Intuitive vs real answer. Say a man had a size 10 d foot and weighed about 160#. My intuitive answer would be about 80 mph. How would you figure this? A close educated estimate would be great.
I would think at 80 mph one could “skip” across the water a little ways, but the OP says “run” … I’d guess exactly one stride.
Assume you had reached a speed of 80 mph or whatever speed was needed before you entered the water.
I am also curious as to what speed others would guess intuitively with no figuring.
I remember water skiing as a kid. At about 45 MPH you could kick off the skis and ski barefooted. Slower, and you went down.
I realize that there are other mechanics involved. And I weighed about 75% less.
I though about that when I posted this. I was just watching a vidio of a lizard running across water. He had big feet!Lots of pictures of motorcycle & snowmobiles doing it online here & there.
This Barefoot skiing - Wikipedia tells us you can barefoot ski at 30-45 mph. But only with two feet constantly in the water.
Ballpark you could barefoot ski on one foot at double that speed. So 60-90 mph depending on weight versus foot size. A bit slower for skinny teen guys with size 12s and much faster for 400lb former ballerinas with an ice cream addiction.
But that’s assuming something else is providing the motion and we can use all the foot’s reaction with the water to hold us up. Running’s not like that at all.
This Running - Wikipedia has a pretty good description of the kinematics of human running. But sadly no numbers. The rest of my Googling was equally number-free.
The key points are that when running, a hefty portion of the total time is spent with both feet in the air. In the case of running on water that means we’re going to need more speed to increase the lift we get while a foot is in/on the water such that the average lift over time is >= body weight.
Lacking better numbers, if we arbitrarily assume a fast runner spends 1/3rd of the time with both feet in the air then we’re gonna need about 1/3rd more speed to offset that. So now we’re up around 80 to 120 mph. Plus the increased impact forces when his foot reenters the water. A 150 lb person “weighs” 300 lbs when his foot hits the pavement running. It’ll be the same when the foot hits the water & needs to support him. Now we’re up around 160-250mph.
In addition to the forces needed to support the runner, the water also has to provide the counterforce to push the runner forward. IOW, he’s going to push backwards with his feet which in turn propels him forward.
Again for lack of numbers I’m going to assume 1/2 again additional force is required to propel the runner forward. Remember we’re talking about pushing against 160-250 mph of wind drag at this point.
So now he we’ve accounted for most of the forces and we see a running speed of 240-375 mph would be required.
That’s also assuming the standard human gait with about a 6 foot running stride. it can’t get longer because our legs aren’t getting any longer. The only way to get a longer stride is to jump harder & fly more between footfalls. Both of which increase the speed required even more. So 6 foot stride it is.
To make 375mph our runner needs to accomplish 105 strides per second. Good luck. 
Actually, it’s worse than all that. Water skiing lift is roughly proportional to the foot speed versus the water speed. When running on land, your foot stops completely each time it touches the ground. If your foot stops on the water you sink just as fast as you would trying to stand on the water. We all know how well that works. And this is true regardless of how fast your body is going.
The way you thrust yourself in water is to accelerate the water backwards to as close to the speed of your foot as possible. IOW, the more effectively you make forward thrust, the less effectively you make lift.
Fun to think about, but bottom line: not gonna work. Not even a little bit.
One of my neighbors is a barefootin’ fool. Despite being 70-something. I’ll ask him about one foot barefooting behind a boat. If it’s possible, he’s done it. If it’s impossible, he’s tried it. Quite the character.
[.
Fun to think about, but bottom line: not gonna work. Not even a little bit.
The physics are interesting. I hit the invisible like button!
You can walk on water … provided you add a lot of custard powder first!
(the walk starts at about 2:45…)
30 m/s, about 67mph.
I guess we’ll have to wait until a human can run that fast.
Pro tip: Use frozen water.
It also works good in very shallow water, say 1/4" or so. 
Have your dad be the creator of all that is, or at least be the creator’s kid’s bestest buddy and you can just walk.
In comic books the Flash could run on water. Even Wally West who could only run the speed of sound.
I would add into the problem that rarely is water “flat”. There are waves and very big ones in the ocean so no way could anyone run across those.
OP, tell me about your intuition. I cannot fathom (heh, not intended) how you came up with 80 MPH.
I think that the difficulty here is that it depends on how you run. And you can’t just say “run the same way that a human does”, because the way that a human runs is “much slower, and not at all on liquid”.
Mythbusters tried it with college track and field athletes. It didn’t work. This was fairly fresh still water. No corn starch or whatever it is you can add to water to make it, um, that weird stuff.
What?!
Did they think it was going to work and just no-one had tried it before (as well as everyone’s understanding of basic physics being flawed)?
No, they often do myths that clearly have no chance of working, then they try to push it to the illogical extreme.
Then they blow shit up.
On “Outrageous Acts of Science” they went over a hoax video of guys running on water (it was a platform just under the surface, of course). During the discussion they mention that you’d have to run 4x as fast as Usain Bolt to generate enough force to stay on top of water.