# 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.

- Authors:

- Publication Date:

- Research Org.:
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1427894

- Report Number(s):
- PNNL-SA-123069

Journal ID: ISSN 2166-2525; KJ0401000

- DOE Contract Number:
- AC05-76RL01830

- Resource Type:
- Journal Article

- Journal Name:
- SIAM/ASA Journal on Uncertainty Quantification

- Additional Journal Information:
- Journal Volume: 6; Journal Issue: 1; Journal ID: ISSN 2166-2525

- Publisher:
- SIAM

- Country of Publication:
- United States

- Language:
- English

- Subject:
- Contaminant transport; advection-reaction equation; stochastic partial differential equations; PDF methods; LED closures; stochastic advection-reaction equations

### Citation Formats

```
Barajas-Solano, David A., and Tartakovsky, Alexandre M.
```*Probability and Cumulative Density Function Methods for the Stochastic Advection-Reaction Equation*. United States: N. p., 2018.
Web. doi:10.1137/16M1109163.

```
Barajas-Solano, David A., & Tartakovsky, Alexandre M.
```*Probability and Cumulative Density Function Methods for the Stochastic Advection-Reaction Equation*. United States. doi:10.1137/16M1109163.

```
Barajas-Solano, David A., and Tartakovsky, Alexandre M. Mon .
"Probability and Cumulative Density Function Methods for the Stochastic Advection-Reaction Equation". United States. doi:10.1137/16M1109163.
```

```
@article{osti_1427894,
```

title = {Probability and Cumulative Density Function Methods for the Stochastic Advection-Reaction Equation},

author = {Barajas-Solano, David A. and Tartakovsky, Alexandre M.},

abstractNote = {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.},

doi = {10.1137/16M1109163},

journal = {SIAM/ASA Journal on Uncertainty Quantification},

issn = {2166-2525},

number = 1,

volume = 6,

place = {United States},

year = {2018},

month = {1}

}