Drainage of power-law fluids from fractured or porous finite domains

We develop a sharp-interface model that captures the coupled effect of spatial heterogeneity and fluid rheology on one-dimensional Newtonian and non-Newtonian buoyancy-driven flow spreading in fractured and porous media over a horizontal impermeable bed. We study the flow in three different geometries: (i) a constant uniform aperture, (ii) an aperture variable along the vertical axis, i.e. perpendicular to the direction of propagation and (iii) an aperture variable along the horizontal axis, i.e. parallel to the direction of propagation. The non-Newtonian rheology is described by the power-law equation of rheological index n and the aperture variation in both directions by a positive number r. The self-similar solutions of the flow obtained at late times allow the transformation of the nonlinear PDEs governing the spreading into nonlinear ODEs. The current shape is affected by the interplay between the rheological index and the spatial variability of the aperture. The residual liquid mass that remains in the fracture at any given time is computed from the current profiles, obtaining a negative power-law behavior in the time of exponent dependent on n and r. In addition, sensitivity analysis is performed to highlight the impact of the model parameters on the current profile and residual mass. The dimensionless analysis outcomes are compared to two real examples of flow within a uniform and a wedge-shaped aperture along the flow direction. The numerical results of the examples confirm that the proposed model can successfully capture the propagation of the gravity current, its profile, and drainage flow rate.