Application of a Bayesian approach to stochastic delineation of capture zones

This paper presents a Bayesian Monte Carlo method for evaluating the uncertainty in the delineation of well capture zones and its application to a wellfield in a heterogeneous, multiaquifer system. In the method presented, Bayes' rule is used to update prior distributions for the unknown parameters of the stochastic model for the hydraulic conductivity, and to calculate probability-based weights for parameter realizations using head residuals. These weights are then assigned to the corresponding capture zones obtained using forward particle tracking. Statistical analysis of the set of weighted protection zones results in a probability distribution for the capture zones. The suitability of the Bayesian stochastic method for a multilayered system is investigated, using the wellfield Het Rot at Nieuwrode, Belgium, located in a three-layered aquifer system, as an example. The hydraulic conductivity of the production aquifer is modeled as a spatially correlated random function with uncertain parameters. The aquitard and overlying unconfined aquifer are assigned random, homogeneous conductivities. The stochastic results are compared with deterministic capture zones obtained with a calibrated model for the area. The predictions of the stochastic approach are more conservative and indicate that parameter uncertainty should be taken into account in the delineation of well capture zones.     

Quantifying uncertainty of the semivariogram of transmissivity of an existing groundwater monitoring network

A method is presented for quantifying the uncertainty of the semivariogram of transmissivity and determining the required number of measurements. In this method, the estimated semivariogram and its 95% confidence limits are first determined from a finite number of measurements. The uncertainty of the estimated semivariogram is then quantified using the random field simulation technique. For a given value of the quantitative index of uncertainty, the required number of measured data can finally be obtained. Actual transmissivity data of an existing groundwater monitoring network are used in the application of the proposed method. The required numbers of measurements of transmissivity for four different values of the quantitative index of uncertainty are provided, from which reliable semivariograms of the transmissivity can be obtained. Copyright © 2004 John Wiley & Sons, Ltd.     

Stochastically conditioned flow paths and travel time distribution in highly heterogeneous synthesized aquifers / Chemins d’écoulement et distribution des temps de parcours au sein d’aquifères synthétiques hautement hétérogènes sous conditions stochastiques

Abstract

Abstract Characterization of heterogeneity at the field scale generally requires detailed aquifer properties such as transmissivity and hydraulic head. An accurate delineation of these properties is expensive and time consuming, and for many if not most groundwater systems, is not practical. As an alternative approach, stochastic representation of random fields is used and presented in this paper. Specifically, an iterative stochastic conditional simulation approach was applied to a hypothetical and highly heterogeneous pre-designed aquifer system. The approach is similar to the classical co-kriging technique; it uses a linear estimator that depends on the covariance functions of transmissivity (T), and hydraulic head (h), as well as their cross-covariances. A linearized flow equation along with a conditional random field generator constitutes the iterative process of the conditional simulation. One hundred equally likely realizations of transmissivity fields with pre-specified geostatistical parameters were generated, and conditioned to both limited transmissivity and head data. The successful implementation of the approach resulted in conditioned flow paths and travel-time distribution under different degrees of aquifer heterogeneity. This approach worked well for fields exhibiting small variances. However, for random fields exhibiting large variances (greater than 1.0), an iterative procedure was used. The results show that, as the variance of the ln[T] increases, the flow paths tend to diverge, resulting in a wide spectrum of flow conditions, with no direct discernable relationship between the degree of heterogeneity and travel time. The applied approach indicates that high errors may result when estimation of particle travel times in a heterogeneous medium is approximated by an equivalent homogeneous medium.     

Incorporating subjective and stochastic uncertainty in an interactive multi-objective groundwater calibration framework

