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

• Published in: Journal of fluid mechanics 2020-11, Vol.903, Article A14
• Main Authors: Chiapponi, L., Petrolo, D., Lenci, A., Di Federico, V., Longo, S.
• Format: Article
• Language: English
• Subjects: Aquatic reptiles; Body size; Computational fluid dynamics; Constitutive equations; Constitutive relationships; Dispersion; Exact solutions; Flow rates; Flow velocity; Fluids; Gravitation; Gravity; Gravity flow; JFM Papers; Layering; Nose; Overpressure; Permeability; Porous media; Power law; Rheological properties; Rheology; Self-similarity; Solutes; Thickness; Upstream; Yield stress
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2020.624

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.