Application of a Hydrological 

Uncertainty Methodology to 

Nuclear Reactor Site Evaluations

Office of Nuclear Regulatory Research

NUREG/CR-7211

PNNL-21589

Application of a Hydrological 

Uncertainty Methodology to 

Nuclear Reactor Site Evaluations

Manuscript Completed: February 2013 

Date Published: August 2016

Prepared by: 

P.D. Meyer and C.D. Johnson

Pacific Northwest National Laboratory

Richland, WA 99352

R.E. Cady, NRC Program Manager

NRC Job Code Number V6068

Office of Nuclear Regulatory Research

NUREG/CR-7211

PNNL-21589

 

iii

ABSTRACT

A variety of surface water and groundwater models are often used in the review of applications 

for new nuclear commercial power reactors. These models are used to evaluate current site 

hydrology and to provide estimates of future site conditions after facility construction has been 

completed. Predicted site conditions such as maximum flood and groundwater elevations may 

be compared to reactor design criteria. The current review guidance emphasizes a conservative 

approach to hydrologic modeling. The limitations of this approach become apparent when 

predicted site conditions exceed a design criterion, or when the margin between the two is 

small. A simple groundwater transport modeling example illustrates how uncertainty concepts 

can be used to inform regulatory decisions under these conditions. An uncertainty analysis 

methodology that can be used to estimate the joint impact of parameter, conceptual model, and 

scenario uncertainties is described, along with available software tools that can be used in a

practical implementation of this methodology. The uncertainty methodology is illustrated using 

an example of reviewing groundwater site characteristics as part of a safety evaluation for a 

new power reactor. The example uses publicly available information from the Bellefonte Nuclear 

Station Units 3 and 4 Combined License application. A detailed description of model 

development is included to illustrate typical modeling decisions that arise in actual reviews. 

Modeling and uncertainty analyses were carried out using the Groundwater Modeling System 

(GMS) software, PEST, UCODE, and custom spreadsheets. The example illustrates that 

following standard procedures for uncertainty analyses can reveal limitations of the hydrologic 

analyses, e.g., unexamined parameter correlations and outstanding data needs, and results in 

more consistent reviews. Explicit consideration of alternative models is likely more valuable than 

quantitative model averaging. However, model averages using linear uncertainty methods can 

be easily completed and should be considered. Simple approaches to uncertainty evaluation 

can have great value, contributing to estimates of the degree of conservatism in the analysis, 

demonstrating an understanding of the system, and providing evidence of due diligence.

v

CONTENTS

ABSTRACT ................................................................................................................... iii

LIST OF FIGURES......................................................................................................... ix

LIST OF TABLES ......................................................................................................... xv

EXECUTIVE SUMMARY ............................................................................................ xvii

ACKNOWLEDGMENTS .............................................................................................. xxi

ABBREVIATIONS AND ACRONYMS....................................................................... xxiii

1 INTRODUCTION.................................................................................................... 1-1

2 MODEL USE IN NEW REACTOR REVIEWS........................................................ 2-1

2.1 Use of Hydrologic Models in Nuclear Power Plant Evaluations................................ 2-1

2.2 What is the Role of a Model?................................................................................... 2-1

2.3 Typical Groundwater Modeling Experience.............................................................. 2-2

2.4 Limitations of the Current Approach......................................................................... 2-3

2.5 Conservative Models or Conservative Decisions? ................................................... 2-4

2.6 Groundwater Transport Example............................................................................. 2-5

2.6.1 A Simple Uncertainty Evaluation ................................................................. 2-7

3 UNCERTAINTY METHODOLOGY AND SOFTWARE.......................................... 3-1

3.1 Uncertainty Methodology Incorporating Parameter, Model, and Scenario 

 Uncertainties............................................................................................................ 3-3

3.2 Implementation of the Uncertainty Methodology ...................................................... 3-6

3.2.1 Software Tools ............................................................................................ 3-7

3.2.2 Software Capabilities and Limitations.......................................................... 3-8

4 EXAMPLE APPLICATION..................................................................................... 4-1

4.1 Building the Groundwater Model.............................................................................. 4-1

4.1.1 Gather Relevant Data.................................................................................. 4-3

4.1.1.1 Site Locale and Topology............................................................. 4-4

4.1.1.2 Site Hydrogeology........................................................................ 4-6

4.1.1.3 Aquifer Properties......................................................................... 4-8
vi

4.1.1.4 Precipitation and Land Usage .................................................... 4-10

4.1.1.5 Hydraulic Information ................................................................. 4-10

4.1.1.6 Conceptual Model Based on the Data ........................................ 4-13

4.1.2 Define the Geological/Model Structure ...................................................... 4-16

4.1.2.1 Groundwater Model Domain....................................................... 4-16

4.1.2.2 Define Hydrogeological Units as Materials ................................. 4-18

4.1.2.3 Build a Solids Model................................................................... 4-18

4.1.2.4 Map the Geological Structure to a MODFLOW Domain.............. 4-22

4.1.3 Configure the MODFLOW Model............................................................... 4-29

4.1.3.1 Temporal Nature of the Model.................................................... 4-29

4.1.3.2 Boundary Conditions on Sides of Model Domain........................ 4-29

4.1.3.3 Boundary Conditions for Surface Water Features....................... 4-29

4.1.3.4 Recharge Boundary Conditions.................................................. 4-31

4.1.3.5 Initial Conditions......................................................................... 4-31

4.1.3.6 Configured Model....................................................................... 4-32

4.2 Model Calibration................................................................................................... 4-32

4.2.1 Define Calibration Targets......................................................................... 4-33

4.2.2 Define Parameters for Estimation.............................................................. 4-33

4.2.3 Calibration Results .................................................................................... 4-34

4.2.4 Define Prior Parameter Information ........................................................... 4-38

4.3 Alternative Models Evaluated for the Bellefonte Site.............................................. 4-41

4.3.1 Hydrogeologic Unit Representation ........................................................... 4-41

4.3.2 Recharge Representation.......................................................................... 4-42

4.3.3 Depth of Active Flow ................................................................................. 4-42

4.3.4 Alternative Model Designations ................................................................. 4-42

4.4 Calibration Implementation .................................................................................... 4-43

4.4.1 Model Evaluation....................................................................................... 4-43

4.5 Alternative Model Probabilities............................................................................... 4-50

4.6 Predictive Modeling ............................................................................................... 4-51
vii

4.6.1 Modifications for Post-Construction Conditions ......................................... 4-52

4.6.2 Post-Construction Model Results .............................................................. 4-55

4.6.3 Predictive Uncertainty ............................................................................... 4-56

4.6.3.1 Software Used in Estimating Predictive Uncertainty ................... 4-57

4.6.3.2 Predictive Uncertainty Results.................................................... 4-58

5 Conclusions.......................................................................................................... 5-1

6 References............................................................................................................ 6-1

APPENDIX A: Dependence of KIC on Log Transformation of Parameters .......... A-1

APPENDIX B: Data Used in Model Development.................................................... B-1

APPENDIX C: Spreadsheets for Calculating Model Selection Criteria and 

 Model-Averaged Quantities .............................................................. C-1

ix

LIST OF FIGURES

Figure 1. Boxplots of (left) Ks estimates from 11 well tests and (right) Sr-90 Kd

estimates from 20 laboratory tests........................................................................ 2-6

Figure 2. Parameter distributions based on data analysis (solid line) and sample 

distributions (symbols) for (left) Ks and (right) Kd parameters................................ 2-7

Figure 3. Travel time results for the 500 random parameter realizations.............................. 2-8

Figure 4. Probability of exceedance for the concentration ratio............................................ 2-9

Figure 5. Framework for application of hydrogeologic models. The relatively short 

history-matching period involves comparison between observations (points) 

and model simulated equivalents (solid line). The much longer predictive 

period involves comparison between model simulation results and 

regulatory or design criteria (e.g., a design maximum groundwater head). ........... 3-2

Figure 6. Location of the Bellefonte site. Lower image taken from FSAR Figure 2.1-

201 (TVA 2009). ................................................................................................... 4-2

Figure 7. Steps in building a MODFLOW groundwater model.............................................. 4-3

Figure 8. Depiction of data inputs used to configure a three-dimensional MODFLOW 

model for simulating groundwater flow in the subsurface...................................... 4-5

Figure 9. DOQQ imagery for the Bellefonte site vicinity. Coordinates are SPCS 

Alabama East, in feet. .......................................................................................... 4-5

Figure 10. Topographic map for the Bellefonte site vicinity.................................................... 4-6

Figure 11. DEM data in the vicinity of the Bellefonte site (as depicted in Global 

Mapper)................................................................................................................ 4-7

Figure 12. Conceptual depiction of hydrogeology in a cross-section of the Sequatchie 

Valley, the location of the Bellefonte site (from Miller 1990).................................. 4-7

Figure 13. Geological units in the vicinity of the Bellefonte site. Adapted from Irvin and 

Dinterman (2009).................................................................................................. 4-8

Figure 14. A geologic cross-section through Units 3 and 4 showing rock units (taken 

from FSAR Figure 2.5-339, TVA 2009)................................................................. 4-9

Figure 15. Total annual precipitation at COOP station 017304 in Scottsboro, Alabama 

from 1955 through 2009. The dashed line is the annual average over this 

time frame. ......................................................................................................... 4-11

Figure 16. Locations of surface water gaging locations. This figure shows a portion of 

FSAR Figure 2.4.12-213 (TVA 2009).................................................................. 4-12

Figure 17. Locations of streams that discharge into “Town Creek” (marked by the 

arrows). .............................................................................................................. 4-13
x

Figure 18. Hydraulic head data and interpreted contours for the 1961 data (blue 

contours at west and north sides) and the September 2005 data (black 

contours in southeast quadrant, all overlain on the Bellefonte area imagery....... 4-14

Figure 19. Example of recent hydraulic head contours interpreted from November 

2006 groundwater headdata............................................................................... 4-15

Figure 20. Conceptual model of karst development at the Bellefonte site (from FSAR 

Figure 2.5-304, TVA 2009). ................................................................................ 4-17

Figure 21. Groundwater head response to a precipitation event: about 10 cm (4 in) of 

precipitation were recorded at Scottsboro, Alabama from 9/22/06 to 9/24/06 

(Scottsboro 2 NE Experimental Station, downloaded 8/27/10 from 

http://www.ncdc.noaa.gov). Groundwater head data from FSAR Table 

2.4.12-204 for monitoring well MW-1202a screened in the soil unit (TVA 

2009).................................................................................................................. 4-17

Figure 22. Extent of the groundwater simulation model grid relative to key features. 

The grid origin is located in the lower left corner of the model at the bottom 

of the domain...................................................................................................... 4-19

Figure 23. Material designations (and corresponding colors) used in GMS to represent 

the stratigraphic units of the subsurface (Section 4.1.1.6). The “dummy” 

material was used to facilitate work in GMS, but was not applied in the 

groundwater model............................................................................................. 4-19

Figure 24. DEM data in the vicinity of the Bellefonte site, looking nominally to the 

NNE. A 10X vertical exaggeration is applied....................................................... 4-20

Figure 25. Set of TINs defining the Soil (between green ground surface and orange) 

and the Weathered Rock (between orange and yellow). The zone of low 

data density represents where ground surface elevations from wells at the 

Bellefonte site were used in place of the DEM data. The orange and yellow 

TINs are offset from ground surface by the nominal thickness of each 

material. View is nominally to the NNE with a 10X vertical exaggeration. ........... 4-21

Figure 26. Solids models for the Soil and weathered rock (WeRo) materials based on 

a uniform thickness of each material from ground surface. View is 

nominally to the NNE with a 10X vertical exaggeration....................................... 4-21

Figure 27. Examination of TINs representing the contacts between bedrock units. The 

left image depicts the TINs based on borehole data for contacts between 

the units, while the right image shows the parallel planes used to represent 

the unit contacts. The view is along a strike of 42.5° with a 10X vertical 

exaggeration....................................................................................................... 4-23

Figure 28. Solids representing the USR, the MSR units A to F, and the contacts with 

the NV and LSR units. View is nominally to the NNE with a 10X vertical 

exaggeration....................................................................................................... 4-24
xi

Figure 29. View of solids used to fill in space such that the groundwater model domain 

is encompassed (left) and the solid used for clipping the top of the solids 

representing the bedrock units (right). The rightmost block representing the 

NV unit has been hidden in the right image so that the variation of the 

clipping solid surface is visible. Note that although the clipping solid is 

green, it does not represent Soil. The view is nominally to the NNE with a 

10X vertical exaggeration but with 25° (left) and –25° (right) dip viewing 

angles................................................................................................................. 4-24

Figure 30. Solids representing bedrock stratigraphic units after being clipped so the 

top elevation matches with the bottom elevation of the WeRo solid. View is 

nominally to the NNE with a 10X vertical exaggeration....................................... 4-25

Figure 31. Final set of solids representing all materials and trimmed to the lateral and 

bottom extent of the groundwater flow model domain. View is nominally to 

the NNE with a 10X vertical exaggeration........................................................... 4-25

Figure 32. Deformation of an originally orthogonal model grid after mapping the solids 

model (Figure 31) to the MODFLOW LPF package using the Grid Overlay 

option. The grid layers follow the topography (using equal grid cell 

thickness for all layers in a given vertical column of grid cells). The water 

table would cut through multiple layers. View is nominally to the NNE with a 

10X vertical exaggeration. .................................................................................. 4-27

Figure 33. MODFLOW model grid for the LPF package, showing the materials 

(hydrogeological units) applied to the grid cells. This model was created by 

splitting the solids model into two portions, mapping (using the Grid 

Overlay option) to two model grids, and then merging both grids into a 

single model grid. Grid cells beyond the shoreline of the Tennessee River 

have been inactivated. View is nominally to the NNE with a 10X vertical 

exaggeration....................................................................................................... 4-28

Figure 34. Fence diagram of MODFLOW model grid for the HUF package, showing 

the hydrogeological units applied to the grid cells along selected cross 

sections. View is nominally to the NNE with a 10X vertical exaggeration............ 4-28

Figure 35. Hydraulic head data from 1961 relative to the model domain boundary. 

Data from numbered wells were used to define the boundary conditions 

along the northwest side. Filled circles indicate wells impacted by pumping 

and thus not used in this analysis. ...................................................................... 4-30

Figure 36. Boundary conditions for the groundwater model. Internal boundary 

conditions (green drain and purple constant head) apply to layer 1, others 

(yellow) to all layers. ........................................................................................... 4-30

Figure 37. Zones for the recharge boundary conditions....................................................... 4-31

Figure 38. MODFLOW groundwater model for the Bellefonte site as configured with 

the hydrogeological units (HUF package), constant head boundary 

conditions (purple points), and drains (white points). .......................................... 4-32
xii

Figure 39. Locations of hydraulic head calibration targets for the four groups of data. 

Points outside the model domain boundary were not included as calibration 

targets. ............................................................................................................... 4-34

Figure 40. Hydraulic head contours (every 2 feet) resulting from the initial PEST 

calibration........................................................................................................... 4-35

Figure 41. Unweighted residuals (observed minus simulated, in feet) for the 1961 

data. Positive (blue) and negative (pink) residual magnitudes are 

proportional to symbol size. ................................................................................ 4-36

Figure 42. Unweighted residuals (observed minus simulated, in feet) for the 

September 2005 data. Positive (blue) and negative (pink) residual 

magnitudes are proportional to symbol size........................................................ 4-36

Figure 43. Unweighted residuals (observed minus simulated, in feet) for the average 

2006/2007 “Bedrock” data. Positive (blue) and negative (pink) residual 

magnitudes are proportional to symbol size........................................................ 4-37

Figure 44. Unweighted residuals (observed minus simulated, in feet) for the average 

2006/2007 “Soil” data. Positive (blue) and negative (pink) residual 

magnitudes are proportional to symbol size........................................................ 4-37

Figure 45. Correlation between hydraulic conductivity and recharge prevents unique 

parameter estimates unless one value is fixed (e.g., recharge here) or prior 

values of parameters are applied. Here the prior mean recharge was 8 

in/yr; the prior mean hydraulic conductivity was 0.438 ft/d. ................................. 4-39

Figure 46. Prior parameter distributions derived from data and assumptions discussed 

in the text: horizontal hydraulic conductivity (top) and recharge (bottom)............ 4-40

Figure 47. Pathlines from points around the Units 3 and 4 reactor buildings computed 

in GMS using MODPATH: plan view (top) and cross-section through the 

reactor building areas (bottom, 10x vertical exaggeration).................................. 4-46

Figure 48. Graphical evaluation of residuals for model H4R1D1: observed vs. 

simulated heads (feet) (top left), unweighted residuals vs. observed heads 

(top right), weighted residuals vs. simulated heads (middle left), quantilequantile plot of weighted residuals (middle right), and boxplot of weighted 

residuals (bottom, with explanatory legend)........................................................ 4-47

Figure 49. Parameter estimates and 95% confidence intervals for hydraulic 

conductivity (top, ft/d) and recharge (bottom, ft/d). See text for explanation 

of parameters. .................................................................................................... 4-49

Figure 50. Composite sensitivities for model H4R1D1 at the final parameter values as 

computed following Hill and Tiedeman (2007). ................................................... 4-50

Figure 51. Model probabilities computed with AICc and KIC, and the weighted sum of 

squared residuals for the alternative models....................................................... 4-51
xiii

Figure 52. Figures depicting the depth of the reactor building excavation (left), 

changes in topography, the locations of reactor buildings, and the 

groundwater model grid zones representing fill material and reactor 

building excavations (right). Figures adapted from FSAR Figure 2.5-348a 

and Figure 2.5-362. ............................................................................................ 4-53

Figure 53. Refined recharge zones for the post-construction scenario. A portion of the 

Lowlands zone was incorporated into the northern corner of the Developed 

Area zone........................................................................................................... 4-53

Figure 54. Details of the post-construction recharge zones around Units 3 and 4. 

Reactor and other significant buildings and the main parking areas are set 

to zero recharge. Note that the colors are simply used to distinguish zones 

and are unrelated to materials. ........................................................................... 4-54

Figure 55. Pre-construction topography and materials (left) and modifications to 

represent the post-construction scenario (right) in the vicinity of Units 3 and 

4. The topography was flattened at the site and the Category 1 building 

foundations were represented by inactive grid cells surrounded by fill 

material. View is nominally to the ENE with a 10X vertical exaggeration. ........... 4-55

Figure 56. Hydraulic head contours for model H4R1D1 with the post-construction 

model configuration modifications. The purple rectangles indicate the 

location of Units 3 and 4, where inactive cells represent building 

foundations and fill material surrounds the foundations. ..................................... 4-56

Figure 57. Comparison of pre-construction (left) and post-construction scenario 

hydraulic head contours for model H4R1D1. Contour intervals are 1 ft............... 4-56

Figure 58. Detail from the post-construction groundwater model showing the locations 

of the predicted heads at the corners of the reactor units. .................................. 4-59

Figure 59. Prior parameter distribution for the hydraulic conductivity of the fill material. ...... 4-60

Figure 60. Predicted heads (top of colored bars) and 95% confidence interval upper 

limits for the individual models and the model average....................................... 4-61

Figure 61. Effect on model-averaged predicted head of including the fill material 

hydraulic conductivity uncertainty. ...................................................................... 4-61

xv

LIST OF TABLES

Table 1. Hydraulic conductivity (Ks) for subsurface materials near the Bellefonte site. 

Minimum and maximum values reported in TVA (2009); midpoints and 

averages were calculated..................................................................................... 4-9

Table 2. Porosity for subsurface materials in the vicinity of the Bellefonte site.................. 4-10

Table 3. Average stage observed at surface water measurement locations. .................... 4-12

Table 4. Initial model grid description. .............................................................................. 4-18

Table 5. Head observation error intervals, standard deviations, and calibration 

weights ............................................................................................................... 4-33

Table 6. Materials represented by hydraulic conductivity parameters in alternative 

models................................................................................................................ 4-48

Table 7. Model selection criteria values and model probabilities for alternative 

models................................................................................................................ 4-51

Table 8. Averaged observed head (calibration target), maximum observed head, 

and simulated equivalent head from the calibrated H4R1D1 model for three 

wells near the desired location of predicted head. All heads in feet. ................... 4-59

xvii

EXECUTIVE SUMMARY

A variety of surface water and groundwater models are often used in the review of applications 

for new nuclear commercial power reactors. Models are used to improve understanding of site 

behavior, e.g., by synthesizing a conceptual understanding of site hydrology with observations 

of the site-specific and regional hydrologic characteristics. Ultimately, however, Nuclear 

Regulatory Commission (NRC) staff need to reach conclusions about site hydrologic 

characteristics and the magnitude of hydrologic hazards and environmental impacts that directly 

support regulatory decisions about facility safety and environmental acceptability. These 

conclusions often require the estimation of some quantitative measure of future site behavior, 

for example, maximum flood and groundwater elevations after plant construction, that can be 

compared to reactor design criteria. To improve the defensibility of decisions informed by the 

results of predictive modeling, it is important to be transparent in describing the development of 

the model and explicit in communicating the uncertainties in model results. This viewpoint is 

consistent with NRC policy in the sense that risk-informed decisions require some 

understanding of uncertainty, which includes the uncertainty of model results used as the partial 

basis for those decisions.

As a practical matter, hydrologic models are not an end in themselves, but are usually part of a 

larger project. Because of this, the way in which hydrological models are developed and used is 

constrained by budget and schedule factors that may be beyond the control of those 

responsible for the hydrologic analysis. For example, groundwater model developers supporting 

a Combined License (COL) applicant and NRC staff reviewing the application share a common 

experience: typically all the data are collected prior to model development. As a result there is 

little (or no) opportunity for the modeling to influence data collection and there are significant 

constraints on the ability to evaluate the model and improve it when model limitations/errors 

become apparent. This situation can lead to relaxation of calibration criteria (i.e., accepting a 

larger error in the calibration) and a greater need for sensitivity analyses and consideration of 

alternative conceptualizations to adequately address uncertainties. Because opportunities for 

feedback from model evaluation are limited, it is important that the NRC staff’s review plans and 

regulatory guidance are explicit about modeling expectations, including the role of uncertainty 

assessment. These expectations should be consistent with established standards and accepted 

procedures.

With regard to hydrology-related review of new reactor applications, the current review criteria 

and technical procedures contained in NRC staff guidance describe a predominantly 

deterministic approach that emphasizes conservatism in the analysis. This approach can be 

problematic when either the acceptance criteria are not satisfied (or the margin of acceptance is 

small), or when the models and technical evaluation methods are not demonstrably 

conservative. Examples are given where these issues have arisen in actual reviews. Instead of 

focusing on the conservatism of models (e.g., bounding assumptions and extreme parameter 

values), it is recommended that the focus be on conservative decisions. The latter implies that 

uncertainties in model outcomes can be properly accounted for; the conservatism in this case 

enters when comparing model outcomes to regulatory or design requirements. For example, a 

decision based on the 95th percentile outcome is more conservative than one based on the 

mean outcome. Focusing on a conservative decision is consistent with a risk-informed 

approach, and it allows for the natural consideration of consequences (e.g., the safety 

consequences of exceeding a design flood elevation). A simple groundwater transport modeling 

example is used to demonstrate the advantage of a risk-informed approach focused on 

conservative decisions with a quantitative consideration of uncertainties.
xviii

An uncertainty analysis methodology that can be used to estimate the joint impact of parameter, 

conceptual model, and scenario uncertainties is described. This methodology is comprehensive 

in that it considers multiple sources of uncertainty while providing a means to focus on those of 

most relevance. It is quantitative, allowing model results to be compared with regulatory criteria 

or design requirements. It is systematic, applicable to a wide range of sites and objectives. And 

the methodology is practical, able to be implemented with standard methods and existing

software tools.

Implementation of the uncertainty methodology requires two basic activities: development of two 

or more models to represent unresolved conceptual and structural uncertainties associated with 

the representation of the site, and calculations to estimate parameter values, parameter 

uncertainties, model probabilities, and predicted values (including predictive uncertainty). Model 

development is a site-specific and labor-intensive process, making it difficult to apply standards 

or generalizable procedures. The Groundwater Modeling System (GMS) package and its 

associated process codes were the primary software used in model development for the 

example discussed in this report. Parameter estimation and uncertainty calculations are more 

amenable than model development to the application of standards and generalizable 

procedures. PEST and UCODE were the principal software packages used to carry out these 

calculations for the example in this report. Due to limitations in software capabilities, calculation 

of model probabilities and model-averaged predictions were carried out using custom 

spreadsheets.

A practical application of the uncertainty methodology is demonstrated in this report through the

review of groundwater site characteristics as part of a safety evaluation for a new power reactor. 

Although the example uses publically available information from the Bellefonte Nuclear Station 

Units 3 and 4 COL application, the groundwater models developed for the example are solely 

the product of the coauthors of this report. Information used in model development included 

digital imagery and topographic data, a regional hydrogeologic description, geologic unit 

contacts from site characterization borehole logs, aquifer properties from site characterization 

hydraulic and physical tests, regional precipitation data, surface water elevation data, and 

observation well groundwater head measurements. 

A detailed description of conceptual model development in GMS is included to illustrate typical 

modeling decisions that arise in actual reviews. The conceptual model was mapped in GMS to a 

MODFLOW groundwater flow model. Alternative models were developed to represent 

uncertainties in the representation of the hydrogeologic units, the variation in recharge, and the 

depth of active flow. Sixteen alternative models were considered. Each model was calibrated to 

a set of groundwater head observations and prior information on parameters for saturated 

hydraulic conductivities and recharge values. Model evaluation was carried out within GMS and 

using custom spreadsheets. With the exception of outliers located in two areas of the model 

domain, the residuals appeared to be normally distributed. Weighted residuals for the calibrated 

models were, however, larger than indicated by the assumed head observation errors. These 

results suggest that, although model error is significant, predictive uncertainties from the models 

can be accurate.

