The lattice Boltzmann method (LBM) is often advocated as an effective simulation tool for multi phase flow in porous media. In this paper, we extended the mainstream Rothman-Keller (R K) multi phase model with multi-relaxation-time (MRT) scheme by adding perturbation in the moment space and a diffusion process (except for two-phase flow) after the recoloring step. The former generates the interfacial tension, and the latter eliminates the over anti diffusion effect of the recoloring step. The method was validated by numerical experiments on the Laplace's law and dynamics of spreading, and the drainage and imbibition processes of real cores were then simulated. The simulations were carried out on graphics processing units (GPUs) using Compute Unified Device Architecture (CUDA) and message passing interface (MPI), which was proven to be very efficient (C) 2013 Elsevier Ltd. All rights reserved.