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• We proposed an analytical method for upscaling hydraulic conductivity using perturbation expansion techniques and Fourier analysis. • The proposed method generates the same results as from the numerical method in a finite difference scheme with periodic boundary conditions efficiently. • The proposed method is validated for various cases considering anisotropy, heterogeneity and geometry in general. Modern geological modeling techniques represent anisotropic heterogeneous formations by high-resolution grids, which can be computationally prohibitive. This motivates the upscaling process that scales-up properties defined at a fine-scale system to equivalent properties defined at a coarse-scale system. In general, analytical methods are very efficient but limited to assumptions and approximations, whereas numerical methods are more robust albeit more time-consuming. In this work, we developed an analytical method to approximate numerical solutions in a finite difference scheme with periodic boundary conditions for two-dimensional problem. Using perturbation expansion techniques and Fourier analysis, the method generates explicit formulas of tensorial equivalent conductivity considering heterogeneity and anisotropy of two-dimensional space, as well as geometry of gridblocks. It is applicable for various cases with different covariance/variagram models and a wide range of log-conductivity variances, correlation lengths, rotation angles, anisotropy ratios of fine grid conductivity, anisotropy ratios of fine grid size, and the number of fine gridblocks in a coarse gridblock. The analytical method matched well with the numerical method for the estimation of the conductivity tensor, hydraulic head, and discharge velocity. The coefficients in the analytical method need to be computed only once for any given statistics, which makes the proposed method much more efficient than the numerical method. [ABSTRACT FROM AUTHOR]

Characterization of flow topology is essential to understand the effects of the heterogeneity and dimensionality of geological formations on the mixing of inert solute clouds in these same geological formations. In this work, we numerically study two indicators of flow topology, the averaged vorticity magnitude | ω | av and the averaged positive second invariant Q av of the deformation tensor ∇ u , in steady Darcy flows through exponentially correlated lognormal hydraulic conductivity fields K. Our numerical results allow us to establish the relationships between the two indicators considered here and the hydraulic conductivity variance σ 2 in 2D and 3D, highlighting the role played by the spatial structure of these porous media on flow topology, and indirectly on mixing. This work leads us to assess the maximum dilution index E max , indicator of mixing, theoretically known to increase monotonically in steady Darcy flows through isotropic heterogeneous porous media. Our numerical results allow us to test this hypothesis by establishing the relationship between the slope a of maximum dilution index E max and the averaged positive second invariant Q av of deformation tensor ∇ u. The parameters of this relationship depend on molecular diffusion and dimensionality of problem considered. [ABSTRACT FROM AUTHOR]

Highlights • Impact of positive porosity-conductivity correlation on probabilistic risk analysis. • Relevance of including porosity variability depends on the metric of interest, the control plane-source distance and the level of conductivity heterogeneity. • A heterogeneous porosity field leads to a reduction of the increased lifetime cancer risk uncertainty. Abstract Heterogeneities in natural porous formations, mainly manifested through the hydraulic conductivity (K) and, to a lesser degree, the porosity (ϕ), largely control subsurface flow and solute transport. The influence of the heterogeneous structure of K on transport processes has been widely studied, whereas less attention is dedicated to the joint heterogeneity of conductivity and porosity fields. Our study employs Monte Carlo simulations to investigate the coupled effect of K − ϕ spatial variability on the transport behavior (and uncertainty) of conservative and reactive plumes within a 3D aquifer domain. We explore multiple scenarios, characterized by different levels of heterogeneity of the geological properties, and compare the computational results from the joint K − ϕ heterogeneous system with the results originating from generally adopted constant ϕ conditions. In our study, the spatially variable K − ϕ fields are positively correlated. We statistically analyze key Environmental Performance Metrics: first arrival times and peak mass fluxes for non-reactive species and increased lifetime cancer risk for reactive chlorinated solvents. The conservative transport simulations show that considering coupled K − ϕ fields decreases the plume dispersion, increases both the first arrival times of solutes and the peak mass fluxes at the observation planes. A positive correlation between aquifer connectivity and peak mass fluxes is identified for both homogeneous and heterogeneous ϕ. Our conservative transport results indicate that the relevance of ϕ variability can depend on the metric of interest, the control plane-source distance as well as the level of heterogeneity of the conductivity field. The analysis on reactive transport shows that ϕ variability only slightly affects the mean increased lifetime cancer risk at the control planes but leads to a considerable reduction of the cancer risk uncertainty. We also see that the sensitivity of cancer risk towards ϕ heterogeneity can be influenced by the level of variability of the conductivity field, the source-to-control plane distance, but is not affected by the manner in which the contaminant concentration is computed. [ABSTRACT FROM AUTHOR]

