Simulating Solute Transport in Porous or Fractured Formations Using Random Walk Particle Tracking: A Review

Since the first attempts some 20 yr ago in the field of hydrology, random walk (RW) particle tracking as applied to solute transport has experienced profound changes. Concepts and mathematical techniques have improved to the point that numerically difficult problems (e.g., advection‐dominated transport in highly heterogeneous media, or reactive transport) are now much easier to address. Random walk has never been widely used for multiphase flow, probably because numerical dispersion is not a major problem for modeling exercises at large scales. However, vadose zone hydrologic studies often point out very strong variations in fluid velocity over relatively short distances. Random walk methods may be well suited for such studies, a possibility which motivated us to write this review. We first give a comprehensive discussion of the theoretical context of the method. The Fokker–Planck–Kolmogorov equation (FPKE) is established for solute transport, as well as the ordinary Langevin equation and its simplifications for transport of small particles (e.g., colloids). Next, numerical methods are developed for the motion of particles in space. An important section is subsequently dedicated to recent RW concepts in the time domain, and to their application to anomalous (non‐Fickian) transport and inverse problems. Adaptations of RW to transport with solute–solid reactions are also provided, as well as several numerical recipes for resolving a few computational difficulties with the RW method. We purposely did not include any comparisons with Eulerian and Lagrangian approaches. These approaches are discussed at length in several references cited in this review. We note, however, that today's computing capabilities provide new incentives to using RW methods for problems where Eulerian methods are potentially unstable or hampered by numerical diffusion.