We study a model of variational inequality of degenerate elliptic-parabolic type. We prove an existence and uniqueness theorem, apply the finite element method and verify the convergence of the approximate solutions.Then we consider a problem with an internal obstacle related to the pseudoparabolic operator governing the heat propagation in non-simple elastic materials([12]),in this case we obtain a more regular solution and an error estimate.At the end of this paper we studyan application to the theory of seepage of homogeneous liquid in fissured rocks ([13]).