It seems the problem was caused by applying power to the device without fully removing the liquid from inside it. The rice is a decent trick with minor or external water damage, but will not always get every little droplet out of the power circuit.
Typically, power failures after a certain amount of time are a result of overheating (there are apps to monitor heat) but could also be electrical problems due to a short or open connection the liquid might temporarily caused.
Take a look at this page:
http://www.ifixit.com/Wiki/Electronics_Water_Damage
Perhaps try a thirsty bag, or even opening the phone and giving the motherboard a good inspection/cleaning. Again I'd like to point out the problem most likely was caused by turning the device on before properly drying out the phone. Good luck, I hope a good cleaning solves your problem!