riser section near the seabed and the riser section being filled with a fluid of a different density than that of the drilling fluid and fig2 is similar to fig1 but here the drilling fluid fills a greater part of the riser section in the drawings reference number 1 denotes a fixed drilling rig comprising a support structure 2 a deck 4 and a derrick 6 the support structure 2 is placed on the seabed 8 and projects above the surface 10 of the sea a riser section 12 of a liner 14 extends from the seabed 8 up to the deck 4 while the liner 14 runs further down into a borehole 15 the riser section 12 is provided with required well head valves not shown a drill string 16 projects from the deck 4 and down through the liner 14 a first pump pipe 17 is coupled to the riser section 12 near the seabed 8 via a valve 18 and the opposite end portion of the pump pipe 17 is coupled to a pump 20 placed near the seabed 8 a second pump pipe 22 runs from the pump 20 up to a collection tank 24 for drilling fluid on the deck 4 a tank 26 for a riser fluid communicates with the riser section 12 via a connecting pipe 28 at the deck 4 the connecting pipe 28 has a volume meter not.