A review and numerical assessment of the random walk particle tracking method

We review the basic mathematical concepts of random walk particle tracking (RWPT) and its advantages and limitations. Three different numerical approaches to overcome the local mass conservation problem of the random walk methodology are examined: (i) the interpolation method, (ii) the reflection principle, and (iii) the generalized stochastic differential equations (GSDE). Analytical solutions of the spatial moments for a two-layer system are compared to model predictions using the different techniques and results demonstrate that the interpolation method reproduces correctly average velocity, but fails to reproduce macrodispersion at higher hydraulic conductivity contrasts between the two layers. On the contrary, the reflection principle and the GSDE approach underestimate average velocity, but reproduce macrodispersion better for high contrasts. The different behavior is based on an artificial shift of mass for increasing heterogeneities for the GSDE approach and the reflection principle, whereas the interpolation method suffers from the smoothing of the dispersion tensor. The behavior of these approaches was furthermore analyzed in two-dimensional heterogeneous hydraulic conductivity fields, which are characterized by different random function models. Solute transport was simulated correctly by all three approaches for the reference fields having Gaussian structures or non-Gaussian structures with an isotropic spatial correlation, even for a variance of the natural log of hydraulic conductivity of σ ln K 2 = 4. However, for the non-Gaussian model with a strong anisotropic spatial correlation and a variance of σ ln K 2 = 2 and higher, the interpolation method was the only technique modelling solute transport correctly. Furthermore, we discuss the general applicability of random walk particle tracking in comparison to the standard transport models and conclude that in advection-dominated problems using a high spatial discretization or requiring the performance of many model runs, RWPT represents a good alternative for modelling contaminant transport.