A linearized approach to flow resistance uncertainty in a 2-D finite volume model of flood flow

A linearized approach to quantifying predictive uncertainty in a 2-D model of shallow water flow in response to uncertainty in friction parameterization is presented. The resulting uncertain finite volume (UFV) method is tested against Monte Carlo simulations for uncertain models over channel only, floodplain only and channel and floodplain meshes. The results show that the UFV model performs well in predicting mean and standard deviations of water depths, for problems with two independent Manning's n values, with standard deviations of up to 0.02 m^1/3 s⁻¹ with a mean value of 0.03 m^1/3 s⁻¹. For depth averaged velocities, mean values are well represented, but standard deviations are poorly predicted by UFV. UFV also performs well when modelling flow over an uneven fractal topography and for a distributed (11 degrees of freedom) parameterization. A computation time advantage of >50 when compared to the Monte Carlo method is observed.     

Probabilistic evaluation of flood hazard in urban areas using Monte Carlo simulation

The goal of the presented research was the derivation of flood hazard maps, using Monte Carlo simulation of flood propagation at an urban site in the UK, specifically an urban area of the city of Glasgow. A hydrodynamic model describing the propagation of flood waves, based on the De Saint Venant equations in two‐dimensional form capable of accounting for the topographic complexity of the area (preferential outflow paths, buildings, manholes, etc.) and for the characteristics of prevailing imperviousness typical of the urban areas, has been used to derive the hydrodynamic characteristics of flood events (i.e. water depths and flow velocities). The knowledge of the water depth distribution and of the current velocities derived from the propagation model along with the knowledge of the topographic characteristics of the urban area from digital map data allowed for the production of hazard maps based on properly defined hazard indexes. These indexes are evaluated in a probabilistic framework to overcome the classical problem of single deterministic prediction of flood extent for the design event and to introduce the concept of the likelihood of flooding at a given point as the sum of data uncertainty, model structural error and parameterization uncertainty. Copyright © 2011 John Wiley & Sons, Ltd.     

Two-stage decision-making under uncertainty and stochasticity: Bayesian Programming

This paper develops a new method for decision-making under uncertainty. The method, Bayesian Programming (BP), addresses a class of two-stage decision problems with features that are common in environmental and water resources. BP is applicable to two-stage combinatorial problems characterized by uncertainty in unobservable parameters, only some of which is resolved upon observation of the outcome of the first-stage decision. The framework also naturally accommodates stochastic behavior, which has the effect of impeding uncertainty resolution. With the incorporation of systematic methods for decision search and Monte Carlo methods for Bayesian analysis, BP addresses limitations of other decision-analytic approaches for this class of problems, including conventional decision tree analysis and stochastic programming. The methodology is demonstrated with an illustrative problem of water quality pollution control. Its effectiveness for this problem is compared to alternative approaches, including a single-stage model in which expected costs are minimized and a deterministic model in which uncertain parameters are replaced by their mean values. A new term, the expected value of including uncertainty resolution, or EVIUR, is introduced and evaluated for the illustrative problem. It is a measure of the worth of incorporating the experimental value of decisions into an optimal decision-making framework. For the illustrative problem, the two-stage adaptive management framework extracted up to approximately 50% of the gains of perfect information. The strength and limitations of the method are discussed and conclusions are presented.     

Probabilistic solution of floodplain inundation equation

Uncertainty in bed roughness is a dominant factor in providing a sufficiently accurate simulation of floodplain flows. This study describes a method to compute the transition probability density distribution of time-varying water elevations where the evolutionary process is based on a conventional one-dimensional storage cell model with governing stochastic differential equation. By including the random inputs (or noise terms) of bed roughness and initial water depth, time-dependent and spatially varying probability density function of the water surface leads to a Fokker–Planck equation. The model’s performance is evaluated by applying it to shallow water flow with a horizontal bed. Sensitivity of model predictions to variations in the bed friction parameters is shown. By comparing the result of the proposed method with that of conventional Monte Carlo simulation, the advantage of the former as a method for density function prediction is confirmed.     

