2001 Ford Expedition water leaking from back right engin ar

Have you look to see if it is the Heater hose going to the heater core.

Does it continue to leak while running if so does it stop when reaches operating temp. If stops leaking at operating temp it is possibly a head gasket.

If you have a telescopic mirror use it to get in and see exactly were it is leaking from.

Head Gaskets are a job to do and can range in cost depending on Labor Rates from $1500 - $2500.

If it is leaking as bad as you say while not running the only thing I would suggest is that if you have to drive it don't take long trips, just short trip to store, work and only neccessary trips and make sure you carry plenty of water with you to keep the radiator filled.

Have you checked the heater hoses? Especially at the heater control valve. If the valve is Vacuum Operated the hose on the firewall side can be leaking. When vehicle is off there is no vacuum to it so will leak and will leak for short time once vehicle is started till vacuum builds. And will start leaking again once you turn vehicle off.

To locate heater hoses look on passenger side there will be 2 hoses about the size of a garden hose going to the firewall. Somewhere between the firewall and engine there will be the heater valve. The hose, if it is a hose, that will be leaking will be one between the valve and the firewall.