# How fast can one mine talc?

A bit of background for this question first. I am working on a high fantasy universe that I have been expanding quite a bit over the months. I have three generic species: Men, Dwarves, and Elves. All of them use their inborn traits and the phenomenon of magic in their own ways. Namely, Dwarves create their stunning stone halls and fortresses using their natural brawn but their own magic, too. Dwarves use magic specifically to soften stone and ore to make it easier to excavate; they soften it down to, say, something like talc.

In order to figure out how long it would take a company of Dwarves to create a mountain hall I need to know how long it takes to excavate talc (either by a m3/hr rate or similar), mostly using older/medieval measures, since that's the comparable level technology readily available in this universe. I hope this isn't too difficult of a query, and I await your answers.

• Have they got explosives? – Willk Jul 15 at 20:14
• We need to know what tools they have and how deep and how wide does the magical talcumisation penetrate etc. Do they have pit ponies and wagons or even rail tracks? How near the surface are they? How do they get rid of the waste? Digging is only part of it. – chasly - supports Monica Jul 15 at 21:37
• What are the physical abilities of the dwarves like? A major component in manual mining is the strength and endurance of your workforce. A dwarf who can work tirelessly shifting stone for 18 hours a day is going to dig a lot faster than one who ends up with pulled muscles after looking the wrong way at a pick axe... – Joe Bloggs Jul 15 at 21:43
• Can they all use the magic, or must they wait for the foreman? Are their picks and shovels magic or is the whole space talcumised at one go? – chasly - supports Monica Jul 15 at 21:53
• How far underground is it? Can they muster high-pressure water jets for the process and pumps for extraction? – Tantalus' touch. Jul 15 at 22:55

From your question, it sounds like Talc was an arbitrary choice whose main purpose was to emphasize the dwarves’ ability to make the rocks soft. So, in this answer, I’ll just assume that breaking the rocks requires a negligible amount of energy, and that the hard part is actually removing them from the mountain.

I’ll also assume that you have lots of dwarves, so that manpower (dwarfpower?) doesn’t become a limiting factor. I’m also neglecting whatever time it takes for dwarves to build safety infrastructure (e.g. wooden supports, pulleys, etc.) because, ideally, they can work on rigging that stuff up in parallel with other dwarves as they mine, leaving the net time unaffected.

Here are some parameters involved in our calculations:

• $$r$$, the radius of a tunnel through which a single rock-carrying dwarf/donkey cart can travel. Given dwarves’ typically small height, $$R\approx 1\space\text{m}$$ should be a good estimate, but you might be able to get it even smaller.
• $$k$$, the number of rock-carrying dwarves that can fit in a unit length of tunnel. I’d guess every dwarf needs about $$2\space\text{m}$$, or maybe $$4$$ to $$5\space\text{m}$$ if they have a mini-donkey-cart as well. So let’s say $$k\approx 0.25\space\text{m}^{-1}$$.
• $$d$$, the horizontal depth (towards the core of the mountain) at which your mountain hall is situated. This is also the approximate distance of a tunnel connecting the mountain’s surface to the underground mountain hall. This really depends on how deep you want to go - perhaps $$D\approx 50\space\text{m}$$ is a reasonable distance.
• $$V_H$$, the volume of your finished mountain hall. A large FIFA soccer field is about $$120\space\text{m}\times 90\space\text{m}$$, so let’s say your mountain hall is about $$120\times 90\times 90$$ or $$9.72\cdot 10^5\space\text{m}^3$$. If you had different dimensions in mind, just plug your volume into the formula at the end of the answer.
• $$V_D$$, the volume of rock that a single dwarf (together with a donkey-cart, perhaps) can carry in one trip. They’re tough little buggers, so I’ll estimate perhaps $$0.5\space\text{m}^3$$.
• $$s$$, the speed with which rock-shuttling dwarves-with-donkey-carts travel. Given that they have short, stubby legs, are carrying rock, and are moving over bumpy terrain, I’d estimate around $$0.3\space\text{m}/\text{s}$$.

