On the separate treatment of mixing and spreading by the reactive-particle-tracking algorithm: An example of accurate upscaling of reactive Poiseuille flow
Introduction
A recent improvement of the reactive-particle-tracking (RPT) method allows mass transfer between particles and subsequent reactions between any number of chemical constituents on the particles (Benson and Bolster, 2016). One of the features of this algorithm is that inter-particle mixing occurs separately from dispersive random walks. True mixing between dissimilar fluids usually occurs on smaller scales and at slower rates than the dispersive spreading (Cirpka, Kitanidis, 2000, Danckwerts, 1953, De Simoni, Carrera, Sánchez-Vila, Guadagnini, 2005, Dentz, Borgne, Englert, Bijeljic, 2011, Dentz, Carrera, 2007, Ding, Benson, Fernández-Garcia, Henri, Hyndman, Phanikumar, Bolster, 2017, Donado, Sánchez-Vila, Dentz, Carrera, Bolster, 2009, Hill, 1976, Le Borgne, Dentz, Villermaux, 2013, Le Borgne, Ginn, Dentz, 2014, Lehwald, Janiga, Thévenin, Zähringer, 2012, Molz, Widdowson, 1988, Nauman, Buffham, 1983, Rezaei, Sanz, Raeisi, Ayora, Vázquez-Suñé, Carrera, 2005). Schmidt et al. (2018a) suggested that the separate simulation of mixing and spreading by the RPT method could provide a way to accurately upscale reactive solute transport, because the smaller-scale true mixing dictates reaction rates, while the random walks simulate the process of particle separation that accompanies sub-grid (upscaled) velocity perturbations. Recent work has further extended the particle methods to allow fluid/solid interaction (Schmidt et al., 2018b). For additional reasons, such as surface area scaling and solubility saturations near mineral grains, dissolution/precipitation reactions also suffer significant scaling effects of reaction rates (see Brantley, Kubicki, White, 2008, White, Peterson, 1990).
Two of the classic examinations of the disparity between mixing and spreading in moving fluids (and the effect on global reaction rates) were performed by Kapoor, Gelhar, Miralles-Wilhelm, 1997, Kapoor, Jafvert, Lyn, 1998. These authors chose a simple system because it can be completely defined at the pore scale: laminar miscible displacement of chemically distinct (and reactive) fluids in Hagen-Poiseuille flow, either in a tube or between plates. In these cases, transport is exactly known, with the well-known parabolic velocity profile between no-slip walls, and random motion solely by molecular diffusion. The higher velocities at the center of the tube cause overlap of the fluids when projected to 1-D, but mixing only occurs along the warped interface. This system exemplifies the lag of mixing behind spreading in non-uniform velocity fields.
The spreading rate was first derived for Poiseuille flow by Taylor (1953), who showed that the 2-D transport of nonreactive tracer in a tube could be upscaled (averaged) to 1-D. Given enough time to sample the entire velocity variability by local diffusion, transport can be effectively described by a one-dimensional advection-dispersion equation with constant velocity and an enhanced macro-dispersion coefficient reflecting subscale advection-induced spreading. The asymptotic (t → ∞) upscaled longitudinal hydrodynamic macro-dispersion coefficient Dmac may be orders-of-magnitude larger than the local-scale molecular diffusion coefficient Dmol, and its functional form depends on the shear velocity distribution and molecular diffusion coefficient (e.g., Bolster et al., 2011).
This has inspired two tacks for upscaled reactive transport. The first tack has derived two dispersion coefficients, one for the effective mixing Deff and another corresponding to Taylor’s Dmac that describes macro-dispersive spreading. In this approach, mixing is assumed to be the dominant mechanism dictating reaction rates (i.e., reactions are nearly instantaneous) so that the statistics of mixing via destruction of concentration gradients give an effective, smaller, dispersion coefficient (Cirpka, Kitanidis, 2000, Cirpka, Kitanidis, 2000, Dentz, Carrera, 2007, Dentz, Kinzelbach, Attinger, Kinzelbach, 2000). This smaller Deff is designed to slow down reactions, but not place solutes in the correct locations, so Cirpka and Kitanidis (2000a) suggests a streamtube approach in which the spreading of the centers of mass for the streamtubes is given by Dmac, while mixing within a streamtube is given by the smaller Deff. The second tack seeks to adjust the reaction rate itself by recognizing that the reactant segregation that results from upscaling should modify the effective reaction rate. This approach has been used on simpler (diffusion-only) problems that allow direct calculation of segregation—as measured by reactant concentration covariance evolution equations (Bolster, de Anna, Benson, Tartakovsky, 2012, Paster, Bolster, Benson, 2014, Schmidt, Pankavich, Benson, 2017, Tartakovsky, de Anna, Le Borgne, Balter, Bolster, 2012). However, in heterogeneous velocity distributions, these equations have yet to be analytically solved, and only simpler expressions based on very fast or very slow reactions end-members have been developed (Porta et al., 2012).
