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Application of fractional differential equation to interpret the dynamics of dissolved heavy-metal uptake in streams at a wide range of scales

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Abstract.

Fractional differential equations (FDEs) provide promising models to simulate non-Fickian transport in heterogeneous systems such as geological media, but the FDEs have seldom been used to model reactive transport across a wide range of spatial scales. To fill the knowledge gap, this study proposed a fractional-order advection-dispersion-reaction (fADR) model to quantify the dynamics of dissolved heavy metals, such as manganese (Mn), moving in streams with oxidative precipitation and sorption in storage zones. The fADR model was applied to fit the Mn concentration profiles documented in the literature at the hyporheic flow path scale (0.30m in length), the reach scale (100-3000m), and the basin scale ( ∼ 20000 m) at Pinal Creek, Arizona. Numerical results showed that compared with standard transport models such as the OTIS model and the single-rate mass transfer model, the fADR model can better capture the observed plumes at all scales. This is because the space fractional-diffusive term in the fADR model can capture super-diffusive jumps of dissolved metals driven by turbulence at the bedform scale and flooding in a decade-long time scale. Meanwhile, the time fractional derivative term in the fADR model describes complex solute retention due to multiple-rate mass exchange between the mobile zone (stream or the hyporheic flux) and various storage regimes with different properties (such as streambed sediments and stagnant portions in the hyporheic zone). This does not rely on the equilibrium chemistry condition for solutes in storage zones assumed by standard hyporheic-exchange models. In addition, the first-order reaction in the fADR model can efficiently characterize the mass decline of Mn downstream resulting from enhanced Mn oxidation (such as oxidation of MN(II) to +3 or +4 oxidation states) due to the input of streamflow with increased pH and dissolved oxygen and/or groundwater recharge with high dissolved metals into the hyporheic zone. The decoupled super-diffusion and retention for Mn can exhibit scale-dependent behaviors due to the evolution of driving mechanisms, which can be characterized by the parsimonious, phenomenological FDE by adjusting the indexes. Therefore, the application of FDEs helps us to interpret the physical and geochemical processes in streams across scales.

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References

  1. R. Metzler, J. Klafter, Phys. Rev. E 339, 1 (2000)

    Google Scholar 

  2. R. Metzler, J. Klafter, J. Phys. 37, R161 (2004)

    ADS  Google Scholar 

  3. Y. Pachepsky, D.A. Benson, W. Rawls, Soil Sci. Soc. Am. J. 64, 1234 (2000)

    Article  ADS  Google Scholar 

  4. H.W. Zhou, S. Yang, S.Q. Zhang, Appl. Math. Model. 68, 603 (2019)

    Article  MathSciNet  Google Scholar 

  5. A. Chang, H.S. Sun, C. Zheng, B. Lu, C. Lu, R. Ma, Y. Zhang, Physica A 502, 356 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  6. D.N. Bradley, G.E. Tucker, D.A. Benson, J. Geophys. Res. 115, F00A09 (2010)

    Article  ADS  Google Scholar 

  7. Y. Zhang, R.L. Martin, D. Chen, B. Baeumer, H.G. Sun, L. Chen, J. Geophys. Res. Earth Surf. 119, 2711 (2014)

    Article  ADS  Google Scholar 

  8. S. Kundu, Physica A 506, 135 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  9. Y. Zhang, D.A. Benson, Geophys. Res. Lett. 35, L07403 (2008)

    ADS  Google Scholar 

  10. A. Atangana, D. Baleanu, J. Eng. Mech. 143, D4016005 (2017)

    Article  Google Scholar 

  11. T. Tu, A. Ercan, M.L. Kavvas, Hydrol. Process. 32, 1406 (2018)

    Article  ADS  Google Scholar 

  12. B.Q. Lu, X.T. Liu, P. Dong, G.R. Tick, C.M. Zheng, Y. Zhang, M. Mahmood-UI-Hassan, H. Bai, E. Lamy, submitted to Appl. Math. Model. (2019)

  13. Y. Zhang, D.A. Benson, D.M. Reeves, Adv. Water Resour. 32, 561 (2009)

    Article  ADS  Google Scholar 

  14. Y. Zhang, H.G. Sun, H.H. Stowell, M. Zayernouri, S.E. Hansen, Chaos Solitons Fractals 102, 29 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  15. Y. Zhang, D.M. Reeves, K. Pohlmann, J.B. Chapman, C.E. Russell, Cent. Eur. J. Phys. 11, 634 (2013)

