We model the formation of fluid escape structures (FES) like pipes, chimneys and pockmarks.• Model predicts pulsed gas flow and morphological features like conical pipe/chimney, annular gas channel, and W-, U-, ring-shaped pockmarks.• Morphology of FES depends on sediment-fluid interactions (e.g. erodibility, anisotropy), not on intrinsic properties (e.g. permeability).