Anomalous transport in disordered fracture networks: Spatial Markov model for dispersion with variable injection modes

Peter K. Kang, Marco Dentz, Tanguy Le Borgne, Seunghak Lee, Ruben Juanes

Research output: Contribution to journalArticlepeer-review

59 Scopus citations


We investigate tracer transport on random discrete fracture networks that are characterized by the statistics of the fracture geometry and hydraulic conductivity. While it is well known that tracer transport through fractured media can be anomalous and particle injection modes can have major impact on dispersion, the incorporation of injection modes into effective transport modeling has remained an open issue. The fundamental reason behind this challenge is that—even if the Eulerian fluid velocity is steady—the Lagrangian velocity distribution experienced by tracer particles evolves with time from its initial distribution, which is dictated by the injection mode, to a stationary velocity distribution. We quantify this evolution by a Markov model for particle velocities that are equidistantly sampled along trajectories. This stochastic approach allows for the systematic incorporation of the initial velocity distribution and quantifies the interplay between velocity distribution and spatial and temporal correlation. The proposed spatial Markov model is characterized by the initial velocity distribution, which is determined by the particle injection mode, the stationary Lagrangian velocity distribution, which is derived from the Eulerian velocity distribution, and the spatial velocity correlation length, which is related to the characteristic fracture length. This effective model leads to a time-domain random walk for the evolution of particle positions and velocities, whose joint distribution follows a Boltzmann equation. Finally, we demonstrate that the proposed model can successfully predict anomalous transport through discrete fracture networks with different levels of heterogeneity and arbitrary tracer injection modes.

Original languageEnglish (US)
Pages (from-to)80-94
Number of pages15
JournalAdvances in Water Resources
StatePublished - Aug 2017

Bibliographical note

Funding Information:
PKK and SL acknowledge a grant (16AWMP-B066761-04) from the AWMP Program funded by the Ministry of Land, Infrastructure and Transport of the Korean government and the support from Future Research Program (2E27030) funded by the Korea Institute of Science and Technology (KIST). PKK and RJ acknowledge a MISTI Global Seed Funds award. MD acknowledges the support of the European Research Council (ERC) through the project MHetScale (617511). TLB acknowledges the support of European Research Council (ERC) through the project ReactiveFronts (648377). RJ acknowledges the support of the US Department of Energy through a DOE Early Career Award (grant DE-SC0009286). The data to reproduce the work can be obtained from the corresponding author.

Publisher Copyright:
© 2017 Elsevier Ltd


  • Anomalous transport
  • Continuous time random walks
  • Discrete fracture networks
  • Injection modes
  • Lagrangian velocity
  • Spatial Markov model
  • Stochastic modeling
  • Time domain random walks


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