The pressure-driven displacement of a non-Newtonian fluid by a Newtonian fluid in a two-dimensional channel is investigated via a multiphase lattice Boltzmann method using a non-ideal gas equation of state well-suited for two incompressible fluids. The code has been validated by comparing the results obtained using different regularized models, proposed in the literature, to model the viscoplasticity of the displaced material. Then, the effects of the Bingham number, which characterizes the behaviour of the yield-stress of the fluid and the flow index, which reflects the shear-thinning/thickening tendency of the fluid, are studied. It was found that increasing the Bingham number and increasing the flow index increases the size of the unyielded region of the fluid in the downstream portion of the channel and increases the thickness of the residual layer of the fluid resident initially in the channel; the latter is left behind on the channel walls by the propagating 'finger' of the displacing fluid. This, in turn, reduces the growth rate of interfacial instabilities and the speed of finger propagation. © 2014 Elsevier Masson SAS. All rights reserved.