Problems of filtration in porous media are of practical importance for studies related to forecasting of pollutants distribution, filtration near the river dams, reservoirs and other hydraulic structures, drainage of footings and basements of buildings, irrigation and drainage of agricultural fields, water, oil and gas production, movement of magma in the crust, etc. The paper investigates the mathematical model of fluid filtration in poroelastic media with predominant elastic properties of deformation, i.e., with high values of medium dynamic viscosity coefficient. The filtration process is defined by the mass conservation laws for fluid and solid phases, Darcy’s law with skeleton movement, the rheological Maxwell law, and the equation of momentum conservation for the system as a whole. Passing to Lagrange variables reduces a system of equations to a parabolic equation for porosity with a degenerate solution. Finite time stabilization of the solution is established by the method of integral energy estimates for low values of volume compressibility coefficient of a solid medium.