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FracFit: A robust parameter estimation tool for fractional calculus models
Author(s) -
Kelly James F.,
Bolster Diogo,
Meerschaert Mark M.,
Drummond Jennifer D.,
Packman Aaron I.
Publication year - 2017
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1002/2016wr019748
Subject(s) - fractional calculus , mathematics , dispersion (optics) , estimation theory , fractional brownian motion , mathematical analysis , statistics , physics , brownian motion , optics
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.