Flow and transport properties of fractured rock masses are a function of geometrical structures across many scales. These structures result from physical processes and states and are highly anisotropic in nature. Fracture surfaces often tend to be shifted with respect to each other, which is generally a result of stress‐induced displacements. This shift controls the fracture's transmissivity through the pore space that forms from the created mismatch between the surfaces. This transmissivity is anisotropic and greater in the direction perpendicular to the displacement. A contact mechanics‐based, first‐principle numerical approach is developed to investigate the effects that this shear‐induced transmissivity anisotropy has on the overall permeability of a fractured rock mass. Deformation of the rock and contact between fracture surfaces is computed in three dimensions at two scales. At the rock mass scale, fractures are treated as planar discontinuities along which displacements and tractions are resolved. Contact between the individual rough fracture surfaces is solved for each fracture at the small scale to find the stiffness and transmissivity that result from shear‐induced dilation and elastic compression. Results show that, given isotropic fracture networks, the direction of maximum permeability of a fractured rock mass tends to be aligned with the direction of the intermediate principal stress. This reflects the fact that fractures have the most pronounced slip in the plane of the maximum and minimum principal stresses, and for individual fractures transmissivity is most pronounced in the direction perpendicular to this slip.