So there are a number of variables involved here, but I'll try to help you out:
- Standard base has 8 mineral patches
- Each mineral patch has 1500 minerals
- If you have 2 SCVs on a patch they harvest ~90 minerals per in game minute (depending on distance)
- Standard base has 2 Vespene Geysers
- Each Geyser has 2500 gas
- 3 SCVs harvest at 101-114 gas per in game minute
Here is the problem: when you have more than 2 SCVs per patch they start to ping pong around between patches. At this time your SCVs harvesting rate starts to become indeterminate. However, we can calculate an upper bound on the time it takes if you have at least 16 SCVs mining minerals. 1500/90 = 16m 40s. We can further provide a better estimate based on the fact that, on average, 3 SCVs harvest at ~102, giving us 1500/102 = 14m 42s. Gas takes understandably longer to mine out because of its higher amount, on average: 2500 / 107.5 = 23m 15s.
Both of these numbers are further influenced by: Distance of minerals to based, number of workers transfered, distribution of workers transfered, build rate of new workers, use of chrono boost and infuse larva, number of worker producers, resource gain rates, and MULES.
Its also worth noting that the influence of workers beyond the 16th is significantly reduced according to this table:
- One SCV on one mineral patch harvests 39-45 minerals per game minute, depending on distance.
- Two SCVs on one mineral patch harvest 78-90 minerals per game minute, depending on distance.
- Three SCVs on one mineral patch harvest ~102 minerals per game minute.
Gas is similarly influenced:
- One SCV on gas harvests 33-42 gas per game minute, depending on distance.
- Two SCVs on gas harvest 67-84 gas per game minute, depending on distance.
- Three SCVs on gas harvest 101-114 gas per game minute, depending on distance. In the case of far-diagonal gas placement you will need 4 SCVs for full saturation
You'll notice that going from 1->2 per patch is a significant increase in gain, but 2->3 is less so.
A longer explanation of how to fully saturate can be found here: Whats the correct number of workers to put on gathering vespene and minerals in?
It is also worth noting that when you expand should not be determined by how long it takes a base to mine out, but rather when you are capable of doing so. This usually requires map control.