The question 

Taken from:
Preliminary Assessment of promising nonaquifer seasonal thermal energy storage
Donald E. Blahnik, Pacific Northwest Laboratory, Washington   

Proceedings of International conference on seasonal thermal energy storage and compressed air energy storage Seattle – Washington, October 19-21, 1981


One calss of materials that could be considered for underwater storages are some types of concrete and clay that can be designed to have the same density as water. Another class of potentially suitable materials are based on bitumen. All these materials have some severe drawbacks for BILS. 

Even if it might look strange: Water seems to be the insulation material of choice. The thermal conductivity of water is about λ=0.6 W/mK which compared to materials like mineral wool (λ=0.03 W/mK) is rather high. But this high conductivity can be compensated with material thickness. Increase the thickness of the water insulation to 2m to reach the same thermal resistance as 10cm of mineral wool. Not reasonable for a small storage, but acceptable if the storage has a diameter of 20m to add 2 m of insulation.

More to come...