Fractional Ensemble Average Governing Equations of Transport by Time-Space Nonstationary Stochastic Fractional Advective Velocity and Fractional Dispersion. II: Numerical Investigation
Publication: Journal of Hydrologic Engineering
Volume 20, Issue 2
Abstract
In this paper, the second in a series of two, the theory developed in the companion paper is applied to transport by stationary and nonstationary stochastic advective flow fields. A numerical solution method is presented for the resulting fractional ensemble average transport equation (fEATE), which describes the evolution of the ensemble average contaminant concentration (EACC). The derived fEATE is evaluated for three different forms: (1) purely advective form of fEATE, (2) moment form of the fractional ensemble average advection-dispersion equation (fEAADE) form of fEATE, and (3) cumulant form of the fractional ensemble average advection-dispersion equation (fEAADE) form of fEATE. The Monte Carlo analysis of the fractional governing equation is then performed in a stochastic flow field, generated by a fractional Brownian motion for the stationary and nonstationary stochastic advection, in order to provide a benchmark for the results obtained from the fEATEs. When compared to the Monte Carlo simulation-based EACCs, the cumulant form of fEAADE gives a good fit in terms of the shape and mode of the ensemble average concentration of the contaminant. Therefore, it is quite promising that the non-Fickian transport behavior can be modeled by the derived fractional ensemble average transport equations either by means of the long memory in the underlying stochastic flow, by means of the time-space nonstationarity of the underlying stochastic flow, or by means of the time and space fractional derivatives of the transport equations.
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Acknowledgments
This work of the first author was supported by the Pukyong National University Research Abroad Fund in 2012(PS-2012-0913).
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Published In
Journal of Hydrologic Engineering
Volume 20 • Issue 2 • February 2015
Copyright
© 2014 American Society of Civil Engineers.
History
Received: Aug 13, 2013
Accepted: Feb 26, 2014
Published online: Feb 28, 2014
Discussion open until: Dec 24, 2014
Published in print: Feb 1, 2015
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