### Abstract

Second-order expressions are derived for the mean and covariance of steady state seepage velocity under mean uniform flows in infinite two- and three-dimensional domains. The order of approximation is defined in terms of the variance σ^{2} of a statistically homogeneous and anisotropic natural log hydraulic conductivity field Y with a Gaussian spatial autocorrelation function. Results show that second-order mean velocity either exceeds or is close to its first-order counterpart, depending on anisotropy. Head fluctuations of order larger than σ affect second-order velocity moments to the same extent as do head fluctuations of order σ in virtually all cases, hence neglecting the former renders the results nonasymptotic. Velocity variances are generally larger when approximated consistently to second than to first order. The ratio between second- and first-order variance approximations is larger in three than in two dimensions, larger for transverse than for longitudinal velocity, and increases with σ^{2}. Anisotropy has a significant effect on second-order velocity variance. Second-order effects have the greatest influence on longitudinal velocity variance at extreme anisotropy ratios and on transverse velocity variance in isotropic domains.

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

Pages (from-to) | 625-637 |

Number of pages | 13 |

Journal | Water Resources Research |

Volume | 33 |

Issue number | 4 |

State | Published - Apr 1997 |

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### ASJC Scopus subject areas

- Aquatic Science
- Environmental Science(all)
- Environmental Chemistry
- Water Science and Technology

### Cite this

*Water Resources Research*,

*33*(4), 625-637.

**Second-order expressions for velocity moments in two- and three-dimensional statistically anisotropic media.** / Hsu, Kuo Chin; Neuman, Shlomo P.

Research output: Contribution to journal › Article

*Water Resources Research*, vol. 33, no. 4, pp. 625-637.

}

TY - JOUR

T1 - Second-order expressions for velocity moments in two- and three-dimensional statistically anisotropic media

AU - Hsu, Kuo Chin

AU - Neuman, Shlomo P

PY - 1997/4

Y1 - 1997/4

N2 - Second-order expressions are derived for the mean and covariance of steady state seepage velocity under mean uniform flows in infinite two- and three-dimensional domains. The order of approximation is defined in terms of the variance σ2 of a statistically homogeneous and anisotropic natural log hydraulic conductivity field Y with a Gaussian spatial autocorrelation function. Results show that second-order mean velocity either exceeds or is close to its first-order counterpart, depending on anisotropy. Head fluctuations of order larger than σ affect second-order velocity moments to the same extent as do head fluctuations of order σ in virtually all cases, hence neglecting the former renders the results nonasymptotic. Velocity variances are generally larger when approximated consistently to second than to first order. The ratio between second- and first-order variance approximations is larger in three than in two dimensions, larger for transverse than for longitudinal velocity, and increases with σ2. Anisotropy has a significant effect on second-order velocity variance. Second-order effects have the greatest influence on longitudinal velocity variance at extreme anisotropy ratios and on transverse velocity variance in isotropic domains.

AB - Second-order expressions are derived for the mean and covariance of steady state seepage velocity under mean uniform flows in infinite two- and three-dimensional domains. The order of approximation is defined in terms of the variance σ2 of a statistically homogeneous and anisotropic natural log hydraulic conductivity field Y with a Gaussian spatial autocorrelation function. Results show that second-order mean velocity either exceeds or is close to its first-order counterpart, depending on anisotropy. Head fluctuations of order larger than σ affect second-order velocity moments to the same extent as do head fluctuations of order σ in virtually all cases, hence neglecting the former renders the results nonasymptotic. Velocity variances are generally larger when approximated consistently to second than to first order. The ratio between second- and first-order variance approximations is larger in three than in two dimensions, larger for transverse than for longitudinal velocity, and increases with σ2. Anisotropy has a significant effect on second-order velocity variance. Second-order effects have the greatest influence on longitudinal velocity variance at extreme anisotropy ratios and on transverse velocity variance in isotropic domains.

UR - http://www.scopus.com/inward/record.url?scp=0030945433&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030945433&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0030945433

VL - 33

SP - 625

EP - 637

JO - Water Resources Research

JF - Water Resources Research

SN - 0043-1397

IS - 4

ER -