In the movies, such as 2004's Troy, we see archers firing arrow upon arrow into an oncoming army. How accurate is this in reality though? Wouldn't they run out of arrows after just a few minutes? What happened when archers ran out of arrows during a siege? And how well could arrows be stockpiled to prevent such a shortage?
I don't have any specific time period in mind, just any time when archers were still in common use. Perhaps the period of the English Longbowman.
When you are talking about the period of the English Longbowmen you are essentially talking about Henry V and his campaign in France culminating with the battle of Agincourt first and foremost, and I can tell you for Henry stocking enough arrows for his archers was a huge task requiring a national effort.
England was bankrupt and haphazardly managed when Henry V took the throne, and he spent much of the first part of his reign reforming every part of the government. The greater part of this involved consolidating his recently reconquered Welsh holdings and doing a massive audit and restructuring of the Royal estates, both of which generated a tremendous amount of ready cash. He also started reforming his military on the supply and organization side.
You are absolutely right that archers could run out of arrows in a few minutes of battle. An archer had to be able to fire ten arrows in a minute, and good archers could fire much quicker. Arrows were carried in sheaves of 24, archers carried 2 sheaves on them and a few spares, so a typical English Longbowman of Henry V's day could expend their entire combat load in between 4 and 8 minutes of steady firing. Realistically they probably would not fire at max rate for a stretch of 4-8 minutes very often, but still, they needed an ample supply to fight a battle lasting hours.
The arrows themselves are difficult. They are made of multiple pieces which each have their own supply train and manufacturing difficulty. For long range arrows the arrow head is made of iron, but two compounds, a hard edge and a softer core to absorb shock on impact, a 30 inch shaft constructed of lighter wood which has to be basically perfectly straight or it's useless. The fletchings are made of goose feathers, and this requires the plucking of a lot of geese. Shorter range arrows are a little shorter, heavier, made of a thicker piece of hard wood like ash and have a head called a bodkin which is very narrow for penetrating armor at close range.
Almost immediately on taking the throne Henry V appointed fletcher (a professional arrow maker) Nicholas Mynot Keeper of the King's Arrows and set him up in the Tower of London with a staff and healthy budget for building up stocks. This was a separate job from the King's Bowyer, responsible for making bows and with the right to commandeer any wood in the country for that purpose. The Keeper of the King's Arrows began production and farmed production out to other fletchers, making orders for arrows in the tens of thousands.
The production of arrows required arrow heads made by smiths by the barrel, tens of thousands of shafts, and goose feathers literally in the millions (there is an order recorded for over 1.1 million goose feathers for the King's arrow making facilities, and that is one single order). Because of all this planning, effort, and expenditure of resources Henry V was exceptionally well equipped when he set off for France. I would caution you though that while Henry V is the obvious example for a question on the logistics of English Longbowmen he was an uncommonly capable administrator and not every monarch would have been so well prepared for a campaign.
Robert Hardy's Longbow: A Social and Military History goes into a lot of detail on all things Longbow related, I find Juliet Barker's Agincourt very valuable for information on Henry V's campaign preparations.
The example that springs to mind for me is the battle of Carrhae in 53 B.C., where the ability of the Parthians to replenish arrows was vital to defeat the Roman army led by Crassus. The Romans formed a square formation, and instead of attacking the Roman army head on, the Parthians surrounded the Romans and began a continual hail of arrows from all sides.
'But the Parthians now stood at long intervals from one another and began to shoot their arrows from all sides at once, not with any accurate aim (for the dense formation of the Romans would not suffer an archer to miss even if he wished it), but making vigorous and powerful shots from bows which were large and mighty and curved so as to discharge their missiles with great force. At once, then, the plight of the Romans was a grievous one; for if they kept their ranks, they were wounded in great numbers, and if they tried to come to close quarters with the enemy, they were just as far from effecting anything and suffered just as much. For the Parthians shot as they fled, and next to the Scythians, they do this most effectively; and it is a very clever thing to seek safety while still fighting, and to take away the shame of flight.'