Model probabilities were calculated using a custom spreadsheet to eliminate the potential effect 

of parameter transformations on the probabilities, something that is not considered in existing 

software. Results showed that only four of the sixteen alternative models were assigned 

substantial model probabilities. High probability models were neither the most complex, nor the 

simplest. 
xix

Predictive models were developed for the high probability models to estimate post-construction 

groundwater head in the reactor power block area. Building foundation locations, the addition of 

fill material, changes in topography due to cut and fill, and changes in recharge were 

implemented in the predictive models to represent post-construction conditions. Parameter 

uncertainty was based on the calibration results with the addition of uncertainty in the hydraulic 

conductivity of the fill material. Predictive uncertainties were estimated using linear methods to 

calculate the mean and variance of the predicted groundwater head values. Results indicated 

that the depth of flow had little impact on the head near the reactor units. The depth of flow likely 

would have a much greater impact on predicted travel time. Model-averaged predicted values 

and predictive uncertainty were found to depend on the model selection criterion used to 

calculate the model probabilities. Since no single criterion has been shown to be superior in all 

cases, multiple criteria should be used.

The upper limit of the 95% confidence interval for the predicted groundwater head was 

increased by the inclusion of uncertainty in the fill material hydraulic conductivity, as much as 

one foot depending on the model selection criterion used. This result demonstrates the potential 

importance of even localized changes to the model under post-construction conditions. The fill 

material occupies a small portion of the model domain, but it is a potentially critical feature given 

that the design criterion for maximum groundwater head is within the excavation. In this case, 

including a reasonable uncertainty estimate for this material of limited extent had a noticeable 

impact on the predicted regulatory value to be compared to the design criterion. Whether or not 

this is significant would depend on the margin between the upper limit and the design criterion.

The example illustrates that following standard procedures for uncertainty analyses can reveal 

limitations of the hydrologic analyses, e.g., unexamined parameter correlations and outstanding 

data needs, and results in more consistent reviews. Explicit consideration of alternative models 

is likely more valuable than quantitative model averaging. However, model averages using 

linear uncertainty methods can be easily completed and should be considered. Simple 

approaches to uncertainty evaluation can have great value, contributing to estimates of the 

degree of conservatism in the analysis, demonstrating an understanding of the system, and 

providing evidence of due diligence.

xxi

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial assistance of the Office of Nuclear Regulatory 

Research of the U.S. Nuclear Regulatory Commission (NRC), the guidance of the NRC Project 

Manager, Ralph E. Cady, and the publication assistance of Mark Fuhrmann. We also appreciate 

the comments of NRC staff members Thomas J. Nicholson and Joseph F. Kanney during 

presentation of the contents of this report, and the comments of NRC staff members Mark S. 

McBride and Nebiyu Tiruneh on the final written version of the report.

xxiii

ABBREVIATIONS AND ACRONYMS

ACRS Advisory Committee on Reactor Safety

AICc Akaike Information Criterion, corrected

ANSI American National Standards Institute

ANS American Nuclear Society

ASTM American Society of Testing and Materials

ASWP Alabama State Water Program

BCF Block-Centered Flow

CFR Code of Federal Regulations

COL Combined Operating License

COOP U.S. Cooperative Network

DEM Digital Elevation Model

DOQQ Digital Orthographic Quarter Quad

DRG Digital Raster Graphic

ECL Effluent Concentration Limit

ER Environmental Report

FSAR Final Safety Analysis Report

GHCN Global Historical Climatology Network

GIS Geographical Information System

GMS Groundwater Modeling System

HCN U.S. Historical Climatology Network

HDF Hierarchical Data Format

HUF Hydrogeologic-Unit Flow

KIC Kashyap Information Criterion

LPF Layer Property Flow

LSR Lower Stones River

MMA Multi-Model Analysis Software

MODFLOW Groundwater Flow Modeling Software

MODPATH Groundwater Flow Path Modeling Software

MSR Middle Stones River

MT3DMS Groundwater Transport Modeling Software

NAD27 North American Datum of 1927

NAD83 North American Datum of 1983

NE Northeast
xxiv

NNE North Northeast

NRC Nuclear Regulatory Commission

NV Nashville

NW Northwest

PCRP Historical precipitation data

PDF Portable Document Format

PEST Parameter Estimation Software

PRA Probabilistic Risk Assessment

RAI Request for Additional Information

RQD Rock Quality Designation

SE Southeast

SPCS State Plane Coordinate System

SPT Standard Penetration Test

SRP Standard Review Plan

SSWR Sum of Squared Weighted Residuals

SW Southwest

TIN Triangulated Irregular Network

TVA Tennessee Valley Authority

UCODE Parameter Estimation Software

USACE U.S. Army Corps of Engineers

USGS U.S. Geological Survey

USR Upper Stones River

UTM Universal Transverse Mercator

WeRo Weathered Rock

WGS84 World Geodetic System of 1984
1-1

1 INTRODUCTION

In its review of applications for new nuclear power plants, U.S. Nuclear Regulatory Commission 

(NRC) staff are responsible for evaluating the hydrologic characteristics of proposed sites and 

assessing the impacts of the facilities on the hydrology and water resources of the surrounding 

area. Current guidance for these staff evaluations (NRC 2007; NRC 1999) describe technical 

procedures without explicit reference to the use of models. In practice, however, NRC staff often 

use hydrologic models as part of their evaluations. Models are used to synthesize a conceptual 

understanding of site hydrology and observations of the site-specific and regional hydrologic 

characteristics. This type of model use, to improve understanding of site behavior, is generally 

uncontroversial. 

Understanding of the hydrology of a site is not an end in itself, however. Ultimately NRC staff 

need to reach conclusions about site hydrologic characteristics and the magnitude of waterrelated hazards and environmental impacts that directly support regulatory decisions about 

facility safety and environmental acceptability. These conclusions often require the estimation of 

some quantitative measure of future site behavior, for example, a flood elevation after plant 

construction. These estimates may be based in part on the results of modeling. This type of 

model use - to make predictions about the future state of a hydrologic system – is not without 

controversy (e.g., Pilkey and Pilkey-Jarvis 2007). It is nonetheless generally viewed as a central 

component of estimating the future effects of regulatory decisions (National Research Council 

2007). To improve the defensibility of decisions informed by the results of predictive modeling, it 

is important to be transparent in describing the development of the model and explicit in 

communicating the uncertainties in model results (National Research Council 2007, 2012). This 

viewpoint is consistent with NRC policy in the sense that risk-informed decisions require some 

understanding of uncertainty, which includes the uncertainty of model results used as the partial 

basis for those decisions.

This report discusses the analysis of uncertainty in hydrologic modeling used in the review of 

applications for new nuclear commercial power reactors. Although the examples provided in the 

report are for groundwater modeling, the methods and principles discussed are applicable to 

other hydrologic models as well.

Chapter 2 of the report describes the types of hydrologic models used in new reactor reviews 

and the role of modeling in the regulatory process. The current review guidance emphasizes a 

conservative approach to modeling. The limitations of this approach are discussed and a simple 

groundwater transport modeling example is provided to illustrate how uncertainty concepts can 

be used to inform regulatory decisions. Chapter 3 describes an uncertainty analysis 

methodology that can be used to estimate the joint impact of parameter, conceptual model, and 

scenario uncertainties. The practical implementation of this methodology is also described, 

including a discussion of the capabilities and limitations of applicable software tools. Chapter 4 

contains an example application of the uncertainty methodology as it might be used in reviewing 

groundwater site characteristics conducted as part of a safety evaluation for a new power 

reactor. The example uses publically available information from the Bellefonte Nuclear Station 

Units 3 and 4 Combined License application. A detailed description of model development is 

included to illustrate typical modeling decisions that arise in actual reviews. The utility and 

limitations of the software for modeling and uncertainty analysis are discussed throughout the 

example. Concluding remarks and observations are provided in the final chapter of the report.

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2 MODEL USE IN NEW REACTOR REVIEWS

2.1 Use of Hydrologic Models in Nuclear Power Plant Evaluations

Hydrologic models are used to support NRC staff reviews of nuclear power plant applications in 

the evaluation of both safety and environmental impacts. For the evaluation of hydrologic site 

characteristics related to safety under Section 2.4 of the Standard Review Plan (SRP) (NRC 

2007), models are often used as part of the determination of flood elevations. Applications of 

models may include determining watershed response to extreme precipitation events, site 

drainage in response to locally intense precipitation, and surge or wave runup due to events 

such as hurricanes and tsunamis. Groundwater flow and transport models are also likely to be 

used under SRP Section 2.4 reviews to evaluate the maximum post-construction groundwater 

elevations on site, to identify and characterize potential transport pathways, and estimate the 

effects of an accidental release of radioactive liquid effluents. 

Hydrologic models may also be used in the review of a nuclear power plant application for the 

evaluation of environmental effects as part of the production of an environmental impact 

statement. For example, models are used to evaluate the impact of plant water use on 

downstream water users and to assess the impact of plant discharge on water quality. Similarly, 

groundwater flow models have been used to evaluate the impacts of plant groundwater use on 

nearby wetlands, on flow in streams on or near the site, and on off-site users of groundwater.

There are also other NRC regulatory actions that may use hydrologic models for similar 

purposes. Models of groundwater flow and transport may be used to evaluate the safety and 

environmental impacts of a proposed in situ uranium recovery operation, for example by 

determining the role of low permeability units and structural faults in controlling flow. Hydrologic 

models may also used in evaluating potential doses resulting from the disposal of radioactive 

waste and in the decommissioning of licensed facilities.

2.2 What is the Role of a Model?

Models of hydrologic systems are generally used for two purposes. The first of these is to 

develop and demonstrate an understanding of the hydrologic behavior of the site. To serve this 

purpose, the model integrates the conceptual understanding of the site (e.g., the geologic 

structure and the active flow and transport processes), observations of site properties and 

behavior (e.g., hydraulic conductivity and head), and the forces acting on the system (e.g., 

precipitation). The model is typically used in this context by comparing model results to 

observations. Inability of the model to reproduce observed conditions indicates a flaw in the 

model, which can be explored by reevaluating assumptions and exploring the sensitivity of 

model outputs to the model structure, parameterization, and boundary conditions. This analysis 

may also suggest where additional data would be useful in improving the understanding of the 

site. Sensitivity analyses can also be used to evaluate the potential impact of design and 

operational changes on site hydrology.

The second purpose for which models are generally used is as a means to make predictions 

about future hydrologic conditions. In this context, the output of a model is compared to a 

quantitative design requirement or regulatory criterion. For example, a flood prediction is 

compared to the design basis flood elevation, maximum groundwater levels are compared to 

the design basis groundwater elevation, and predicted concentration from the accidental release 

of radioactive liquid effluents are compared to the concentration limits identified in 10 CFR 20, 

Appendix B. While the models generating these predictions can be based on prior knowledge of
2-2

the hydrologic system and site-specific observations, the predictions are made under postconstruction conditions that can be substantially different that those used to develop the models. 

To compensate for the resulting uncertainties and to ensure some margin of safety, the models 

are typically based on conservative assumptions.

2.3 Typical Groundwater Modeling Experience

As a practical matter, hydrologic models are not an end in themselves, but are usually part of a 

larger project. This is certainly the case for a nuclear power plant application. Because of this, 

the way in which hydrologic models are developed and used by an applicant is constrained by 

budget and schedule factors that may be beyond the control of those responsible for the 

hydrologic analysis. This has implications for the review of modeling carried out by NRC staff as 

part of the safety and environmental evaluations.

Bredehoeft (2010) recently described a typical “recipe” for groundwater model development and 

use, consisting of the following steps.

• Collect all available data.

• Create cross-sections and isopach maps.

• Decide the information is insufficient.

• Drill more wells.

• Build a groundwater model.

• Run the model. Discover that it is sensitive to data that was not collected, or that the 

concept of the system is in error.

• Out of money/time. “Further work is indicated.”

The primary flaw in this approach as identified by Bredehoeft (2010) is that the 

conceptualization is not tested (by running the groundwater model) until the end of the process, 

leaving little or no opportunity for (1) the modeling to influence data collection, and (2) the 

conceptualization and groundwater model to be modified. 

It is instructive to compare the “recipe” described by Bredehoeft (2010) to the typical experience 

of groundwater modeling associated with new reactor applications. From the perspective of

reviewers following procedures in SRP 2.4.12 as part of a safety evaluation, the typical 

experience can be described as follows. 

• The Final Safety Analysis Report (FSAR) and supporting documents contain all the data that 

will be available for the review.

• The FSAR describes a single groundwater conceptualization.

• Analysis of post-construction conditions is initially not supported by modeling or is supported 

with a simple model with minimal discussion of limitations and uncertainty.

• One or more rounds of additional requests are required to obtain sufficient information on 

alternative groundwater conceptualizations, assumptions, and uncertainties.

• Schedules are often impacted by repeated requests for additional information.

From the perspective of the model developer (supporting applicants), Findikakis and Waterman 

(2010) described a typical situation as follows.

• Development of a groundwater model occurs after data collection.

• Different project teams are responsible for data collection and modeling.

• Data are collected to satisfy geotechnical needs and minimum site characterization 

requirements, not groundwater modeling needs.
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• Drilling and well development techniques may be inappropriate for aquifer characterization 

methods.

Model developers supporting an applicant and application reviewers thus share a common 

experience that refects the flaws identified by Bredehoeft (2010). Typically, all the data are

collected prior to model development. As a result there is little (or no) opportunity for the 

modeling to influence data collection and there are significant constraints on the ability to 

evaluate the model and improve it when model limitations/errors become apparent. Findikakis 

and Waterman (2010) stated that this situation leads to relaxation of calibration criteria (i.e., 

accepting a larger error in the calibration) and a greater need for sensitivity analyses and 

consideration of alternative conceptualizations to adequately address uncertainties.

The experience described above suggests that the groundwater modeling process associated 

with new reactor applications could be improved. Because the process of model development 

as part of the FSAR preparation and the review of that modeling as part of the safety evaluation 

are performed sequentially with limited opportunities for feedback, it is important that the NRC 

staff’s review plans and regulatory guidance are explicit about modeling expectations, including 

the role of uncertainty assessment. These expectations should be consistent with established 

standards such as ANSI/ANS-2.17-2010 (ANS 2010) and accepted procedures such as the 

guidelines for effective modeling discussed in Hill and Tiedeman (2007). Each of these 

referenced documents contains a detailed discussion of uncertainty in the context of 

groundwater modeling.

2.4 Limitations of the Current Approach

The NRC has a policy to include risk information in the review process. But what does “riskinformed” mean in the context of hydrologic analyses for new reactor safety evaluations? The 

current review criteria and technical procedures contained in the hydrology section of the SRP 

(NRC 2007) describe a predominantly deterministic approach that emphasizes conservatism in 

the analysis. There are, in fact, no explicit probability or risk concepts identified. For example, 

the flooding evaluation refers to probable maximum precipitation, flood, hurricane, storm surge, 

and tsunami. The “probable maximum” concept is a conservative and deterministic construct 

with a nonzero, but unquantified probability of exceedance. 

Groundwater analyses are similarly described in deterministic, conservative terms. For example, 

the maximum groundwater elevation expected at the site must be determined. Accidental 

releases of radioactive liquid effluents are determined using the “bounding set of plausible 

surface and subsurface pathways” and assuming “worst-case contamination.”

Two conditions are required for a conservative approach to be adequate: (1) the acceptance 

criteria must be satisfied, and (2) the models and technical evaluation methods must be

demonstrably conservative. When one of these conditions is not met, a conservative approach 

is inconclusive, thereby raising additional questions. For example, what is the path forward 

when the acceptance criteria are not met or when the margin is unacceptably small? This 

question is of practical importance as the issue has arisen in multiple reviews of new reactor 

applications, including with respect to storm surge exceeding the design basis flood elevation 

and locally intense precipitation exceeding the capacity of the site drainage system.

One approach is to relax the conservatism in the evaluation, moving gradually from a 

demonstrably conservative analysis to a less conservative, but presumably more realistic, 

analysis. This is sometimes refered to as a hierarchical approach. As an example of this 

approach, the analysis of the accidental release of radioactive liquid effluents might begin by 
2-4

assuming purely advective groundwater transport, but introduce dispersion to the transport 

evaluation if radionuclide concentrations exceed the limits. This will tend to cause the 

acceptance criteria to be satisfied, but by reducing the conservatism of the analysis, the second 

condition above may not be satisfied, or at the least the degree of conservatism in the modified 

approach may need to be addressed.

Another approach when the acceptance criteria are violated or the margin is small is to 

introduce mitigating features in the facility design. For example, additional flood protection 

measures could be included, or the plant elevation could be raised. The potential danger in this 

approach is that it may result in over-design and unnecessary costs.

The second condition required for a conservative approach to be adequate, that the models and 

technical evaluation methods must be demonstrably conservative, may fail to be satisfied when 

the model and methods are based on limited data. 

This issue is of practical concern as it has been raised during Advisory Committee on Reactor 

Safety (ACRS) review of a safety evaluation in the context of the analysis of accidental release 

of radioactive liquid effluents in groundwater. Specifically, the ACRS staff questioned whether 

selecting minimum measured values of the adsorption coefficients ( Kd ) could be justified given 

the observed variability in Kd and the limited number of samples. The specific question from the 

transcript was, 

“These Kd values vary quite a bit. When you ask for the minimum value, then you have to 

think about whether the sample is big enough for the minimum to be reasonable. If you 

have two values, taking the lower one is not very sensible. If you have 100 values, taking 

the minimum is probably excessive. So do they have enough samples in order for taking 

the minimum to be a meaningful thing to do?” (NRC 2009, p. 138).

The question indicates not only a concern that the method for selecting a model parameter 

(using the extreme observation as the parameter value) could be nonconservative when data 

are limited, but also a concern that the method could be overly conservative when data are 

abundant. 

Another situation in which the conservatism of a model may be questioned is when a model is

used for predictions under unobserved conditions. For example, a groundwater model that is 

developed using data for current site conditions, may be used to predict groundwater elevations 

and transport pathways under post-construction conditions. This issue has also been raised by 

the ACRS in reviewing draft interim staff guidance on implementing SRP Section 2.4.12,

“You characterize the site before, let‘s say, presubstantial construction. Now you have a 

predictive model which takes into account the effective construction...How do you know at 

the end that your predictive model is right?” (NRC 2010, p. 47).

While the concern in the question may have been whether a model remains accurate in 

predicting post-construction conditions, a “right” model could also be one that remains 

conservative.

2.5 Conservative Models or Conservative Decisions?

A focus on conservative models, i.e., bounding assumptions and extreme parameter values, 

can lead to an exaggeration of uncertainties (in the sense of relying on overly conservative 

results) and often implies a failure to realistically quantify uncertainties. This approach is not 
2-5

risk-informed. In contrast, a focus on conservative decisions implies that uncertainties in model 

outcomes can be properly accounted for; the conservatism in this case enters when comparing 

model outcomes to regulatory or design requirements. For example, a decision based on the

95th percentile outcome is more conservative than one based on the mean outcome. Similarly, 

tolerating a failure probability of one per thousand is more conservative than accepting a one 

percent failure rate. Focusing on a conservative decision thus is consistent with a risk-informed 

approach. In addition, it allows for the natural consideration of consequences (e.g., the safety 

consequences of exceeding a design flood elevation). 

The following section describes the application of a groundwater transport model to illustrate the 

limitations of the current deterministic approach focused on conservative models and to

demonstrate the advantage of a risk-informed approach focused on conservative decisions with 

a quantitative consideration of uncertainties.

2.6 Groundwater Transport Example

Consider a simple transport model that might be relevant to the review under SRP Section 

2.4.13 (NRC 2007) of an accidental release of radionuclide liquid effluents to groundwater.

Under a conservative release scenario, liquid effluent containing a radionuclide mixture is 

assumed to be released from a tank directly to groundwater, with the initial concentration 

(activity) of any specific radionuclide given by C0. Transport is assumed to occur via advection, 

with radionuclides subject to decay and adsorption. Under these conditions, the concentration at 

any downgradient location is given by 

0

t C Ce−λ = (1)

where λ is the radionuclide-specific decay coefficient and t is the travel time. This concentration 

can be compared to a radionuclide-specific effluent concentration limit (ECL) from 10 CFR 20 

Appendix B, Table 2. 

Travel time is given by

( ) f

s

D t R

K h

θ = ∇ (2)

where D is the travel distance, θ is the effective porosity, Ks is the saturated hydraulic 

conductivity, ∇h is the magnitude of the hydraulic head gradient, and Rf is the retardation 

factor, defined as

1 b d

f

K

R ρ

θ

= + (3)

with ρb the bulk density and Kd the adsorption coefficient. This model is conservative in that the 

only processes affecting transport are advection and adsorption.

Site data available to support the accidental release analysis included measurements of 

groundwater head (to estimate the gradient), measurements of the physical properties of soil 

and rock (to estimate the bulk density and porosity), well tests for estimating hydraulic 

conductivity, and laboratory measurements of adsorption on core samples for estimating 

adsorption coefficients. Measures of the latter two parameters exhibited significant variability, 

with a range of [0.08, 2.64] m/d for the 11 Ks estimates and a range of [3.6, 166.4] ml/g for the 
2-6

Kd estimates (Sr-90 is considered here). Figure 1 contains boxplots of the Ks and Kd

estimates, showing the median, interquartile range, limits (excluding outliers), and estimated 

outliers. 

Figure 1. Boxplots of (left) Ks estimates from 11 well tests and (right) Sr-90 Kd estimates 

from 20 laboratory tests

From the transport perspective, a conservative approach would result in concentration values 

that are maximized to some extent. For the model of Eq. (1), this corresponds to a minimum 

travel time, which occurs when using large values of Ks and small values of Kd . The 

conservative data in this case are the maximum Ks of 2.64 m/d and the minimum Kd of 3.6 ml/g 

for Sr-90. By assigning these conservative data values to the parameters of Eq. (2), and 

choosing representative (average) values for the remaining parameters (that exhibit less 

variability than Ks and Kd ), a deterministic, conservative evaluation of the concentration from 

an accidental release to groundwater was completed. The resulting concentration ratio was 

C/ECLSr-90 = 3.25

As in any conservative analysis in which the result exceeds or is close to the design or 

regulatory criterion, the question immediately arises: what is the degree of conservatism in this 

result? In particular, is this value conservative enough? For example, are there sufficient data to 

substantiate that the most conservative data represent a bounding case? Conversely, is it 

possible that the analysis is overly conservative? For example, is it unreasonable to 

characterize a long transport pathway with a minimum Kd value when a broad range of values 

was observed at the site? These questions can be only partially addressed by carefully 

examining the assumptions made in deriving the model and determining the parameter values.

In this example the concentration exceeded the regulatory limit defined by the ECL. Given that 

the result represents some unknown degree of conservatism, how should the analysis proceed? 

Would it be best to relax some of the conservative model assumptions? For example, dispersion 

could be included in the transport model. In that case, however, a dispersion coefficient would 

have to be chosen. As a result, there is a risk that the analysis moves from something that can 

be defended (e.g., an analysis based on data) to something that is more difficult to defend (e.g., 

an analysis based on a dispersivity value without data support). Would it be better to relax the 

assumptions on the parameter values, for example, by assuming that the transport pathway is 

better represented by a less extreme Kd value? This would require justifying the apparently 

nonconservative parameter value. Whichever approach is taken, the question still remains -

what is the degree of conservatism in the analysis? The fundamental issue is that this question 

cannot be adequately addressed without some examination of uncertainty in the transport 

analysis.

0

1

2

3

Ks (m/d)

0

50

100

150

Kd (ml/g)
2-7

2.6.1 A Simple Uncertainty Evaluation

To illustrate how consideration of uncertainties can, with relatively little effort, provide useful 

information on the degree of conservatism in the transport evaluation, the conservative model of 

Eq. (1) was combined with an assessment of the uncertainties in the model parameters. 

Estimates of the probability distributions of the parameters were based on an analysis of the 

data. Ks and Kd were best fit with lognormal distributions; the parameters of the distributions 

were estimated from the sample mean and variance of the data sets. The hydraulic gradient and 

porosity were assumed to follow normal distributions with standard deviations equal to ten 

percent of the mean values. In the absence of any information to the contrary, the parameters 

were assumed to be uncorrelated with one another.

Five-hundred random realizations of the parameters were generated based on the data 

analysis. The resulting parameter probability distributions for Ks and Kd are shown in Figure 2

along with the sample distributions for the measured data. Additional analyses could be 

completed, such as nonparametric tests for the assumption of lognormality in the parameters 

and application of a Bayesian method to include uncertainties in the estimated moments of the 

parameter probability distributions. For this example, parameter distributions were considered 

reasonable given the qualitative comparison of Figure 2 and the conservative assumptions of 

the transport model.

Figure 2. Parameter distributions based on data analysis (solid line) and sample 

distributions (symbols) for (left) Ks and (right) Kd parameters

Travel time was computed using Eq. (2) for each of the 500 random realizations of parameter 

values. The resulting distribution of travel time (in years) is shown in Figure 3 for each of the 

random parameters used in the calculation. As illustrated in the apparent relationships (or lack 

thereof) between travel time and the parameter values, the relatively small coefficient of 

variation of the hydraulic head gradient and effective porosity result in the domination of travel 

time variability by the hydraulic conductivity and the retardation coefficient. This has some 

implications for site characterization. Although travel time is sensitive to the gradient and the 

porosity, additional data characterizing these parameters would be expected to have little 

impact on the distribution of travel time.

