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Geostatistics as a subfield of statistics accounts for the spatial correlations encountered in many applications of, for example, earth sciences. Valuable information can be extracted from these correlations, also helping to address the often encountered burden of data scarcity. Despite the value of additional data, the use of geostatistics still falls short of its potential. This problem is often connected to the lack of user-friendly software hampering the use and application of geostatistics. We therefore present GSTools, a Python-based software suite for solving a wide range of geostatistical problems. We chose Python due to its unique balance between usability, flexibility, and efficiency and due to its adoption in the scientific community. GSTools provides methods for generating random fields; it can perform kriging, variogram estimation and much more. We demonstrate its abilities by virtue of a series of example applications detailing their use.

Thermal retardation and dispersion are important processes affecting advective heat transport in sedimentary aquifers, yet little is known how they are influenced by heterogeneity of hydraulic conductivity. We investigate the effect of macro‐scale heterogeneity on transient heat transport in a three‐dimensional domain through direct numerical Monte‐Carlo simulations. The model describes the evolution of a heat plume in a heterogeneous aquifer generated by a borehole heat exchanger. We characterize the transport by calculating the dispersion coefficient and effective thermal retardation factor as ensemble average of the heterogeneous realizations. In addition to different degrees of heterogeneity, we examine the influence of the thermal Péclet number on the effective thermal retardation factor. Simulations reveal that for homogeneous hydraulic conductivity, the effective thermal retardation factor equals the predicted, apparent thermal retardation factor. However, in heterogeneous cases, the effective thermal retardation factor is substantially lower than the apparent value at early times, with this effect becoming more pronounced as the Péclet number increases. We attribute the deviation of the effective thermal retardation factor from the apparent value to preferential flow through zones with higher hydraulic conductivity and delayed local heat diffusion into zones with lower hydraulic conductivity. Assuming that the effective thermal retardation factor differs from the apparent value in the presence of local thermal non‐equilibrium (LTNE) effects, we call the observed effect “field‐scale LTNE.” Finally, we derive a formula estimating effective thermal retardation as a function of log‐conductivity variance and the Péclet number. Our results can improve heat tracer techniques in hydraulically heterogeneous environments.

Injection of silicate grouting materials is widely used to create temporary horizontal layers for reducing inflow of groundwater at construction sites, in regions with shallow water tables. The erosion of a grouting layer was investigated by means of analytical solutions for groundwater flow and transport within a pit after construction finished. Erosion is assumed to occur by dissolution of the temporary injection layer and subsequent advective transport. Thereby, the hydraulic conductivity changes with time. This paper presents novel analytical solutions and approximate solutions for the major fluxes in the construction pit as a function of the domain settings, aquifer gradient and hydraulic conductivity. In addition, the mass flux and the dilution ratio of erosion-related components leaving the construction pit and entering the aquifer are quantified. Derived solutions are verified against numerical simulations. A sensitivity study shows the impact of domain settings on fluxes and dilution ratio. The results confirm that mass flux of grout components increases with ongoing erosion. Thus, its effect on groundwater quality increases with time after construction ceased.

We investigate the upscaling of diffusive transport parameters using a stochastic framework. At sub-REV (representative elementary volume) scale, the complexity of the pore space geometry leads to a significant scatter of the observed diffusive transport. We study a large set of volumes reconstructed from focused ion beam-scanning electron microscopy data. Each individual volume provides us sub-REV measurements on porosity and the so-called transport-ability, being a dimensionless parameter representing the ratio of diffusive flux through the porous volume to that through an empty volume. The detected scatter of the transport-ability is mathematically characterized through a probability distribution function (PDF) with a mean and variance as function of porosity, which includes implicitly the effect of pore structure differences among sub-REV volumes. We then investigate domain size effects and predict when REV scale is reached. While the scatter in porosity observations decreases linearly with increasing sample size as expected, the observed scatter in transport-ability does not converge to zero. Our results confirm that differences in pore structure impact transport parameters at all scales. Consequently, the use of PDFs to describe the relationship of effective transport coefficients to porosity is advantageous to deterministic semiempirical functions. We discuss the consequences and advocate the use of PDFs for effective parameters in both continuum equations and data interpretation of experimental or computational work. The presented statistics-based upscaling technique of sub-REV microscopy data provides a new tool in understanding, describing and predicting macroscopic transport behavior of microporous media.

