How deep could dwarves actually delve?

Question

Given an arbitrarily long time to do so, what is the maximum depth too which a dwarven civilization could practically extend their mines / cities? And what would be the final limiting factor preventing further expansion of the downwards frontier?

I figure eventually they'll hit the Mohorovicic discontinuity and be prevented from going further by the fact that the rock will not remain sufficiently solid (and that's if they aren't killed by the heat at that depth first), which puts an absolute upper limit around 32km. But what insurmountable problems would they hit before that?

Assume that over sufficient spans of time, any necessary technologies can be developed or acquired; these can be fully modern dwarves, with advanced knowledge of fancy materials and engineering. They need not be stuck in a Tolkienian pseudo-Medievalist culture. Additionally, assume sufficient time for the dwarves themselves to evolve to adapt to conditions at depth, to whatever extent biological adaptation remains plausible.

Answer 1

The best way to determine this is to look at the real life maximum depth, which is the Russian Kola Superdeep Borehole, which reached a depth of 12 km at which point it was considered infeasible due to problem and was called off. And the problem? Heat, mainly. Once it got deeper than 10km it was a staggering 350 F, at which point they stopped drilling, because they projected the temperature at 212 F. Additionally, the rocks were porous and permeable from the heat, which gave it a rubbery texture, making drilling difficult bordering on impossible. Also, the hole was 9 inches in diameter.

That being the case, they could have gone deeper, but the project was aiming at a depth of 15,000 meters, and called it off because they considered that infeasible. Given that, a depth of 13,000 meters seems plausible for your fully modern dwarves.

Answer 2

The logistics of a super-deep city is very non-trivial.

A normal car engine uses 9m3 of CO2 per minute, or 0.15m3 per second. If you have a pipe that has a diameter of 0.55m (an area of ~1m2), the air will move at a leisurely 0.5kph. That's not to bad, right? But it's a single engine. If you have a whole city underground, you need a lot of air. 10 cars and your pipe now runs at 5kph. 1000 vehicles and your 0.5m pipe is at 50kph. In a modern city, cars are super-common (80% of adults have a car). So if you have a city of 10,000, you've got 8,000 cars, and you need 1200m3 per second. If you have a 10m wide ventilation shaft, the airflow will be about 3.8kph.

There is a mass difference between CO2 and O2, and this means that you need to pump that mass difference up your ventilation system (an extra 0.6kg per m3 of CO2), and where that mass comes from? Burning the fuel in engines introduces the carbon into the air. By the time you're 10km underground, you have a pressure of 58kPa of pressure from the mass difference. This is about the same as a soccer ball - not to much, but way more than a simple fan can produce. You actually need a pump that can provide pressure.

A long pipe has fair amount of resistance. Blowing air through a drinking straw is easy enough. Blowing air through a 50m hosepipe is another matter entirely. By the time you've got 10km of depth, you're going to need a fairly enormous amount of pressure just to drive air down there. You're going to need stages of air-pumps. But those air-pumps need engines to drive them, so you end up with a situation similar to the rocket-law. For every extra pump you install near the bottom, you need a whole bunch of pumps leading up to the surface.

You are moving literal tonnes of air under reasonably high pressures up to and down from the surface. I honestly can't imagine the scale of a system needed to drive air down to a city 10km underground. It would involve power plants, HUGE ventilation shafts, and endless arrays of turbines after turbines.

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Then there's the other logistics: water, food, transport. How do you get people to the surface? Elevators are limited to about 500m by the tensile strength of the cables (a long cable will snap under it's own weight), so you'd have a chain of 20 elevators to go 10km underground. At each stop you'd have to get out, wait ages for the next one to arrive and ride it down. Seriously, just build a 100km road with a gentle gradient and set up a bus system. Great, more area that needs ventilation.

Supplying water is another interesting one. A normal human can use way over 100 liters of water per day. Assuming a society where water is precious, you can get way lower usages, but probably 30 liters per day is pushing it for a good level of hygiene. So to supply 10,000 people, you need to provide 300,000 liters of water per day. Underground streams? I consider that unlikely at 10km depth (but in all honesty I know nothing about geology). Running a pipe vertically for 10km will create a pressure of 100MPa. That sort of pressure is lower than the tensile strength of steel, but you'd probably want a bunch of stop-off dams mid-way down to avoid having ridiculous wall-thickness pipes. More amazingly massive underground infrastructure

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For reference, the deepest mine us humans have built is only 4km deep. You can read about it here: https://en.wikipedia.org/wiki/TauTona_Mine but they need air-conditioning to cool the environment, and it takes an hour to get to/from the surface. When reading on other deep mines, you find quotes like:

The mine also built the world's largest ice factory which produced up to 8,000t of ice daily to cool wall rock temperatures (50-60 deg Celsius).

