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• Representative hydraulic conductivities in saturated groundwater flow

  X. Sánchez-Vila, A. Guadagnini, J. Carrera

  Reviews of geophysics (2006). Vol. 44 (3), pp. 1-46

  
  

  Abstract

  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

  Abstract
  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 [...]

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• Travel time and trajectory moments of conservative solutes in two-dimensional convergent flows

  M. Riva, X. Sánchez-Vila, A. Guadagnini, M. Willmann, M. Simoni

  Journal of Contaminant Hydrology (2006). Vol. 82 (1-2), pp. 23-43

  
  

  Abstract

  We address advective transport of a solute traveling toward a single pumping well in a two-dimensional randomly heterogeneous aquifer. The two random variables of interest are the trajectory followed by an individual particle from the injection point to the well location and the particle travel time under steady-state conditions. Our main objective is to derive the predictors of trajectory and travel time and the associated uncertainty, in terms of their first two statistical moments (mean and variance). We consider a solute that undergoes mass transfer between a mobile and an immobile zone. Based on Lawrence et al. [Lawrence, A.E., Sánchez-Vila, X., Rubin, Y., 2002. Conditional moments of the breakthrough curves of kinetically sorbing solute in heterogeneous porous media using multirate mass transfer models for sorption and desorption. Water Resour. Res. 38 (11), 1248, doi:10.1029/2001WR001006.], travel time moments can be written in terms of those of a conservative solute times a deterministic quantity. Moreover, the moments of solute particles trajectory do not depend on mass transfer processes. The resulting mean and variance of travel time and trajectory for a conservative species can be written as functions of the first, second moments and cross-moments of trajectory and velocity components. The equations are developed from a consistent second order expansion in σY (standard deviation of the natural logarithm of hydraulic conductivity). Our solution can be completely integrated with the moment equations of groundwater flow of Guadagnini and Neuman [Guadagnini, A., Neuman, S.P., 1999a. Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly non uniform domains 1. Theory and computational approach. Water Resour. Res. 35(10), 2999–3018.,Guadagnini, A., Neuman, S.P., 1999b. Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly non uniform domains 2. Computational examples. Water Resour. Res. 35(10), 3019–3039.], it is free of distributional assumptions regarding the log conductivity field, and formally includes conditioning. We present analytical expressions for the unconditional case by making use of the results of Riva et al. [Riva, M., Guadagnini, A., Neuman, S.P., Franzetti, S., 2001. Radial flow in a bounded randomly heterogeneous aquifer. Transport in Porous Media 45, 139–193.]. The quality of the solution is supported by numerical Monte Carlo simulations. Potential uses of this work include the determination of aquifer reclamation time by means of a single pumping well, and the demarcation of the region potentially affected by the presence of a contaminant in the proximity of a well, whenever the aquifer is very thin and Dupuit–Forchheimer assumption holds.

  Abstract
  We address advective transport of a solute traveling toward a single pumping well in a two-dimensional randomly heterogeneous aquifer. The two random variables [...]

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• A procedure for the solution of multicomponent reactive transport problems

  J. Carrera, M. Simoni, X. Sánchez-Vila, A. Guadagnini

  Water Resources Research (2005). Vol. 41 (W11410), pp. 1-16

  
  

  Abstract

  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, ∇TuD ∇u, 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.

  Abstract
  Modeling transport of reactive solutes is a challenging problem, necessary for understanding the fate of pollutants and geochemical processes occurring in aquifers, rivers, [...]

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• Travel time and trajectory moments of conservative solutes in three dimensional heterogeneous porous media under mean uniform flow

  X. Sánchez-Vila, A. Guadagnini

  Advances in Water Resources (2005). Vol. 28 (5), pp. 429-439

  
  

  Abstract

  We present expressions satisfied by the first statistical moments (mean and variance–covariance) of travel time and trajectory of conservative solute particles advected in a three-dimensional heterogeneous aquifer under uniform in the mean flow conditions. Closure of the model is obtained by means of a consistent second-order expansion in σY (standard deviation of the log hydraulic conductivity) of (statistical) moments of quantities of interest. As such, the results obtained are nominally limited to mildly non-uniform fields, with σY < 1. Resulting mean and variance of particles travel time and trajectory are functions of first and second moments and cross-moments of trajectory and velocity components. Our solution is applicable to infinite domains and is free of distributional assumptions. As an important application of the methodology we obtain closed-form expressions for the unconditional mean and variance of travel time and particle trajectory for isotropic log-conductivity domain characterized by an exponential variogram. This allows us to recover the non linear behavior of mean travel time versus distance, in agreement with numerical results published in the literature, as well as a non-linear effect in the mean trajectory. The analysis of trajectory variance allows recovering some known results regarding transverse macro-dispersion, evidencing some limitations typical of perturbation theory.

