Catenary Arch Built into a Hill?

A mock-up of my latest idea for a wood-fired kiln (above) is an extended catenary arch chamber built into a hill in which I dig steps so that the...

...front of the chamber (above) where the firebox would be is about 4 feet wide and four feet tall, and the...

...rear of the chamber (above) where the chimney would be is a shorter and narrower because of the step up. This mock design would be about three feet wide and three feet high at the back of the kiln.

I built this mock-up with three forms, each one shorter than the first by the thickness of a refractory block that I will use to construct the kiln. I suppose I could design the arch to be a little higher and wider. My shelves are 24 x 24.