We present a numerical method for the simulation of misci- ble and immiscible multiphase flow in porous media, with emphasis on the advection-dominated case. A fractional flow formulation is adopted, resulting in a "pressure" equa- tion and a "saturation" equation. The key idea of the pro- posed methodology is a multiple scale decomposition of the variable of interest into resolved and unresolved scales. This acknowledges the presence of fine scales which can- not be captured by any grid, but whose influence on the coarse scales is not negligible. The multiscale approach leads to a stabilized finite element formulation, which pre- vents global spurious oscillations of the numerical solution without introducing excessive dissipation. The method is further improved by incorporating a novel shock-capturing technique based on a nonlinear dissipation mechanism pro- portional to the absolute value of the subscales. We be- lieve this approach is entirely new in the context of flow in porous media. Numerical simulations of tracer injection (miscible flow) and waterflood (immiscible flow) are presented. The proposed subgrid scale method with shock- capturing shows exceptional performance in all test cases studied. These test cases illustrate the potential and appli- cability of the proposed formulation for solving multiphase compositional flows in porous media.