This paper is the outcome of a community initiative to identify major unsolved scientific problems in hydrology motivated by a need for stronger harmonisation of research efforts. The procedure involved a public consultation through online media, followed by two workshops through which a large number of potential science questions were collated, prioritised, and synthesised. In spite of the diversity of the participants (230 scientists in total), the process revealed much about community priorities and the state of our science: a preference for continuity in research questions rather than radical departures or redirections from past and current work. Questions remain focused on the process-based understanding of hydrological variability and causality at all space and time scales. Increased attention to environmental change drives a new emphasis on understanding how change propagates across interfaces within the hydrological system and across disciplinary boundaries. In particular, the expansion of the human footprint raises a new set of questions related to human interactions with nature and water cycle feedbacks in the context of complex water management problems. We hope that this reflection and synthesis of the 23 unsolved problems in hydrology will help guide research efforts for some years to come. ARTICLE HISTORY

[1] Many details about the flow of water in soils in a hillslope are unknowable given current technologies. One way of learning about the bulk effects of water velocity distributions on hillslopes is through the use of tracers. However, this paper will demonstrate that the interpretation of tracer information needs to become more sophisticated. The paper reviews, and complements with mathematical arguments and specific examples, theory and practice of the distribution(s) of the times water particles injected through rainfall spend traveling through a catchment up to a control section (i.e., "catchment" travel times). The relevance of the work is perceived to lie in the importance of the characterization of travel time distributions as fundamental descriptors of catchment water storage, flow pathway heterogeneity, sources of water in a catchment, and the chemistry of water flows through the control section. The paper aims to correct some common misconceptions used in analyses of travel time distributions. In particular, it stresses the conceptual and practical differences between the travel time distribution conditional on a given injection time (needed for rainfall-runoff transformations) and that conditional on a given sampling time at the outlet (as provided by isotopic dating techniques or tracer measurements), jointly with the differences of both with the residence time distributions of water particles in storage within the catchment at any time. These differences are defined precisely here, either through the results of different models or theoretically by using an extension of a classic theorem of dynamic controls. Specifically, we address different model results to highlight the features of travel times seen from different assumptions, in this case, exact solutions to a lumped model and numerical solutions of the 3-D flow and transport equations in variably saturated, physically heterogeneous catchment domains. Our results stress the individual characters of the relevant distributions and their general nonstationarity yielding their legitimate interchange only in very particular conditions rarely achieved in the field. We also briefly discuss the impact of oversimple assumptions commonly used in analyses of tracer data.

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We consider steady flow of water in a confined aquifer toward a fully penetrating well of radius rw (Figure 1). The hydraulic conductivity K is modeled as a three‐dimensional stationary random space function. The two‐point covariance of Y = In (K/KG) is of axisymmetric anisotropy, with I and Iυ, the horizontal and vertical integral scales, respectively, and KG, the geometric mean of K. Unlike previous studies which assumed constant flux, the well boundary condition is of given constant head (Figure 1). The aim of the study is to derive the mean head 〈H〉 and the mean specific discharge 〈q〉 as functions of the radial coordinate r and of the parameters σy2, e = I/Iυ and rw/I. An approximate solution is obtained at first‐order in σy2, by replacing the well by a line source of strength proportional to K and by assuming ergodicity, i.e., equivalence between , , space averages over the vertical, and 〈H〉 〈q〉, ensemble means. An equivalent conductivity Keq is defined as the fictitious one of a homogeneous aquifer which conveys the same discharge Q as the actual one, for the given head Hw in the well and a given head in a piezometer at distance r from the well. This definition corresponds to the transmissivity determined in a pumping test by an observer that measures Hw, , andQ. The main result of the study is the relationship (19) Keq = KA(1 − λ) + Kefuλ, where KA is the conductivity arithmetic mean and Kefu is the effective conductivity for mean uniform flow in the horizontal direction in the same aquifer. The weight coefficient λ < 1 is derived explicitly in terms of two quadratures and is a function of e, rw/I and r/I . Hence Keq unlike Kefu, is not a property of the medium solely. For rw/I < 0.2 and forr/I > 10, λ has the simple approximate expression λ* = ln (r/I)/ In )r/rw). Near the well, λ ≅ 0 and Keq ≅ KA, which is easily understood, since for rw/I ≪ 1 the formation behaves locally like a stratified one. In contrast, far from the well λ ≅ 1 and Keq ≅ Kefu the flow being slowly varying there. Since KA > Kefu, our result indicates that the transmissivity is overestimated in a pumping test in a steady state and it decreases with the distance from the well. However, the difference between KA and Kefu is small for highly anisotropic formations for which e ≪ 1 . A nonlocal effective conductivity, which depends only on the heterogeneous structure, is derived in Appendix A along the lines of Indelman and Abramovich [1994].

