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I'd avoid the oil drums because of low capacity and small infiltration surface per pit, even if you do a deep pit. And when they start to fail they are unlikely to resist the swelling and shrinking of the black cotton soil for long. I think Tim is right about the investment in a brick lining if, as Andy points out, the latrines are likely to be used for some time. Latrines on black cotton soils in Malawi in the late 80s were brick lined down to the limestone base about 2 m deep, with the base carefully cleaned of clay before commissioning to ensure maximum infiltration downwards, as horizontal movement of liquid was virtually nil.
I had thought about weldmesh and geotextile or even woven bamboo if available, but I'm not sure that a flexible lining like this would handle the soil movement well over time. Maybe if there was a band of sand and gravel, say 50 cm thick all around the flexible lining? This would have the advantage of increasing the infiltration surface but would start to get expensive too.
Try to find another place to put the latrines?????