  Abstract
  We present expressions satisfied by the first statistical moments (mean and variance–covariance) of travel time and trajectory of conservative solute particles advected [...]

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• A methodolgy to compute mixing ratios with uncertain end-members

  J. Carrera, E. Vázquez-Suñé, X. Sánchez-Vila

  Water Resources Research (2004). Vol. 40 (W12101), pp. 1-11

  
  

  Abstract

  Mixing calculations involve computing the ratios in which two or more end‐members are mixed in a sample. Mixing calculations are useful for a number of tasks in hydrology, such as hydrograph separation, water or solute mass balances, and identification of groundwater recharge sources. Most methods available for computing mixing ratios are based on assuming that end‐member concentrations are perfectly known, which is rarely the case. Often, end‐members cannot be sampled, and their concentrations vary in time and space. Still, much information about them is contained in the mixtures. To take advantage of this information, we present here a maximum likelihood method to estimate mixing ratios, while acknowledging uncertainty in end‐member concentrations. Maximizing the likelihood of concentration measurements with respect to both mixing ratios and end‐member concentrations leads to a general constrained optimization problem. An algorithm for solving this problem is presented and applied to two synthetic examples of water mixing problems. Results allow us to conclude that the method outperforms traditional approaches, such as least squares or linear mixing, in the computation of mixing ratios. The method also yields improved estimates of end‐member concentrations, thus enlarging the potential of mixing calculations. The method requires defining the reliability of measurements, but results are quite robust with respect to the assumed standard deviations. A nice feature of the method is that it allows for improving the quality of computations by increasing the number of samples and/or analyzed species.

  Abstract
  Mixing calculations involve computing the ratios in which two or more end‐members are mixed in a sample. Mixing calculations are useful for a number of tasks in hydrology, [...]

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• Introductory review of specific factors influencing urban groundwater, an emerging branch of hydrogeology, with reference to Barcelona, Spain

  E. Vázquez-Suñé, X. Sánchez-Vila, J. Carrera

  Hydrogeology Journal (2005). Vol. 13, pp. 522-533

  
  

  Abstract

  A number of specific factors must be considered when dealing with groundwater in urban areas. Urbanization significantly affects the natural water cycle, both in terms of quantity and quality. In particular, the main contributors to recharge and discharge clearly differ from those in natural systems. Moreover, water can affect underground structures and infrastructure characteristics of cities such as basements, public transport services (trains, underground railways, etc.), and utility conduits. As a result, urban groundwater is emerging as a distinct branch of hydrogeology. The objective of this paper is to review some of the topics that are specific to urban groundwater. These include (1) fluctuations in groundwater levels caused by changes in land and water uses; (2) pollution problems caused by point or non-point sources in urban areas; (3) characterization and quantification of the components contributing to groundwater recharge and discharge; (4) specific characteristics of groundwater flow and solute transport models in urban areas; and (5) integration of data for sustainable urban water management. Some of these issues are illustrated for the particular case of Barcelona, where a comprehensive hydrogeological study has been carried out during the last few years by both the public and the private sectors.

  Abstract
  A number of specific factors must be considered when dealing with groundwater in urban areas. Urbanization significantly affects the natural water cycle, both in terms of [...]

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• On the striking similarity between the moments of breakthrough curves for a heterogeneous medium and a homogeneous medium with a matrix diffusion term

  X. Sánchez-Vila, J. Carrera

  Journal of Hydrology (2004). Vol. 294 (1-3), pp. 164-175

  
  

  Abstract

  A usual method to obtain aquifer parameters is to analyze the moments of the breakthrough curves (BTCs) in tracer tests. The parameters to be estimated in this analysis would depend on the conceptual model adopted. Intuitively, if different processes were considered, the shape of the BTCs should be quite different, and one would tend to think that the time and space evolution of the temporal moments should also be quite different. Contrarily, in this paper, we show that two very different conceptual models of solute transport lead to virtually identical moments of the BTC. The two models selected for this study are the classical advection–dispersion equation with a Fickian macrodispersive term and a homogeneous medium advection model with mass-transfer between mobile and immobile matrix phases, for three different models of matrix shape. In both models, the first three moments are linear with travel distance, while the fourth moment is a second order polynomial. This agreement allows us to choose parameters yielding the same moments in the two models. As we consider two fitting parameters, we select them to match the second and third moment. Match in the first moment is obtained from physical arguments. It turns out that the resulting leading term of the fourth moment is identical for both models. As a direct consequence of this work, it follows that for large travel distances it would not be possible to discriminate between conceptual models using data from a single BTC.

