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Estimates of effective permeability for non-Newtonian fluid flow in randomly heterogeneous porous media

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Abstract

An understanding of the interplay between non-Newtonian effects in porous media flow and field-scale domain heterogeneity is of great importance in several engineering and geological applications. Here we present a simplified approach to the derivation of an effective permeability for flow of a purely viscous power–law fluid with flow behavior index n in a randomly heterogeneous porous domain subject to a uniform pressure gradient. A standard form of the flow law generalizing the Darcy’s law to non-Newtonian fluids is adopted, with the permeability coefficient being the only source of randomness. The natural logarithm of the permeability is considered a spatially homogeneous and correlated Gaussian random field. Under the ergodic hypothesis, an effective permeability is first derived for two limit 1-D flow geometries: flow parallel to permeability variation (serial-type layers), and flow transverse to permeability variation (parallel-type layers). The effective permeability of a 2-D or 3-D isotropic domain is conjectured to be a power average of 1-D results, generalizing results valid for Newtonian fluids under the validity of Darcy’s law; the conjecture is validated comparing our results with previous literature findings. The conjecture is then extended, allowing the exponents of the power averaging to be functions of the flow behavior index. For Newtonian flow, novel expressions for the effective permeability reduce to those derived in the past. The effective permeability is shown to be a function of flow dimensionality, domain heterogeneity, and flow behavior index. The impact of heterogeneity is significant, especially for shear-thinning fluids with a low flow behavior index, which tend to exhibit channeling behavior.

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Acknowledgements

This work was supported by the Università di Bologna ‘RFO (Ricerca Fondamentale Orientata)’ 2007 and 2008 funds. We thank the reviewers for their comments, which improved the quality of the final manuscript.

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  1. Dipartimento di Ingegneria delle Strutture, dei Trasporti, delle Acque, del Rilevamento e del Territorio (DISTART), Università di Bologna, Viale Risorgimento 2, 40136, Bologna, Italy

    Vittorio Di Federico, Marco Pinelli & Rita Ugarelli

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  1. Vittorio Di Federico
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  2. Marco Pinelli
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  3. Rita Ugarelli
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Correspondence to Vittorio Di Federico.

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Di Federico, V., Pinelli, M. & Ugarelli, R. Estimates of effective permeability for non-Newtonian fluid flow in randomly heterogeneous porous media. Stoch Environ Res Risk Assess 24, 1067–1076 (2010). https://doi.org/10.1007/s00477-010-0397-9

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