Water Science and Engineering 2018, 11(2) 101-113 DOI:   https://doi.org/10.1016/j.wse.2018.07.006  ISSN: 1674-2370 CN: 32-1785/TV

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Lagrangian framework
Chemical reaction
Diffusion-limited process
Multi-step reactions
Interaction radius
Bing-qing Lu
Yong Zhang
Hong-guang Sun
Chun-miao Zheng
Article by Bing-qing Lu
Article by Yong Zhang
Article by Hong-guang Sun
Article by Chun-miao Zheng

Lagrangian simulation of multi-step and rate-limited chemical reactions in multi-dimensional porous media

Bing-qing Lu a, Yong Zhang a,*, Hong-guang Sun b, Chun-miao Zheng c

a Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487, USA
b Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 210098, China
c School of Environmental Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China


Management of groundwater resources and remediation of groundwater pollution require reliable quantification of contaminant dynamics in natural aquifers, which can involve complex chemical dynamics and challenge traditional modeling approaches. The kinetics of chemical reactions in groundwater are well known to be controlled by medium heterogeneity and reactant mixing, motivating the development of particle-based Lagrangian approaches. Previous Lagrangian solvers have been limited to fundamental bimolecular reactions in typically one-dimensional porous media. In contrast to other existing studies, this study developed a fully Lagrangian framework, which was used to simulate diffusion-controlled, multi-step reactions in one-, two-, and three-dimensional porous media. The interaction radius of a reactant molecule, which controls the probability of reaction, was derived by the agent-based approach for both irreversible and reversible reactions. A flexible particle tracking scheme was then developed to build trajectories for particles undergoing mixing-limited, multi-step reactions. The simulated particle dynamics were checked against the kinetics for diffusion-controlled reactions and thermodynamic well-mixed reactions in one- and two-dimensional domains. Applicability of the novel simulator was further tested by (1) simulating precipitation of calcium carbonate minerals in a two-dimensional medium, and (2) quantifying multi-step chemical reactions observed in the laboratory. The flexibility of the Lagrangian simulator allows further refinement to capture complex transport affecting chemical mixing and hence reactions.

Lagrangian framework
   Chemical reaction   Diffusion-limited process   Multi-step reactions   Interaction radius  
Received 2017-05-17 Revised 2018-02-25 Online: 2018-04-30 
DOI: https://doi.org/10.1016/j.wse.2018.07.006

This work was supported by the National Natural Science Foundation of China (Grants No. 41330632, 41628202, and 11572112).

Corresponding Authors: Yong Zhang
Email: yzhang264@ua.edu
About author:


Andrews, S.S, Bray, D., 2004. Stochastic simulation of chemical reactions with spatial resolution and single molecule detail. Physical Biology, 1(3–4), 137–151. https://doi.org/10.1088/1478-3967/1/3/001.

Barnard, J.M., 2017. Simulation of mixing-limited reactions using a continuum approach. Advances in Water Resources, 104(6), 15–22. https://doi.org/10.1016/j.advwatres.2017.03.012.

Benson, D.A., Meerschaert, M.M., 2008. Simulation of chemical reaction via particle tracking: Diffusion-limited versus thermodynamic rate-limited regimes. Water Resources Research, 44(12), W12202. https://doi.org/10.1029/2008WR007111.

Benson, D.A., Aquino, T., Bolster, D., Engdahl, N., Henri, C.V., Fernàndez-Garcia, D., 2017. A comparison of Eulerian and Lagrangian transport and non-linear reaction algorithms. Advances in Water Resources, 99, 15–37. https://doi.org/10.1016/j.advwatres.2016.11.003.

Berkowitz, B., Cortis, A., Dentz, M., Scher, H., 2006. Modeling non-Fickian transport on geological formations as a continuous time random walk. Review of Geophysics, 44, RG2003, https://doi.org/10.1029/2005RG000178.

Bolster, D., Benson, D.A., Singha, K., 2017. Upscaling chemical reactions in multicontinuum systems: When might time fractional equations work? Chaos, Solitons & Fractals, 102, 414–425. https://doi.org/10.1016/j.chaos.2017.04.028.

Cirpka, O.A., 2002. Choice of dispersion coefficients in reactive transport calculations on smoothed fields. Journal of Contaminant Hydrology, 58(3), 261–282. https://doi.org/10.1016/S0169-7722(02)00039-6.

Cirpka, O.A., Valocchi, A.J., 2016. Debates-stochastic subsurface hydrology from theory to practice: Does stochastic subsurface hydrology help solving practical problems of contaminant hydrogeology? Water Resources Research, 52(12), 9218–9227. https://doi.org/10.1002/2016WR019087.

