Here we apply asymptotic averaging technique in order to solve the problem of filtration of viscous compressible barotropic fluid in the rigid porous medium with periodic structure of pores. As an equation of motion we consider a generalization of Brinkman’s equation governing the transient flow of a fluid through a porous medium to take into account the variations of the viscosity and drag coefficients with the pressure. In order to determine the components of effective permeability tensor we solve the boundary value problem in a unit cell (and perform averaging of the solutions). We present the results of analytical and numerically-analytical solutions of the problem in a cell in one- and three-dimensional cases respectively. Pressure and velocity distributions are found as solutions of homogenized macroscopic equations. We investigate the effect of nonlinearity associated with the dependence of the viscosity and drag coefficients on the pressure. We investigate the influence of different types of barotropic relations between density and pressure.