FracFit: A robust parameter estimation tool for fractional calculus models

James F. Kelly*, Diogo Bolster, Mark M. Meerschaert, Jennifer D. Drummond, Aaron I. Packman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

38 Scopus citations


Anomalous transport cannot be adequately described with classical Fickian advection-dispersion equations (ADE) with constant coefficients. Rather, fractional calculus models may be used, which capture salient features of anomalous transport (e.g., skewness and power law tails). FracFit is a parameter estimation tool based on space-fractional and time-fractional models used by the hydrology community. Currently, four fractional models are supported: (1) space-fractional advection-dispersion equation (sFADE), (2) time-fractional dispersion equation with drift (TFDE), (3) fractional mobile-immobile (FMIM) equation, and (4) temporally tempered Lévy motion (TTLM). Model solutions using pulse initial conditions and continuous injections are evaluated using stable distributions or subordination integrals. Parameter estimates are extracted from measured breakthrough curves (BTCs) using a weighted nonlinear least squares (WNLS) algorithm. Optimal weights for BTCs for pulse initial conditions and continuous injections are presented. Two sample applications are analyzed: (1) pulse injection BTCs in the Selke River and (2) continuous injection laboratory experiments using natural organic matter. Model parameters are compared across models and goodness-of-fit metrics are presented, facilitating model evaluation.

Original languageEnglish (US)
Pages (from-to)2559-2567
Number of pages9
JournalWater Resources Research
Issue number3
StatePublished - Mar 1 2017


  • CTRW
  • fractional derivative
  • parameter estimation

ASJC Scopus subject areas

  • Water Science and Technology


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