The invention relates to self-supporting capillary membranes which are longitudinally reinforced by continuous reinforcing fibres incorporated in the wall of the capillary membrane, which fibres extend in the longitudinal direction of the capillary membrane. The capillary membranes according to the invention have an improved ultimate tensile strength and a reduced elongation at break as a result of which capillary rupture caused by flow transverse to the capillary membranes occurs considerably fewer times.