### Abstract

Starting from the premise that Darcy's law applies with a random hydraulic conductivity field K(x) defined on a support ω, we consider steady state flow in a larger domain Ω driven by statistically independent random source and boundary functions. Writing K(x) as the sum of a slowly varying unbiased estimate κ(x) and a zero mean estimation error K′(x), the conditional ensemble moments 〈h(x)〉_{κ}≡〈h(x)|κ(x)〉 and 〈q(x)〉_{κ}≡〈q(x)|κ(x)〉 become conditionally unbiased predictors of the hydraulic head h(x) and Darcy flux q(x), respectively; 〈q(x)〉_{κ} satisfies a standard continuity equation driven by ensemble mean source and boundary functions; and 〈q(x)〉_{κ} = -κ(x)▽〈h(x)〉_{κ}+r_{κ}(x) where r_{κ}(x) is a residual flux. We present an exact formal expression for 〈r(x)〉_{κ} which demonstrates that 〈q(x)〉_{κ} is nonlocal (depends on head gradients at points other than x) with a well-defined kernel, and non-Darcian (there is no effective or equivalent conductivity valid for arbitrary directions of conditional mean flow), except in special cases. In some of these cases the effective conductivity is κ(x), in some it is a symmetric or a nonsymmetric second-rank tensor, and in yet other situations it exists only as a set of directional scalars but not as a tensor. We describe a weak approximation for 〈r(x)〉_{κ} which improves with the quality of the estimate κ(x), and an ancillary approximation for the effective hydraulic conductivity tensor in an infinite domain. We then use high-resolution finite element Monte Carlo simulation to verify aspects of these approximations under uniform 3-D and radial 2-D flows in strongly heterogenous, statistically homogeneous and isotropic porous media.

Original language | English (US) |
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Title of host publication | Finite Elements in Water Resources, Proceedings of the International Conference |

Publisher | Publ by Computational Mechanics Publ |

Pages | 117-136 |

Number of pages | 20 |

Volume | 2 |

State | Published - 1992 |

Event | Proceedings of the 9th International Conference on Computational Methods in Water Resources - Denver, CO, USA Duration: Jun 1 1992 → Jun 1 1992 |

### Other

Other | Proceedings of the 9th International Conference on Computational Methods in Water Resources |
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City | Denver, CO, USA |

Period | 6/1/92 → 6/1/92 |

### Fingerprint

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*Finite Elements in Water Resources, Proceedings of the International Conference*(Vol. 2, pp. 117-136). Publ by Computational Mechanics Publ.

**Theory and high-resolution finite element analysis of 2-D and 3-D effective permeabilities in strongly heterogeneous porous media.** / Neuman, Shlomo P; Orr, S.; Levin, O.; Paleologos, E.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Finite Elements in Water Resources, Proceedings of the International Conference.*vol. 2, Publ by Computational Mechanics Publ, pp. 117-136, Proceedings of the 9th International Conference on Computational Methods in Water Resources, Denver, CO, USA, 6/1/92.

}

TY - GEN

T1 - Theory and high-resolution finite element analysis of 2-D and 3-D effective permeabilities in strongly heterogeneous porous media

AU - Neuman, Shlomo P

AU - Orr, S.

AU - Levin, O.

AU - Paleologos, E.

PY - 1992

Y1 - 1992

N2 - Starting from the premise that Darcy's law applies with a random hydraulic conductivity field K(x) defined on a support ω, we consider steady state flow in a larger domain Ω driven by statistically independent random source and boundary functions. Writing K(x) as the sum of a slowly varying unbiased estimate κ(x) and a zero mean estimation error K′(x), the conditional ensemble moments 〈h(x)〉κ≡〈h(x)|κ(x)〉 and 〈q(x)〉κ≡〈q(x)|κ(x)〉 become conditionally unbiased predictors of the hydraulic head h(x) and Darcy flux q(x), respectively; 〈q(x)〉κ satisfies a standard continuity equation driven by ensemble mean source and boundary functions; and 〈q(x)〉κ = -κ(x)▽〈h(x)〉κ+rκ(x) where rκ(x) is a residual flux. We present an exact formal expression for 〈r(x)〉κ which demonstrates that 〈q(x)〉κ is nonlocal (depends on head gradients at points other than x) with a well-defined kernel, and non-Darcian (there is no effective or equivalent conductivity valid for arbitrary directions of conditional mean flow), except in special cases. In some of these cases the effective conductivity is κ(x), in some it is a symmetric or a nonsymmetric second-rank tensor, and in yet other situations it exists only as a set of directional scalars but not as a tensor. We describe a weak approximation for 〈r(x)〉κ which improves with the quality of the estimate κ(x), and an ancillary approximation for the effective hydraulic conductivity tensor in an infinite domain. We then use high-resolution finite element Monte Carlo simulation to verify aspects of these approximations under uniform 3-D and radial 2-D flows in strongly heterogenous, statistically homogeneous and isotropic porous media.

AB - Starting from the premise that Darcy's law applies with a random hydraulic conductivity field K(x) defined on a support ω, we consider steady state flow in a larger domain Ω driven by statistically independent random source and boundary functions. Writing K(x) as the sum of a slowly varying unbiased estimate κ(x) and a zero mean estimation error K′(x), the conditional ensemble moments 〈h(x)〉κ≡〈h(x)|κ(x)〉 and 〈q(x)〉κ≡〈q(x)|κ(x)〉 become conditionally unbiased predictors of the hydraulic head h(x) and Darcy flux q(x), respectively; 〈q(x)〉κ satisfies a standard continuity equation driven by ensemble mean source and boundary functions; and 〈q(x)〉κ = -κ(x)▽〈h(x)〉κ+rκ(x) where rκ(x) is a residual flux. We present an exact formal expression for 〈r(x)〉κ which demonstrates that 〈q(x)〉κ is nonlocal (depends on head gradients at points other than x) with a well-defined kernel, and non-Darcian (there is no effective or equivalent conductivity valid for arbitrary directions of conditional mean flow), except in special cases. In some of these cases the effective conductivity is κ(x), in some it is a symmetric or a nonsymmetric second-rank tensor, and in yet other situations it exists only as a set of directional scalars but not as a tensor. We describe a weak approximation for 〈r(x)〉κ which improves with the quality of the estimate κ(x), and an ancillary approximation for the effective hydraulic conductivity tensor in an infinite domain. We then use high-resolution finite element Monte Carlo simulation to verify aspects of these approximations under uniform 3-D and radial 2-D flows in strongly heterogenous, statistically homogeneous and isotropic porous media.

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M3 - Conference contribution

VL - 2

SP - 117

EP - 136

BT - Finite Elements in Water Resources, Proceedings of the International Conference

PB - Publ by Computational Mechanics Publ

ER -