Jacobi collocation method for the fractional advection‐dispersion equation arising in porous media

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.

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Application of a Legendre collocation method to the space–time variable fractional-order advection–dispersion equation

Journal of Taibah University for Science ◽

10.1080/16583655.2019.1576265 ◽

2019 ◽

Vol 13 (1) ◽

pp. 324-330 ◽

Cited By ~ 7

Author(s):

F. Mallawi ◽

J. F. Alzaidy ◽

R. M. Hafez

Keyword(s):

Dispersion Equation ◽

Collocation Method ◽

Fractional Order ◽

Space Time ◽

Time Variable ◽

Advection Dispersion ◽

Legendre Collocation Method

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

WEENTECH Proceedings in Energy ◽

10.32438/wpe.222021 ◽

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.

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