Now, before we start calculating, we have a choice: are we willing to dig lots of tunnels in the mountain in order to speed up excavation? This might be disadvantageous later on, because there will be lots of different entry-points through which enemies might sneak into your hall. However, you should be able to fill up the tunnels afterwards, or guard them with dwarves, or maybe you actually like the prospect of having many entrances/exits. We’ll discuss this again at the end.

A good strategy would be to build these tunnels in adjoined pairs - one for entering the hall with an empty cart, and one for dwarves to shuttle rock away (like two lanes on a street). Entry and exit can’t take place in the same tunnel, because entering and exiting dwarves are traveling in opposite directions.

Let $$n$$ be the number of such tunnel pairs. The volume of rock that we must remove in order to excavate $$n$$ tunnel-pairs with radii of $$r$$ and depths of $$d$$ is equal to $$8\pi ndr^2$$. However, since you have unlimited manpower, you can excavate all of these tunnels simultaneously. The rate at which dwarves remove rock from the tunnel will be equal to

$$V_Dsk$$

Which means the amount of time it takes (in seconds) to finish the tunnels is equal to

$$\frac{4\pi dr^2}{V_D sk}=\color{green}{\frac{4\pi dr^2}{V_D sk}}$$

Hang on to that expression - we’ll come back to it later.

Now we get to the fun part - actually carving out the rock for the mountain hall. With $$n$$ tunnel-pairs, we can now remove rock from the main hall at a rate of

$$nV_D sk$$

which means that the amount of time required is about

$$\frac{V_H}{nV_D sk}=\color{green}{\frac{V_H }{nV_D sk}}$$

Add this to the amount of time needed to excavate the tunnels, and we get

$$\frac{4\pi dr^2}{V_D sk}+\frac{V_H k}{nV_D sk} = \color{green}{\frac{(4\pi ndr^2 + V_H)}{nV_D sk}}$$

Let’s plug in the estimated values of these variables that we gave earlier:

$$\frac{(4\pi n(50\space\text{m})(1\space\text{m})^2+9.72\cdot 10^5\space\text{m}^3)}{n(0.5\space\text{m}^3)(0.3\space\text{m}/\text{s}) (0.25\space\text{m}^{-1})} \\ \approx 1.68\cdot 10^4\space\text{s} + \frac{2.60\cdot 10^7\space\text{s}}{n}$$

Let’s convert that to hours. There are $$3600\space\text{s}$$ in an hour, so we have

$$4.67\space\text{h} + \frac{7.22\cdot 10^3\space\text{h}}{n}$$

If you only want to use one pair of tunnels (i.e. $$n=1$$), that works out to about $$7.22\cdot 10^3$$ hours, or $$301$$ days. Keep in mind that this is solid, uninterrupted worktime, not counting breaks. If you have enough dwarves, you can keep the line constantly moving by letting dwarves alternate shifts, but $$301$$ straight days of continuous work is still quite a bit.

If you’re willing to have two tunnel-pairs, this drops to about $$150$$ days. Five tunnel-pairs gets it down to $$60$$ days, and ten brings it to only $$30$$ days. This is where manpower limitations might be a problem, but even for $$n=10$$ you actually only need a couple hundred to a thousand dwarves to keep this process running uninterrupted (even with shifts taken into account).

Here’s a table, so that you can decide how you want to make the trade-off between tunnels and time:

$$\begin{array}{c|c} \text{# tunnels} & \text{days} \\\hline 1 & 301 \\\hline 2 & 150 \\\hline 3 & 100 \\\hline 4 & 75 \\\hline 5 & 60 \\\hline 10 & 30 \\\hline 20 & 15 \\\hline 30 & 10 \\\hline \end{array}$$

So if you want to get it done really fast, dig lots of different tunnels (but your mountain hall will look like swiss cheese when you’re done, unless you fill most of them in afterwards). Otherwise, use a few tunnels and it will take a couple of months.

Like I said before - if any of my parameter estimates at the beginning seemed unrealistic, feel free to tweak them yourself and reevaluate. And keep in mind this is all only a ballpark estimate, and it could go a lot slower depending on any logistical/plot hiccups that occur.

• A data-point that may be useful: According to a Wikipedia article, a pit-pony could haul 30 tons of coal in an 8 hour shift, in tubs on an underground railway. See – Patricia Shanahan Jul 16 at 0:09