For something like 9 months in the Civil War, the Union and Confederate armies engaged in a stalemated trench warfare that was a preview of the western front in WWI (a preview that no one learned from). Only Grant's ability to keep flanking the Confederate line and stretching it out until Lee faced thinning his troops too much eventually broke the stalemate.
One interesting parallel with WWI-- the Union in the Civil War had miners dig tunnels under the Confederate lines and packed them with explosives. When they blew, it created a great gap in the line and an opportunity for the Union, an opportunity that was lost when Union soldiers went racing into the crater rather than around it. Trapped in the crater, they were slaughtered by the Confederates. The mistake was apparently the result of a last minute change of plans. A group of black soldiers was supposed to lead the attack and had been trained to not go into the hole, but they were replaced at the last minute with white soldiers who had not been similarly briefed.
Anyway, it is odd how history repeats itself. In WWI, the British tried the same trick, blowing a huge hole in German lines and eventually making a little headway against the German army, though the advantage was, as so many such things were on the Western front, short-lived.