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This paper employs stochastic analysis to investigate the combined effect of temporal and spatial variability on the temporal variance of baseflow in large catchments. The study makes use of the well‐known aggregated reservoir model, representing the catchment as a network of parallel linear reservoirs. Each reservoir models a sub‐catchment as an independent unit whose discharge temporal variation is characterized by a response time. By treating the rainfall‐generated recharge and the sub‐catchment response times as random variables, the statistical temporal moments of total baseflow are quantified. Comparisons are made between the temporal variance of baseflow in the aggregated reservoir model and that of a single homogeneous reservoir to define an upscaled response time. The analysis of the statistical moments of the random baseflow reveals that the number of reservoirs Nhas a weak impact on baseflow variance, with ergodic conditions achieved even with a small number of reservoirs. The study highlights that the ratio between the recharge correlation time and the geometric mean of the sub‐catchment response times plays a critical role in baseflow damping and the upscaled response. The results indicate that the dynamics of baseflow generation depend not only on the catchment hydro‐geological structure but also on the variability of the input signal. This research underscores the importance of understanding the combined influences of hydro‐geological factors and recharge input variability for baseflow prediction under uncertainty. The present study should be regarded as a first step, setting the theoretical framework for future research toward incorporating field data. Stochastic model to investigate the effect of temporal and spatial variability on the baseflow statistics in large heterogeneous catchmentsBaseflow statistics depend strongly on the ratio of recharge timescale and the geometric mean of reservoir timescalesThe temporal variance of baseflow is used to define an equivalent reservoir with its characteristic timescale Stochastic model to investigate the effect of temporal and spatial variability on the baseflow statistics in large heterogeneous catchments Baseflow statistics depend strongly on the ratio of recharge timescale and the geometric mean of reservoir timescales The temporal variance of baseflow is used to define an equivalent reservoir with its characteristic timescale

The drawdown time is important for the design and assessment of infiltration basins. This paper investigates its dependence on soil heterogeneity, and simple analytical solutions for the mean and the variance are given. The solutions are tested through Monte Carlo simulations in various realistic scenarios, using a modified Green Ampt model for layered soils and a model based on the integration of the 1‐D Richards equation. The impact of the employed approximations is assessed, and the leading role of the spatial distribution of the saturated hydraulic conductivity is emphasized. The effects of other relevant design and natural factors, including the initial water level and water content distribution, are also investigated. Infiltration basins are receiving an increasing interest in water resources management as an alternative to surface water storage in the managed aquifer recharge and as a green infrastructure at the urban scale. One of the main parameters in the design of the infiltration basins is the drawdown time, that is, the time needed to empty the basin after it has been filled with water, for instance, after a flood event or by a managed treatment plant outlet. This work provides simple analytical solutions for the evaluation along a stochastic approach of the drawdown time, accounting for the soil heterogeneity and leading to a probabilistic design of the infiltration structure. The time needed to empty infiltration basins, or drawdown time τ, is an important quantity for their design and assessmentThis work proposes analytical solutions for the mean and variance of τconsidering the soil heterogeneitySolutions have been tested through a set of Monte Carlo numerical simulations, showing that the probability density function is lognormal and investigating the effects of design and natural factors The time needed to empty infiltration basins, or drawdown time τ, is an important quantity for their design and assessment This work proposes analytical solutions for the mean and variance of τconsidering the soil heterogeneity Solutions have been tested through a set of Monte Carlo numerical simulations, showing that the probability density function is lognormal and investigating the effects of design and natural factors

Natural attenuation and in situ oxidation are commonly considered as low‐cost alternatives to ex situ remediation. The efficiency of such remediation techniques is hindered by difficulties in obtaining good dilution and mixing of the contaminant, in particular if the plume deformation is physically constrained by an array of wells, which serves as a containment system. In that case, dilution may be enhanced by inducing an engineered sequence of injections and extractions from such pumping system, which also works as a hydraulic barrier. This way, the aquifer acts as a natural mixer, in a manner similar to the industrialized engineered mixers. Improving the efficiency of hydrogeological mixers is a challenging task, owing to the need to use a 3‐D setup while relieving the computational burden. Analytical solutions, though approximated, are a suitable and efficient tool to seek the optimum solution among all possible flow configurations. Here we develop a novel physically based model to demonstrate how the combined spatiotemporal fluctuations of the water fluxes control solute trajectories and residence time distributions and therefore, the effectiveness of contaminant plume dilution and mixing. Our results show how external forcing configurations are capable of inducing distinct time‐varying groundwater flow patterns which will yield different solute dilution rates. The subsurface is viewed as a natural mixer due to the spatiotemporal variability of the flow fieldSystematic analysis of the interplay between parameters controlling the spatial and temporal variability of the flow field on dilutionNew analytical first‐order solution to predict dilution enhancement in transient and spatially heterogeneous 3‐D subsurface flows

