In the paper a Barenblatt-Biot consolidation model for flows in porous elastic media is derived by homogenization. The starting Biot micro model assumes a two-component material in a domain \(\Omega \) with inclusions \(\Omega _\varepsilon ^{(2)}\) \(\varepsilon \)-periodically distributed in the matrix \(\Omega _\varepsilon ^{(1)}\). Both volumes are assumed to be saturated with a slightly compressible viscous fluid. All the coefficients of the constitutive relations are assumed to be \(\varepsilon \)-periodic. The system consists of the vector equilibria equation for displacement coupled with the scalar evolution equation for pressure on both \((0,T)\times \Omega _\varepsilon ^{(i)}\). The system is completed with Deresiewicz-Skalak condition on the \(\varepsilon \)-periodic interface of the components. Using the two-scale convergence technique homogenization of this micro model leads to the Aifantis double porosity model on \((0,T)\times \Omega \). Comparison and remarks on various models are included.