Trajectory‐based modeling of fluid transport in a medium with smoothly varying heterogeneity

Abstract

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.

Calculating Trajectories Associated With Solute Transport in a Heterogeneous Medium

“…In this paper we present a trajectory‐based technique for calculating solute transport. The trajectories are influenced by the diffusive and dispersive properties of the medium, similar to the asymptotic paths developed in Vasco et al (). Moreover, the paths presented below are valid for an arbitrary porous medium and rapid spatial variations in medium properties are not an issue, as long as the macroscopic heterogeneity is several times larger than the representative elementary volume used to average over microscopic variations in properties (de Marsily, , p. 15).…”

“…Such methods are restrictive in the sense that they assume that some coefficient or parameter, such as frequency, attains large or small values (Chapman et al, ; Knessl & Keller, ; Smith, ) or that the properties of the medium are smoothly varying in comparison to the length scale of the propagating tracer front (Vasco & Finsterle, ). The analysis of Vasco et al () produced a set of ordinary differential equations, the characteristic equations for the path x ( t ) and gradient vector p ( t ). These equations, valid for a medium with smoothly varying properties, are generalizations of those given by Smith () and Chapman et al () that allow for spatial variations in the medium properties and a full dispersion tensor.…”

“…Second, the derivatives in equation are projected onto the vector p , while the expression on the right‐hand side of equation may contain contributions that are not in the direction of the vector p . Finally, equations and are only valid when the heterogeneity is smoothly varying in comparison to the length scale of the tracer front (Vasco et al, ). That is, the heterogeneity must vary smoothly in comparison to the width of the transition zone from the background solute concentration to the concentration behind the tracer front.…”

“…In essence, the asymptotic approach is equivalent to neglecting the right‐hand side in equation . If the medium, flow field, and concentration distribution are smoothly varying, then the divergence of the term on the right‐hand side may be neglected and an eikonal equation results (Vasco & Datta‐Gupta, ; Vasco et al, ). The characteristic equations associated with the eikonal equation are given by and .…”

“…A hybrid mixed Eulerian‐Lagrangian method, known as the method of characteristics (Konikow & Bredehoft, ), combines particle tracking and Eulerian numerical methods by averaging the concentration of particles within the cells of the fixed grid. Asymptotic techniques (Chapman et al, ; Smith, ; Vasco & Finsterle, ; Vasco et al, ) provide an alternative, grid‐free means for modeling transport, though the approach is restricted to porous media with smoothly varying properties.…”