    Google Scholar 

  16. M. Czikkely, E. Neubauer, I. Fekete, P. Ymeri, C. Fogarassy, Water 10, 1377 (2018)

    Article  Google Scholar 

  17. P. Govind, S. Madhuri, Res. J. Animal Vet. Fish. Sci. 2, 17 (2014)

    Google Scholar 

  18. K. Hudson-Edwards, Science 352, 288 (2016)

    Article  ADS  Google Scholar 

  19. A. Buccolieri, G. Buccolieri, A. Dell’Atti, M. Perrone, A. Turnone, Ann. Chim. 96, 167 (2006)

    Article  Google Scholar 

  20. R. Garrett, Hum. Ecol. Risk Assess. 6, 945 (2010)

    Article  Google Scholar 

  21. F. Boano, J.W. Harvey, A. Marion, A.I. Packman, R. Revelli, L. Ridolfi, A. Wörman, Rev. Geophys. 52, 603 (2006)

    Article  ADS  Google Scholar 

  22. C.J. Gandy, J.W.N. Smith, A.P. Jarvis, Sci. Total Environ. 373, 435 (2007)

    Article  ADS  Google Scholar 

  23. C.C. Fuller, J.W. Harvey, Environ. Sci. Technol. 34, 1150 (2000)

    Article  ADS  Google Scholar 

  24. F.J. Triska, J.H. Duff, R.J. Avanzino, Hydrobiologia 251, 167 (1993)

    Article  Google Scholar 

  25. G.F. Birch, S.E. Taylor, C. Matthai, Environ. Pollut. 113, 357 (2001)

    Article  Google Scholar 

  26. B. Palumbo-Roe, J. Wragg, V.J. Banks, J. Soils Sed. 12, 1633 (2012)

    Article  Google Scholar 

  27. R.P. Schwarzenbach, P.M. Gschwend, D.M. Imboden, Environmental Organic Chemistry (John Wiley, New York, 2002)

  28. K.E. Bencala, R. Walters, Water Resour. Res. 19, 718 (1983)

    Article  ADS  Google Scholar 

  29. B.A. Kimball, R.E. Broshears, K.E. Bencala, D.M. McKnight, Environ. Sci. Technol. 28, 2065 (1994)

    Article  ADS  Google Scholar 

  30. A. Wörman, Water Resour. Res. 34, 2703 (1998)

    Article  ADS  Google Scholar 

  31. R.L. Runkel, K.E. Bencala, R.E. Broshears, S.C. Chapra, Water Resour. Res. 32, 409 (1996)

    Article  ADS  Google Scholar 

  32. R.L. Runkel, D.M. McKnight, K.E. Bencala, S.C. Chapra, Water Resour. Res. 32, 419 (1996)

    Article  ADS  Google Scholar 

  33. R.L. Runkel, B.A. Kimball, D.M. McKnight, K.E. Bencala, Water Resour. Res. 35, 3829 (1999)

    Article  ADS  Google Scholar 

  34. M.F. Horstemeyer, Practical Aspects of Computational Chemistry, Vol. 87 (Springer, Dordrecht, 2009)

  35. M. Karttunen, SIAM Conference on Computational Science and Engineering, 27 February -- 3 March 2017 Atlanta, Georgia, USA CSE17 (SIAM, 2017)

  36. M. Andersson, J. Yuan, B. Sunden, Appl. Energ. 87, 1461 (2010)

    Article  Google Scholar 

  37. C. Frayret, A. Villesuzanne, M. Pouchard, S. Matar, Int. J. Quan. Chem. 101, 826 (2005)

    Article  Google Scholar 

  38. N. Asproulis, M. Kalweit, D. Drikakis, 2nd Micro and Nano Flows Conference West London, UK MNF 2009, edited by T. Karayannis, M. Colling (Brunel University, London, 2009)