The interactive multi-objective genetic algorithm (IMOGA) combines traditional optimization with an interactive framework that considers the subjective knowledge of hydro-geological experts in addition to quantitative calibration measures such as calibration errors and regularization to solve the groundwater inverse problem. The IMOGA is inherently a deterministic framework and identifies multiple large-scale parameter fields (typically head and transmissivity data are used to identify transmissivity fields). These large-scale parameter fields represent the optimal trade-offs between the different criteria (quantitative and qualitative) used in the IMOGA. This paper further extends the IMOGA to incorporate uncertainty both in the large-scale trends as well as the small-scale variability (which can not be resolved using the field data) in the parameter fields. The different parameter fields identified by the IMOGA represent the uncertainty in large-scale trends, and this uncertainty is modeled using a Bayesian approach where calibration error, regularization, and the expert’s subjective preference are combined to compute a likelihood metric for each parameter field. Small-scale (stochastic) variability is modeled using a geostatistical approach and added onto the large-scale trends identified by the IMOGA. This approach is applied to the Waste Isolation Pilot Plant (WIPP) case-study. Results, with and without expert interaction, are analyzed and the impact that expert judgment has on predictive uncertainty at the WIPP site is discussed. It is shown that for this case, expert interaction leads to more conservative solutions as the expert compensates for some of the lack of data and modeling approximations introduced in the formulation of the problem.     

Effects of uncertainty in rock-physics models on reservoir parameter estimation using seismic amplitude variation with angle and controlled-source electromagnetics data

This paper investigates the effects of uncertainty in rock-physics models on reservoir parameter estimation using seismic amplitude variation with angle and controlled-source electromagnetics data. The reservoir parameters are related to electrical resistivity by the Poupon model and to elastic moduli and density by the Xu-White model. To handle uncertainty in the rock-physics models, we consider their outputs to be random functions with modes or means given by the predictions of those rock-physics models and we consider the parameters of the rock-physics models to be random variables defined by specified probability distributions. Using a Bayesian framework and Markov Chain Monte Carlo sampling methods, we are able to obtain estimates of reservoir parameters and information on the uncertainty in the estimation. The developed method is applied to a synthetic case study based on a layered reservoir model and the results show that uncertainty in both rock-physics models and in their parameters may have significant effects on reservoir parameter estimation. When the biases in rock-physics models and in their associated parameters are unknown, conventional joint inversion approaches, which consider rock-physics models as deterministic functions and the model parameters as fixed values, may produce misleading results. The developed stochastic method in this study provides an integrated approach for quantifying how uncertainty and biases in rock-physics models and in their associated parameters affect the estimates of reservoir parameters and therefore is a more robust method for reservoir parameter estimation.     

A combinatorial optimization scheme for parameter structure identification in ground water modeling

This research develops a methodology for parameter structure identification in ground water modeling. For a given set of observations, parameter structure identification seeks to identify the parameter dimension, its corresponding parameter pattern and values. Voronoi tessellation is used to parameterize the unknown distributed parameter into a number of zones. Accordingly, the parameter structure identification problem is equivalent to finding the number and locations as well as the values of the basis points associated with the Voronoi tessellation. A genetic algorithm (GA) is allied with a grid search method and a quasi-Newton algorithm to solve the inverse problem. GA is first used to search for the near-optimal parameter pattern and values. Next, a grid search method and a quasi-Newton algorithm iteratively improve the GA's estimates. Sensitivities of state variables to parameters are calculated by the sensitivity-equation method. MODFLOW and MT3DMS are employed to solve the coupled flow and transport model as well as the derived sensitivity equations. The optimal parameter dimension is determined using criteria based on parameter uncertainty and parameter structure discrimination. Numerical experiments are conducted to demonstrate the proposed methodology, in which the true transmissivity field is characterized by either a continuous distribution or a distribution that can be characterized by zones. We conclude that the optimized transmissivity zones capture the trend and distribution of the true transmissivity field.     

Bayesian inference of the Cole–Cole parameters from time- and frequency-domain induced polarization

The inversion of induced‐polarization parameters is important in the characterization of the frequency electrical response of porous rocks. A Bayesian approach is developed to invert these parameters assuming the electrical response is described by a Cole–Cole model in the time or frequency domain. We show that the Bayesian approach provides a better analysis of the uncertainty associated with the parameters of the Cole–Cole model compared with more conventional methods based on the minimization of a cost function using the least‐squares criterion. This is due to the strong non‐linearity of the inverse problem and non‐uniqueness of the solution in the time domain. The Bayesian approach consists of propagating the information provided by the measurements through the model and combining this information with a priori knowledge of the data. Our analysis demonstrates that the uncertainty in estimating the Cole–Cole model parameters from induced‐polarization data is much higher for measurements performed in the time domain than in the frequency domain. Our conclusion is that it is very difficult, if not impossible, to retrieve the correct value of the Cole–Cole parameters from time‐domain induced‐polarization data using standard least‐squares methods. In contrast, the Cole–Cole parameters can be more correctly inverted in the frequency domain. These results are also valid for other models describing the induced‐polarization spectral response, such as the Cole–Davidson or power law models.     

