One problem related to water pump has been reported for the 1999 Audi A4. The most recently reported issues are listed below.
Rapid and catastrophic failure of water pump disrupted timing chain and led to severe damage to engine, requiring its replacement. This incident occurred while I was driving at night on a limited access road without shoulders. I needed to go several hundred yards, at most, to pull off road for safety. Audi's 3-year warranty had expired, and so it offered no help, despite low mileage and regular maintenance of the car.