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I have some experience of both using trapezoidal blocks in Kenya and of black cotton soils. the un-mortared lining in trapezoidal blocks worked fine, but it didn't catch on because the method for making them by hand was not very satisfactory. I even designed a superstructure using the same blocks with a Zimbabwean style 'snail shell' plan.
Pit linings in black cotton soil are a huge problem and I have not seen anything very convincing published on the subject. However, if you have a supply of reasonably strong trapezoidal blocks, this would have a good chance of working. A word of caution though, the movement of black cotton soils will lift things out of the ground! For instance, concrete manhole rings tend to be lifted in the wet season, but don't seat back in place during the dry and so you get terrible infiltration problems with the next rains.
I would suggest that placing a layer of sand or gravel around the lining may be sufficient to prevent the movement causing the lining to fail. The only problem with this is that suitable granular material is usually hard to come by in areas with black cotton soil. It will certainly be a great advantage structurally having circular-ish pits over square or rectangular ones. In Zimbabwe school latrines are built with a row of individual circular pits, but not in Black cotton soils.