Nonlinear waves and pattern formation in multiphase flows in porous media

Abstract

The paper analyzes pattern formation in initially homogeneous one-dimensional two-phase flows in porous medium. It is shown that generally these flows are unstable. The mechanism of the instabilities is associated with inertial effects. Such instabilities are of explosive type and are probably important in various engineering applications and natural phenomena. In small-amplitude finite approximation the evolution of patterns is governed by the Korteweg-de Vries-Burgers equation. Pattern formation occurs when the coefficient multiplying the Burgers term becomes negative. During nonlinear evolution a soliton with a tail is formed. The amplitude of the soliton increases while the tail decreases. These results can be regarded as a generalization of results by Harris and Crighton (1994) to the case of two-phase flows in porous medium. The obtained solution in form of soliton with a tail can be interpreted as initial phase of formation of the phase composition inhomogeneities in porous media. In the case of fluidized beds this pattern can be regarded as initial phase of bubble formation in a fluidized bed of granular material. The characteristic size of bubbles and time of its formation are estimated.