In one aspect, a method of connecting a riser from an offshore rig to wellhead equipment that includes a first connector is disclosed. The method, in one non-limiting embodiment, includes lowering a riser of sufficient length having a second connector at a lower end thereof adapted to connect to the first connector of the wellhead equipment, wherein the riser is in fluid communication with and is filled with the sea water; closing the riser proximate to the lower end of the riser after lowering the riser; displacing the sea water in the riser with a fluid that is heavier than the sea water to straighten the riser and the second connector; and connecting the second connector to the first connector.