Wasser läuft / fließt in die Küche - water comes from outside or from another room. (Storm, water pipe break)
Wasser läuft in der Küche - is very uncommon. But says that the water tap or the water connet supply works. (Ich habe den Wasserhahn repariert. Das Wasser läuft jetzt wieder in der Küche.)
As I understand it, it's only dative when it's happening in that place: The cat is jumping (in place) on the table, the person is sitting (still) on the chair, the water is running inside the kitchen (say, in the sink). It's accusative when it's going from one place to another: The cat is jumping onto the table (from the floor), the person is sitting down onto the chair (from standing), the water is running into the kitchen (from outside). In this example it's accusative, running into the kitchen, not within the kitchen.
Thanks for this. I'm seeing a trend that i had never considered before that must be painfully obvious to native speakers -- the use of the word TO in the English translation in the accusative indicating what is acted upon by the subject of the sentence. Thus:
The cat jumps onTO the table. The person is sitting down onTO the chair. The water is running inTO the kitchen.
I just went back and reviewed the definition that The Accusative is the Direct Object in the sentence - or the person or thing being acted UPON / the object TO which the action is being done / directed in a sentence. So,
The cat lands on the table. The person puts their self on top of the chair. The water flows across the threshold into the kitchen.
Am I correct in this? I'm a new learner and want to be sure I'm interpreting this correctly
I wouldn't really relate it to direct objects; it's more of a separate idea on its own. If you used another two-way preposition like "unter," there's not really a sense of acting on the object ("Die Katze läuft unter den Tisch" isn't really acting on the table). We also can't say "-to" with all prepositions ("underto the table" is obviously nonsense), so you can't always rely on that either. ("Into" and "onto" are really the only ones.)
But if it helps you to remember "in" and "auf" that way, go for it. The general rule, as moiraslana said, is that destinations (going from somewhere not in/on/under/etc. something, to somewhere that is in/on/under it) use accusative, while locations (remaining in/on/under the thing the whole time) use dative.
So you can memorize a list of two-way prepositions (like this list) and pretty much apply that rule to all of them.