Using laboratory experiments on a network scale together with numerical simulations on a granular scale, we investigate the displacement process as air invades a highly compacted granular material. Experiments in a vertically placed Hell-Shaw cell reveal a non-monotonic behavior of branching formation as air injection rate Q increases from 0.1 to 50 ml min–1 when the liquid viscosity is less than 22.5 mPa s. In the low-injection-rate region where Q < 1 ml min–1, fractures grow in random directions, and the number of branches increases as the air injection rate decreases. However, after the transition to the high-injection-rate region where Q≥ 1 ml min–1, the number of branches increases with increasing air injection rate. At a given air injection rate, increasing the liquid viscosity from 1.01 to 219 mPa s leads to an increasingly concentrated air flow. The numerical simulations exhibit good agreement with the experimental results. More importantly, they shed light on the physics underlying the growth of the fractures by capturing the distribution of the magnitude of velocity, as well as computing the inter-grain force chains in the granular material. The simulations suggest that a high liquid viscosity concentrates the velocity field and force chains and reduces the speeds and inter-grain forces of the grains adjacent to fractures, while a higher air injection rate increases the grain speeds and inter-grain forces. In addition, the distribution of the forces chains behaviors non-monotonically as the air injection rate decreases, which explains the non-monotonic behavior of branching formation observed in the experiments.

We investigate the morphology and dynamics of the pattern of immiscible invasion by injecting a high-viscosity liquid into a granular suspension consisting of movable solid grains in a low-viscosity liquid. Laboratory experiments conducted in a Hele–Shaw cell shed light on how the frictional forces of the grains and the viscous forces of the liquids affect the instability of the liquid–liquid interface and the formation of viscous fingers. The frictional force increases with an increase in either the volume fraction or the size of the grains, leading to higher resistance to the invading pattern. Upon changing the grain shape from spherical to irregular, both the frictional force and the rotational energy of the grains increase, resulting in more numerous but narrower fingers. Increasing either the injection rate or the viscosity of the injected liquid increases the viscous pressure within the fingers, promoting the splitting of the pattern. Although the defending liquid always has a lower viscosity than the invading liquid in this study, the former's viscous force becomes non-negligible as the viscosity ratio of the invading liquid to the defending liquid decreases to near unity, which destabilizes the fluid–fluid interface and causes a transition to an asymmetric pattern.

Multiphase flow involving fluids and particles is inherently complex due to the coupled effects of viscosity, capillary forces, and particle-scale friction. The fluids exhibit viscous behavior, which manifests as instabilities at the interface; changes in fluid viscosity also significantly affect the interface. In this study, we injected aqueous solutions of varying viscosities into a layer of dry, hydrophobic particles. As the viscous and frictional forces changed, we observed alterations in the number and width of the finger-like invasion patterns. When the pressure gradient exceeds a critical threshold, the finger-like structures expand radially outward. The fractal dimension of the invasion patterns was calculated using the standard box-counting method applied to binarized invasion masks, allowing the space-filling degree and morphological transition of the patterns to be quantified under different particle layer filling ratios, injection rates, particle sizes, and liquid viscosities.