In this paper, the one-dimensional mathematical model of joint motion of two immiscible fluids in a poroelastic medium is considered. This model is a generalization of the Muskat-Leverett classical model in which porosity is considered to be a given function of spatial coordinates. The consideration of porous medium compressibility is the basic moment. The proposed model is based on the mass conservation equation for liquids and a porous skeleton, Darcy’s law for liquids with consideration of a porous skeleton motion, the Laplace formula for capillary pressure, rheological equation for porosity and equilibrium condition “of the system as a whole”. Paragraph 1 provides the formulation of the onedimensional model and the conversion of system of equations written in Euler variables. The transition to Lagrange variables leads to a closed system of equations that does not contain solid phase velocity. Paragraph 2 deals with the problem of piston-like displacement of fluids in poroelastic soil. A self-similar analogue of the Verigin’s problem is considered. In case of a porosity dependant special type filtration coefficient, the self-similar solution of the problem of piston-like displacement of fluids in quadrature for an elastic medium is obtained.