How Long to Leak 40,000 Gallons of Water?

[TonyUC]

So I recently got a water bill that was a little higher than expected...

I got charged for using 40,000 gallons of water over an 8 month period of time. The kicker is that I haven't been using that water supply. So, after a day of digging I found a copper section of the pipe, with a small hole in it. I'm thinking if the water company would have just regularly checked my water usage, I could have fixed this a long time ago, and avoided paying such a high bill. So my question to you guys is, how long would it take for 40,000 gallons to leak out of the hole?

Let’s assume that the hole has been same size the whole time.

Here are the specs:
Hole was 3/32 of an inch.
Pipe is an inch in diameter.
From what I've read, I believe the pressure in the pipe would be between 40 and 50 psi.
40,000g of water leaked.

Thats about all the information I have, I'm just looking for a ball park estimate.
Your help is much appreciated!

 

[tiny-tim]

Welcome to PF!

Hi TonyUC! Welcome to PF!

The maximum possible speed would be v, with P = ρv²/2, and the volume per second would be πr²v

ρ is the density of water, P is the pressure, and r is the radius of the hole (half the diameter)

(i can't do the actual figures, since I'm not familiar with the imperial unit system … i don't even know how many gallons there are in a ball park )

If it was leaking into the ground, I suspect the speed would actually be a lot less than that.

 

[AlephZero]

Without messing about with Bernouilli's equation, you can start from the other end.

8 months is about 8 x 30 x 24 = 5760 hours. So you were leaking about 7 gallons an hour. If it had the full mains pressure behind it, leaking a bucket full or water every 15 or 20 minutes doesn't sound crazy.

FWIW Tiny Tim's formula gives about 60 gallons an hour as a maximum flow rate.

 

[Q_Goest]

I typically use the orifice flow equation given in API 520 for liquid flow. Using 45 psi dropping to 0 psi across the pipe wall and assuming a discharge coefficient of 0.8 (about right for a flat plate orifice), I get a flow rate of about 1.4 GPM, so that's ~ 20 days to leak 40,000 gallons. But the outlet pressure on the pipe isn't likely to be 0 psi. If underground, it has soil around it that will certainly restrict the flow. As Tiny Tim mentions, "If it was leaking into the ground, I suspect the speed would actually be a lot less than that." That's exactly right. It could be a fairly small drop in pressure across the pipe wall because the soil restricts flow. It could be the flow rate is only 10% or less of the 1.4 GPM value. It's very difficult to say without some specialized analysis.

I'd agree with the other posters that it seems reasonable to suggest that the hole you located is responsible for the leakage over the 8 month period.

 

[256bits]

For the op, just collect the water dripping from a tap into a conatainer and you will see how quickly it adds up over time.