Case Study: American Civil War
The American Civil War happens to be quite a good case study for numerical analysis of supply requirements, and this is for two reasons.
The first is that, in the Civil War, there were only three modes of transport: rail, water and wagon.
The second is that there has been actual mathematical analysis of the required transport, and it helps to explain a lot about the tactical and (especially) strategic movements of the Civil War.
Part 1: To the Supply Dump
For now we'll assume that there's unlimited supplies available
somewhere in the country. This is more true for the Union mid-war than anyone else in the ACW, but it's useful to show how much the transport issues are a bottleneck.
The supply dump (or base of supply, depot, or any of a number of other terms) is the place where an army draws their supplies for the last short distance (though that's a relative term). It can be a waterside landing (as was used in the Peninsular War) or a rail head, or ideally a full port (as a port has so much more capacity).
The first kind of problem that can occur is that not enough supplies can reach the supply dump. For this it's worth considering the scale of the problem:
Roughly speaking, a combined-arms army requires 1 ton of supplies every day for every 200 men, or 50 tons per 10,000 men (and the count is total men present, not merely men holding muskets/rifles in the firing line). This is gunpowder, lead, food, shells, but the vast majority is animal feed (because an army without horses is an army without artillery, cavalry or mobility, and an army without any of those things is promptly dead.)
This is quite a lot for a train line to support, more than one might think - a typical single train is 40 cars, and travels at 10-15 miles per hour in this period (smaller trains with fewer cars are faster but carry less).
Each car is about 5-10 tons, depending on gauge, and the real devil is in the details of the railway - a double tracked line can handle it all quite easily, but a single tracked line with sidings is much harder. A single tracked line without sidings is awful - in some cases you have single tracked lines 150 miles long without sidings (though few if any were used in the OTL ACW, one might have to be used for a Trent war) and that basically means all you can do is run a train there and back once a day, allowing for some pretty frantic unloading at the far end... and that means you're delivering enough for a couple of divisions.
With riverine support you can move supplies nearly as fast, but more importantly you can move them in far greater bulk. A few barges arriving per hour is very useful.
Part 2: From the Supply Dump
This is where we can get really mathematical.
There are two ways of supplying an army at this time, and both involve wagons.
1) Circuit supply.
This is where the wagons pick up their supplies at the depot, head to the army, and drop off their supplies before heading back home again in a circuit. What matters here is the distance, and the rate of advance the wagons can manage - since overworking horses kills them very quickly, this is not very high. (An epidemic of foot-in-mouth hit both the armies of McClellan and Lee in quick succession during 1862 - it's during this time that Lincoln made his famous quip about borrowing the army for a short time (a little unfair to McClellan, whose transportation was crippled and thus couldn't move) while Jefferson Davis just purchased more horses.)
For good roads and good weather, the wagons can move fifteen miles per day (one "daily march"). For poor weather it can get much worse, down to a few miles per day at most cross-country. The reason why this matters is that, if you have half the speed of your wagons, you need twice as many wagons to deliver the same supplies to the fighting front... and you need more fodder for those wagons, too.
It turns out to be a mathematical relationship, worked out empirically (and found in
Hagerman). For an army of 100,000 (500 tons per day) operating M days march from the depot, W wagons are required:
M W
2 1440
3 2260
4 3140
5 4105
6 5150
7 6280
8 7500
9 8815
10 10230
Why does this matter, particularly, apart from the titanic number of wagons you need by the time you're ten days out?
Well, each of those wagons requires about six horses (IIRC) and, much more importantly, six men to run it on a long-term basis. By the time an army's operating five days from base, 25% of it is on the supply chain alone - and in bad weather, that can be as little as 20 miles. (This happened OTL on the Peninsula, and is something important to consider in, say, a Trent War counterfactual because Canada is very muddy indeed during the thaw).
It also indicates why you really, really need to be careful with disease. Diseased men still consume, but the requirements of the wagon train demand healthy men.
So much for circuit supply, for which the problems scale with the size of the army. What about how smaller forces are more mobile?
The answer is simple... you take the supply wagons with you, and forage.
2) Self Contained Supply.
This method has several downsides, and one of them is that you can't use it long term - the American countryside is simply less well populated than the European one - while another is that you can't go back over the area you used previously. The most famous one, Sherman's March to the Sea, relied on moving during the harvest and on dividing into four columns too far apart for practical mutual support - fortunately there was no-one in the way. In the end he made it to the coast (where he quite gratefully resumed conventional supply) and if he'd been blocked from getting there his army would probably have disintegrated.
For this, one assumes that you start with a given number of wagons, and that you send them back to base as they're emptied. (This means you don't have to keep feeding the draft animals).
The relationship here is more complex, and in Hagerman is simulated based on certain assumptions (one of them a wagon size a little over the typical, at 3,000 lbs per wagon instead of 2,000-2,500, and another that half the animal feed can be found by forage). The results are below, again for an army of 100,000 men:
Days march/ Wagons remaining/ Wagons detached
4.5/ 1300/ 1255
9.5/ 2920/ 2500
14.3/ 4980/ 4080
19/ 7780/ 5740
23.8/ 11670/ 7570
This means that setting off for a 24-day march out of contact for an army of 100,000 men basically means starting with 19,000 wagons - your army is pretty much a wagon train. It also means that you're out of supply at the end of the whole thing.
For a very small force of a few thousand (the kind of thing that happened in the Revolutionary War) it's much, much easier to move without a supply line - it's just so much easier to forage, with an army of 2,000 requiring maybe two tons of food per day (and with four months' winter supplies for a village of 50 people consisting of about a week's food for the army).
Why does this all matter?
Well, it's interesting. And it also explains just why McClellan had to retreat from Richmond in the Seven Days, for example - his supply line, already long (he'd been ordered to fix his supply dump) was blown away and he had to retreat to get back in supply.
It also explains why the movement of most civil war armies is quite slow - they're simply having to re-establish a new base of supply every couple of dozen miles.
Put this way, it's easy to see why the petrol motor was such a massive advance for armies. It made them far less dependent on fodder tonnage, and faster to boot - and it tied up fewer men, with less need to care for horses. The Red Ball express consumed 6,000 vehicles to deliver 12,500 tons of supplies per day, to distances of up to 250 miles - to do the same with Civil War logistics would take, under ideal conditions, 527,000 wagons (or in other words about three million horses).
