It is well known that random $ k $-CNF formulas over $ n $ variables with $ cn $ clauses are unsatisfiable (i.e. they are contradictions) with high probability, for sufficiently large constant $ c $. Thus, random $ k $-CNF formulas (for $ c $ large enough) constitute a natural distribution over unsatisfiable Boolean formulas (or dually, over tautologies, i.e. negations of contradictions). This distribution has been studied extensively.
My question is the following: are there any other established distributions over propositional tautologies or contradictions, that can be considered as capturing the "average-case" of tautologies or unsatisfiable formulas? Have these distributions been intensively studied?