Solution of a porous medium problem using laplace transform and general similarity technique

The present paper describes and studies the flow of two immiscible fluids in a cylindrical porous medium. Initially, the porous medium is partially filled with oil and then water is passed through it. Since oil and water are non-mixing nature of fluids, the instabilities occur at the interface. In this paper, the phenomenon of instabilities in the polyphasic flow through homogeneous porous medium is discussed under the assumption of mean capillary pressure. The resulting governing equation is a linear partial differential equation which gives the saturation of water at any point in the cylinder at any time. The analytical solution has been obtained using two different methods - the Laplace transform and the similarity solutions via infinitesimal Lie group of transformations. The solution is also analysed graphically.