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A dipole solution for power-law gravity currents in porous formations

Published online by Cambridge University Press:  04 August 2015

S. Longo Open the ORCID record for S. Longo [Opens in a new window] ,
V. Di Federico  and
L. Chiapponi
S. Longo*
Affiliation:
Dipartimento di Ingegneria Civile, Ambiente Territorio e Architettura (DICATeA), Università di Parma, Parco Area delle Scienze, 181/A, 43124 Parma, Italy
V. Di Federico
Affiliation:
Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali (DICAM), Università di Bologna, Viale Risorgimento, 2, 40136 Bologna, Italy
L. Chiapponi
Affiliation:
Dipartimento di Ingegneria Civile, Ambiente Territorio e Architettura (DICATeA), Università di Parma, Parco Area delle Scienze, 181/A, 43124 Parma, Italy
*
†Email address for correspondence: sandro.longo@unipr.it
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Abstract

A theoretical and experimental analysis of non-Newtonian gravity currents in porous media with variable properties is presented. A mound of a power-law fluid of flow behaviour index $n$ is released into a semi-infinite saturated porous medium above a horizontal bed, and can drain freely out of the formation at the origin. The porous medium permeability varies along the vertical as $z^{({\it\omega}-1)}$, porosity varies along the vertical as $z^{({\it\gamma}-1)}$, $z$ being the vertical coordinate and ${\it\omega}$ and ${\it\gamma}$ constant numerical coefficients. A self-similar solution describing the space–time evolution of the resulting gravity current is derived for shear-thinning fluids with $n<1$, generalizing earlier results for Newtonian fluids. The solution conserves a generalized dipole moment of the mound. The spreading of the current front is proportional to $t^{{\it\gamma}n/(2+{\it\omega}(n+1))}$. Expressions for the time evolution of the outgoing flux at the origin and of the current volume are derived in closed form. The Hele-Shaw analogue is derived for flow of a power-law fluid in a porous medium with vertically variable properties. Results from laboratory experiments conducted in two Hele-Shaw cells confirm the constancy of the dipole moment, and compare well with the theoretical formulation.

JFM classification

Geophysical and Geological Flows: Gravity currents Low-Reynolds-number flows: Hele-Shaw flows Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 778 , 10 September 2015 , pp. 534 - 551
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Copyright
© 2015 Cambridge University Press 

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