Assessment of radionuclide transport uncertainty in the unsaturated zone of Yucca Mountain

The present study assesses the uncertainty of flow and radionuclide transport in the unsaturated zone at Yucca Mountain using a Monte Carlo method. Matrix permeability, porosity, and sorption coefficient are considered random. Different from previous studies that assume distributions of the parameters, the distributions are determined in this study by applying comprehensive transformations and rigorous statistics to on-site measurements of the parameters. The distribution of permeability is further adjusted based on model calibration results. Correlation between matrix permeability and porosity is incorporated using the Latin Hypercube Sampling method. After conducting 200 Monte Carlo simulations of three-dimensional unsaturated flow and radionuclide transport for conservative and reactive tracers, the mean, variances, and 5th, 50th, and 95th percentiles for quantities of interest (e.g., matrix liquid saturation and water potential) are evaluated. The mean and 50th percentile are used as the mean predictions, and their associated predictive uncertainties are measured by the variances and the 5th and 95th percentiles (also known as uncertainty bounds). The mean predictions of matrix liquid saturation and water potential are in reasonable agreement with corresponding measurements. The uncertainty bounds include a large portion of the measurements, suggesting that the data variability can be partially explained by parameter uncertainty. The study illustrates propagation of predictive uncertainty of percolation flux, increasing downward from repository horizon to water table. Statistics from the breakthrough curves indicate that transport of the reactive tracer is delayed significantly by the sorption process, and prediction on the reactive tracer is of greater uncertainty than on the conservative tracer because randomness in the sorption coefficient increases the prediction uncertainty. Uncertainty in radionuclide transport is related to uncertainty in the percolation flux, suggesting that reducing the former entails reduction in the latter.     

Assessment of uncertainty associated with the estimation of well catchments by moment equations

Non-local stochastic moment equations are used successfully to analyze groundwater flow in randomly heterogeneous media. Here we present a moment equations-based approach to quantify the uncertainty associated with the estimation of well catchments. Our approach is based on the development of a complete second order formalism which allows obtaining the first statistical moments of the trajectories of conservative solute particles advected in a generally non-uniform groundwater flow. Approximate equations of moments of particles’ trajectories are then derived on the basis of a second order expansion in terms of the standard deviation of the aquifer log hydraulic conductivity. Analytical expressions are then obtained for the predictors of locations of mean stagnation points, together with their associated uncertainties. We implement our approach on heterogeneous media in bounded two-dimensional domains, with and without including the effect of conditioning on hydraulic conductivity information. The impact of domain size, boundary conditions, heterogeneity and non-stationarity of hydraulic conductivity on the prediction of a well catchment is explored. The results are compared against Monte Carlo simulations and semi-analytical solutions available in the literature. The methodology is applicable to both infinite and bounded domains and is free of distributional assumptions (and so applies to both Gaussian and non-Gaussian log hydraulic conductivity fields) and formally includes the effect of conditioning on available information.     

Development of probability distributions for urban hydrologic model parameters and a Monte Carlo analysis of model sensitivity

This paper proposes an approach to estimating the uncertainty related to EPA Storm Water Management Model model parameters, percentage routed (P_R) and saturated hydraulic conductivity (K_sat), which are used to calculate stormwater runoff volumes. The methodology proposed in this paper addresses uncertainty through the development of probability distributions for urban hydrologic parameters through extensive calibration to observed flow data in the Philadelphia collection system. The established probability distributions are then applied to the Philadelphia Southeast district model through a Monte Carlo approach to estimate the uncertainty in prediction of combined sewer overflow volumes as related to hydrologic model parameter estimation. Understanding urban hydrology is critical to defining urban water resource problems. A variety of land use types within Philadelphia coupled with a history of cut and fill have resulted in a patchwork of urban fill and native soils. The complexity of urban hydrology can make model parameter estimation and defining model uncertainty a difficult task. The development of probability distributions for hydrologic parameters applied through Monte Carlo simulations provided a significant improvement in estimating model uncertainty over traditional model sensitivity analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Quantification of uncertainty in design water levels due to uncertain bed form roughness in the Dutch river Waal

Hydrodynamic river models are applied to design and evaluate measures for purposes such as safety against flooding. The modelling of river processes involves numerous uncertainties, resulting in uncertain model results. Knowledge of the type and magnitude of these uncertainties is crucial for a meaningful interpretation of the model results. Uncertainty in the hydraulic roughness due to bed forms is one of the main contributors to the uncertainty in the modelled water levels. The aim of this study was to quantify the uncertainty in the bed form roughness under design conditions and quantify the effect on the design water levels in the Dutch river Waal. Five roughness models that predict bed form roughness based on measured bed form and flow characteristics were extrapolated to design conditions. The results show that the 95% confidence interval of the predicted Nikuradse roughness values under design conditions ranges from 0.32 to 1.03 m. This uncertainty was propagated through the two‐dimensional hydrodynamic model, WAQUA, by means of a Monte Carlo simulation for an idealized schematization of the Dutch river Waal. The uncertain bed form roughness results in an uncertainty in the design water levels, with a 95% confidence interval of 0.53 m, which is significant for Dutch river management practice. The uncertainty in the bed form roughness was mainly caused by a lack of knowledge about the physical process of bed form evolution that causes roughness. An improved estimation of bed form roughness can significantly reduce the uncertainty in the design water levels. Copyright © 2012 John Wiley & Sons, Ltd.     

