Interface dynamics of two-phase flow, with relevance for leakage of oil retained by mechanical oil barriers, is studied by means of a two-dimensional (2D) lattice-Boltzmann method (LBM) combined with a phase-field model for interface capturing. A multirelaxation-time (MRT) model of the collision process is used to obtain a numerically stable model at high Reynolds number flow. In the phase-field model, the interface is given a finite but small thickness, where the fluid properties vary continuously across a thin interface layer. Surface tension is modeled as a volume force in the transition layer. The numerical model is implemented for simulations with the graphic processing unit (GPU) of a desktop personal computer. Verification tests of the model are presented. The model is then applied to simulate gravity currents (GCs) obtained from a lock-exchange configuration, using fluid parameters relevant for those of oil and water. Interface instability phenomena are observed, and obtained numerical results are in good agreement with theory. This work demonstrates that the numerical model presented can be used as a numerical tool for studies of stratified shear flows with relevance to oil-boom failure.