MTL is the logic of all left-continuous t-norms and their residua. Its algebraic semantics is constituted by the variety V(MTL) of MTL-algebras. Among schematic extensions of MTL there are infinitevalued logics L such that the finitely generated free algebras in the corresponding subvariety V(L) of V(MTL) are finite. In this paper we focus on Gödel and Nilpotent Minimum logics. We give concrete representations of their associated free algebras in terms of finite algebras of sections over finite posets.