What is the leakage rate of my underground piping system?

[russ_watters]

TL;DR Summary

Trying to find the leakage rate of a pipe that loses 1psi per minute.

This is a situation I don't often deal with; I have an underground piping system that is failing a pressure test and losing pressure at a rate of about 1psi/min from 80psi. I'd like to calculate the leakage rate.

The pipe is a simple 10" diameter, schedule 40 steel pipe, 500' long (0.365" th). That's roughly 2,038.8 gal.

First, I calculate the compressibility of water. Googling, I'm seeing 2.8% compression for 1,000 psi (taken from a graph). Or 0.06 gal/psi. This feels reasonable.

Next, I use the hoop stress equation and 29 MPsi Young's Modulus to calculate the expansion of the pipe. I get a volume difference of 0.0019 gal. This feels surprisingly low.

Could someone double-check me please.

 

[anorlunda]

Just a small quantity of off gassing of that liquid might skew the calculations. Especially if pressure is decreasing, some quantity of dissolved gasses might form bubbles. Can you re-check the calculation if 1% of the volume was gaseous?

 

[jrmichler]

Double check coming up:

Compressibility (densities from NBS/NRC Steam Tables):
Water at 20 deg C and 1 bar absolute = 998.23 kg/m^3
Water at 20 deg C and 6.5 bar absolute = 998.48 kg/m^3
Water at 20 deg C and 70 bar absolute = 1001.37 kg/m^3

Change in volume = $2038.8 gal * (1 - 998.23/998.48) = 0.51 gal / 80 PSI$
$0.51 / 80 = 0.0064 gal / PSI$
We differ by a factor of 10.

I used the hoop stress equation, 0.37 inch wall, and 30E6 modulus, and got 0.00185 gal/PSI. This agrees nicely with your calculation.

 

[mfb]

Summary:: Trying to find the leakage rate of a pipe that loses 1psi per minute.

I'm seeing 2.8% compression for 1,000 psi (taken from a graph). Or 0.06 gal/psi.

That's far too high. Even at the bottom of the Mariana trench the compression is just several percent, at ~1000 times atmospheric pressure or ~13000 psi. A factor 10 as error looks plausible.

I share the concern about gas being present somewhere.

 

[russ_watters]

Thanks guys; yep, I slipped a decimal place in reading off this graph:

That's 10,000 psi, not 1,000 psi for 2.8%.

Yes, it is possible there is a significant trap of air somewhere and that could cause a much higher volumetric leakage for the same pressure drop. I didn't measure, but the air vent was maybe a foot below the cap on the pipe, so presumably at least that half a cubic foot was full of air. 1 psi of expansion should be about 0.04 gallons. And there's no real telling if there is a significant air trap somewhere along the 500' length.

 

[hutchphd]

And there's no real telling if there is a significant air trap somewhere along the 500' length.

Can you instead inject measured water sufficient to regain the original pressure after, say, 10 minutes of leak? If the entrapped air doesn't change that would give a pretty good number.

 

[russ_watters]

Why did I post this in aerospace engineering instead of mechanical engineering? Fixed...

 

[russ_watters]

Can you instead inject measured water sufficient to regain the original pressure after, say, 10 minutes of leak? If the entrapped air doesn't change that would give a pretty good number.

I could put a flow meter on the inlet to measure the fill, but that isn't really the critical issue; on its own it doesn't inform much to what the leak rate is.

 

[mfb]

If water leaks and you refill with water, shouldn't the refill rate match the leak rate? That would eliminate the volume considerations as you end up with the same pressure and volume as before.

 

[russ_watters]

If water leaks and you refill with water, shouldn't the refill rate match the leak rate?

No...we're filling from a garden hose at city water pressure, and the leak is likely a few mm in diameter, mostly at close to zero pressure. It leaks for hours or days and then we refill to retest in minutes.

I've only witnessed one of the tests and we haven't tried an exact timed interval between tests. Nobody has tried to estimate the leak rate before I just did; it was just a pass/fail test.

 

[hutchphd]

No...we're filling from a garden hose at city water pressure, and the leak is likely a few mm in diameter, mostly at close to zero pressure. It leaks for hours or days and then we refill to retest in minutes.

The timing is not an issue (measured refill=total leak over previous interval). The injection under pressure is a different matter: you would need an external water feed tank capable of feeding water under under pressure.
Actually if you wanted to keep the injection water pressure lower, you could inject, at lower pressure and "calibrate" the compressibility: Put known amounts of water into the sealed system and monitor the resultant pressure increase. Extrapolate this (presumed linear?) line to the pressure region of interest and that slope gives you the result you need.

 

[Baluncore]

Next, I use the hoop stress equation and 29 MPsi Young's Modulus to calculate the expansion of the pipe.

The extension stress will be about half the hoop stress. How much does the pipe elongate?

 

[jrmichler]

You can measure the total amount of leakage by adding a tee fitting and valve to the fill line. Fill, close the fill valve, and open the tee into a bucket.

 

[hutchphd]

That is not complicated enough! (It does assume the fill line is submerged...)