[1] Modeling transport of reactive solutes is a challenging problem, necessary for understanding the fate of pollutants and geochemical processes occurring in aquifers, rivers, estuaries, and oceans. Geochemical processes involving multiple reactive species are generally analyzed using advanced numerical codes. The resulting complexity has inhibited the development of analytical solutions for multicomponent heterogeneous reactions such as precipitation/dissolution. We present a procedure to solve groundwater reactive transport in the case of homogeneous and classical heterogeneous equilibrium reactions induced by mixing different waters. The methodology consists of four steps: (1) defining conservative components to decouple the solution of chemical equilibrium equations from species mass balances, (2) solving the transport equations for the conservative components, (3) performing speciation calculations to obtain concentrations of aqueous species, and (4) substituting the latter into the transport equations to evaluate reaction rates. We then obtain the space-time distribution of concentrations and reaction rates. The key result is that when the equilibrium constant does not vary in space or time, the reaction rate is proportional to the rate of mixing, r T u D ru, where u is the vector of conservative components concentrations and D is the dispersion tensor. The methodology can be used to test numerical codes by setting benchmark problems but also to derive closed-form analytical solutions whenever steps 2 and 3 are simple, as illustrated by the application to a binary system. This application clearly elucidates that in a three-dimensional problem both chemical and transport parameters are equally important in controlling the process.Citation: De Simoni, M., J. Carrera, X. Sánchez-Vila, and A. Guadagnini (2005), A procedure for the solution of multicomponent reactive transport problems, Water Resour. Res., 41, W11410,

Anomalous (or ''non-Fickian'') transport is ubiquitous in the context of tracer migration in geological formations. We quantitatively identify the origin of anomalous transport in a representative model of a heterogeneous porous medium under uniform (in the mean) flow conditions; we focus on anomalous transport which arises in the complex flow patterns of lognormally distributed hydraulic conductivity (K) fields, with several decades of K values. Transport in the domains is determined by a particle tracking technique and characterized by breakthrough curves (BTCs). The BTC averaged over multiple realizations demonstrates anomalous transport in all cases, which is accounted for entirely by a power law distribution $t 212b of local transition times. The latter is contained in the probability density function w(t) of transition times, embedded in the framework of a continuous time random walk (CTRW). A unique feature of our analysis is the derivation of w(t) as a function of parameters quantifying the heterogeneity of the domain. In this context, we first establish the dominance of preferential pathways across each domain, and characterize the statistics of these pathways by forming a particle-visitation weighted histogram, H w ðKÞ, of the hydraulic conductivity. By converting the ln(K) dependence of H w ðKÞ into time, we demonstrate the equivalence of H w ðKÞ and w(t), and delineate the region of H w ðKÞ that forms the power law of w(t). This thus defines the origin of anomalous transport. Analysis of the preferential pathways clearly demonstrates the limitations of critical path analysis and percolation theory as a basis for determining the origin of anomalous transport. Furthermore, we derive an expression defining the power law exponent b in terms of the H w ðKÞ parameters. The equivalence between H w ðKÞ and w(t) is a remarkable result, particularly given the nature of the K heterogeneity, the complexity of the flow field within each realization, and the statistics of the particle transitions.

Heterogeneity is the single most salient feature of hydrogeology. An enormous amount of work has been devoted during the last 30 years to addressing this issue. Our objective is to synthesize and to offer a critical appraisal of results related to the problem of finding representative hydraulic conductivities. By representative hydraulic conductivity we mean a parameter controlling the average behavior of groundwater flow within an aquifer at a given scale. Three related concepts are defined: effective hydraulic conductivity, which relates the ensemble averages of flux and head gradient; equivalent conductivity, which relates the spatial averages of flux and head gradient within a given volume of an aquifer; and interpreted conductivity, which is the one derived from interpretation of field data. Most theoretical results are related to effective conductivity, and their application to real world scenarios relies on ergodic assumptions. Fortunately, a number of results are available suggesting that conventional hydraulic test interpretations yield (interpreted) hydraulic conductivity values that can be closely linked to equivalent and/or effective hydraulic conductivities. Complex spatial distributions of geologic hydrofacies and flow conditions have a strong impact upon the existence and the actual values of representative parameters. Therefore it is not surprising that a large body of literature provides particular solutions for simplified boundary conditions and geological settings, which are, nevertheless, useful for many practical applications. Still, frequent observations of scale effects imply that efforts should be directed at characterizing well-connected stochastic random fields and at evaluating the corresponding representative hydraulic conductivities

where the subscript c implies that conditioning is done on the same data as those used to obtain (K(x))c. These conditional moments constitute optimum unbiased predictors of the random, and therefore generally unknown, system states q(x) and h(x), respectively. We are interested here in computing the state predictors (q(X))c and (h(x))c together with the conditional second moments of the associated prediction errors, q'(x) and h'(x), in 12 and on F. Obviously, the same line of reasoning applies to the unconditional counterparts of the above set of statistical quantities. As will soon become obvious, merely replacing the parameters of standard deterministic models by their (conditional or unconditional) ensemble mean values leads to biased and suboptimal results. One way to render deterministic models more suitable for randomly nonuniform media is to use "effective" or "equivalent" parameters traditionally obtained by some method of "upscaling." Among the more rigorous theoretical criteria of equivalence for hydraulic conductivity are those proposed by Indelman and Dagan [1993] which, however, are not easy to implement in practice. A major conceptual difficulty with upscaling is that it postulates a local relationship between (conditional) mean driving force and flux (in the form of Darcy's law) when, in fact, this relationship is generally nonlocal, as we shall soon see. We describe in this paper a formal method of localizing the above relationship (thereby allowing an effective hydraulic conductivity to be defined in an ensemble sense) and explore its performance numerically. Another conceptual difficulty with traditional upscaling is that it requires the a priori definition of a numerical grid even in the absence of firm theoretical guidelines for its selection. Both the nonlocal and localized approaches we describe in this paper are independent of grid specifications; in our approach a grid is introduced only after, not before, problem formulation.Deterministic alternatives to (conditional or unconditional) Monte Carlo simulation seek to predict flow and transport under uncertainty without having to generate random fields or variables. One approach is to write a system of partial differential equations satisfied approximately by the first two (conditional or unconditional) ensemble moments of a quantity such as hydraulic head [Zhang, 1998] In this paper we describe a numerical method derived rigorously from the exact conditional moment equations for steady state flow originally developed by Neuman and Orr [1993a]. The extension of the methodology to unconditional moment equations is straightforward. Taking the conditional ensemble mean (expectation) of (1) where Gc is the deterministic Green's function associated with Exact Second Conditional Moment EquationsThe (54) and I is the identity tensor. Whereas the first term on the right-hand-side of (53) is isotropic, the second term is generally anisotropic. Since Kc(x) is defined on the support scale to, it is a local rather than an upscaled effective para...