POD-based Monte Carlo approach for the solution of regional scale groundwater flow driven by randomly distributed recharge

We present a methodology conducive to the application of a Galerkin model order reduction technique, Proper Orthogonal Decomposition (POD), to solve a groundwater flow problem driven by spatially distributed stochastic forcing terms. Typical applications of POD to reducing time-dependent deterministic partial differential equations (PDEs) involve solving the governing PDE at some observation times (termed snapshots), which are then used in the order reduction of the problem. Here, the application of POD to solve the stochastic flow problem relies on selecting the snapshots in the probability space of the random quantity of interest. This allows casting a standard Monte Carlo (MC) solution of the groundwater flow field into a Reduced Order Monte Carlo (ROMC) framework. We explore the robustness of the ROMC methodology by way of a set of numerical examples involving two-dimensional steady-state groundwater flow taking place within an aquifer of uniform hydraulic properties and subject to a randomly distributed recharge. We analyze the impact of (i) the number of snapshots selected from the hydraulic heads probability space, (ii) the associated number of principal components, and (iii) the key geostatistical parameters describing the heterogeneity of the distributed recharge on the performance of the method. We find that our ROMC scheme can improve significantly the computational efficiency of a standard MC framework while keeping the same degree of accuracy in providing the leading statistical moments (i.e. mean and covariance) as well as the sample probability density of the state variable of interest.     

A comparative study of numerical approaches to risk assessment of contaminant transport

In risk analysis, a complete characterization of the concentration distribution is necessary to determine the probability of exceeding a threshold value. The most popular method for predicting concentration distribution is Monte Carlo simulation, which samples the cumulative distribution function with a large number of repeated operations. In this paper, we first review three most commonly used Monte Carlo (MC) techniques: the standard Monte Carlo, Latin Hypercube sampling, and Quasi Monte Carlo. The performance of these three MC approaches is investigated. We then apply stochastic collocation method (SCM) to risk assessment. Unlike the MC simulations, the SCM does not require a large number of simulations of flow and solute equations. In particular, the sparse grid collocation method and probabilistic collocation method are employed to represent the concentration in terms of polynomials and unknown coefficients. The sparse grid collocation method takes advantage of Lagrange interpolation polynomials while the probabilistic collocation method relies on polynomials chaos expansions. In both methods, the stochastic equations are reduced to a system of decoupled equations, which can be solved with existing solvers and whose results are used to obtain the expansion coefficients. Then the cumulative distribution function is obtained by sampling the approximate polynomials. Our synthetic examples show that among the MC methods, the Quasi Monte Carlo gives the smallest variance for the predicted threshold probability due to its superior convergence property and that the stochastic collocation method is an accurate and efficient alternative to MC simulations.     

Estimation of flood inundation probabilities using global hazard indexes based on hydrodynamic variables

In this paper a new procedure to derive flood hazard maps incorporating uncertainty concepts is presented. The layout of the procedure can be resumed as follows: (1) stochastic input of flood hydrograph modelled through a direct Monte-Carlo simulation based on flood recorded data. Generation of flood peaks and flow volumes has been obtained via copulas, which describe and model the correlation between these two variables independently of the marginal laws involved. The shape of hydrograph has been generated on the basis of a historical significant flood events, via cluster analysis; (2) modelling of flood propagation using a hyperbolic finite element model based on the DSV equations; (3) definition of global hazard indexes based on hydro-dynamic variables (i.e., water depth and flow velocities). The GLUE methodology has been applied in order to account for parameter uncertainty. The procedure has been tested on a flood prone area located in the southern part of Sicily, Italy. Three hazard maps have been obtained and then compared.     