0.01 0.1 1 10

0

0.2

0.4

0.6

0.8

1

Ks (m/d)

Prob(X<x)

0.1 1 10 100 1 103 ×

0

0.2

0.4

0.6

0.8

1

Kd (ml/g)
2-8

The distribution of travel time is highly skewed (note the use of a log scale in Figure 3), with a 

minimum travel time of 13.2 years, a median of 882 years, and a geometric mean travel time of 

857 years. In this simulation, the minimum travel time did not occur at either the maximum 

hydraulic conductivity or the minimum adsorption coefficient. The extreme parameter values 

were unlikely to occur in the same realization because the parameters were assumed to be 

uncorrelated.

Figure 3. Travel time results for the 500 random parameter realizations

The distribution of travel times results in a distribution of concentration values (or C/ECL 

concentration ratios) when applied to Eq. (1). The concentration ratio distribution is shown in 

Figure 4 as a complementary cumulative distribution function for those C/ECL ratios greater 

than about 0.03. This figure shows the probability of exceeding a given concentration ratio; only 

the probabilities for the largest ratios are shown as these would be of most concern in a safety 

evaluation. 

As discussed above, the maximum observed Ks value and the minimum observed Kd value 

resulted in a concentration ratio of 3.25, which falls at the 98th percentile of the distribution of 

Figure 4. That is, there is an estimated two percent probability that the concentration ratio will be 

larger than 3.25. 

0.03 0.04 0.05 0.06 0.07

10

100

1 103 ×

1 104 ×

1 105 ×

Gradient

Travel Time (yr)

0.15 0.2 0.25 0.3 0.35

10

100

1 103 ×

1 104 ×

1 105 ×

Porosity

0.01 0.1 1 10

10

100

1 103 ×

1 104 ×

1 105 ×

Ks (m/d)

Travel Time (yr)

0.1 1 10 100 1 103 ×

10

100

1 103 ×

1 104 ×

1 105 ×

Kd (ml/g)
2-9

Figure 4. Probability of exceedance for the concentration ratio

From the measurements and fitted parameter distributions shown in Figure 2, the most 

conservative measured Ks and Kd values (2.64 m/d and 3.6 ml/g, respectively) correspond to 

the 96th and 8th percentiles. That is, there is a 4% (for Ks ) and 8% (for Kd ) probability of values 

leading to worse outcomes – higher concentrations. These probabilities are less conservative 

than the outcome from using the most conservative measured parameters in the transport 

model, which resulted in a 2% chance of a worse outcome as described above.

There are a number of factors that could affect these results. Parameter correlations would 

modify the way in which conservative parameter values combine to produce outcomes with a 

comparable, or greater, degree of conservatism. In this example, a negative correlation between 

Ks and Kd would result in more comparable values. In addition, the example results were based 

on a fit to the sampling distibutions, which represent the population of parameter values. This 

reflects a judgment that the bulk of transport is most likely to occur along high-velocity, lowreactivity pathways. In some cases, it may be more justifiable to use distributions representing 

the probability density of the mean values, reflecting a judgment that the outcome is more 

responsive to average conditions. 

Irrespective of the particular assumptions or methods used in conducting the analysis, however, 

the explicit modeling of parameter uncertainty provides a quantitative estimate of how these 

uncertainties affect the outcome of interest. This information can be used to estimate the degree 

of conservatism in the decision, given the assumptions inherent in the model (a conservative 

transport model in the example) and what is known about the site (measurements of the site 

groundwater flow and transport characteristics in the example). This focus on the outcome is 

appropriate. As seen in the example, a focus on the degree of conservatism in the model 

parameters can lead to an overly conservative outcome.

The consideration of the degree of conservatism contributes to a risk-informed analysis.

Consistent values of the degree of conservatism can be used as a matter of policy. As shown in 

Figure 4, for example, the concentration ratio corresponding to a 5% probability of exceedance 

(the 95th percentile) is 1.09. The ratio corresponding to a 10% probability of exceedance (the 

90th percentile) is 0.25. One of these values may be appropriate for use in a safety evaluation. 

Given a mixture of radionuclides, each with its own travel time distribution, the probability of 

exceedance for the entire mixture could be easily computed, from which the probability of 

exceeding a concentration ratio of 1.0 is easily determined. This result could be evaluated for 

acceptability with due consideration of an acceptable degree of conservatism. 

0.01 0.1 1 10

0

0.15

0.1

0.05

Concentration/ECL

Prob Exceedance
2-10

The example discussed here was for an accidental release of radioactive liquid effluents in 

groundwater, but the concepts apply to other hydrologic analyses and outcomes. For example, 

the probability of exceeding the design basis flood level or groundwater elevation could be 

similarly addressed in the safety evaluation. Ultimately, probabilistic outcomes and 

consequences could be combined to include hydrology in a probabilistic risk assessment (PRA). 

For flood analyses, including uncertainty in the probability distribution used to represent flood 

frequencies is likely to be important. This is analogous to including uncertainty in the 

distributions of Figure 2 from the groundwater transport example.
3-1

3 UNCERTAINTY METHODOLOGY AND SOFTWARE

The previous section illustrated the potential benefits of a relatively simple consideration of 

uncertainty in the application of a groundwater transport model. A more general methodology for 

uncertainty evaluation has the following desirable characteristics.

• Comprehensive: it should consider many sources of uncertainty while providing a means to 

focus on those of most relevance.

• Quantitative: it should be possible to compare results with regulatory criteria or design 

requirements.

• Systematic: it should be applicable to a wide range of sites and objectives and enable the 

common application of computer codes and methods.

• Practical: it should allow for the use of a simple approach first and it should use existing 

software tools when possible.

Meyer et al. (2007) described an uncertainty evaluation methodology that exhibited the first 

three characteristics. In this report we demonstrate that the methodology is also practical for 

application to groundwater modeling for environmental and safety evaluations associated with 

nuclear power plant licensing. This is accomplished by illustrating some simplifications in the 

methodology and describing the capabilities of available software that can be helpful in 

implementing certain aspects of the methodology.

The framework in which hydrogeologic models are applied to nuclear power plant evaluations is 

illustrated in Figure 5. Site characterization and data gathering provide information in the 

history-matching period that can be used to develop and evaluate models. While some historical

data may be applicable, it is not uncommon for site-specific measurements of hydrologic 

behavior (such as groundwater heads) to cover a period of time as short as one year, as 

indicated in the figure. Uncertainty is assessed in the history-matching period based on a 

combination of expert judgment and a quantitative measure of the ability of the model to 

reproduce observations. Many of these uncertainties can be reduced to some extent by 

collecting additional data, but as noted previously this is not always a practical option.

Models are applied in the predictive period to evaluate site behavior under post-construction 

conditions with regard to regulatory requirements and design criteria. Models used in the 

predictive period are derived from the models developed in the history-matching period, but may 

have significant differences. For example, history matching is conducted under pre-construction 

conditions, but estimates of maximum groundwater head are required under post-construction 

conditions. Construction of a nuclear power plant will substantially modify local conditions, likely 

including site topography, recharge, and subsurface materials.

The modifications may affect site groundwater heads and transport pathways. The predictive 

period generally extends over the regulatory period of the facility, potentially up to 80 years for a 

nuclear power plant. The predictive period may be longer if this is required to capture important 

temporal effects: for example, the impact of plant operations on downstream water use under 

variable climatic conditions.
3-2

Figure 5. Framework for application of hydrogeologic models. The relatively short 

history-matching period involves comparison between observations (points) 

and model simulated equivalents (solid line). The much longer predictive 

period involves comparison between model simulation results and regulatory 

or design criteria (e.g., a design maximum groundwater head).

The uncertainties determined in the history-matching period are propagated into the predictive 

period with the addition of uncertainties that apply only in the predictive period (for example, 

uncertainties in the properties of the fill materials used in construction). Because the predictive 

period characterizes unobservable future conditions, some of these uncertainties are irreducible 

characteristics of the system being modeled (for example, recharge with modified surface 

conditions and subsurface materials).

The uncertainties addressed by Meyer et al. (2007) fell into three categories.

Conceptual model uncertainty. As described by Neuman and Wierenga (2003), the 

conceptual model is a hypothesis about the behavior of the system being modeled and the 

relationships between the components of the system. It is typically represented mathematically 

to render quantitative predictions (conceptual-mathematical model). Uncertainties in the 

conceptual model are generally assessed in the history-matching period and applied in the 

predictive period. However, there may be unique features of the predictive model that introduce 

additional uncertainties to the predictive period conceptualization.

Parameter uncertainty. Parameters are quantities required to obtain a solution from the 

mathematical model and are thus model-specific. They generally represent a characteristic of 

the site (such as the saturated hydraulic conductivity), but may be only indirectly related to 

measurements of those characteristics due to differences between the scale of measurements 

and the scale represented by parameters, and due to the potential dependence of parameter 

values on model assumptions. Parameter uncertainties are generally assessed in the historymatching period and applied in the predictive period. However, there may be additional 

uncertainties that only apply to parameters of the predictive model.

Scenario uncertainty. A scenario is a future state or condition assumed for a system that 

affects the hydrology (e.g., post-construction changes in surface topography, recharge, and 

ground-water extraction). These types of uncertainties apply only to the predictive period.
3-3

3.1 Uncertainty Methodology Incorporating Parameter, Model, and Scenario 

Uncertainties 

This section summarizes a method, described fully in Meyer et al. (2007), to provide an optimal 

way of assessing predictive uncertainty with consideration of parameter, conceptual model, and 

scenario uncertainties. This method relies on the specification of a set of alternative models and 

scenarios and weights the alternative model results by a measure of the model probabilities and 

the alternative scenarios by a measure of the scenario probabilities. The method closely follows 

the description of Draper (1995).

The contribution of parameter uncertainty to a model-predicted output, ∆ , can be written as

( )

( )

, ,

,, , ( ,, )

k i

k ik k k i k

pM S

p M S p M Sd

∆ =

∆ ∫

D

D θθ D θ

(4)

where pM S (∆ k i , , D ) is the probability density function of ∆ conditioned on a specific model,

Mk , a set of measurements of system behavior (e.g., hydraulic head), D, and a specific 

scenario, i S . The model is characterized by a set of uncertain parameters, θk , with a distribution 

of ( ,, ) kk i pM S θ D .

Referring to the modeling framework of Figure 5, ∆ , Mk , θk , and i S apply in the predictive 

period. The data, D, are obtained from the history-matching period. Mk is possibly modified 

from a corresponding model in the history-matching period, Mk

′ , whose parameters, k θ′ , are

contained in θk . ∆ is conditioned on D because Mk

′ and k θ′ are conditioned on D in the historymatching period.

Model uncertainty is included by postulating a set of K alternative models that represent 

conceptual and structural uncertainties, and then calculating the weighted average of the 

models’ predictions of ∆ from Eq. (4). In doing so, each model must be capable of predicting 

the same output, ∆ , and each model’s parameter set, θk , in Eq. (4) must be conditioned 

(through history matching) on the same data set, D. The weighted average prediction is written 

as

( ) ( ) ( ) 1

, ,, ,

K

i ki ki

k

p S p M S pM S =

∆ =∆ D DD ∑ (5)

where the weights, pM S ( k i D, ), are the posterior probabilities of each model k conditioned on 

the data and the scenario. 

Scenario uncertainty is included in a similar manner by postulating a set of I alternative 

scenarios and calculating a weighted average of the predicted output.

( ) ( ) ( ) 1

,

I

i i

i

p p S pS =∆= ∆ D D ∑ (6)

Here the weights are prior scenario probabilities; they are not conditioned on the data because 

they represent a primarily subjective judgment regarding the likelihood of the specific scenario’s

conditions occurring in the future (Meyer et al. 2007).
3-4

The posterior model probabilities of the history-matching models are calculated using Bayes’ 

theorem. 

( ) ( ) ( )

( ) ( ) 1

| ,

|

k ki

k i K

l li

l

p M pM S pM S

p M pM S =

′ ′ ′ =

∑ ′ ′

D

D

D

(7)

where pM S ( k i ′ | ) is the subjective, prior model probability for model Mk

′ (for a given scenario i S ) 

and p M (D k

′) is the model likelihood for Mk

′ (which is independent of the scenario as explained 

in Meyer et al. [2007]). The posterior model probabilities of the predictive models, pM S ( k i D, ) in 

(5), are then assumed to be equivalent to the corresponding values in (7),

pM S pM S ( ki ki D D , , ) = ( ′ ). (8)

When Eqs. (7) and (8) are applied to Eq. (4), the resulting method is called Bayesian Model 

Averaging (Draper 1995; Hoeting et al. 1999).

The calculations involved in Eqs. (4) to (7) were simplified for this report. First, Eq. (4) is 

approximated as

( ) ( ) ˆ , , ,,, k i k ik pM S pM S ∆ ≈∆ D D θ (9)

where ˆ

k θ is an estimate of θk and can be written as

ˆ ˆ ˆ

k

k

k

+

 ′ =      

θ

θ

θ . (10)

Here ˆ

k θ′ is the least squares estimate of k θ′ (the parameters of the history-matching model Mk

′ ) 

and ˆ

k

+ θ is the prior estimate of any additional parameters applying only to the predictive period 

model Mk . Second, instead of calculating complete probability distributions, results were 

obtained only for the mean and variance of the predicted value. Given Eq. (9), for an individual 

model, the estimated moments are 

( ) ˆ , , k

EM S ∆ =∆ k i θ D (11)

, ,

T

k i

k k

LS

k Var M S   ∂∆ ∂∆ ∆  =     ∂ ∂  

D Σ

θ θ (12)

(see Hill and Tiedeman 2007). The mean predicted value is the prediction evaluated at ˆ

k θ . The 

variance is estimated from prediction sensitivities k

  ∂∆   ∂  θ

evaluated at ˆ

k θ and the parameter 

covariance matrix 
3-5

0

0

LS

LS k

k LS

k

+

  ′ =    

Σ

Σ

Σ . (13)

Here

2 1 ˆ ( ) T

LS k k

LS

k σ − Σ J ωJ ′ = (14)

is the parameter covariance matrix from the regression used to estimate the parameters of the 

history-matching model,

T

2

ˆ

ˆ

k k

LS N P z k

σ

=

= − θ θ

e ωe is the least squares estimate of the calculated error 

variance, T e ωe is the sum of squared weighted residuals (SSWR) from the regression ( e being 

the vector of residuals and ω being the weight matrix), Nz is the number of data in D, Pk is the 

number of estimated hydrologic parameters in k θ′ , and k J is the Jacobian, or sensitivity, matrix 

from the regression as evaluated at ˆ

k θ . LS

k

+ Σ in Eq. (13) is the prior covariance for the additional 

parameters of the predictive model, k

+ θ .

The first two moments of Eq. (5) are given by

[ ] [ ] ( ) 1

, ,, ,

K

i ki k i

k

E S E M S pM S

=

∆ =∆ DD D ∑ (15)

[ ] [ ] ( )

( [ ] [ ]) ( )

1

2

1

, ,, ,

,, , ,

K

i ki k i

k

K

k i i k i

k

Var S Var M S p M S

E M S E S pM S

=

=

∆= ∆

+ ∆ −∆

∑

∑

D DD

D DD

(16)

while the moments of Eq. (6) are given by

( ) ( ) ( ) 1

,

I

i i

i

E E S pS

=

∆= ∆ D D ∑ (17)

( ) ( ) ( )

[ ( ) ( )] ( )

1

2

1

,

, .

I

i i

i

I

i i

i

Var Var S p S

E S E pS

=

=

∆= ∆

+ ∆ −∆

∑

∑

D D

D D

(18)

Calculations are also simplified by estimating the model likelihoods in Eq. (7) using 

( ) 1

exp 2 k k p M IC   ′ = −    D (19)

where k IC is one of a number of model selection criteria that have been proposed and 

evaluated for use in hydrologic analyses (Neuman 2003; Ye et al. 2004; Poeter and Anderson 

2005; Tsai and Li 2008; Ye et al. 2008, Singh et al. 2010). For this report, two model selection 

criteria were used,

2 2 ( 1) ln 2 ˆ 1

k k

k z ML k

z k

N N AICc N P

N N

σ

+

= ++ − − (20)
3-6

2 ˆ ln 2ln ( ) ˆ

ln 2 ln | |

k z ML k

ML

k k

KIC N p

P

σ

π

= − ′

− − ′

θ

Σ

(21)

as presented in Ye et al. (2008). Here, 1 N P k k = + , 2 2 ˆ ˆ z k

ML LS

z

N P

N

σ σ − = is the maximum likelihood 

estimate of the calculated error variance, ˆ( ) k p θ′ is the prior parameter probability density 

evaluated at ˆ

k θ′ , and z k

z

ML LS

k k

N P

N

− Σ Σ ′ ′ = is the maximum likelihood parameter covariance matrix.

In computing the model likelihoods and model probabilities using Eqs. (19) and (7), only one of 

the model selection criteria in Eqs. (20) and (21) should be used. The absolute values of the 

criteria are not meaningful; it is only the differences between values that are important. Because 

of the exponential form of Eq. (19), two alternative models with a difference in AICc (or KIC) 

values of more than six or seven implies dominance of one model over the other (i.e., the 

probability of the less likely model will be a small fraction of the high likelihood model).

3.2 Implementation of the Uncertainty Methodology

To implement the uncertainty methodology described above requires two basic activities: model 

development and parameter estimation/uncertainty calculation.

Model development. Two or more alternative models must be developed, that together

represent the unresolved conceptual and structural uncertainties associated with the 

representation of the site. Model development in the context of the uncertainty methodology 

described above involves the following.

• Each model must describe hydrologic conditions in the history matching period and must be 

capable of simulating the observations used in the parameter estimation.

• Each model, possibly with modification, must be capable of representing the predictive 

period and must be capable of simulating the desired predictive outputs. 

• Each model, possibly with modification, must be capable of representing each alternative 

scenario.

• A common dataset of observations from the history-matching period must be used for 

parameter estimation. Errors in these observations must be characterized by a (possibly 

diagonal) covariance matrix, which is used to determine weights for the parameter 

estimation.

• Prior model and scenario probabilities (possibly uniform) must be specified, usually based 

on expert judgment.

• If prior parameter estimates are used, the prior distributions for these estimates must be 

specified.

Parameter estimation & uncertainty calculation. Following model development and 

specification of scenarios, the following calculations are required to implement the uncertainty 

methodology.

• Parameter estimates for the history-matching models are calculated for Eq. (9) using 

weighted, least squares regression and parameter uncertainties are calculated using Eq. 

(14).

• Regression results for each model are used in Eq. (19) and (20) or (21) to calculate model 

likelihoods. Model probabilities are calculated from Eq. (7) using the model likelihoods (and 
3-7

prior model probabilities) from the set of alternative models characterizing the historymatching period.

• Predictive period models are used to calculate predictive values and uncertainties from Eqs. 

(11) and (12). The latter involves calculation of prediction sensitivities.

• Model- and scenario-averaged predictions and prediction uncertainties are calculated using 

Eqs. (15) to (18).

3.2.1 Software Tools

This section primarily discusses software tools that were used in the application described in the 

following chapter; this section is not intended to be a review of available software. Failure to 

describe any particular software tool here does not imply a judgment about its applicability or 

usefulness. 

Model development is site-specific and labor-intensive, requiring the interjection of experience

and subjective decisions at many points in the process. This makes it difficult to apply standards 

or generalizable procedures, although some guidance does exists (e.g., ANS 2010; ASTM 

2008; ASTM 2010). As discussed previously, for best results model development should be 

integrated with site characterization. However, specific tools or methods to complete this 

integration (e.g., to determine what additional data would be most valuable) are not discussed 

here.

Model development typically is based on a set of diverse data obtained during site 

characterization and monitoring. In addition, modeling and other analyses can generate 

substantial amounts of data. Management of this data involves documentation of sampling and 

analysis methods, data quality, and data uncertainties. With respect to a nuclear power plant 

application, data management is primarily the responsibility of the applicant. To clearly 

document the technical basis for their conclusions, however, reviewers should carefully manage 

any data used or generated in the application review. Spreadsheet software1 was the primary

data management tool used in the application described in the following chapter.

Conceptual model development typically involves preparation of figures to aid in interpretation of 

data (e.g., stratigraphic fence diagrams based on borehole data, time series of groundwater 

head measurements, etc.) and data analysis to derive parameter estimates (e.g., saturated 

hydraulic conductivity estimates from pump or slug test data) and to characterize uncertainties. 

Again, documentation of the conceptual model is primarily the responsibility of the applicant. For 

the application described here, some multipurpose software was used in developing the 

conceptual model (spreadsheet and Mathcad2

), as was the specialized modeling software, 

Groundwater Modeling System (GMS3

). Global Mapper (GM 2005) was used to prepare spatial 

GIS data for use in GMS. The Corpscon utility (USACE 2004) was used to convert point data 

from one coordinate system to another, also for use in GMS.

The conceptual model serves as the basis for a mathematical/numerical model. Implementation 

of a numerical model requires specification of boundaries, spatial and temporal discretization, 

parameter values, and initial conditions, among other activities. GMS was used as the principal 

tool to translate the conceptual model into a numerical model. GMS was also used as the 

 

1 Excel 2007/2010, ©2006/2010 Microsoft Corporation, Redmond, Washington. 2 Mathcad v. 15.0, ©2010 Parametric Technology Corporation, Needham, Massachusetts.

3 GMS v. 7.1.10, ©2012 Aquaveo LLC, Provo, Utah.
3-8

primary user interface to the model codes, several of which were used in the application: 

MODFLOW-2000 (Harbaugh et al. 2000; McDonald and Harbaugh 1988) for groundwater flow, 

MODPATH (Pollock 1994) to evaluate pathways, and MT3DMS (Zheng and Wang 1999; Zheng 

et al. 2001; Zheng 2006) for groundwater transport. 

Model development also includes a process of model evaluation whereby the characteristics of 

the model (e.g., model assumptions, discretization, boundary conditions, etc.) are evaluated and 

modified to better represent what is known about the site and to eliminate or reduce model 

errors (e.g., mass balance errors, unrealistic heads, etc.). Model evaluation was primarily 

carried out within the GMS software.

Parameter estimation and uncertainty calculation are more amenable than model 

development to the application of generalizable procedures. Nonetheless, there are many 

subjective considerations that arise in their implementation, as discussed in the guidelines of Hill 

and Tiedeman (2007). Parameter estimation and uncertainty calculations can also be 

computationally demanding, which sometimes generates a tension between completing these 

tasks and increasing the complexity of a model. 

A number of software tools were used in the application described in the following chapter to 

carry out parameter estimation and uncertainty calculations. Both PEST (Doherty 2004) and 

UCODE (Poeter et al. 2005) were used to estimate parameters and parameter uncertainties (as 

linear confidence intervals and parameter covariances). These codes also provide the model 

selection criteria values, although for reasons explained in Appendix A, these values were 

computed separately using a spreadsheet.

Predictive uncertainties were evaluated using utility software associated with PEST and 

UCODE. As described in Doherty (2010) a variety of utilities are distributed with PEST that 

implement the predictive error methods of Moore and Doherty (2005). For the application here, 

the utility “predunc6” was used. For UCODE, the LINEAR_UNCERTAINTY companion code 

(Poeter et al. 2005) was used to compute predictive uncertainty.

Poeter and Hill (2007) document MMA, a code that uses output produced by UCODE and 

LINEAR_UNCERTAINTY to compute model probabilities and model-averaged predicted values.

Because MMA does not directly read PEST output, and for reasons related to the computation 

of model selection criteria (see Appendix A), model probabilities and model-averaged predicted 

values were calculated in a spreadsheet for this report. 

3.2.2 Software Capabilities and Limitations

Some of the capabilities and limitations of the software used in the application described in the 

following chapter are summarized here. Those features of the software most relevant to the 

analysis of model and parameter uncertainty are listed here. 

GMS

Capabilities

• The GMS Map module facilitates alternative conceptual model development.

• PEST is integrated.

• A Monte Carlo simulation capability is integrated.

Limitations

• PEST is only implemented for use with MODFLOW.

• The set of MODFLOW parameters that can be estimated with PEST in GMS is limited.
3-9

• ParallelPEST, the multi-core/multi-processor version of PEST is not implemented in the 

federal agency distribution of GMS v. 7.1.

• The use of an HDF file for storing model input makes it more difficult to work with files 

outside of GMS.

• There are limited Monte Carlo simulation options (e.g., correlations between parameters 

is not implemented).

PEST

Capabilities

• Nonlinear regression uses the Levenberg-Marquardt method.

• Singular value decomposition and regularization techniques for highly-parameterized 

systems are available.

• Global optimization methods are implemented.

• A large number of utilities are available for file preparation and manipulation, 

groundwater-specific applications, and predictive error and uncertainty analysis.

• Model selection criteria values are calculated.

Limitations

• The large number of utilities and extensive capabilities can be overwhelming; 

documentation is currently not tightly integrated.

• Predictive uncertainty is not well-integrated with the basic operation of PEST.

• KIC values don’t consider the impact of parameter transformation on the parameter 

covariance term in Eq. (21) (see Appendix A).

UCODE

Capabilities

• Nonlinear regression uses the Levenberg-Marquardt method.

• Utilities for residual analysis, model linearity evaluation, and linear uncertainty estimates

are tightly integrated.

• Utilities are available for analysis of highly nonlinear models.

• Model selection criteria values are calculated.

• Results are readable by MMA for multi-model analysis.

Limitations

• A limited set of methods is currently implemented.

• KIC values don’t consider the impact of parameter transformation on the parameter 

covariance term in Eq. (21) (see Appendix A).

