A Hierarchical Bayesian Model Averaging Framework for Groundwater Prediction under Uncertainty

Groundwater prediction models are subjected to various sources of uncertainty. This study introduces a hierarchical Bayesian model averaging (HBMA) method to segregate and prioritize sources of uncertainty in a hierarchical structure and conduct BMA for concentration prediction. A BMA tree of models is developed to understand the impact of individual sources of uncertainty and uncertainty propagation to model predictions. HBMA evaluates the relative importance of different modeling propositions at each level in the BMA tree of model weights. The HBMA method is applied to chloride concentration prediction for the “1,500‐foot” sand of the Baton Rouge area, Louisiana from 2005 to 2029. The groundwater head data from 1990 to 2004 is used for model calibration. Four sources of uncertainty are considered and resulted in 180 flow and transport models for concentration prediction. The results show that prediction variances of concentration from uncertain model elements are much higher than the prediction variance from uncertain model parameters. The HBMA method is able to quantify the contributions of individual sources of uncertainty to the total uncertainty.     

The Effect of Undetected Barriers on Groundwater Drawdown and Recovery

In large-scale pumping projects, such as mine dewatering, predictions are often made about the rate of groundwater level recovery after pumping has ceased. However, these predictions may be impacted by geological uncertainty—including the presence of undetected impermeable barriers. During pumping, an impermeable barrier may be undetected if it is located beyond the maximum extent of the cone of depression; yet it may still control drawdown during the recovery phase. This has implications for regional-scale modeling and monitoring of groundwater level recovery. In this article, non-dimensional solutions are developed to show the conditions under which a barrier may be undetected during pumping but still significantly impact groundwater level recovery. The magnitude of the impact from an undetected barrier will increase as the ratio of pumping rate to aquifer transmissivity increases. The results are exemplified for a hypothetical aquifer with an unknown barrier 3 km from a pumping well. The difference in drawdown between a model with and without a barrier may be <1 m in the 10 years while pumping is occurring, but up to 50 m after pumping has ceased.     

Identification of Groundwater Parameters Using an Adaptative Multiscale Method

The identification of groundwater parameters in heterogeneous systems is a major challenge in groundwater modeling. Flexible parameterization methods are needed to assess the complexity of the spatial distributions of these parameters in real aquifers. In this article, we introduce an adaptative parameterization to identify the distribution of hydraulic conductivity within the large‐scale (4400 km²) Upper Rhine aquifer. The method is based on adaptative multiscale triangulation (AMT) coupled with an inverse problem procedure that identifies the parameters' distributions by reducing the error between measured and simulated heads. The AMT method has the advantage of combining both zonation and interpolation approaches. The AMT method uses area‐based interpolation rather than an interpolation based on stochastic features. The method is applied to a standard 2D groundwater model that takes into account the interactions between the aquifer and surface water bodies, groundwater recharge, and pumping wells. The simulation period covers 204 months, from January 1986 to December 2002. Recordings at 109 piezometers are used for model calibration. The simulated heads are globally quite accurate and reproduce the main dynamics of the system. The local hydraulic conductivities resulting from the AMT method agree qualitatively with existing local experimental observations across the Rhine aquifer.     

Practical Postcalibration Uncertainty Analysis: Yucca Mountain, Nevada

The values of parameters in a groundwater flow model govern the precision of predictions of future system behavior. Predictive precision, thus, typically depends on an ability to infer values of system properties from historical measurements through calibration. When such data are scarce, or when their information content with respect to parameters that are most relevant to predictions of interest is weak, predictive uncertainty may be high, even if the model is "calibrated." Recent advances help recognize this condition, quantitatively evaluate predictive uncertainty, and suggest a path toward improved predictive accuracy by identifying sources of predictive uncertainty and by determining what observations will most effectively reduce this uncertainty. We demonstrate linear and nonlinear predictive error/uncertainty analyses as applied to a groundwater flow model of Yucca Mountain, Nevada, the United States' proposed site for disposal of high-level radioactive waste. Linear and nonlinear uncertainty analyses are readily implemented as an adjunct to model calibration with medium to high parameterization density. Linear analysis yields contributions made by each parameter to a prediction's uncertainty and the worth of different observations, both existing and yet-to-be-gathered, toward reducing this uncertainty. Nonlinear analysis provides more accurate characterization of the uncertainty of model predictions while yielding their (approximate) probability distribution functions. This article applies the above methods to a prediction of specific discharge and confirms the uncertainty bounds on specific discharge supplied in the Yucca Mountain Project License Application.     

