They were on the defensive. Grant had got them pinned down, and was simply batering away at their lines till they grew too thin to hold.
The south used strategies like trying to cut off the railroad. Also the trench welfare.
ANSWERThe Union army of the Potomac was the victor at the battle of Petersburg.
Siege of Petersburg Siege of Richmond
Ulysses S. Grant was in command of Union forces at Petersburg.
Total War: Shogun 2 "Fall of the Samurai" Expansion Trailer
Visit the Aurora Battle Cruiser in Saint Petersburg, Russia
Under Siege Preparing For Battle
The First Stage of the Battle of Stalingrad in 1942