Review of strategies for handling geological uncertainty in groundwater flow and transport modeling
Highlights
► Uncertainty from geological structure; model parameters; local scale heterogeneity. ► Lessons learned from previous studies on relative importance of these uncertainties. ► Challenges in model averaging.
Introduction
The foundation for a numerical groundwater model is a conceptual model comprising the modeler’s understanding of the groundwater system in question as well as the simplifications deemed justifiable for the specific modeling study. The backbone of the conceptual model is a geological model consisting of a number of structural elements, typically derived from stratigraphical reasoning, and a strategy for how to handle the geological heterogeneity within the structural elements. In addition, the conceptual model includes decisions on physical and chemical process equations, spatial and temporal discretisation, boundary and initial conditions. There are uncertainties related to all these elements of the conceptual model. In this paper we will only deal with the geologically related sources of uncertainty.
Traditionally, groundwater models have been constructed on the basis of a single geological model structure with the assumed best possible geological representation of the unknown reality [1]. A commonly used approach is then to assume the existence of so-called effective parameter values characterizing the large-scale variation of the hydraulic properties, e.g., piece-wise constant values within the structural elements. These parameters are defined so that they, when optimized, are expected to reproduce the average behavior of the heterogeneous properties within the structural elements. Parameter assessment through inverse methods provides parameter estimates as well as information about the uncertainty of these parameters. This information may be used to assess prediction uncertainty caused by uncertainty of the effective model parameters, e.g., by regression-based methods [19], [50], [58], or by Monte Carlo analyses [35], [38]. However, as we will show later, the inverse method normally used to quantify the uncertainty (covariance) of estimated effective parameter values undervalues the actual effective parameter uncertainty and the related prediction uncertainty.
Due to local scale geological heterogeneity the hydraulic properties exhibit generally unknown spatial heterogeneity. This local scale heterogeneity is often neglected in groundwater models, for example when parameter values are assumed constant over a structural element (a zone) that typically includes many computational grids. This generates uncertainty additional to that caused by model parameter uncertainty. For idealised aquifer conditions Gelhar [24] showed analytically that the uncertainty on hydraulic head predictions due to unaccounted-for local scale heterogeneity is a function of the hydraulic gradient as well as the variance and correlation length scale of the local scale hydraulic conductivity. The most commonly used methodology today for assessing the predictive uncertainty due to local scale heterogeneity is high-resolution calibration-constrained Monte Carlo simulation (e.g., [22], [25], [54]). In the following we use the short term “constrained” simulation for “calibration-constrained” simulation, while simulation that is not constrained by calibration is termed “unconstrained”. As a new alternative, which is not yet widely applied, Cooley [17] and Cooley and Christensen [18] presented and tested a generic theory to examine the influence of nonlinearity and local scale heterogeneity on parameter estimates and predictions of a groundwater model, when the model parameters are lumped parameters calibrated by regression. Application of the theory to quantify uncertainty of parameters and predictions typically combines generalized regression analysis and high-resolution unconstrained Monte Carlo simulation.
In recent years it has been recognized that geological structural uncertainty often is the most important source of uncertainty [4], [33], [37], [56]. The most widely used methodology to assess uncertainty of model predictions due to conceptual geological uncertainty is to apply multiple geological models in a scenario modeling approach [38], [40]. A similar strategy is pursued in other fields of environmental modeling [2], [20], [34], [45], [49]. An evaluation of the total prediction uncertainty comprising both the prediction uncertainty of each of the multiple models and the differences in predictions among the multiple models can be made by use of Bayesian Model Averaging [32], [37], [46].
The objectives of the present paper are (a) to review our previous studies based on different uncertainty assessment methodologies and synthesize the key findings in relation to the state-of-the-art; and (b) to present a generic classification of uncertainty strategies for groundwater modeling and discuss future perspectives and research challenges. Collectively, the present group of authors has conducted studies of all three sources of uncertainty: conceptual geological uncertainty [29], [33], [51], [56]; parameter uncertainty [7], [8], [9], [10], [15], [33] and local scale heterogeneity [9], [11], [14], [18], but none of our studies have included more than one or two of these sources at the same time.
Section snippets
Classification of methodologies for assessing uncertainty in groundwater modeling
The geologically related uncertainty originates from two main sources: geological structures and hydraulic parameter values within these structures. Within a geological structural element, the parameter values will always exhibit local scale heterogeneity. This heterogeneity can be accounted for or neglected, dependent on the adopted modeling strategy. This leads to three main categories of strategies for uncertainty assessments: (a) geological structures, focussing on uncertainty due to the
General approach
We have conducted four real-world case studies in Denmark with a focus on conceptual geological uncertainty: (a) Esbjerg [29]; (b) Copenhagen west [33]; Eggeslevmagle [55], [56]; and Bording [51]. The location of the case studies and key characteristics of the four studies are illustrated in Fig. 2. Methodologically, the four studies had much in common:
- 1.
Multiple geological models were established. The two geological models in Harrar et al. [29] and the two models in Seifert et al. [51] were
General approach
We have conducted two field case studies [9] and three synthetic case studies [11], [14], [18] with a focus on quantification of prediction uncertainty caused by parameter uncertainty and local-scale heterogeneity. The field cases were conducted in Tude Å and Gjern Å (Fig. 2). In most studies, we used regression-based analysis including the extended theory and the software developed by Cooley [17], [18], and Christensen and Cooley [12]. Methodologically, these studies had much in common:
- 1.
It was
Which source of uncertainty will dominate prediction uncertainties?
Uncertainty assessments considering all three main geologically related sources of uncertainty (Fig. 1) will seldom be feasible in practice, because the analyses inevitably become very comprehensive and very computationally demanding. Hence, in order to prioritise the available limited study resources and select the most suitable methodologies, it is necessary to evaluate which source(s) of uncertainty will dominate the prediction uncertainties. As all studies are unique with respect to
Procedures and strengths
A severe limitation of the multiple modeling approach advocated by Refsgaard et al. [45] and applied in several studies [29], [33], [51], [52], [56] is that the results from all the models cannot be integrated into an optimal model prediction and a single measure of the total uncertainty.
It is well known that simple averaging of predictions made by several different models is a very robust model prediction. Cavadias and Morin [6] showed more than two decades ago that weighting of discharge
Conclusions
The geological model has always been recognized as a very important element in groundwater modeling [1]. Our results show that the geological models are relatively less important for flow modeling, if calibration against head and discharge data is performed, and if model predictions are confined (i) to the same types of variables as the data used for calibration, and (ii) to similar situations with respect to climate, groundwater abstractions, etc. In such case the inevitable (unknown) errors
Acknowledgements
The present study was funded by a grant from the Danish Strategic Research Council for the project HYdrological Modelling for Assessing Climate Change Impacts at differeNT Scales (HYACINTS – www.hyacints.dk) under contract no: DSF-EnMi 2104-07-0008. Constructive comments from two anonymous reviewers are acknowledged.
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