Reducing Uncertainty in Calibrating Aquifer Flow Model with Multiple Scales of Heterogeneity

Modeling and calibration of natural aquifers with multiple scales of heterogeneity is a challenging task due to limited subsurface access. While computer modeling plays an essential role in aquifer studies, large uncertainty exists in developing a conceptual model of an aquifer and in calibrating the model for decision making. Due to uncertainties such as a lack of understanding of subsurface processes and a lack of techniques to parameterize the subsurface environment (including hydraulic conductivity, source/sink rate, and aquifer boundary conditions), existing aquifer models often suffer nonuniqueness in calibration, leading to poor predictive capability. A robust calibration methodology is needed that can address the simultaneous estimations of aquifer parameters, source/sink, and boundary conditions. In this paper, we propose a multistage and multiscale approach that addresses subsurface heterogeneity at multiple scales, while reducing uncertainty in estimating the model parameters and model boundary conditions. The key to this approach lies in the appropriate development, verification, and synthesis of existing and new techniques of static and dynamic data integration. In particular, based on a given set of observation data, new inversion techniques can be first used to estimate aquifer large‐scale effective parameters and smoothed boundary conditions, based on which parameter and boundary condition estimation can be refined at increasing detail using standard or highly parameterized estimation techniques.     

A framework for dealing with uncertainty due to model structure error

Although uncertainty about structures of environmental models (conceptual uncertainty) is often acknowledged to be the main source of uncertainty in model predictions, it is rarely considered in environmental modelling. Rather, formal uncertainty analyses have traditionally focused on model parameters and input data as the principal source of uncertainty in model predictions. The traditional approach to model uncertainty analysis, which considers only a single conceptual model, may fail to adequately sample the relevant space of plausible conceptual models. As such, it is prone to modelling bias and underestimation of predictive uncertainty.     

Estimating River Conductance from Prior Information to Improve Surface‐Subsurface Model Calibration

Most groundwater models simulate stream‐aquifer interactions with a head‐dependent flux boundary condition based on a river conductance (CRIV). CRIV is usually calibrated with other parameters by history matching. However, the inverse problem of groundwater models is often ill‐posed and individual model parameters are likely to be poorly constrained. Ill‐posedness can be addressed by Tikhonov regularization with prior knowledge on parameter values. The difficulty with a lumped parameter like CRIV, which cannot be measured in the field, is to find suitable initial and regularization values. Several formulations have been proposed for the estimation of CRIV from physical parameters. However, these methods are either too simple to provide a reliable estimate of CRIV, or too complex to be easily implemented by groundwater modelers. This paper addresses the issue with a flexible and operational tool based on a 2D numerical model in a local vertical cross section, where the river conductance is computed from selected geometric and hydrodynamic parameters. Contrary to other approaches, the grid size of the regional model and the anisotropy of the aquifer hydraulic conductivity are also taken into account. A global sensitivity analysis indicates the strong sensitivity of CRIV to these parameters. This enhancement for the prior estimation of CRIV is a step forward for the calibration and uncertainty analysis of surface‐subsurface models. It is especially useful for modeling objectives that require CRIV to be well known such as conjunctive surface water‐groundwater use.     

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.     

Screening‐Level Sensitivity Analysis for the Design of Pump‐and‐Treat Systems

Pump‐and‐treat systems can prevent the migration of groundwater contaminants and candidate systems are typically evaluated with groundwater models. Such models should be rigorously assessed to determine predictive capabilities and numerous tools and techniques for model assessment are available. While various assessment methodologies (e.g., model calibration, uncertainty analysis, and Bayesian inference) are well‐established for groundwater modeling, this paper calls attention to an alternative assessment technique known as screening‐level sensitivity analysis (SLSA). SLSA can quickly quantify first‐order (i.e., main effects) measures of parameter influence in connection with various model outputs. Subsequent comparisons of parameter influence with respect to calibration vs. prediction outputs can suggest gaps in model structure and/or data. Thus, while SLSA has received little attention in the context of groundwater modeling and remedial system design, it can nonetheless serve as a useful and computationally efficient tool for preliminary model assessment. To illustrate the use of SLSA in the context of designing groundwater remediation systems, four SLSA techniques were applied to a hypothetical, yet realistic, pump‐and‐treat case study to determine the relative influence of six hydraulic conductivity parameters. Considered methods were: Taguchi design‐of‐experiments (TDOE); Monte Carlo statistical independence (MCSI) tests; average composite scaled sensitivities (ACSS); and elementary effects sensitivity analysis (EESA). In terms of performance, the various methods identified the same parameters as being the most influential for a given simulation output. Furthermore, results indicate that the background hydraulic conductivity is important for predicting system performance, but calibration outputs are insensitive to this parameter (K_BK). The observed insensitivity is attributed to a nonphysical specified‐head boundary condition used in the model formulation which effectively “staples” head values located within the conductivity zone. Thus, potential strategies for improving model predictive capabilities include additional data collection targeting the K_BK parameter and/or revision of model structure to reduce the influence of the specified head boundary.     

