  When modelling fluid flow in fractured reservoirs, it is common to represent
the fracturesas lower-dimensional inclusions embedded in the host medium.
Existing discretizationsof flow in porous media with thin inclusions assume
that the principal directions of theinclusion permeability tensor are aligned
with the inclusion orientation. While this mod-elling assumption works well
with tensile fractures, it may fail in the context of faults,where the damage
zone surrounding the main slip surface may introduce anisotropy thatis not
aligned with the main fault orientation. In this paper, we introduce a
generalizeddimensional reduced model which preserves full-tensor permeability
effects also in theout-of-plane direction of the inclusion. The governing
equations of flow for the lower-dimensional objects are obtained through
vertical averaging. We present a framework fordiscretization of the resulting
mixed-dimensional problem, aimed at easy adaptation ofexisting simulation
tools. We give numerical examples that show the failure of existingformulations
when applied to anisotropic faulted porous media, and go on to show
theconvergence of our method in both 2D and 3D
