[For the entire collection see Zbl 0604.00019.] We propose a numerical method for a type of degenerate variational inequality arising in the axisymmetric porous flow well problems which have been studied by the authors. We use the finite element method to obtain the discrete problem. The convergence of the solution of the discrete problem to the solution of the degenerate variational inequality is proved. The solution of the physical problem depends upon the unknown discharge q. We give a numerical method for finding q which has been found to converge rapidly.