8,000 tonnes of ice? Wow. It takes a serious amount of energy to do that.

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So, how deep? I have no idea, but the infrastructure your dwarves would build in their pursuit of depth could easily dwarf anything we've built today (pun fully intended).

Answer 3

Depth, probably a few kilometers. Barrier to further digging: lack of motivation.

Given that you've removed the classical/stereotypical depiction of dwarven civilisation, what you're actually going to end up with is technology which bears more resemblance to modern skyscraper construction and naval/space engineering. Why? Because expansion has a cost. Whereas surface humans generally prefer to take the path of least resistance and expand over the surface to a height of no more than a few tens of metres, expansion of your dwarven city in any direction is logistically difficult and expensive. That's likely to result in a lot of efficiency and compression in the architecture and engineering.

(Modern) Dwarves won't exchange air with the surface, they'll scrub and reprocess it in situ, with 'local' utility stations serving a small section of the city (more like the earliest power stations which were geographically localised rather than modern national-level networks). This provides both efficiency (balancing economy-of-scale against the costs of distribution of fresh air) and redundancy, allowing areas to be evacuated in the event of equipment failure. Access and transportation of people and materials is always a problem in large systems (see for instance the Elevator Conundrum in skyscrapers); but you have much more three-dimensional space to get clever with logistical solutions: if the -500m arterial highway is always congested, building a bypass parallel to it 20m deeper is not noticeably more expensive than the original.

As such, the question of how deep you could technically build is basically equivalent to how extreme an environment you could inhabit, and given an unlimited supply of clean energy (nuclear fusion or another source of energy with minimal volume of inputs/outputs is a must) and some fancy materials science you can manage quite a lot. Certain construction techniques would be different - dwarves would probably steer away from concrete because it's very voluminous and energy-intensive to move around and managing the carbon dioxide flux as it's made and cured would be a pain, but unlike a skyscraper, a shaft wall doesn't have to support all its own weight as it descends, so could be much more heavily-built.

Eventually, heat will become an overwhelming obstacle, but moving increasingly vast amounts of heat away from your structure is fundamentally an engineering problem, not a physical limit. Heat pumps which transfer the heat from a small, deep area to a larger, shallower area (with further pumps to take the heat ultimately to the surface), or generators which convert the thermal energy into electrical which can be transferred far away by superconducting wires. Digging deeper will be the same fuzzy frontier as building taller skyscrapers is for modern humans: there is no fundamental limit in what we could build, we are limited by what we choose to build (or rather, what our economic system chooses for us).

Hence, a dwarven society will stop digging deeper when it becomes uneconomic to do so, because the increasing rewards cease to outweigh the increasing engineering costs. While there will always be intrepid members of any society wanting to push the boundaries (and dare I say possess the biggest shaft? :p), a frontier mindset will only take you so far when heavy engineering is required. For the majority of society, in stereotypical places-where-you-find-dwarves geology there's no major advantage to going deeper rather than wider, and so there'll come a point where they just won't need to.

If you take a very generous space estimate of 0.1ha to support a dwarf and an appropriately grandiose 4m average cavern high to translate that into 4000 cubic metres per dwarf, add another zero for inefficiency and for the caverns to be relatively sparsely distributed through the volume of the rock, you find that a dwarven civilisation could comfortably have a density of 25,000 people per cubic kilometre. This compares quite favourably with the most densely-populated human cities assuming only one kilometre depth, where only the top ten or so exceed that density. Digging just two kilometres down would give the city world-record-holding surface density, and this assumes that the civilisation is entirely self-sustaining, with no importation of space-hungry food from the surface.

Answer 4

About 4km after that it is too hot.

The biggest hard limit on how deep humans can mine is heat, the deeper you dig the hotter it gets, below a certain depth it is just too hot to keep humans alive for more than a few minutes. we can dig further but humans can't go down them, they have to be automated.

The heat limiting depth averages around 3-4km down on continental crust, quite a bit less on oceanic or particularly volcanic continental crust. Cooling is ridiculously impractical because the rock acts a huge heat sink and conduction of heat form rock to rock is too efficient, you basically need to be cool billions of tones of rock constantly because more heat from the core is being conducted out.