Combined effect of rheology and confining boundaries on spreading of gravity currents in porous media

• We model gravity currents of power-law fluid in porous channels of given shape.
• Self-similar solutions are derived for motion in horizontal and inclined channels.
• Physical behaviour governed by current volume α, channel shape β and behaviour index n.
• Two critical values of α modulate the dependency for horizontal channels.
• Theoretical results are validated by experiments in triangular and semi-circular channels.

One-dimensional flows of gravity currents within horizontal and inclined porous channels are investigated combining theoretical and experimental analysis to evaluate the joint effects of channel shape and fluid rheology. The parameter ββ governs the shape of the channel cross section, while the fluid rheology is characterised by a power-law model with behaviour index n  . Self-similar scalings for current length and height are obtained for horizontal and inclined channels when the current volume increases with time as tαtα.For horizontal channels, the interplay of model parameters α,nα,n, and ββ governs the front speed, height, and aspect ratio of the current (ratio between the average height and the length). The dependency is modulated by two critical values of α,αβ=n/(n+1)α,αβ=n/(n+1) and αn=(2β+1)/βαn=(2β+1)/β. For all channel shapes, αβαβ discriminates between currents whose height decreases (α<αβα<αβ) or increases (α>αβα>αβ) with time at a particular point. For all power-law fluids, αnαn discriminates between decelerated currents, with time-decreasing aspect ratio (α<αnα<αn), and accelerated currents, with time-increasing aspect ratio (α>αnα>αn). Only currents with time-decreasing height (α<αβα<αβ) and aspect ratio (α<αnα<αn) respect model assumptions asymptotically; the former constraint is more restrictive than the latter.For inclined channels, a numerical solution in self-similar form is obtained under the hypothesis that the product of the channel inclination θθ and the slope of the free-surface is much smaller than unity; this produces a negligible error for θ>2°θ>2°, and is acceptable for θ>0.5°θ>0.5°. The action of gravity in inclined channels is modulated by both the behaviour index n   and the shape factor ββ. For constant flux, the current reaches at long times a steady state condition with a uniform thickness profile. In steep channels and for sufficiently long currents, the free-surface slope becomes entirely negligible with respect to channel inclination, and the constant thickness profile depends only on n.Theoretical results are validated by comparison with experiments (i) in horizontal and inclined channels with triangular or semicircular cross-section, (ii) with different shear-thinning fluids, and (iii) for constant volume and constant flux conditions. The experimental results show good agreement with theoretical predictions in the long-time regime.Our analysis demonstrates that self-similar solutions are able to capture the essential long-term behaviour of gravity currents in porous media, accounting for diverse effects such as non-Newtonian rheology, presence of boundaries, and channel inclination. This provides a relatively simple framework for sensitivity analysis, and a convenient benchmark for numerical studies.