s u m m a r yDescribing the complex structures that exist in many sedimentary aquifers is crucial for reliable groundwater flow and transport simulation. However, hardly any aquifer can be inspected in such detail that all decimeter to meter heterogeneity is resolved. Aquifer analogs serve as surrogates to construct models of equivalent heterogeneity, and thus imitate those features relevant for flow or transport processes. Gravel pits found in excavation show excellent sections of the sedimentary sequence and thus offer direct insight into the structural and textural composition of the subsoil. This paper describes an approach to also inspect the third dimension: by mapping during the ongoing excavation it is possible to obtain a three-dimensional representation of the subsurface within a short period of time. A detailed description of a case study is presented and the findings from sedimentological, hydrogeological and geophysical analyses are compared. The gravel pit is located near the town of Herten in southwest Germany, where relatively young unconsolidated fluvio-glacial and fluvial sediments in the Rhine basin are mined. The excavated gravel body is built up by architectural elements typical for braided river deposits. The study generated a high-resolution data set of lithofacies, hydrofacies and ground penetrating radar (GPR) profiles. It represents the basis for a full three-dimensional geostatistical reconstruction presented in the second part

a b s t r a c tOne of the main issues in the application of multiple-point statistics (MPS) to the simulation of threedimensional (3D) blocks is the lack of a suitable 3D training image. In this work, we compare three methods of overcoming this issue using information coming from bidimensional (2D) training images. One approach is based on the aggregation of probabilities. The other approaches are novel. One relies on merging the lists obtained using the impala algorithm from diverse 2D training images, creating a list of compatible data events that is then used for the MPS simulation. The other (s2Dcd) is based on sequential simulations of 2D slices constrained by the conditioning data computed at the previous simulation steps. These three methods are tested on the reproduction of two 3D images that are used as references, and on a real case study where two training images of sedimentary structures are considered. The tests show that it is possible to obtain 3D MPS simulations with at least two 2D training images. The simulations obtained, in particular those obtained with the s2Dcd method, are close to the references, according to a number of comparison criteria. The CPU time required to simulate with the method s2Dcd is from two to four orders of magnitude smaller than the one required by a MPS simulation performed using a 3D training image, while the results obtained are comparable. This computational efficiency and the possibility of using MPS for 3D simulation without the need for a 3D training image facilitates the inclusion of MPS in Monte Carlo, uncertainty evaluation, and stochastic inverse problems frameworks.

s u m m a r yThe heterogeneity of sedimentary structures at the decimeter scale is crucial to the understanding of groundwater flow and transport. In a series of two papers, we provide a detailed analysis of a fluvioglacial aquifer analog: the Herten site. The geological data along a series of 2D sections in a quarry, the corresponding GPR measurements, and their sedimentological interpretation are described in the companion paper. In this paper, we focus on the three-dimensional reconstruction of the heterogeneity. The resulting numerical model is provided as an electronic supplementary material for further studies. Furthermore, the geostatistical parameters derived from this analysis and the methodology described in the paper could be used in the future for the simulation of similar deposits where less data would be available. To build the 3D model, we propose a hierarchical simulation method which integrates various geostatistical techniques. First, we model the subdivision of the domain into regions corresponding to main sedimentological structures (e.g. a sedimentation event). Within these volumes, we use multiplepoint statistics to describe the internal heterogeneity. What is unusual here is that we do not try to use a complex training image for the multiple-point algorithm accounting for all the non-stationarity and complexity, but instead use a simple conceptual model of heterogeneity (ellipsoidal shapes as a training image) and constrain the multiple point simulations within the regions by a detailed interpolation of orientation data derived from the 2D sections. This method produces realistic geological structures. The analysis of the flow and transport properties (hydraulic conductivity and tracer breakthrough curves) of the resulting model shows that it is closer to the properties estimated directly from the 2D geological observations rather than those estimated from a model of heterogeneity based on probability of transitions and not including the modeling of the large-scale structures.

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Calibration of a SIR (Susceptibles–Infected–Recovered) model with official international data for the COVID-19 pandemics provides a good example of the difficulties inherent in the solution of inverse problems. Inverse modeling is set up in a framework of discrete inverse problems, which explicitly considers the role and the relevance of data. Together with a physical vision of the model, the present work addresses numerically the issue of parameters calibration in SIR models, it discusses the uncertainties in the data provided by international authorities, how they influence the reliability of calibrated model parameters and, ultimately, of model predictions.

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Sand lenses at various spatial scales are recognized to add heterogeneity to glacial sediments. They have high hydraulic conductivities relative to the surrounding till matrix and may affect the advective transport of water and contaminants in clayey till settings. Sand lenses were investigated on till outcrops producing binary images of geological cross-sections capturing the size, shape and distribution of individual features. Sand lenses occur as elongated, anisotropic geobodies that vary in size and extent. Besides, sand lenses show strong non-stationary patterns on section images that hamper subsequent simulation. Transition probability (TP) and multiple-point statistics (MPS) were employed to simulate sand lens heterogeneity. We used one cross-section to parameterize the spatial correlation and a second, parallel section as a reference: it allowed testing the quality of the simulations as a function of the amount of conditioning data under realistic conditions. The performance of the simulations was evaluated on the faithful reproduction of the specific geological structure caused by sand lenses. Multiple-point statistics offer a better reproduction of sand lens geometry. However, two-dimensional training images acquired by outcrop mapping are of limited use to generate three-dimensional realizations with MPS. One can use a technique that consists in splitting the 3D domain into a set of slices in various directions that are sequentially simulated and reassembled into a 3D block. The identification of flow paths through a network of elongated sand lenses and the impact on the equivalent permeability in tills are essential to perform solute transport modeling in the low-permeability sediments.

Geological structures are by nature inaccessible to direct observation. This can cause difficulties in applications where a spatially explicit representation of such structures is required, in particular when modelling fluid migration in geological formations. An increasing trend in recent years has been to use analogs to palliate this lack of knowledge, i.e., exploiting the spatial information from sites where the geology is accessible (outcrops, quarry sites) and transferring the observed properties to a study site deemed geologically similar. While this approach is appealing, it is difficult to put in place because of the lack of access to well-documented analog data. In this paper we present comprehensive analog data sets which characterize sedimentary structures from important groundwater hosting formations in Germany and Brazil. Multiple 2-D outcrop faces are described in terms of hydraulic, thermal and chemical properties and interpolated in 3-D using stochastic techniques. These unique data sets can be used by the wider community to implement analog approaches for characterizing reservoir and aquifer formations.