Inference of geostatistical hyperparameters with the correlated pseudo-marginal method

We consider non-linear Bayesian inversion problems targeting the geostatistical hyperparameters of a random field describing hydrogeological or geophysical properties given hydrogeological or geophysical data. This problem is of particular importance in the non-ergodic setting as there are no analytical upscaling relationships linking the data to the hyperparameters, such as, mean, standard deviation, and integral scales. Full inversion of the hyperparameters and the local properties of the field (typically involving many thousands of unknowns) brings substantial computational challenges, such that simplifying model assumptions (e.g., homogeneity or ergodicity) are typically made. To prevent the errors resulting from such simplified assumptions while also circumventing the burden of high-dimensional full inversions, we use a pseudo-marginal Metropolis–Hastings algorithm that treats the random field as latent variables. In this random effects model, the intractable likelihood of observing the data given the hyperparameters is estimated by Monte Carlo averaging over realizations of the random field. To increase the efficiency of the method, low-variance approximations of the likelihood ratio are obtained by using importance sampling and by correlating the samples used in the proposed and current steps of the Markov chain. We assess the performance of this correlated pseudo-marginal method by considering two representative inversion problems involving diffusion-based and wave-based physics, respectively, in which we infer the hyperparameters of (1) hydraulic conductivity fields using apparent hydraulic conductivity data in a data-poor setting and (2) fracture aperture fields using borehole ground-penetrating radar (GPR) reflection data in a more data-rich setting. For the first test case, we find that the correlated pseudo-marginal method generates similar estimates of the geostatistical mean as classical rejection sampling, while an inversion assuming ergodicity provides biased estimates. For the second test case, we find that the correlated pseudo-marginal method estimates the hyperparameters well, while rejection sampling is computationally unfeasible and a simplified model assuming homogeneity leads to biased estimates

Advancing measurements and representations of subsurface heterogeneity and dynamic processes: towards 4D hydrogeology

Essentially all hydrogeological processes are strongly influenced by the subsurface spatial heterogeneity and the temporal variation of environmental conditions, hydraulic properties, and solute concentrations. This spatial and temporal variability generally leads to effective behaviors and emerging phenomena that cannot be predicted from conventional approaches based on homogeneous assumptions and models. However, it is not always clear when, why, how, and at what scale the 4D (3D + time) nature of the subsurface needs to be considered in hydrogeological monitoring, modeling, and applications. In this paper, we discuss the interest and potential for the monitoring and characterization of spatial and temporal variability, including 4D imaging, in a series of hydrogeological processes: (1) groundwater fluxes, (2) solute transport and reaction, (3) vadose zone dynamics, and (4) surface–subsurface water interactions. We first identify the main challenges related to the coupling of spatial and temporal fluctuations for these processes. We then highlight recent innovations that have led to significant breakthroughs in high-resolution space–time imaging and modeling the characterization, monitoring, and modeling of these spatial and temporal fluctuations. We finally propose a classification of processes and applications at different scales according to their need and potential for high-resolution space–time imaging. We thus advocate a more systematic characterization of the dynamic and 3D nature of the subsurface for a series of critical processes and emerging applications. This calls for the validation of 4D imaging techniques at highly instrumented observatories and the harmonization of open databases to share hydrogeological data sets in their 4D components

