- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Continuous time random walks for the evolution of Lagrangian velocities

Marco Dentz1,2,*, Peter K. Kang3, Alessandro Comolli1,2,4, Tanguy Le Borgne5, and Daniel R. Lester6

  • 1Spanish National Research Council (IDAEA-CSIC), 08034 Barcelona, Spain
  • 2Hydrogeology Group (CSIC-UPC), 08034 Barcelona, Spain
  • 3Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea
  • 4Department of Civil Engineering, Technical University of Catalonia, 08034 Barcelona, Spain
  • 5Université de Rennes 1, CNRS, Geosciences Rennes, UMR No. 6118, Rennes, France
  • 6School of Civil, Environmental and Chemical Engineering, RMIT University, Melbourne, Victoria 3000, Australia

  • *marco.dentz@csic.es
PDF

Phys. Rev. Fluids 1, 074004 – Published 23 November, 2016

DOI: https://doi.org/10.1103/PhysRevFluids.1.074004

Abstract

We develop a continuous time random walk (CTRW) approach for the evolution of Lagrangian velocities in steady heterogeneous flows based on a stochastic relaxation process for the streamwise particle velocities. This approach describes the persistence of velocities over a characteristic spatial scale, unlike classical random walk methods, which model the persistence over a characteristic time scale. We first establish the relation between Eulerian and Lagrangian velocities for both equidistant and isochrone sampling along streamlines, under transient and stationary conditions. Based on this, we develop a space-continuous CTRW approach for the spatial and temporal dynamics of Lagrangian velocities. While classical CTRW formulations have nonstationary Lagrangian velocity statistics, the proposed approach quantifies the evolution of the Lagrangian velocity statistics under both stationary and nonstationary conditions. We provide explicit expressions for the Lagrangian velocity statistics and determine the behaviors of the mean particle velocity, velocity covariance, and particle dispersion. We find strong Lagrangian correlation and anomalous dispersion for velocity distributions that are tailed toward low velocities as well as marked differences depending on the initial conditions. The developed CTRW approach predicts the Lagrangian particle dynamics from an arbitrary initial condition based on the Eulerian velocity distribution and a characteristic correlation scale.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (33)

