Question

Apply the hydrostatic formula to calculate pressure forces in oil wells. To make calculations tractable, some gross simplifications will be made. The realism of these simplifications is discussed below. Results will be sufficient to explain how very large forces can develop. The Deepwater Horizon oil well was about 10700 m deep (measured from the sea surface).

Assume the cross-sectional area of the well was 0.2 m2. The density of crude oil is about 870 kg/m3, and the density of saltwater is about 1030 kg/m3 (this is slightly more than the value of 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 well can differ.

2. the pores in the sediments surrounding the well are filled with saltwater.

3. the well is filled with oil.

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

Let pa=atmospheric pressure (neglect variations of atmospheric pressure with elevation), pb= pressure of fluid in the sediments adjacent to the base of the well, pw=pressure inside the well, at the base of the well.

(i) calculate pb–pa

(ii) calculate pw–pa

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

(iv) calculate the net force on the well seal

[hint: since the pressure of oil inside the well (pw) pushes down on the seal, while the pressure outside (pb) pushes up, the net force depends on the difference between the two pressures. The net force also depends on the cross-section of the well. You should get 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), but only 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 is placed – for example, if the seal were placed at the top of the well, at sea level (instead of at the base of the well), the net force on the seal would be the same as calculated above

Answer 1