Thaw / Brine partially frozen turkey at the same time
I believe the reasoning for changing the water every 30 minutes is to speed up defrosting.   I imagine the defrost time would be reduced drastically. However, that seems like a mess and a lot of work. If you're brining a partially frozen turkey, from a safety perspective, I would not change the water out. I'd just leave it in the water. The main thing you are looking for is that the water temp never goes above 4C/40F. You can do this by doing in the fridge, or cooler, or outside if you live in the right place.
See these related questions for more info:

How can I effectively time a short thawing and brining of a a turkey?
How cold can a brining turkey get before freezing?