I’ll preface with my qualifications, so if a more qualified person comes along you can disregard me. I’m an engineer who has taken a few thermodynamics courses and has worked as an engineer for a hvac manufacturing plant. I’ve never done anything strictly related to geothermal, but I’ve read a decent bit about it (and watched Technology Connections’ video on the subject, it’s a good entry point)
You may want to call up a company who does geothermal cooling and see what options you have, they’ve gotten pretty creative on how to bury the cooling lines. (See the video mentioned before)
Going the route of just sticking a large water tank underground probably won’t do a ton. I expect that you will have a poor surface to volume ratio, which means poor heat transfer, which means you’ll saturate your thermal mass fairly quickly. What this may allow you to do is run your HVAC system during the night/morning when it’s much more efficient, and ‘charge’ your thermal mass for the hottest part of the day.
Assuming you use 300kg of water in a day, and you can get a 10°C delta, my very rough back of the napkin math says you’re only going to have about 3 kWh of cooling from just the cold water, which is a decent bit, but it’s not a ton. Best case scenario you cut your cooling needs by around 10-20%.
I’m too lazy to do the math of the heat exhange with the ground, but my bet would be you’re better off spending any money you have set aside for this on better insulation techniques and/or a proper geothermal cooling system.
I do like your creative idea though
Well there is plenty of reason to not do it, but I’m assuming you’ve thought about what tampering with your water supply means (and that car radiators are not food safe, and could contain lead or other nasty metals). I think it goes without saying you’re also running the risk of leaks, a high water bill, and mornings without water if your system has issues.
I’d also like to cover my ass a little and do the typical engineer thing and remind you that an idealized number like this is never realized. You will have to account for losses due to inadvertent and incomplete heat transfer. But you may also get a higher reduction due to the ground heat exhange, which I am still too lazy to work out. ¯\_(ツ)_/¯
All of that said, I really hope you succeed at getting cheaper cooling and have fun doing it. There are houses which use water piped through concrete flooring to regulate temperature, so you’re not very far in uncharted territory. Doing big projects like this are also a really good way to learn a lot very quickly.
I also still highly recommend getting a quote for a reputable geothermal cooling system. If nothing else but to brag about how much you saved by DIYing