Felipe P. J. de Barros

On the link between contaminant source release conditions and plume prediction uncertainty.

The initial width of contaminant plumes is known to have a key influence on expected plume development, dispersion and travel time statistics. In past studies, initial plume width has been perceived identical to the geometric width of a contaminant source or injection volume. A recent study on optimal sampling layouts (Nowak et al., 2009) showed that a significant portion of uncertainty in predicting plume migration stems from the uncertain total hydraulic flux through the source area. This result points towards a missing link between source geometry and plume statistics, which we denote as the effective source width. We define the effective source width by the ratio between the actual and expected hydraulic fluxes times the geometric source width. The actual hydraulic flux through the source area is given by individual realizations while the expected one represents the mean over the ensemble. It is a stochastic quantity that may strongly differ from the actual geometric source width for geometrically small sources, and becomes identical only at the limit of wide sources (approaching ergodicity). We derive its stochastic ensemble moments in order to explore the dependency on source scale. We show that, if the effective source width is known rather than the geometric width, predictions of plume development can greatly increase in predictive power. This is illustrated on plume statistics such as the distribution of plume length, average width, transverse dispersion, total mass flux and overall concentration variance. The analysis is limited to 2D depth-averaged systems, but implications hold for 3D cases.

Stochastic evaluation of mixing-controlled steady-state plume lengths in two-dimensional heterogeneous domains

We study plumes originating from continuous sources that require a dissolved reaction partner for their degradation. The length of such plumes is typically controlled by transverse mixing. While analytical expressions have been derived for homogeneous flow fields, incomplete characterization of the hydraulic conductivity field causes uncertainty in predicting plume lengths in heterogeneous domains. In this context, we analyze the effects of three sources of uncertainty: (i) The uncertainty of the effective mixing rate along the plume fringes due to spatially varying flow focusing, (ii) the uncertainty of the volumetric discharge through (and thus total mass flux leaving) the source area, and (iii) different parameterizations of the Darcy-scale transverse dispersion coefficient. The first two are directly related to heterogeneity of hydraulic conductivity. In this paper, we derive semi-analytical expressions for the probability distribution of plume lengths at different levels of complexity. The results are compared to numerical Monte Carlo simulations. Uncertainties in mixing and in the source strength result in a statistical distribution of possible plume lengths. For unconditional random hydraulic conductivity fields, plume lengths may vary by more than one order of magnitude even for moderate degrees of heterogeneity. Our results show that the uncertainty of volumetric flux through the source is the most relevant contribution to the variance of the plume length. The choice of different parameterizations for the local dispersion coefficient leads to differences in the mean estimated plume length.

A hypothesis‐driven approach to optimize field campaigns

Most field campaigns aim at helping in specified scientific or practical tasks, such as modeling, prediction, optimization, or management. Often these tasks involve binary decisions or seek answers to yes/no questions under uncertainty, e.g., Is a model adequate? Will contamination exceed a critical level? In this context, the information needs of hydro(geo)logical modeling should be satisfied with efficient and rational field campaigns, e.g., because budgets are limited. We propose a new framework to optimize field campaigns that defines the quest for defensible decisions as the ultimate goal. The key steps are to formulate yes/no questions under uncertainty as Bayesian hypothesis tests, and then use the expected failure probability of hypothesis testing as objective function. Our formalism is unique in that it optimizes field campaigns for maximum confidence in decisions on model choice, binary engineering or management decisions, or questions concerning compliance with environmental performance metrics. It is goal oriented, recognizing that different models, questions, or metrics deserve different treatment. We use a formal Bayesian scheme called PreDIA, which is free of linearization, and can handle arbitrary data types, scientific tasks, and sources of uncertainty (e.g., conceptual, physical, (geo)statistical, measurement errors). This reduces the bias due to possibly subjective assumptions prior to data collection and improves the chances of successful field campaigns even under conditions of model uncertainty. We illustrate our approach on two instructive examples from stochastic hydrogeology with increasing complexity.

An indirect assessment on the impact of connectivity of conductivity classes upon longitudinal asymptotic macrodispersivity

Solute transport takes place in heterogeneous porous formations, with the log conductivity, Y = ln K, modeled as a stationary random space function of given univariate normal probability density fu ...

Probabilistic human health risk assessment of degradation‐related chemical mixtures in heterogeneous aquifers: Risk statistics, hot spots, and preferential channels

An edited version of this paper was published by AGU. Copyright (2015) American Geophysical Union.

Probability density function of steady state concentration in two‐dimensional heterogeneous porous media

Spatial variability of hydraulic aquifer parameters causes meandering, squeezing, stretching, and enhanced mixing of steady state plumes in concentrated hot-spots of mixing. Because the exact spatial distribution of hydraulic parameters is uncertain, the spatial distribution of enhanced mixing rates is also uncertain. We discuss how relevant the resulting uncertainty of mixing rates is for predicting concentrations. We develop analytical solutions for the full statistical distribution of steady state concentration in two-dimensional, statistically uniform domains with log-hydraulic conductivity following an isotropic exponential model. In particular, we analyze concentration statistics at the fringes of wide plumes, conceptually represented by a solute introduced over half the width of the domain. Our framework explicitly accounts for uncertainty of streamline meandering and uncertainty of effective transverse mixing (defined at the Darcy scale). We make use of existing low-order closed-form expressions that lead to analytical expressions for the statistical distribution of local concentration values. Along the expected position of the plume fringe, the concentration distribution strongly clusters at its extreme values. This behavior extends over travel distances of up to tens of integral scales for the parameters tested in our study. In this regime, the uncertainty of effective transverse mixing is substantial enough to have noticeable effects on the concentration probability density function. At significantly larger travel distances, intermediate concentration values become most likely, and uncertainty of effective transverse mixing becomes negligible. A comparison to numerical Monte Carlo simulations of flow and solute transport show excellent agreement with the theoretically derived expressions.

