTCs exhibited early arrivals in the clay layer, possibly resulting from preferential flow pathways. These findings can provide hints for contamination remediation works in aquifer-aquitard systems.

ANALYTICAL SOLUTION FOR THE TWO-DIMENSIONAL LINEAR ADVECTION-DISPERSION EQUATION IN POROUS MEDIA VIA THE FOKAS METHOD

<abstract> We present the analytical solution of the two-dimensional linear advection-dispersion equation ($ 2 $-D LAD) in the quarter plane and the semi-infinite domain for two-dimensional solute transport in a porous medium. The governing equation includes terms describing advection, longitudinal and transverse dispersions and linear equilibrium adsorption. The analytical solution in terms of integrals in the complex plane is established by utilizing the unified transform method, also known as the Fokas method. The method hinges upon analysis of the divergence form of the governing equation and the so-called global relation, which is an algebraic relation coupling all known and unknown initial and boundary values. Particularly, the integral representation of the solution yields an accurate and fast numerical evaluation of the solution for the $ 2 $-D LAD equation. We demonstrate examples as an application of the developed solution and compare the analytical solution with numerical results. </abstract>

Soil water/salt balance and water productivity of typical irrigation schedules for cotton under film mulched drip irrigation in northern Xinjiang

Film mulched drip irrigation (FMDI) has clear benefits for saving water and promoting cotton production in Xinjiang, China. Due to the complicated transport processes for soil water and salt under FMDI, it is still challenging to systematically answer the questions such as the soil water and salt balance in the root zone, cotton yield and water productivity (WP) under typical planting modes and irrigation schedules. Previously, we proposed a HYDRUS-2D model with modified root water uptake (RWU) function and simulated the two-dimensional soil water flow and solute transport in cotton fields under FMDI, but we ignored the effects of the cotton yield and WP. From the perspective of smart irrigation, it is necessary to evaluate the effects of irrigation on crop production, WP, and soil environments. Thus, in the present study, our HYDRUS-2D model coupled with a validated crop water production function was applied to evaluate the effects of typical irrigation schedules and the antecedent soil salinity distributions on the soil water/salt balance, and plant water status, and leaching fraction, as well as the cotton yield and WP under FMDI in 36 scenarios. These scenarios included two antecedent soil salinity distributions (AS), six irrigation amounts (IA) and three irrigation intervals (IF). Compared with the lower AS scenarios, the higher AS scenarios obtained lower cotton transpiration and evapotranspiration but higher evaporation, drainage, and soil water storage, and the crop water status and leaching fraction were also increased due to the soil salt stress restricting RWU. The salt balance results indicated the current irrigation scheduling under FMDI leads to desalination processes in different scenarios. Moreover, the cotton yield and the corresponding WP for cotton (WPETa) and irrigation (WPI) were estimated. Under different AS scenarios, binary quadratic regression equations were developed based on IA and IF data to predict the cotton yield, WPETa, and WPI. Appropriate irrigation strategies are recommended based on the results. Our results could provide a scientific basis for optimizing irrigation schedules and improving the understanding of irrigation management under FMDI among local farmers in Xinjiang.

Two-Dimensional Solute Transport with Multiple Point Sources in Semi-infinite Porous Media

Two-Dimensional Solute Transport with Multiple Point Sources in Semi-infinite Porous Media

Optimization Design of Groundwater Pollution Monitoring Scheme and Inverse Identification of Pollution Source Parameters Using Bayes’ Theorem

