Probability and Cumulative Density Function Methods for the Stochastic Advection-Reaction Equation

Abstract

We present a cumulative density function (CDF) method for the probabilistic analysis of $$d$$-dimensional advection-dominated reactive transport in heterogeneous media. We employ a probabilistic approach in which epistemic uncertainty on the spatial heterogeneity of Darcy-scale transport coefficients is modeled in terms of random fields with given correlation structures. Our proposed CDF method employs a modified large-eddy-diffusivity (LED) approach to close and localize the nonlocal equations governing the one-point PDF and CDF of the concentration field, resulting in a $$(d + 1)$$ dimensional PDE. Compared to the classsical LED localization, the proposed modified LED localization explicitly accounts for the mean-field advective dynamics over the phase space of the PDF and CDF. To illustrate the accuracy of the proposed closure, we apply our CDF method to one-dimensional single-species reactive transport with uncertain, heterogeneous advection velocities and reaction rates modeled as random fields.

BibTeX entry

@article{barajassolano-2018-probability,
    title =     { { Probability and Cumulative Density Function Methods for the Stochastic Advection-Reaction Equation } },
    author =    {Barajas-Solano, D. A. and Tartakovsky, A. M.},
    journal =   {SIAM/ASA J. Uncert. Quantif.},
    volume =    {6},
    issue =     {1},
    pages =     {180-212},
    year =      {2018},
    doi =       {10.1137/16M1109163},
}