The Localization Error in the Multi-scale Finite Volume Method for Incompressible flow in Porous media

In the present study, the pressure equation associated with two-phase, incompressible and immiscible flow in porous media is solved by the multi-scale finite volume method (MsFV) for 2D problems. The MsFV method along with its main source of errors is mathematically and physically described. Associated with the computational grids used in the MsFV method, a set of two-scale isotropic permeability domains is designed. These permeability domains are produced to show how and where the errors are initiated in the pressure domain of the MsFV method. For each permeability domain, the pressure and velocity solutions obtained by the multi-scale method are compared with those of the standard finite volume method (as the reference solutions). The numerical results indicate that the MsFV method is sensitive to the fine cells with low permeability data located at the faces and corners of the dual grid blocks. The most errors is observed when the corners of the dual blocks are located on fine cells with low permeability value. In addition. By introducing the adjusted boundary condition, the effects of the permeability averaging for the edges and corners of the dual blocks on the MsFV errors are also investigated.