Using CFD in a GLUE framework to model the flow and dispersion characteristics of a natural fluvial dead zone

Monte Carlo simulations of a two‐dimensional depth‐averaged distributed bed‐roughness flow model, TELEMAC‐2D, are used to model a detailed tracer dispersion test in a complex reach of the River Severn in the Generalized Likelihood Uncertainty Estimation (GLUE) framework. A time efficient, zero equation, spatially distributed eddy viscosity model is derived from physical reasoning and used to close the flow equations. It is shown to have the property of low numerical diffusion, avoiding recourse to a globally large value of the eddy viscosity. For models of complex river flows, there are typically so many degrees of freedom in the specification of distributed parameters owing to the limitations of field data collection, that the identification of a unique model structure is unlikely. The data used here to constrain the model structure come from a continuous tracer injection experiment, comprising six spatially distributed time series of concentration measurements. Several hundred Monte‐Carlo simulations of different model structures were investigated and it was found that multiple model structures produced feasible simulations of the tracer mixing, giving rise to the phenomenon of equifinality. Rather than optimizing the model structure on the basis of the constraining data, we derive relative possibility measures that express our relative degree of belief in each model structure. These measures can then be used as weights for assessing predictive uncertainty when using a range of model structures, to estimate the flow distribution under varying stages, or for providing maps indicating fully distributed confidence limits in the risk assessments process. Such an approach is used here, and helps to identify the circumstances under which two‐dimensional modelling can be useful. The framework is not limited to the model structures that are developed herein, and more advanced process representation techniques can be included as computational efficiency increases. Copyright © 2001 John Wiley & Sons, Ltd.     

Stochastic overland flows

A theoretical solution framework to the nonlinear stochastic partial differential equations (SPDE) of the kinematic wave and diffusion wave models of overland flows under stochastic inflows/outflows, stochastic surface roughness field and stochastic state of flows was obtained. This development was realized by means of an eigenfunction representation of the time-space overland flow depths, and by transforming the problem into the phase space. By using Van Kampen's lemma and the cumulant expansion theory of Kubo-Van Kampen-Fox, the deterministic partial differential equation (PDE) for the evolutionary probability density function (pdf) of overland flow depths was finally obtained. Once this deterministic PDE is solved for the time-varying pdf of overland flow depths, then the time-space varying pdf of overland flow depths can be obtained by a transformation given in the text. In this solution framework it is possible to incorporate the stochastic dynamic behavior of the parameters and of the forcing functions of the overland flow process. For example, not only the individual rainfall duration and fluctuating rain intensity characteristics but also the sequential behavior of rainfall patterns is incorporated into the evolutionary probability density function of overland flow depths.     

Stochastic overland flows Part 1: Physics-based evolutionary probability distributions

A theoretical solution framework to the nonlinear stochastic partial differential equations (SPDE) of the kinematic wave and diffusion wave models of overland flows under stochastic inflows/outflows, stochastic surface roughness field and stochastic state of flows was obtained. This development was realized by means of an eigenfunction representation of the time-space overland flow depths, and by transforming the problem into the phase space. By using Van Kampen's lemma and the cumulant expansion theory of Kubo-Van Kampen-Fox, the deterministic partial differential equation (PDE) for the evolutionary probability density function (pdf) of overland flow depths was finally obtained. Once this deterministic PDE is solved for the time-varying pdf of overland flow depths, then the time-space varying pdf of overland flow depths can be obtained by a transformation given in the text. In this solution framework it is possible to incorporate the stochastic dynamic behavior of the parameters and of the forcing functions of the overland flow process. For example, not only the individual rainfall duration and fluctuating rain intensity characteristics but also the sequential behavior of rainfall patterns is incorporated into the evolutionary probability density function of overland flow depths.     

Flow in unsaturated random porous media,nonlinear numerical analysis and comparison to analytical stochastic models

This work presents a rigorous numerical validation of analytical stochastic models of steady state unsaturated flow in heterogeneous porous media. It also provides a crucial link between stochastic theory based on simplifying assumptions and empirical field and simulation evidence of variably saturated flow in actual or realistic hypothetical heterogeneous porous media. Statistical properties of unsaturated hydraulic conductivity, soil water tension, and soil water flux in heterogeneous soils are investigated through high resolution Monte Carlo simulations of a wide range of steady state flow problems in a quasi-unbounded domain. In agreement with assumptions in analytical stochastic models of unsaturated flow, hydraulic conductivity and soil water tension are found to be lognormally and normally distributed, respectively. In contrast, simulations indicate that in moderate to strong variable conductivity fields, longitudinal flux is highly skewed. Transverse flux distributions are leptokurtic. the moments of the probability distributions obtained from Monte Carlo simulations are compared to modified first-order analytical models. Under moderate to strong heterogeneous soil flux conditions (σ²_y≥1), analytical solutions overestimate variability in soil water tension by up to 40% as soil heterogeneity increases, and underestimate variability of both flux components by up to a factor 5. Theoretically predicted model (cross-)covariance agree well with the numerical sample (cross-)covarianaces. Statistical moments are shown to be consistent with observed physical characteristics of unsaturated flow in heterogeneous soils.©1998 Elsevier Science Limited. All rights reserved     

