Thought it would be worth ballpark estimating the pressures in the pipe in the BP oil spill. The pipe is supposed to be about 19.5 inches (internal diameter? – other sources say 20 inches) and occluded by about 30%. A barrel of oil is 158.984 liters. The oil is supposedly about 50% methane. Flow rates are 5000-10000 barrels/day. (These are culled from various online sources).
If the pipe were circular and 19.5″ then the cross section would be 7707 cm^2 or 7.707×10-3 square meters. Occluding by 30% gives an apparent cross section of 5.4×10-3 square meters (5400 cm^2).
A barrel/day is 0.00184 liters per second so a 1000 barrels/day is just under 2 liters per second or 1840 cm^3/second. Since the cross section is about 5400 cm^2, the velocity is 0.34 cm/second. With 5000 bpd it would be 1.7cm/second and at 10000 bpd 3.4 cm/second.
Since impulse equals momentum, mv = (integral) f dt, knowing the density of the material and its velocity allows us to estimate the force, and the force/area is the pressure. If we assume crude oil & gas is about the same density as water (pessimistic, and will over estimate the forces needed – but in a linear manner) then we have about 3.4 g-cm/second/second mean force on each cm^3 of oil at 10000 bpd. Thus the total force on a putative cork would be 3.4 x 18400 gcm/second/second = 62560 ergs or 6.26 Newtons. The pressure would be 11,600 Newtons/square meter. One atmosphere is about 1000×100Newton/square meter or 100,000 newton/square meter giving about a 0.1 atmosphere pressure.
Is this sensible?
Apparently BP was able to cut about halfway through the pipe before the force became high enough to bind it. I don’t know the torque of their motors, but 3 Newtons is enough to bind a blade. If it were a lot higher they couldn’t have made it that far.
Since they are now capturing 15000 or so bpd (and the flow has to be bigger than this) it looks like these estimates are a bit low. Pressures are probably of the order of 0.5 atm. They have probably been slowly ramping up the flow rates in order to stop a blockage in the pipe due to methane clatherates. If the pipe gets blocked the pressure is enough to just pop it off the top of the riser.