  In early study of Engel manifolds from R. Montgomery, the Cartan prolongation
and the development map are central figures. However, the development map can
be globally defined only if the characteristic foliation is "nice". In this
paper, we introduce the Cartan prolongation of a contact 3-orbifold and the
development map associated to a more general Engel manifold, and we give
necessary and sufficient condition for the Cartan prolongation to be a
manifold. Moreover, we explain the Cartan prolongation of a 3-dimensional
contact \'etale Lie groupoid and the development map associated to any Engel
manifold, and we proof that all Engel manifolds obtained as the Cartan
prolongation of a "space" with contact structure are obtained from a contact
3-orbifold.
