Simulating Solute Transport in Porous or Fractured Formations Using Random Walk Particle Tracking: A Review
چکیده
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 techmerical 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 solseveral alternative Eulerian-Lagrangian schemes based ute transport, as well as the ordinary Langevin equation and its simplion 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; Younès, 2004). solute–solid reactions are also provided, as well as several numerical We 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 methods numerical applications. In the hydrologic sciences they for problems where Eulerian methods are potentially unstable or hamwere first applied in the late 1970s and early 1980s to pered by numerical diffusion. transport in saturated alluvial formations (e.g., Ahlstrom et al., 1977; Prickett et al., 1981). Despite the encouraging results during the last two decades for simulating T is an increasing need for accurate numerical transport in saturated media, RW methods have been solutions to solute transport problems in porous used only very rarely for unsaturated media. This is surand/or fractured formations. This need is predicated by prising since no inherent incompatibility exists between increased concerns about soil and water quality, the longthe RW method and unsaturated flow. On the contrary, term impact of industrial and agricultural contaminants soils are often characterized by strong contrasts in local on the environment, and the feasibility of underground flow velocities when a moisture front propagates during repository sites for nuclear or other wastes. While convariably saturated flow. While Eulerian methods often siderable efforts during the past several decades have experience numerical difficulties when simulating such resulted in vastly improved numerical techniques for velocity contrasts over short distances, Lagrangian modsolving subsurface transport problems, accurate simulaels are much easier and more accurately to implement, tions of seemingly straightforward advection–dispersion as we shall demonstrate later. problems often remain a challenge when transport is In attempts to focus our review primarily on RW advection dominated. This problem becomes even more methods, we decided not to provide detailed comparichallenging when attempts are made to accurately account sons between discrete Eulerian approaches (such as fifor the enormous heterogeneity of soil and groundwater nite volume and finite element methods) and Lagransystems (e.g., de Marsily et al., 1998, 2004). gian methods. Abundant literature already exists on such comparisons (we also provide several useful references). F. Delay, Univ. of Poitiers, Earth Sciences Building, 40 Av. du Recteur An extensive comparison between Eulerian and LagranPineau, 86022 Poitiers, Cedex, France; P. Ackerer and C. Danquigny, gian approaches would have required a much longer Fluid and Solid Mechanics Institute of Strasbourg, 2 rue Boussingault, review. For these same reasons, we will not discuss all 67000 Strasbourg, France. Received 9 Sept. 2004. Review and Analysis. *Corresponding author (fred.delay@hydrasa.univ-poitiers.fr). Abbreviations: ADE, advection–dispersion equation; CTRW, continuous time random walk; ELLAM, Eulerian Lagrangian localized adPublished in Vadose Zone Journal 4:360–379 (2005). doi:10.2136/vzj2004.0125 joint method; EPT, enhanced particle tracking; FPKE, Fokker–Planck– Kolmogorov equation; LEA, local equilibrium assumption; RW, ran© Soil Science Society of America 677 S. Segoe Rd., Madison, WI 53711 USA dom walk; TDRW, time domain random walk. 360 Published online May 13, 2005