Heterogeneity can be dealt with by defining homogeneous equivalent properties, known as averaging, or by trying to describe the spatial variability of the rock properties from geologic observations and local measurements. The techniques available for these descriptions are mostly continuous Geostatistical models, or discontinuous facies models such as the Boolean, Indicator or Gaussian-Threshold models and the Markov chain model. These facies models are better suited to treating issues of rock strata connectivity, e.g. buried high permeability channels or low permeability barriers, which greatly affect flow and, above all, transport in aquifers. Genetic models provide new ways to incorporate more geology into the facies description, an approach that has been well developed in the oil industry, but not enough in hydrogeology. The conclusion is that future work should be focused on improving the facies models, comparing them, and designing new in situ testing procedures (including geophysics) that would help identify the facies geometry and properties. A world-wide catalog of aquifer facies geometry and properties, which could combine site genesis and description with methods used to assess the system, would be of great value for practical applications.RØsumØ On peut aborder le problme de l'hØtØrogØnØitØ en s'efforçant de dØfinir une permØabilitØ Øquivalente homogne, par prise de moyenne, ou au contraire en dØ-crivant la variation dans l'espace des propriØtØs des roches à partir des observations gØologiques et des mesures locales. Les techniques disponibles pour une telle description sont soit continues, comme l'approche GØosta-tistique, soit discontinues, comme les modles de facis, BoolØens, ou bien par Indicatrices ou Gaussiennes SeuillØes, ou enfin Markoviens. Ces modles de facis sont mieux capables de prendre en compte la connectivitØ des strates gØologiques, telles que les chenaux enfouis à forte permØabilitØ, ou au contraire les facis fins de barrires de permØabilitØ, qui ont une influence importante sur les Øcoulement, et, plus encore, sur le transport. Les modles gØnØtiques rØcemment apparus ont la capacitØ de mieux incorporer dans les modles de facis les observations gØologiques, chose courante dans l'industrie pØ-trolire, mais insuffisamment dØveloppØe en hydrogØo-logie. On conclut que les travaux de recherche ultØrieurs devraient s'attacher à dØvelopper les modles de facis, à les comparer entre eux, et à mettre au point de nouvelles mØthodes d'essais in situ, comprenant les mØthodes gØo-physiques, capables de reconnaître la gØomØtrie et les propriØtØs des facis. La constitution d'un catalogue mondial de la gØomØtrie et des propriØtØs des facis aquifres, ainsi que des mØthodes de reconnaissance utilisØes pour arriver à la dØtermination de ces systmes, serait d'une grande importance pratique pour les applications.Resumen La heterogeneidad se puede manejar por medio de la definición de características homogØneas equivalentes, conocidas como promediar o tratando de describir la variabilidad espacia...

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...

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