A prototype reservoir simulator is described which combines three new simulation technologies to create a simulator of increased speed, accuracy and versatility. These new technologies are: New finite difference equations for the determination of reservoir pressures that incorporate the sharp pressure gradients that occur around the wells. Streamline simulation. The reservoir pressures are used to determine the location of the streamlines. Saturations are then determined by I D solutions along each streamline. Dynamic gridding. The saturations on each streamline are determined by solving 1D finite difference approximations to the saturation equation. However, the locations of specific saturation values are determined rather than the usual determination of saturations at specific locations. The resulting dynamic spatial grid is finely spaced in areas where saturations are changing most rapidly. The resulting method rigorously accounts for gravity, capillary pressures and all other phenomena that can be incorporated into traditional, non-streamline, finite difference equations. The results of a 2D, 2-well waterflood simulation illustrate the substantial detail and understanding that can be gained from a relatively coarsely gridded simulation.