In the present study, the incompressible flow through highly heterogeneous porous media is modeled by the Multi-resolution Multi-scale Finite Volume (MrMsFV) method. In order to focus on the effects of the absolute permeability structure on the accuracy and performance of the MrMsFV method, the single phase flow is considered and the effects of the gravity and variation of fluid viscosity and density are ignored. The accuracy of the MrMsFV method is examined by comparing its numerical results with those of the standard finite volume method. These permeability fields are extracted from the tenth comparative study problem of the society of petroleum engineering. For the permeability fields in which the permeability varies smoothly, it is shown that the MrMsFV method produces acceptable results. On the other hand, the numerical results along with mathematical analyses show that the MrMsFV method may produce pressure fields with unphysical peaks for channelized permeability fields. In these cases sufficient conditions for the monotonicity and boundedness of the solution are violated. In fact, the coarse scale transmissibilities may be computed in such a way that the coefficient matrix of the coarse scale pressure equation not to be a so-called M-matrix.