These upscaling approaches point out the fundamental problem associated with an Eulerian advection, dispersion, and reaction equation (ADRE). The dispersion coefficient valid for conservative transport at some scale will over-predict fluid mixing and reaction at the same scale, but under-predict spreading at some larger scale (or volume averaged to fewer spatial dimensions). We seek to correct this problem with a Lagrangian framework. For M species undergoing Fickian dispersion in incompressible flow, the coupled ADREs arewhere ci is the concentration of each of the species labeled v is a velocity vector, D is a dispersion tensor, and R() is a reaction function among the M constituents with N reaction channels. The ADRE assumes that D describes mixing and spreading in exactly the same way. For continuously varying v, this is only true at the molecular scale. In practice, however, all variables and parameters in (1) have some finite support scale, and the discrepancy between mixing and spreading grows with support scale (Dentz, Borgne, Englert, Bijeljic, 2011, Le Borgne, Dentz, Villermaux, 2013, Le Borgne, Ginn, Dentz, 2014). Indeed, if is given by Darcy’s Law and a hydraulic conductivity parameter, then this type of upscaling has already occurred. As briefly reviewed above, this discrepancy may be accounted for by adjusting the only remaining equation parameters that are held in the reaction term R(), or by solving the equation separately with larger and smaller to figure out mixing versus proper positions of reactants. If the perturbations of ci, v, and D are known, as well as their auto- and cross-correlations in time and space, then the adjusted R() can be approximated with closure assumptions (Dentz, Borgne, Englert, Bijeljic, 2011, Porta, Chaynikov, Thovert, Riva, Guadagnini, Adler, 2013, Porta, Thovert, Riva, Guadagnini, Adler, 2012, Tartakovsky, de Anna, Le Borgne, Balter, Bolster, 2012) that may not be particularly accurate for some values of coefficients. We include a brief comparison of the two most notable analytic upscaling approximations to our numerical method in this paper.
On the other hand, the micro-scale physics of particle motion and interaction may already carry all information neglected by the analytic upscaling. Here we show that, for the simplest case, the RPT method does indeed automatically track the necessary small-scale information and performs a natural upscaling.
Section snippets
Hagen-Poiseuille flow
We simulate an identical problem of flow, transport, and kinetic bimolecular reaction between two parallel plates as did Kapoor et al. (1997) (Fig. 1). The concentration units are arbitrary, but we will use moles/L (molar). The local thermodynamic reaction rate is given by the law of mass action for constituents A and B, where k [molar T] is a rate coefficient. Without loss of generality we assume unit activity coefficients. This type of reaction has been studied
Results and discussion
The rates of C production and late-time A decline agree quite well in both log-log and linear coordinates, when the lower number of initial A particles () is used to represent concentration fluctuation distances on the order of one-half pore width (Fig. 4). We expect that an even better fit could be achieved by adjusting the particle number, but we have not changed our original visual estimate of particle support volume equal to one-half pore width (see also the appendix). This level of
Conclusions
In this technical note we show that the RPT method can accurately simulate dimensionally upscaled transport and reaction for pre-asymptotic times in Poiseuille flow. We used the time-dependent, upscaled effective Taylor macro-dispersion coefficient Dmac for random walks and the isotropic molecular diffusion for locally diffusive mass transfer between particles. An accurate model using finite differences would require full specification of the velocity in 2-D and take tens of millions of nodes
Acknowledgements
We thank the editor, Graham Sander, reviewers Olaf Cirpka, Giovanni Porta, and one anonymous reviewer for extremely helpful comments. This material is based upon work supported by, or in part by, the US Army Research Office under Contract/Grant number W911NF-18-1-0338. The authors were also supported by the National Science Foundation under awards EAR-1417145, DMS-1211667, DMS-1614586, EAR-1351625, EAR-1417264, EAR-1446236, and CBET-1705770.
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