4-1

4 EXAMPLE APPLICATION

This chapter presents an example application of the uncertainty methodology presented above. 

The example is for groundwater analyses related to the safety review of a nuclear power plant 

application, such as might be carried out in review of the Final Safety Analysis Report (FSAR) 

Section 2.4.12. The example uses publically available information from the Bellefonte Nuclear 

Power Plant Units 3 and 4 Combined Operating License (COL) application, namely, the FSAR 

and the Environmental Report (ER) submitted as part of the application. 

The objectives of the example are to demonstrate the application of the uncertainty 

methodology to a relevant site and to illustrate the type of results that can be expected. We also 

show what is required to evaluate uncertainties using existing software tools and we comment 

on areas for potential improvement. The example uses a set of groundwater models. No 

groundwater model was submitted as part of the Bellefonte application and no model is referred 

to in FSAR 2.4.12. The groundwater models developed for this example and described below 

are solely the product of the coauthors of this report. Their use here implies neither suitability of 

the models for use in an actual review of the Bellefonte application nor approval by the NRC 

staff for such a purpose.

While the uncertainty methodology described above addresses the joint consideration of 

parameter, model, and scenario uncertainties, the implementation described here is limited to 

the consideration of parameter and model uncertainties. Including alternative scenarios requires 

additional simulation in the predictive period and averaging across scenarios [see Eqs. (17) and 

(18)], but does not require any other calculations since the scenario probabilities are subjective, 

prior values. See Meyer et al. (2007) for additional discussion.

4.1 Building the Groundwater Model

The site of the proposed Bellefonte Nuclear Power Plant Units 3 and 4 is located along the 

Tennessee River near Scottsboro, Alabama, in Jackson County (referred to hereafter as the 

Bellefonte site). The site description and available data are documented in the COL application 

located on the NRC website (http://www.nrc.gov/reactors/new-reactors/col/bellefonte.html): 

specifically, the FSAR, Rev. 1, and the ER, Rev. 1. The proposed Units 3 and 4 are located 

adjacent to existing Units 1 and 2 reactor buildings, which were partially constructed in the 

1980s, but never completed. The location of the site is shown in Figure 6.

The first question to ask when preparing to build a groundwater model is, “What will this model 

be used for?” Identifying the purpose of the model and the kinds of questions that model 

simulations will be expected to address is important for defining the overall model configuration.

If a model is used for regional-scale assessments, then the inclusion of local-scale specifics will 

be less important. If the model is for designing an in-situ remediation system, then a fine-scale 

model resolution may be needed for describing the flow field specifics and for solute transport.

For a groundwater review following SRP Sections 2.4.12 and 2.4.13 (NRC 2007), the model will 

be used to evaluate groundwater heads, transport pathways, and possibly contaminant 

concentrations. This example emphasizes the use of a model to estimate the maximum postconstruction groundwater heads in the power block area.

The process of building a groundwater model is depicted in Figure 7. Initially, one must 

determine the objectives of the modeling and assemble the set of data that will be used to 
4-2

Figure 6. Location of the Bellefonte site. Lower image taken from FSAR Figure 2.1-201 

(TVA 2009).

Bellefonte

Alabama 
4-3

Figure 7. Steps in building a MODFLOW groundwater model

construct the model. A basic conceptual site model is developed from the data. The model 

framework (lateral grid, layers, domain extent, etc.) is defined based on knowledge of 

hydrogeology and the modeling objectives. Once the framework is determined, the groundwater 

model parameters and boundary conditions can be specified. The groundwater model then 

requires calibration with either trial and error adjustment of parameter values or an automated 

optimization process. For this example, automated calibration using least squares regression 

was used to provide the required parameter uncertainty estimates. The calibrated model can 

then be used in predictive simulations, typically after making adjustments (e.g., to reflect postconstruction conditions). The process of implementing each of these steps with respect to 

modeling the Bellefonte site is described in subsequent sections of this report.

4.1.1 Gather Relevant Data

The first step in building and configuring a numerical model for simulating groundwater flow with 

MODFLOW is to gather the relevant site-specific data. Figure 8 depicts the types of data that go 

into defining a groundwater flow model. Many of the necessary data are quantitative (elevations, 

distances, flow rates, aquifer properties, etc.), but some of the input is qualitative.

The nature of the model (lateral grid resolution, number of model layers, domain extent, 

temporal nature, etc.) must be balanced between meeting the objectives of the modeling, the 

computational constraints associated with the simulations (i.e., memory, run time) and the 

resolution of the available data. The extent of the model domain may also be influenced by the 

locations of natural boundary conditions. Keeping this in mind, the data elements of Figure 8

were gathered and examined for the Bellefonte site.

Gather Relevant Data

Solids Model

MODFLOW

Domain

Definition

MODFLOW Model Structure

(Materials, Elevations, Grid Discretization)

Triangulated

Irregular

Networks

Boreholes

& Cross

Sections

Configure the MODFLOW Model

Calibrate the Model

(Boundary & Initial Conditions, Hydraulic Stresses, Temporal 

Discretization, Run Options [calculation, solver, output, etc.])

Run Predictive Simulations
4-4

4.1.1.1 Site Locale and Topology

Photographic images provide a basis of reference for locating geographic features and 

buildings. Digital orthographic quarter quad (DOQQ) imagery from 2006 was downloaded from 

the Alabama State Water Program (ASWP 2006) website for the following named quads (with 

quarter quad indicated):

 Henagar (NW and SW)

 Hollywood (NW, NE, SW, and SE)

 Stevenson (SW)

 Wannville (SW and SE)

The DOQQ imagery (Figure 9) was provided in the Universal Transverse Mercator (UTM) 

projection for zone 16 relative to the North American Datum of 1983 (NAD83) in units of meters.

The State Plane Coordinate System (SPCS) for Alabama East (Zone 101) relative to NAD83 in 

units of feet was selected for this work, so Global Mapper was used to transform the imagery to 

the SPCS prior to exporting a zone around the Bellefonte Nuclear Plant. The export process 

creates a world file with a file extension of *.jgw corresponding to the current coordinate system 

(combination of projection, datum, and units). The exported image was read into GMS, which 

used the world file to automatically register the image to the appropriate coordinates. Images 

are found listed in the Images module of GMS. Note that multiple images can be loaded into 

GMS and are treated as layered images with the first image loaded at the back and the most 

recent image loaded on top. The georeferenced DOQQ provides a basis for registering nongeoreferenced image files (e.g., maps copied from PDF documents) in that coordinates of 

specific features (e.g., buildings, road inter-sections) can be determined and mapped to specific 

pixels of the non-georeferenced image. This approach was used to register several images 

(e.g., hydraulic head contours, construction diagrams) in GMS.

A USGS topographic map (digital raster graphic [DRG]; Figure 10) of the Hollywood quad, 

georeferenced to the Universal Transverse Mercator (UTM) projection relative to the North 

American Datum of 1927 (NAD27) in units of meters, was also downloaded from the ASWP

website. Topographic maps provide additional detail about surface features, including streams, 

lakes/ponds, and topography. However, the topography information is not in a format conducive 

to applying in a numerical model.

Digital elevation model (DEM) data are a representation of topographic elevations as an array of 

numbers, and thus are well suited for use in numerical models. USGS 1/3 arc-second (10 m) 

DEM data were retrieved from the AlabamaView website (USGS 2009). The DEM data were

loaded into Global Mapper (Figure 11) and were automatically converted from a UTM projection 

relative to the World Geodetic System of 1984 (WGS84) in units of meters to the same 

coordinate system as the DOQQ imagery.
4-5

Figure 8. Depiction of data inputs used to configure a three-dimensional MODFLOW 

model for simulating groundwater flow in the subsurface

Figure 9. DOQQ imagery for the Bellefonte site vicinity. Coordinates are SPCS Alabama 

East, in feet

Geological Information

(borehole locations, 

geological unit contacts)

Number

Top of 

Weathered 

Rock Elev (ft)

Top of 

Competent 

Rock Elev (ft) NV/USR USR/A A/B B/C C/D D/E E/F F/LSR

B-1005 598 592.9 465 444 394

B-1032 592 581.8 513 492 443

B-1034 608 601 579 454 385

B-1059 605.5 593.5 569 500

B-1066 616.9 587.4 583 518

B-1067 605.6 604.9 574 506

Stratigraphic Unit Contacts Elev from Borehole Logs (ft)

0

10

20

30

40

50

60

70

80

90

1955 1965 1975 1985 1995 2005

Precipitation (in./yr)

Year

Recharge Information

(precipitation, 

land cover/use)

Monitoring

Point

Northing

(ft)

Easting

(ft)

Reference

Elevation

(ft. msl)

Ground

Elevation

(ft. msl)

Well 

Depth

(ft. bre)

Screen 

Length

(ft.)

Top of

Screen

(ft. msl)

Bottom

of Screen

(ft. msl)

MW-1201a 1532950 628276.6 613.91 611.05 12.91 5 606.45 601.45

MW-1201b 1532950 628276.6 613.78 611.04 77.81 10 546.42 536.42

MW-1201c 1532950 628276.6 613.65 610.91 119 20 515.1 495.1

MW-1202a 1532571.1 629135.1 617.52 614.99 15.42 5 607.55 602.55

MW-1202c 1532571.1 629135.1 617.62 614.93 53 10 575.07 565.07

MW-1203a 1532718.7 628698.8 621.93 619.02 12.57 5 614.81 609.81

MW-1203b 1532718.7 628698.8 621.86 619.14 32.9 10 599.41 589.41

MW-1203c 1532718.7 628698.8 621.7 619.04 121 20 521.15 501.15

Hydraulic Information

(well locations, hydraulic head,

river flow, river/lake stage)

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-4

-2

0

2

4

6

8

10

0 1000 2000 3000 4000

Obs. Well Drawdown (ft)

Pumped Well Drawdown (ft)

Elapsed Time (min)

Aquifer Information

(conductivity, porosity, 

pump tests, sediment analysis)

Topography Information

(DEM data)

Reference Information

(DOQQ/Map Imagery)

Groundwater Model

Bellefonte
4-6

4.1.1.2 Site Hydrogeology

Site hydrogeology is a key component for a groundwater flow model in that it delineates 

geological features and hydraulic units for groundwater movement. The hydrogeologic setting of 

the Bellefonte site is the Valley and Ridge aquifer system in the southern portion of the 

Sequatchie Valley, a northeast trending, eroded anticline (Miller 1990). The thrust faulting and 

folding that produced the anticline resulted in fracturing of the sedimentary rock formations and 

led to the erosion of the overlying sandstone-shale units and karst formation in the underlying 

limestone units. A conceptual depiction of the hydrogeology of the Sequatchie Valley is shown 

in Figure 12, taken from Miller (1990). A geologic map of the region surrounding the Bellefonte 

site and a representative cross-section taken south of the site, both shown in Figure 13, 

illustrate the location of the site on the southeasterly dipping side of the Sequatchie anticline.

The Sequatchie fault is far enough to the northwest to be outside the area of interest.

The relevant formation descriptions from Irvin and Dinterman (2009) are:

• Nashville Group: partly argillaceous wackestone/packstone with interbedded lime mudstone. 

Some intervals are abundantly fossiliferous. 

• Stones River Group: Lower part is partly argillaceous lime mudstone and wackestone that is 

locally fossiliferous. Upper part is thin- to thick-bedded, fossiliferous limestone with rare very 

fine grained calcareous sandstone. Laminated bentonitic shale is near the upper contact. At 

the base, locally, is sandy chert conglomerate (Pond Spring Formation).

• Knox Group—finely crystalline, siliceous dolomite and minor locally silty limestone. At the 

surface, the Knox weathers to residuum with abundant, predominantly dense chert that 

generally preserves primary sedimentary features.

Figure 10. Topographic map for the Bellefonte site vicinity
4-7

Figure 11. DEM data in the vicinity of the Bellefonte site (as depicted in Global Mapper)

Figure 12. Conceptual depiction of hydrogeology in a cross-section of the Sequatchie 

Valley, the location of the Bellefonte site (from Miller 1990)

Additional information pertaining to the geology of the site is contained in TVA (2009). FSAR 

Figures 2.5-307 and 2.5-308 show photos of the inclined geology taken during construction of 

Bellefonte Units 1 and 2 (not shown here). FSAR Figures 2.5-339 to 2.5-341 show cross 

sections of interpreted geologic bedding planes through the location proposed for Units 3 and 4 

(part of Figure 2.5-339 is shown in Figure 14 below). In a cross section perpendicular to a strike 

of N45°E, FSAR Fig. 2.5-306 shows the stratigraphic contacts between the Lower Stones River, 

Middle Stones River (which has multiple subunits termed A through F), and Upper Stones River 

units. For this work, contacts were based on the borehole logs in FSAR Appendix 2BB (listed in 

Table B.1 of Appendix B of this report). Elevations of the top of the weathered rock come from 

the Standard Penetration Test (SPT) borehole data; top of competent rock was based on the 

Rock Quality Designation (RQD) and borehole log notes on rock weathering.

Bellefonte
4-8

Figure 13. Geological units in the vicinity of the Bellefonte site. Adapted from Irvin and 

Dinterman (2009).

4.1.1.3 Aquifer Properties

The key aquifer property for modeling groundwater flow is the hydraulic conductivity of the 

subsurface materials. Table 1 lists data as reported in TVA (2009) for the Bellefonte Units 1 and 

2 site and from site investigation tests conducted in 2006 for Units 3 and 4. Average values 

(both mean and geometric mean) were calculated for that data. The materials of interest are 

Soil, Weathered Rock, and Competent Rock (i.e., bedrock, in the form of the Nashville, Upper 

Stones River, Middle Stones River, and Lower Stones River units).

To simulate solute transport, the porosity of the subsurface materials (Table 2) and adsorptionrelated properties are required. Potential radionuclide contaminants and their linear equilibrium

Scale 1:24,000, no vertical exaggeration in cross section

Surficial deposits not shown in cross section

A

A'

A A'

Knox Group

Sequatchie Formation (and other)

Red Mountain Formation Bangor/Monteagle Limestones

Nashville and

Stones River Groups
4-9

Figure 14. A geologic cross-section through Units 3 and 4 showing rock units (taken 

from FSAR Figure 2.5-339, TVA 2009)

Table 1. Hydraulic conductivity (Ks) for subsurface materials near the Bellefonte site. 

Minimum and maximum values reported in TVA (2009); midpoints and 

averages were calculated

Material Min. Ks

(cm/s)

Max.Ks

(cm/s)

Midpoint of 

Range

(cm/s [ft/d])

Geometric 

Mean of 

Max./Min.

(cm/s [ft/d])

Data Source

Soil 1.00E-08 1.00E-06 5.05 E-07

[1.43 E-3]

1.00 E-07

[2.83 E-4]

FSAR for Bellefonte Units 

1 and 2

Bedrock

4.00E-06 3.80E-03 1.90 E-03

[5.39]

1.23 E-04

[0.349]

FSAR for Bellefonte Units 

1 and 2, borehole packer 

tests

2.50E-05 4.20E-03 2.11 E-03

[5.99]

3.24 E-04

[0.919]

Units 3 and 4, 2006 site 

investigation, borehole 

packer tests

6.11E-07 3.95E-03 1.98 E-03

[5.60]

4.91 E-05

[0.139]

Units 3 and 4, 2006 site 

investigation, MW-1203b 

pump test

Bedrock (All Data) 2.00 E-03

[5.66]

1.25 E-04

[0.355]

Mean/Geometric Mean of 

all six Max./Min. values for 

Bedrock

Weathered Rock 9.99 E-04

[2.83]

3.54 E-06

[0.010]

Mean/Geometric Mean of 

the Soil

and Bedrock (All Data) 

Averages
4-10

Table 2. Porosity for subsurface materials in the vicinity of the Bellefonte site

Material Interval Below 

Bedrock Surface Void Ratio Total Porosity Data Source

Soil — — 0.45 TVA (2009)

Bedrock — — 0.005 TVA (2009)

Bedrock

0-5 ft 0.01378 0.0136 Void ratios computed from information in Attachment A of Request for Additional Information 

(RAI) Response (ML082070075a

) 

on cavity and total core lengths; 

porosities computed from void ratios

5-10 ft 0.02712 0.0264

10-20 ft 0.01504 0.0148

20-30 ft 0.00191 0.00191

a

Available at http://adams.nrc.gov/wba/

partitioning coefficients (Kd) are described in FSAR Section 2.4.13 (TVA 2009). The average dry 

bulk density (ρb) of the soil is given in FSAR Section 2.4.12 (TVA 2009) as 90 lb/ft3 (1.44 g/cm3

), 

which can be used with the Kd and porosity values to calculate the retardation factor for solute 

transport.

4.1.1.4 Precipitation and Land Usage

Historical precipitation data (PRCP) were retrieved (from 

http://www1.ncdc.noaa.gov/pub/data/ghcn/daily/, or alternatively, from 

www.ncdc.noaa.gov/cdoweb/search#t=secondTabLink) for the Scottsboro, Alabama Global 

Historical Climatology Network (GHCN) (Menne et al. 2012) station USC00017304 (U.S. 

Cooperative Network [COOP] station 017304), which is part of the U.S. Historical Climatology 

Network (HCN) and is located at latitude 34.6736° and longitude -86.0536° and an elevation of 

187.5 m above sea level. The average annual total precipitation for the period from 1955 to 

2009 is 53.4 in./yr (Figure 15).

FSAR Section 2.4.12.1.2 (TVA 2009) states that precipitation at the Bellefonte site “averages 

about 50 in. per year” and that “approximately 8 in. of this precipitation is estimated to enter 

groundwater storage.” The average precipitation per year is consistent with that calculated from 

the Scottsboro COOP station 017304 data.

Land use can be inferred from the satellite imagery (Section 4.1.1.1).

4.1.1.5 Hydraulic Information

A variety of hydraulic information for a site can be used to define boundary conditions or 

calibration targets. Where there are induced stresses, such as pumping wells or injection wells, 

the flow rate and time frames of the hydraulic stresses must be assembled. Similarly, river flow 

data (i.e., from river gages) are important for defining boundary conditions and/or for use as a 

flow calibration target. Certain rivers/streams or surface features may act as drains, resulting in 

loss of groundwater from the subsurface to the surface water (i.e., gaining streams, potentially a 

seasonal effect). Bodies of water often represent constant head boundary conditions (although 

the constant head may vary seasonally, which is a time-varying constant head). The regional 

hydraulic head distribution may also be used as a constant head boundary condition. Hydraulic 

head data from monitoring wells can be used to help establish boundary conditions, but are

most often used as calibration target data. For the Bellefonte site, the hydraulic information of 
4-11

interest pertains to gaining streams, constant hydraulic head regions, and observed monitoring 

well hydraulic head data.

The major surface water features at the Bellefonte site are Town Creek and the Tennessee 

River (Figure 6, Figure 10). Town Creek actually starts as a river/stream, but the portion most 

relevant to this work is essentially a lake (backwater of the Tennessee River) on the northwest 

side of the Bellefonte site. The Tennessee River runs along the southeast side of the Bellefonte 

site and constitutes, at this location, the Guntersville Reservoir. Figure 16 shows locations of 

surface water gaging locations on both Town Creek and the Tennessee River. Data are 

available in FSAR Table 2.4.12-204 for monthly observed surface water elevations from June 

2006 through May 2007. Average stage elevations based on this data for SW-1, SW-2, SW-3, 

and SW-4 are listed in Table 3 (SW-5 and SW-6 represent on-site ponds, not Town Creek). The 

overall stage elevation for Town Creek and the Tennessee river is 594.5 ft, which is essentially 

constant over a seasonal cycle.

Examining the topographic map of the Bellefonte vicinity (Figure 17), five streams are observed 

to discharge into Town Creek. Elevations of these streams were estimated from the topographic 

map and were used to define drainage features in the groundwater model. 

Figure 15. Total annual precipitation at COOP station 017304 in Scottsboro, Alabama 

from 1955 through 2009. The dashed line is the annual average over this time 

frame.

0

10

20

30

40

50

60

70

80

90

1955 1965 1975 1985 1995 2005

Total Annual Precipitation (in./yr)

Year

Average for 1955-2009: 53.36 in/yr = 0.0122 ft/d
4-12

Figure 16. Locations of surface water gaging locations. This figure shows a portion of 

FSAR Figure 2.4.12-213 (TVA 2009).

Table 3. Average stage observed at surface water measurement locations

Surface Water Gage

Location

Average Stage (ft) for

June 2006-May 2007 Data a

SW-1 594.5

SW-2 594.4

SW-3 594.6

SW-4 594.6

a Based on data in FSAR Table 2.4.12-204. The standard 

deviation for stage data at each location is about 0.3 to 0.4 ft.
4-13

Figure 17. Locations of streams that discharge into “Town Creek” (marked by the 

arrows)

Three data sets of observed groundwater hydraulic heads at monitoring wells were used for 

model calibration and to assist with boundary condition definition. These data sets included data 

from 1961 (FSAR Figure 2.4.12-204), from September 2005 (FSAR Figure 2.4.12-210), and 

from monthly observations (plus two extra events) between June 2006 and May 2007 (FSAR 

Table 2.4.12-204, TVA 2009). Bedrock wells that exhibited a slow response during development 

and wells listed as “dry” in FSAR Table 2.4.12-204 were excluded. Figure 18 shows a 

composite of portions of FSAR Figures 2.4.12-204 and 2.4.12-210, which indicate flow towards 

Town Creek. The 2006-2007 data are shown as a portion of FSAR Figure 2.4.12-215 (Figure 

19), also showing groundwater flow from the Bellefonte site to Town Creek. The groundwater 

head data used in the calibration process are listed in Table B.3 in Appendix B for these three 

data sets.

Because of the well locations, some of the data from 1961 were also used to define the 

boundary conditions along the northwest side of the model.

4.1.1.6 Conceptual Model Based on the Data

According to the conceptual model presented in FSAR Section 2.5 (TVA 2009), groundwater 

flow occurs in three stratigraphic regions: the overburden (residual soil), the epikarst zone 

(upper weathered rock), and the karst zone (bedrock). In this model, flow is predominately 

downward in the overburden with horizontal flow occurring in the epikarst and through 

solutionally enlarged joints and bedding plane fractures in the karst zone. Karst features 
4-14

identified in the site investigation were soft zones and cavities in soil, cavities in boreholes, and 

an irregular rock surface. A conceptual depiction of the karst environment is shown in Figure 20. 

The occurrence of observed cavities (those greater than 0.1 ft in thickness) decreased with 

depth. Most cavities were observed within 10 to 20 ft. of the top of rock. The deepest cavity was 

observed at 63 ft. below ground surface. Many of the cavities were soil-filled, which was 

attributed to the argillaceous character of the limestone. Thin shale layers observed in the 

boreholes were also stated to be a lithologic control on the flow of groundwater. 

Figure 18. Hydraulic head data and interpreted contours for the 1961 data (blue contours 

at west and north sides) and the September 2005 data (black contours in 

southeast quadrant), all overlain on the Bellefonte area imagery

Under the conceptual model of groundwater flow presented in FSAR Section 2.4.12 (TVA 

2009), during normal (dry) conditions, lateral groundwater flow is primarily through the fractures 

and joints of the partially-filled epikarst region. During periods of significant recharge (following 

heavy or frequent rainfall events), the epikarst fills and significant lateral flow takes place in the 

overburden soils. Flow in the karst bedrock zone is generally small. Groundwater occurrence in 
4-15

wells was inconsistent across the site, with dry wells or wells with minimal flow observed in each 

of the zones depicted in Figure 20.

Bedrock stratigraphy consists of basically parallel inclined units along a strike of nominally 

N45°E with a dip of about 15°SE. Units progress with depth (and from southeast to northwest) in 

the sequence of Nashville (NV), Upper Stones River (USR), Middle Stones River (MSR) 

subunits A through F, and Lower Stones River (LSR). While the elevated River Ridge hills to the 

southeast are comprised of Sequatchie, Red Mountain, and other limestones and shale, these 

units are represented here as Nashville material. This approach is reasonable because there is 

little basis for distinguishing properties between these River Ridge units, the groundwater flow of 

interest at the site is expected to be towards Town Creek, and representing the groundwater 

table in a layered hill provides challenges in model configuration (discussed in Section 4.1.2).

Figure 19. Example of recent hydraulic head contours interpreted from November 2006 

groundwater head data
4-16

Groundwater flow is expected to be sensitive to recharge and site topography and thus would 

be expected to flow towards Town Creek (from both the northwest and southeast) except for the 

southeast side of the River Ridge hills, where groundwater would flow towards the Tennessee 

River. The Tennessee River and Town Creek are natural hydraulic boundaries. However, the 

Sequatchie anticline was selected as the boundary location to the northwest instead of Town 

Creek to allow the model to address potential interaction between the site and groundwater 

users on the northwest side of Town Creek. 

Because of the regulation of the reservoir (Tennessee River), elevations in the reservoir and 

Town Creek are fairly constant in time. As a result, temporal variation in groundwater heads is 

primarily due to variation in recharge. Groundwater heads would be expected to vary with 

seasonal changes in recharge; such variation can be seen in some of the historical data (FSAR 

Figure 2.4.12-211, TVA 2009), but is not clearly evident in the monthly data collected more 

recently (FSAR Figure 2.4.12-218, TVA 2009) due to the short period of data collection.

Groundwater heads also respond to precipitation events. A rapid response to a precipitation 

event was observed between September 21st and 26th, 2006 in many of the monitoring wells at 

the site. Figure 21, the head response in well MW-1202a, screened in the soil unit, is typical. 

