Numerical simulation results are presented for the displacement of a drop in a porous medium. The drop is surrounded by a more viscous fluid with which it is fully miscible. The simulations are based on a set of augmented Hele-Shaw equations that account for nonconventional, so-called Korteweg stresses resulting from locally steep concentration gradients. Globally, these stresses tend to stabilize the displacement. However, there are important distinctions between their action and the effects of surface tension in an immiscible flow. Since the Korteweg stresses depend on the concentration gradient field, the effective net force across the miscible interface region is not just a function of the drop's geometry, but also of the velocity gradient tensor. Locally high strain at the leading edge of the drop generates steep concentration gradients and large Korteweg stresses. Around the rear of the drop, the diffusion layer is much thicker and the related stresses smaller. The drop is seen to form a tail, which can be explained based on a pressure balance argument similar to the one invoked to explain tail formation in Hele-Shaw flows with surfactant. The dependence of such flows on the Peclet number is complex, as steeper concentration gradients amplify the growth of the viscous fingering instability, while simultaneously generating larger stabilizing Korteweg forces.