please, can you be so kind and tell me where l can fill the flask? l try many times in the beach and nothing happened! thank you

Face the creek on the side of the mine.
Go close to the bridge and you will have the hotspot to get water by the left side of the bridge close to the shore.

Walkthrough:
Take the empty flask from inventory and click it on lone rock by edge of beach or at start of the plank bridge and Ariane will fill the flask with water.