Admittedly, the concept of entropy density has been used somewhat successfully for description of continuum close to equilibrium. But there are problems with extending this to general non-equilibrium case. I think a mix of oil with water is rather awkward to describe thermodynamically, because it is unstable and it is hard to introduce thermodynamic description for a fluid consisting of many drops of oil of different sizes scattered in water.
Second law of thermodynamics can be phrased in terms of entropy, but it does not really involve time at all. All it says is after process involving no heat transfer is finished, entropy of final equilibrium state is not lower than entropy of initial equilibrium state.