Mixing Cell Model Applied to Transport in Porous Media

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The mixing cell model is used to solve the diffusion-advection equation with complex initial and boundary conditions.

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The equation describing the one-dimensional transport of a reacting component in porous media is

,

where is the fluid phase concentration, is the effective dispersion coefficient, is distance, is time, is the interstitial fluid velocity, and is a reaction constant. A function similar to the gamma distribution is used as the initial condition, and a time-dependent boundary condition is applied at .

,

, and

.

Here , , , , , and are constants.

The mixing cell model is an explicit finite-difference approximation to the diffusion-advection equation in which the dispersion term is neglected and a pure advection equation given by is solved.

Finite-difference solutions of this equation can produce numerical dispersion, which is controlled to simulate solutions to the advection-diffusion equation [1]. The time derivative is approximated with a forward difference, while a backward difference is used to approximate the space derivative. For the cell, this approximation is written as

.

The numerical dispersion introduced is .

To solve the advection-diffusion equation, this numerical dispersion is equated to the physical dispersion coefficient . Thus for any choice of , . This model greatly simplifies the computation, making solutions for complex convective-dispersive phenomena possible; complexities such as additional transport phenomena, chemical reactions, or flow profiles can be added to the cell model without changing its mathematical form or markedly increasing the difficulty of obtaining solutions.

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Contributed by: Clay Gruesbeck (October 2015)
Open content licensed under CC BY-NC-SA


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Reference

[1] H. C. Van Ommen, "The 'Mixing-Cell' Concept Applied to Transport of Non-reactive and Reactive Components in Soils and Groundwater," Journal of Hydrology, 78(3–4), 1985 pp. 201–213. doi:10.1016/0022-1694(85)90101-5.



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