About 10 cm (4 in) of precipitation were recorded at Scottsboro, Alabama between September 

22nd and 24th, and the observed groundwater head increased about 5 ft (1.5 m) between 

September 21st and the 26th. For evaluating the maximum groundwater head around the power 

block, these data indicate that it is important to evaluate the unsteady behavior of the flow. 

Completing such an evaluation using the existing data would be difficult, however. Better aquifer 

pump test data, on-site precipitation data, and more frequent groundwater head measurements 

would be required. For this example, however, there is no opportunity to obtain this data (a 

situation often encountered in practice, as discussed above). As a result, a steady-state 

groundwater flow model was developed for this example. The uncertainty in the maximum 

groundwater head induced by this steady-state assumption was evaluated qualitatively, as 

described below in the section on predictive modeling.

4.1.2 Define the Geological/Model Structure

After the necessary data are gathered, the next step in building a model for groundwater 

simulation is to define the geological structure and the model grid structure.

4.1.2.1 Groundwater Model Domain

Based on the modeling objectives and the conceptual model, an initial model grid was selected 

with the properties shown in Table 4. The uniformly spaced model grid was aligned nominally 

orthogonal to the strike of the bedrock units, at a 45° rotation (Figure 22), with boundaries at the 

Tennessee River, the Sequatchie anticline, and covering most of the extent of Town Creek to 

the northeast and southwest. This placed the location of Bellefonte Units 3 and 4 nominally in 

the middle of the model domain. The vertical origin was (somewhat arbitrarily) selected at an 

elevation of 450 ft. The vertical distribution of layers was determined later in the process, but 

this depth was believed to be sufficiently large, while allowing fine enough vertical resolution in 

the grid. The selected number of grid cells in the three directions resulted in a total grid cell 

count of about a half million, which is reasonably detailed, while still allowing feasible simulation 

run times. This initial definition of the model domain provides a basis for building the geological 

model structure in terms of lateral and vertical extents.
4-17

Figure 20. Conceptual model of karst development at the Bellefonte site (from FSAR 

Figure 2.5-304, TVA 2009)

Figure 21. Groundwater head response to a precipitation event: about 10 cm (4 in) of 

precipitation were recorded at Scottsboro, Alabama from 9/22/06 to 9/24/06 

(Scottsboro 2 NE Experimental Station, downloaded 8/27/10 from 

http://www.ncdc.noaa.gov). Groundwater head data from FSAR Table 2.4.12-

204 for monitoring well MW-1202a screened in the soil unit (TVA 2009).
4-18

Table 4. Initial model grid description

Grid Information Units X direction

(columns)

Y direction

(rows)

Z direction

(layers)

Number of Cells — 135 120 31

Total Grid Length ft 13500 12000 148

Grid Cell Size

(i.e., width, thickness) ft 100 100 4.8

Origin Coordinates a,b ft 625800.0 1525300.0 450.0

a State Plane Coordinate System (Alabama East), NAD83 for lateral coordinates. Vertical 

elevations are relative to NAVD88. b Angle of rotation for the model grid = 45°.

4.1.2.2 Define Hydrogeological Units as Materials

Given the conceptual model of the site lithology/stratigraphy, a set of materials was defined in 

GMS (Figure 23). When materials are used, the properties for each material (e.g., hydraulic 

conductivity, porosity) and the zones where each material is present are specified. Working with 

the model is easier in that property values can readily be changed at a material level rather than 

on a grid cell-by-cell level. GMS translates the material properties into parameter values for 

each grid cell when creating the input files used by the MODFLOW code.

4.1.2.3 Build a Solids Model

There are two typical approaches to start the process of defining the geological structure of a 

site. One approach is to enter lithologic/stratigraphic contact information from borehole logs as 

“borehole data,” which can be organized in Excel then placed in an ASCII text file. It is often 

convenient to define a “dummy” stratigraphic material as the bottom-most material and to have 

a uniform bottom elevation (rather than the varying actual borehole bottom elevations). The 

borehole data text file can be imported into GMS via the file import wizard and the data sent to 

the GMS Boreholes module. Depending on the density and location of borehole data, it may be 

necessary to add “control point” boreholes (i.e., estimated/extrapolated lithology) such that the 

boreholes extend at least to the boundaries of the groundwater model domain (if not beyond). 

Once in GMS, the boreholes need to be connected by creating cross sections between pairs of 

boreholes. The cross sections should be created around the groundwater model domain 

boundaries and in multiple (preferably orthogonal) directions within the groundwater model 

domain. Creating borehole cross sections involves matching up stratigraphic units and 

estimating how a unit that is present in one borehole but not the other “pinches out” between the 

two locations. GMS is particular in requiring no unmatched stratigraphic units and interpretation 

of a given cross section is challenging at times. However, once all the cross sections are 

defined, the “horizons -> solids” command can be applied to have GMS interpolate the contacts 

across all the cross sections and automatically create a solids model from these interpolations.
4-19

Figure 22. Extent of the groundwater simulation model grid relative to key features. The 

grid origin is located in the lower left corner of the model at the bottom of the 

domain.

Figure 23. Material designations (and corresponding colors) used in GMS to represent 

the stratigraphic units of the subsurface (Section 4.1.1.6). The “dummy” 

material was used to facilitate work in GMS, but was not applied in the 

groundwater model.

The second approach to defining the geological structure involves working with scatter data sets 

and triangulated irregular networks (TINs) to form a solids model. This approach offers more 

flexibility at the expense of being more involved. In this approach, the user defines relevant 

surfaces, such as ground surface and contacts between units, based on the gathered data and 

conceptual model. Because the data density was low outside of the Bellefonte site and there 

exists a good conceptual model, this second approach was applied in this work.

Nominal

Location

of Anticline 

Model

Domain

Boundary Grid Origin 

Bellefonte

Site

N
4-20

The USGS DEM (Figure 11) data were exported from Global Mapper as a set of coordinate

points (X, Y, elevation) for an area encompassing the model domain. The exported DEM data 

were opened in GMS as a tab-delimited text file with a header row, which was designated as a 

2D Scatter Point data set in the file import wizard (thus the data showed up in the 2D Scatter 

Data module of GMS). The scatter data set was converted to a TIN with the GMS “Scatter 

Points -> TIN” command. This DEM TIN thus represents the ground surface elevations across 

the model domain (Figure 24). However, the DEM was refined by manually deleting scatter data 

points within the boundaries of available well data near the Bellefonte site and points 

representing the ground surface elevation at those wells were inserted. Although coarser, sitespecific well data on ground surface elevations were deemed to be more useful than the generic 

DEM data. The resultant impact to the ground surface elevation TIN produced from the modified 

scatter data is shown in Figure 25.

Little information on the vertical extent of the Soil and Weathered Rock (WeRo) materials is 

available outside of the borehole data at the Bellefonte site. Thus, an average thickness for 

each of these two units was applied uniformly where borehole data did not exist. Based on the 

borehole data (Table B.1), the average thickness of the Soil is 13.0 ft and the average thickness 

of the Weathered Rock material is 10.2 ft. The ground surface TIN was duplicated and 

translated downward vertically by 13 ft and by 23.2 ft to create TINs for the bottom of the Soil 

and Weathered Rock units, respectively (Figure 25). These three TINs (DEM, Soil, Weathered 

Rock) were placed together in a folder of the GMS TIN module and the “Horizons -> Solids” 

command was applied on that folder using the DEM TIN as the topmost elevation and the 

Weathered Rock TIN as the bottommost elevation. The resulting solids model for these two 

materials is shown in Figure 26.

Figure 24. DEM data in the vicinity of the Bellefonte site, looking nominally to the NNE. A

10X vertical exaggeration is applied
4-21

Figure 25. Set of TINs defining the Soil (between green ground surface and orange) and 

the Weathered Rock (between orange and yellow). The zone of low data 

density represents where ground surface elevations from wells at the 

Bellefonte site were used in place of the DEM data. The orange and yellow 

TINs are offset from ground surface by the nominal thickness of each 

material. View is nominally to the NNE with a 10X vertical exaggeration.

Figure 26. Solids models for the Soil and Weathered Rock (WeRo) materials based on a 

uniform thickness of each material from ground surface. View is nominally to 

the NNE with a 10X vertical exaggeration.
4-22

The borehole data (Table B.1) for the USR, MSR, and LSR bedrock units were read into GMS 

as separate 2D scatter data sets (x, y, height) and then converted to individual TINs as 

described above. Examination of these TINs (Figure 27) in three-dimensions reveals that they 

are nearly parallel planes. However, if each contact data set is fit to a plane and those planes 

are extended, it is seen that some of them intersect. Because the existing data are of limited 

spatial extent and need to be extrapolated (such that the TINs extend somewhat beyond the 

groundwater model domain) and because the contacts lay in nearly parallel planes, it was 

decided to use parallel planes to represent the contacts. The plane equations for the seven sets 

of contact data were averaged to arrive at a consistent plane equation, with an appropriate 

offset to the average center of each data set. The Nashville/USR contact consisted of a single 

data point, so the plane equation was applied through that point. The resulting set of TINs is 

shown in Figure 27. Flat TINs above and below the model domain extents were added and the 

entire set of 10 TINs was processed with the GMS “Horizons -> Solids” command to obtain a 

set of 9 solids (Figure 28).

Although the bedrock units have been defined along their contacts, additional solids are 

required to represent bedrock throughout the rest of the groundwater model domain. GMS 

allows the user to define “prisms” for solids with a drawn plan-view shape and specified 

top/bottom elevations. Prisms were drawn to represent the LSR and NV units to the northwest 

and southeast, respectively. This gave a set of solids representing the bedrock units (Figure 29) 

that then needed to be merged with the Soil and Weathered Rock solids. The GMS Solids 

module has an undocumented feature for performing set operations (union, difference, 

intersection) on two selected solids. Thus, a solid can be defined (using the prism tool or from 

TINs) to act as a clipping solid when performing a difference operation. The Weathered Rock 

TIN was used with a flat TIN of arbitrary high elevation to create a solid having the Weathered 

Rock solid bottom elevation. This clipping solid (Figure 29) was then differenced with each of 

the bedrock unit solids to produce solids that mesh precisely with the overlying weathered rock 

solid (Figure 30). Clipping solids were also used around the sides and the bottom to trim the 

solids to match the groundwater model domain, which resulted in the final set of solids shown in 

Figure 31.

4.1.2.4 Map the Geological Structure to a MODFLOW Domain

At this point the conceptual model, solids model, and initial model grid definition need to be 

brought together to map the hydrogeology to the MODFLOW groundwater model. Particular 

attention must be given to the expected water table elevation across the model domain; it is 

preferred to avoid grid cells going dry during the simulation, if possible.

GMS offers several approaches for mapping the solids model to the MODFLOW model grid, 

depending on which groundwater flow package is being used in MODFLOW: BCF/LPF or HUF. 

The Layer Property Flow (LPF) package is used most often in MODFLOW groundwater models; 

BCF (Block-Centered Flow) is essentially an older version of LPF and can be handled in a 

similar fashion. The GMS “Solids -> MODFLOW” command applies to the LPF package and can 

be done in three ways:

• Boundary Matching. Gives a deformed grid that matches the boundaries of the solids 

model; may result in very thin layers in places; cells have aquifer properties of the material 

intersecting the grid cell center.

• Grid Overlay. Gives a (potentially deformed) grid with uniform vertical cell thickness 

across all layers for any given vertical column of cells; cells have aquifer properties of the 

material intersecting the grid cell center.
4-23

• Grid Overlay with K Equivalent. Gives a (potentially deformed) grid with uniform vertical 

thickness for any given vertical column of cells across all layers; aquifer properties are 

specific to each grid cell (i.e., uncoupled from material definitions) and are effective values 

based on the proportions of materials intersecting the cell.

The choice of one of these three methods depends on the nature of the model and the modeling 

objectives.

Figure 27. Examination of TINs representing the contacts between bedrock units. The 

left image depicts the TINs based on borehole data for contacts between the 

units, while the right image shows the parallel planes used to represent the 

unit contacts. The view is along a strike of 42.5° with a 10X vertical 

exaggeration.

LSR NV

 

 

NV

 

 
4-24

Figure 28. Solids representing the USR, the MSR units A to F, and the contacts with the 

NV and LSR units. View is nominally to the NNE with a 10X vertical 

exaggeration.

Figure 29. View of solids used to fill in space such that the groundwater model domain is 

encompassed (left) and the solid used for clipping the top of the solids 

representing the bedrock units (right). The rightmost block representing the 

NV unit has been hidden in the right image so that the variation of the clipping 

solid surface is visible. Note that although the clipping solid is green, it does 

not represent Soil. The view is nominally to the NNE with a 10X vertical 

exaggeration but with 25° (left) and –25° (right) dip viewing angles.
4-25

Figure 30. Solids representing bedrock stratigraphic units after being clipped so the top 

elevation matches with the bottom elevation of the Weathered Rock (WeRo)

solid. View is nominally to the NNE with a 10X vertical exaggeration.

Figure 31. Final set of solids representing all materials and trimmed to the lateral and 

bottom extent of the groundwater flow model domain. View is nominally to the 

NNE with a 10X vertical exaggeration.
4-26

GMS also includes a “Solids -> HUF” command, which is applicable when the MODFLOW 

model uses the Hydrogeologic-Unit Flow (HUF) package. This process translates the solids 

models of hydrogeological units to appropriate HUF package definitions, retaining the elevations 

of the solids but not altering the MODFLOW grid. Thus, a given model grid layer could contain 

more than one hydrogeological unit. This approach requires that the user has chosen a 

meaningful grid definition (including the top elevation of the top layer), since the grid is not 

deformed to match the solids model. Additionally, the user must take care to avoid misleading 

images in GMS, because the HUF hydrogeological units do not affect the material definitions of 

the grid cells (i.e., displaying the materials assigned to each grid cell doesn’t reflect the 

hydrogeological units of the HUF package).

For the model of the Bellefonte site, there are boundary conditions for Town Creek and the 

Tennessee River at an elevation of 594.5 ft (Section 4.1.1.5). Thus, it is expected that the water 

table will progress from some elevation around 630 ft in the River Ridge hills and hills northwest 

of Town Creek down to 594.5 ft.

An attempt to apply the GMS “Solids -> MODFLOW” command for the LPF package using the 

Boundary Matching option failed, either because the hydrogeological units are not horizontal 

layers or because of slight irregularities in the matching of surfaces between hydrogeological 

units. When the mapping to the LPF package is applied with either Grid Overlay option, the 

originally orthogonal grid becomes deformed as shown in Figure 32. This provides a good 

match to the solids model, but the water table will cut through multiple layers and the simulation 

will have dry cells (which may require the option of cell rewetting). It is challenging for most (if 

not all) MODFLOW solvers to converge to a solution even when grid cells that go dry are 

allowed to rewet (hence the preference for a model configuration that avoids dry cells).

To use the LPF package in MODFLOW and avoid dry cells, a single layer should contain the 

water table, but this cannot be accomplished with the mapping to the initial grid definition (Table 

4). To obtain a single layer containing the water table, the solids model would have to be split, 

each portion mapped to a separate grid, then the grids manually combined. Selecting an 

elevation of 594 ft as the plane to slice the solids model into two pieces leaves 1⁄2 ft below the 

presumed lowest elevation of the water table (i.e., Town Creek/Tennessee River). The upper 

portion of this sliced solids model can be mapped to a 1-layer groundwater flow model grid. This 

will incorporate the topography and (when using the Grid Overlay method) assign the material 

for each grid cell based on the material at the center of the grid cell. The bottom half of the 

sliced model can be mapped to a 30-layer groundwater flow model grid, which will result in an 

orthogonal grid. Both of these MODFLOW groundwater models then need to be exported as 

“Native Text” (from the MODFLOW menu). To merge the files, data from the one-layer model 

need to be manually copied from the *.mfs, *.lpf, *.dis, and *.ba6 files (assuming that 

MODFLOW 2000 is being used) to the corresponding files for the 30-layer model (with 

appropriate adjustments to indices). The merged “native text” files can be loaded into GMS. The 

resulting MODFLOW groundwater model grid is shown in Figure 33. This approach is 

complicated, requiring a reasonable knowledge of MODFLOW input files, and the top layer 

material designations should be assessed to ensure that the appropriate material type (i.e., the 

dominant material through which groundwater flows) is represented.

The alternative to the above-described manipulations is to use the HUF package instead of the 

LPF (or BCF) package. When using the HUF package, a single model layer can be used for the 

water table, while the hydrogeological units defined in the solids model are applied for 

calculating groundwater flow. The initial model grid definition (Table 4) needs to be modified to 

apply the topography of the top of the solids model (i.e., the ground surface shown in Figure 25) 

to the top elevation of the uppermost grid layer. Then the GMS “Solids -> HUF” command can 
4-27

be applied to map the solids onto the MODFLOW groundwater model. Selected cross sections 

depicting the hydrogeological units in the MODFLOW model grid are shown in Figure 34.

The HUF package was used because it not only affords a straightforward method of mapping 

the hydrogeological units to the MODFLOW model, but also because it facilitates modeling of 

alternative conceptual representations of the site. For the HUF package, the representation of 

the soil and rock units, derived from geotechnical borehole data and physically reasonable 

assumptions (as represented by the solids model), is independent of the grid discretization. 

Thus, changing the conceptual model can easily be done by either altering the solids model and 

re-mapping it to the MODFLOW model or by simply altering material parameter values. For 

example, to combine the soil and weathered rock units into a single unit entails nothing more 

than assigning identical values for the materials properties. Use of the HUF package also avoids 

potential anomalies in hydraulic head results that may occur with the LPF approach (Figure 33) 

that stem from assigning material properties using a categorization approach based on the 

material present at the center of a grid cell. The HUF package accounts for all units present 

within a grid cell (e.g., transmissivity is a sum of unit conductivities weighted by thickness of 

units within the cell).

Figure 32. Deformation of an originally orthogonal model grid after mapping the solids 

model (Figure 31) to the MODFLOW LPF package using the Grid Overlay 

option. The grid layers follow the topography (using equal grid cell thickness 

for all layers in a given vertical column of grid cells). The water table would 

cut through multiple layers. View is nominally to the NNE with a 10X vertical 

exaggeration.
4-28

Figure 33. MODFLOW model grid for the LPF package, showing the materials 

(hydrogeological units) applied to the grid cells. This model was created by 

splitting the solids model into two portions, mapping (using the Grid Overlay 

option) to two model grids, and then merging both grids into a single model 

grid. Grid cells beyond the shoreline of the Tennessee River have been 

inactivated. View is nominally to the NNE with a 10X vertical exaggeration.

Figure 34. Fence diagram of MODFLOW model grid for the HUF package, showing the 

hydrogeological units applied to the grid cells along selected cross sections.

View is nominally to the NNE with a 10X vertical exaggeration.
4-29

4.1.3 Configure the MODFLOW Model

With the groundwater flow model material types and grid structure defined, the temporal nature, 

boundary conditions, and initial conditions can be defined to complete the model configuration.

4.1.3.1 Temporal Nature of the Model

There are two choices for the temporal nature of a MODFLOW groundwater model: steady state 

or transient. A steady state model represents a set of constant conditions whereas a transient 

model will include variations in boundary conditions (e.g., river stage, recharge, pumping at 

wells, etc.) over time. For the Bellefonte site, there are no induced hydraulic stresses (e.g., 

wells) and the conceptual model (Section 4.1.1.6) assumed steady-state flow due to limitations 

in the data available to support a time-dependent numerical model.

4.1.3.2 Boundary Conditions on Sides of Model Domain

Grid cells on the southeast side of the model domain were inactivated such that the boundary 

followed the Tennessee River shoreline. This “downgradient” boundary was set to a constant 

hydraulic head of 594.5 ft to represent the river.

Constant head boundary conditions were assigned to the “upgradient” boundary at the 

northwest side of the model domain based on professional judgment in approximating 

interpolation of the 1961 data (Figure 35), ignoring wells impacted by pumping. The northwest 

model boundary nominally follows the Sequatchie anticline and a “valley” in the hydraulic head, 

except the northwest corner where the topography and groundwater hydraulic head rise. The 

set of boundary condition hydraulic head values for the northwest side is shown in Figure 36.

Between specified points the boundary conditions are set by linear interpolation.

The northeast and southwest sides of the model domain are left as no-flow boundary conditions 

to match the conceptual model.

4.1.3.3 Boundary Conditions for Surface Water Features

Boundary conditions were set in layer 1 for surface water features inside the groundwater model 

domain. The entirety of Town Creek and inlet for the nuclear plant water intake were set to the 

same constant head as the Tennessee River (594.5 ft). The five streams identified as draining 

into Town Creek were represented as drain boundary conditions, which allow withdrawal of 

water from the model domain when the groundwater hydraulic head is above the bottom 

elevation of the drain. The bottom elevation of each drain was specified at the head (or model 

boundary) for each stream and was linearly interpolated to the elevation of Town Creek (594.5 

ft) along the line representing the path of the stream. The conductance (i.e., proportionality 

factor for flow through the bottom of the drain that accounts for hydraulic conductivity and drain 

dimensions) of each drain was given a large value of 100 ft2/d per ft, and GMS calculated the 

length of the stream through each grid cell. The stream paths and the drain elevations at the 

head of each stream are shown in Figure 36.
4-30

Figure 35. Hydraulic head data from 1961 relative to the model domain boundary. Data 

from numbered wells were used to define the boundary conditions along the 

northwest side. Filled circles indicate wells impacted by pumping and thus 

not used in this analysis.

Figure 36. Boundary conditions for the groundwater model. Internal boundary conditions 

(green drain and purple constant head) apply to layer 1, others (yellow) to all 

layers.

1

Model

Domain

Boundary

2

3

5

4

11

14

13

12

9

10

8

7

6

Nominal

Location

of Anticline

609.0

611.0

616.0

615.0

610.0

630.0

594.5

643.8 608.4

603

616.6

612.9

594.5

Model

Domain

Boundary

Inactive

Grid Cells

Active

Grid Cells
4-31

4.1.3.4 Recharge Boundary Conditions

The plan view of the model domain was divided into several zones for specifying the recharge 

(from precipitation) based on nominal land usage. The zones, shown in Figure 37, included:

• The northwest portion of the model, a mixture of forest, farmland, and sparse residential 

development,

• The lowlands on the southeast side of Town Creek, consisting of forest and wetlands, 

• The developed area encompassing the existing Bellefonte Nuclear Plant and associated 

facilities, and

• The River Ridge highlands along the Tennessee River, primarily forest with a few clearings.

No recharge was applied over the area of Town Creek itself. The initial recharge rate was taken 

to be 8 in/yr (0.00182 ft/d) based on TVA (2009).

Figure 37. Zones for the recharge boundary conditions

4.1.3.5 Initial Conditions

Initial conditions for a MODFLOW model consist of the hydraulic heads for each active grid cell. 

Where constant head boundary conditions are specified, the boundary condition is taken as the 

initial condition. For a purely transient simulation, a good set of initial conditions is critical to 

obtaining meaningful results. For a steady state simulation, the closer the initial conditions are to 

the solution, the more easily the solver will converge. However, it is reasonable to begin a 

steady state simulation with initial conditions that are relatively far from the final solution. One 

approach is to guess the water table based on observed data. Another approach is to set the 

initial hydraulic heads to some high value and let the heads settle down to the converged 
4-32

solution. Reasonable high values would be ground surface or some constant value that is above 

the expected water table across the model domain. For this work, an initial hydraulic head of 

750 ft was specified for all grid cells that did not have a constant head boundary condition. Once 

a reasonable head solution was obtained, this solution was used as the initial condition for 

subsequent simulations.

4.1.3.6 Configured Model

The boundary and initial conditions described above were applied to the MODFLOW 

groundwater model grid/hydrogeological structure defined based on using the HUF package.

The resulting model, showing materials and boundary conditions (except recharge), is 

presented in Figure 38. Alternative models were formulated from this initial model configuration

to represent model uncertainties.

4.2 Model Calibration

Groundwater flow model calibration is the process of adjusting model parameters to obtain a 

hydraulic head distribution that best matches the observed values for hydraulic head and (if 

available) the observed discharge of ground or surface water. There are two basic approaches 

to calibration: directed trial and error or automated parameter estimation. The trial and error 

approach involves manual adjustment of parameter values based on the conceptual model and 

knowledge of the site data. Automated parameter estimation is a process using optimization 

techniques (e.g., the Levenberg-Marquardt algorithm) to find the hydraulic head solution with 

the minimum error in the calibration targets, and was used here.

Figure 38. MODFLOW groundwater model for the Bellefonte site as configured with the 

hydrogeological units (HUF package), constant head boundary conditions 

(purple points), and drains (white points)
4-33

4.2.1 Define Calibration Targets

The three data sets of observed hydraulic heads at monitoring wells discussed in Section 

4.1.1.5 were used for hydraulic head calibration targets. Despite the age of the 1961

observations relative to the other data sets, the older data set provided important information at 

points in the northwest area of the model domain at a distance from the actual Bellefonte site.

The September 2005 data provided information across much of the area between Town Creek 

and the reservoir. The June 2006 through May 2007 data set, which primarily includes locations 

in the vicinity of the proposed reactors, was split into two groups (Soil and Bedrock) for display 

purposes. To reflect the steady-state flow assumption, the average head values for the 

2006/2007 period were used as the calibration targets. The average range in head over the 

observation time period was 5.3 ft (1.6 m) for the soil wells and 5.7 ft (1.7 m) for the bedrock 

wells (see Figure 21 for an example).

Calibration weights were based on assumed error intervals for the head observations of ±5 ft for 

the 1961 data, ±2 ft for the September 2005 data, and ±1 ft for the June 2006 to May 2007 data. 