Towards optimal allocation of computer resources: Trade-offs between uncertainty quantification, discretization and model reduction

The computational complexity of numerical models can be broken down into contributions ranging from spatial, temporal and stochastic resolution, e.g., spatial grid resolution, time step size and number of repeated simulations dedicated to quantify uncertainty. Controlling these resolutions allows keeping the computational cost at a tractable level whilst still aiming at accurate and robust predictions. The objective of this work is to introduce a framework that optimally allocates the available computational resources in order to achieve highest accuracy associated with a given prediction goal. Our analysis is based on the idea to jointly consider the discretization errors and computational costs of all individual model dimensions (physical space, time, parameter space). This yields a cost-to-error surface which serves to aid modelers in finding an optimal allocation of the computational resources (ORA). As a pragmatic way to proceed, we propose running small cost-efficient pre-investigations in order to estimate the joint cost-to-error surface, then fit underlying complexity and error models, decide upon a computational design for the full simulation, and finally to perform the designed simulation at near-optimal costs-to-accuracy ratio. We illustrate our approach with three examples from subsurface hydrogeology and show that the computational costs can be substantially reduced when allocating computational resources wisely and in a situation-specific and task-specific manner. We conclude that the ORA depends on a multitude of parameters, assumptions and problem-specific features and, hence, ORA needs to be determined carefully prior to each investigation.

Vertical dispersion in vegetated shear flows

Canopy layers control momentum and solute transport to and from the overlying water surface layer. These transfer mechanisms strongly dependent on canopy geometry, affect the amount of solute in the river, the hydrological retention and availability of dissolved solutes to organisms located in the vegetated layers, and are critical to improve water quality. In this work, we consider steady state transport in a vegetated channel under fully developed flow conditions. Under the hypothesis that the canopy layer can be described as an effective porous medium with prescribed properties, i.e., porosity and permeability, we model solute transport above and within the vegetated layer with an advection-dispersion equation with a spatially variable dispersion coefficient (diffusivity). By means of the Generalized Integral Transform Technique, we derive a semianalytical solution for the concentration field in submerged vegetated aquatic systems. We show that canopy layers permeability affects the asymmetry of the concentration profile, the effective vertical spreading behavior, and the magnitude of the peak concentration. Due to its analytical features, the model has a low computational cost. The proposed solution successfully reproduces previously published experimental data.

Pictures of blockscale transport: Effective versus ensemble dispersion and its uncertainty

Abstract Solute transport models tend to use coarse numerical grid blocks to alleviate computational costs. Aside from computational issues, the subsurface environment is usually characterized over a coarse measurement network where only large scale fluctuations of the flow field are captured. Neglecting the subscale velocity fluctuations in transport simulators can lead to erroneous predictions with consequences in risk analysis and remediation. For such reasons, upscaled dispersion coefficients in spatially heterogeneous flow fields must (1) account for the subscale variability that is filtered out by homogenization and (2) be modeled as a random function to incorporate the uncertainty associated with non-ergodic solute bodies. In this work, we examine the low order statistical properties of the blockscale dispersion tensor. The blockscale is defined as the scale upon which the spatially variable flow field is homogenized (e.g. the numerical grid block). Using a Lagrangian framework, we discuss different conceptualizations of the blockscale dispersion tensor. We distinguish effective and ensemble blockscale dispersion, which measure the impact of subscale velocity fluctuations on solute dispersion. Ensemble dispersion quantifies subscale velocity fluctuations between realizations, which overestimates the actual velocity variability. Effective dispersion on the other hand quantifies the actual blockscale velocity variability and thus reflects the impact of subscale velocity fluctuations on mixing and spreading. Based on these concepts, we quantify the impact of subscale velocity fluctuations on solute particle spreading and determine the governing equations for the coarse-grained concentration distributions. We develop analytical and semi-analytical expressions for the average and variance of the blockscale dispersion tensor in 3D flow fields as a function of the structural parameters characterizing the subsurface. Our results illustrate the relevance of the blockscale, the initial scale of the solute body and local-scale dispersion in controlling the uncertainty of the plume’s dispersive behavior. The analysis performed in this work has implications in numerical modeling (i.e. grid design) and allows to quantify the uncertainty of the blockscale dispersion tensor.

Dynamic interactions between hydrogeological and exposure parameters in daily dose prediction under uncertainty and temporal variability

We study the time dependent interaction between hydrogeological and exposure parameters in daily dose predictions due to exposure of humans to groundwater contamination. Dose predictions are treated stochastically to account for an incomplete hydrogeological and geochemical field characterization, and an incomplete knowledge of the physiological response. We used a nested Monte Carlo framework to account for uncertainty and variability arising from both hydrogeological and exposure variables. Our interest is in the temporal dynamics of the total dose and their effects on parametric uncertainty reduction. We illustrate the approach to a HCH (lindane) pollution problem at the Ebro River, Spain. The temporal distribution of lindane in the river water can have a strong impact in the evaluation of risk. The total dose displays a non-linear effect on different population cohorts, indicating the need to account for population variability. We then expand the concept of Comparative Information Yield Curves developed earlier (see de Barros et al. [29]) to evaluate parametric uncertainty reduction under temporally variable exposure dose. Results show that the importance of parametric uncertainty reduction varies according to the temporal dynamics of the lindane plume. The approach could be used for any chemical to aid decision makers to better allocate resources towards reducing uncertainty.