We use five established, but conceptually different artificial intelligence algorithms for analysing clogging and quantifying colloid transport at pore scale: artificial neural networks, decision tree, random forest, linear regression, and support vector regression. We test how these algorithm can predict clogging by interpolating physics based simulation data. Our training and test data set is based on results from Lattice Boltzmann simulations reproducing the physics of colloid transport through a typical pore throat present in glass beads or medium sized sand. We perform hyperparameter optimization through cross validation for all algorithms. The tree based methods have the highest Nash–Sutcliffe efficiencies among all tested algorithms with values mostly above 0.9 for the independent test data. The event of clogging can be predicted even with 100% accuracy. Our results indicate a non-linear, rather categorial nature of the (simulation) data. This is in contrast to the typical use of neural network algorithms for simulation data while tree based methods are often applied to observational data. We partly link this to the small size of our dataset. Our application of artificial intelligence in porous media research shows that time-consuming Lattice Boltzmann simulations can be easily supplemented and extended at small computational costs while predictability of clogging and quantitative effects of process specific parameters on colloidal transport are given with high reliability.

Aquifer heterogeneity in combination with data scarcity is a major challenge for reliable solute transport prediction. Velocity fluctuations cause non-regular plume shapes with potentially long-tailing and/or fast-travelling mass fractions. High monitoring cost and a shortage of simple concepts have limited the incorporation of heterogeneity into many field transport models up to now. We present an easily applicable hierarchical conceptualization strategy for hydraulic conductivity to integrate aquifer heterogeneity into quantitative flow and transport modelling. The modular approach combines large-scale deterministic structures with random substructures. Depending on the modelling aim, the required structural complexity can be adapted. The same holds for the amount of monitoring data. The conductivity model is constructed step-wise following field evidence from observations, seeking a balance between model complexity and available field data. The starting point is a structure of deterministic blocks, derived from head profiles and pumping tests. Then, subscale heterogeneity in the form of random binary inclusions is introduced to each block. Structural parameters can be determined, for example, from flowmeter measurements or hydraulic profiling. As proof of concept, we implemented a predictive transport model for the heterogeneous MADE site. The proposed hierarchical aquifer structure reproduces the plume development of the MADE-1 transport experiment without calibration. Thus, classical advection–dispersion equation (ADE) models are able to describe highly skewed tracer plumes by incorporating deterministic contrasts and effects of connectivity in a stochastic way without using uni-modal heterogeneity models with high variances. The reliance of the conceptual model on few observations makes it appealing for a goal-oriented site-specific transport analysis of less well investigated heterogeneous sites.

A new method is presented which allows interpreting steady-state pumping tests in heterogeneous isotropic transmissivity fields. In contrast to mean uniform flow, pumping test drawdowns in heterogeneous media cannot be described by a single effective or equivalent value of hydraulic transmissivity. An effective description of transmissivity is required, being a function of the radial distance to the well and including the parameters of log-transmissivity: mean, variance, and correlation length. Such a model is provided by the upscaling procedure radial coarse graining, which describes the transition of near-well to far-field transmissivity effectively. Based on this approach, an analytical solution for a steady-state pumping test drawdown is deduced. The so-called effective well flow solution is derived for two cases: the ensemble mean of pumping tests and the drawdown within an individual heterogeneous transmissivity field. The analytical form of the solution allows inversely estimating the parameters of aquifer heterogeneity. For comparison with the effective well flow solution, virtual pumping tests are performed and analysed for both cases, the ensemble mean drawdown and pumping tests at individual transmissivity fields. Interpretation of ensemble mean drawdowns showed proof of the upscaling method. The effective well flow solution reproduces the drawdown for two-dimensional pumping tests in heterogeneous media in contrast to Thiem's solution for homogeneous media. Multiple pumping tests conducted at different locations within an individual transmissivity field are analysed, making use of the effective well flow solution to show that all statistical parameters of aquifer heterogeneity can be inferred under field conditions. Thus, the presented method is a promising tool with which to estimate parameters of aquifer heterogeneity, in particular variance and horizontal correlation length of log-transmissivity fields from steady-state pumping test measurements.

