Skip to main content
U.S. flag

An official website of the United States government

Bounded fractional diffusion in geological media: Definition and Lagrangian approximation

November 3, 2016

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.

Publication Year 2016
Title Bounded fractional diffusion in geological media: Definition and Lagrangian approximation
DOI 10.1002/2016WR019178
Authors Yong Zhang, Christopher T. Green, Eric M. LaBolle, Roseanna M. Neupauer, Hong-Guang Sun
Publication Type Article
Publication Subtype Journal Article
Series Title Water Resources Research
Index ID 70177969
Record Source USGS Publications Warehouse
USGS Organization National Research Program - Western Branch; Toxic Substances Hydrology Program