Integrated water resource management in a major canal command in eastern India

The present rice‐dominated cropping system in the Hirakud canal command (eastern India) is under severe threat due to imbalance between irrigation water supply and demand. The canal water supply, which is the only source of irrigation, only meets 54% of the demand at 90% probability of exceedance (PE). In order to mitigate the irrigation water deficit from canal water, groundwater is considered as a supplemental source. Quasi‐three‐dimensional groundwater flow simulation modelling was, therefore, carried out by using Visual MODFLOW to detect the change in hydraulic head due to transient pumping stresses. The simulation model was calibrated and validated satisfactorily. Sensitivity analysis of the model parameters shows that groundwater recharge is most sensitive followed by aquifer hydraulic conductivity at almost all the sites of the command area, whereas the model is comparatively less sensitive to specific storage and specific yield. Enhanced pumping scenarios showed that groundwater extraction can be increased up to 50 times of the existing pumping without causing any adverse effect to the aquifer but the aquifer does not permit to exploit water in order to fulfill the irrigation water demand even at 10% PE. Hence, it is imperative to develop an optimal land and water resources management plan of the command area. Copyright © 2011 John Wiley & Sons, Ltd.     

A Model-Averaging Method for Assessing Groundwater Conceptual Model Uncertainty

This study evaluates alternative groundwater models with different recharge and geologic components at the northern Yucca Flat area of the Death Valley Regional Flow System (DVRFS), USA. Recharge over the DVRFS has been estimated using five methods, and five geological interpretations are available at the northern Yucca Flat area. Combining the recharge and geological components together with additional modeling components that represent other hydrogeological conditions yields a total of 25 groundwater flow models. As all the models are plausible given available data and information, evaluating model uncertainty becomes inevitable. On the other hand, hydraulic parameters (e.g., hydraulic conductivity) are uncertain in each model, giving rise to parametric uncertainty. Propagation of the uncertainty in the models and model parameters through groundwater modeling causes predictive uncertainty in model predictions (e.g., hydraulic head and flow). Parametric uncertainty within each model is assessed using Monte Carlo simulation, and model uncertainty is evaluated using the model averaging method. Two model-averaging techniques (on the basis of information criteria and GLUE) are discussed. This study shows that contribution of model uncertainty to predictive uncertainty is significantly larger than that of parametric uncertainty. For the recharge and geological components, uncertainty in the geological interpretations has more significant effect on model predictions than uncertainty in the recharge estimates. In addition, weighted residuals vary more for the different geological models than for different recharge models. Most of the calibrated observations are not important for discriminating between the alternative models, because their weighted residuals vary only slightly from one model to another.     

Evaluating Groundwater Interbasin Flow Using Multiple Models and Multiple Types of Data

The groundwater interbasin flow, Q_y, from the north of Yucca Flat into Yucca Flat simulated using the Death Valley Regional Flow System (DVRFS) model greatly exceeds assessments obtained using other approaches. This study aimed to understand the reasons for the overestimation and to examine whether the Q_y estimate can be reduced. The two problems were tackled from the angle of model uncertainty by considering six models revised from the DVRFS model with different recharge components and hydrogeological frameworks. The two problems were also tackled from the angle of parametric uncertainty for each model by first conducting Morris sensitivity analysis to identify important parameters and then conducting Monte Carlo simulations for the important parameters. The uncertainty analysis is general and suitable for tackling similar problems; the Morris sensitivity analysis has been utilized to date in only a limited number of regional groundwater modeling. The simulated Q_y values were evaluated by using three kinds of calibration data (i.e., hydraulic head observations, discharge estimates, and constant‐head boundary flow estimates). The evaluation results indicate that, within the current DVRFS modeling framework, the Q_y estimate can only be reduced to about half of the original estimate without severely deteriorating the goodness‐of‐fit to the calibration data. The evaluation results also indicate that it is necessary to develop a new hydrogeological framework to produce new flow patterns in the DVRFS model. The issues of hydrogeology and boundary flow are being addressed in a new version of the DVRFS model planned for release by the U.S. Geological Survey.     

