A Practical Upscaling of Mixing-Limited Reactive Transport Using the Concept of Lamellae.
Résumé
Understanding reactive transport of solutes in porous media underlies problems ranging from in-situ remediation or natural attenuation, to quantifying redox processes and/or precipitation rates in natural biogeochemical cycling, to design of geothermal systems. Despite decades of research, we remain essentially unable to predict multicomponent reactive transport in natural porous media. We report on a new approach to upscaling mixing-limited reactive transport, that combines a Lagrangian focus on transport of the moving interface between the mixing solutions, with kinetics of mixing-limitation governed by scalar dissipation rates. We treat the moving interface between reacting solutions as a chain of individual strips, or lamellae, after Ranz, and Villermaux, and upscale mixing-limited reaction extent using the scalar dissipation of an analogous passive solute after de Simoni. Given the statistical description of how the entire set of lamella behave (achievable in principle for any given rendition of a porous media), the problem reduces to one of predicting the reactions on a given lamella as it evolves kinematically due to stretching and diffusion. This simplification reduces much of the complexity while honoring the mixing on the lamella during its advection along with the moving front. The resulting framework for upscaling reaction extent is: a. characterize the deformation of the individual lamellae, b. determine the reaction extent on each one, and c. sum the reaction extents on each lamella to obtain an expression for global reaction extent. We describe details of the development and demonstrate its utility in several idealized problems by comparison to explicit numerical simulation of the reactive transport processes. The Figure below shows a linear flood with flow left-to-right in a 2D heterogeneous flow field,with solution B displacing solution A. The left panel shows influent solution boundary at initial time to as dashed line on left with an individual lamella shown in zoom, original length rho=1 and width so. The right panel shows same lamella at later time after stretching and diffusion dictate new length rho=1 and width s.