Particle tracking in time-dependent two-phase flows

Describing the transport of nutrients and contaminants as well as temperature and other scalars in hydrogeological systems presents both computational and modeling challenges due to the heterogeneity of the media themselves, of the resulting flows, and of the flowing phase distributions. Random walk particle tracking methods involve discretizing transported plumes into point masses that undergo random motion, such that the probability density of particle positions corresponds to the concentration field that typical grid-based Eulerian methods solve for. Particle tracking methods for transport are not affected by the instabilities that Eulerian methods are prone to in advection-dominated systems, and they mitigate numerical dispersion because they do not implicitly homogenize concentrations over an underlying grid.  From a computational standpoint, since particles represent possible physical trajectories, computational power is naturally localized where mass is present, and locally-adaptive time steps can be employed. These reasons mean particle tracking methods are well suited for resolving plume structures for scalar concentration fields that are relatively localized in space but exhibit complex structure.So far, the application of random walk particle tracking methods to heterogeneous media has been mainly restricted to time-independent conditions. In the presence of more than one fluid phase, such as air and water, if a chemical species is restricted to a specific phase, moving phase configurations lead to moving interfaces that present challenges for particle tracking. We propose an extension of particle tracking methods to fully time-dependent, two-phase flow conditions, where the restriction of a transported species to one of the fluid phases is handled through the application of a chemical potential that takes a lower value in the carrier phase. Particles feel an effective drift near the fluid-fluid interface that is proportional to the potential difference between the two phases, leading to a concentration ratio that follows Henry's law at equilibrium. By increasing this potential difference, the amount of mass that crosses the interface can be made arbitrarily small. This formulation avoid explicit reconstruction of phase boundaries and does not require direct computation of particle reflection at fluid-fluid interfaces. We illustrate the application of the method to the simulation of solute fronts in heterogeneous media under two-phase-flow conditions where the solute is restricted to a single phase, and we discuss the possibility of extending the method to more complex interactions between the transported scalar and the fluid-fluid interface.