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Apply the hydrostatic formula to calculate pressure forces inoil wells. To make calculations tractable, some grosssimplifications will be made. The realism of these simplificationsis discussed below. Results will be sufficient to explain how verylarge forces can develop. The Deepwater Horizon oil well was about10700 m deep (measured from the sea surface).

Assume the cross-sectional area of the well was 0.2 m2. Thedensity of crude oil is about 870 kg/m3, and the density ofsaltwater is about 1030 kg/m3 (this is slightly more than the valueof 1000 kg/m3 that I want you to memorize for water density,because the presence of salt slightly increases water density).

Consider the case shown below. Assume

1. the well base is sealed, so pressure inside and outside wellcan differ.

2. the pores in the sediments surrounding the well are filledwith saltwater.

3. the well is filled with oil.

4. the elevation of the oil surface in the well is the same asthe elevation of the sea surface.

Let pa=atmospheric pressure (neglect variations of atmosphericpressure with elevation), pb= pressure of fluid in the sedimentsadjacent to the base of the well, pw=pressure inside the well, atthe base of the well.

(i) calculate pb–pa

(ii) calculate pw–pa

(iii) calculate pb–pw (hint: your answer should be more than 100times atmospheric pressure).

(iv) calculate the net force on the well seal

[hint: since the pressure of oil inside the well (pw) pushesdown on the seal, while the pressure outside (pb) pushes up, thenet force depends on the difference between the two pressures. Thenet force also depends on the cross-section of the well. You shouldget a force exceeding the weight of a 300,000kg object].

•In fact, some of the surrounding sediments are filled with oil,which reduces the net force below the value calculated in (iv), butonly slightly. •To avoid the buildup of such large forces,engineers often pour dense, mud-laden water into the well. Aside:the net force on the seal doesn’t depend on where the seal isplaced – for example, if the seal were placed at the top of thewell, at sea level (instead of at the base of the well), the netforce on the seal would be the same as calculated above