Selection of noise parameters for Kalman filter

The Bayesian probabilistic approach is proposed to estimate the process noise and measurement noise parameters for a Kalman filter. With state vectors and covariance matrices estimated by the Kalman filter, the likehood of the measurements can be constructed as a function of the process noise and measurement noise parameters. By maximizing the likelihood function with respect to these noise parameters, the optimal values can be obtained. Furthermore, the Bayesian probabilistic approach allows the associated uncertainty to be quantified. Examples using a single-degree-of-freedom system and a ten-story building illustrate the proposed method. The effect on the performance of the Kalman filter due to the selection of the process noise and measurement noise parameters was demonstrated. The optimal values of the noise parameters were found to be close to the actual values in the sense that the actual parameters were in the region with significant probability density. Through these examples, the Bayesian approach was shown to have the capability to provide accurate estimates of the noise parameters of the Kalman filter, and hence for state estimation.     

Worth of secondary data compared to piezometric data for the probabilistic assessment of radionuclide migration

A common approach for the performance assessment of radionuclide migration from a nuclear waste repository is by means of Monte-Carlo techniques. Multiple realizations of the parameters controlling radionuclide transport are generated and each one of these realizations is used in a numerical model to provide a transport prediction. The statistical analysis of all transport predictions is then used in performance assessment. In order to reduce the uncertainty on the predictions is necessary to incorporate as much information as possible in the generation of the parameter fields. In this regard, this paper focuses in the impact that conditioning the transmissivity fields to geophysical data and/or piezometric head data has on convective transport predictions in a two-dimensional heterogeneous formation. The Walker Lake data based is used to produce a heterogeneous log-transmissivity field with distinct non-Gaussian characteristics and a secondary variable that represents some geophysical attribute. In addition, the piezometric head field resulting from the steady-state solution of the groundwater flow equation is computed. These three reference fields are sampled to mimic a sampling campaign. Then, a series of Monte-Carlo exercises using different combinations of sampled data shows the relative worth of secondary data with respect to piezometric head data for transport predictions. The analysis shows that secondary data allows to reproduce the main spatial patterns of the reference transmissivity field and improves the mass transport predictions with respect to the case in which only transmissivity data is used. However, a few piezometric head measurements could be equally effective for the characterization of transport predictions.     

A comparison of two stochastic inverse methods in a field-scale application

Inverse modeling is a useful tool in ground water flow modeling studies. The most frequent difficulties encountered when using this technique are the lack of conditioning information (e.g., heads and transmissivities), the uncertainty in available data, and the nonuniqueness of the solution. These problems can be addressed and quantified through a stochastic Monte Carlo approach. The aim of this work was to compare the applicability of two stochastic inverse modeling approaches in a field-scale application. The multi-scaling (MS) approach uses a downscaling parameterization procedure that is not based on geostatistics. The pilot point (PP) approach uses geostatistical random fields as initial transmissivity values and an experimental variogram to condition the calibration. The studied area (375 km2) is part of a regional aquifer, northwest of Montreal in the St. Lawrence lowlands (southern Québec). It is located in limestone, dolomite, and sandstone formations, and is mostly a fractured porous medium. The MS approach generated small errors on heads, but the calibrated transmissivity fields did not reproduce the variogram of observed transmissivities. The PP approach generated larger errors on heads but better reproduced the spatial structure of observed transmissivities. The PP approach was also less sensitive to uncertainty in head measurements. If reliable heads are available but no transmissivities are measured, the MS approach provides useful results. If reliable transmissivities with a well inferred spatial structure are available, then the PP approach is a better alternative. This approach however must be used with caution if measured transmissivities are not reliable.     

