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

Abstract

Traditional Eulerian approaches to solute transport generally require very fine discretization of the transport Since the first attempts some 20 yr ago in the field of hydrology, domain to overcome recurrent problems of unstable nurandom walk (RW) particle tracking as applied to solute transport has experienced profound changes. Concepts and mathematical tech-merical solutions and/or artificial diffusion. Solving such niques have improved to the point that numerically difficult problems problems generally imposes a heavy computational bur-(e.g., advection-dominated transport in highly heterogeneous media, den. If the discretization is too coarse, oscillations and or reactive transport) are now much easier to address. Random walk numerical diffusion of standard Eulerian methods may has never been widely used for multiphase flow, probably because yield poor or even incorrect solutions, particularly when numerical dispersion is not a major problem for modeling exercises nonlinear processes such as multiphase flow or adsorpat large scales. However, vadose zone hydrologic studies often point tion-desorption are considered in conjunction with stanout very strong variations in fluid velocity over relatively short disdard advective-dispersive transport. This problem proved tances. Random walk methods may be well suited for such studies, to be crucial in safety calculations of underground reposa possibility which motivated us to write this review. We first give a comitory sites for nuclear wastes. To address these problems, prehensive discussion of the theoretical context of the method. The Fokker-Planck-Kolmogorov equation (FPKE) is established for sol-several alternative Eulerian-Lagrangian schemes based ute transport, as well as the ordinary Langevin equation and its simpli-on particle tracking have been developed, such as the fications for transport of small particles (e.g., colloids). Next, numerimethod of characteristics (Konikow et al., 1996). Partical methods are developed for the motion of particles in space. An cle tracking has also been used for numerical integration important section is subsequently dedicated to recent RW concepts in more complicated schemes, such as the Eulerian Lain the time domain, and to their application to anomalous (non-Fickian) grangian localized adjoint method (ELLAM) (Russell transport and inverse problems. Adaptations of RW to transport with and Celia, 2002; Younè s, 2004). solute-solid reactions are also provided, as well as several numericalWe provide a review of the principles of fully Lagranrecipes for resolving a few computational difficulties with the RW gian schemes based on RW particle tracking for solving method. We purposely did not include any comparisons with Eulerian advection-dispersion problems. Lagrangian approaches and Lagrangian approaches. These approaches are discussed at length in several references cited in this review. We note, however, that today's have long been used widely in physics for a variety of computing capabilities provide new incentives to using RW...

Implementation of a Time-Domain Random-Walk Method into a Discrete Element Method to Simulate Nuclide Transport in Fractured Rock Masses

“…According to former researches [30,31], the difficulty of discovering nuclide transport mechanism lies in solving the partial differential transport equation. As mentioned by Delay et al [32], Lagrangian schemes (particle tracking methods) are more accurate and less computationally intensive than Eulerian schemes. The main Lagrangian schemes include random-walk particle tracking (RWPT) method [33,34], convolution-based particle tracking (CBPT) method [35,36], and continuous-time random-walk (CTRW) method [37,38].…”

“…The trace of the fracture was 10 m, fluid velocity was 4.0 × 10 −4 m/s, hydrodynamic dispersion coefficient in the fracture was 2.0 × 10 −5 m 2 /s, and the aperture of the fracture was 2.5 × 10 −4 m. Thus, the Peclet number (Pe = / ) was 200. The analytical solution can be obtained using (32). The corresponding numerical solution was obtained using the TDRW method implemented into UDEC, which is compared in Figure 5 with the analytical solution.…”

Transient and asymptotic dispersion in confined sphere packings with cylindrical and non-cylindrical conduit geometries

“…Advective-diffusive mass transport of conservative tracers in the interparticle void space was simulated using the random-walk particle-tracking technique [19]. The idea of the method is to distribute a large amount of tracers (N = 4 × 10 6 in our study) in the volume of interest and to track target quantities (tracer trajectories, statistical moments of the tracer ensemble, etc.)…”

From analytical solutions of solute transport equations to multidimensional time‐domain random walk (TDRW) algorithms

“…The particle displacement during a given time step includes a deterministic displacement along flow-path lines to simulate advection and may also include an additional stochastic displacement to simulate dispersion (or diffusion), as in the well-known random walk (RW) method. The reader is referred to Delay et al [2005] and Salamon et al [2006] for comprehensive reviews of space-based particle-tracking methods, including theoretical, conceptual, and numerical developments in the field of subsurface hydrology since the early works of Ahlstrom et al [1977] and Prickett et al [1981]. Additional recent publications of relevance include Bechtold et al [2011] and Lejay and Pichot [2012].…”