Asymptotic model of the mobile interface between two liquids in a thin porous stratum

Abstract

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.