This is effectively an addition to other answers:
An original Archimedes screw for liquids does not work as most people imagine it does,
and the diagram does NOT show how an original Archimedes screw works for fluids. The diagram is valid - it's just of something else which works in a similar manner and has become synonymous with the original design in peoples' minds. And both versions do not work as most people imagine them to.
In an original Archimedes Screw the outer cylinder is integral with the "screw" and turns with the screw - there is no moving seal.
When a marble or solid object that cannot fit through the sealing gap is used the two screws work the same. When a fluid is used the difference is important. From the reference below:
- An analysis, using the lifting of marbles instead of water, is used in almost all nineteenth century texts. The lower end of the helical tube dips into a dish of marbles and scoops one up. The helix continues to revolve, and the marble is continually being lifted a short distance up an inclined plane. The frictional forces are small, and the marble keeps rolling down an infinite succession of inclined planes formed by the revolving helix. At the same time the marble resides at the local low spot on the helix, and is carried up the slope by forces perpendicular to its local motion.
The key point is that the "payload" sess a downwards ramp the whole way and simply "runs downhill.
A good illustration of the principle is given on this page
They use tubing so there is no doubt about the "seal".
This image demonstrates what the fluid or other payload "sees".

Effectively the fluid sits in "buckets at all times and there is no opportunity to escape.
In typical original style screws the whole outer casing is sealed to the "screw" and the outer casing rotates with the screw. Consideratuiomn of a cross section shows that as with the tubing each batch of water sits in a container and NOT in a sloping tube.
Even in the stationary casing systems mentioned below the payload still usually sits in a "bucket" and runs forever downhill down a ramp as it ascends (!).
The crucial point of the original system is that the screw can be turned at any speed and even stopped and there is no leakage (apart from any that results from poor construction). Here is a "modern" real world device that works in this manner. This allows the device to be turned by hand at slow speed, or rotated intermittently, or with pauses, with no loss of fluid. Modern sealing techniques and relatively fast and consistent speeds of rotation allow systems where the casing is stationary.

Image from here
From this very good page
The nature of the original design is clearly explained by the Wikipedia - Archimedes Screw page.
While they suggest that whether the inner and outer were sealed in original designs, even a brief consideration of the available facts shows that it was indeed the case. viz
- "Depictions of Greek and Roman water screws show them being powered by a human treading on the outer casing to turn the entire apparatus as one piece, which would require that the casing be rigidly attached to the screw." - Wikipedia