Abstract A systematic understanding of hyporheic flux (HF) and residence times (RT) is important as they are a major control of biogeochemical processing in streambeds. Previous studies addressing the effect of heterogeneity in streambed hydraulic conductivity (K) on HF and RT have come to deviating conclusions depending on the specific study design and the selection of heterogeneity cases being investigated. To more systematically evaluate the effects of conductivity heterogeneity on HF and RT, we simulated hyporheic exchange induced by idealized streambed ripples over a large range of heterogeneities. Conductivity heterogeneity was represented in the simulations in terms of 10,000 different heterogeneity realizations from a geostatistical model based on continuous Gaussian and discrete indicator random fields. We demonstrate that any isotropic homogeneous K-field, as an average of a heterogeneous K-field, can only match RT or HF of the respective heterogeneous K-field, but never both. We found exponential correlations of RT and HF with the variance of heterogeneous conductivity. Based on these correlations, an equivalent anisotropic homogeneous conductivity tensor K can be derived. This equivalent anisotropic K efficiently accounts for the effects of small scale heterogeneity on HF and RT. It can be calculated from the median and variance of the hydraulic conductivity distribution of the targeted heterogeneous sediment, without explicitly characterizing the sediment texture. [ABSTRACT FROM AUTHOR]

Quantifying the uncertainty in solute mass discharge at an environmentally sensitive location is key to assess the risks due to groundwater contamination. Solute mass fluxes are strongly affected by the spatial variability of hydrogeological properties as well as release conditions at the source zone. This paper provides a methodological framework to investigate the interaction between the ubiquitous heterogeneity of the hydraulic conductivity and the mass release rate at the source zone on the uncertainty of mass discharge. Through the use of perturbation theory, we derive analytical and semi-analytical expressions for the statistics of the solute mass discharge at a control plane in a three-dimensional aquifer while accounting for the solute mass release rates at the source. The derived solutions are limited to aquifers displaying low-to-mild heterogeneity. Results illustrate the significance of the source zone mass release rate in controlling the mass discharge uncertainty. The relative importance of the mass release rate on the mean solute discharge depends on the distance between the source and the control plane. On the other hand, we find that the solute release rate at the source zone has a strong impact on the variance of the mass discharge. Within a risk context, we also compute the peak mean discharge as a function of the parameters governing the spatial heterogeneity of the hydraulic conductivity field and mass release rates at the source zone. The proposed physically-based framework is application-oriented, computationally efficient and capable of propagating uncertainty from different parameters onto risk metrics. Furthermore, it can be used for preliminary screening purposes to guide site managers to perform system-level sensitivity analysis and better allocate resources. [ABSTRACT FROM AUTHOR]

Detecting where and when a contaminant entered an aquifer from observations downgradient of the source is a difficult task; this identification becomes more challenging when the uncertainty about the spatial distribution of hydraulic conductivity is accounted for. In this paper, we have implemented an application of the restart normal-score ensemble Kalman filter (NS-EnKF) for the simultaneous identification of a contaminant source and the spatially variable hydraulic conductivity in an aquifer. The method is capable of providing estimates of the spatial location, initial release time, the duration of the release and the mass load of a point-contamination event, plus the spatial distribution of hydraulic conductivity together with an assessment of the estimation uncertainty of all the parameters. The method has been applied in synthetic aquifers exhibiting both Gaussian and non-Gaussian patterns. The identification is made possible by assimilating in time both piezometric head and concentration observations from an array of observation wells. The method is demonstrated in three different synthetic scenarios that combine hydraulic conductivities with unimodal and bimodal histograms, and releases in high and low conductivity zones. The results prove that the specific implementation of the EnKF is capable of recovering the source parameters with some uncertainty and of recovering the main patterns of heterogeneity of the hydraulic conductivity fields by assimilating a sufficient number of state variable observations. The proposed approach is an important step towards contaminant source identification in real aquifers, which may have logconductivity spatial distributions with either Gaussian or non-Gaussian features, yet, it is still far from practical applications since the transport parameters, the external sinks and sources and the initial and boundary conditions are assumed known. [ABSTRACT FROM AUTHOR]

