ABSTRACT Reservoir studies are often directed at specifically evaluating, amongst other things: The Sweep of Injected Water in a Waterflood, Tracing Injected Gas Through a Field, The Efficiency of a Polymer or Miscible Component Flood. Each of these problems can be completely obscured by the presence of numerical diffusion, which dominates and masks out true, physical saturation changes. No finite-difference method yet developed which relies on a fixed grid can claim to accurately model any sharp saturation front without degrading it eventually into a smooth, uniform distribution. It is pointless to try and remedy this by increasing the truncation accuracy of the scheme, since it is the physical dynamics of the approximation which are important. The basic difficulties in capturing and tracing any sharp saturation front with time are best understood by considering the fundamental property laws which are necessary for a physical solution. These are: Conservation Properties, Boundedness Properties, Transport Properties and Shock Capture. We describe a method based on the finite-element scheme which allows these laws to be closely approximated. In addition, its formulation is such that it can be incorporated easily into existing IMPES black-oil and compositional simulators. The method is validated by comparison with one- and two-dimensional laboratory experiments and by numerical experiments designed to trap shock waves.