Explicit numerical schemes are popular in multiphysics and multiscale simulations, yet their use in stiff problems often requires time steps to be so small as to render simulations over large time horizons infeasible. Exponential time differencing (ETD) has proved to be an efficient scheme for tackling differential operators with linear stiff and nonlinear non-stiff parts. Such natural separation, however, is absent in many important applications, including multiphase flow and transport in porous media. We introduce a strategy for using ETD in such problems and demonstrate its efficiency in numerical experiments. We also compare the ETD performance to that of an explicit scheme. We conclude that the best outcome is achieved by combining ETD with a fourth-order Runge-Kutta method. Although our methodology is demonstrated on two-dimensional multiphase flow in porous media, it is equally applicable to other applications described by parabolic differential equations of this kind.