Cutting concrete patio?

Here's the situation. I have a stamped concrete patio where
the concrete apparently was not finished off with the proper
consisten pitch so that water will drain. There are two areas
where pools of water remain after the rain. Both of these
happen to be in areas where there are relief cuts in the
concrete. It appears that these cuts only go about half way
through the concrete. Which makes sense I guess, since
for the purpose of relief cuts that is sufficient.

So, my idea is to cut them the rest of the way through in the
area of the puddles so the water can then drain. I'm thinking
about 10 ft or so is all I need in total and I'm thinking about
using a circular saw with either a diamond blade or else one
of those abrasive saw blades that also say they will cut
concrete. The difference is the diamond blade is $40, while
the abrasive ones are $6. I obviously don't care if it takes
a bit longer with the abrasive blade, but I don't want to have
to go through 4 blades and have it take 3 hours either. I think
we can assume the remaining concrete is about 2" thick.

Any thoughts? Anyone do anything similar with one of
the abrasive type blades? Thanks.