I usually start my Water Fasts with a dry fast to get things going a little faster. Usually only 2-3 days. I usually get tremendous thirst by then because of physical labor. Also, your mouth is so gross tasting, it's hard to suffer it.
I'm not sure I agree that 1 day of dry fasting equals 3 days of water fasting. That would be nice.
Your body is like 70% water. If you are overweight, every 8 lbs of fat is roughly 1 gallon of water, give or take, which means you have a healthy reserve of water.
Moses fasted for 40 days on Mt. Sinai with no food or water. When he returned to the camp and found the Israelites worshiping the golden calf, he punished them and then returned to the mountain to do it again. It doesn't appear that he had any refeeding in between.
Following that, the prophet Elijah and also Christ were recorded as doing 40 day fasts, and it is assumed that these also were dry just as Moses' fast.
I probably wouldn't attempt a 40 day dry fast, though, unless you're on really good terms with your Creator.