  1. S. B. Pope, Turbulent Flows (Cambridge University Press, Cambridge, 2000).
  2. D. W. Meyer and F. Saggini, Testing the Markov hypothesis in fluid flows, Phys. Rev. E 93, 053103 (2016).
  3. P. de Anna, T. Le Borgne, M. Dentz, A. M. Tartakovsky, D. Bolster, and P. Davy, Flow Intermittency, Dispersion, and Correlated Continuous Time Random Walks in Porous Media, Phys. Rev. Lett. 110, 184502 (2013).
  4. P. K. Kang, P. de Anna, J. P. Nunes, B. Bijeljic, M. J. Blunt, and R. Juanes, Pore-scale intermittent velocity structure underpinning anomalous transport through 3-D porous media, Geophys. Res. Lett. 41, 6184 (2014).
  5. M. Holzner, V. L. Morales, M. Willmann, and M. Dentz, Intermittent lagrangian velocities and accelerations in three-dimensional porous medium flow, Phys. Rev. E 92, 013015 (2015).
  6. E. W. Montroll and G. H. Weiss, Random walks on lattices. II. J. Math. Phys. 6, 167 (1965).
  7. H. Scher and M. Lax, Stochastic transport in a disordered solid. I. Theory, Phys. Rev. B 7, 4491 (1973).
  8. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339, 1 (2000).
  9. B. Berkowitz, A. Cortis, M Dentz, and H. Scher, Modeling non-Fickian transport in geological formations as a continuous time random walk, Rev. Geophys. 44, RG2003 (2006).
  10. B. Berkowitz and H. Scher, Anomalous Transport in Random Fracture Networks, Phys. Rev. Lett. 79, 4038 (1997).
  11. R. Benke and S. Painter, Modeling conservative tracer transport in fracture networks with a hybrid approach based on the Boltzmann transport equation, Water Resour. Res. 39, 1324 (2003).
  12. S. Painter and V. Cvetkovic, Upscaling discrete fracture network simulations: An alternative to continuum transport models, Water Resour. Res. 41, W02002 (2005).
  13. T. Le Borgne, M. Dentz, and J. Carrera, Spatial Markov processes for modeling Lagrangian particle dynamics in heterogeneous porous media, Phys. Rev. E 78, 026308 (2008).
  14. T. Le Borgne, M. Dentz, and J. Carrera, A Lagrangian Statistical Model for Transport in Highly Heterogeneous Velocity Fields, Phys. Rev. Lett. 101, 090601 (2008).
  15. P. K. Kang, M. Dentz, T. Le Borgne, and R. Juanes, Spatial Markov Model of Anomalous Transport Through Random Lattice Networks, Phys. Rev. Lett. 107, 180602 (2011).
  16. B. Bijeljic, P. Mostaghimi, and M. J. Blunt, Signature of Non-Fickian Solute Transport in Complex Heterogeneous Porous Media, Phys. Rev. Lett. 107, 204502 (2011).
  17. Y. Edery, A. Guadagnini, H. Scher, and B. Berkowitz, Origins of anomalous transport in heterogeneous media: Structural and dynamic control, Water Resour. Res. 50, 1490 (2014).
  18. P. K. Kang, S. Brown, and R. Juanes, Emergence of anomalous transport in stressed rough fractures, Earth Planet. Sci. Lett. 454, 46 (2016).
  19. M. F. Shlesinger, B. J. West, and J. Klafter, Lévy Dynamics of Enhanced Diffusion: Application to Turbulence, Phys. Rev. Lett. 58, 1100 (1987).
  20. S. Thalabard, G. Krstulovic, and J. Bec, Turbulent pair dispersion as a continuous-time random walk, J. Fluid Mech. 755, R4 (2014).
  21. I. M. Sokolov, Models of anomalous diffusion in crowded environments, Soft Matter 8, 9043 (2012).
  22. G. Dagan, Flow and Transport in Porous Formations (Springer, New York, 1989).
  23. J. D. Hyman, S. L. Painter, H. Viswanathan, N. Makedonska, and S. Karra, Influence of injection mode on transport properties in kilometer-scale three-dimensional discrete fracture networks, Water Resour. Res. 51, 7289 (2015).
  24. A. Frampton and V. Cvetkovic, Significance of injection modes and heterogeneity on spatial and temporal dispersion of advecting particles in two-dimensional discrete fracture networks, Adv. Water Resour. 32, 649 (2009).
  25. P. K. Kang, M. Dentz, T. Le Borgne, S. Lee, and R. Juanes (unpublished).
  26. V. D. Cvetkovic, G. Dagan, and A. M. Shapiro, An exact solution of solute transport by one-dimensional random velocity fields, Stoch. Hydrol. Hydraul. 5, 45 (1991).
  27. V. Cvetkovic, H. Cheng, and X.-H. Wen, Analysis of nonlinear effects on tracer migration in heterogeneous aquifers using Lagrangian travel time statistics, Water Resour. Res. 32, 1671 (1996).
  28. A. Baule and R. Friedrich, Joint probability distributions for a class of non-Markovian processes, Phys. Rev. E 71, 026101 (2005).
  29. P. K. Kang, M. Dentz, T. Le Borgne, and R. Juanes, Anomalous transport on regular fracture networks: Impact of conductivity heterogeneity and mixing at fracture intersections, Phys. Rev. E 92, 022148 (2015).
  30. R. Kubo, M Toda, and N. Hashitsume, Statistical Physics II, Non-Equilibrium Statistical Mechanics (Springer, Berlin, 1991).
  31. G. Margolin and B. Berkowitz, Spatial behavior of anomalous transport, Phys. Rev. E 65, 031101 (2002).
  32. M. Dentz, A. Cortis, H. Scher, and B. Berkowitz, Time behavior of solute transport in heterogeneous media: Transition from anomalous to normal transport, Adv. Water Resour. 27, 155 (2004).
  33. A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations (CRC, Boca Raton, 1998).

Outline

Information

Phys. Rev. Fluids 1, 074004– Published 23 November, 2016
  • Received 6 August 2016

©2016 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation