The impact of boundary on the fractional advection–dispersion equation for solute transport in soil: Defining the fractional dispersive flux with the Caputo derivatives

2007 ◽  
Vol 30 (5) ◽  
pp. 1205-1217 ◽  
Author(s):  
Xiaoxian Zhang ◽  
Mouchao Lv ◽  
John W. Crawford ◽  
Iain M. Young
Keyword(s):  
Solute Transport ◽  
Dispersion Equation ◽  
Caputo Derivatives ◽  
Advection Dispersion ◽  
The Impact ◽  
Dispersive Flux

scholarly journals An Experimental Study on Solute Transport in One-Dimensional Clay Soil Columns

Geofluids ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-17 ◽  
Author(s):  
Muhammad Zaheer ◽  
Zhang Wen ◽  
Hongbin Zhan ◽  
Xiaolian Chen ◽  
Menggui Jin
Keyword(s):  
Solute Transport ◽  
Dispersion Equation ◽  
Transport Process ◽  
Breakthrough Curve ◽  
Small Scale ◽  
Low Permeability ◽  
Late Time ◽  
Time Behavior ◽  
One Dimensional ◽  
Advection Dispersion

Solute transport in low-permeability media such as clay has not been studied carefully up to present, and we are often unclear what the proper governing law is for describing the transport process in such media. In this study, we composed and analyzed the breakthrough curve (BTC) data and the development of leaching in one-dimensional solute transport experiments in low-permeability homogeneous and saturated media at small scale, to identify key parameters controlling the transport process. Sodium chloride (NaCl) was chosen to be the tracer. A number of tracer tests were conducted to inspect the transport process under different conditions. The observed velocity-time behavior for different columns indicated the decline of soil permeability when switching from tracer introducing to tracer flushing. The modeling approaches considered were the Advection-Dispersion Equation (ADE), Two-Region Model (TRM), Continuous Time Random Walk (CTRW), and Fractional Advection-Dispersion Equation (FADE). It was found that all the models can fit the transport process very well; however, ADE and TRM were somewhat unable to characterize the transport behavior in leaching. The CTRW and FADE models were better in capturing the full evaluation of tracer-breakthrough curve and late-time tailing in leaching.

Review of non-equilibrium flow and transport models in saturated porous media

2021 ◽  
pp. 228-245
Author(s):  
Aman Chandel ◽  
Deepak Swami
Keyword(s):  
Porous Media ◽  
Solute Transport ◽  
Dispersion Equation ◽  
Chemical Interaction ◽  
Joint Probability ◽  
Saturated Porous Media ◽  
Soil Matrix ◽  
Lumped Model ◽  
Advection Dispersion ◽  
Non Equilibrium

This study deals with review of different improvements done in the formulation of the governing equations to simulate accurate solute transport in saturated porous media over the years. The traditional advection-dispersion equation (ADE) model is the simplest lumped model founded on the assumptions of Fick’s law of diffusion. But it typically underestimates the breakthrough concentration in leading and/or tailing region due to non-fickian transport. It is modified into mobile-immobile model (MIM) considering the medium having micropores with stagnant water pockets but allowing solute exchange by diffusion between mobile and immobile zone which is quantified by mass transfer coefficient. Multi-process non-equilibrium (MPNE) model further simulates for a system with both physical and chemical non-equilibrium by assuming instantaneous and rate-limited sorption in advective and non-advective domains. Using the concept of dual permeability, slow fast transport (SFT) model divides the liquid phase in the domain into three zones i.e. fast, slow and immobile. Here chemical interaction between the fluid and soil matrix takes place only in slow and immobile zones. Non-fickian solute transport does not follow Brownian motion rules so a random variable is required to explain it. Hence continuous time random walk (CTRW) model is used where solute transport is characterized by joint probability variable. Special case of CTRW with solute having considerable probability of moving long distances and follow power law gives Fractional advection-dispersion equation (FADE) model. These models varying from relatively simple to more complex formulations and assumptions are discussed here highlighting the merits and demerits of each.

scholarly journals How to use solutions of Advection-Dispersion Equation to describe reactive solute transport through porous media

2021 ◽  
Vol 60 (3) ◽  
pp. 229-240
Author(s):  
Jetzabeth Ramírez Sabag ◽  
Dennys Armando López Falcón
Keyword(s):  
Boundary Condition ◽  
Porous Media ◽  
Solute Transport ◽  
Dispersion Equation ◽  
Analytical Solutions ◽  
Peclet Number ◽  
Reactive Solute Transport ◽  
Péclet Number ◽  
Advection Dispersion ◽  
Outlet Boundary Condition

ResumenLas soluciones de la Ecuación de Advección-Dispersión son usadas frecuentemente para describir el transporte de solutos a través de medios porosos, considerando adsorción en equilibrio, de tipo lineal y reversible. Para indicar algunas sugerencias acerca de este tema, se hizo una revisión de las soluciones analíticas disponibles. Hay soluciones para Problemas con Condiciones de Frontera, de primer y tercer-tipo en la entrada así como de primer y segundo-tipo a la salida. Se analiza el comportamiento de las soluciones equivalentes, para sistemas finitos y semi-infinitos, observando que las soluciones de los sistemas semi-infinitos se aproximan a las correspondientes de los sistemas finitos conforme la condición de frontera de salida en el infinito se aproxima a la ubicación de medición del sistema finito. Solamente se presentan las soluciones analíticas con condiciones de frontera de segundo-tipo a la salida, ya que son iguales a las correspondientes soluciones analíticas con frontera de primer-tipo a la salida, para ambos tipos de condiciones de frontera de entrada usadas. Un análisis paramétrico, basado en el número de Peclet, muestra que todas las soluciones convergen cuando el número de Peclet es mayor que veinte. Los sistemas investigados deben tener un número de Peclet mayor que cinco para usar con confianza las soluciones de la Ecuación de Advección-Dispersión para describir el transporte de soluto en medios porosos.Palabras Clave: Ecuación de Advección-Difusión, Soluciones Analíticas, Transporte de Solutos Reactivos, Medios Porosos.AbstractThe solutions of Advection-Dispersion Equation are frequently used to describe solute transport through porous media when considering lineal and reversible equilibrium adsorption. To notice some warnings about this item, a review of analytical solutions available was done. There are solutions for Boundary Value Problems with first and third-type inlet boundary conditions as well as first and second-type outlet boundary condition. The behavior of equivalent solutions for finite and semi-infinite systems are analyzed, observing that semi-infinite system solutions approximates to the corresponding finite ones as the “infinite” outlet boundary condition approach to the finite measurement location. Because the analytical solutions with a first-type outlet boundary condition are equal to the corresponding analytical solutions with a second-type one, for both inlet boundary condition type used, only the latter is presented. A parametric analysis based on Peclet number shows that all solutions converge for Peclet number greater than twenty. Systems under research must have Peclet number greater than five to use confidently the solutions of Advection-Dispersion Equation to describe reactive solute transport through porous media.Keywords: Advection-Diffusion Equation, Analytical solutions, Reactive Solute Transport, Porous Media.

scholarly journals Modelling solute transport in homogeneous and heterogeneous porous media using spatial fractional advection-dispersion equation

2018 ◽  
Vol 13 (No. 1) ◽  
pp. 18-28 ◽  
Author(s):  
G. Moradi ◽  
B. Mehdinejadiani
Keyword(s):  
Sensitivity Analysis ◽  
Solute Transport ◽  
Dispersion Equation ◽  
Dispersion Coefficient ◽  
Breakthrough Curves ◽  
Heterogeneous Porous Media ◽  
Average Pore ◽  
Advection Dispersion ◽  
Estimated Parameters ◽  
Heterogeneous Soil

This paper compared the abilities of advection-dispersion equation (ADE) and spatial fractional advection-dispersion equation (sFADE) to describe the migration of a non-reactive contaminant in homogeneous and heterogeneous soils. To this end, laboratory tests were conducted in a sandbox sizing 2.5 × 0.1 × 0.6 m (length × width × height). After performing a parametric sensitivity analysis, parameters of sFADE and ADE were individually estimated using the inverse problem method at each distance. The dependency of estimated parameters on distance was examined. The estimated parameters at 30 cm were used to predict breakthrough curves (BTCs) at subsequent distances. The results of sensitivity analysis indicated that average pore-water velocity and dispersion coefficient were, respectively, the most and least sensitive parameters in both mathematical models. The values of fractional differentiation orders (α) for sFADE were smaller than 2 in both soils. The scale-dependency of the dispersion coefficients of ADE and sFADE was observed in both soils. However, the application of sFADE to describe solute transport reduced the scale effect on the dispersion coefficient, especially in the heterogeneous soil. For the homogeneous soil, the predicting results of ADE and sFADE were nearly similar, while for the heterogeneous soil, the predicting results of sFADE were more satisfactory in comparison with those of ADE, especially when the transport distance increased. Compared to ADE, the sFADE simulated somewhat better the tailing parts of BTCs and showed the earlier arrival of tracer. Overall, the solute transport, especially in the heterogeneous soil, was non-Fickian and the sFADE somewhat better described non-Fickian transport.

Modelling of Convective Melt Flow and Interface Shape in Commercial Bridgman-Stockbarger Growth of CdZnTe

2000 ◽  
Author(s):  
Toby D. Rule ◽  
Ben Q. Li ◽  
Kelvin G. Lynn
Keyword(s):  
Heat Transfer ◽  
Thermal Stress ◽  
Solute Transport ◽  
Latent Heat ◽  
Thermal Stresses ◽  
Radiation Detector ◽  
Time History ◽  
High Quality ◽  
Melt Convection ◽  
The Impact

Abstract CdZnTe single crystals for radiation detector and IR substrate applications must be of high quality and controlled purity. The growth of such crystals from a melt is very difficult due to the low thermal conductivity and high latent heat of the material, and the ease with which dislocations, twins and precipitates are introduced during crystal growth. These defects may be related to solute transport phenomena and thermal stresses associated with the solidification process. As a result, production of high quality material requires excellent thermal control during the entire growth process. A comprehensive model is being developed to account for radiation and conduction within the furnace, thermal coupling between the furnace and growth crucible, and finally the thermal stress fields within the growing crystal which result from the thermal conditions imposed on the crucible. As part of this effort, the present work examines the heat transfer and fluid flow within the crucible, using thermal boundary conditions obtained from experimental measurements. The 2-D axisymetric numerical model uses the deforming finite element method, with allowance made for melt convection, solidification with latent heat release and conjugate heat transfer between the solid material and the melt. Results are presented for several stages of growth, including a time-history of the solid-liquid interface (1365 K isotherm). The impact of melt convection, thermal end conditions and furnace temperature gradient on the growth interface is evaluated. Future work will extend the present model to include radiation exchange within the furnace, and a transient analysis for studying solute transport and thermal stress.

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