[1] In parts 1 and 2 a multi-indicator model of heterogeneous formations is devised in order to solve flow and transport in highly heterogeneous formations. The isotropic medium is made up from circular (2-D) or spherical (3-D) inclusions of different conductivities K, submerged in a matrix of effective conductivity. This structure is different from the multi-Gaussian one, even for equal log conductivity distribution and integral scale. A snapshot of a two-dimensional plume in a highly heterogeneous medium of lognormal conductivity distribution shows that the model leads to a complex transport picture. The present study was limited, however, to investigating the statistical moments of ergodic plumes. Two approximate semianalytical solutions, based on a self-consistent model (SC) and on a first-order perturbation in the log conductivity variance (FO), are used in parts 1 and 2 in order to compute the statistical moments of flow and transport variables for a lognormal conductivity pdf. In this paper an efficient and accurate numerical procedure, based on the analytic-element method [Strack, 1989], is used in order to validate the approximate results. The solution satisfies exactly the continuity equation and at high-accuracy the continuity of heads at inclusion boundaries. The dimensionless dependent variables depend on two parameters: the volume fraction n of inclusions in the medium and the log conductivity variance s Y 2 . For inclusions of uniform radius, the largest n was 0.9 (2-D) and 0.7 (3-D), whereas the largest s Y 2 was equal to 10. The SC approximation underestimates the longitudinal Eulerian velocity variance for increasing n and increasing s Y 2 in 2-D and, to a lesser extent, in 3-D, as compared to numerical results. The FO approximation overestimates these variances, and these effects are larger in the transverse direction. The longitudinal velocity pdf is highly skewed and negative velocities are present at high s Y 2 , especially in 2-D. The main results are in the comparison of the macrodispersivities, computed with the aid of the Lagrangian velocity covariances, as functions of travel time. For the longitudinal macrodispersivity, the SC approximation yields results close to the numerical ones in 2-D for n = 0.4 but underestimates them for n = 0.9. The asymptotic, large travel time values of macrodispersivities in the SC and FO approximations are close for low to moderate s Y 2 , as shown and explained in part 1. However, while the slow tendency to Fickian behavior is well reproduced by SC, it is much quicker in the FO approximation. In 3-D the SC approximation is closer to numerical one for the highest n = 0.7 and the different s Y 2 = 2, 4, 8, and the comparison improves if molecular diffusion is taken into account. Transverse macrodispersivity for small travel times is underestimated by SC in 2-D and is closer to numerical results in 3-D, whereas FO overestimates them. Transverse macrodispersivity asymptotically tends to zero in 2-D for large travel times. In 3-D the numerical simulations l...

[1] Flow of uniform mean velocity U takes place in a formation of spatially variable, random conductivity K(x). Advective transport of a plume of an inert solute is investigated by the Lagrangean approach. The aim of the study is to determine the spatial moments of the plume, i.e., of fluid particle trajectories, for highly heterogeneous aquifers, for which s Y > 1, where Y = ln K. A multi-indicator model of the permeability structure, which is different from the common multi-Gaussian one, is proposed: the formation is modeled as a collection of N blocks of different K ( j) . The structure is defined by the distribution of K ( j) , the blocks' shape, and the coordinates of their centroids. The following simplifications are adopted: the blocks are inclusions of a regular shape (circles, spheres for isotropic media investigated here) defined by the radius A, and the inclusions are not overlapping, and their centroids are distributed uniformly and independently in space. At the continuous limit the model is characterized by the joint pdf f(Y, A). The model is shown to be quite general and to comprise binary, bimodal, indicator variograms and unimodal distributions of Y as particular cases. The study is focused on the latter case, with Y normal N [hY i, s Y 2 ] and stationary covariance of given integral scale I Y ; these are the parameters commonly estimated for sedimentary formations. This leaves still freedom in selecting the pdf f(A). The simple model selected for semianalytical and numerical analysis is that of inclusions of radius R and volume fraction n, submerged in a matrix of effective conductivity K ef . The latter represents the effect of inclusions of much smaller radius, which appear as a nugget in the log conductivity two-point covariance. An approximate analytical solution of the flow is obtained by using a self-consistent approximation, while a fully numerical one is derived in part 3 [Janković et al., 2003a]. Transport is solved by particle tracking, and the time-dependent spatial moments (trajectories variance, skewness, kurtosis) are presented in part 2 [Fiori et al., 2003]. In the self-consistent approximation the asymptotic longitudinal macrodispersivity a L , which is a function of Y, shows strong nonlinear effects: inclusions of large positive Y lead to a finite a L , whereas a L grows unbounded for those of negative Y. This effect is not captured by the common first-order approximation in s Y , which is symmetrical and overestimates a L for Y > 0 and underestimates it for Y < 0. As a result, the second spatial moment is predicted accurately by the first-order approximation, by cancellation of errors, provided that f(Y) is symmetrical. However, the transient regime and higher-order moments are not captured by the first-order approximation.INDEX TERMS: 1829 Hydrology: Groundwater hydrology; 1831 Hydrology: Groundwater quality; 1832 Hydrology: Groundwater transport; 1869 Hydrology: Stochastic processes; KEYWORDS: transport, self-consistent, inclusion, dispersion, analytic element, first-order C...

