Abstract The paper deals with homogenization of a double porosity fluid‐saturated periodic medium. At the mesoscopic level, dynamic behaviour of the medium is described by the Biot model featured by high contrasts in the permeability and the poroelastic coefficients. The fluid flow is governed by the Darcy flow model extended by inertia terms and by the mass conservation equation. To respect the high contrasts, some of the material properties are scaled by the small parameter which characterizes the size of the heterogeneities being subject of the asymptotic analysis. The macroscopic model of the medium is obtained using the two‐scale homogenization based on the periodic unfolding method. For this, the Laplace transformation in time is used to introduce the local autonomous problems for characteristic responses defining the effective medium properties. In comparison with the low contrast heterogeneous medium, the microflow in the double porosity gives rise to the fading memory effects involved also in the macroscopic poroviscoelastic constitutive law. The problem treated in the paper is an extension of previously studied problems with either rigid skeleton part, or deformable Biot ‘medium without high contrasts in material properties. Numerical illustrations of the homogenized effective model parameters are given. The derived two‐scale model is a convenient tool for studying wave propagation in many natural media and provides a basis for material research.