The brief
The A3 had been going through coolant. Not pouring it out, but needing top-ups often enough that the owner knew something was leaking.
Two other clues came with it. A faint whining noise from the front of the engine at idle, the kind you only catch with the radio off. And the temperature gauge had started wandering higher than its usual settled spot.
Coolant loss, a whine, and a restless gauge. Three things, one likely cause: the water pump.
The diagnosis
A pressurised cooling-system test showed exactly where the fluid was going. The leak traced to the water pump's shaft seal, the seal that holds coolant back while the pump shaft turns. When it weeps, coolant escapes around the shaft and runs down the engine, which matched the dried brown trail staining the block.
A stethoscope on the pump with the engine running picked up the whine, and it was coming from the pump bearing, which had gone rough.
So both the seal and the bearing were on the way out. When a pump is failing on two fronts at once you replace it; a reseal would leave the worn bearing in place to fail later.
The work
Drained the cooling system into a clean catch. Released the drive belt section to get at the pump.
The old pump came out. On this engine the pump and the thermostat housing are one integrated module, so the whole assembly gets swapped together rather than in pieces. A new VAG-spec replacement went in with a fresh seal.
Refilled with the correct coolant at the right ratio, then ran the bleed cycle to clear the trapped air pockets. A modern cooling system always traps air after a refill, and topping up alone will not shift it; the system has to be bled properly or the gauge wanders and you get a hot spot.
Then held pressure on the system to confirm no weeps anywhere before the car left the bay.
The outcome
No drips. Coolant level holding steady between checks. No whine from the front of the engine. And the temperature gauge back to sitting where it should and staying there.
The A3 went home with the cooling system back to spec. For the owner, that means a car that holds its coolant and runs at the right temperature, which is the difference between a planned repair now and an overheating event later.
A water pump that fails completely can leave you stranded, or worse, take the engine with it. Catching it at the top-ups-and-whine stage kept it simple.