What would the pressure be in a 70m high pipe filled with water at hydrostatic pressure after closing a ball valve part way down the height? To make things simple if we assume for water:
g = 10 m/s^2
Rho = 1000 kg/m^3
P = Rho x g x h
P = Hydrostatic Pressure, Rho = mass density, g = gravitational acceleration, h = height of fluid column.
Before closing ball valve:
Pressure @ P1 = 700 kPa
Pressure @ P2 = 500 kPa
Pressure @ P3 = 0 kPa
After closing ball valve:
Pressure @ P1 = ?
Pressure @ P2 = 500 kPa?
Pressure @ P3 = 0 kPa
Is the hydrostatic pressure trapped below the ball valve after it is closed? In other words is the pressure at P1 200 kPa or 700 kPa?

 A: Assume the water is incompressible, and the valve is closed slowly. Just before the valve is completely closed, but almost closed, the two heads of water (P1 to P2, and P2 to P3) are not completely blocked off and the pressure at the water interface is P2.  The pressure remains at P2 when the valve completes closing. The pressure at P1 is from head of water from P1 to P2 plus the pressure at P2.
If the pipe is not filled to the top so that no water is expelled from the top at P3 due to closing the valve, the pressure at P1 is the same with the valve closed as before it was closed, because the pressure at the bottom of the valve seal remains at P2, the same as before the valve is closed.
If the pipe is initially full of water up to P3 such that closing the valve pushes a small amount of essentially incompressible water out the pipe, the pressure at P1 drops slightly due the pressure at P2 decreasing due to loss  of water in the section of pipe from P2 to P3.
This same question was asked and answered on this exchange. See Pressure change when closing valve inside a column of water, and the answer by @docscience.
As this answer points out, the pressure at P1 after the valve is closed does not change even if all water is removed in the section from P2 to P3; this assumes the valve makes a perfectly leak tight seal.  If the seal leaks water and air enters, with no head of water in P2 to P3, the pressure at the bottom of the valve seal will decrease to the atmospheric pressure and the pressure at P1 will decrease to the head of water from P1 to P2 plus the pressure of the atmosphere.
Also see Pressure inside when a container is closed on this exchange.
A: Nothing is perfect in this world.
The water has a very small compressibility, but it is not incompressible.
When one closes the valve, the water does not "know" anymore what happens above.
The situation is not different from that of a closed horizontal recipient containing a given volume of water. If it is filled with water at a given pressure, then hermetically closed, will it keep indefinitely its exact volume and therefore water at the same pressure ? I don't believe it. How does the pressure vary with time ? I won't try to guess.
What I think is that one way or another, the pressure will decrease with time.
So the pressure just under the valve at P$_2$ is very hard to guess. In fact, depending how the valve is closed, if it managed to inject even a very very small quantity of water into the lower part during the process of closing, the pressure might even be higher than $500$ kPA (or rather, more than $600$ kPa, see the bottom of my post). The one thing one can be sure of, is that the pressure at P$_1$ will be $200$ kPA more than the value et P$_2$.
Incidentally, I usually think in terms of hectopascals, hPa, which is not equal to, but very close to, a now obsolete unit (but with which I followed weather forecasts for decades) called the "millibar".
Since the pipe is open to the atmosphere, you have already just about 1000 hPa at P$_3$.
So before the valve is closed, the pressure is 6000 hPa at P$_2$ and 8000 hPa at P$_1$.
A: The pressure at P1 was 700 kPa before you closed the ball valve, due to the water column above that point.  The pressure at the ball valve was 500 kPa, for the same reason.  The pressure isn't "trapped" by the ball valve when it is closed.  The water column above the ball valve will ensure that it remains at 500 kPa.  The water below the ball valve can't leak out and lower the pressure there because the same water column exists in the full length of the pipe as existed before you closed the ball valve, and all pressures were in equilibrium before you closed the ball valve (e.g., you specified hydrostatic conditions).  This means that the pressure at P1 remains at 700 kPa after the ball valve is closed.
