Trajectory-based Modeling of Fluid Transport in a Medium with Smoothly Varying Heterogeneity

Using an asymptotic methodology, valid in the presence of smoothly-varying heterogeneity and prescribed boundaries, we derive a trajectory-based solution for tracer transport. The analysis produces a Hamilton-Jacobi partial differential equation for the phase of the propagating tracer front. The trajectories follow from the characteristic equations that are equivalent to the Hamilton-Jacobi equation. The paths are determined by the fluid velocity field, the total porosity, and the dispersion tensor. Due to their dependence upon the local hydrodynamic dispersion, they differ from conventional streamlines. This difference is borne out in numerical calculations for both uniform and dipole flow fields. In an application to the computational x-ray imaging of a saline tracer test, we illustrate that the trajectories may serve as the basis for a form of tracer tomography. In particular, we use the onset time of a change in attenuation for each volume element of the x-ray image as a measure of the arrival time of the saline tracer. The arrival times are used to image the spatial variation of the effective hydraulic conductivity within the laboratory sample.