These assumed errors reflect the differences in time of the observations and the relative

uncertainty in the 1961 measurements. Better estimates of observation errors could be obtained 

with more information about the head measurements. Observation standard deviations were 

computed by assuming normally distributed head errors and that the error intervals represented 

95% confidence values. Calibration weights were then entered as the inverse of the head 

observation standard deviations. Values are given in Table 5.

Table 5. Head observation error intervals, standard deviations, and calibration weights

Dataset 95% 

Interval (ft)

Std.

Dev. (ft)

Weight

(ft-1)

1961 5 2.55 0.39

2005 2 1.02 0.98

2006-07 1 0.51 1.96

Locations of the head data points used as calibration targets are shown in Figure 39; head 

values are listed in Table B.3 of Appendix B. There are no flow calibration targets for this site.

The implications of this are discussed in Section 4.2.4.

4.2.2 Define Parameters for Estimation

The parameters selected for estimation during calibration were the horizontal hydraulic 

conductivity of each material type and the recharge. For calibration purposes, the bedrock units 

identified at the site were divided into two groups, with each unit in a group assigned the same 

parameter values. One group included the bedrock units characterized as prominently 

silty/argillaceous limestone: the Middle Stones River subunits A, C, and E (shaded in Figure 14). 

The other group included all other bedrock units. In addition to these two bedrock groups, 

parameters were estimated for the soil material and for the weathered rock material. For each 

material type, vertical hydraulic conductivity was tied to the estimated value of the horizontal 

hydraulic conductivity and assumed to be a factor of 10 smaller (i.e., anisotropy ratio of 10). 

Initial parameter values were based on the geometric mean values provided in Table 1 and the 

average recharge discussed in Section 4.1.3.4. Calibrated values were determined to be 

insensitive to the initial parameter values. Minimum and maximum parameter limits were set 

sufficiently small/large such that they did not restrict the final estimated parameter values. 
4-34

Figure 39. Locations of hydraulic head calibration targets for the four groups of data. 

Points outside the model domain boundary were not included as calibration 

targets.

4.2.3 Calibration Results

An initial calibration was carried out to evaluate the ability of the calibration code to produce a 

reasonable solution. To improve the robustness of the parameter estimation, all recharge zones 

except Town Creek were grouped together and given the same recharge rate. Calibration was 

completed using PEST with the PEST input files constructed by GMS. Hydraulic head results for 

the optimal solution returned by PEST are shown in Figure 40. Flooded cells indicate locations 

where the hydraulic head of the groundwater exceeded the ground surface elevation, which are 

interpreted here to represent expected conditions (e.g., along the Tennessee River), and 

artifacts of the resolution of the ground elevation data or minor errors in the assigned drain 

elevations. Flooded cells could also indicate problems with the model configuration, such as 

streams that should have been included or drain conductances that should be increased. 

• 1961 Data (red)

• Sept. 2005 data (cyan)

• Data for head in Soil (pink)

• Data for head in Bedrock (yellow)
4-35

Head residuals (the difference between the observed value and the simulation result) are shown 

in Figures 41 to 44. The residuals in these figures are shown as colored circles where the area 

of each circle is proportional to the magnitude of the residual. Positive residuals (blue) indicate a 

simulation result that is lower than the observed hydraulic head; negative residuals (pink) 

indicate a simulation result higher than the observed head. The majority of the residuals in the 

area of the proposed reactors were negative, except for the soil wells shown in Figure 44. The 

four soil wells with the largest positive residuals were all indicated to be nonresponsive in FSAR 

Table 2.4.12-204, which suggests that these residuals should be discounted when evaluating 

the model.

Different scales are used for the display of residuals in Figure 41 to Figure 44 so that they are 

not directly comparable. It can be seen, however, that the residuals frequently are larger in 

magnitude than the estimated error intervals of the head observations. In fact, the least squares 

standard error for the calibration [ ˆσ LS , see discussion after Eq. (14)] was 8.0, indicating that the 

fit is inconsistent with the observation weighting (Hill and Tiedeman 2007). This may indicate 

some structural errors in the model. Additional discussion of the residuals can be found in 

Section 4.4.1.

Figure 40. Hydraulic head contours (every 2 feet) resulting from the initial PEST 

calibration
4-36

Figure 41. Unweighted residuals (observed minus simulated, in feet) for the 1961 data.

Positive (blue) and negative (pink) residual magnitudes are proportional to 

symbol size.

Figure 42. Unweighted residuals (observed minus simulated, in feet) for the September 

2005 data. Positive (blue) and negative (pink) residual magnitudes are 

proportional to symbol size.

+20.8

–10.8

+3.5

–28

–10
4-37

Figure 43. Unweighted residuals (observed minus simulated, in feet) for the average 

2006/2007 “Bedrock” data. Positive (blue) and negative (pink) residual 

magnitudes are proportional to symbol size.

Figure 44. Unweighted residuals (observed minus simulated, in feet) for the average 

2006/2007 “Soil” data. Positive (blue) and negative (pink) residual magnitudes 

are proportional to symbol size.

+7

–5.2

+12.4

–5.3

+7.9

–2.6
4-38

4.2.4 Define Prior Parameter Information

At the Bellefonte site, the data available for use in calibration are limited to hydraulic heads at 

wells. This is an issue because the groundwater flow at the site is governed by the amount of 

recharge and the hydraulic conductivity. Recharge and hydraulic conductivity will be positively 

correlated in any calibration carried out using heads alone as the basis for the parameter 

estimation. This is illustrated for the Bellefonte site model in Figure 45 which shows calibrated 

values of hydraulic conductivity when recharge was fixed at three values (4, 8, and 12 in/yr). 

(This is for a model with a single hydraulic conductivity parameter for all hydrogeologic units.) 

The calibrated hydraulic conductivity value depends linearly on the value of recharge. The 95% 

confidence limits shown on the figure indicate that hydraulic conductivity can be fit with little 

uncertainty when the recharge is known. However, no unique solution can be determined when 

both hydraulic conductivity and recharge are estimated using head data only.

Ideally, there would be some flow observations at the site that would constrain the parameters

and allow a unique solution. At the Bellefonte site, however, there are no significant streams on 

site at which flows could be measured. The boundaries of Town Creek and the reservoir are 

relatively constant due to the reservoir control and are not responsive to groundwater 

conditions. Flow parameters could also be constrained by transport observations (e.g., results of 

a tracer test), but such data are not available at the Bellefonte site and are typically not collected 

during site characterization for a nuclear power plant.

In the absence of flow observations, prior information on the values of hydraulic conductivity and 

recharge were used to constrain the calibration. This required prior information equations to be 

used in the objective function with suitable weights representing the errors in the prior 

estimates. Hill and Tiedeman (2007) recommend caution in using prior information (Chapter 11, 

Guideline 5), preferring instead to initially estimate parameters without any prior information to 

understand the information that is directly available from the observations alone. For the 

Bellefonte site, that approach requires that either the recharge or the hydraulic conductivity be 

estimated, but not both due to the correlation. Fixing a parameter during the calibration is 

equivalent to applying a prior estimate with an infinite weight. This will cause the uncertainty of 

the fitted parameter to be underestimated. Thus prior information was applied to both the 

hydraulic conductivity and the recharge. 

The optimization balances (via user-supplied weights) the prior information and the head 

observations to provide unique estimates and uncertainties that reflect the combination of head 

data and prior information. Figure 45 shows the solution for the homogeneous model when prior 

information on recharge and hydraulic conductivity were included. The 95% confidence intervals 

on the parameter estimates reflect the parameter correlation and the relatively large standard 

deviations (small weights) used for the prior information. The large magnitude of the confidence 

intervals indicates that, although the prior information has provided unique parameter estimates, 

other combinations of parameter values are almost equally likely.
4-39

Figure 45. Correlation between hydraulic conductivity and recharge prevents unique 

parameter estimates unless one value is fixed (e.g., recharge here) or prior 

values of parameters are applied. Here the prior mean recharge was 8 in/yr; 

the prior mean hydraulic conductivity was 0.438 ft/d.

At the Bellefonte site, prior information for the hydraulic conductivity (as provided in the FSAR) 

included a range for the hydraulic conductivity of the soil and a range for the bedrock plus a 

single point on the cumulative distribution function for the bedrock pressure test data. That is, 

the application did not provide the set of hydraulic conductivity measurements, but a summary 

of those measurements consisting of the minimum and maximum values measured and a 

statement that a specific percentage of measured values were less than a specified value. In an 

actual review, the full set of actual measurements would be obtained through a request for 

additional information. Statistical analysis of these data could then be completed to develop the 

prior information mean and weight. For this example, however, an assumption was made that 

the hydraulic conductivity followed a lognormal distribution and that the ranges provided 

represented 95% probability intervals. From this assumption the parameters of the hydraulic 

conductivity distribution for each dataset listed in Table 1 were determined. The mean value 

was used as the prior estimate and the inverse of the standard deviation was used as the prior 

equation weight. Results for the bedrock datasets were arbitrarily assigned to the three 

hydrogeologic parameter groups: Unit 3 and 4 packer test data to the weathered rock, Unit 1 

and 2 packer test data to the silty limestone units, and Unit 3 and 4 pump test data to the 

remaining rock units.

Prior information for the Bellefonte site recharge included only a point estimate (8 in/yr). It was 

assumed that this represented the geometric mean of a lognormal distribution, and that there 

was a 1⁄2×95% probability that the recharge was between 4 and 8 in/yr. From these assumptions

a weight for the prior equation was determined. Prior parameter distributions are shown in 

Figure 46.
4-40

Figure 46. Prior parameter distributions derived from data and assumptions discussed in

the text: horizontal hydraulic conductivity (top) and recharge (bottom)

As discussed by Ye et al. (2008), when prior parameter estimates are available, they can be 

incorporated in the calibration dataset (Cooley 1983; Carrera and Neuman 1986) and the 

observation weight matrix. Use of prior parameter estimates is implemented in both PEST and 

UCODE. The uncertainty methodology described in Section 3.1 requires that the same set of 

0 5 10 15 20

0

0.2

0.4

0.6

0.8

1

Recharge (in/yr)

P(Recharge<=x)
4-41

data be applied to the calibration of each alternative model. This condition will not hold, 

however, when prior parameter estimates are included in the least squares objective function for 

alternative models with differing numbers of estimated parameters. This can have a significant 

impact on the calculation of model selection criteria such as AIC and KIC. Resolution of this 

issue has not been addressed in the literature. Possible approaches are to:

• Calculate the model selection criteria using only the system state observations (groundwater 

head in this example). In this option, the prior parameter estimates are used in the 

calibration, but not in the calculation of 2 ˆσ ML in Eqs. (20) and (21).

• Include prior parameter estimates only for parameters common to all the alternative models.

• Include extra prior parameter equations in those models with fewer estimated parameters so 

that the effective number of prior data contributing to the objective function in each model is 

the same.

For this example, the last option was used. It was implemented by increasing the weight of a

prior estimate, to provide the effect of adding one or more duplicate prior data. For example, a 

model with one recharge parameter for each recharge zone identified in Figure 37 would have 

four prior parameter estimates. An alternative model with a single parameter characterizing 

recharge across the domain would have one prior parameter estimate. By increasing the weight 

of this single prior estimate by a factor of four (the weight being the inverse variance of the 

estimate) the effective number of prior recharge estimates contributing to the objective function 

was increased to four. (In PEST and UCODE the prior weights are entered as the inverse 

standard deviation; these were increased by a factor of two to get the equivalent result.)

4.3 Alternative Models Evaluated for the Bellefonte Site

The uncertainty methodology represents model uncertainty using a set of alternative models. 

For this example application, the conceptual issues considered were the degree of karst 

development, the importance of the argillaceous bedrock layers, the depth of active flow, and 

the variation in recharge. Since no karst-specific flow features (e.g., springs or large conduits) 

were identified in the site characterization, a distinct weathered rock layer is the only model 

element representing karst development. Similarly, the importance of the argillaceous layers is 

incorporated in the model through their explicit representation as separate hydrogeologic units. 

Variation in recharge is represented through the use of multiple recharge zones.

The alternative models were formed by considering combinations of alternatives for three 

aspects of site conceptualization and parameterization. These three aspects were the 

representation of the individual hydrogeologic units, the recharge, and the depth of active flow. 

The hydrogeologic units (via their respective hydraulic conductivity values) and the recharge 

were expected to control groundwater flow at the site. The depth of significant porosity 

development (through weathering and cavity formation) is a key aspect of the conceptual model 

of groundwater flow described above.

4.3.1 Hydrogeologic Unit Representation

The following options were considered for the representation of the identified hydrogeologic 

units.

1. A homogeneous domain in which all units had the same properties.

2. A two-zone domain in which the upper zone included the soil and weathered rock units 

and the lower zone included the other rock units.
4-42

3. A three-zone domain in which the upper zone included the soil and weathered rock units 

and the remaining rock units were represented by one of two sets of properties, those for 

the limestone units (NV/USR, MSR-B, MSR-D, MSR-F, and LSR) and those for the 

silty/argillaceous limestone units (MSR-A, MSR-C, and MSR-E).

4. A four-zone domain in which the soil and weathered rock units were represented 

distinctly and the remaining rock units were represented as in option 3.

Alternatives were implemented in GMS by modifying parameters of the hydrogeologic units on 

the Materials page.

4.3.2 Recharge Representation

Two options were considered for recharge.

1. Recharge was homogeneous across the site.

2. Recharge was varied across four zones representing (a) the surface across Town 

Creek, (b) the vegetated lowlands on the plant side of Town Creek, (c) the area 

developed for the plant with minimal vegetation, and (d) the uplands between the plant 

and the river.

Alternatives were implemented in GMS by changing the recharge zone assignments in the Map 

module.

4.3.3 Depth of Active Flow

Two options were considered for the depth of active flow.

1. A deep domain extending to an elevation of 450 ft. msl (the existing 31 layer MODFLOW 

model).

2. A shallow domain extending to an elevation of about 541.52 ft. msl. This depth is 54 ft. 

below the elevation of the river and Town Creek. 

The shallow domain option represents a case in which the significant rock porosity is limited to 

the upper portion of the rock units. This conceptualization was implemented in the MODFLOW 

groundwater model by inactivating all but the top 12 layers. All head observation points are 

located in the top 12 model layers except for MW-1204c, which has a screen mid-point elevation 

of 509.35 ft.msl. Because the multi-model comparison requires that all models use the same 

data, the elevation of the MW-1204c observation point was set to 542 ft. msl to include this data 

point in the active model domain for this conceptualization. This was not anticipated to 

significantly affect the model comparison since it involved a single observation’s contribution to 

the regression objective function, the vertical variation in head at that depth was small in the 

deep domain models (particularly for the more homogeneous models), and the residual for the 

MW-1204c observation was relatively small in the deep domain models.

4.3.4 Alternative Model Designations

There are 16 possible combinations of the three conceptual model aspects described above, 

leading to 16 alternative models. Each of the alternative models was designated by a sixcharacter name in the form of H?R?D?, where the ? characters were replaced by the option 

numbers as listed above. For example, model H4R2D1 was the model with four hydrogeologic 

zones, four zones of recharge, and a deep domain.
4-43

4.4 Calibration Implementation

The primary calibrations completed for the example were conducted using GMS and PEST. 

Because the implementation of PEST within GMS has some limitations, the PEST input files 

were generated from within GMS, but the actual execution of PEST was completed as a standalone code (i.e., from the Windows OS environment). The basic steps required to complete the 

calibrations were as follows.

1. Develop the groundwater flow model within GMS. Enter key values (usually negative 

numbers) for those parameters to be estimated using PEST. In the example, key values 

were entered in the Map module for the recharge zones and in the Materials window for the 

hydraulic conductivity.

2. Enter parameter information in the GMS-MODFLOW-Parameter window. This includes the 

initial parameter values, minimum and maximum values, selections for the log 

transformation option, and designating tied values, among other options.

3. Execute a forward run of MODFLOW from GMS and evaluate the flow solution for the initial 

parameter values.

4. Generate the PEST input files (control, template, and instruction files) by executing a 

parameter estimation run of MODFLOW from within GMS. Terminate this run after initiation

is complete and PEST execution begins.

5. Using a text editor, modify the PEST control file (*.pst) generated by GMS to include the 

prior information equations.

6. Execute PEST. In the example, Parallel PEST was executed from the Windows OS using a 

custom script file. 

7. Import the optimal parameters determined from the PEST execution (the *.par file) back into 

GMS from the GMS-MODFLOW-Parameters window.

8. Execute a forward run of MODFLOW from GMS using the optimal parameter values and 

evaluate the results.

Because the calibration is executed from outside GMS, a similar set of steps could be 

completed using UCODE instead of PEST. This would require a couple of extra steps, however,

to translate PEST input files to the UCODE format and to get the optimal parameters back into a 

GMS-readable file.

4.4.1 Model Evaluation

Two of the calibration steps above involve evaluating the model results, once with the initial 

parameter values to ensure that the calibration begins at a reasonable solution and once with 

the optimal parameter values to ensure that the results are acceptable. GMS was used in a 

visual evaluation of heads (Figure 40) and in evaluating flow pathways computed using 

MODPATH as shown in Figure 47 (model H4R1D1) for a plan view and across-section through 

the Units 3 and 4 reactor areas. All pathways terminate in Town Creek and some extend to a 

significant depth (note 10x vertical exaggeration. This suggests that the depth of the flow 

domain has a significant impact on flow pathways and potentially travel times.

Some model evaluations were more appropriately completed outside the GMS environment 

because GMS has limited statistical analysis and graphing capabilities. These evaluations were 

completed using custom spreadsheets (see Figure C-1 of Appendix C for an example). Text file 

output from PEST was either copied or imported into the spreadsheets. Output used included 

the list of measured values, simulated equivalents, calibration weights, and calculated residuals. 

Estimated parameter values, 95% linear confidence limits, and the reported parameter 
4-44

estimation covariance matrix were also included in the spreadsheets. A statistical evaluation of 

the residuals was completed using the integrated spreadsheet graphics capabilities. 

Results were evaluated for each of the alternative models discussed above. The standard error 

of regression varied from 7.9 to 10.8 with the results shown below being typical for the majority 

of the alternative models. There were only small differences in the head residuals between the 

deep and the shallow domain models. 

Several of the plots used for model evaluation are shown in Figure 48 for the H4R1D1 model, 

including plots of measured heads and weighted residuals versus the simulated (modeled) 

equivalent heads, a boxplot of weighted residuals, and a quantile-quantile plot of weighted 

residuals (constructed following Helsel and Hirsch [2002]). These plots were used to evaluate 

the quality of the model fit and randomness of the residuals to determine whether the predicted 

values and predictive uncertainty could be accurate. The magnitudes of the unweighted head 

residuals were generally less than 10 feet, although these exceeded 20 feet in the extreme 

cases. Residuals appeared to increase with the magnitude of measured heads, indicating a 

possible structural error in the model. The quantile-quantile plot indicates that the weighted 

residuals are not normally distributed and suggests that the distribution is influenced by a 

number of outliers. The boxplot in Figure 48 shows that five of these outliers are positive (i.e., 

the simulated equivalents were less than the measured heads), with a single negative outlier. 

Three of the positive outliers were measured in observation wells completed in the soil zone and 

located near each other between the proposed units (see Figure 44). According to footnotes in 

FSAR Table 2.4.12-204 (TVA 2009), these wells exhibited a slow response during monitoring; 

observed heads may be unrepresentative of actual conditions. They were nonetheless included 

in the calibration because the data suggested that they were somewhat responsive to site 

conditions and it was desired to have relatively shallow head observations in the calibration. 

The remaining two positive outliers were from bedrock wells that were part of a multi-level 

cluster located near the base of River Ridge (see Figure 43). The large negative outlier was also 

from a well located near the base of River Ridge (see Figure 42). These outliers suggest that

either there are unknown errors in the observed data, or, more likely, that structural errors in the

model result in a poor representation of the high groundwater ridge coincident with River Ridge 

(see Figure 47). 

Model fit was inconsistent with the assumed head observation errors as reflected in the large 

standard error of the regression. This was interpreted to indicate that model errors were 

significant. With the six outliers removed, however, the weighted residuals satisfied conditions of 

normality, as determined by the correlation coefficient between the ordered weighted residuals 

and the standard normal quantiles (Figure 48). This suggests that a number of sources may be 

contributing to the model error and that predicted values and predictive uncertainties derived 

from the model may be accurate (Hill and Tiedeman 2007; Hill et al. 1998), at least in the areas 

of the model domain not associated with the outliers. 

Parameter estimates and their uncertainties were evaluated by examining composite 

sensitivities (printed in the output files) and by comparison with prior estimates. Estimated 

parameter values were reasonably close to initial values except for the hydraulic conductivity of 

the soil material, which was generally estimated to be about four orders of magnitude greater 

than the initial value of 3.5×10-4 ft/d. The discrepancy between the estimated value and the prior 

information indicates either that there is a significant structural error in the model, or that the 

prior information for the soil hydraulic conductivity does not represent conditions at the site. 

Hydraulic heads collected at the site prior to and shortly after a significant rainfall (discussed in 
4-45

Section 4.1.1.6 with respect to Figure 21) indicated a widespread and rapid response of 

groundwater head to the precipitation. This suggests that the range given for the hydraulic 

conductivity of the soil in TVA (2009) is unrepresentative of actual site conditions and that the 

estimates from the calibration may be more reasonable.

Parameter estimates from the calibrations and 95% confidence intervals are shown in Figure 

49. In this figure, the hydraulic conductivity parameters are shown as hk_11 to hk_14 and the 

recharge parameters as rch_31 to rch_34. As a result of the way the various alternative models 

represented the hydraulic conductivity and recharge zones, a given parameter did not always 

represent the same region of the model. For example, all models names beginning with “H1” 

were homogeneous and the hk_11 parameter represented the entire domain. For the remaining 

models, the materials represented by the hydraulic conductivity parameters are given in Table 

6. For the recharge parameters, the models with “R1” in the name had a single recharge zone 

represented by the parameter rch_34. For the “R2” models, the four recharge zones were 

represented by the four parameters: rch_34 for the surface across Town Creek, rch_33 for the 

vegetated lowlands on the plant side of Town Creek, rch_32 for the area developed for the 

plant, and rch_31 for the River Ridge uplands.

In general, the parameter results shown in Figure 49 indicate that the estimates were fairly 

stable across the alternative models. For a given parameter, differences in values estimated for 

the alternative models generally varied by less than one order of magnitude. Parameter 

confidence intervals were two orders of magnitude in nearly every cases and were significantly 

larger for many of the parameters. These large parameter uncertainties are indicative of a low 

composite sensitivity (e.g., the measured heads were much more sensitive to hk_11 than to 

hk_14). Composite sensitivities (Hill and Tiedeman 2007) for model H4R2D1 are shown 

graphically in Figure 50. 
4-46

Figure 47. Pathlines from points around the Units 3 and 4 reactor buildings computed in 

GMS using MODPATH: plan view (top) and cross-section through the reactor 

building areas (bottom, 10x vertical exaggeration)

A’ A
4-47

Figure 48. Graphical evaluation of residuals for model H4R1D1: observed vs. simulated

heads (feet) (top left), unweighted residuals vs. observed heads (top right), 

weighted residuals vs. simulated heads (middle left), quantile-quantile plot of 

weighted residuals (middle right), and boxplot of weighted residuals (bottom, 

with explanatory legend)

− 30 − 20 − 10 0 10 20 30

Weight*Residual

75%

median (50%)

25%

+ outliers

IQR

- outlier

max datum 

≤ 75% + 1.5*IQR

min datum 

≥ 25% - 1.5*IQR
4-48

Table 6. Materials represented by hydraulic conductivity parameters in alternative 

models

Model Name Parameter

hk_11 hk_12 hk_13 hk_14

H1... entire domain n/a n/a n/a

H2... soil & weathered rock n/a all limestone units n/a

H3... soil & weathered rock n/a limestone silty limestone

H4... soil weathered rock limestone silty limestone
4-49

Figure 49. Parameter estimates and 95% confidence intervals for hydraulic conductivity 

(top, ft/d) and recharge (bottom, ft/d). See text for explanation of parameters.
4-50

Figure 50. Composite sensitivities for model H4R1D1 at the final parameter values as 

computed following Hill and Tiedeman (2007) 

4.5 Alternative Model Probabilities

As discussed previously, the values for KIC reported by PEST and UCODE do not take into 

account the possible log transformation of parameters, which will affect the KIC values as 

described in Appendix A. Because of this the model selection criteria were computed within a 

spreadsheet to account for the log transformation of parameters (Figure C-1 of Appendix C). 

The model probabilities follow directly from the model selection criteria and were calculated in 

the same spreadsheet (Figure C-2 of Appendix C). 

The model selection criteria for each of the alternative models are shown in Table 7. These 

values provide a balance between model fit and model complexity. The resulting model 

probabilities are also provided in the table and shown graphically in Figure 51. For comparison, 

values of the weighted sum of squared residuals, a measure of fit only, are provided in Table 7

and shown in Figure 51. 

The use of AICc resulted in three models with significant probability. Two of these models were 

those with four hydraulic conductivity zones and one recharge zone (H4R1D1 and H4R1D2). 

The former used the deep domain and was somewhat preferred by AICc, while the latter used 

the shallow domain. The other model with significant probability assigned by AICc was the bestfitting model with the minimum SSWR (H3R2D1). This model lumped the soil and weathered 

rock materials together and had four recharge zones. One other model (H3R2D2) was assigned 

a relatively small, but noticeable probability (5%). 

