In order to promote an Adventurer from 4 stars to 5 stars, a total of 25,000 Eldwater is required, compared to the 2,500 required to promote from 3 stars to 4 stars. The only ways I've found that you can obtain Eldwater is through getting duplicate Adventurers in summons and the roughly 100 you get from daily endeavors.
Now the issue is that you get very little Eldwater from duplicate summons (500-1000 if I remember correctly) and only 100 from daily endeavors. Without spending any money and only doing the daily endeavors, that's 250 days worth of endeavors. If we include summons through Wyrmite, in this example tenfold summons, you maybe get 2 Adventurers per summon. Assuming both of those are duplicates, that's still only 1000-2000 Eldwater per summon. To reach the amount of Eldwater required for a promotion using only summons, that's still 13-25 tenfold summons.
The amount of Eldwater required to promote from 4 stars to 5 stars is so extreme that I feel like I'm missing some important or simpler way to obtain the resource.