Mixing Ratios With Age: Application to Preasymptotic One‐Dimensional Equilibrium Bimolecular Reactive Transport in Porous Media

Abstract

Analysis of reactive transport in natural and engineered porous media has benefited from the concept of mixing ratios, in particular as a basis for mathematical separation of transport and reactions processes. General use of solute age has also been recently explored as a way to describe solute mass transfer and/or as a proxy for reaction extent. Age here is defined as exposure time to the flow field. Pairing these concepts, we develop mixing ratio models that are structured on age. One‐dimensional transport is cast in terms of age‐structured mixing ratios in general and compared with conventional formulations of mixing ratio models, demonstrating that age is often a more natural independent variable than absolute time. Using this modeling framework, we then apply age‐structured mixing ratios to the problem of mixing‐limited reactive transport in one‐dimension by explicitly considering unmixed and mixed phases. In order to address mixing limitations under the entirety of transport including the preasymptotic dispersion timeframe, we use the same age variable to define the dispersion coefficient value in a local formulation of transport. Establishing an age‐dependent dispersion coefficient allows simulation of the whole transport time including both preasymptotic and asymptotic dispersion conditions with one model. In our application we explore use of this modeling approach to both synthetic preasymptotic data and experimental asymptotic data pertaining to one famous experiment.

A Closer Look: High-Resolution Pore-Scale Simulations of Solute Transport and Mixing Through Porous Media Columns

“…Yet, the observed degree of incomplete reaction is ostensibly larger than in our results. Multiple factors (or a combination of them) could explain this discrepancy, including the subtle differences in initial and boundary conditions at the interface (Gurung and Ginn, 2020), the possible influence of the lateral walls of the column on the flow and transport in G02, or the irregular geometry of the grains (as opposed to the spherical grains employed here). Below we explore the possible influence on the results of yet another element.…”

“…Recognizing the importance of the mixing process and the need for upscaled reactive transport models that implicitly account for it, numerous modeling approaches have been proposed in recent years to model mixing-limited reactive transport from a variety of perspectives (e.g., Oates, 2007;Benson and Meerschaert, 2008;Edery et al, 2009;Sanchez-Vila et al, 2010;Ding et al, 2013;Chiogna and Bellin, 2013;Hochstetler and Kitanidis, 2013;Paster et al, 2013Paster et al, , 2014Porta et al, 2016;Schmidt et al, 2017Schmidt et al, , 2018Schmidt et al, , 2019Ginn, 2018;Sole-Mari et al, 2020;Gurung and Ginn, 2020). In order to validate these new formulations, most authors test their models against laboratory experiments.…”

Towards Understanding Factors Affecting Arsenic, Chromium, and Vanadium Mobility in the Subsurface

“…Formulation of an accurate macroscale representation of the complex nonlinear reactions that occur at the microscale is one of the biggest challenges facing field-scale mechanistic modeling [ 124 , 137 , 157 ]. Currently, no one macroscale approach has been found that can produce reliable simulations of complex nonlinear geochemical fate and transport in all porous medium systems of interest [ 124 , 136 , 137 , 156 , 157 ]. What has been well established is that macroscale models do not follow a similar form to their microscale counterparts, and assuming so can introduce large errors [ 135 – 137 , 157 , 177 ].…”

“…What has been well established is that macroscale models do not follow a similar form to their microscale counterparts, and assuming so can introduce large errors [ 135 – 137 , 157 , 177 ]. Mathematically, the issue arises when attempting to represent averages of nonlinear microscale quantities as a non-linear combination of macroscale averaged quantities that is of the same functional form as the microscale form [ 114 , 135 , 136 , 177 ]. Put another way, assuming that each region of a macroscale model can be modeled as a batch system with known inputs and outputs introduces significant errors.…”

“…The standard macroscale ADRE is given by where is the assumed Fickian form of hydrodynamic dispersion experienced in the averaging region, and superscripts indicate macroscale quantities. Equation (5) relies on the standard assumption that the averaging region for the macroscale model is well mixed [ 135 , 136 , 155 , 159 ] and the porosity is constant in space and time. A result of this assumption is that the reaction rate may be expressed for species AB in our case as where would be the same as in Equation (3) .…”