I have a deep well jet pump (2 lines) and it keeps turning on and off, it reaches the shutoff pressure and then within a minute or two the pressure drops and it turns on again. I just had the foot valve replaced the spring, the pressure tank is a few years old. When I shut the valve at the outlet of the pump, before the pressure tank, it rapidly turns on and off. I haven't checked the pressure in the tank bladder yet.

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When I shut the valve at the outlet of the pump

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Not sure what you mean by that. Do you mean the shutoff valve leading to the house?

Could be a waterlogged tank. Check the air valve at the top of the tank. Depress the stem a few seconds. If you get water, then the tank is bad. If not, then turn off the system and drain pressure to zero by opening a low level faucet. Set tank pressure to 2# below the cut in pressure. For instance, if the cut in pressure is 30, the set the tank to 28.

One way or the other, you are losing pressure somewhere. If the footvalve is good, then you have to start thinking upstream of the tank. That's why I'm asking about the valve at the pump outlet and where it leads.