Bounded fractional diffusion in geological media: Definition and Lagrangian approximation

Water Resources Research
By: , and 



Spatiotemporal Fractional-Derivative Models (FDMs) have been increasingly used to simulate non-Fickian diffusion, but methods have not been available to define boundary conditions for FDMs in bounded domains. This study defines boundary conditions and then develops a Lagrangian solver to approximate bounded, one-dimensional fractional diffusion. Both the zero-value and non-zero-value Dirichlet, Neumann, and mixed Robin boundary conditions are defined, where the sign of Riemann-Liouville fractional derivative (capturing non-zero-value spatial-nonlocal boundary conditions with directional super-diffusion) remains consistent with the sign of the fractional-diffusive flux term in the FDMs. New Lagrangian schemes are then proposed to track solute particles moving in bounded domains, where the solutions are checked against analytical or Eularian solutions available for simplified FDMs. Numerical experiments show that the particle-tracking algorithm for non-Fickian diffusion differs from Fickian diffusion in relocating the particle position around the reflective boundary, likely due to the non-local and non-symmetric fractional diffusion. For a non-zero-value Neumann or Robin boundary, a source cell with a reflective face can be applied to define the release rate of random-walking particles at the specified flux boundary. Mathematical definitions of physically meaningful nonlocal boundaries combined with bounded Lagrangian solvers in this study may provide the only viable techniques at present to quantify the impact of boundaries on anomalous diffusion, expanding the applicability of FDMs from infinite do mains to those with any size and boundary conditions.

Additional publication details

Publication type Article
Publication Subtype Journal Article
Title Bounded fractional diffusion in geological media: Definition and Lagrangian approximation
Series title Water Resources Research
DOI 10.1002/2016WR019178
Volume 52
Issue 11
Year Published 2016
Language English
Publisher American Geophysical Union
Publisher location Washington, D.C.
Contributing office(s) Toxic Substances Hydrology Program, National Research Program - Western Branch
Description 17 p.
First page 8561
Last page 8577