How can I estimate how far into a cave network breathing is possible?

Question

I want two populations to coexist in a world with extensive tunnels carved into the rock. One population lives in the tunnels, coming out to trade and gather food. The other is aware of the tunnels but is wary of entering them.

I want the tunnels to go far enough that after a certain point there is insufficient air flow to maintain enough oxygen to breathe. The tunnel dwellers are well aware of this effect and know to only venture that far for short periods to avoid asphyxiation. To the surface dwellers they spread rumours that only the tunnel dwellers are capable of breathing the tunnel air, despite the fact that the regions where the tunnel dwellers live has perfectly breathable air for both populations (which are both human).

For tunnels that are mostly just over head height and arm span width, occasionally widening into rooms, after what distance into the tunnels would breathing no longer be sustainable? The tunnels are carved by humans, not by underground rivers, so they have no other end to create airflow from pressure differences.

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Following Monty Wild's answer here is some further detail. The tunnels are carved horizontally into sandstone. The humans carry candles and oil lamps with them when travelling further into the tunnels. There are over 50 separate entrance tunnels each leading to its own main chamber after about 40 metres, in which small wood fires are kept burning for heating and cooking. Further tunnels continue from the main chamber, branching and opening into chambers for storage. Some of these tunnels connect the main chambers to each other, but there is no interconnection between tunnels further in than the main chambers. Each of the main chambers is about twice human height and about 300 metres square in floor area (a circle roughly 20 metres in diameter).

I'm looking to estimate whether a group of humans could survive in the main chambers, and if so how much further into the tunnels beyond they could travel before breathing became impossible.

Answer 1

This is a problem of mining air flow. When tunnels exist where there is no circular path allowing air flow, the air can become contaminated by outgassing from the walls of the mine and the activities of the inhabitants. The length of tunnels as described that would result in the air becoming unbreathable would depend on the nature of the material being mined, the processes that were taking place within the mine (i.e. human activity and/or combustion-powered machines), the frequency of that activity, and the distance from ventilated spaces, as well as the speed of airflow through those ventilated spaces (i.e. the potential for eddies to cause air exchange).

From soldiers' experiences in World War two escape tunnel mining, problems with air supply where humans are working in areas where there is only one exit can become significant over distances of a few tens of meters such that it was necessary to install ventilation apparatus to provide workers at the head of the tunnel with fresh air.

In a deep mine through material prone to outgassing, a lack of human activity and artificial ventilation could result in an area becoming uninhabitable within as little as a few days.

The sustainability of mining depends then upon the material being mined, the nature of the activities being performed in the mine, and the technological level of its creators. This is too broad a question to answer with a definitive distance given that these factors are neither stated in the question nor likely to be a constant over time.

EDIT

Given the edit to the question specifying the nature of the caves:

Since we are talking about sandstone that is presumably not prone to outgassing, we can discount that as part of the problem.

Since many of the main chambers described appear to be interconnected, each of these could be described as existing in a loop connected to outside air at at least two points. This would mean that outside wind is likely to cause a draught and air exchange, meaning that these main chambers would likely be habitable in the long-term by a small population without any particular ventilation apparatus. With a larger population, something as simple as sails to catch the wind and direct it into the tunnels would probably suffice.

Any main chambers which did not have a tunnel loop may suffer from air stagnation, as would the dead-end tunnels. This all depends on the size of the tunnels to the main chambers, as larger tunnels allows air eddies to penetrate more deeply. The smaller the tunnels, the lower the population the main chamber would support. However, the traversal of the tunnels by humans would to some degree provide impetus for ventilation, but it would not be as effective as wind ventilation.

The simple provision of a chimney in each main chamber would probably provide all the ventilation needed - the fire maintained in each chamber would cause the hot air at the top of the chamber to rise up the chimney and draw in fresh air from the tunnels.

My estimate would be that the main chambers that exist in a loop would be occupiable by perhaps 20-30 humans. However, dead-end tunnels more than ten metres or so long would be suitable only for occasional use without artificial ventilation. Main chambers that exist as a dead-end would be occupiable by far fewer humans, probably less than 10. Chimneys and fires would make all these main chambers occupiable by perhaps 30 humans.

It would probably be possible to make a dead-end tunnel several hundred metres long before lack of air exchange made breathing there impractical without artificial means of providing ventilation. The exact distance would depend on the activity to be performed in the tunnel. However, these tunnels could be made habitable simply by boring a small shaft to the surface.

Answer 2

Problem: If the tunnels are human-carved, how did the workers carve them absent oxygen? Especially if they're going to be using fire for light?

Forced airflow so that tunnels can be dug deeper than they can be supported normally (without artificial airflow) is a possible solution. But unless that tech is lost, whoever built the tunnels can get to the end of the tunnels whenever they want.

The builders would also have to have a reason to want to build/dig the structure. Digging is hard and removing the spoil is difficult, time-consuming, and expensive - especially as your tunnels get longer/deeper. What is/was worth that cost?