Signatures Géoélectriques de la Propagation et du Mélange

Effective assessment of the fate of water-soluble contaminants leaked to aquifers is crucial for the management and preservation of groundwater resources. The arrival time of a contaminant plume to a given location depends on its mean velocity and spreading rate, whereas the associated sanitary risk is strongly correlated to the plume's degree of mixing upon its arrival. Mixing in heterogeneous media results from the competition between local-scale diffusion and the spreading of the solute, the latter in turn governed by flow velocity heterogeneity spanning a wide range of spatial scales. As a result, contaminant solute transport unfolds over multiple spatial and temporal scales and is, thus, very challenging to characterize using conventional sparsely-sampled and local hydrological measurement techniques alone. Complementarily, the time-lapse direct-current (DC) geophysical method provides spatially- and temporally-distributed information on equivalent DC electrical conductivity, which is sensitive to the transport of electrically-conductive solutes and carries information on solute heterogeneity below the measurement support scale. Combined with conventional fluid sampling, the DC methodology holds promise as a means to quantitatively characterize the state and evolution of solute spreading and mixing. However, this requires establishing quantitative links between spreading and mixing measures and average DC electrical conductivity under general transport conditions. This calls first for quantification of the uncertainty of electrically-inferred solute transport measures and second for the development of an upscaling framework for predictive purposes. Since both tasks remain largely unresolved, there is a risk for systematic errors in the interpretations. Here, we present numerical, experimental and theoretical investigations aiming at advancing in both directions. Relying on a Bayesian inference framework, we quantify to what extent time-series of the equivalent electrical conductivity tensor observed during tracer tests can constrain geostatistical parameters of hydraulic conductivity fields. We find that the most and least informative data are the flow-aligned component of the tensor and the solute mass breakthrough, respectively. The variance of the field, controlling the spreading rate, is the best constrained parameter for all test cases and data types, followed by the integral scale in the direction perpendicular to the mean flow field. As an experimental contribution, we report on an optically- and electrically monitored milli-fluidic saline tracer test aimed at understanding electrical signatures of the diffusion-limited mixing of an initially layered tracer distribution. We show that the different diffusion rates of the optical and electrical tracers must be taken into account to reach quantitative agreement between the optically-inferred and measured time-series of equivalent DC electrical conductivity. We find that the electrical data can constrain the initial layers' widths and associated diffusion transport time-scales as well as the degree of mixing of the tracer upon its arrival to the electrode positions. As a theoretical contribution, we introduce a new petrophysical parameter, the mixing factor M, to account for the impact of fluid conductivity heterogeneity below the measurement support scale. When considering anisotropic media, the diagonal components of the M-tensor depends on the variance and anisotropy ratio of heterogeneous conductivity fields, while the nature of the mapping is affected by the connectivity of the conductivity field. We derive an expression for M, and thus for the equivalent conductivity, in terms of the expected value of the conductivity field fluctuations and the secondary electric field. Then, we study numerically the mapping linking the statistical properties of either field.Une évaluation quantitative du devenir des contaminants hydrosolubles déversés dans les aquifères est cruciale pour la gestion et la préservation des ressources en eau souterraine. Le temps d'arrivée d'un