Aquitards are common hydrogeological features in the subsurface. Typically, pumping tests are used to parameterize the hydraulic conductivity of heterogeneous aquitards. However, they do not take spatial variability and uncertainty into account. Alternatively, core-scale measurements of hydraulic conductivity are used in geostatistical upscaling methods, for which their correlation lengths are needed, but this information is extremely difficult to obtain. This study investigates whether a pumping test can be used to obtain the correlation lengths needed for geostatistical upscaling and account for the uncertainty about heterogeneous aquitard conductivity. Random realizations are generated from core-scale data with varying correlation lengths and inserted into a groundwater flow model which simulates the outcome of an actual pumping test. The realizations yielded a better fit to the pumping test data than the traditional pumping test result, assuming homogeneous layers are selected. Ranges of horizontal and vertical correlation lengths that fit the pumping-test well are found. However, considerable uncertainty regarding the correlation lengths remains, which should be considered when parameterizing a regional groundwater flow model.

In this study we present a formula for the hydraulic head describing the mean drawdown of a three dimensional steady state pumping test in heterogeneous anisotropic porous media effectively. By modeling the hydraulic conductivity K(x→) as spatial random function and using the upscaling method Coarse Graining we succeed in deriving a closed form solution hefw (r) which we understand as an extension of Thiem's formula to heterogeneous media. The solution hefw (r) does not only depend on the radial distance r but accounts also for the statistics of K(x→) , namely geometric mean KG, variance σ2, horizontal correlation length ℓ and anisotropy ratio e. We perform a sensitivity analysis on the parameters of hefw (r) and implement an inverse estimation strategy. Using numerical pumping tests we show the applicability of hefw (r) on the interpretation of drawdown data. This will be done for both, an ensemble of as well as for single pumping tests. Making use of the inverse estimation method we find excellent agreement of estimated parameters with initial values, in particular for the horizontal correlation length. Key Points Formula for effective description of hydraulic head of steady state pumping test Analysis of 3D numerical pumping tests in highly heterogeneous anisotropic media Direct estimation method to infer on statistics of log normal conductivity

Most large-scale hydrologic models fall short in reproducing groundwater head dynamics and simulating transport process due to their oversimplified representation of groundwater flow. In this study, we aim to extend the applicability of the mesoscale Hydrologic Model (mHM v5.7) to subsurface hydrology by coupling it with the porous media simulator OpenGeoSys (OGS). The two models are one-way coupled through model interfaces GIS2FEM and RIV2FEM, by which the grid-based fluxes of groundwater recharge and the river–groundwater exchange generated by mHM are converted to fixed-flux boundary conditions of the groundwater model OGS. Specifically, the grid-based vertical reservoirs in mHM are completely preserved for the estimation of land-surface fluxes, while OGS acts as a plug-in to the original mHM modeling framework for groundwater flow and transport modeling. The applicability of the coupled model (mHM–OGS v1.0) is evaluated by a case study in the central European mesoscale river basin – Nägelstedt. Different time steps, i.e., daily in mHM and monthly in OGS, are used to account for fast surface flow and slow groundwater flow. Model calibration is conducted following a two-step procedure using discharge for mHM and long-term mean of groundwater head measurements for OGS. Based on the model summary statistics, namely the Nash–Sutcliffe model efficiency (NSE), the mean absolute error (MAE), and the interquartile range error (QRE), the coupled model is able to satisfactorily represent the dynamics of discharge and groundwater heads at several locations across the study basin. Our exemplary calculations show that the one-way coupled model can take advantage of the spatially explicit modeling capabilities of surface and groundwater hydrologic models and provide an adequate representation of the spatiotemporal behaviors of groundwater storage and heads, thus making it a valuable tool for addressing water resources and management problems.