FracFit: A Robust Parameter Estimation Tool for Fractional Calculus Models

Abstract

Anomalous transport cannot be adequately described with clas- sical Fickian advection-dispersion equations (ADE) with constant coefficients. Rather, fractional calculus models may be used, which capture salient fea- tures of anomalous transport (e.g. skewness and power-law tails). FracFit is a parameter estimation tool based on space- and time-fractional models used by the hydrology community. Currently, four fractional models are sup- ported: 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 L evy motion (TTLM). Model solutions using pulse initial conditions and continuous injections are eval- uated using stable distributions or subordination integrals. Parameter esti- mates 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 sam- ple applications are analyzed: 1)pulse injection BTCs in the Selke river and 2) continuous injection laboratory experiments using natural organic mat- ter. Model parameters are compared across models and goodness-of- t met- rics are presented, facilitating model evaluation.