We present a second-order computational homogenization framework for the multiscale simulation of heterogeneous porous media. Our model is based on the ideas of Kouznetsova et al. [1] on second-order computational homogenization of mechanical problems and Sciarra et al. [2] on the modeling of second-order effects in poroelastic solids. Motivated by the works [1, 3], we connect a second-order macro-continuum with a first-order continuum at the microscale. The poroelastic character of the material will be accounted for by a transient Darcy-Biot-type framework across length scales. In conjunction with the higher-order macroscopic formulation this yields to a combined size effect, which depends on material and structural properties of the porous solid [4, 5, 6]. The interconnections and consequences of this size effect as well as its dominating features will be analyzed in a number of numerical examples. References [1] V.G. Kouznetsova, M.G.D. Geers and W.A.M. Brekelmans. Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Computer methods in Applied Mechanics and Engineering, 193, 5525–5550 (2020). [2] G. Sciarra, F. dell’Isola, and O. Coussy. Second gradient poromechancis. International Journal of Solids and Structures, 44, 6607–6629 (2007). [3] I. A. Rodrigues Lopez, and F. M. Andrade Pires. Unlocking the potential of second order computational homogenisation: An overview of distinct formulations and a guide for their implementation. Archives of Computational Methods in Engineering, 1–55 (2021). [4] F. Larsson, K. Runesson, and F. Su. Variationally consistent computational homogenization of transient heat flow. International Journal for Numerical Methods in Engineering, 81(13), 1659–1686 (2010). [5] S. Kaessmair and P. Steinmann. Computational first-order homogenization in chemo-mechanics. Archive of Applied Mechanics, 88, 271–286 (2018). [6] E. Polukhov and M.-A. Keip. Computational homogenization of transient chemo-mechanical processes based on a variational minimization principle. Advanced Modeling and Simulation in Engineering Sciences, 7, 1–26 (2020).