We propose a new generalized model to describe deformations of the mobile interface
separating two immiscible liquids in a porous medium. The densities and the viscosities
of the fluids can have any value. The horizontal size of the interface is much
greater than the vertical size of the domain. Unlike the classical theory, the new model
describes gravitational non-equilibrium processes, including the Rayleigh–Taylor
instability which appears in the form of a negative apparent diffusion parameter. Several
flow regimes are established depending on the ratio between gravity and the elastic
fluid/medium forces, and between the vertical and horizontal flow rates. The model is
used to analyse the evolution of the interface during the free spreading of one liquid
over another. This is characterized by the presence of interface degeneration points.
The explicit solution to the problem of oil and water flow towards a well is presented
as an application to oil reservoirs.