This paper addresses a study of the controllability of a class of Newtonian filtration equations, with nonnegative constraints on the control and state variables. When the control enters the system through the whole domain where the equation evolves, we characterize the set of nonnegative targets which are approximately controllable at any time T > 0. The proof combines the Fenchel-Rockafellar duality theory and a fixed point argument. When the control is restricted to be active in a proper open subset of the whole domain, we prove a negative controllability result by means of a localization technique which reflects the underlying obstruction phenomenon in the system.