panache de contaminants à un endroit donné dépend de sa vitesse moyenne et de sa taux de étalement, alors que le risque sanitaire associé est fortement corrélé au degré de mélange du panache à son arrivée. Le mélange dans les milieux hétérogènes résulte de la compétition entre la diffusion à l'échelle locale et l'étalement du soluté, cette dernière étant à son tour régie par l'hétérogénéité de la vitesse d'écoulement sur une large gamme d'échelles spatiales. Par conséquent, le transport de solutés contaminants se déroule sur de multiples échelles spatiales et temporelles et il est donc très difficile de le caractériser en s'appuyant uniquement sur des techniques de mesure hydrologiques locales et à faible échantillonnage. En complément, la méthode géophysique à courant continu fournit des informations réparties dans l'espace et dans le temps sur la conductivité électrique équivalente, qui est sensible au transport des solutés conducteurs d'électricité et fournit des informations sur l'hétérogénéité des solutés sous l'échelle du support de mesure. Combinée à l'échantillonnage conventionnel des fluides, la méthodologie est prometteuse comme moyen de caractériser quantitativement l'état et l'évolution de la propagation et du mélange des solutés. Cependant, cela nécessite d'établir des liens quantitatifs entre les mesures d'étalement et de mélange et la conductivité électrique moyenne dans des conditions générales de transport. Cela nécessite d'abord de quantifier l'incertitude des mesures de transport de solutés déduites électriquement et ensuite de développer un cadre de mise à l'échelle à des fins de prédiction. Comme ces deux tâches restent largement non résolues, il existe un risque d'erreurs systématiques dans les interprétations. Nous présentons ici des études numériques, expérimentales et théoriques visant à faire progresser ces objectifs de recherche. En nous appuyant sur un cadre d'inférence bayésienne, nous quantifions dans quelle mesure les séries temporelles du tenseur de conductivité électrique équivalent observés pendant les essais de traçage peuvent contraindre les paramètres géostatistiques des champs de conductivité hydraulique. Nous constatons que les données les plus et les moins informatives sont respectivement la composante du tenseur alignée sur le flux et la percée de la masse de soluté. La variance du champ, qui contrôle le taux d'étalement, est le paramètre le mieux contraint pour tous les cas d'essai et tous les types de données, suivi par l'échelle intégrale dans la direction perpendiculaire au champ d'écoulement moyen. En guise de contribution expérimentale, nous présentons un essai de traceur salin milli-fluidique contrôlé optiquement et électriquement, visant à comprendre les signatures électriques du mélange limité par la diffusion d'une distribution de traceur initialement stratifiée. Nous montrons que les différents taux de diffusion des traceurs optiques et électriques doivent être pris en compte pour obtenir une correspondance quantitative entre les séries temporelles de conductivité électrique équivalente déduites par voie optique et mesurées. Nous constatons que les données électriques peuvent contraindre les largeurs des couches initiales et les échelles de temps de transport et de diffusion associées, ainsi que le degré de mélange du traceur à son arrivée aux électrodes. Comme contribution théorique, nous introduisons un nouveau paramètre pétrophysique, le facteur de mélange M, pour tenir compte de l'impact de l'hétérogénéité de la conductivité du fluide sous l'échelle du support de mesure. En considérant les milieux anisotropes, les composantes diagonales du tenseur N dépendent de la variance et du rapport d'anisotropie des champs de conductivité hétérogènes, tandis que la nature de la cartographie est affectée par la connectivité du champ de conductivité. Nous dérivons une expression pour M, et donc pour la conductivité équivalente, en termes de valeur attendue des fluctuations du champ de conductivité et du champ électrique secondaire. Ensuite, nous étudions numériquement la cartographie reliant les propriétés statistiques de l'un ou l'autre champ