In the process of identifying groundwater pollution sources, in order to solve the problem that the monitoring data of monitoring wells was insufficient or the correlation between monitoring data and model parameters was weak, a monitoring well optimization method based on Bayesian formula and information entropy was proposed. Two-dimensional phreatic groundwater solute transport model was built and solved by using GMS software. To reduce the computational load of calling the numerical model repeatedly in the optimization design of the monitoring schemes and the identification process of the pollution sources, the Kriging method was used to establish the surrogate model of the numerical model. Under the condition of single well monitoring and determined monitoring frequency, with the target of optimization of monitoring position number D and monitoring time interval ∆t, both the single-objective monitoring scheme with the minimum information entropy of the model parameter posterior distribution and the multi-objective monitoring scheme with the minimum information entropy and the shortest monitoring time were optimized respectively. According to the above-optimized monitoring schemes, the delayed rejection adaptive Metropolis algorithm was used to identify the pollution source parameters. The case study results showed that under the condition of pre-set single well monitoring with monitoring frequency of 10 times, the single-objective optimized monitoring scheme was D = 37 and Δt = 20 days. Under this monitoring scheme, the mean errors of inversion pollution source parameters α = (XS, YS, T1, T2, QS) were 0.09%, 0.4%, 4.72%, 2.43%, and 9.29%, respectively. The multi-objective optimized monitoring scheme was D = 37 and Δt = 2 days. Under this monitoring scheme, the mean errors of the inversion parameters α = (XS, YS, T1, T2, QS) were 12.76%, 3.77%, 5.13%, 1.36%, and 7.68%, respectively. Compared with the monitoring scheme based on the single-objective optimization, although the inversion mean error of the five parameters based on the multi-objective optimized monitoring scheme increased by 2.75%, the monitoring time significantly reduced from 180 to 18 days.

Probabilistic solution to two-dimensional stochastic solute transport model by the Fokker-Planck equation approach

The study of solute transport in rivers is of environmental significance, since solute transported in river flows may affect the river water quality conditions. Solute transport in natural flows is a challenging process to be investigated, given natural flows usually indicate uncertainty in space and time. Two-dimensional models are more preferable to one-dimensional models, especially when the underlying flow field is complex. In this study, a stochastic horizontally two-dimensional solute transport model is developed. The proposed model can account for the effect of uncertain two-dimensional flow field and solute loading conditions on the uncertainty of the solute transport. The governing equation of the proposed model is in the form of Fokker-Planck equation, which characterizes the spatiotemporal evolution of the probability density function of the state variables of the solute transport process. Consequently, the mean and standard deviation behavior of the ensemble solute transport under uncertain conditions can be quantified. The numerical solutions of the proposed model are validated with the Monte Carlo simulations under uncertain flow fields. The comparison of the results by the two methods indicate the developed two-dimensional stochastic solute transport model can sufficiently express the spatial variability of the solute transport process and adequately capture the ensemble solute transport behavior by the complete probabilistic description of the solute transport process under uncertain conditions.

Mass-transfer impact on solute mobility in porous media: A new mobile-immobile model

The theory for modeling non-equilibrium solute transport in porous media is still based on approximations to a model proposed by Lapidus and Amundson in 1952 that has not been updated. This Mobile–Immobile Model (MIM) is based on the definition of a mass-transfer coefficient (α), which has been proven subject to some severe limitations. Measurements at both laboratory and field scales have demonstrated the scale-dependency of α values. This means that the MIM theory fails in real applications, since α is not constant, as defined in the kinetic model theory, but is a time-residence (or distance) dependent coefficient. Multi-rate mass-transfer models have been proposed in recent literature to capture real-world solute transport with a multiple mass transfer. In this study, we propose a novel model, which implements the analytical solution of Fick's second law of diffusion directly in the nonequilibrium advection/dispersion equation of solute transport in porous media. New model solutions properly fitted data collected during tracer tests carried out at the CNR-IRSA Laboratory (Bari, Italy) in a horizontal sandbox, 2 m of length, by using sodium chloride as the conservative tracer. Selected breakthrough curves at specific positions were used to validate the proposed model solution and estimate both conventional and proposed coefficients of mass transfer. Results have shown a decreasing trend of α from 0.09 to 0.04 h−1 after about 1.2 m of filtration for the investigated sand, whereas new solutions provide two scale-invariant tracer coefficients of rate of tracer mass-transfer (0.004 ± 0.005 h−1) and of tracer time delay (1.19 ± 0.01). The proposed model performs very well, since it provides a readily solved analytical solution with respect to the conventional MIM. Results of the proposed MIM are very similar to those provided by the conventional MIM. The new model solution can be implemented in particle tracking or random walk software in order to solve two-dimensional nonequilibrium solute transport in groundwater.

