A problem of penetration of a viscous liquid into a porous medium with simultaneous squeezing along the surface of this medium is formulated and discussed. The movement of a liquid is initiated by compressing a viscous drop positioned between the upper plate and the porous medium. The system of equations determining the changing of pressure and evolution of an impregnation front was formulated and discussed in dimensionless variables. Typical solutions were obtained for different sets of characteristic parameters including the case of an anisotropic porous medium. Varying the determining parameters leads to different relationships between impregnation and surface squeezing. It was proven that the formulation of the problem under discussion and solutions obtained in dimensionless form are valid for a "rheokinetic" liquid (viscosity changing due to chemical reactions during flow), if an isothermal situation is considered. However flow of a rheokinetic liquid ceases due to unlimited growth of its viscosity and necessity to increase pressure if we want to maintain the preset velocity of the upper pressing plate.