This is a fairly easy problem. The annulus is 10,000 ft of 10 ppg mud which weighs in at 5,200 psi.
The drillpipe has three layers: 4,000 ft of 8 ppg (1,664 psi), 4,000 ft 9 ppg (1,872 psi), and 2,000 ft 10 ppg mud for a total 4,576 psi. Thus we need another 625 psi for pressure balance.
This problem is easy, but hey not every problem can be hard.
This question was incomplete, it did not state that the remaining 2000ft was filled with the 10 lb/gal mud, rather it was requesting for the Pipe pressure needed to completely displace the 10 lb/gal fluid into the annulus with the other two fluid in the drill pipe. Pipe pressure required should be 1664 psi since the question did not state that the remaining 2000ft was occupied by 10 lb/gal fluid.
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