Dentz, M., Borgne, T.L., Englert, A., Bijeljic, B., 2011. Mixing, spreading and reaction in heterogeneous media: A brief review. Journal of Contaminant Hydrology, 120–121, 1–17. https://doi.org/10.1016/j.jconhyd.2010.05.002.

Ding, D., Benson, D.A., Paster, A., Bolster, D., 2013. Modeling bimolecular reactions and transport in porous media via particle tracking. Advances in Water Resources, 53, 56–65. https://doi.org/10.1016/j.advwatres.2012.11.001.

Edery, Y., Scher, H., Berkowitz, B., 2009. Modeling bimolecular reactions and transport in porous media. Geophysical Research Letters, 36(2), L02407. https://doi.org/10.1029/2008GL036381.

Edery, Y., Scher, H., Berkowitz, B., 2010. Particle tracking model of bimolecular reactive transport in porous media. Water Resources Research, 46(7), W07524. https://doi.org/10.1029/2009WR009017.

Engdahl, N.B., Benson, D.A., Bolster, D., 2017. Lagrangian simulation of mixing and reactions in complex geochemical systems. Water Resources Research, 53(4), 3513-3522. https://doi.org/10.1002/2017WR020362.

Erban, R., Chapman, S.J., 2009. Stochastic modeling of reaction-diffusion processes: Algorithms for bimolecular reactions. Physical Biology, 6(4), 046001. https://doi.org/10.1088/1478-3975/6/4/046001.

Gillespie, D.T., 1977. Exact stochastic simulation of coupled chemical react ions. The Journal of Physical Chemistry, 81(25), 2340–2361. https://doi.org/10.1021/j100540a008.

Gillespie, D.T., 2009. A diffusional bimolecular propensity function. Journal of Chemical Physics, 131(16), 164109. https://doi.org/10.1063/1.3253798.

Gramling, C.M., Harvey, C.F., Meigs, L.C., 2002. Reactive transport in porous media: A comparison of model prediction with laboratory visualization. Environmental Science & Technology, 36(11), 2508–2514. https://doi.org/10.1021/es0157144.

Ham, P.A.S., Schotting, R.J., Prommer, H., Davis, G.B., 2004. Effects of hydrodynamic dispersion on plume lengths for instantaneous bimolecular reactions. Advances in Water Resources, 27(8), 803–813. https://doi.org/10.1016/j.advwatres.2004.05.008.

Hattne, J., Fange, D., Elf, J., 2005. Stochastic reaction-diffusion simulation with MesoRD. Bioinformatics, 21(12), 2923–2924. https://doi.org/10.1093/bioinformatics/bti431.

Isaacson, S.A., Peskin, C.S., 2006. Incorporating diffusion in complex geometries into stochastic chemical kinetics simulations. SIAM Journal on Scientific Computing, 28(1), 47–74. https://doi.org/10.1137/040605060.

Kang, K., Redner, S., 1984. Scaling approach for the kinetics of recombination processes. Physical Review Letters, 52(12), 955–958. https://doi.org/10.1103/PhysRevLett.52.955.

Kang, K., Redner, S., 1985. Fluctuation-dominated kinetics in diffusion-controlled reactions. Physical Review A: Atomic, Molecular and Optical Physics, 32(1), 435–447. https://doi.org/10.1103/PhysRevA.32.435.

Kapoor, V., Gelhar, L.W., Miralles-Wilhelm, F., 1997. Bimolecular second-order reactions in spatially varying flows: Segregation induced scale-dependent transformation rates. Water Resources Research, 33(4), 527–536. https://doi.org/10.1029/96WR03687.

Kopelman, R., 1988. Fractal reaction kinetics. Science, 241(4873), 1620–1626. https://doi.org/10.1126/science.241.4873.1620.

LaBolle, E.M., Fogg, G.E., Tompson, A.F.B., 1996. Random-walk simulation of transport in heterogeneous porous media: Local mass conservation problem and implementation methods. Water Resources Research, 32(3), 583–593. https://doi.org/10.1029/95WR03528.

LaBolle, E.M., Quastel, J., Fogg, G.E., Gravner, J., 2000. Diffusion processes in composite porous media and their numerical integration by random walks: Generalized stochastic differential equations with discontinuous coefficients. Water Resources Research, 36(3), 651–662. https://doi.org/10.1029/1999WR900224.

Luo, J., Dentz, M., Carrera, J., Kitanidis, P., 2008. Effective reaction parameters for mixing controlled reactions in heterogeneous media. Water Resources Research, 44(2), W02416. https://doi.org/10.1029/2006/WR005658.

