We hear a slight noise of water running on the outlet of the water heater & when we close the cold water supply valve it stops so it sounds like it's down stream of the water heater.
The water meter turns very slowly when this happens & stops when we close the valve. There are no faucets leaking & have shut the hot water supply valve at each sink & the noise is still heard. We live on a slab foundation so there is a possability that its under there. We have copper pipes & the house is 26 years old. We see no water anywhere inside or outside of the house, is it possible that the water could not appear because its going into the ground? It has run all day on Friday. Any tips would be appreciated thanks, Jay in Phoenix, Az

It sounds like you have done a very good job of diagnosing the problem. I would think it is a water leak under the slab and it is just soaking into the ground.
The first thing you should do is to call your insurance company to see if you are covered or ( if you are good at reading policies read and find out yourself).

Now to try to help you locate the leak, do you have a pet? Pets like to lay on the warm floor over where the leak might be. Walk around in your bare feet and try to notice any warm spots on the floor, even on the carpeted areas, they will get warm also after a time.

Just because you hear this noise in one area louder than another area does not mean that the leak is closer to the noisy spot. Another way to maybe locate the leak is to turn the water off to the water heater for a few minutes, then turn it back on and go to each hot water faucet, one at a time, open the hot faucet and see if any air or dirty water comes out. Do this to each one.

Next thing to do if you are not going to try to repair it yourself is to call a plumber. Let me know what happens, and if this answer helps please rate it.

When you say you close the cold water valve. Do you mean the cold water valve by the meter that shuts of the entire house or the one on the hot water tank?

If it is the valve on the water heater then don't use any hot water for a couple of hours. Then feel any exposed pipes downstream of the hot water tank. If one is warm you can follow along the pipe to try to track the leak down.

It is a good possibility you have a slab leak. If so there are company's can locate it's direct spot by hooking some equipment up to the pipes in your home. After the leak is located you need to break up the floor, uncover the pipe and cut out the leak and replace it. Follow the previous advice and try to find unusually warm spots on your floor. Your hot lines will run from your heater generally in a direct route towards your sinks, and any other warm water fixtures. Typically it will be a hot line leaking underground because the warm water causes the pipe to expand and contract.