### Abstract

Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1238-1255) developed exact first and second nonlocal moment equations for advective-dispersive transport in finite, randomly heterogeneous geologic media. The velocity and concentration in these equations are generally nonstationary due to trends in heterogeneity, conditioning on site data and the influence of forcing terms. Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) solved the Laplace transformed versions of these equations recursively to second order in the standard deviation σ_{y} of (natural) log hydraulic conductivity, and iteratively to higher-order, by finite elements followed by numerical inversion of the Laplace transform. They did the same for a space-localized version of the mean transport equation. Here we recount briefly their theory and algorithms; compare the numerical performance of the Laplace-transform finite element scheme with that of a high-accuracy ULTIMATE-QUICKEST algorithm coupled with an alternating split operator approach; and review some computational results due to Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) to shed light on the accuracy and computational efficiency of their recursive and iterative solutions in comparison to conditional Monte Carlo simulations in two spatial dimensions.

Original language | English (US) |
---|---|

Pages (from-to) | 131-161 |

Number of pages | 31 |

Journal | Communications in Computational Physics |

Volume | 6 |

Issue number | 1 |

State | Published - Jul 2009 |

### Fingerprint

### Keywords

- Conditioning
- Finite elements
- Geologic media
- Laplace transform
- Localization
- Moment equations
- Nonlocality
- Porous media
- Random heterogeneity
- Stochastic transport
- Ultimate-quickest

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

*Communications in Computational Physics*,

*6*(1), 131-161.

**Laplace-transform finite element solution of nonlocal and localized stochastic moment equations of transport.** / Morales-Casique, Eric; Neuman, Shlomo P.

Research output: Contribution to journal › Article

*Communications in Computational Physics*, vol. 6, no. 1, pp. 131-161.

}

TY - JOUR

T1 - Laplace-transform finite element solution of nonlocal and localized stochastic moment equations of transport

AU - Morales-Casique, Eric

AU - Neuman, Shlomo P

PY - 2009/7

Y1 - 2009/7

N2 - Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1238-1255) developed exact first and second nonlocal moment equations for advective-dispersive transport in finite, randomly heterogeneous geologic media. The velocity and concentration in these equations are generally nonstationary due to trends in heterogeneity, conditioning on site data and the influence of forcing terms. Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) solved the Laplace transformed versions of these equations recursively to second order in the standard deviation σy of (natural) log hydraulic conductivity, and iteratively to higher-order, by finite elements followed by numerical inversion of the Laplace transform. They did the same for a space-localized version of the mean transport equation. Here we recount briefly their theory and algorithms; compare the numerical performance of the Laplace-transform finite element scheme with that of a high-accuracy ULTIMATE-QUICKEST algorithm coupled with an alternating split operator approach; and review some computational results due to Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) to shed light on the accuracy and computational efficiency of their recursive and iterative solutions in comparison to conditional Monte Carlo simulations in two spatial dimensions.

AB - Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1238-1255) developed exact first and second nonlocal moment equations for advective-dispersive transport in finite, randomly heterogeneous geologic media. The velocity and concentration in these equations are generally nonstationary due to trends in heterogeneity, conditioning on site data and the influence of forcing terms. Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) solved the Laplace transformed versions of these equations recursively to second order in the standard deviation σy of (natural) log hydraulic conductivity, and iteratively to higher-order, by finite elements followed by numerical inversion of the Laplace transform. They did the same for a space-localized version of the mean transport equation. Here we recount briefly their theory and algorithms; compare the numerical performance of the Laplace-transform finite element scheme with that of a high-accuracy ULTIMATE-QUICKEST algorithm coupled with an alternating split operator approach; and review some computational results due to Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) to shed light on the accuracy and computational efficiency of their recursive and iterative solutions in comparison to conditional Monte Carlo simulations in two spatial dimensions.

KW - Conditioning

KW - Finite elements

KW - Geologic media

KW - Laplace transform

KW - Localization

KW - Moment equations

KW - Nonlocality

KW - Porous media

KW - Random heterogeneity

KW - Stochastic transport

KW - Ultimate-quickest

UR - http://www.scopus.com/inward/record.url?scp=67650983044&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67650983044&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:67650983044

VL - 6

SP - 131

EP - 161

JO - Communications in Computational Physics

JF - Communications in Computational Physics

SN - 1815-2406

IS - 1

ER -