People have had secret passages out of fortification in the past. I'm not sure anybody used them to continually supply the siege defenders but you might, say, send a messenger out at night or something infrequent like that. The problem is that if it is discovered, you are in big trouble. And having regular supplies dropped off involves logistics that are unlikely to remain hidden for a year.

If the defenders can dig a tunnel, the attackers can do so, too. In fact, they'd be stupid not to do so, since once you remove the stuff the walls are standing on, they wont be standing very much longer.

The besieging army, if human, must constantly send out foraging parties in all directions to keep their army fed. 3-4 km is not nearly far enough to avoid prompt discovery. Organized logistics chains are expensive, and didn't replace foraging parties for food until the mid-1800s.

@user535733 foraging, raiding, scouting, patrolling. The besieging army will not just be huddled around the walls, they'd be trying to control the nearby lands. Assuming they are large enough and a many weeks long siege does suggest they are that big. In fact, a large enough army might leave a besieging regiment that is there to keep the fortified army behind the walls, while other attacking armies go around the rest of the countryside land and conquer it. Then it wouldn't really matter that the king (or whoever) is still safe behind the fortifications.

If you are besieged for a whole year and your king hasn't come to relieve the siege in all that time, this means that the entire country is controlled by the enemy. There is no point in moving two miles left or two miles right -- you will still be in enemy controlled territory. You may as well just open the gates.

Note that the 1 year worth of provisions was stockpiled by the entire city over the course of many months. Some 80-90% of the medieval workforce was dedicated to food production, so there's no way you'll keep up with food production by sending out anything less than 80% of the town, and that assumes they have access to their farmland as they left it, which they certainly will not. Even if people make it outside the walls, where is all the food supposed to come from?

You are very narrow minded in your aproach, if you get out of the city with your army overnight you can get a nice head's start and attempt to make it to allied territory, or you can atleast do a pincer night attack, because if you pop up half your army near the enemy camp and give them a nasty surprise and the retreat back through your own gates if you don't manage to rout them, you can at least scare them.

As for the supplies part, it would depend on how well the tunnel entrance is hidden, because I plan to set mine in some wood near the city and one year is a whole lot of time for an besieging army to survive on foraging alone, meaning that the forest nearby would be long deprived of any foodstuffs and they would have to rely on their supply chains or supplies reaped from occupied villages. So if you send out a few bands of trusted and discrete men they might start smuggling in bags of grain or rice or things like that.

How is this not a real-world Question, please? Either way I don't happen to have heard of a city, but many medieval castles have secret tunnels. What makes 10 meters a day a realistic assumption? Again, castles or citadels might but why would a town or city have enough provisions to last a year?

If a moat surrounds the castle, water will be a problem.

Another problem is the siege army -- medieval armies were not supplied well and it's reasonable to assume soldiers will pillage nearby villages, then hunt in the forest; thus the tunnel outlet is likely to be discovered. In addition, siege force will have scouts and lookouts spread over wide area to warn of potential relief force arrival. That all being said, it makes sense to try and build a tunnel. Usually, in the medieval times, smugglers would bribe the opposing army to let the trade continue.

The recent novel "How to rule an Empire and get away with it" is centred on this issue of long sieges versus tunneling , if not very realistically.

If I was writing this story, I'd put the tunnel there in the first place as an act of god: Limestone grottos beneath the city and the surrounding landscape. These things can stretch out for miles upon miles and be terribly complicated. The issue is then perhaps not so much the digging process, but the "finding a way"-process...

Digging a tunnel creates a lot of dirt. Assuming your tunnel is something like 2x2m (anything narrower and you will have a traffic problem moving the dirt out), this is 4 cubic meters of dirt per meter or 14 000 cubic meters fo a 3,5km tunnel. Where would all this dirty go unnoticed to the besiegers?

Is your city a minig city? Somehow I just got the vision of a city located on a mountain range which, under siege, makes a desperate effort to dig THROUGH the mountain. They succeed and manage to connect two big regions with a safe tunnel, that were previously barely connected by treacherous passes. They get help and, after the siege, rich from the newfound possibilities.

You underestimate how long it takes for a large army to escape unnoticed through a tunnel.

Worth pointing out that tunneling is a common method of ending a siege - from the attackers' point of view. Dig under the walls, have a small force break in to open the gates whilst distracting the defenders by attacking on the opposite side to the tunnel. An attack tunnel that intercepts an escape tunnel might make an interesting plot point.

@RonJohn. Forget about the unnoticed part. Assuming 30sec/person in a narrow tunnel, that's 8.3 hours per thousand people just to leave the tunnel.

@Erik : and many longer lasting sieges featured a whole war underground with both sides digging tunnels and counter-tunnels to intercept the other side's tunnelers. youtube.com/watch?v=HxgjeF5UHYw