Dynamics of Sharp Interfaces in One-, Two-Phase Flows in Porous Media: Asymmetry in the Boussinesq and Charny Equations

For the Boussinesq and Charny nonlinear diffusion equations we employ the known similarity solutions and derive new ones using the Adomian decomposition method and separation of variables. For a periodic water-drive regime with a heavier fluid sweeping a lighter one from a porous formation we arrive at an explicit analytical expression for the interface and describe the phenomena of “superpropagation” and “counterslumping” that date back to tidal “superelevation” in coastal unconfined aquifers described by J.R. Philip. Similarly, for a two-phase flow with a straight sharp interface separating two fluids of contrasting viscosity the interface in a periodic drive regime propagates deeper than in a constant rate sweep. Applications to groundwater hydrology and petroleum engineering are discussed.