Gasflooding in oil reservoirs leads to bypassing of the oil as a result of gravitational, viscous, and/or heterogeneity effects. A part of the bypassed oil can be recovered by the flowing solvent because of crossflow/mass transfer. In this work, we studied the effect of the orientation of the bypassed region and the enrichment of the solvent on the mass transfer. Laboratory-scale mass transfer and coreflood experiments were conducted. Numerical simulation was used to identify the role of the different mechanisms. Results indicate that the mass transfer is the least for the vertical orientation, intermediate for the inverted orientation, and the highest for the horizontal orientation. The mass transfer increases with enrichment for all orientations. Liquid-phase diffusion controls vertical-orientation mass transfer for the fluids studied. Gravity-driven flow contributes the most to the mass transfer in the horizontal and inverted orientations. Oil recovery in horizontal gasfloods can be nonmonotonic with enrichment. Multiphase flow in the near-miscible floods leads to less gravity override compared with the first-contact-miscible (FCM) floods.