Effective Dispersion Equations for Reactive Flows Involving Free Boundaries at the Microscale

Abstract

We consider a pore-scale model for reactive flow in a thin 2-D strip, where the convective transport dominates the diffusion. Reactions take place at the lateral boundaries of the strip (the walls), where the reaction product can deposit in a layer with a non-negligible thickness compared to the width of the strip. This leads to a free boundary problem, in which the moving interface between the fluid and the deposited (solid) layer is explicitly taken into account. Using asymptotic expansion methods, we derive an upscaled, one-dimensional model by averaging in the transversal direction. The result is consistent with (Taylor dispersion) models obtained previously for a constant geometry. Finally, numerical computations are presented to compare the outcome of the effective (upscaled) model with the transversally averaged, two dimensional solution.

Analysis and Upscaling of a Reactive Transport Model in Fractured Porous Media with Nonlinear Transmission Condition

“…to [9,11,19] for a non-dimensionalization step. Let T > 0 be a maximal time and Ω p (the porous block) and Ω ε f (the fracture) be two adjacent domains separated by the interface Γ :…”

“…Specifically, we let the fracture thickness ε go to zero and reduce the fracture model to a boundary condition. We refer to [9,25] for a general procedure applied to convection dominated regimes, and to [19,30,31] for more specific applications related to precipitation-dissolution models, or to biofilm growth in porous media. However, these papers refer strictly to the fracture region and do not consider the coupling with a porous block.…”

“…Our approach is inspired from similar exercises carried out for precipitation-dissolution type models in [19,25], but uses the drift model discussed e.g. in [1,2,3], and assumes that the velocity is now very high, i.e.…”

Effective Behavior Near Clogging in Upscaled Equations for Non-isothermal Reactive Porous Media Flow

“…These papers assume the pore geometry to be fixed, which is a valid assumption if the mineral layer is not changed much compared to the pore aperture. Upscaled pore scale models honoring porosity changes are found in (Kumar et al 2011van Noorden 2009a,b). In these papers, the position of the interface between grain and void space is tracked, giving a problem with a free boundary.…”

Rigorous upscaling of rough boundaries for reactive flows

“…In other words, the precipitate layer is very thin, so it does not change the boundary Γ ε . An alternative situation, when Γ ε and hence Ω ε evolve in time depending on the solution is considered in [31,32,41,43].…”