The interaction of fluid layers from simultaneous injection of two gravity currents

We study the displacement and interaction process of multiple fluid layers from simultaneous injection of two gravity currents into a confined porous medium, partly inspired by the practice of enhanced oil recovery and geological CO2 sequestration. Two coupled nonlinear advective-diffusive equations are derived to describe the time evolution of the interface shape of both the upper (lighter) and lower (heavier) intrusive layers. At early times, the governing PDEs decouple and reduce to two nonlinear diffusion equations for the spreading of two gravity currents along either the top or bottom boundary of the porous medium. As time progresses, the upper and lower currents approach each other and start to interact; the flow becomes mainly advective in the bulk, with non-negligible diffusive (buoyancy) effects near the propagating front. We also find that there are many sub-regimes of gravity current interaction at late times, impacted by four dimensionless parameters, representing the flow rate partition, ratio of buoyancy over injection forces, and the viscosity contrasts between the injecting and displaced fluids. Finally, by defining appropriate similarity variables at early or late times, the governing PDEs reduce to ODEs, corresponding to a class of self-similar solutions in different asymptotic regimes.