A non-negative and high-resolution finite volume method for the depth-integrated solute transport equation using an unstructured triangular mesh

This paper proposes a new high-resolution finite volume method for solving the two-dimensional (2D) solute transport equation using an unstructured mesh. A new simple r-factor algorithm is introduced into the Total Variation Diminishing flux limiter to achieve a more efficient yet accurate high-resolution scheme for solving the advection term. To avoid the physically-meaningless negative solutions resulted from using the Green–Gauss theorem, a nonlinear two-point flux approximation scheme is adopted to deal with the anisotropic diffusion term. The developed method can be readily coupled with a two-dimensional finite-volume-based flow models under unstructured triangular mesh. By integrating with the ELCIRC flow model, the proposed method was verified using three idealized benchmark cases (i.e., advection of a circle-shaped solute field, advection in a cyclogenesis flow field and transport of a initially square-shaped solute plume), and further applied to simulate the non-reactive solute transport process driven by irregular tides in the Deep Bay, eastern Pearl River Estuary of China. These cases are also simulated by models using other existing methods, including different r-factor for advection term and the Green–Gauss theorem for diffusion term. The comparison between the results from the new method and those from other existing methods demonstrated the new method could describe advection induced concentration shock and discontinuities, and anisotropic diffusion at high resolution without providing spurious oscillations and negative values.

Analysis of two-dimensional solute transport through heterogeneous porous medium

Analysis of two-dimensional solute transport through heterogeneous porous medium

A Green's function method for two-dimensional reactive solute transport in a parallel fracture-matrix system

The reactive solute transport in a single fracture bounded by upper and lower matrixes is a classical problem that captures the dominant factors affecting transport behavior beyond pore scale. A parallel fracture-matrix system which considers the interaction among multiple paralleled fractures is an extension to a single fracture-matrix system. The existing analytical or semi-analytical solution for solute transport in a parallel fracture-matrix simplifies the problem to various degrees, such as neglecting the transverse dispersion in the fracture and/or the longitudinal diffusion in the matrix. The difficulty of solving the full two-dimensional (2-D) problem lies in the calculation of the mass exchange between the fracture and matrix. In this study, we propose an innovative Green's function approach to address the 2-D reactive solute transport in a parallel fracture-matrix system. The flux at the interface is calculated numerically. It is found that the transverse dispersion in the fracture can be safely neglected due to the small scale of fracture aperture. However, neglecting the longitudinal matrix diffusion would overestimate the concentration profile near the solute entrance face and underestimate the concentration profile at the far side. The error caused by neglecting the longitudinal matrix diffusion decreases with increasing Peclet number. The longitudinal matrix diffusion does not have obvious influence on the concentration profile in long-term. The developed model is applied to a non-aqueous-phase-liquid (DNAPL) contamination field case in New Haven Arkose of Connecticut in USA to estimate the Trichloroethylene (TCE) behavior over 40 years. The ratio of TCE mass stored in the matrix and the injected TCE mass increases above 90% in less than 10 years.

Numerical Solution of Two-Dimensional Solute Transport System Using Operational Matrices

In this study, the numerical solution of the two-dimensional solute transport system in a homogeneous porous medium of finite length is obtained. The considered transport system has the terms accounting for advection, dispersion and first-order decay with first-type source boundary conditions. Initially, the aquifer is considered solute free, and a constant input concentration is considered at inlet boundary. The solution is describing the solute concentration in rectangular inflow region of the homogeneous porous media. The numerical solution is derived using a powerful method, viz. spectral collocation method. The numerical computation and graphical presentations exhibit that the method is effective and reliable during the solution of the physical model with complicated boundary conditions even in the presence of reaction term.