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

Shlomo P Neuman, S. Orr, O. Levin, E. Paleologos

Research output: Chapter in Book/Report/Conference proceedingConference contribution

23 Citations (Scopus)

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+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.

Original languageEnglish (US)
Title of host publicationFinite Elements in Water Resources, Proceedings of the International Conference
PublisherPubl by Computational Mechanics Publ
Pages117-136
Number of pages20
Volume2
StatePublished - 1992
EventProceedings of the 9th International Conference on Computational Methods in Water Resources - Denver, CO, USA
Duration: Jun 1 1992Jun 1 1992

Other

OtherProceedings of the 9th International Conference on Computational Methods in Water Resources
CityDenver, CO, USA
Period6/1/926/1/92

Fingerprint

Tensors
Porous materials
Hydraulic conductivity
Finite element method
Fluxes
Error analysis
Hydraulics
Monte Carlo simulation

ASJC Scopus subject areas

  • Engineering(all)

Cite this

Neuman, S. P., Orr, S., Levin, O., & Paleologos, E. (1992). Theory and high-resolution finite element analysis of 2-D and 3-D effective permeabilities in strongly heterogeneous porous media. In 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.

Finite Elements in Water Resources, Proceedings of the International Conference. Vol. 2 Publ by Computational Mechanics Publ, 1992. p. 117-136.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Neuman, SP, Orr, S, Levin, O & Paleologos, E 1992, Theory and high-resolution finite element analysis of 2-D and 3-D effective permeabilities in strongly heterogeneous porous media. in 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.
Neuman SP, Orr S, Levin O, Paleologos E. Theory and high-resolution finite element analysis of 2-D and 3-D effective permeabilities in strongly heterogeneous porous media. In Finite Elements in Water Resources, Proceedings of the International Conference. Vol. 2. Publ by Computational Mechanics Publ. 1992. p. 117-136
Neuman, Shlomo P ; Orr, S. ; Levin, O. ; Paleologos, E. / Theory and high-resolution finite element analysis of 2-D and 3-D effective permeabilities in strongly heterogeneous porous media. Finite Elements in Water Resources, Proceedings of the International Conference. Vol. 2 Publ by Computational Mechanics Publ, 1992. pp. 117-136
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