When using KIC, H4R1D1 and H4R1D2 were preferred, similar to AICc, but KIC preferred the 

shallow domain model by a probability ratio of almost four to one. No other models received 

significant probabilities. This result is consistent with some other comparisons of model 

selection criteria in the literature, which have found that the use of KIC can tend to emphasize a 

smaller number of models than AICc (e.g., Singh et al. 2010).
4-51

Table 7. Model selection criteria values and model probabilities for alternative models

Model Probability

Model Nz Pk SSWR AICc KIC ln ML − Σk AICc KIC

H1R1D1 71 2 8010.42 341.89 349.90 9.51 0.00% 0.00%

H2R1D1 71 3 6026.49 323.93 328.31 5.71 0.00% 0.01%

H3R1D1 71 4 5590.22 320.92 323.02 3.33 0.01% 0.20%

H4R1D1 71 5 4244.35 303.75 313.78 11.22 39.59% 20.25%

H1R2D1 71 5 5137.42 317.31 354.44 38.32 0.05% 0.00%

H2R2D1 71 6 4932.50 316.88 344.02 28.37 0.06% 0.00%

H3R2D1 71 7 4021.04 304.92 337.15 33.58 22.03% 0.00%

H4R2D1 71 8 4176.33 310.24 349.18 40.49 1.54% 0.00%

H1R1D2 71 2 7767.93 339.71 347.91 9.71 0.00% 0.00%

H2R1D2 71 3 6056.45 324.29 321.71 -1.25 0.00% 0.38%

H3R1D2 71 4 5725.10 322.61 321.30 -0.08 0.00% 0.47%

H4R1D2 71 5 4276.54 304.29 311.07 7.97 30.28% 78.68%

H1R2D2 71 5 5227.29 318.54 354.05 36.70 0.02% 0.00%

H2R2D2 71 6 4941.69 317.02 337.98 22.20 0.05% 0.00%

H3R2D2 71 7 4184.85 307.76 337.38 30.98 5.34% 0.00%

H4R2D2 71 8 4222.75 311.03 349.85 40.38 1.04% 0.00%

Figure 51. Model probabilities computed with AICc and KIC, and the weighted sum of 

squared residuals for the alternative models

4.6 Predictive Modeling

Having characterized parameter uncertainty in the history-matching period and having obtained 

model probabilities for the alternative models, these results were used with a predictive model to 
4-52

estimate predictive uncertainties. This required modification of the basic model, used in the 

calibration discussed above, to reflect the conditions of the predictive period. For this example, 

involving review of a nuclear power plant application, that involved modifying the model to 

represent the groundwater conditions after the plant has been constructed. These modifications 

included structural changes in the models (excavations, topography), modification of parameter 

values (recharge), and the addition of new parameters (to represent the fill material not present 

in the history-matching models). In many cases, an entirely new model may also be required to 

simulate desired regulatory outcomes. For example, while an evaluation of contaminant 

transport was not considered in the history matching period, it may be required in the postconstruction period.

4.6.1 Modifications for Post-Construction Conditions

The primary changes from construction of Units 3 and 4 that would be expected to affect 

groundwater at the Bellefonte site are the removal of existing subsurface materials in the 

excavations of building foundations, the placement of engineered fill materials within the 

excavation and around building foundations, surface grading that affects the site topography, 

and land use changes that affect recharge.

The FSAR (TVA 2009) contains several figures (2.5-347, 2.5-348a, and 2.5-348b) that identify 

the extent of the excavations for Units 3 and 4. Based on this information, and considering 

constraints of the groundwater model grid cell size, the lateral extent of the reactor building 

foundations (Category 1 Structure Excavation) and the surrounding fill material were identified in 

the groundwater model, as shown in Figure 52. The vertical extent of the building foundations is

down to an elevation of 588.6 ft, which puts it into model grid layer 3 (which actually extends to 

an elevation of 584.9 ft). The building foundation zone was set to inactive grid cells (thus no-flow 

boundary conditions). The backfill material around the foundations was assumed to encompass 

layers 1 and 2 (from ground surface to 589.7 ft elevation). A new “Fill” material was defined to 

represent the backfill, with a horizontal hydraulic conductivity of 2.83 ft/d (10-3 cm/s). 

The construction plan for Units 3 and 4 includes modifications to the land use and topography, 

denoted in Figure 52 (see FSAR Figure 2.5-362 for details). Site plans for buildings, parking 

lots, and other developed areas are also apparent, which means that the land use will alter. 

Thus, the recharge zones required modification in the newly developed area, as shown in 

Figure 53. The major buildings and parking lots are distinguished as zones of zero recharge 

(Figure 54). Recharge must be set in MODFLOW to apply to Layer 1 only, otherwise recharge 

might inappropriately be applied in lower layers due to inactive grid cells.

Changes to topography included a cut from part of the hillside to the southeast and fill across 

the reactor area to flatten the site, with a gradual slope away from the power block buildings to 

promote drainage. Several approaches can be used to implement the revised topography 

contours (Figure 52) in the MODFLOW groundwater model. If the MODFLOW model was using 

material definitions with the LPF package, then the changes could be implemented by drawing 

arcs/polygons representing the elevation contours in a GMS Map module coverage, and 

mapping the coverage to layer 1 of the MODFLOW grid. This would provide the correct 

topography and would not change the material designations, which should be appropriate given 

the original approach to mapping the solids model to the top layer (but an assessment to 

confirm the material designations would be appropriate).
4-53

Figure 52. Figures depicting the depth of the reactor building excavation (left), changes 

in topography, the locations of reactor buildings, and the groundwater model 

grid zones representing fill material and reactor building excavations (right).

Figures adapted from FSAR Figure 2.5-348a and Figure 2.5-362.

Figure 53. Refined recharge zones for the post-construction scenario. A portion of the 

Lowlands zone was incorporated into the northern corner of the Developed 

Area zone.

Reactor building 

excavations, with 

surrounding backfill

Topography

Changes
4-54

Figure 54. Details of the post-construction recharge zones around Units 3 and 4. Reactor 

and other significant buildings and the main parking areas are set to zero 

recharge. Note that the colors are simply used to distinguish zones and are 

unrelated to materials.

When using the HUF package for the MODFLOW groundwater model, the process is 

complicated by the uncoupled nature of the HUF hydrogeological units and the model grid 

elevations, so a multi-step approach is required. One approach is to define the new topography 

elevations as a scatter data set, Map module points/arcs, or directly as a TIN. The goal is to 

define a TIN that represents the post-construction ground surface elevations. This TIN can be 

used with flat planes above and below to create solids for clipping/filling the existing solids 

model (Section 4.1.2.3, Figure 31). The adjusted solids model can then be mapped to the HUF 

package of the MODFLOW groundwater model as described in Section 4.1.2.4. A second 

approach, applied in this example, is to manipulate the HUF package data arrays for potentially 

affected units and the grid elevation data arrays. Data arrays for grid layer elevations (i.e., the 

top elevation of layer 1) and for the HUF unit top elevations and thicknesses can be exported to

the GMS 2D Grid module. Data arrays can be manipulated directly in GMS using the GMS Data 

Calculator. The Data Calculator could be used to merge the Soil unit top elevation with the new 

topography (i.e., mapped to the MODFLOW grid as described above for the LPF package). That 

said, it may be more efficient to copy the data arrays into an external spreadsheet, perform 

manipulation/calculations in the spreadsheet, then copy the adjusted arrays back to GMS. The 

adjusted data arrays should be transferred (internally using the “2D Grid -> Layer” command or 

by pasting data from an external source) to the hydrogeological unit definitions for the HUF 

package. Note that both top elevations and thicknesses of the hydrogeological units must be 

adjusted. The fill material (discussed above) must be added as a new HUF hydrogeological unit. 

The resulting changes for the post construction model topography are depicted in Figure 55.

Town Creek

Lowlands

Developed Area

Addition to

Developed

Area

Units 3 & 4 Parking

Maintenance & Admin. Bldgs. Access Portal

Highlands

near River
4-55

Figure 55. Pre-construction topography and materials (left) and modifications to 

represent the post-construction scenario (right) in the vicinity of Units 3 and 

4. The topography was flattened at the site and the Category 1 building 

foundations were represented by inactive grid cells surrounded by fill 

material. View is nominally to the ENE with a 10X vertical exaggeration.

4.6.2 Post-Construction Model Results

Simulations were conducted with the post-construction groundwater model configuration to 

assess the impact of the topographic and subsurface changes on the hydraulic heads and the 

groundwater flow directions. Figure 56 shows the resulting hydraulic head contours for the postconstruction H4R1D1 simulation (compare to the pre-construction results in Figure 40). Figure 

57 shows a zoomed view and compares pre-construction results to post-construction results. 

The differences are small and are localized around the Bellefonte site. The significance of the 

differences with respect to estimating maximum groundwater head (and its uncertainty) is 

evaluated in the following section.
4-56

Figure 56. Hydraulic head contours for model H4R1D1 with the post-construction model 

configuration modifications. The purple rectangles indicate the location of 

Units 3 and 4, where inactive cells represent building foundations and fill 

material surrounds the foundations.

Figure 57. Comparison of pre-construction (left) and post-construction scenario 

hydraulic head contours for model H4R1D1. Contour intervals are 1 ft.

4.6.3 Predictive Uncertainty

Predictive uncertainties for this example were estimated using linear methods to calculate the 

mean and variance of the predicted values resulting from parameter uncertainties. As discussed 
4-57

in Section 3.2, GMS has a Monte Carlo simulation capability, but it was not used due to the 

computational requirements of Monte Carlo simulation and the limitations of the GMS 

implementation (a limited set of applicable parameters, limited distribution types, and no 

parameter correlations). The linear uncertainty estimates used here [Eq. (12)] are 

computationally quick, requiring only the parameter covariance (available from the calibration of 

the history-matching model) and prediction sensitivities computed with the post-construction 

predictive model. 

4.6.3.1 Software Used in Estimating Predictive Uncertainty

The PEST predunc6 utility implements the methods of Moore and Doherty (2005). The utility 

calculates the parameter covariance matrix using observation weights from a PEST control file 

and the Jacobian matrix file associated with that control file. Both of these files have to reflect 

the calibration conditions. Prediction sensitivities are read by predunc6 from a separate 

Jacobian file produced by a PEST execution under the predictive conditions. One of the 

requirements of predunc6 is that the parameter sets for all three files (calibration control file, 

calibration Jacobian file, and prediction Jacobian file) must be consistent. This means that the 

predictive conditions can’t include parameters that are not also included in the calibration. As 

discussed above, this condition is likely to be violated for a nuclear power plant application due 

to the introduction of new materials (e.g., excavation fill) and simulation requirements (e.g.,

contaminant transport) under post-construction conditions. This limits the applicability of 

predunc6. 

It is also worth noting that one of the inputs required by the predunc6 utility is the prior 

parameter covariance [the C(p) matrix in the PEST documentation] supplied by the user. A 

result equivalent to Eq. (12) will be returned by predunc6 when the prior parameter variances

are much larger than the parameter estimation variances resulting from the calibration [i.e., 

when the diagonal elements of C(p) are sufficiently large]. 

Predictive uncertainty results presented below were computed using the UCODE 

Linear_Uncertainty utility. This utility reads the parameter covariance matrix directly from 

UCODE calibration results. Because the calibration was completed using PEST, this required

translating the PEST control file from the calibration to a UCODE input file and executing a 

UCODE sensitivity run at the optimal parameter values. This produced the needed UCODE 

parameter covariance matrix for the calibration (pre-construction) model. A UCODE sensitivity 

run was also completed with the post-construction model to generate the prediction sensitivities. 

For this example, the UCODE input file for the post-construction predictive sensitivity run was 

translated from a PEST control file generated in GMS. UCODE can easily accommodate 

parameters in the predictive model that were not used in the calibration model. It does this by 

supplementing the parameter covariance matrix read from the calibration output with the 

variances of any additional parameters used in the prediction sensitivity run. UCODE uses prior 

information contained in the input file for the prediction sensitivity run to determine the variances 

of the additional parameters.

There are some limitations of GMS when working with predicted quantities for which 

uncertainties are desired. When working within GMS, contributions to uncertainty are limited to 

parameters that are allowed in MODFLOW as designated parameters (i.e., that can be specified 

with key values). Similarly, predicted values must be quantities that can be specified as 

observations in MODFLOW. In particular, results from other codes available in GMS (e.g., 

MODPATH and MT3DMS) can’t be used with PEST and therefore can’t be easily used to 

calculate predictive uncertainty. Using models prepared within GMS, but working outside GMS

to estimate predictive uncertainty, one could use any model parameter as a contributor to 
4-58

uncertainty and any results as a predicted value. However, this would require the user to access

the parameter input files and model output files to create the needed template and instruction 

files by hand instead of relying on GMS to do so.

4.6.3.2 Predictive Uncertainty Results

For this example, the predicted quantities were groundwater heads at three locations at the 

corners of each proposed reactor unit. These locations are indicated on Figure 58 as the yellow 

dots. Note that the desired predicted heads are within the fill material (colored gray in the figure) 

adjacent to the buildings. These locations are appropriate for estimating maximum groundwater 

levels required as part of a review under SRP 2.4.12. Based on the head contours shown in 

Figure 58, the predicted head at the location indicated by the arrow will be the largest of the six 

predicted values.

The assumption of steady-state flow with the use of average observed heads as calibration 

targets for the history-matching models is non-conservative when the modeling objective is to 

estimate maximum groundwater head. The average observed heads used as calibration targets

were about two to three feet (0.6 to 0.9 m) less than the maximum observed heads for the 

2006/2007 observation period. The results from the calibration, however, indicated that the 

simulated heads were generally larger than the average observed heads by about the same 

amount, at least in the area of the desired predicted head; Table 8 provides results for several 

wells in this area.

In this case, the errors in the model fortuitously provide an unquantified degree of conservatism 

countering the steady-state assumption. To account for an additional quantifiable degree of 

conservatism, the upper limit of an estimated 95% confidence interval (with an exceedance 

probability of 2.5%) for the predicted head was evaluated. This value could be compared to the 

design limit in a safety evaluation. If the limit was exceeded or if the margin was small, a more 

reliable analysis could be carried out by obtaining additional site data and completing an 

unsteady flow analysis. 

For the estimation of predictive head uncertainty, the hydraulic conductivity and recharge 

uncertainties from the calibration were included (through the parameter estimation covariance). 

Two sets of prediction sensitivity runs were completed. In one, no additional post-construction 

parameters contributed to the predictive uncertainty. In the second, the hydraulic conductivity of 

the fill material contributed to the predictive variance through the use of a prior parameter 

equation in the UCODE input file. The fill hydraulic conductivity was assumed to be lognormally 

distributed with a geometric mean value of 2.8 ft/d (10-3 cm/s) and a 95% probability interval of 

plus and minus one order of magnitude from the mean. The resulting distribution is shown in 

Figure 59. Predictive uncertainties for each of the alternative models with significant model 

probabilities were computed with the UCODE Linear_Uncertainty utility. Model averaged 

predictive uncertainties were calculated in Excel (see Figure C-3 in Appendix C).

Predicted heads at the location indicated with the arrow in Figure 58 and the prediction 

uncertainties are shown in Figure 60. In this figure the top of the colored bars are the predicted 

head values for the four alternative models assigned a probability of at least 5% by either the 

AICc or KIC model selection criterion. Models H4R1D1 and H4R1D2 predict similar values 

indicating that the depth of flow has little impact on the head near the reactor units. As noted 

previously, the depth of flow may have a much greater impact on a different predicted quantity, 

for example travel time from the reactor area to Town Creek. H3R2D1 and H3R2D2 also have 

similar predicted head values. However, these models with three hydraulic conductivity zones 

and four recharge zones predict a head more than two feet higher than models with four 
4-59

hydraulic conductivity zones and one recharge zone. The model-averaged predicted values are 

shown for model probabilities calculated from both AICc and KIC. Because H3R2D1 and 

H3R2D2 were given negligible probabilities by KIC, the larger predicted values do not contribute 

to the KIC model-averaged predicted head. This is not the case for the AICc model-averaged 

head since the two H3 models have a combined 27% probability. 

Figure 58. Detail from the post-construction groundwater model showing the locations 

of the predicted heads at the corners of the reactor units

Table 8. Averaged observed head (calibration target), maximum observed head, and 

simulated equivalent head from the calibrated H4R1D1 model for three wells 

near the desired location of predicted head. All heads in feet.

MW-1205c MW-1213c MW-1202a

Average Observed Head 609.7 609.8 607.2

Maximum Observed Head 613.3 612.6 610.7

Calibrated Head 614.7 612.6 609.8
4-60

Figure 59. Prior parameter distribution for the hydraulic conductivity of the fill material

The error bars in Figure 60 are the upper limits of the 95% confidence intervals computed from 

the predicted value variances. The three different colors indicate the contribution to that 

variance: parameter uncertainty only as given by Eq. (12) (shown in blue), or parameter 

uncertainty plus model uncertainty as given by the squared quantity on the right hand side of 

Eq. (16) (shown in red and green). The contribution from model uncertainty is the variance of 

the individual model prediction about the model-averaged mean prediction. The modelaveraged mean prediction [Eq. (15)] used in Eq. (16) can be calculated using either AICc or

KIC; results from both are shown in Figure 60. For the H4 models, model uncertainty has a 

minor effect when using AICc and no effect when using KIC. For the H3 models, the effect of 

model uncertainty is significant. This reflects the fact that the model-averaged predicted values 

are much closer to the H4 individual model predictions than to the H3 individual model 

predictions. 

Model-average uncertainty (indicated by the error bars on the rightmost group of Figure 60) is 

equivalent to the H4 individual model results when using KIC. For AICc, the model-average 

uncertainty is influenced by the larger uncertainties of the H3 models since 27% of the 

probability is assigned to these models, as discussed above. The upper limit of the 95% 

confidence interval is 3.5 or 4.0 ft. larger than the model-average predicted value (depending on 

whether KIC or AICc is used). The inclusion of uncertainty in the groundwater head value for 

comparison with the design criterion thus adds a considerable degree of conservatism to the 

evaluation.

Figure 61 shows the model-averaged predictions and uncertainty measure for the case in which 

the uncertainty of the fill material hydraulic conductivity was included (w/ Fill, equivalent to the

rightmost group of Figure 60) and the case in which the hydraulic conductivity was assumed to 

be known and assigned its geometric mean value (w/o Fill). The predicted values (top of the 

colored bars) are unaffected by the inclusion of uncertainty in the fill conductivity since both 

results use the mean value. The upper limit of the 95% confidence interval is increased by the 

inclusion of fill conductivity uncertainty, as much as by one foot when using KIC. Whether or not 

this is significant would depend on the margin between the upper limit and the design criterion. 

The results of the example demonstrate the potential importance of even localized changes to 

the model under post-construction conditions. The fill material occupies a small portion of the 

model domain, but it is a potentially critical feature given that the design criterion for maximum 

groundwater head is within the excavation. In this case, including a reasonable uncertainty 

0.1 1 10 100 0

0.2

0.4

0.6

0.8

Fill Ks (ft/d)

P(Fill Ks <=x)
4-61

estimate for this material of limited extent had a noticeable impact on the predicted regulatory 

value to be compared to this design criterion.

Figure 60. Predicted heads (top of colored bars) and 95% confidence interval upper

limits for the individual models and the model average

Figure 61. Effect on model-averaged predicted head of including the fill material 

hydraulic conductivity uncertainty

5-1

5 CONCLUSIONS

This report describes a methodology to assess predictive uncertainty in hydrologic model 

analyses carried out in support of safety and environmental evaluations of nuclear reactor 

applications. Current review criteria and technical procedures contained in NRC staff guidance

(e.g., NRC, 2007) describe a predominantly deterministic approach that emphasizes 

conservatism in the analysis. It is recommended in this report that the focus be on making 

conservative decisions instead of focusing on the conservatism of models (e.g., bounding 

assumptions and extreme parameter values). A focus on conservative decisions requires that

uncertainties in model outcomes be properly accounted for when comparing these outcomes to 

regulatory or design requirements, as illustrated in the report using a simple groundwater 

transport modeling example. This is consistent with a risk-informed approach, and it allows for 

the natural consideration of consequences (e.g., the safety consequences of exceeding a 

design flood elevation).

The uncertainty methodology described in this report considers the combined impact of 

uncertainties associated with the conceptual-mathematical basis of a model, the model 

parameters, and the scenarios to which the model is applied. The methodology was applied to 

groundwater issues commonly arising in the review of nuclear power plant applications under 

Section 2.4.12 of the Standard Review Plan (NRC 2007). This example used publicly available 

data from a combined license application. Some of the conclusions from the example are 

provided here.

Modeling expectations should be explicitly discussed in regulatory guidance and review plans, 

including the assessment of uncertainties and the consistent application of procedures, e.g., the 

discussion in ANS 2.17 (ANS 2010) and the guidelines for effective modeling in Hill and 

Tiedeman (2007). This may result in better integration of site characterization and modeling, 

and fewer requests for additional information.

Following standard procedures for uncertainty analyses can reveal limitations of the hydrologic 

analyses, e.g., unexamined parameter correlations and outstanding data needs. In addition, the 

use of standard procedures promotes consistent reviews.

To the extent possible, existing software was used to implement the uncertainty methodology. 

Parameter estimation and uncertainty calculations were primarily carried out using UCODE and 

PEST. UCODE provides a well-integrated set of essential tools. PEST has advanced 

capabilities, but is less well integrated and the version used here has a limitation when adding 

uncertain parameters to the predictive model. Uncertainty software is incompletely integrated 

with modeling platforms such as GMS and thus requires hand manipulation of files and 

completion of tasks outside of GMS. Calculation of model selection criteria for multi-model 

analysis was completed using custom spreadsheets because, for one of the criteria, the 

UCODE and PEST calculations depend on whether parameters are log transformed in the 

model calibrations. Calculation of model probabilities and model-averaged predictions were also 

completed in these spreadsheets.

Predictive models should include not only the uncertainties evaluated in the history-matching 

period (through model calibration), but also any additional uncertainties that might apply only in 

the predictive period. The example demonstrated the potential importance of even localized 

changes to the model under post-construction conditions. Including a reasonable uncertainty 

estimate for the hydraulic conductivity of the fill material, which occupied a very limited extent of 
5-2

the model domain, had a noticeable impact on the predicted regulatory value to be compared to 

the design criterion. 

The example also demonstrated that not all model uncertainties can be quantified. Data 

limitations led to the use of a steady-state groundwater flow model, a non-conservative choice 

when evaluating maximum groundwater head. It was important in this case to acknowledge the 

additional uncertainty this decision introduced to the analysis and to qualitatively evaluate its

effect on the uncertainty in the predicted value. 

Out of the sixteen alternative models considered in the multi-model analysis, four were assigned 

substantial model probabilities by the methodology. Model-averaged predicted values and 

predictive uncertainty were found to depend on the model selection criterion used to calculate 

the model probabilities. These results are consistent with other studies. Since no single criterion 

has been shown to be superior in all cases, multiple criteria should be used.

Explicit consideration of alternative models is likely more valuable than quantitative model 

averaging. However, model averaging has been shown in other studies to improve model 

predictive performance over the use of a single model. Using linear uncertainty methods, model 

averages can be easily completed and should be considered. 

Relatively simple and practical approaches to uncertainty evaluation should be integrated into 

the hydrologic analyses of NRC staff. These approaches contribute to estimates of the degree 

of conservatism in the analysis, demonstrate an understanding of the systems being evaluated, 

and provide evidence of the staff’s due diligence.
6-1

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APPENDIX A

Dependence of KIC on Log Transformation of Parameters

A-1

APPENDIX A

Dependence of KIC on Log Transformation of Parameters

When calculating the model selection criterion KIC from least squares optimization results, 

parameter transformations need to be considered. PEST and UCODE allow parameters to be 

log-transformed during the regression. Both codes report the parameter covariance matrix in the 

transformed space using log base 10. As indicated by Meyer et al. (2007), multi-model 

comparisons using KIC must use untransformed parameter covariances. PEST and UCODE, 

however, report KIC values computed with transformed parameters. One solution to this 

problem is to recompute the covariance matrix with all parameters untransformed and fixed at 

the optimal values (obtained with some or all of the parameters log-transformed). Alternatively, 

the untransformed KIC values can be calculated directly from the transformed parameter 

covariance matrix as described below.

Ye et al. (2008) give an expression for calculating KIC,

2 ˆ ln 2ln ( ) ˆ

ln 2 ln | |

z ML

ML

KIC N p

P

σ

π

= −

− −

θ

Σ

(A.1)

where Nz is the number of observations, P is the number of estimated hydrologic parameters, ˆ

θ

is the vector of maximum likelihood parameter estimates,

T

2

ˆ

ˆ

k k

ML Nz

σ

=

=

θ θ

e ωe is the maximum 

likelihood estimate of the common observation error variance, = −ˆ * e zz is the vector of residuals 

(the difference between zˆ , the simulated observation evaluated at ˆ

θ , and the observed values, 

), and ML Σ is the parameter covariance matrix evaluated at ˆ

θ . Because ML Σ is a function of 

the parameter transformations under which regression is carried out, it must be evaluated 

without transformations for KIC calculation. Since 

2 ln ln ˆ ln T

ML k k

ML Σ = P σ − J ωJ (A.2)

this entails evaluating the Jacobian matrix, J , without transformations. 
A-2

For a model characterized by parameters , 1,..., j θ j P = , J has elements i

ij

j

z J

θ

∂ = ∂

. For a 

parameter that is log-transformed (base 10) during regression1

, ln10

log

i i

ij j

j j

z z J θ

θ θ

∂ ∂ = = ∂ ∂

 . As a 

result,

2 ln(ln10) 2

ln

ˆ ln ln

T

k k

T

j k k

j

L θ ∈= + ∑ +

J ωJ

J ωJ

 

L

(A.3)

where L is the set of L parameters that are log-transformed during regression. As a result,

2 ln(ln10) 2

ln

ˆ ln ln

ML

ML

j

j

L θ ∈

= + + ∑

Σ

Σ

L

(A.4)

and

2 ln(ln10) 2 ˆ ln j

j

KIC

KIC L θ ∈

=  + + ∑

L

(A.5)

where ML Σ and KIC are the parameter covariance matrix and KIC value reported by PEST or 

UCODE with log-transformed parameters. 

