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A continuous time random walk (CTRW) integro-differential equation with chemical interaction

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Abstract

A nonlocal-in-time integro-differential equation is introduced that accounts for close coupling between transport and chemical reaction terms. The structure of the equation contains these terms in a single convolution with a memory function M (t), which includes the source of non-Fickian (anomalous) behavior, within the framework of a continuous time random walk (CTRW). The interaction is non-linear and second-order, relevant for a bimolecular reaction A + B → C. The interaction term ΓP A  (s, t) P B  (s, t) is symmetric in the concentrations of A and B (i.e. P A and P B ); thus the source terms in the equations for A, B and C are similar, but with a change in sign for that of C. Here, the chemical rate coefficient, Γ, is constant. The fully coupled equations are solved numerically using a finite element method (FEM) with a judicious representation of M (t) that eschews the need for the entire time history, instead using only values at the former time step. To begin to validate the equations, the FEM solution is compared, in lieu of experimental data, to a particle tracking method (CTRW-PT); the results from the two approaches, particularly for the C profiles, are in agreement. The FEM solution, for a range of initial and boundary conditions, can provide a good model for reactive transport in disordered media.

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References

  1. B. Berkowitz, A. Cortis, M. Dentz, H. Scher, Rev. Geophys. 44, RG2003 (2003)

    ADS  Google Scholar 

  2. B. Bijeljic, A.H. Muggeridge, M.J. Blunt, Water Resour. Res. 40, W11501 (2004)

    Article  ADS  Google Scholar 

  3. B. Bijeljic, P. Mostaghimi, M.J. Blunt, Phys. Rev. Lett. 107, 204502 (2011)

    Article  ADS  Google Scholar 

  4. A.E. Scheidegger, An evaluation of the accuracy of the diffusivity equation for describing miscible displacement in porous media, in Proceedings of the Theory of Fluid Flow in Porous Media Conference, University of Oklahoma (1959), pp. 101–116

  5. S.E. Silliman, E.S. Simpson, Water Resour. Res. 23, 1667 (1987)

    Article  ADS  Google Scholar 

  6. B. Berkowitz, J. Klafter, R. Metzler, H. Scher, Water Resour. Res. 38, 1191 (2002)

    Article  ADS  Google Scholar 

  7. M. Dentz, A. Cortis, H. Scher, B. Berkowitz, Adv. Water Res. 27, 155 (2004)

    Article  Google Scholar 

  8. Y. Edery, A. Guadagnini, H. Scher, B. Berkowitz, Adv. Water Res. 51, 86 (2013)

    Article  Google Scholar 

  9. B. Berkowitz, I. Dror, S.K. Hansen, H. Scher, Rev. Geophys. 54, 930 (2016)

    Article  ADS  Google Scholar 

  10. Y. Edery, H. Scher, B. Berkowitz, Water Resour. Res. 46, W07524 (2010)

    Article  ADS  Google Scholar 

  11. Y. Edery, H. Scher, B. Berkowitz, Water Resour. Res. 47, W08535 (2011)

    Article  ADS  Google Scholar 

  12. Y. Edery, I. Dror, H. Scher, B. Berkowitz, Phys. Rev. E 91, 052130 (2015)

    Article  MathSciNet  ADS  Google Scholar 

  13. S. Raveh-Rubin, Y. Edery, I. Dror, B. Berkowitz, Water Resour. Res. 51, 7702 (2015)

    Article  ADS  Google Scholar 

  14. I.M. Sokolov, M.G.W. Schmidt, F. Sagués, Phys. Rev. E 73, 31102 (2006)

    Article  ADS  Google Scholar 

  15. S. Eule, R. Friedrich, F. Jenko, I.M. Sokolov, Phys. Rev. E 78, 060102(R) (2008)

    Article  ADS  Google Scholar 

  16. C.N. Angstmann, I.C. Donnelly, B.I. Henry, Math. Model. Nat. Phenom. 8, 17 (2013)

    Article  MathSciNet  Google Scholar 

  17. S.K. Hansen, B. Berkowitz, Phys. Rev. E 91, 32113 (2015)

    Article  MathSciNet  ADS  Google Scholar 

  18. R. Ben-Zvi, H. Scher, S. Jiang, B. Berkowitz, Transp. Porous Media 115, 239 (2016)

    Article  MathSciNet  Google Scholar 

  19. R. Ben-Zvi, H. Scher, B. Berkowitz, Int. J. Numer. Methods Eng. 112, 459 (2017)

    Article  Google Scholar 

  20. H. Scher, M. Lax, Phys. Rev. B 7, 4502 (1973)

    Article  ADS  Google Scholar 

  21. R. Ben-Zvi, Comput. Struct. 34, 881 (1990)

    Article  Google Scholar 

  22. L. Greengard, J. Strain, Commun. Pure Appl. Math. 43, 949 (1990)

    Article  Google Scholar 

  23. B. Alpert, L. Greengard, T. Hagstrom, SIAM J. Numer. Anal. 37, 1138 (2000)

    Article  MathSciNet  Google Scholar 

  24. S. Jiang, L. Greengard, Comput. Math. Appl. 47, 955 (2004)

    Article  MathSciNet  Google Scholar 

  25. J. Li, SIAM J. Numer. Anal. 31, 4696 (2010)

    Google Scholar 

  26. S. Jiang, L. Greengard, S. Wang, Adv. Comput. Math. 41, 529 (2015)

    Article  MathSciNet  Google Scholar 

  27. A. Cortis, B. Berkowitz, Ground Water 43, 947 (2005)

    Article  Google Scholar 

  28. K. Xu, S. Jiang, J. Sci. Comput. 55, 16 (2013)

    Article  MathSciNet  Google Scholar 

  29. G.M. Porta, M. Riva, A. Guadagnini, Adv. Water Res. 35, 151 (2012)

    Article  Google Scholar 

  30. G.M. Porta, G. Ceriotti, J.-F. Thovert, J. Contam. Hydrol. 185–186, 1 (2016)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Earth and Planetary Sciences, Weizmann Institute of Science, Rehovot, 7610001, Israel

    Rami Ben-Zvi, Alon Nissan, Harvey Scher & Brian Berkowitz

Authors
  1. Rami Ben-Zvi
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  2. Alon Nissan
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  3. Harvey Scher
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  4. Brian Berkowitz
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Corresponding author

Correspondence to Brian Berkowitz.

Additional information

Contribution to the Topical Issue “Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook”, edited by Ryszard Kutner and Jaume Masoliver.

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Ben-Zvi, R., Nissan, A., Scher, H. et al. A continuous time random walk (CTRW) integro-differential equation with chemical interaction. Eur. Phys. J. B 91, 15 (2018). https://doi.org/10.1140/epjb/e2017-80417-8

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  • DOI: https://doi.org/10.1140/epjb/e2017-80417-8

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