We study a model of variational inequality of degenerate ellipticparabolic 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 (Ting T. W. 1974), in this case we obtain a more regular solution and an error estimate. At the end of this paper we study an application to the theory of seepage of homogeneous liquid in fissured rocks (Zheltov I. V., Kochina I. N., Barenblatt G. I. 1960).