We describe the numerical computations of permeability and dispersivities of solute and heat for a periodic porous medium by solving certain boundary value problems for a unit cell. These cell problems are derived from the asymptotic theory of homogenization which systematically accounts for the effects of micro(pore) scale mechanics on the macroscale processes. Variational principles are devised to replace the cell boundary value problems and are then used for finite-element computations. The geometry chosen consists of a cubic array of uniform grains of Wigner-Seitz shape. Comparisons of numerical results with available experimental data and with other theories are discussed.