Universal dynamics for unsaturated flows in a confined porous layer

We study the dynamics of unsaturated flows from fluid injection into a confined porous layer. A partial differential equation of the evolution type is derived to describe the time evolution of the interface shape, defined as the location where the saturation of the injected fluid is zero. The saturation field can then be computed once the interface evolution is obtained. We provide an example calculation and demonstrate how the flow behaviour evolves from early-time unconfined to late-time confined behaviours. In particular, at early times, the influence of capillary forces indicates the existence of a new similarity solution in the unconfined limit, which is different from the gravity current solution. At late times, we obtain two new similarity solutions, a modified shock and a compound wave, in addition to the rarefaction and shock solutions in the sharp-interface limit. A regime diagram is also provided, which summarizes all possible similarity solutions and the time transitions between them for the unsaturated flows resulting from fluid injection into a confined porous layer. The influence of the dimensionless control parameters are also discussed, including the effects of viscosity ratio, pore-scale heterogeneity and relative contribution of capillary over buoyancy forces.