Seriously, I’m afraid that I don’t have an answer for you; I’m just feeling a bit anal-retentive, and wanted to point out that they’re tear ducts, not ducks. Also, tears don’t come from your tear ducts; they are actually produced by the lacrimal glands underneath your upper eyelid. Tears drain into your tear ducts (and from there, into your nasal cavity).
Well, you said that it happens when you lie down, so one guess (and it’s just a guess) is that the normal tears aren’t able to reach your tear duct because of the angle. Of course, most people don’t seem to leak tears when they lie down, but perhaps you’re suffering from minor allergies that increase your tear flow slightly.
I know that, with my allergies, I will sometimes have one of my eyes start watering profusely for no apparant reason, and then dry up again for just as little reason.
I note that when I lie on my side for a while, the eye closer to the ground tears. Methinks this is fluid normally that stays on your eye (I have contacts- liquid lubrication is only necessary for me in the early morning) but drains to the outer side, and your eye is not wet enough so more tear is made and agian and again until enough is formed to actually leak out onto your face.
Don’t know what happens in people, but one of my cats has scar tissue on his right tear duct and can’t properly drain tears from his eye, so the inside and underside of his eye is always wet. IANAD, but possible a blockage in the duct, or something that causes it not to drain.