Plutarch, Life of Crassus 24.5-6
Crassus counted on being able to hold the Roman formation intact with minimal casualties until the Parthians ran out of arrows, but his plan led to disaster as the Parthian general Surena had planned his supply chain with great foresight, and had numerous camels available to carry arrows to the archers.
'Now as long as they had hopes that the enemy would exhaust their missiles and desist from battle or fight at close quarters, the Romans held out; but when they perceived that many camels laden with arrows were at hand, from which the Parthians who first encircled them took a fresh supply, then Crassus, seeing no end to this, began to lose heart, and sent messengers to his son with orders to force an engagement with the enemy before he was surrounded; for it was his wing especially which the enemy were attacking and surrounding with their cavalry, in the hope of getting in his rear.'
Plutarch, Life of Crassus, 25.1
So in this case, the Parthians were ready to replenish the arrow supply for the front line via camel. While the Parthians were only resupplied once in Plutarch's account, it is likely that the camels could have returned to the supply train to collect more.
One thing to keep in mind was the reuse of arrows. Arrows were commonly gathered after battles and retooled or simply re used if they were not in poor shape. This lead to a couple things, first it lowered the cost of needing new arrows, second, if the arrow struck ground or other unclean things, it allowed for a build up of bacteria on the arrow, that way, even if this arrow hit someone in a non vital area in a future battle, it could cause an infection and ultimately disable the unit it struck.
To answer your question further, there was a similar question here: http://www.reddit.com/r/AskHistorians/comments/1e7xa3/
I'm surprised no-one has mentioned the "straw boats borrow arrows" incident from the pre-Three Kingdoms period in China, c. 208. In short, one side found itself short of arrows, and by a long way: they needed something on the order of 100,000 arrows, in anticipation of the largest naval battle ever fought (by number of ships). So, choosing a foggy evening, one side loaded straw and straw figures onto numerous boats, tied to ropes. The other side, seeing the boats emerging from the mist, and thinking they were being attacked, naturally called for their archers, who shot tens of thousands of arrows at the 'attackers.' The arrows stuck in the straw, and the 'attackers (who eventually won the battle), retrieved the boats, and all those arrows.
The incident, and the entire Battle of Chibi, was dramatized in the movie Red Cliff. You can read more about it here: http://www.theepochtimes.com/n2/china-news/chinese-traditional-culture-borrowing-arrows-6542.html
While the battle was real, its exact setting is still a bit of a mystery. Was this incident romanticized? Possibly... perhaps in the way that incidents at Agincourt were romanticized in Henry V. But as in Shakespeare, most of the incidents are grounded in some reality. Did Cao Cao have over 800,000 men in his (eventually defeated) force? Also difficult to nail down. But we do know that many of the players were actual historical figures: Cao Cao, Liu Bei, Zhou Yu, etc.
The Japanese had a fascinating approach to battlefield archery. The archer would have several quivers, some were lightweight frames, but for fixed battles sometimes a quiver with 50 small compartments was used.
There were several classes of arrows, some for armor, some for tearing flesh. All made of different woods for different purposes like distance or penetration. The heads varied as well.
The samurai would have a silk parachute hidden on their back, and if they had to flee could deploy it and it would fill with air and act as armor, depleting the energy of the arrow.
The arrows were a considerable expense.
http://www.ncjsc.org/gloss_yanone.htm
In early periods before long range bows, the archer was the foremost part of any formation and would engage at less than 30 feet. As bows evolved they made their way further back into the formation and the tactics changed.
I can imagine this was especially hard for armies that consisted almost solely on archers. Take the Mongols for example, does anyone have a source for them?
I'm going to link you to this previous discussion that elaborates a bit on both English and Byzantine archers. http://www.reddit.com/r/AskHistorians/comments/1j7111/how_many_arrows_did_the_average_archer_carry/
One thing you might find interesting is that according to the Praecepta Militaria of Emperor Nikephoros II Phokas, the Byzantine Imperial Armories were expected to equip expeditionary armies with upwards of 720,000 arrows.
Additionally, De Ceremoniis Aulae Byzantinae lists in the manifest for the Crete expedition, a supply of 10,000 arrows for a mere 50 "Roman bows". So, it would appear that archers were rather well supplied by the Byzantine military.