When the permeability field of a given porous medium domain is heterogeneous by the existence of randomly distributed fractures such that numerical investigation becomes cumbersome, another level of upscaling may be required. That is such complex permeability field could be relaxed (i.e., smoothed) by constructing an effective permeability field. The effective permeability field is an approximation to the real permeability field that preserves certain quantities and provides an overall acceptable description of the flow field. In this work, the effective permeability for a fractured rock system is obtained for different coarsening scenarios starting from very coarse mesh all the way towards the fine mesh simulation. In all these scenarios, the effective permeability as well as the pressure at each cell is obtained. The total flux at the exit boundary is calculated in all these cases, and very good agreement is obtained.