Abstract
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 language | English (US) |
|---|---|
| Pages (from-to) | 2559-2567 |
| Number of pages | 9 |
| Journal | Water Resources Research |
| Volume | 53 |
| Issue number | 3 |
| DOIs | |
| State | Published - Mar 1 2017 |
| Externally published | Yes |
Funding
Kelly was partially supported by ARO MURI grant W911NF-15-1-0562 and NSF grant EAR-1344280. Meerschaert was partially supported by ARO MURI grant W911NF-15-1-0562 and NSF grants DMS-1462156 and EAR-1344280. Bolster was partially supported by NSF grants EAR-1351625 and EAR-1417264. Packman was supported by NSF grant EAR-1344280 and ARO grant W911NF-15-1-0569. John Nolan (Department of Mathematics and Statistics, American University, Washington, DC) graciously provided the STABLE toolbox (www.RobustAnalysis.com). We acknowledge Noah Schmadel and Adam S. Ward (Department of Environmental Engineering, Indiana University) for providing the Selke River data. Financial support for the Selke experiment was provided by The Leverhulme Trust through the project “Where rivers, groundwater and disciplines meet: A hyporheic research network.” Insightful comments by Yong Zhang, Department of Geological Sciences, University of Alabama, are also acknowledged. The Selke River data are available from Adam S. Ward ([email protected]). The NOM data are available from Bolster ([email protected]).
Keywords
- CTRW
- fractional derivative
- parameter estimation
ASJC Scopus subject areas
- Water Science and Technology