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In this study, geomechanical behavior of fractures and faults in rocks as a saturated deformable discrete fracture-porous medium has been evaluated using coupled fluid flow-geomechanics numerical modeling. The purpose of this paper is to observe and evaluate the effects of fractures and faults on the pore pressures during fluid flow through reservoirs. This issue involves solving the equations that have been derived from the Biot consolidation theory such as fluid mass balance equation, Darcy law and momentum balance equations. Govern coupled equations were solved using the standard Galerkin finite element method for continuous porous medium. Elements called "zero-thickness elements” were also used to discretize the fault as a discontinuous part of the porous medium. In compared with the previous and similar methods, the method introduced in this paper, made modifications in either the choice of the element and the method of solving the governing equations. The main advantage of this paper is providing clear precise formulations of the double node zero-thickness element in hydro-mechanical modeling of fractures and faults. Verification of the proposed process and models presented in this paper were done by providing three index problems which their analytical and numerical solutions are available. The results of our model provide a good agreement to these reference solutions which indicates the accuracy of the method presented in this paper.