Worth of secondary data compared to piezometric data for the probabilistic assessment of radionuclide migration

A common approach for the performance assessment of radionuclide migration from a nuclear waste repository is by means of Monte-Carlo techniques. Multiple realizations of the parameters controlling radionuclide transport are generated and each one of these realizations is used in a numerical model to provide a transport prediction. The statistical analysis of all transport predictions is then used in performance assessment. In order to reduce the uncertainty on the predictions is necessary to incorporate as much information as possible in the generation of the parameter fields. In this regard, this paper focuses in the impact that conditioning the transmissivity fields to geophysical data and/or piezometric head data has on convective transport predictions in a two-dimensional heterogeneous formation. The Walker Lake data based is used to produce a heterogeneous log-transmissivity field with distinct non-Gaussian characteristics and a secondary variable that represents some geophysical attribute. In addition, the piezometric head field resulting from the steady-state solution of the groundwater flow equation is computed. These three reference fields are sampled to mimic a sampling campaign. Then, a series of Monte-Carlo exercises using different combinations of sampled data shows the relative worth of secondary data with respect to piezometric head data for transport predictions. The analysis shows that secondary data allows to reproduce the main spatial patterns of the reference transmissivity field and improves the mass transport predictions with respect to the case in which only transmissivity data is used. However, a few piezometric head measurements could be equally effective for the characterization of transport predictions.     

Issues in characterizing heterogeneity and connectivity in non-multiGaussian media

The performances of kriging, stochastic simulations and sequential self-calibration inversion are assessed when characterizing a non-multiGaussian synthetic 2D braided channel aquifer. The comparison is based on a series of criteria such as the reproduction of the original reference transmissivity or head fields, but also in terms of accuracy of flow and transport (capture zone) forecasts when the flow conditions are modified. We observe that the errors remain large even for a dense data network. In addition some unexpected behaviours are observed when large transmissivity datasets are used. In particular, we observe an increase of the bias with the number of transmissivity data and an increasing uncertainty with the number of head data. This is interpreted as a consequence of the use of an inadequate multiGaussian stochastic model that is not able to reproduce the connectivity of the original field.     

Dimensionality reduction for efficient Bayesian estimation of groundwater flow in strongly heterogeneous aquifers

We focus on the Bayesian estimation of strongly heterogeneous transmissivity fields conditional on data sampled at a set of locations in an aquifer. Log-transmissivity, Y, is modeled as a stochastic Gaussian process, parameterized through a truncated Karhunen–Loève (KL) expansion. We consider Y fields characterized by a short correlation scale as compared to the size of the observed domain. These systems are associated with a KL decomposition which still requires a high number of parameters, thus hampering the efficiency of the Bayesian estimation of the underlying stochastic field. The distinctive aim of this work is to present an efficient approach for the stochastic inverse modeling of fully saturated groundwater flow in these types of strongly heterogeneous domains. The methodology is grounded on the construction of an optimal sparse KL decomposition which is achieved by retaining only a limited set of modes in the expansion. Mode selection is driven by model selection criteria and is conditional on available data of hydraulic heads and (optionally) Y. Bayesian inversion of the optimal sparse KLE is then inferred using Markov Chain Monte Carlo (MCMC) samplers. As a test bed, we illustrate our approach by way of a suite of computational examples where noisy head and Y values are sampled from a given randomly generated system. Our findings suggest that the proposed methodology yields a globally satisfactory inversion of the stochastic head and Y fields. Comparison of reference values against the corresponding MCMC predictive distributions suggests that observed values are well reproduced in a probabilistic sense. In a few cases, reference values at some unsampled locations (typically far from measurements) are not captured by the posterior probability distributions. In these cases, the quality of the estimation could be improved, e.g., by increasing the number of measurements and/or the threshold for the selection of KL modes.     

Comparison of parameter uncertainty analysis techniques for a TOPMODEL application