  39. K.A. Williams, S. Saini, T.M. Wick, Biotechnol. Progr. 18, 951 (2002)

    Article  Google Scholar 

  40. G. Yadigaroglu, Nucl. Eng. Des. 235, 153 (2005)

    Article  Google Scholar 

  41. B. Sanderse, S.P. Pijl, B. Koren, Wind Energy 14, 799 (2011)

    Article  ADS  Google Scholar 

  42. F.J. Higuera, J. Jimenez, Europhys. Lett. 9, 663 (1989)

    Article  ADS  Google Scholar 

  43. S. Succi, The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond (Oxford University Press, 2001) p. 281

  44. Y. Zhang, D.A. Benson, M.M. Meerschaert, E.M. LaBolle, H.P. Scheffler, Phys. Rev. E 74, 026706 (2006)

    Article  ADS  Google Scholar 

  45. Y. Zhang, M.M. Meerschaert, B. Baeumer, E.M. LaBolle, Water Resour. Res. 51, 6311 (2015)

    Article  ADS  Google Scholar 

  46. B.Q. Lu, J. Song, S.Y. Li, G.R. Tick, W. Wei, J.T. Zhu, C.M. Zheng, Y. Zhang, Soil Sci. Soc. Am. J. 82, 1057 (2018)

    Article  ADS  Google Scholar 

  47. J.W. Harvey, C.C. Fuller, Water Resour. Res. 34, 623 (1998)

    Article  ADS  Google Scholar 

  48. H.M. Valett, J.A. Morrice, C.N. Dahm, M.E. Campana, Limnol. Oceanogr. 41, 333 (1996)

    Article  ADS  Google Scholar 

  49. R.G. Storey, K.W.F. Howard, D.D. Williams, Water Resour. Res. 39, 1034 (2003)

    Article  ADS  Google Scholar 

  50. L.S. Balistrieri, J.W. Murray, Geochim. Cosmochim. Acta 50, 2235 (1986)

    Article  ADS  Google Scholar 

  51. J.G. Catts, D.L. Langmuir, Appl. Geochem. 1, 255 (1986)

    Article  Google Scholar 

  52. A. Tessier, D. Fortin, N. Belzile, R.R. DeVitre, G.G. Leppard, Geochim. Cosmochim. Acta 60, 387 (1996)

    Article  ADS  Google Scholar 

  53. J.D. Hem, C.E. Roberson, C.J. Lind, Geochim. Cosmochim. Acta 49, 801 (1985)

    Article  ADS  Google Scholar 

  54. R.L. Runkel, R.E. Broshears, Tech. Rep. 91-01, Centre for Advanced Decision Support in Water and Environmental System, University of Colorado, Boulder (1991)

  55. R.L. Runkel, U.S. Geol, Surv. Water Resour. Invest. Rep. 98, 4018 (1998)

    Google Scholar 

  56. M. Th. van Genuchten, P.J. Wierenga, Soil Sci. Soc. Am. J. 40, 473 (1976)

    Article  Google Scholar 

  57. C. Harvey, S.M. Gorelick, Water Resour. Res. 36, 637 (2000)

    Article  ADS  Google Scholar 

  58. C.E. Feehley, C.M. Zheng, F.J. Molz, Water Resour. Res. 36, 2501 (2000)

    Article  ADS  Google Scholar 

  59. M.L. Brusseau, Z.L. Guo, J. Contam. Hydrol. 208, 17 (2018)

    Article  ADS  Google Scholar 

  60. R. Haggerty, S.M. Gorelick, Water Resour. Res. 31, 2383 (1995)

    Article  ADS  Google Scholar 

  61. M. Zaramella, A. Marion, A.I. Packman, J. Contam. Hydrol. 84, 21 (2006)

    Article  ADS  Google Scholar 

  62. J. Klafter, A. Blumen, M.F. Shlesinger, Phys. Rev. A 35, 3081 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  63. T.H. Solomon, E.R. Weeks, H.L. Swinney, Phys. Rev. Lett. 71, 3975 (1993)

    Article  ADS  Google Scholar 

  64. I. Park, W. Seo II, Adv. Water Resour. 111, 105 (2018)

    Article  ADS  Google Scholar 

  65. Y. Zhang, B. Baeumer, C. Chen, D.M. Reeves, H.G. Sun, Water Resour. Res. 53, 3491 (2017)

    Article  ADS  Google Scholar 

  66. Y. Zhang, M.M. Meerschaert, A.I. Packman, Geophy. Res. Lett. 39, L20404 (2012)

    ADS  Google Scholar 

  67. M.M. Meerschaert, D.A. Benson, B. Baeumer, Phys. Rev. E 63, 021112 (2001)

    Article  ADS  Google Scholar 

  68. B. Baeumer, Y. Zhang, R. Schumer, Ground Water 53, 699 (2015)

    Article  Google Scholar 

  69. H.G. Sun, Y. Zhang, W. Chen, D.M. Reeves, J. Contam. Hydrol. 157, 47 (2014)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Geological Sciences, University of Alabama, 35487, Tuscaloosa, AL, USA

    Mary Hastings Puckett, Yong Zhang, Bingqing Lu & YueHan Lu

  2. State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, Nanjing, China

    HongGuang Sun

  3. School of Environmental Science & Engineering, Southern University of Science and Technology, Shenzhen, 518055, Guangdong, China

    Chunmiao Zheng

  4. Guangdong Provincial Key Laboratory of Soil and Groundwater Pollution Control & Jiangsu Provincial Key Laboratory of Materials Cycling and Pollution Control, Jiangsu Engineering Laboratory of Water and Soil Eco-remediation, School of Environment, Nanjing Normal University, 210023, Nanjing, China

    Wei Wei

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  1. Mary Hastings Puckett
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Correspondence to Yong Zhang.

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Hastings Puckett, M., Zhang, Y., Lu, B. et al. Application of fractional differential equation to interpret the dynamics of dissolved heavy-metal uptake in streams at a wide range of scales. Eur. Phys. J. Plus 134, 377 (2019). https://doi.org/10.1140/epjp/i2019-12897-1

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