I know this may sound crazy, but when your eyes water like that it could be dry eyes.....https://www.nei.nih.gov/health/dryeye/dryeye.asp
I go thru periods where mine water no matter what position I am in, my Dr told me dry eye....I was shocked as they feel very wet...lol...but if you look at the symptoms in the link above you will see watery or tearing is part of it....it can be med related.....
I have a thyroid condition, and take antihistamines...and those are just 2 causes of dry eye.....
My eyes water a lot, even when I am just carrying on normal activity. I also have some burning sometimes like when you have sunscreen on and it melts into your eyes. I am not aware of any specific cause, but I hope yours gets better. I can get quite embarrassed sometimes - people think I am crying. Interesting, that you experience it when you lie down. It must wake you from sleep.
My eyes always water when I lay my head down. Like I am crying. I doubt it is a chiari symptom though.