How long can creatures with no swim speed travel in deep water before they die?

Question

After answering this question about how deep can PCs go underwater, I started wondering how long can you keep swimming before you die. Here are the constraints:

• You can breathe underwater (ex, via magic items or spells)

• Your speed is 30 ft and you have no swim speed

• You are diving straight down, and the water is deep enough to accommodate your travel

• You are not benefiting from Adv/D.Adv beyond what Exhaustion grants you

• You die only when you reach an Exhaustion level of 6

• The RAW is followed, note we are not trying to be simulationist

Asking for a 100% chance of death is typically not useful, and in this case it is intuitively not (a +30 minimum for Con save guarantees survival, but that isn't interesting because nobody can normally achieve that).

So, to be useful, how long can a creature swim according to the constraints above before they have a probability of death equal to 95% or greater?

Answer 1

They will not die

Once they reach exhaustion Level 5 their speed drops to 0 and they can no longer swim and therefore no longer risk more exhaustion.

How far can they go before this happens?

You really need to change your handle if you can't work out a simple markov chain probability calculation like this one :-)

Assume your modifier to your exhaustion check is \$n\$. Your chance of passing a DC\$X\$ check is \${21+n-X}\over{20}\$.

Once you fail your third check you have disadvantage on your checks. This reduces your chance of passing to \$\left({21+n-X}\over{20}\right)^2\$.

You cannot go any further once you fail 5 checks as your speed equals 0.

You can Dash giving you a speed of 60 feet per round, however, water is difficult terrain for a non-swimmer so this means you can only move 30 feet.

If you swim straight down you spend 3 rounds at less than 100 feet, another 3 rounds at less than 200 feet and after that you are deeper than 200 feet.

You need to check based on swimming each hour at DC10, that is every 600 rounds.

You need to check for a forced march at DC10 after 8 hours equivalent. This is after 3 rounds at less than 100 feet, 3 rounds at less than 200 feet and 1198 rounds deeper than 200 feet, that is, after 2 hours 24 seconds.

You then need to make forced march checks every 150 rounds (15 minutes) at DC11, DC12, DC13, ...

This is a Markov chain with 1 starting state and 2 intermediate states (that are all have the same transition probabilities), 2 more intermediate states (that are all have the same but different from the first group transition probabilities) and 1 terminal state. The only complication is the transition probabilities change over time.

Putting this altogether and calculating to 2 significant figures (i.e. a less that 1 in 200 chance of doing better), a person with a +0 Constitution modifier and no proficiency in Constitution saving throws can go 14.5 miles in 4:15:24. A person with +5 Constitution modifier and +6 proficiency bonus can go 24.7 miles in 7:15:24.

Given that the Mariana Trench (the deepest part of the Earth's ocean) is just under 7 miles deep this is not a major impediment. Indeed, even the +0 wimp has a better than a 50% chance of getting to the bottom with 1 or less levels of exhaustion.

Spreadsheet

Answer 2

Diving straight down could take no effort

I didn't find RAW about diving rules, and I know you didn't want a simulation, but D&D is modelled approximately on real life

Without protection, we will die at around 300 feet

So, will we die or not? If we breathe and have no protection against pressure, according to this link without protection from this pressure, Humans cap out at around 300 feet in depth:

At depths beginning around 100 feet, it gives a sense of euphoria or drunkenness. By 300 feet, it causes loss of consciousness and death.

Someone in heavy armor would have the easiest time diving straight down, because they are more dense than someone in no armor. Weights could also help make a dive easier.

See this source for information about Ocean water density.

We also need to know whether we're dealing with salt water or not.

The density of pure water is 1000 kg/m3. Ocean water is more dense because of the salt in it. Density of ocean water at the sea surface is about 1027 kg/m3.

There are two main factors that make ocean water more or less dense than about 1027 kg/m3: the temperature of the water and the salinity of the water.

Any swimming effort would be counteracted by the density difference. Real-life divers use Buoyancy compensators to control this.

Anyone who fought against a very large buoyancy difference would quickly tire out, but someone able to control their buoyancy could use next to zero effort to dive or ascend.

RAW - absolute cold and crushing pressureh

Plane of Water DMG 56

... the absolute cold and crushing pressure mean a swift end to creatures accustomed to the surface or the Sea of Light.

But...

If you still think we should ignore pressure/temperature since very deep water is not the Plane of Water, then I point you over to Dale M's answer with the caveat that movement speed would have to be 15 per DMG 252:

Dashing: A chase participant can freely use the Dash action a number of times equal to 3 + its Constitution modifier. Each additional Dash action it takes during the chase requires the creature to succeed on a DC 10 Constitution check at the end of its turn or gain 1 level of exhaustion.

Also, Dash is only mentioned in the PHB in the combat actions section. I'd like to see how far people could get at half the speed based on his calculations and on the forced march rules. So perhaps we could even allow speed of 20 (40 / 2).