In this thesis the mimetic finite difference method for grids with curved faces is presented, implemented and tested with an emphasis on applications in reservoir simulation. The thesis gives a brief introduction to reservoir modeling and introduce the mimetic method for flat and for curved faces. Then the continuity condition for the curved mimetic method is discussed. It is shown that the suggested continuity condition is not valid for cases with a difference in permeability between two cells separated by a curved face. An alternative continuity condition is discussed and implemented. Numerical examples confirm that the original continuity condition is incorrect for general examples with heterogeneous permeability. Numerical examples for the alternative continuity condition shows that it is correct for simple cases, and that it gives no gain in accuracy compared to the mimetic method. In conclusion the curved mimetic method is primarily of academic interest.