Elsevier

Advances in Water Resources

Volume 123, January 2019, Pages 40-53
Advances in Water Resources

On the separate treatment of mixing and spreading by the reactive-particle-tracking algorithm: An example of accurate upscaling of reactive Poiseuille flow

https://doi.org/10.1016/j.advwatres.2018.11.001Get rights and content

Highlights

  • Accurate upscaling of reactive transport Poiseuille flow with Reactive Particle Tracking (RPT) model.

  • RPT model separately simulates mixing with local molecular diffusion and spreading by Taylor macro-dispersion.

  • Comparison to two semi-analytic upscaling techniques: Volume-averaging and Ensemble Streamtube.

  • Separate treatment of mixing and spreading makes Lagrangian RPT model more representative than Eulerian advection-dispersion-reaction equation.

Abstract

The Eulerian advection-dispersion-reaction equation (ADRE) suffers the well-known scale-effect of reduced apparent reaction rates between chemically dissimilar fluids at larger scales (or dimensional averaging). The dispersion tensor in the ADRE must equally and simultaneously account for both solute mixing and spreading. Recent reactive-particle-tracking (RPT) algorithms can, by separate mechanisms, simulate 1) smaller-scale mixing by inter-particle mass transfer, and 2) mass spreading by traditional random walks. To test the supposition that the RPT can accurately track these separate mechanisms, we upscale reactive transport in Hagen-Poiseuille flow between two plates. The simple upscaled 1-D RPT model with one velocity value, an upscaled Taylor macro-dispersivity, and the local molecular diffusion coefficient matches the results obtained from a detailed 2-D model with fully described velocity and diffusion. Both models use the same thermodynamic reaction rate, because the rate is not forced to absorb the loss of information upon upscaling. Analytic and semi-analytic upscaling is also performed using volume averaging and ensemble streamtube techniques. Volume averaging does not perform as well as the RPT, while the streamtube approach (using an effective dispersion coefficient along with macro-dispersion) performs almost exactly the same as RPT.

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 arecit=v·ci+·(Dci)+R(cA,cB,,cM,k1,,kN);i=A,B,,Mwhere ci is the concentration of each of the species labeled i=A,B,,M, 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 v 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 D 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 A+BC 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 R=kcAcB for constituents A and B, where k [molar1 T1] 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 (N=260) 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.

References (45)

  • M.J. Schmidt et al.

    A kernel-based Lagrangian method for imperfectly-mixed chemical reactions

    J. Comput. Phys.

    (2017)
  • M.J. Schmidt et al.

    A Lagrangian method for reactive transport with solid/aqueous chemical phase interaction

    (2018)
  • D.A. Benson et al.

    Arbitrarily complex chemical reactions on particles

    Water Resour. Res.

    (2016)
  • D.A. Benson et al.

    Simulation of chemical reaction via particle tracking: diffusion-limited versus thermodynamic rate-limited regimes

    Water Resour. Res.

    (2008)
  • D. Bolster et al.

    A particle number conserving Lagrangian method for mixing-driven reactive transport

    Water Resour. Res.

    (2016)
  • O.A. Cirpka et al.

    An advective-dispersive stream tube approach for the transfer of conservative-tracer data to reactive transport

    Water Resour. Res.

    (2000)
  • O.A. Cirpka et al.

    Characterization of mixing and dilution in heterogeneous aquifers by means of local temporal moments

    Water Resour. Res.

    (2000)
  • M. De Simoni et al.

    A procedure for the solution of multicomponent reactive transport problems

    Water Resour. Res.

    (2005)
  • M. Dentz et al.

    Mixing and spreading in stratified flow

    Phys. Fluids

    (2007)
  • M. Dentz et al.

    Temporal behavior of a solute cloud in a heterogeneous porous medium: 1. point-like injection

    Water Resour. Res.

    (2000)
  • D. Ding et al.

    Modeling bimolecular reactions and transport in porous media via particle tracking

    Adv. Water Resour.

    (2012)
  • Cited by (0)

    View full text