We analyze solute transport in a radially converging 3‐D flow field in a porous medium with spatially heterogeneous hydraulic conductivity (K). The aim of the paper is to analyze the impact of heterogeneity and the mode of injection on BreakThrough Curves (BTCs) detected at a well pumping a contaminated aquifer. The aquifer is conceptualized as an ensemble of blocks of uniform but contrasting Kand the analysis makes use of the travel time approach. Despite the approximations introduced, the model reproduces a laboratory experiment without calibration of transport parameters. Our results also show excellent agreement with numerical simulations for different levels of heterogeneity. We focus on the impact on the BTC of both heterogeneity in Kand solute release conditions. It is shown that the injection mode matters, and the differences in the BTCs between uniform and flux‐proportional injection increase with the heterogeneity of the K‐field. Furthermore, we study the effect of heterogeneity and mode of injection on early and late arrivals at the well. Our model reproduces experimental BTCs using the statistical description of the aquifer structure, with no parameters to be fittedInjection mode impacts the BTC shape, with adverse consequences on management of groundwater contamination (risk and remediation)Ergodicity in the BTC is obtained only for very thick aquifers, in particular for highly heterogeneous media

Heterogeneity of the hydraulic properties is one of the main causes of the seemingly random distribution of solute concentration observed in contaminated aquifers, with macrodispersivity providing a global measure of spreading. Earlier studies on transport of solutes in heterogeneous formations, either theoretical or numerical, expressed dispersivity as a function of the geostatistical properties of the hydraulic conductivity K. In most cases, Kfollows a second‐order statistical characterization, which may not be adequate when heterogeneity is high. In this work, we adopt the Multi‐Indicator Model–Self Consistent Approach (MIMSCA) to compute the longitudinal and transverse macrodispersivity. This methodology enables to model the Kfield by using geological inclusions of different shapes and orientation (defined here as the microstructure), while replicating the heterogeneous macrostructure obtained by the second‐order statistics. The above scheme attempts to reproduce the effect on macrodispersion of different distribution and orientation of local facies, and for instance it may represent the orientation and spatial features of the layers that are often observed in aquifers. The relevant impact of the microstructure on effective conductivity, longitudinal and transverse macrodispersivities is analyzed and discussed, for both binary and lognormally distributed Kfields. Macrodispersion is influenced by the microstructure of hydraulic conductivity when heterogeneity is highA Multi‐Indicator method with spheroidal inclusions allows to capture higher‐order effects on macrodispersionThe asymptotic value of transverse macrodispersivity may vary of two orders of magnitude depending on microstructure

A uniform and horizontal head gradient J is applied to a stratified formation whose given random conductivity K is function of the vertical coordinate x$\_3$ only. K is assumed to be stationary and of finite integral scale I. By Darcy's law, the velocity field V$\_1$(x$\_3$) = JK depicts a fluctuating shear flow. A solute body is injected instantaneously in the formation. In a Lagrangean framework, the second spatial moment of the mean concentration $\$\backslash langleC(\backslash mathbf\{x,\}\; t)\backslash rangle\$$ can be related to the one-particle trajectories variance $\$X\_\{11\}(t,\backslash text\{\; Pe\})\$$ where $\$\backslash text\{Pe\}\$\backslash ,\$=\$\backslash ,\$\backslash langle\; V\_\{1\}\backslash rangleI\_v/D\_\{dT\}\$$ and $\$D\_\{dT\}\$$ is the transverse pore-scale dispersion coefficient. X$\_11$ was determined in the past by Matheron and de Marsily (1980). The present study is concerned with determining the local concentration variance $\$\backslash sigma\_C^2\$$, that depends on the two-particles trajectories covariance Z$\_11$(t). The latter is derived exactly and langle Crangle and $\$\backslash sigma\_C^2\$$ are determined by assuming normal or lognormal probability distribution of trajectories. The results are illustrated for small and very large (ergodic) solute plumes. For large travel time the concentration coefficient of variation at the center of the plume tends asymptotically to a constant value, unlike formations with finite horizontal correlation length of the hydraulic conductivity. The results may serve for benchmarking of numerical codes and in applications for short travel distances in highly anisotropic formations.

The spreading of a contaminant in a heterogeneous aquifer depends on the scales of variability effectively explored by the plume. In particular, we observe two major contributions of the fluctuating velocity field in the contaminant movement: (i) the spreading caused by velocity variations of scales lesser than that of the plume size, which we will call “relative” spreading, and (ii) the meander-like movement of the plume as a whole caused by velocity variations of scale larger than that of the plume size. The aim of this work is to consider the effects of the finite size of the contaminant plume on the local concentration moments and σC. In particular a “relative” concentration, which depends on the scales of variability effectively explored by the plume, is defined. First, the mathematical formulation of the problem is developed along the Lagrangian framework. In particular, the expressions for the relative mean concentration and its variance are presented. Then, the methodology is applied to the regional transport problem, where the influence of the size of the plume and the pore-scale dispersion are quantitatively assessed.