Parameter uncertainty in hydrologic modeling is crucial to the flood simulation and forecasting. The Bayesian approach allows one to estimate parameters according to prior expert knowledge as well as observational data about model parameter values. This study assesses the performance of two popular uncertainty analysis (UA) techniques, i.e., generalized likelihood uncertainty estimation (GLUE) and Bayesian method implemented with the Markov chain Monte Carlo sampling algorithm, in evaluating model parameter uncertainty in flood simulations. These two methods were applied to the semi-distributed Topographic hydrologic model (TOPMODEL) that includes five parameters. A case study was carried out for a small humid catchment in the southeastern China. The performance assessment of the GLUE and Bayesian methods were conducted with advanced tools suited for probabilistic simulations of continuous variables such as streamflow. Graphical tools and scalar metrics were used to test several attributes of the simulation quality of selected flood events: deterministic accuracy and the accuracy of 95 % prediction probability uncertainty band (95PPU). Sensitivity analysis was conducted to identify sensitive parameters that largely affect the model output results. Subsequently, the GLUE and Bayesian methods were used to analyze the uncertainty of sensitive parameters and further to produce their posterior distributions. Based on their posterior parameter samples, TOPMODEL’s simulations and the corresponding UA results were conducted. Results show that the form of exponential decline in conductivity and the overland flow routing velocity were sensitive parameters in TOPMODEL in our case. Small changes in these two parameters would lead to large differences in flood simulation results. Results also suggest that, for both UA techniques, most of streamflow observations were bracketed by 95PPU with the containing ratio value larger than 80 %. In comparison, GLUE gave narrower prediction uncertainty bands than the Bayesian method. It was found that the mode estimates of parameter posterior distributions are suitable to result in better performance of deterministic outputs than the 50 % percentiles for both the GLUE and Bayesian analyses. In addition, the simulation results calibrated with Rosenbrock optimization algorithm show a better agreement with the observations than the UA’s 50 % percentiles but slightly worse than the hydrographs from the mode estimates. The results clearly emphasize the importance of using model uncertainty diagnostic approaches in flood simulations.     

Estimation of spatial variability of aquifer parameters from geophysical methods: a case study of Sindhudurg district,Maharashtra, India

Accurate estimation of aquifer parameters, especially from crystalline hard rock area, assumes a special significance for management of groundwater resources. The aquifer parameters are usually estimated through pumping tests carried out on water wells. While it may be costly and time consuming for carrying out pumping tests at a number of sites, the application of geophysical methods in combination with hydro-geochemical information proves to be potential and cost effective to estimate aquifer parameters. Here a method to estimate aquifer parameters such as hydraulic conductivity, formation factor, porosity and transmissivity is presented by utilizing electrical conductivity values analysed via hydro-geochemical analysis of existing wells and the respective vertical electrical sounding (VES) points of Sindhudurg district, western Maharashtra, India. Further, prior to interpolating the distribution of aquifer parameters of the study area, variogram modelling was carried out using data driven techniques of kriging, automatic relevance determination based Bayesian neural networks (ARD-BNN) and adaptive neuro-fuzzy neural networks (ANFIS). In total, four variogram model fitting techniques such as spherical, exponential, ARD-BNN and ANFIS were compared. According to the obtained results, the spherical variogram model in interpolating transmissivity, ARD-BNN variogram model in interpolating porosity, exponential variogram model in interpolating aquifer thickness and ANFIS variogram model in interpolating hydraulic conductivity outperformed rest of the variogram models. Accordingly, the accurate aquifer parameters maps of the study area were produced by using the best variogram model. The present results suggest that there are relatively high value of hydraulic conductivity, porosity and transmissivity at Parule, Mogarne, Kudal, and Zarap, which would be useful to characterize the aquifer system over western Maharashtra.     

Issues and Options in the Delineation of Well Capture Zones under Uncertainty

The delineation of wellhead protection areas (WHPAs) under uncertainty is still a challenge for heterogeneous porous media. For granular media, one option is to combine particle tracking (PT) with the Monte Carlo approach (PT‐MC) to account for geologic uncertainties. Fractured porous media, however, require certain restrictive assumptions under this approach. An alternative for all types of media is the capture probability (CP) approach, which is based on the solution of the standard advection‐dispersion equation in a backward mode, making use of the analogy between forward and backward transport processes. Within this context, we review the current controversy about the correct form of the conceptual model for transport, finding that the advection‐diffusion model, which represents the diffusive interchange between streamtubes with differing velocities, is more physically realistic than the conventional advection‐dispersion model. For mildly to moderately heterogeneous materials, stochastic theories and simulation experiments show that this process converges at the field scale to an effective advection‐dispersion process that can be simulated with conventional transport models using appropriate macrodispersivity values. For highly heterogeneous materials, stochastic theories do not yet exist but there is no reason why the process should not converge naturally as well. Macrodispersivities appear to be formation‐specific. The advection‐dispersion model can be used for capture zone delineation in heterogeneous granular media. For fractured porous systems, hybrid equivalent porous medium and discrete fracture network or CP‐based approaches may have potential. In general, capture zones delineated by PT without MC will always be too small and should not be used as a basis for land‐use decisions.     

