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# Cooling a liquid in a rigid container (constant volume)

## Cooling a liquid in a rigid container (constant volume)

(OP)

Dear forum,

This question is on the relationship between temperature and pressure in a fixed volume in the context of a sub-sea oil well.

The wellhead is 2,000m below sea level, ambient (external) temperature is 4 deg C.

The reservoir is a further 1,000m below the wellhead, reservoir temperature is 40 deg C.

When the well is put on production, gas flows up from the reservoir to the wellhead through 7" production tubing which is housed concentrically in a 13-5/8" tubing. The annulus between the two pipes is isolated at the reservoir end has a valve at the wellhead end.

Before the well was put on production, the annulus was filled with a NaCl Brine (1.02 SG). At this point, the pressure at the wellhead and the reservoir can easily be determined by calculating the hydro static head of fluid. The valve is closed and the well starts to flow. As the well flows, the annulus heats up which will cause the pressure to increase. Some volume is bled off by opening and closing the valve to reduce the pressure at the wellhead back down to the pre-production value (essentially brine hydrostatic). Next the well is shut down and the annulus cools.

When it cools back down to ambient, what will happen to the pressure in the annulus at both the wellhead and reservoir? I can get to the annulus temperature profile, how much the pressure increases and how much volume is bled off but where I become stuck is what happens on cooling and the annulus is no longer full.

How might I calculate the wellhead and reservoir pressures at the shut down condition?

Best regards - Simon

### RE: Cooling a liquid in a rigid container (constant volume)

This is a very complex problem and impossible to solve analytically without a lot of information.

If it is unsafe to simply allow the system to cool and measure the final pressure, I suggest you use the data you have, to establish the relationship between pressure and volume and extrapolate eg:

- Each time you release liquid during the heating process you get a corresponding pressure reduction. Each of these events will give you a p/v slope (delta p/delta v). Probably not worth correcting for the change in density of released water with temperature - 36* is not a huge temperature change. It would help if you have data for both small and large releases. This will help establish the linearity (or otherwise) of the relationship.

- The P-V relationship during cooling will initially have the same slope as a small release. It may even be "close enough" to assume a linear relationship to extrapolate out to the total delta V (the total volume released during heating). Remember that the final pressure will not go much below 1 kPa abs because even at 4*C the water will start to flash at about 0.8 kPa (7.4 kPa @ 40*C)

je suis charlie

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