Calculating Trajectories Associated With Solute Transport in a Heterogeneous Medium

We present a trajectory‐based technique for calculating solute transport in a porous medium that has several advantages over existing methods. Unlike streamlines, the extended trajectories are influenced by each of the important parameters governing transport, including molecular diffusion and transverse dispersion. The approach is complete and does not require any additional techniques, such as operator splitting or particle tracking, in order to account for the full dispersion tensor. The semianalytic expressions make it clear how the flow field, the concentration distribution, and the dispersion tensor contribute to the velocity field of an injected solute. The equations are valid for an arbitrary porous medium, including those with rapid spatial variations in properties, overcoming limitations faced by previous approaches based upon asymptotic techniques. A test on a layered model with sharp boundaries indicates that the extended trajectories are compatible with the results of a numerical simulator and differ from streamlines. We also describe a new form of the dispersion tensor that incorporates a known asymmetry. The trajectories indicate that the modifications of the dispersion tensor lead to more focused transport within regions of high conductivity. Finally, the trajectories are used to define a semianalytic relationship between solute travel times and variations in solute velocities along a path that may be used for tomographic imaging. In an application to the injection of a radioactive tracer into a Berea sandstone core, monitored using micropositron emission tomographic (micro‐PET) observations, the sensitivities are used to map the spatial variations of permeability within the core. Plain Language Summary We describe a new approach to calculating the trajectories of solute moving in a flow field. The paths reflect the effects of the physical processes influencing the solute migration. The trajectories are useful for visualization of solute transport and for efficient imaging approaches that resemble medical tomography. Key Points We describe a technique for computing the trajectory of a transported solute The trajectories are influenced by the flow field, diffusion, and the dispersion tensor, a generalization of streamlines The trajectories form the basis for an efficient tomography‐like inversion algorithm, as illustrated by an application to laboratory data