The extended finite element method (X-FEM) has been developed to minimize requirements on the mesh in a problem with a displacement discontinuity. We present the development carried out to take advantage of the X-FEM approach in simplifying the meshing of complex 3D networks of discontinuities with junctions. Contact with large sliding along the branched discontinuities is discussed. Solutions are proposed and discussed to solve some matrix conditioning issues. Several examples are presented in this paper in order to prove the efficiency of the proposed approach.