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

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 k for HB fluids, number of layers N, and upstream overpressure. It is seen that layering produces i) a relatively modest increase of 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 t2n=(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 macrodispersion due to the velocity difference between layers, typically over-predicting the former and under-predicting the latter.