I would like to know if there is any way to make water rise gradually in a certain area.
Yes, albeit not without mods or highly complex commands.
The best you can get is by putting a waterfall high up, and letting it spread, although this won't make it rise, only spread for 8 blocks or so.
If you had a mod then it would be easy, but I'm presuming you're wanting to do this on vanilla. I can't think of any mods which would do this off the top of my head, but I'm sure there are some out there.
With commands, it could be possible if you set up a command block timer, and then get it to set water tiles going up in the X axis on the blocks you want it to rise on.
If however, commands aren't your strong point, then I suggest you learn or install mods.