A Potential-Based Inversion of Unconfined Steady-State Hydraulic Tomography

The importance of estimating spatially variable aquifer parameters such as transmissivity is widely recognized for studies in resource evaluation and contaminant transport. A useful approach for mapping such parameters is inverse modeling of data from series of pumping tests, that is, via hydraulic tomography. This inversion of field hydraulic tomographic data requires development of numerical forward models that can accurately represent test conditions while maintaining computational efficiency. One issue this presents is specification of boundary and initial conditions, whose location, type, and value may be poorly constrained. To circumvent this issue when modeling unconfined steady-state pumping tests, we present a strategy that analyzes field data using a potential difference method and that uses dipole pumping tests as the aquifer stimulation. By using our potential difference approach, which is similar to modeling drawdown in confined settings, we remove the need for specifying poorly known boundary condition values and natural source/sink terms within the problem domain. Dipole pumping tests are complementary to this strategy in that they can be more realistically modeled than single-well tests due to their conservative nature, quick achievement of steady state, and the insensitivity of near-field response to far-field boundary conditions. After developing the mathematical theory, our approach is first validated through a synthetic example. We then apply our method to the inversion of data from a field campaign at the Boise Hydrogeophysical Research Site. Results from inversion of nine pumping tests show expected geologic features, and uncertainty bounds indicate that hydraulic conductivity is well constrained within the central site area.     

Identifying groundwater pumping source information using simulated annealing

This study develops an approach referred to as SA‐MF using simulated annealing (SA) and a three‐dimensional groundwater flow model (MODFLOW‐2000) to determine the pumping source information, namely the pumping source location, the pumping rate and the pumping period. Eight scenarios, each with various cases for aquifers with homogeneous or heterogeneous conductivities, are used to test the applicability and accuracy of SA‐MF in determining the pumping source information. The results show that at least five measured heads should be used to analyse the pumping source identification problem and at least three observation wells are required to effectively determine the pumping source information. The SA‐MF gives good estimated results in both synthetic and real problems even if the measured heads contain measurement errors. Copyright © 2008 John Wiley & Sons, Ltd.     

Applying a Regional Transport Modeling Framework to Manage Nitrate Contamination of Groundwater

Regional nitrate contamination in groundwater is a management challenge involving multisector benefits. There is always conflict between restricting anthropogenic activities to protect groundwater quality and prioritizing economic development, especially in productive agriculture dominated areas. To mitigate the nitrate contamination in groundwater, it is necessary to develop management alternatives that simultaneously support environmental protection and sustainable economic development. A regional transport modeling framework is applied to evaluate nitrate fate and transport in the Dagu Aquifer, a shallow sandy aquifer that supplies drinking water and irrigation water for a thriving agricultural economy in Shandong Province in east coastal China. The aquifer supports intensive high-value vegetable farms and nitrate contamination is extensive. Detailed land-use information and fertilizer use data were compiled and statistical approaches were employed to analyze nitrogen source loadings and the spatiotemporal distribution of nitrate in groundwater to support model construction and calibration. The evaluations reveal that the spatial distribution and temporal trends of nitrate contamination in the Dagu Aquifer are driven by intensive fertilization and vertical water exchange, the dominant flow pattern derived from intensive agricultural pumping and irrigation. The modeling framework is employed to assess the effectiveness of potentially applicable management alternatives. The predictive results provide quantitative comparisons for the trend and extent of groundwater quality mitigation under each scenario. Recommendations are made for measures that can both improve groundwater quality and sustain productive agricultural development.     

Groundwater modelling and marine intrusion in the semi‐arid systems (Cap‐Bon,Tunisia)

This article studies the effect of drought and pumping discharge on groundwater supplies and marine intrusion in the Korba aquifer (Cap‐Bon peninsula, Tunisia). The Groundwater Modelling System has been used to model the groundwater flow and to simulate the seawater intrusion. The calibration is based on the groundwater levels in the steady state from 1963, and in the transient state from the groundwater levels from 1963 to 2005. The main objective is to quantify the components of the groundwater mass balance and to estimate the hydraulic conductivity distribution. The impact of pumping discharge on the groundwater level evolution has been examined by two pumping scenarios P1 (no. 8420) and P2 (no. 8862) wells. The hydrodynamic modelling shows the increasing drawdowns after 14 years of pumping: 4 m in P1 well and about 5 m in P2 well below sea level. The drawdowns are accompanied by the inverse hydraulic gradient. The numerical model was used to discuss the management of the groundwater resources of Cap‐Bon. As the population continues to grow and the demand for groundwater pumping intensifies beyond the 1963 level, it can be expected that the actual extent of seawater intrusion in the future would be more severe than the model prediction. Better strategies for groundwater development and management will be necessary to protect the freshwater aquifers to the marine intrusion. Copyright © 2011 John Wiley & Sons, Ltd.     

