How do you keyframe curve bevel depth?

I want to be able to make vines grow in high speed, like a time lapse.

What I wanted to try is make a curve, and add bevel depth and have it look like teh end of a vine, and have it scale so it looks like it is growing longer and longer.

How would one do that? Is there any other way of making it look like a fast growing vine.

There are many ways, the easiest one I think is having a cylinder -maybe with a subsurf modifier- and a curve modifier. Then you just scale the cylinder to fill the length of the reference curveā€¦
You could even give a try to the new rotobezier addon to animate the curves, just be sure to give it enough resolution in the first frame.
Other way would be animating a taper curve.