Geological heterogeneity, i.e. the spatial variability of discrete hydrogeological units, is investigated in an aquifer analog of glacio-fluvial sediments to determine how such a geological structure can be simplified for characterization needs. The aquifer analog consists of ten hydrofacies whereas the scarcity of measurements in typical field studies precludes such detailed spatial models of hydraulic properties. Of particular interest is the role of connectivity of the hydrofacies structure, along with its effect on the connectivity of mass transport, in site characterization for predicting early arrival times. Transport through three realizations of the aquifer analog is modeled with numerical particle tracking to ascertain the fast flow channel through which early arriving particles travel. Three simplification schemes of two-facies models are considered to represent the aquifer analogs, and the velocity within the fast flow channel is used to estimate the apparent hydraulic conductivity of the new facies. The facies models in which the discontinuous patches of high hydraulic conductivity are separated from the rest of the domain yield the closest match in early arrival times compared to the aquifer analog, but assuming a continuous high hydraulic conductivity channel connecting these patches yields underestimated early arrivals times within the range of variability between the realizations, which implies that the three simplification schemes could be advised but pose different implications for field measurement campaigns. Overall, the results suggest that the result of transport connectivity, i.e. early arrival times, within realistic geological heterogeneity can be conserved even when the underlying structural connectivity is modified. [ABSTRACT FROM AUTHOR]

Aquifer heterogeneity presents a primary challenge in predicting the movement of solutes in groundwater systems. The problem is particularly difficult on very large scales, across which permeability, chemical properties, and pumping rates may vary by many orders of magnitude and data are often sparse. An example is the fluvio-deltaic aquifer system of Bangladesh, where naturally-occurring arsenic (As) exists over tens of thousands of square kilometers in shallow groundwater. Millions of people in As-affected regions rely on deep (≥150 m) groundwater as a safe source of drinking water. The sustainability of this resource has been evaluated with models using effective properties appropriate for a basin-scale contamination problem, but the extent to which preferential flow affects the timescale of downward migration of As-contaminated shallow groundwater is unknown. Here we embed detailed, heterogeneous representations of hydraulic conductivity (K), pumping rates, and sorptive properties (K d ) within a basin-scale numerical groundwater flow and solute transport model to evaluate their effects on vulnerability and deviations from simulations with homogeneous representations in two areas with different flow systems. Advective particle tracking shows that heterogeneity in K does not affect average travel times from shallow zones to 150 m depth, but the travel times of the fastest 10% of particles decreases by a factor of ∼2. Pumping distributions do not strongly affect travel times if irrigation remains shallow, but increases in the deep pumping rate substantially reduce travel times. Simulation of advective-dispersive transport with sorption shows that deep groundwater is protected from contamination over a sustainable timeframe (>1000 y) if the spatial distribution of K d is uniform. However, if only low-K sediments sorb As, 30% of the aquifer is not protected. Results indicate that sustainable management strategies in the Bengal Basin should consider impacts of both physical and chemical heterogeneity, as well as their correlation. These insights from Bangladesh show that preferential flow strongly influences breakthrough of both conservative and reactive solutes even at large spatial scales, with implications for predicting water supply vulnerability in contaminated heterogeneous aquifers worldwide. [ABSTRACT FROM AUTHOR]

