The bull I shot this year was a one lung hit. I shot him at 40 yards, the shot looked perfect 1" behind the shoulder half-way up and the arrow sunk in to the fletchings. He ran off about 60 yards and stood there. After about 15 seconds be started to wable and fell head-over-heals down the mountain. We gathered our things and started down the hill after him. When we crested the edge of the hill my bull was standing up. We waited there and watched him bed down about 10 yards from where he fell. So we backed out to let him die.
2 hours later I snuck in thinking for sure he was dead but when I got there he was standing up in his bed. Not wanting to push the animal I decide to back out and wait a couple hours longer. When I got back the second time the bull was gone. I found about a 6 inch circle of blood in his bed but after he got up he left no blood. I ended up doing circles until I finally glassed the bull 150 yards away in some nasty thick trees, STILL ALIVE. He was pretty much done because he watched me walk up to him and put an arrow 3" under my first shot. The bull died 10 minutes later.
After dressing the bull I found that I had split a rib in half, which caused my arrow to deflect towards his liver. It went through one lung and ended up in his liver. I know this would of killed him but as you can see it would of taken a long time. All I can say is that elk are amazingly tough animals.
If you did hit one lung and are positive you didn't push him, then he should be very close. My bull traveled about 200 yards from where I first hit him.
I hope you find the bull and Good luck for the rest of the season.