Neuman, S.P., Tartakovsky, D.M., 2009. Perspective on theories of non-Fickian transport in heterogeneous media. Advances in Water Resources, 32(5), 670–780.

Oates, P.M., Harvey, C.F., 2006. A colorimetric reaction to quantify fluid mixing. Experiments in Fluids, 41(5), 673–683. https://doi.org/10.1007/s00348-006-0184-z.

Paster, A., Bolster, D., Benson, D.A., 2013. Particle tracking and the diffusion-reaction equation. Water Resources Research, 49(1), 1–6. https://doi.org/10.1029/2012WR012444.

Pogson, M., Smallwood, R., Qwarnstrom, E., Holcombe, M., 2006. Formal agent-based modelling of intracellular chemical interactions. BioSystems, 85(1), 37–45. https://doi.org/10.1016/j.biosystems.2006.02.004.

Raje, D.S., Kapoor, V., 2000. Experimental study of bimolecular reaction kinetics in porous media. Environmental Science & Technology, 34(7), 1234–1239. https://doi.org/10.1021/es9908669.

Scheibe, T.D., Tartakovsky, A.M., Tartakovsky, D.M., Redden, G.D., Meakin, P., 2007. Hybrid numerical methods for multiscale simulations of subsurface biogeochemical processes. Journal of Physics: Conference Series, 78(1), 012063. https://doi.org/10.1088/1742-6596/78/1/012063.

Smoluchowski, M.V., 1918. Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Zeitschrift für Physikalische Chemie, 92(1), 129–168. https://doi.org/10.1515/zpch-1918-9209. (in German)

Sung, B.J., Yethiraj, A., 2005. Molecular-dynamics simulations for nonclassical kinetics of diffusion-controlled bimolecular reactions. The Journal of Chemical Physics, 123(11), 114503. https://doi.org/10.1063/1.2035081.

Tartakovsky, A.M., Tartakovsky, G.D., Scheibe, T.D., 2009. Effects of incomplete mixing on multicomponent reactive transport. Advances in Water Resources, 32(11), 1674–1679. https://doi.org/10.1016/j.advwatres.2009.08.012.

Tournier, A.L., Fitzjohn, P.W., Bates, P.A., 2006. Probability-based model of protein-protein interactions on biological timescales. Algorithms for Molecular Biology, 1, 25. https://doi.org/10.1186/1748-7188-1-25.

Toussaint, D., Wilczek, F., 1983. Particle-antiparticle annihilation in diffusive motion. The Journal of Chemical Physics, 78, 2642. https://doi.org/10.1063/1.445022.

Trautz, M., 1916. Das Gesetz der Reaktionsgeschwindigkeit und der Gleichgewichte in Gasen. Bestätigung der Additivität von Cv3/2R. Neue Bestimmung der Integrationskonstanten und der Moleküldurchmesser. Zeitschrift für Anorganische und Allgemeine Chemie, 96(1), 1–28. https://doi.org/10.1002/zaac.19160960102. (in German)

Willingham, T.W., Werth, C.J., Valocchi, A.J., 2008. Evaluation of the effects of porous media structure on mixing-controlled reactions using pore-scale modeling and micromodel experiments. Environmental Science & Technology, 42(9), 3185–3193. https://doi.org/10.1021/es7022835.

Zhang, Y., Papelis, C., Sun, P.T., Yu, Z.B., 2013. Evaluation and linking of effective parameters in particle-based models and continuum models for mixing-limited bimolecular reactions. Water Resources Research, 49(8), 4845–4865. https://doi.org/10.1002/wrcr.20368.

Zhang, Y., Qian, J.Z., Papelis, C., Sun, P.T., Yu, Z.B., 2014. Improved understanding of bimolecular reactions in deceptively simple homogeneous media: From laboratory experiments to Lagrangian quantification. Water Resources Research, 50(2), 1704–1715. https://doi.org/10.1002/2013WR014711.

Zhang, Y., Meerschaert, M.M., Baeumer, B., LaBolle, E.M., 2015. Modeling mixed retention and early arrivals in multidimensional heterogeneous media using an explicit Lagrangian scheme. Water Resources Research, 51(8), 6311–6337. https://doi.org/10.1002/2015WR016902.

Zhang, Y., Green, C.T., LaBolle, E.M., Neupauer, R.M., Sun, H.G., 2016. Bounded fractional diffusion in geological media: Definition and Lagrangian approximation. Water Resources Research, 52(11), 8561–8577. https://doi.org/10.1002/2016WR019178.

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