Modeling Bimolecular Reactive Transport With Mixing‐Limitation: Theory and Application to Column Experiments

Abstract

The challenge of determining mixing extent of solutions undergoing advective‐dispersive‐diffusive transport is well known. In particular, reaction extent between displacing and displaced solutes depends on mixing at the pore scale, that is, generally smaller than continuum scale quantification that relies on dispersive fluxes. Here a novel mobile‐mobile mass transfer approach is developed to distinguish diffusive mixing from dispersive spreading in one‐dimensional transport involving small‐scale velocity variations with some correlation, such as occurs in hydrodynamic dispersion, in which short‐range ballistic transports give rise to dispersed but not mixed segregation zones, termed here ballisticules. When considering transport of a single solution, this approach distinguishes self‐diffusive mixing from spreading, and in the case of displacement of one solution by another, each containing a participant reactant of an irreversible bimolecular reaction, this results in time‐delayed diffusive mixing of reactants. The approach generates models for both kinetically controlled and equilibrium irreversible reaction cases, while honoring independently measured reaction rates and dispersivities. The mathematical solution for the equilibrium case is a simple analytical expression. The approach is applied to published experimental data on bimolecular reactions for homogeneous porous media under postasymptotic dispersive conditions with good results.

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

“…The irreversible one-dimensional bimolecular reaction experiments of Gramling et al (2002) and Raje and Kapoor (2000) illustrate the error that occurs when dispersive spreading and mixing are assumed contemporaneous. Numerous studies addressing these data (e.g., Alhashmi et al, 2015;Barnard, 2017;Chiogna & Bellin, 2013;Dentz et al, 2011;Ginn, 2018;Porta et al, 2012Porta et al, , 2013Zhang et al, 2013) have explored different paths in distinguishing spreading from mixing. Most can be classified into continuum Eulerian models (Barnard, 2017;Chiogna & Bellin, 2013;Hochstetler & Kitanidis, 2013;Porta et al, 2012Porta et al, , 2016Rubio et al, 2008;Sanchez-Vila et al, 2010) or Lagrangian particle tracking approaches (Alhashmi et al, 2015;Ding et al, 2013;Edery et al, 2010;Sole-Mari et al, 2020;Zhang et al, 2013).…”

Upscaling Mixing in Highly Heterogeneous Porous Media via a Spatial Markov Model

“…Incomplete mixing typically results in an overestimation of the amount of reaction that will occur, presenting the need to artificially, and often non-physically, alter the effective reaction rate used in the model to better match observations [19]. This has led to the development of upscaled models that aim to account for subscale concentration fluctuations that limit reaction e.g., [20][21][22]. The correct prediction of reaction rates has many practical implications in the context of porous media and aquifers, e.g., the prediction of contaminant migration [23,24], the remediation of contaminated groundwater [25,26], and the fundamental prediction of naturally occurring geochemical reactions that shape the subsurface below us.…”

A Solution for Mixing‐Limited Equilibrium Reactions in Idealized Radial Flow

“…Thus far, we have assumed that dispersive spreading quantified through the hydrodynamic dispersion coefficient is concomitant with complete mixing, which has been demonstrated to be in error at least for linear (column) flows (Gramling et al, 2002;Kapoor et al, 1997), leading to potentially significant overestimation of reaction rates in either analytical or numerical solutions (e.g., Jose & Cirpka, 2004;Perez et al, 2019), and unknown impacts for coupled transport and chemical reactions at larger scales in nonuniform, for example, radial flows (e.g., Luo et al, 2008). Ginn (2018) gives a brief review of the numerous approaches this challenge and proposes the concept of formation and eventual destruction of "ballisticules" as a theoretical basis for mixing limitations at the Darcy scale. Ballisticules are described as segregation zones (Raje & Kapoor, 2000) that occur at the displacement front between an incoming and displaced fluid.…”

“…End-member solutions correspond to each boundary or initial solution, and mixing ratios are the fractions of boundary or initial condition solutions present in the flow field at any given time (e.g., Cirpka & Valocchi, 2007). Recent developments distinguishing mixing from dispersion using the concept of "ballisticules" (Ginn, 2018) are then applied to extend the de Simoni analytical approach to devise one pathway to dealing with the third challenge in radial flow in exploratory simulations. Solutions obtained are quasi-analytical.…”