The inverse problem and ground water management

The response of groundwater basins to natural and anthropogenic inputs depends on many interrelated factors such as the values of groundwater flow and mass transport parameters. This work presents a theoretical analysis of the impact of parameter uncertainty on groundwater management decisions. It is shown that under classical, Bayesian, and deterministic assumptions about the parameter structure, the resulting management decisions could be very different. This underscores the importance of adopting the proper parameter structure and the need for using consistent methods to solve the inverse problem.     

Hydrologic drought prediction under climate change: Uncertainty modeling with Dempster–Shafer and Bayesian approaches

Representation and quantification of uncertainty in climate change impact studies are a difficult task. Several sources of uncertainty arise in studies of hydrologic impacts of climate change, such as those due to choice of general circulation models (GCMs), scenarios and downscaling methods. Recently, much work has focused on uncertainty quantification and modeling in regional climate change impacts. In this paper, an uncertainty modeling framework is evaluated, which uses a generalized uncertainty measure to combine GCM, scenario and downscaling uncertainties. The Dempster–Shafer (D–S) evidence theory is used for representing and combining uncertainty from various sources. A significant advantage of the D–S framework over the traditional probabilistic approach is that it allows for the allocation of a probability mass to sets or intervals, and can hence handle both aleatory or stochastic uncertainty, and epistemic or subjective uncertainty. This paper shows how the D–S theory can be used to represent beliefs in some hypotheses such as hydrologic drought or wet conditions, describe uncertainty and ignorance in the system, and give a quantitative measurement of belief and plausibility in results. The D–S approach has been used in this work for information synthesis using various evidence combination rules having different conflict modeling approaches. A case study is presented for hydrologic drought prediction using downscaled streamflow in the Mahanadi River at Hirakud in Orissa, India. Projections of n most likely monsoon streamflow sequences are obtained from a conditional random field (CRF) downscaling model, using an ensemble of three GCMs for three scenarios, which are converted to monsoon standardized streamflow index (SSFI-4) series. This range is used to specify the basic probability assignment (bpa) for a Dempster–Shafer structure, which represents uncertainty associated with each of the SSFI-4 classifications. These uncertainties are then combined across GCMs and scenarios using various evidence combination rules given by the D–S theory. A Bayesian approach is also presented for this case study, which models the uncertainty in projected frequencies of SSFI-4 classifications by deriving a posterior distribution for the frequency of each classification, using an ensemble of GCMs and scenarios. Results from the D–S and Bayesian approaches are compared, and relative merits of each approach are discussed. Both approaches show an increasing probability of extreme, severe and moderate droughts and decreasing probability of normal and wet conditions in Orissa as a result of climate change.     

Worth of head data in well-capture zone design: deterministic and stochastic analysis

In this study, we examine the effects of conditioning spatially variable transmissivity fields using head and/or transmissivity measurements on well-capture zones. In order to address the challenge posed by conditioning a flow model with spatially varying parameters, an innovative inverse algorithm, the Representers method, is employed. The method explicitly considers this spatial variability.

A number of uniform measurement grids with different densities are used to condition transmissivity fields using the Representers method. Deterministic and stochastic analysis of well-capture zones are then examined. The deterministic study focuses on comparison between reference well-capture zones and their estimated mean conditioned on head data. It shows that model performance due to head conditioning on well-capture zone estimation is related to pumping rate. At moderate pumping rates transmissivity observations are more crucial to identify effects arising from small-scale variations in pore water velocity. However, with more aggressive pumping these effects are reduced, consequently model performance, through incorporating head observations, markedly improves. In the stochastic study, the effect of conditioning using head and/or transmissivity data on well-capture zone uncertainty is examined. The Representers method is coupled with the Monte Carlo method to propagate uncertainty in transmissivity fields to well-capture zones. For the scenario studied, the results showed that a combination of 48 head and transmissivity data could reduce the area of uncertainty (95% confidence interval) in well-capture zone location by over 50%, compared to a 40% reduction using either head or transmissivity data. This performance was comparable to that obtained through calibrating on three and a half times the number of head observations alone.