Spreading of conservative solutes in groundwater due to aquifer heterogeneity is quantified by the macrodispersivity, which was found to be scale dependent. It increases with travel distance, stabilizing eventually at a constant value. However, the question of its asymptotic behavior at very large scale is still a matter of debate. It was surmised in the literature that macrodispersivity scales up following a unique scaling law. Attempts to define such a law were made by fitting a regression line in the log‐log representation of an ensemble of macrodispersivities from multiple experiments. The functional relationships differ among the authors, based on the choice of data. Our study revisits the data basis, used for inferring unique scaling, through a detailed analysis of literature marcodispersivities. In addition, values were collected from the most recent tracer tests reported in the literature. We specified a system of criteria for reliability and reevaluated the reliability of the reported values. The final collection of reliable estimates of macrodispersivity does not support a unique scaling law relationship. On the contrary, our results indicate, that the field data can be explained as a collection of macrodispersivities of aquifers with varying degree of heterogeneity where each exhibits its own constant asymptotic value. Our investigation concludes that transport, and particularly the macrodispersivity, is formation‐specific, and that modeling of transport cannot be relegated to a unique scaling law. Instead, transport requires characterization of aquifer properties, e.g., spatial distribution of hydraulic conductivity, and the use of adequate models.

[1] Solute transport in a three-dimensional, heterogeneous hillslope is analyzed through a series of detailed, three-dimensional numerical simulations. The investigation focuses on the transport of a pulse of a passive solute, taking into account realistic features of the relevant flow domain (i.e., the spatial heterogeneity of the soil hydraulic properties and records of the time-dependent meteorological data and evapotranspiration). The scope of the present work is to analyze a few issues regarding the travel time probability density function (pdf) f(t) of solute, with particular reference to the following: (1) the suitability of the time invariance assumption and the circumstance under which it may represent a valid approximation, (2) the shape of the resulting travel time pdf f(t), and (3) the difference between f(t) and the instantaneous unit hydrograph. It is found that in many cases of practical interest the transport of contaminants in catchments can be analyzed by assuming the time-invariant approximation for f(t), provided that the calendar time is replaced by a flow-corrected time. Here f(t) is calculated through the analysis of the breakthrough curve represented in terms of flux-averaged concentration versus cumulate discharge. The rescaling of time with respect to the cumulated outflow takes care in an approximate way of the transient processes occurring in the porous medium. The derived f(t) is weakly dependent on various attributes, like the level of heterogeneity, presence of evaporation or transpiration, and injection period. The main exception regards the cases in which plant transpiration is intense in the vicinity of the channel after relatively long periods of low rain. The impact of the parameters' heterogeneity on f(t) is generally quite limited, the dispersion being ruled by the distribution of length paths within the hillslope. The derived f(t) seems compatible with the Gamma distribution, being characterized by both fast and slow responses, with a pronounced power law early peak and an exponential-like tail. Comparison of f(t) with the often employed Gamma instantaneous unit hydrograph emphasizes the differences between water and solute dynamics after rainfall events. Citation: Fiori, A., and D. Russo (2008), Travel time distribution in a hillslope: Insight from numerical simulations, Water Resour.