Normalizing data of different observation points

I want to normalize my data, but I am too stuck with the specifics of it.

I have observations in weekly intervals from different groundwater wells, and as expected they have a different range of values throughout the whole time series. Currently I have a vanishing gradient problem in my model, so I thought maybe normalization/scaling will work.

However the problem is, I do not know how to scale my data. I will either use min-max scaling or z-score. Should I use a separate scaler for each well, or should I use one global scaler for the whole data? And if yes, how do I make sure that the model behaves correct when there is an observation from a well with a totally different range? Or this question is just irrelevant?

A little bit more information about what I want to do: I have ~780 weeks of data, and I would like to predict the next weeks’ groundwater level given the previous n weeks.