Evaluation of Bias Associated with Capture Maps Derived from Nonlinear Groundwater Flow Models

The impact of groundwater withdrawal on surface water is a concern of water users and water managers, particularly in the arid western United States. Capture maps are useful tools to spatially assess the impact of groundwater pumping on water sources (e.g., streamflow depletion) and are being used more frequently for conjunctive management of surface water and groundwater. Capture maps have been derived using linear groundwater flow models and rely on the principle of superposition to demonstrate the effects of pumping in various locations on resources of interest. However, nonlinear models are often necessary to simulate head‐dependent boundary conditions and unconfined aquifers. Capture maps developed using nonlinear models with the principle of superposition may over‐ or underestimate capture magnitude and spatial extent. This paper presents new methods for generating capture difference maps, which assess spatial effects of model nonlinearity on capture fraction sensitivity to pumping rate, and for calculating the bias associated with capture maps. The sensitivity of capture map bias to selected parameters related to model design and conceptualization for the arid western United States is explored. This study finds that the simulation of stream continuity, pumping rates, stream incision, well proximity to capture sources, aquifer hydraulic conductivity, and groundwater evapotranspiration extinction depth substantially affect capture map bias. Capture difference maps demonstrate that regions with large capture fraction differences are indicative of greater potential capture map bias. Understanding both spatial and temporal bias in capture maps derived from nonlinear groundwater flow models improves their utility and defensibility as conjunctive‐use management tools.     

Evaluation of Hydraulic Conductivity Estimates from Various Approaches with Groundwater Flow Models

Significant efforts have been expended for improved characterization of hydraulic conductivity (K) and specific storage (S_s) to better understand groundwater flow and contaminant transport processes. Conventional methods including grain size analyses (GSA), permeameter, slug, and pumping tests have been utilized extensively, while Direct Push-based Hydraulic Profiling Tool (HPT) surveys have been developed to obtain high-resolution K estimates. Moreover, inverse modeling approaches based on geology-based zonations, and highly parameterized Hydraulic Tomography (HT) have also been advanced to map spatial variations of K and S_s between and beyond boreholes. While different methods are available, it is unclear which one yields K estimates that are most useful for high resolution predictions of groundwater flow. Therefore, the main objective of this study is to evaluate various K estimates at a highly heterogeneous field site obtained with three categories of characterization techniques including: (1) conventional methods (GSA, permeameter, and slug tests); (2) HPT surveys; and (3) inverse modeling based on geology-based zonations and highly parameterized approaches. The performance of each approach is first qualitatively analyzed by comparing K estimates to site geology. Then, steady-state and transient groundwater flow models are employed to quantitatively assess various K estimates by simulating pumping tests not used for parameter estimation. Results reveal that inverse modeling approaches yield the best drawdown predictions under both steady and transient conditions. In contrast, conventional methods and HPT surveys yield biased predictions. Based on our research, it appears that inverse modeling and data fusion are necessary steps in predicting accurate groundwater flow behavior.     

Geostatistical mapping of geomorphic variables in the presence of trend

Mapping geomorphic variables geostatistically, specifically by kriging, runs into difficulties when there is trend. The reason is that the variogram required for the kriging must be of residuals from any trend, which in turn cannot be estimated optimally by the usual method of trend surface analysis because the residuals are correlated. The difficulties can be overcome by the use of residual maximum likelihood (REML) to estimate both the trend and the variogram of the residuals simultaneously. We summarize the theory of REML as it applies to kriging in the presence of trend. We present the equations to show how estimates of the trend are combined with kriging of residuals to give empirical best linear unbiased predictions (E‐BLUPs). We then apply the method to estimate the height of the sub‐Upper‐Chalk surface beneath the Chiltern Hills of southeast England from 238 borehole data. The variogram of the REML residuals is substantially different from that computed by ordinary least squares (OLS) analysis. The map of the predicted surface is similar to that made from kriging with the OLS variogram. The variances, however, are substantially larger because (a) they derive from a variogram with a much larger sill and (b) they include the uncertainty of the estimate of the trend. Copyright © 2006 John Wiley & Sons, Ltd.