### Abstract

We consider steady state unsaturated flow in bounded, randomly heterogeneous soils under the influence of random boundary and source terms. Our aim is to predict pressure heads and fluxes without resorting to Monte Carlo simulation, upscaling, or linearization of the constitutive relationship between unsaturated hydraulic conductivity and pressure head. We represent this relationship through Gardner's exponential model while treating its exponent α as a random constant and saturated hydraulic conductivity Ks as a spatially correlated random field. We linearize the steady state unsaturated flow equations by means of the Kirchhoff transformation and integrate them in probability space to obtain exact integro-differential equations for the conditional mean and variance-covariance of transformed pressure head and flux. After approximating these equations recursively to second order in the standard deviation σ_{Y} of Y = ln K_{s}, we solve them by finite elements for superimposed mean uniform and divergent flows in the vertical plane, with and without conditioning on measured Y values. Comparison with Monte Carlo solutions demonstrates that whereas our nonlocal solution is nominally restricted to mildly nonuniform media with σ_{Y}^{2} ≪ 1, it yields remarkably accurate results for strongly nonuniform media with σ_{Y}^{2} at least as large as 2. This accords well with a previous theoretical analysis, which shows that the solution may remain asymptotic for values of σ_{Y}^{2} as large as 2.

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

Pages (from-to) | 91-916 |

Number of pages | 826 |

Journal | Water Resources Research |

Volume | 38 |

Issue number | 4 |

State | Published - Apr 2002 |

### Fingerprint

### Keywords

- Conditioning
- Heterogeneity
- Randomness
- Uncertainty
- Unsaturated flow

### ASJC Scopus subject areas

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

### Cite this

*Water Resources Research*,

*38*(4), 91-916.

**Conditional moment analysis of steady state unsaturated flow in bounded, randomly heterogeneous soils.** / Lu, Zhiming; Neuman, Shlomo P; Guadagnini, Alberto; Tartakovsky, Daniel M.

Research output: Contribution to journal › Article

*Water Resources Research*, vol. 38, no. 4, pp. 91-916.

}

TY - JOUR

T1 - Conditional moment analysis of steady state unsaturated flow in bounded, randomly heterogeneous soils

AU - Lu, Zhiming

AU - Neuman, Shlomo P

AU - Guadagnini, Alberto

AU - Tartakovsky, Daniel M.

PY - 2002/4

Y1 - 2002/4

N2 - We consider steady state unsaturated flow in bounded, randomly heterogeneous soils under the influence of random boundary and source terms. Our aim is to predict pressure heads and fluxes without resorting to Monte Carlo simulation, upscaling, or linearization of the constitutive relationship between unsaturated hydraulic conductivity and pressure head. We represent this relationship through Gardner's exponential model while treating its exponent α as a random constant and saturated hydraulic conductivity Ks as a spatially correlated random field. We linearize the steady state unsaturated flow equations by means of the Kirchhoff transformation and integrate them in probability space to obtain exact integro-differential equations for the conditional mean and variance-covariance of transformed pressure head and flux. After approximating these equations recursively to second order in the standard deviation σY of Y = ln Ks, we solve them by finite elements for superimposed mean uniform and divergent flows in the vertical plane, with and without conditioning on measured Y values. Comparison with Monte Carlo solutions demonstrates that whereas our nonlocal solution is nominally restricted to mildly nonuniform media with σY2 ≪ 1, it yields remarkably accurate results for strongly nonuniform media with σY2 at least as large as 2. This accords well with a previous theoretical analysis, which shows that the solution may remain asymptotic for values of σY2 as large as 2.

AB - We consider steady state unsaturated flow in bounded, randomly heterogeneous soils under the influence of random boundary and source terms. Our aim is to predict pressure heads and fluxes without resorting to Monte Carlo simulation, upscaling, or linearization of the constitutive relationship between unsaturated hydraulic conductivity and pressure head. We represent this relationship through Gardner's exponential model while treating its exponent α as a random constant and saturated hydraulic conductivity Ks as a spatially correlated random field. We linearize the steady state unsaturated flow equations by means of the Kirchhoff transformation and integrate them in probability space to obtain exact integro-differential equations for the conditional mean and variance-covariance of transformed pressure head and flux. After approximating these equations recursively to second order in the standard deviation σY of Y = ln Ks, we solve them by finite elements for superimposed mean uniform and divergent flows in the vertical plane, with and without conditioning on measured Y values. Comparison with Monte Carlo solutions demonstrates that whereas our nonlocal solution is nominally restricted to mildly nonuniform media with σY2 ≪ 1, it yields remarkably accurate results for strongly nonuniform media with σY2 at least as large as 2. This accords well with a previous theoretical analysis, which shows that the solution may remain asymptotic for values of σY2 as large as 2.

KW - Conditioning

KW - Heterogeneity

KW - Randomness

KW - Uncertainty

KW - Unsaturated flow

UR - http://www.scopus.com/inward/record.url?scp=0036547497&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036547497&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0036547497

VL - 38

SP - 91

EP - 916

JO - Water Resources Research

JF - Water Resources Research

SN - 0043-1397

IS - 4

ER -