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The viscous flow over a thick permeable circular disk in the Reynolds number (Re) range of 10 to 130 and in the Darcy number (Da) range of 10^−9 to 1 is examined. Direct numerical simulations are performed on a 2D grid with axisymmetric boundary conditions. Three flow regimes are observed: I, II, and III. In regime I (effectively impervious; Da<10^−6), the wake is characterized by the presence of a toroidal vortex whose length is approximately equal to that of an impervious disk. In regime II (transition; 10^−6<Da<10^−3), the increase in Da causes the vortex to shorten and move downstream and eventually vanishes at a critical Darcy number Dac. Regime III (Da>10^−3) is the highly permeable regime, in which there is no recirculation. In I, good agreement with existing experimental data for impervious disks is found. In III, an analytical expression for the drag force on the disk is derived, showing good agreement with the numerical results. A global upper limit of Dac=Damax above which the disk is unable to maintain a recirculating wake for any Re is identified. Finally, in regime II, it is demonstrated that increasing the permeability can lead to large variations in the length of the recirculating wake but with minimal effect on the drag coefficient even when Da>Damax. This has important implications in our understanding of the locomotive strategies adopted by organisms that use porous bodies for movement.