The grid orientation effect is a phenomenon which leads to the computation of fundamentally different solutions on grids oriented diagonal and parallel to the principal flow direction. Grid orientation remains an important consideration for many practical simulation studies, and renewed interest in gas injection processes motivates the revisiting of this classical problem. In this article, we show that there are aspects of the grid orientation effect that can be traced back directly to the treatment of early-time, near-well flow and therefore have a major impact on adverse mobility ratio displacements such as miscible or immiscible gas injection. The details of this effect mean that any uncertainty quantification study should account for the interaction of the near-well heterogeneity and the grid orientation effect. We also show how two possible methods-a well-sponge method and a local embedding technique-are able to produce a solution largely independent of grid orientation for single phase two-component miscible flow. Both methods are versatile in that they can be implemented on general grid topologies. They are illustrated on Cartesian grids for both the standard quarter five spot problem with two different grid orientations, and a problem with a single injection well and two producing wells at different angles to the grid lines. Our results show that it is possible to reduce grid-orientation effects for challenging adverse mobility ratio miscible displacements with only local treatments around the injection wells.