Study of immiscible displacements in porous media using a color-gradient-based multiphase lattice Boltzmann method

•The MRT R–K LBM ensures better numerical stability and reduces spurious currents.•The non-equilibrium bounce back is extended to two-phase boundary condition.•Three typical flow patterns for immiscible displacement in porous media are obtained.•The first time to obtain a completed ‘M–Ca’ phase-diagram using LBM in literature. A multiple-relaxation-time (MRT) Rothman and Keller (R–K) lattice Boltzmann model is presented for two phase flows with kinematic viscosity contrast. For two-phase flows in porous media, the numerical stability may be reduced due to the presence of complex wall boundaries. The MRT R–K model is shown to be able to ensure better numerical stability and reduce spurious currents significantly. The non-equilibrium bounce back scheme is extended to handle the pressure and velocity boundary condition in two-phase flow simulations. Immiscible displacement in complex heterogeneous media is investigated and three typical flow patterns are obtained, stable displacement, viscous fingering and capillary fingering. Cases with both capillary number Ca and viscosity ratio M ranging from 10-3 to 103 are simulated. The three typical flow patterns correspond to the three domains in the M–Ca phase-diagram. The boundaries that separate the three domains in the model results are qualitatively consistent with previous experimental studies. The MRT R–K model coupled with the developed boundary condition is a good tool for the study of two-phase flows in porous media.