Also, related question: 1) how many days could an army last without supplies IRL? I'm talking completely, 100% out of supplies. Would the army simply starve to death/surrender within a week or two? In other words, were WW2 armies capable of foraging (or overrunning another army and taking their supplies) effectively?
question 2) is this the time frame that's measured in HOI3?
The two should completely match up for HOI4. If someone muffs up an invasion and fails to supply a landed army, that army should die (or surrender) exactly as swiftly as it would have in real life. I haven't played HOI3 in a while, but if I remember correctly it feels like armies last longer without supplies than it feels like they should.
I think 30 days is too much too.
Well, playing with stuff that came around after World War 2, the Vietnam Era Amphibious Assault/Support Ships(some of which dated back to WW2), which are actual combat ships, rather than simple supply ships, could either carry troops to the shore, or carried the means of getting to shore with them, and all the supplies necessary to fight for up to approximately 1 month of "significant combat" before needing a resupply. During which time the ship they attacked from would be serving as a command post and supply depot for the troops. They wouldn't actually have most of those supplies on them in the field, at least to start with the ship would still be holding them, until the situation was such that they deemed things as stable enough to offload them to a secure location(at which point those ships could presumably leave to resupply and bring back another month's worth of supplies, presumably in much less than a month's time).
So the answer to #1 is "it depends" with the answer being contingent on how large is the defending force vs what are they being attacked with, and how much time did the defenders/attackers have to "stockpile" supplies prior to their supply line getting cut? What kind of facility do they have to take refuge in? There are historical examples of sieges that literally lasted for decades. For a less outrageous one, British forces held up against a combined siege from France and Spain for roughly 3.5 YEARS at Gibraltar with only the very rare chance for any supplies to get through from 24 June 1779 to 7 February 1783.
Basically, if the Navy is able to land troops somewhere, they're going to be able to get supplies to them in sufficient quantities to support them. Even if the means of getting supplies to those troops lies outside the scope of more traditional methods(a seaport or road/railways). Modern "untraditional" supply methods range from airtransports doing airdrops(or landing and unloading, kind of an option in WW2), the large cargo helicopters performing ferry services(not an option for WW2 era), to LCAC's(again, not really an option for WW2, although possible with a jet turbine engine), to the WW2 version of the LCU(Landing Craft Utility, which according to wiki would have been the LCT, LCA, LCM and NL), the LST(Landing Ship Tank --which were predominately WW2 era beasts designed to be run aground on sandy beaches for the purpose of unloading tanks(and transporting the previously mentioned earlier iterations of the LCU) and serving as a beachfront command and control post), Amy Ducks(many of which are still used as aquatic touring craft in many locales today), and the various and sundry hodgepodge of (shallow) watercraft they employed during D-Day and other invasions to shuttle material between ships and shore.
It should be noted that the LST, unlike the others mentioned above, was a ship in its own right, and not dependent on another ship in order to make an oceanic transit, although it's ability to defend itself at sea would be obviously limited, so escorts would be a good idea when loaded if nothing else.
http://en.wikipedia.org/wiki/Landing_Ship,_Tank
So to be more "historically accurate" it would be "if an ampibious assault ship is operating off the coast of a territory with a shoreline of _____ types, the assault ship(s) should be able to provide supplies to the adjoining troops on land." (up to its rated "carry capacity" anyhow, so long as it is likewise in range of its own resupply chain) But that goes back to naval and air supremacy kind of being important. That assault ship isn't going to last very long if an enemy battleship or enemy Air Wing starts gunning for it and nobody else is around to protect it.