  Abstract
  A usual method to obtain aquifer parameters is to analyze the moments of the breakthrough curves (BTCs) in tracer tests. The parameters to be estimated [...]

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• A perturbation solution to the transient Henry problem for seawater intrusion

  D. Tartakovsky, A. Guadagnini, X. Sánchez-Vila, M. Dentz, J. Carrera

  Development in Water Science (2004). Vol. 55 (2), pp. 1573-1581

  
  

  Abstract

  Henry's formulation of seawater intrusion in coastal aquifers consists of a fully coupled system of flow and transport equations with variable density. While the Henry problem often serves as a benchmark for numerical codes, the accuracy of the existing analytical solutions is hard to gauge. We use a perturbation expansion in a small parameter, the ratio between the densities of seawater and freshwater, to derive an analytical solution for the transient Henry problem, which describes the saline intrusion caused by a sudden change in fresh groundwater discharge. This approach is effective for other variable density flow scenarios, since it allows one to decouple the flow and transport equations.

  Abstract
  Henry's formulation of seawater intrusion in coastal aquifers consists of a fully coupled system of flow and transport equations with variable density. While the Henry [...]

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• Sobre la no identificabilidad de los momentos estadísticos del tiempo de llegada de un soluto en un medio heterogéneo

  X. Sánchez-Vila, J. Carrera

  Boletín geológico y minero (2003). Vol. 114 (3), pp. 287-298

  
  

  Abstract

  El análisis de las curvas de llegada procedentes de ensayos de trazadores es una de las metodologías habituales para la obtención de parámetros hidráulicos. En cualquier interpretación de un ensayo, loe parámetros estimados dependerán del modelo conceptual adoptado. Intuitivamente, la inclusión conceptual de diferentes modelos debería dar lugar a curvas con una apariencia muy distinta, lo que a su vez supondría unos momentos estadísticos de la curva de llegada muy distintos. Por el contrarío, en este articulo se muestra que la utilización de algunos modelos conceptuales habituales y en apariencia muy distintos dan lugar e momentos estadísticos virtualmente idénticos en cuanto e su variación con respecto e la distancia o el tiempo recorrido por el soluto. Loe modelos presentados corresponden a: (1) la ecuación de advección dispersión clásica IADEI con un término macrodispersivo de tipo Fickiano: 121 un medio homogéneo con un término de transferencia de masa entre la fase móvil y la matriz rocosa (fase inmóvil) dentro de la cual el soluto presenta transporte difusivo; y 131 el transporte de un soluto que experimente reacciones de adsorción con cinética de equilibrio. En los tres modelos estudiados loe primeros tres momentos de la curva de llegada son lineales con la distancia, mientras que el término dominante del momento de cuarto orden es cuadrático. Este similar comportamiento permite encontrar pare cede modelo conceptual un conjunto de parámetros de modo que reproduzcan de manera similar une curva de llegada concrete. Esto supone un problema importante de identificabilidad de parámetros cuando se usen datos procedentes de une única curve de llegada, puesto que por le apariencia de le curve no seré posible deducir cuáles han sido los procesos dominantes o, por lo menos, descartar alguno de loe procesos posibles. En cualquier ejemplo real será necesario recurrir e información externa pera identificar loe procesos relevantes que han dedo forme e le curva de llegada

  Abstract
  El análisis de las curvas de llegada procedentes de ensayos de trazadores es una de las metodologías habituales para la obtención de parámetros [...]

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• Mean travel time of conservative solutes in randomly heterogeneous unbounded domains under mean uniform flow

  A. Guadagnini, X. Sánchez-Vila, M. Riva, M. Simoni

  Water Resources Research (2003). Vol. 39 (3), pp. 1-5

  
  

  Abstract

  We derive a closed‐form expression for mean travel time of a conservative solute migrating under uniform in the mean flow conditions within an infinite stationary field with simple exponential correlation of the natural logarithm of hydraulic conductivity. Our expression is developed from a consistent second‐order expansion in σY (standard deviation of the log hydraulic conductivity) of the equations for moments of travel time and trajectories of conservative solutes in two‐dimensional randomly nonuniform flows of Guadagnini et al. [2001]. As such, it is nominally valid for moderately heterogeneous fields, with σY2 < 1. Its validity for larger heterogeneity degrees is tested against numerical Monte Carlo simulations. Our results clarify the nonlinear effect in the mean travel time with respect to distance that has been observed numerically (and modeled empirically) in the literature.

  Abstract
  We derive a closed‐form expression for mean travel time of a conservative solute migrating under uniform in the mean flow conditions within an infinite stationary field [...]

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