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Dispersion induced by non-Newtonian gravity flow in a layered fracture or formation

Published online by Cambridge University Press:  21 September 2020

L. Chiapponi Open the ORCID record for L. Chiapponi [Opens in a new window] ,
D. Petrolo Open the ORCID record for D. Petrolo [Opens in a new window] ,
A. Lenci ,
V. Di Federico Open the ORCID record for V. Di Federico [Opens in a new window]  and
S. Longo Open the ORCID record for S. Longo [Opens in a new window]
L. Chiapponi
Affiliation:
Department of Engineering and Architecture, Università degli Studi di Parma, Parco Area delle Scienze 181/A, 43124Parma, Italy
D. Petrolo
Affiliation:
Department of Engineering and Architecture, Università degli Studi di Parma, Parco Area delle Scienze 181/A, 43124Parma, Italy
A. Lenci
Affiliation:
Department of Civil, Chemical, Environmental, and Materials Engineering, Alma Mater Studiorum Università di Bologna, Viale Risorgimento 2, 40136Bologna, Italy
V. Di Federico
Affiliation:
Department of Civil, Chemical, Environmental, and Materials Engineering, Alma Mater Studiorum Università di Bologna, Viale Risorgimento 2, 40136Bologna, Italy
S. Longo*
Affiliation:
Department of Engineering and Architecture, Università degli Studi di Parma, Parco Area delle Scienze 181/A, 43124Parma, Italy
*
†Email address for correspondence: [email protected]
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Abstract

Models are developed to grasp the combined effect of rheology and spatial layering on buoyancy-driven dispersion in geologic media. We consider a power-law (PL) or Herschel–Bulkley (HB) constitutive equation for the fluid, and an array of $N$ independent layers in a vertical fracture or porous medium subject to the same upstream overpressure. Under these assumptions, analytical solutions are derived in self-similar form (PL) or based on an expansion (HB) for the nose of single-phase gravity currents advancing into the layers ahead of a pressurized body. The position and size of the body and nose and the shape of the latter are significantly influenced by the interplay of model parameters: flow behaviour index $n$, dimensionless yield stress $\kappa$ for HB fluids, number of layers $N$ and upstream overpressure. It is seen that layering produces (i) a relatively modest increase of the total flow rate with respect to the single layer of equal thickness, and (ii) macro-dispersion at the system scale in addition to local dispersion. The second longitudinal spatial moment of the solute cloud scales with time as $t^{2n/(n+1)}$ for power-law fluids. The macro-dispersion induced by the layering prevails upon local dispersion beyond a threshold time. Theoretical results for the fracture are validated against a set of experiments conducted within a Hele-Shaw cell consisting of six layers. Comparison with experimental results shows that the proposed model is able to capture the propagation of the current and the macro-dispersion due to the velocity difference between layers, typically over-predicting the former and under-predicting the latter.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Low-Reynolds-number flows: Hele-Shaw flows Non-Newtonian Flows: Non-Newtonian Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , A14
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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