Percolation losses in paddy fields with a dynamic soil structure: model development and applications

The hydraulic characteristics of the plough pan of paddy fields provide continuous ponding conditions during the growing season and control the water use efficiency in wet rice production. Its saturated hydraulic conductivity K_s, however, exhibits a large spatiotemporal variability as a consequence of a highly dynamic soil structure involving temporary shrinkage cracks. Water flow through the earthen bunds surrounding the fields further contributes to the uncertainty in water flux calculations. The objective of this study was to develop a simple deterministic model with stochastic elements (‘PADDY‐FLUX’) for depiction of deep percolation, and to assess the effect of different water management scenarios on percolation in two channel command areas. Darcy's law is used as the fundamental equation for water flow calculations with the ponding water depth h as a time‐dependent variable. Flux uncertainty is estimated by a Monte‐Carlo‐type implementation. K_s is treated as a random variable of a bimodal probability density function (PDF), which is the weighted sum of two Gaussian PDFs (accounting for a matrix and a preferential flow domain). The weighing factor α is a function of h, reflecting an increasing risk for preferential flow situations after desiccation and the development of shrinkage cracks. Under‐bund percolation is calculated using transfer functions. The results demonstrate that percolation losses increase in the following order: continuous soil saturation < continuous flooding (CF) < mid‐season drainage and intermittent irrigation (MD + II) < mid‐season drainage and continuous flooding. The bunds contribute up to 54 and 17% to total fluxes under CF and MD + II, respectively. Preferential water fluxes are responsible for the major part of water losses as soon as desiccation causes the formation of shrinkage cracks. As a conclusion, continuous soil saturation should be promoted as the least water‐intensive irrigation regime, while intermittent irrigation is recommended only in case that irreversible shrinkage cracks have already developed. Copyright © 2010 John Wiley & Sons, Ltd.     

Comparison of Ensemble Kalman Filter groundwater-data assimilation methods based on stochastic moment equations and Monte Carlo simulation

Traditional Ensemble Kalman Filter (EnKF) data assimilation requires computationally intensive Monte Carlo (MC) sampling, which suffers from filter inbreeding unless the number of simulations is large. Recently we proposed an alternative EnKF groundwater-data assimilation method that obviates the need for sampling and is free of inbreeding issues. In our new approach, theoretical ensemble moments are approximated directly by solving a system of corresponding stochastic groundwater flow equations. Like MC-based EnKF, our moment equations (ME) approach allows Bayesian updating of system states and parameters in real-time as new data become available. Here we compare the performances and accuracies of the two approaches on two-dimensional transient groundwater flow toward a well pumping water in a synthetic, randomly heterogeneous confined aquifer subject to prescribed head and flux boundary conditions.     

Flow structure over square bars at intermediate submergence: Large Eddy Simulation study of bar spacing effect

Turbulent open-channel flow over 2D roughness elements is investigated numerically by Large Eddy Simulation (LES). The flow over square bars for two roughness regimes (k-type roughness and transitional roughness between d-type and k-type) at a relative submergence of H/k = 6.5 is considered, where H is the maximum water depth and k is the roughness height. The selected roughness configurations are based on laboratory experiments, which are used for validating numerical simulations. Results from the LES, in turn, complement the experiments in order to investigate the time-averaged flow properties at much higher spatial resolution. The concept of the double-averaging (DA) of the governing equations is utilized to quantify roughness effects at a range of flow properties. Double-averaged velocity profiles are analysed and the applicability of the logarithmic law for rough-wall flows of intermediate submergence is evaluated. Momentum flux components are quantified and roughness effect on their vertical distribution is assessed using an integral form of the DA-equations. The relative contributions of pressure drag and viscous friction to the overall bed shear stress are also reported.