Six conceptually different transport models were applied to the macrodispersion experiment (MADE)‐1 field tracer experiment as a first major attempt for model comparison. The objective was to show that complex mass distributions in heterogeneous aquifers can be predicted without calibration of transport parameters, solely making use of structural and flow data. The models differ in their conceptualization of the heterogeneous aquifer structure, computational complexity, and use of conductivity data obtained from various observation methods (direct push injection logging, DPIL, grain size analysis, pumping tests and flowmeter). They share the same underlying physical transport process of advection by the velocity field solely. Predictive capability is assessed by comparing results to observed longitudinal mass distributions of the MADE‐1 experiment. The decreasing mass recovery of the observed plume is attributed to sampling and no physical process like mass transfer is invoked by the models. Measures like peak location and strength are used in comparing the modeled and measured plume mass distribution. Comparison of models reveals that the predictions of the solute plume agree reasonably well with observations, if the models are underlain by a few parameters of close values: mean velocity, a parameter reflecting log‐conductivity variability, and a horizontal length scale related to conductivity spatial correlation. The robustness of the results implies that conservative transport models with appropriate conductivity upscaling strategies of various observation data provide reasonable predictions of plumes longitudinal mass distribution, as long as key features are taken into account. Aquifer transport is predicted by six models of different heterogeneity conceptualization and field data inputKey model inputs are mean velocity and heterogeneity representation including spatial correlationThe macrodispersion experiment (MADE) site plume longitudinal mass distribution is a robust transport measure Aquifer transport is predicted by six models of different heterogeneity conceptualization and field data input Key model inputs are mean velocity and heterogeneity representation including spatial correlation The macrodispersion experiment (MADE) site plume longitudinal mass distribution is a robust transport measure

Flow of an inert solute in an heterogeneous aquifer is usually considered as dominated by large-scale advection. As a consequence, the pore-scale dispersion, i.e. the pore scale mechanism acting at scales lower than that characteristic of the heterogeneous field, is usually neglected in the computation of global quantities like the solute plume spatial moments. Here the effect of pore-scale dispersion is taken into account in order to find its influence on the longitudinal asymptotic dispersivity D11we examine both the two-dimensional and the three-dimensional flow cases. In the calculations, we consider the finite size of the solute initial plume, i.e. we analyze both the ergodic and the nonergodic cases. With Pe the Péclat number, defined as Pe=Uλ/D, where U, λ, D are the mean fluid velocity, the heterogeneity characteristic length and the pore-scale dispersion coefficient respectively, we show that the infinite Péclat approximation is in most cases quite adequate, at least in the range of Péclat number usually encountered in practice (Pe > 102). A noteworthy exception is when the formation log-conductivity field is highly anisotropic. In this case, pore-scale may have a significant impact on D11, especially when the solute plume initial dimensions are not much larger than the heterogeneities' lengthscale. In all cases, D11appears to be more sensitive to the pore-scale dispersive mechanisms under nonergodic conditions, i.e. for plume initial size less than about 10 log-conductivity integral scales.

In this Reply we further comment on the main assumptions and the results of our recent manuscript “Advective transport in heterogeneous aquifers: Are proxy models predictive?” Markovianity and relevance of the assumptions are discussed

First‐order analysis was used to investigate the effect of a few characteristics of the vadose zone and the solute source at the soil surface on the expected breakthrough curves (BTCs) of tracer solutes in a combined unsaturated‐saturated spatially heterogeneous formation with a three‐dimensional, statistically anisotropic structure. The analysis is based on a stochastic continuum representation of the stationary and nonstationary Eulerian velocity vectors in the unsaturated and saturated domains, respectively, and a general Lagrangian description of the solute mass flux in steady state flow. Results of the present analysis suggest that the length scales and the flow characteristics of the spatially heterogeneous, unsaturated zone that affect the travel time probability density function between the soil surface and the water table also affect the expected solute BTC at a control plane located downstream in the groundwater. The flow in the unsaturated zone might enhance the early arrival time and the spreading of the expected solute BTC in the groundwater, especially when the vertical extent of the unsaturated zone is relatively large; when the unsaturated zone consists of relatively dry, coarse‐textured, stratified soil material; and when the horizontal extent of the solute source in the direction of the mean groundwater flow is relatively large. The effect of the characteristics of the vadose zone and the solute source at the soil surface on the expected solute BTC in the groundwater decreases as the distance to the control plane increases and as the process of the solute breakthrough is more advanced. The relatively good agreement between the results of the present first‐order, stochastic analysis with the results of detailed, numerical analyses of flow and transport in a three‐dimensional heterogeneous, combined vadose zone–groundwater flow domain suggests that the first‐order stochastic analyses may be sufficiently reliable to show appropriate trends which might occur in realistic combined vadose zone–groundwater flow systems.