Pneumatic fractures in confined granular media

We perform experiments where air is injected at a constant overpressure P-in, ranging from 5 to 250 kPa, into a dry granular medium confined within a horizontal linear Hele-Shaw cell. The setup allows us to explore compacted configurations by preventing decompaction at the outer boundary, i.e., the cell outlet has a semipermeable filter such that beads are stopped while air can pass. We study the emerging patterns and dynamic growth of channels in the granular media due to fluid flow, by analyzing images captured with a high speed camera (1000 images/s). We identify four qualitatively different flow regimes, depending on the imposed overpressure, ranging from no channel formation for P-in below 10 kPa, to large thick channels formed by erosion and fingers merging for high P-in around 200 kPa. The flow regimes where channels form are characterized by typical finger thickness, final depth into the medium, and growth dynamics. The shape of the finger tips during growth is studied by looking at the finger width w as function of distance d from the tip. The tip profile is found to follow w(d)alpha d(beta), where beta = 0.68 is a typical value for all experiments, also over time. This indicates a singularity in the curvature d(2)d/dw(2) similar to kappa similar to d(1-2 beta), but not of the slope dw/dd similar to d(beta-1), i.e., more rounded tips rather than pointy cusps, as they would be for the case beta > 1. For increasing P-in, the channels generally grow faster and deeper into the medium. We show that the channel length along the flow direction has a linear growth with time initially, followed by a power-law decay of growth velocity with time as the channel approaches its final length. A closer look reveals that the initial growth velocity v(0) is found to scale with injection pressure as v(0) alpha P-in(3/2), while at a critical time t(c) there is a cross-over to the behavior v(t) alpha t(-alpha), where a is close to 2.5 for all experiments. Finally, we explore the fractal dimension of the fully developed patterns. For example, for patterns resulting from intermediate P-in around 100-150 kPa, we find that the box-counting dimensions lie within the range D-B is an element of[1.53,1.62], similar to viscous fingering fractals in porous media.