IP signature of metallic particles: lessons learnt from field and laboratory experiments

International audiencePast metallurgical sites and deposits account for a significant proportion of potentially contaminated sites in the European Union (EU): about 100,000 have been identified only in the North West regions of the EU. While recent wastes from sites still in operation are commonly recovered, this is not the case for old aggregated materials with a high content of ferrous (and other) metals, white and black slag, etc., which are considered to be sources of pollution and are costly to manage or dispose of. These sites could be considered as opportunities to recover large volumes of resources (metals, materials and land) using urban mining techniques if they were better characterized

IP signature of metallic particles: lessons learnt from field and laboratory experiments

International audiencePast metallurgical sites and deposits account for a significant proportion of potentially contaminated sites in the European Union (EU): about 100,000 have been identified only in the North West regions of the EU. While recent wastes from sites still in operation are commonly recovered, this is not the case for old aggregated materials with a high content of ferrous (and other) metals, white and black slag, etc., which are considered to be sources of pollution and are costly to manage or dispose of. These sites could be considered as opportunities to recover large volumes of resources (metals, materials and land) using urban mining techniques if they were better characterized

Electrical Signatures of Diffusion-Limited Mixing: Insights from a Milli-fluidic Tracer Experiment

We investigate how diffusion-limited mixing of a layered solute concentration distribution within a porous medium impacts bulk electrical conductivity. To do so, we perform a milli-fluidic tracer test by injecting a fluorescent and electrically conductive tracer in a quasi two-dimensional (2D) water-saturated porous medium. High resolution optical- and geoelectrical monitoring of the tracer is achieved by using a fluorimetry technique and equipping the flow cell with a resistivity meter, respectively. We find that optical and geoelectrical outputs can be related by a temporal re-scaling that accounts for the different diffusion rates of the optical and electrical tracers. Mixing-driven perturbations of the electrical equipotential field lines cause apparent electrical conductivity time-series, measured perpendicularly to the layering, to peak at times that are in agreement with the diffusion transport time-scale associated with the layer width. Numerical simulations highlight high sensitivity of such electrical data to the layers’ degree of mixing and their distance to the injection electrodes. Furthermore, the electrical data correlate well with time-series of two commonly used solute mixing descriptors: the concentration variance and the scalar dissipation rate

Inferring geostatistical properties of hydraulic conductivity fields from saline tracer tests and equivalent electrical conductivity time-series

International audienceWe use Approximate Bayesian Computation and the Kullback–Leibler divergence measure to quantify to what extent horizontal and vertical equivalent electrical conductivity time-series observed during tracer tests constrain the 2-D geostatistical parameters of multivariate Gaussian log-hydraulic conductivity fields. Considering a perfect and known relationship between salinity and electrical conductivity at the point scale, we find that the horizontal equivalent electrical conductivity time-series best constrain the geostatistical properties. The variance, controlling the spreading rate of the solute, is the best constrained geostatistical parameter, followed by the integral scales in the vertical direction. We find that horizontally layered models with moderate to high variance have the best resolved parameters. Since the salinity field at the averaging scale (e.g., the model resolution in tomograms) is typically non-ergodic, our results serve as a starting point for quantifying uncertainty due to small-scale heterogeneity in laboratory-experiments, tomographic results and hydrogeophysical inversions involving DC data

Scientific deliverable Enigma ITN:Report on process-based geophysical methodologies to monitoring subsurface processes

The identified activities are to (i) develop an upscaling framework for quantifying the impact of spreading and mixing on geophysical signals (ESR9), (ii) develop quantitative inversion of SIP signals induced by biochemical processes (ESR12) and (iii) enhance resolution of time lapse geophysical imaging of transport with new experimental and inversion strategies (ESR10, ESR11). In the following report, we start by discussing the ongoing laboratory experiments and associated theoretical developments (ESR9 and 10), before moving to crosshole time-lapse GPR (ESR10) and finally to the use of innovative traces (ESR11) to image transport processes. In this way, we will naturally move from the pore scale and the scale of representative elementary volumes (REV), to metric scale and finally to larger field scales

Advancing measurements and representations of subsurface heterogeneity and dynamic processes: towards 4D hydrogeology

Abstract. Essentially all hydrogeological processes are strongly influenced by the subsurface spatial heterogeneity and the temporal variation of environmental conditions, hydraulic properties, and solute concentrations. This spatial and temporal variability generally leads to effective behaviors and emerging phenomena that cannot be predicted from conventional approaches based on homogeneous assumptions and models. However, it is not always clear when, why, how, and at what scale the 4D (3D + time) nature of the subsurface needs to be considered in hydrogeological monitoring, modeling, and applications. In this paper, we discuss the interest and potential for the monitoring and characterization of spatial and temporal variability, including 4D imaging, in a series of hydrogeological processes: (1) groundwater fluxes, (2) solute transport and reaction, (3) vadose zone dynamics, and (4) surface–subsurface water interactions. We first identify the main challenges related to the coupling of spatial and temporal fluctuations for these processes. We then highlight recent innovations that have led to significant breakthroughs in high-resolution space–time imaging and modeling the characterization, monitoring, and modeling of these spatial and temporal fluctuations. We finally propose a classification of processes and applications at different scales according to their need and potential for high-resolution space–time imaging. We thus advocate a more systematic characterization of the dynamic and 3D nature of the subsurface for a series of critical processes and emerging applications. This calls for the validation of 4D imaging techniques at highly instrumented observatories and the harmonization of open databases to share hydrogeological data sets in their 4D components. </jats:p