Assessing the spatial distribution of hydraulic conductivity ( K ) in natural aquifers is fundamental to predict the spatio-temporal evolution of solutes, a process that is mainly controlled by the heterogeneity of K . In sedimentary aquifers, the vertical variations of K are typically more relevant than the horizontal ones in controlling the plume evolution at the local scale; such K vertical distributions can be inferred by combining the Lagrangian formulation of transport with the assimilation of tracer test data via the ensemble Kalman filter (EnKF). In this work, the data for the assimilation procedure are provided by monitoring tracer tests with electrical resistivity tomography (ERT). Our main objective is to show the possibility of directly using ERT data by assimilating the solute travel times, instead of the concentration values, thus avoiding the need for a petrophysical law. The methodology is applied to both a synthetic and a real test case and gives a satisfactory retrieval of the K field distribution, as well as of the solute evolution. [ABSTRACT FROM AUTHOR]

The knowledge of subsurface heterogeneity is a prerequisite to describe flow and transport in porous media. Of particular interest are the variance and the correlation scale of hydraulic conductivity. In this study, we present how these aquifer parameters can be inferred using empirical steady state pumping test data. We refer to a previously developed analytical solution of “effective well flow” and examine its applicability to pumping test data as under field conditions. It is examined how the accuracy and confidence of parameter estimates of variance and correlation length depend on the number and location of head measurements. Simulations of steady state pumping tests in a confined virtual aquifer are used to systematically reduce sampling size while determining the rating of the estimates at each level of data density. The method was then applied to estimate the statistical parameters of a fluvial heterogeneous aquifer at the test site Horkheimer Insel, Germany. We conclude that the “effective well flow” solution is a simple alternative to laboratory investigations to estimate the statistical heterogeneity parameter using steady state pumping tests. However, the accuracy and uncertainty of the estimates depend on the design of the field study. In this regard, our results can help to improve the conceptual design of pumping tests with regard to the parameter of interest. [ABSTRACT FROM AUTHOR]

Nitrate contamination of aquifers is threatening the sustainability of water resource production around the globe. Denitrification is an ubiquitous process in groundwater systems and needs to be represented in managerial and predictive efforts. Yet, fully understanding the temporal and spatial dynamics of all biogeochemical and physical properties controlling nitrate reduction at the basin scale remains a challenging task. Conceptual simplifications such as homogenization are used, but their implications on model performance are often not well assessed. This paper (1) analyzes the importance of denitrification spatial variability on the fate of non-point source nitrate contamination, and (2) assesses the performance of upscaled denitrification rate coefficient values. Heterogeneity in the hydraulic conductivity of a typical unconsolidated alluvial aquifer system and its associated uncertainty is systematically represented in our 3D numerical simulations of flow and transport. The spatial variability in the nitrate reduction capacities of the aquifer is accounted for by 3 different conceptual models: a positive and a negative correlation with the aquifer hydraulic properties and a uniform random distribution. Our results indicate that heterogeneity in the denitrification rate coefficient – as well as the model describing spatial variability – has a significant impact on concentration statistics observed at a series of extraction wells. However, we also show that, under the modeling setting used in this study (e.g., stationarity of reactivity), upscaled values of denitrification rate coefficients allow a virtually perfect reproduction of nitrate concentration statistics, a promising outcome for employing parameter estimation (calibration) of effective rate coefficients of denitrification: A detailed description of the spatial variability of the nitrate reduction capacity of the alluvial aquifer is not necessary. Yet, these homogeneous, upscaled denitrification rate coefficients result from a complex interaction between the biochemistry and the physics of the aquifer, which makes their direct estimation a challenging task. [Display omitted] • Spatial variability structure of denitrification rate coefficients influence nitrate dynamics. • Upscaled, effective denitrification rate coefficients reproduce well concentration statistics. • Upscaling denitrification rates is complex due to interplays between key processes. [ABSTRACT FROM AUTHOR]

