Modeling of big faults or weak planes of strong and weak discontinuities is of major importance to assess the Geomechanical behaviour of mining/civil tunnel, reservoirs etc. For modelling fractures in Geomechanics, prior art has been limited to Interface Elements which suffer from numerical instability and where faults are required to be aligned with element edges. In this paper, we consider comparative study on finite elements for capturing strong discontinuities by means of elemental (EFEM) and nodal enrichments (XFEM) for Geomechanical applications. For general fracture scenarios, Oliver et al. (2008) have demonstrated advantages of EFEM over XFEM for including: (a) ease of modeling in any standard finite element code due to the possibility of static condensation, (b) higher accuracy in some cases for coarse meshes, and, (c) a reduced computational cost. However, there has been no work that considers cohesive fractures in Geomechanics. We present a numerical analysis based on a linear constitutive model and linear triangular elements for the bulk and a linear constitutive behaviour for the fault for a representative set of fault planes in a circular tunnel made in rock mass. In the numerical examples we consider both well-traced and bad-traced meshes. We focus on traction profile analysis for both EFEM and XFEM, since that is crucial to understand the cohesive fracture behaviour specially rock joint. Therefore in each case, shear, normal and stress ratio are compared across the two methods along the fault plane. This is the first such work that compares XFEM and EFEM behaviour for Geomechanics problems. For the considered class of problems and the sharp penalization approach without regularizations adopted, it can be observed that accuracy and robustness of EFEM may strongly depend on the fault orientation with respect to the element edges, while, XFEM is independent of fault alignment with element edges. It has been also observed that, with mesh refinement model, stress oscillation completely removed with XFEM however EFEM fails to provide oscillation free results with finer mesh in case bad-traced model. We also performed error analysis and found that in case of bad-traced mesh the average error in the XFEM is 84% smaller than of EFEM and for XFEM the error goes to zero with increasing mesh resolution, while for EFEM the error increases with mesh resolution. Thus, we experimentally demonstrate that, XFEM performs better than EFEM in terms of accuracy and rate of convergence with marginal increase in computational cost. This work hence provides novel insights on the application of XFEM and EFEM in Geomechanics.