Time of flight transport equation in Eulerian reference frame

A general advection equation of time of flight (TOF) is developed in the Eulerian frame for unsteady state, which can be fully coupled with governing equations for primary variables like pressure, velocity, saturation and temperature. Both initial and boundary conditions are provided for TOF in a general form. The outlet boundary condition is more robust compared with published methods. The influence of dispersion on TOF is included for mixing or diffusion phenomena in both non-porous media flow and porous media flow. Arrival times can be solved by the same set of equations by assigning inlet boundary conditions on producer wells and outlet boundary conditions on injector wells. Neutral fluid particle velocity for multiphase flow is discussed. Published saturation velocities can only track water saturation at water fronts. A more general form is developed, based on neutral fluid particle velocity that can track water saturation at both water fronts and swept regions. Several examples from non-porous media flow with and without dispersion to porous media flow are demonstrated. The advection equation of TOF with corresponding boundary conditions is very similar to traditional governing transport equations which can be solved by the same method. Time dependent properties like fluid rheology during hydraulic fracturing and catalyst particle activity can be coupled with other transport variables, such as temperature, for optimization.