'siege' is an interesting term and one we tend to use lightly.
Fort Donelson - Grant surrounded the CS forces and prepares to attack immediately. The CS surrendered the same day.
Vicksburg - Grant surrounded the CS forces, attacked and then used regular approaches to constrict the defenders. The CS surrendered.
Petersburg - The CS forces were not surrounded. They were in very close quarters for 9 months with various flanking attempts and other remote actions attempting to gain advantage (for either side). eventually the CS line was stretched too thin...
Grant used the word "siege" in my post above.