A new incompressible gas-liquid two-phase flow model for non-Newtonian power-law fluid is proposed based on an incompressible lattice Boltzmann model. And the fundamental physical mechanism of Newtonian fluid displacing non-Newtonian power-law fluid liquid in porous medium is studied by using the proposed model. The effects of capillary number Ca, dynamic viscosity ratio M, surface wettability θ, porous medium geometry, and power law index n on the displacement process are investigated. The comprehensive results show that with the increase of capillary number, the displacement process turns faster, the fingering phenomenon becomes more obvious and the displacement efficiency decreases. However, for different values of power-law index n, the effects of the Ca on the displacement process have some differences. Specially, the decrease rate of displacement efficiency becomes slow if the displaced fluid is shear thickening fluid as compared with that if the displaced fluid is shear thinning fluid. On the other hand, the displacement efficiency decreases as dynamic viscosity ratio M increases. And the effect of the viscosity ratio on the displacement process becomes more obvious for the low value of the power-law index n. Moreover, the effect of the surface wettability of the porous medium on the displacement process is also related to the size of the power-law index. With the increase of the contact angle of the porous medium, the fingering phenomenon turns less obvious, and the displacement efficiency increases. However, with the increase of power-law index n, the influence of the contact angle on the displacement process decreases. Besides, the displacement processes with different geometric types of the porous media are also studied in the work. The results show that comparing with the case of porous medium denoted by circle shape and square shape, the fingering phenomenon obtained by the case of triangular shape is most obvious, and the displacement efficiency is lowest.