A system for handling drill cuttings conveys cuttings slurry into bulk tanks via a conduit. The bulk tanks have an un-pressurized interior volume that receives the slurry. A conveyance member positioned inside the bulk tank forces the slurry out of a discharge port at the bottom of the bulk tank. One suitable conveyance member is a screw-type conveyor coupled to a motor that applies a vertical motive force to the slurry. The bulk tanks hold the cuttings slurry until it can be discharged via the discharge port to a transport vessel for processing or disposal. For offshore operations, the system includes a separation unit on the rig that forms the cuttings slurry from fluid returning from the wellbore and a cuttings flow unit that conveys the slurry effluent from the separation unit to the bulk tanks. In one arrangement, a controller and sensors control the flow of slurry into the bulk tanks.