• Log-conductivity (Y) is modeled as a generalized sub-Gaussian (GSG) field. • Average longitudinal spreading is smaller for GSG than for Gaussian Y fields. • Plume stretching in GSG fields is enhanced with respect to Gaussian fields. • Local dispersion masks identification of impacts of non-Gaussianity on transport. • The GSG nature of Y can be difficult to identify relying only on transport metrics. There are increasing evidences that probability distributions and associated statistical moments of a variety of hydrogeological and soil science variables and their spatial increments display distinctive scale-dependent features that are not captured by a typical Gaussian model. A Generalized Sub-Gaussian (GSG) model is able to capture key aspects of this pattern. We present the results of a suite of computational analyses set in a Monte Carlo framework and aimed at assessing the impact of a GSG structure of log hydraulic conductivity (Y) on transport of a conservative solute through a three-dimensional bounded porous medium under steady-state saturated Darcy flow. Our results indicate that the longitudinal spreading of a plume is on average significantly smaller for Sub-Gaussian than for Gaussian Y fields. Otherwise, the velocity field arising from a Sub-Gaussian Y field induces enhanced plume stretching with respect to what can be observed in a Gaussian Y setting, this aspect potentially influencing the strength of solute mixing within these two types of conductivity domains. We also find that, in some cases, it may be difficult to identify the nature of the underlying conductivity field relying solely on observations of solute concentrations migrating within the system. In this regard, we show that the action of local dispersion tends to mask the influence of Sub-Gaussianity on major transport metrics. [ABSTRACT FROM AUTHOR]

• An integro-differential equation for upscaling reactive transport in a correlated velocity field is derived. • The proposed approach provides a way to predict reactive-transport based on the geostatistical properties of the media. • Breakthrough curves of a non-linearly decaying solute from the upscaled model are compared with the ones from a 2D model. While commonly non-local transport models have been shown to reproduce breakthrough curves resulting from transport in heterogeneous aquifers successfully, open questions include the formal link between the upscaled governing equations and the sub-scale heterogeneity, and the ability to account for the effect of heterogeneity on effective chemical reaction rates in the presence of non-linear multi-component reactions. Time-domain random walk approaches based on velocity Markov models provide a framework to resolve these issues by incorporating the spatial correlation of velocity of a solute particle in consecutive locations along its trajectory. These approaches often rely on particle tracking approaches, which, however, can be computationally burdensome especially when non-linear reactions are sought to be modeled. In this paper, an integro-differential equation is proposed for upscaling multi-component reactive transport in heterogeneous media that relies on copulas for representing the velocity correlation structure. For this purpose, we express concentration or flux of solutes as a distribution over their velocity. We then derive the integro-differential equation that governs the evolution of concentration distribution over a quantity defined as velocity-rank. In this way, the spatial evolution of breakthrough curves away from the source is predicted based on ergodic cross-sectional velocity distributions and a parameterized copula function which expresses the correlation between velocity-ranks of a solute particle along its trajectory. We demonstrate the validity of the proposed model by comparing breakthrough curves for conservative and non-linearly reacting solutes based on realizations of hydraulic conductivity fields to the results of the upscaled model. [ABSTRACT FROM AUTHOR]

• A convolutional neural network (CNN) is built to estimate macro-dispersivity from hydraulic conductivity field directly. • The validation of the CNN shows that most estimations fall within ± 20% error range for conductivity fields with moderate heterogeneity. • For a given size of training datasets, the performance of estimating drops with an increase in the heterogeneity of conductivity field. • The CNN trained by conductivity fields with high heterogeneity can achieve superior performance in estimations for fields with less heterogeneity. Building a quantitative relation between the spatial heterogeneity of the hydraulic conductivity fields and the macroscale behavior of solute transport is fundamental for groundwater environment problem. In this work, the deep learning technique is explored to build the functional mapping between the hydraulic conductivity field and the longitudinal macro-dispersivity. We examine the capability of the deep neural network in estimating macro-dispersivities of conductivity fields with different variances. The universality of the trained deep neural network is investigated. Comparisons of the neural network results and the reference values (macro-dispersivities from transport simulation) suggest the promising potential of deep learning technique in porous media with moderate heterogeneity. For a given size of training datasets, the deep neural network produces better macro-dispersivity estimation for the conductivity field with smaller variance. The trained neural network by conductivity fields with larger variance has stronger universality for macro-dispersivity estimation. This study demonstrates that deep neural network can be an effective alternative for estimating macroscale behavior of solute transport by directly interpreting hydraulic conductivity fields. [ABSTRACT FROM AUTHOR]