 

1 if β θ = log , then β θ ln10 ln = , and ( ) 1

β θθ θ log ln10 −∂ =∂ = ∂
APPENDIX B

Data Used in Model Development

B-1

Table B-1. List of borehole data used to define the surface of contacts between 

geological units

Elevation of Contact Between Specified Units e 

(ft)

Borehole 

Name a

Total 

Depth

(ft)

Northing b

(ft)

Easting b

(ft)

Surface 

Elevation

(ft)

Elev. of 

Top of 

Weathered 

Rock c (ft)

Elev. of 

Top of 

Competent 

Rock d (ft)

NV/

USR

USR/

A A/B B/C C/D D/E E/F F/LSR

B-1000 185.9 1533129.7 628373.5 611.9 595.4 590.9 503 483 434

B-1001A 17.2 1533081.9 628359.4 611.3 594.5 583.3

B-1002 120.5 1532906.7 628471.5 613.3 597.3 592.8 583

B-1003 120.4 1532838.5 628359.6 613.6 598.8 587.4 591

B-1004 150.2 1532950.0 628276.6 611.1 598.0 597.9 490 469

B-1005 251 1532943.5 628407.1 608.9 598.0 592.9 465 444 394

B-1006 176 1532755.0 628259.7 614.5 595.7 588.9 460

B-1007 120 1533041.2 628430.9 608.4 600.3 591.9 D

B-1008 121.3 1533048.9 628211.6 614.1 598.1 592.6 519 498

B-1009 75.8 1532957.3 628130.1 613.9 595.2 595.1 D

B-1010 74 1532836.3 628178.5 611.7 598.2 595.5 D

B-1011 75.3 1532610.4 628132.4 615.6 599.1 599.0 591

B-1012 50 1532678.0 628247.8 616.7 602.8 600.2 585

B-1013 50 1532758.8 628315.6 614.8 603.2 593.8 584

B-1014 121.2 1532984.8 628354.3 609.1 604.1 598.6 D

B-1015 75.4 1533020.6 628276.3 611.5 599.0 598.5 D

B-1016 20.7 1532867.3 628224.3 611.6 595.3 594.1 D

B-1017 50.6 1532937.4 628225.4 611.9 592.9 588.4 D

B-1018 16.5 1532827.3 628217.6 612.7 603.1 596.5 D

B-1019 35 1532716.2 628178.6 613.7 597.3 596.9 D

B-1020 35 1532590.2 628188.0 616.7 606.7 595.7 C

B-1021 35.4 1532997.4 628462.5 608.6 597.3 583.3 D
B-2

Elevation of Contact Between Specified Units e 

(ft)

Borehole 

Name a

Total 

Depth

(ft)

Northing b

(ft)

Easting b

(ft)

Surface 

Elevation

(ft)

Elev. of 

Top of 

Weathered 

Rock c (ft)

Elev. of 

Top of 

Competent 

Rock d (ft)

NV/

USR

USR/

A A/B B/C C/D D/E E/F F/LSR

B-1022 76.1 1532633.2 628366.9 617.4 598.9 595.9 549

B-1023 60.1 1532853.9 628421.0 613.4 602.3 597.4 582

B-1024 175.4 1532702.4 628492.0 617.1 602.6 593.7 540

B-1025 75 1532823.2 628567.9 616.9 600.9 590.6 547

B-1026 50 1533363.2 628319.4 608.6 589.9 562.6 D

B-1027 50 1532755.9 627608.5 608.6 596.2 592.3 578

B-1028 88.1 1532272.0 628314.6 624.9 612.6 589.9 566

B-1029 35.9 1533050.8 628111.8 612.6 607.6 603.8 D

B-1031 35 1532538.5 628081.8 616.9 601.9 590.5 594

B-1032 175.6 1533054.2 628254.8 613.0 592.0 581.8 513 492 443

B-1033 121.5 1532169.8 628888.6 625.9 613.4 609.4 556

B-1034 249.5 1532442.6 629037.0 622.0 608.0 601.0 579 454 385

B-1035 171.1 1532549.5 629112.9 616.1 603.5 600.1 585 461

B-1036 55.3 1532263.9 628856.1 625.2 609.2 594.0 582 457

B-1037 149.7 1532329.0 628980.8 621.4 607.4 606.6 569

B-1038 65.8 1532091.5 628812.5 629.4 616.7 611.0 A

B-1039 120 1532472.2 628859.9 620.2 610.2 610.1 B

B-1040 119.7 1532554.0 629061.0 616.9 605.7 603.5 596

B-1041 120.3 1532544.9 628845.0 620.1 608.6 605.1 510

B-1042 124.3 1532377.2 628935.9 622.0 599.0 593.0 585

B-1043A 75.6 1532424.3 628756.8 622.2 607.7 606.7 B

B-1044 76 1532399.8 629086.6 618.7 610.7 602.7 563

B-1045 75.1 1532482.4 628986.7 622.4 603.9 596.3 596

B-1046 49.4 1532242.9 628945.3 625.5 601.9 599.1 A

B-1047 120 1532622.2 628995.6 619.2 607.7 604.2 B
B-3

Elevation of Contact Between Specified Units e 

(ft)

Borehole 

Name a

Total 

Depth

(ft)

Northing b

(ft)

Easting b

(ft)

Surface 

Elevation

(ft)

Elev. of 

Top of 

Weathered 

Rock c (ft)

Elev. of 

Top of 

Competent 

Rock d (ft)

NV/

USR

USR/

A A/B B/C C/D D/E E/F F/LSR

B-1048 76.3 1532210.9 628709.9 627.8 608.3 602.3 601

B-1049 128 1532539.2 628917.1 617.3 607.9 592.3 496

B-1050 75 1532300.7 628780.2 624.1 610.1 603.9 603

B-1051A 51.9 1532159.7 628729.8 629.1 608.5 590.4 586

B-1052 79 1532330.9 628912.5 621.8 595.1 567.8 580

B-1053 35 1532489.4 628910.4 620.2 608.3 599.6 B

B-1054 35 1532232.5 628800.2 628.2 611.3 593.2 A

B-1055 71.6 1532492.0 629080.8 619.6 608.1 600.8 581

B-1056 45.7 1532231.0 628882.6 626.9 606.9 581.2 A

B-1057 35.2 1532372.8 628836.1 621.6 608.0 603.1 606

B-1058 65.2 1532351.3 629027.4 620.2 612.4 584.9 564

B-1059 250 1532712.7 628706.2 619.0 605.5 593.5 569 500

B-1060 176.3 1532444.1 629267.9 621.7 605.3 602.7 600 534

B-1061 75.2 1532124.2 629092.1 630.2 622.6 620.6 573

B-1062 150.8 1532323.8 629378.9 627.7 618.8 556.7 555 489

B-1063 50.2 1531925.4 628928.3 634.7 630.7 630.6 USR

B-1064 50 1532447.5 628574.8 623.9 609.7 602.6 B

B-1065 41.5 1532496.4 629393.6 628.1 610.6 587.8 USR

B-1066 124.4 1532293.3 629209.1 626.4 616.9 587.4 583 518

B-1067 174.6 1532868.7 628839.9 614.4 605.6 604.9 574 506

B-1068 50 1532280.8 627961.6 617.2 605.0 586.0 C

B-1069 50 1531732.6 628733.3 635.8 626.5 594.8 USR

B-1070 51 1530825.5 628348.8 647.2 644.0 596.2 601

B-1071 75 1531172.7 626508.3 614.5 596.5 588.3 D

B-1072 50.1 1530152.0 627504.4 627.1 620.3 596.0 USR
B-4

Elevation of Contact Between Specified Units e 

(ft)

Borehole 

Name a

Total 

Depth

(ft)

Northing b

(ft)

Easting b

(ft)

Surface 

Elevation

(ft)

Elev. of 

Top of 

Weathered 

Rock c (ft)

Elev. of 

Top of 

Competent 

Rock d (ft)

NV/

USR

USR/

A A/B B/C C/D D/E E/F F/LSR

B-1073 75.3 1530642.8 627276.0 629.2 604.5 597.1 B

B-1074 50 1530685.8 627635.9 637.4 631.7 596.4 621

B-1075 75 1530444.8 627150.2 626.2 612.7 611.2 B

B-1076 50 1532710.0 628084.9 612.7 604.4 592.0 D

B-1077 15.8 1532870.6 626759.1 602.1 590.4 575.6 LSR

B-1078 75 1533819.2 627721.5 603.6 591.5 588.6 LSR

B-1079 150 1533742.4 628277.0 604.8 593.0 592.9 547

B-1080 35 1532389.9 630169.0 615.8 605.9 580.8 NV

B-1081 33.5 1532026.1 628657.8 625.7 614.2 597.2 A

B-1082 75 1532535.4 628502.6 621.0 613.6 584.3 574

B-1083 75 1532571.1 629135.1 615.0 601.6 600.0 585

B-1084 50.3 1532586.4 628991.9 614.8 607.7 602.1 B

B-1085 6.9 1532807.8 627935.7 608.5 601.6 600.0 D

B-1086 75.3 1533155.7 628520.7 607.2 599.7 599.6 D

B-1087 35.4 1532677.2 629071.3 613.0 609.0 600.5 B

B-1088 71.2 1533043.3 628497.2 608.1 599.6 597.1 D

B-1089 40 1531668.7 628473.1 630.5 614.3 595.0 A

B-1090 128 1533200.9 628455.8 609.9 603.4 586.9 D

B-1091 35 1531925.1 629403.7 641.9 627.8 620.9 NV

B-1092 56 1532370.7 628902.4 621.6 604.8 600.6 592

B-1094 35 1533085.6 628571.3 607.6 596.0 591.6 D

B-1095 50.1 1532077.1 628968.9 630.0 617.4 599.0 588

B-1096 50.1 1532615.5 629217.0 616.5 607.5 605.9 577

B-1097 50 1531841.7 628482.3 625.3 614.1 612.9 A

B-1098 50 1532350.7 627838.6 616.4 601.0 589.9 D
B-5

Elevation of Contact Between Specified Units e 

(ft)

Borehole 

Name a

Total 

Depth

(ft)

Northing b

(ft)

Easting b

(ft)

Surface 

Elevation

(ft)

Elev. of 

Top of 

Weathered 

Rock c (ft)

Elev. of 

Top of 

Competent 

Rock d (ft)

NV/

USR

USR/

A A/B B/C C/D D/E E/F F/LSR

B-1099 50 1533127.1 627932.7 604.8 595.6 578.8 573

a Boring information (names, depth, coordinates) was taken from FSAR Table 2.5-227.

b State Plane Coordinate System (Alabama East), NAD83.

c Top of Weathered Rock is the elevation of the Standard Penetration Test (SPT) refusal.

d Top of Competent Rock is the elevation at which Rock Quality Designation (RQD) > 70, weathering = Fresh, and 

below which no significant weathered intervals exist. In the absence of such data, this elevation was arbitrarily set 

to 0.1 ft. below the top of the weathered rock.

e The elevation of the contact between stratigraphic units is based on the borehole logs (see FSAR Appendix 2BB 

and Fig. 2.5-306). A text entry of a unit name indicates that the unit was encountered, but a contact between units 

could not be discerned. NV = Nashville Unit; USR = Upper Stones River Unit; A-F = Middle Stones River (MSR) 

Units; LSR = Lower Stones River Unit.
B-6

Table B-2. List of monitoring wells and their characteristics (based on FSAR Table 

2.4.12-203)

Monitoring

Well Name

Northing a

(ft)

Easting a

(ft)

Reference 

Elevation b

(ft)

Ground 

Elevation b

(ft)

Well

Depth c

(ft bre)

Screen

Length (ft)

Top of 

Screen b (ft)

Bottom of 

Screen b,d

(ft)

Boring

Depth c

(ft bgs)

MW-1201a 1532950 628276.6 613.91 611.05 12.91 5 606.45 601.45 10.05

MW-1201b 1532950 628276.6 613.78 611.04 77.81 10 546.42 536.42 75.07

MW-1201c 1532950 628276.6 613.65 610.91 119 20 515.1 495.1 116.26

MW-1202a 1532571.1 629135.1 617.52 614.99 15.42 5 607.55 602.55 12.89

MW-1202c 1532571.1 629135.1 617.62 614.93 53 10 575.07 565.07 50.31

MW-1203a 1532718.7 628698.8 621.93 619.02 12.57 5 614.81 609.81 9.66

MW-1203b 1532718.7 628698.8 621.86 619.14 32.9 10 599.41 589.41 30.18

MW-1203c 1532718.7 628698.8 621.7 619.04 121 20 521.15 501.15 118.34

MW-1204a 1532472.2 628859.9 623.1 620.45 12.95 5 615.6 610.6 10.3

MW-1204b 1532472.2 628859.9 623.16 620.48 53.2 10 580.41 570.41 50.52

MW-1204c 1532472.2 628859.9 623.1 620.49 124.2 20 519.35 499.35 121.59

MW-1205a 1532290.8 629211.7 629.42 627.04 13 5 621.87 616.87 10.62

MW-1205b 1532290.8 629211.7 629.34 627.01 33.16 10 606.63 596.63 30.83

MW-1205c 1532290.8 629211.7 629.14 626.89 49.11 10 590.48 580.48 46.86

MW-1206b 1530825.5 628348.8 650.35 647.57 27 10 633.8 623.8 24.22

MW-1206c 1530825.5 628348.8 649.95 647.4 52.8 10 607.6 597.6 50.25

MW-1207a 1532280.8 627961.6 619.78 617.09 14.8 5 610.43 605.43 12.11

MW-1207b 1532280.8 627961.6 619.8 617.24 21 5 604.25 599.25 18.44

MW-1207c 1532280.8 627961.6 619.9 617.11 53.45 10 576.9 566.9 50.66

MW-1208a 1531172.7 626508.3 617.33 614.79 18.73 10 609.05 599.05 16.19

MW-1208b 1531172.7 626508.3 617.22 614.72 32.15 5 590.52 585.52 29.65

MW-1208c 1531172.7 626508.3 617.26 614.69 57.95 10 569.76 559.76 55.38

MW-1209b 1530685.8 627635.9 640.39 637.78 28 10 622.84 612.84 25.39

MW-1209c 1530685.8 627635.9 640.44 637.84 53.05 10 597.84 587.84 50.45
B-7

Monitoring

Well Name

Northing a

(ft)

Easting a

(ft)

Reference 

Elevation b

(ft)

Ground 

Elevation b

(ft)

Well

Depth c

(ft bre)

Screen

Length (ft)

Top of 

Screen b (ft)

Bottom of 

Screen b,d

(ft)

Boring

Depth c

(ft bgs)

MW-1210a 1533742.4 628277 607.88 605.03 14.3 5 599.03 594.03 11.45

MW-1210b 1533742.4 628277 608.01 605.04 33.79 10 584.67 574.67 30.82

MW-1210c 1533742.4 628277 607.96 605.15 69.72 10 548.69 538.69 66.91

MW-1211a 1532389.9 630169 618.87 615.89 11.53 5 612.79 607.79 8.55

MW-1211c 1532389.9 630169 618.66 615.76 38.1 10 591.01 581.01 35.2

MW-1212a 1533819.2 627721.5 607 603.98 15.28 5 597.17 592.17 12.26

MW-1212b 1533819.2 627721.5 606.86 604.07 33.55 10 583.76 573.76 30.76

MW-1212c 1533819.2 627721.5 606.79 603.94 64 10 553.24 543.24 61.15

MW-1213b 1532159.7 628729.8 632.02 629.21 40 10 602.47 592.47 37.19

MW-1213c 1532159.7 628729.8 632.2 629.42 50 10 592.65 582.65 47.22

MW-1214a 1532755.9 627608.5 612.23 609.53 14 5 603.68 598.68 11.3

MW-1214b 1532755.9 627608.5 612.09 609.74 22.8 5 594.74 589.74 20.45

MW-1214c 1532755.9 627608.5 612.08 609.54 43.5 10 579.03 569.03 40.96

MW-1215a 1532957.7 628666.65 635.64 632.79 13.25 5 627.84 622.84 10.4

MW-1215b 1531700.65 628603.2 635.63 632.77 33 10 613.08 603.08 30.14

MW-1215c 1531700.65 628603.2 635.6 632.79 52.5 10 593.55 583.55 49.69

MW-1216a 1532870.6 626759.1 604.56 602.57 25.1 10 589.91 579.91 23.11

MW-1216c 1532870.6 626759.1 604.64 602.23 63 10 552.09 542.09 60.59

MW-1217a 1532752.4 628266.8 617.32 614.27 13.34 5 609.43 604.43 10.29

MW-1217b 1532752.4 628266.8 617.1 614.15 33.3 10 594.25 584.25 30.35

MW-1217c 1532752.4 628266.8 617.08 614.14 52.8 10 574.73 564.73 49.86

OW-1 1532758 628739.6 623.33 620.55 82.89 20 560.89 540.89 80.11

OW-2 1532723.1 628659.5 621.2 618.43 92.9 20 548.75 528.75 90.13

OW-3 1532669 628708 622.89 620.13 82.87 20 560.47 540.47 80.11

OW-4 1532751.6 628804.2 623.23 620.4 13.03 5 615.65 610.65 10.2

OW-5 1532819.5 628708.5 621.2 618.34 12.8 5 613.85 608.85 9.94
B-8

Monitoring

Well Name

Northing a

(ft)

Easting a

(ft)

Reference 

Elevation b

(ft)

Ground 

Elevation b

(ft)

Well

Depth c

(ft bre)

Screen

Length (ft)

Top of 

Screen b (ft)

Bottom of 

Screen b,d

(ft)

Boring

Depth c

(ft bgs)

OW-6 1532631.7 628659.6 623.23 620.45 12.9 5 615.78 610.78 10.12

OW-7 1532793.7 628292.6 617.46 614.78 52.85 10 575.06 565.06 50.17

OW-8 1532706.9 628239.1 618 615.33 54.2 10 574.25 564.25 51.53

OW-9 1532802.3 628203.2 615.58 613.1 52.7 10 573.33 563.33 50.22

OW-10 1532835.4 628236.5 616.24 613.31 33.45 10 593.24 583.24 30.52

OW-11 1532839.7 628305.6 616.39 613.71 32.92 10 593.92 583.92 30.24

OW-12 1532780.6 628335.7 617.45 614.76 32.95 10 594.95 584.95 30.26

a State Plane Coordinate System (Alabama East), NAD83. Coordinates are from the nearest borehole 

according to FSAR Figure 2.5-327.

b Relative to the North American Vertical Datum of 1988 (NADV88).

c “bre” = below reference elevation; “bgs” = below ground surface

d A 0.45 ft-long cap is threaded onto the bottom of the screen. Thus the bottom of screen 

elevation equals the reference elevation minus the well depth plus 0.45 ft.

Table B-3. List of the observed hydraulic heads in monitoring wells

Well Name Easting a

(ft)

Northing a

(ft)

Screen Center Elevation b,c

(ft)

Observed 

Head 

Elevation b (ft)

Well Category d

1 627442.0 1540804.0 Assumed to be within Layer 1 609.0 1961

2 627255.0 1540308.0 Assumed to be within Layer 1 615.0 1961

3 627067.0 1539759.0 Assumed to be within Layer 1 617.0 1961

4 626397.0 1536021.0 Assumed to be within Layer 1 605.0 1961

5 625593.0 1537441.0 Assumed to be within Layer 1 619.0 1961

11 623382.0 1534667.0 Assumed to be within Layer 1 620.0 1961

12 620167.0 1533810.0 Assumed to be within Layer 1 609.0 1961

13 621198.0 1533046.0 Assumed to be within Layer 1 595.0 1961

14 623423.0 1531881.0 Assumed to be within Layer 1 591.0 1961

W9 632225.0 1535986.0 Assumed to be within Layer 1 593.9 2005

W19 632188.0 1535312.0 Assumed to be within Layer 1 596.0 2005
B-9

Well Name Easting a

(ft)

Northing a

(ft)

Screen Center Elevation b,c

(ft)

Observed 

Head 

Elevation b (ft)

Well Category d

W22 632075.0 1534726.0 Assumed to be within Layer 1 607.4 2005

W20 631439.0 1533540.0 Assumed to be within Layer 1 597.9 2005

W12 629330.0 1531581.0 Assumed to be within Layer 1 595.8 2005

W30 628843.0 1534102.0 Assumed to be within Layer 1 595.3 2005

WT6 628281.0 1533066.0 Assumed to be within Layer 1 597.3 2005

WT5 628107.0 1532367.0 Assumed to be within Layer 1 605.1 2005

WT2 627333.0 1530919.0 Assumed to be within Layer 1 611.4 2005

WT3 627121.0 1532442.0 Assumed to be within Layer 1 598.3 2005

W16 627121.0 1528723.0 Assumed to be within Layer 1 625.8 2005

WT4 626946.0 1533815.0 Assumed to be within Layer 1 594.6 2005

P2 626572.0 1529659.0 Assumed to be within Layer 1 613.8 2005

P4 626073.0 1528823.0 Assumed to be within Layer 1 614.2 2005

P1 625973.0 1530770.0 Assumed to be within Layer 1 602.8 2005

WT1 625711.0 1531094.0 Assumed to be within Layer 1 603.7 2005

W17 625349.0 1529035.0 Assumed to be within Layer 1 610.8 2005

P3 625074.0 1530470.0 Assumed to be within Layer 1 597.5 2005

MW-1203b 628698.8 1532718.7 594.4 609.4 Average, Bedrock

MW-1204c 628859.9 1532472.2 509.4 609.6 Average, Bedrock

MW-1205b 629211.7 1532290.8 601.6 609.3 Average, Bedrock

MW-1205c 629211.7 1532290.8 585.5 609.7 Average, Bedrock

MW-1206b 628348.8 1530825.5 628.8 639.9 Average, Bedrock

MW-1206c 628348.8 1530825.5 602.6 634.5 Average, Bedrock

MW-1207c 627961.6 1532280.8 571.9 602.7 Average, Bedrock

MW-1208b 626508.3 1531172.7 588.0 607.7 Average, Bedrock

MW-1208c 626508.3 1531172.7 564.8 607.0 Average, Bedrock
B-10

Well Name Easting a

(ft)

Northing a

(ft)

Screen Center Elevation b,c

(ft)

Observed 

Head 

Elevation b (ft)

Well Category d

MW-1209b 627635.9 1530685.8 617.8 619.2 Average, Bedrock

MW-1210b 628277.0 1533742.4 579.7 597.9 Average, Bedrock

MW-1211c 630169.0 1532389.9 586.0 609.9 Average, Bedrock

MW-1212b 627721.5 1533819.2 578.8 596.8 Average, Bedrock

MW-1212c 627721.5 1533819.2 548.2 595.5 Average, Bedrock

MW-1213b 628729.8 1532159.7 597.5 609.5 Average, Bedrock

MW-1213c 628729.8 1532159.7 587.7 609.8 Average, Bedrock

MW-1214b 627608.5 1532755.9 592.2 600.7 Average, Bedrock

MW-1214c 627608.5 1532755.9 574.0 600.7 Average, Bedrock

MW-1215b 628603.2 1531700.7 608.1 614.7 Average, Bedrock

MW-1215c 628603.2 1531700.7 588.6 614.4 Average, Bedrock

MW-1216c 626759.1 1532870.6 547.1 597.3 Average, Bedrock

MW-1217b 628266.8 1532752.4 589.3 600.8 Average, Bedrock

OW-11 628305.6 1532839.7 588.9 600.7 Average, Bedrock

OW-12 628335.7 1532780.6 590.0 600.9 Average, Bedrock

OW-8 628239.1 1532706.9 569.3 601.3 Average, Bedrock

MW-1202a e 629135.1 1532571.1 605.1 607.2 Average, Soil

MW-1208a 626508.3 1531172.7 604.1 607.8 Average, Soil

MW-1210a 628277.0 1533742.4 596.5 599.7 Average, Soil

MW-1212a 627721.5 1533819.2 594.7 597.5 Average, Soil

MW-1214a 627608.5 1532755.9 601.2 602.7 Average, Soil

MW-1216a 626759.1 1532870.6 584.9 597.4 Average, Soil

MW-1204a 628859.9 1532472.2 610.6 613.0 Average, Soil

MW-1217a 628266.8 1532752.4 604.4 604.8 Average, Soil

OW-4 628804.2 1532751.6 613.2 613.6 Average, Soil
B-11

Well Name Easting a

(ft)

Northing a

(ft)

Screen Center Elevation b,c

(ft)

Observed 

Head 

Elevation b (ft)

Well Category d

OW-5 628708.5 1532819.5 611.4 612.5 Average, Soil

OW-6 628659.6 1532631.7 613.3 616.0 Average, Soil

a State Plane Coordinate System (Alabama East), NAD83. Coordinates are from the nearest borehole 

according to FSAR Figure 2.5-327.

b Relative to the North American Vertical Datum of 1988 (NADV88).

c Because of the data source for the 1961 and September 2005 data, no screen elevation information was 

available. Hence these wells were assigned to model Layer 1. The Average Bedrock and Average Soil 

data points are assigned to the appropriate model layer based on the screen center elevation, thus the 

categorization may differ from the effective location (i.e., a well may be in the “Bedrock” category, but the 

screen center could potentially fall within a model Layer 1 grid cell whose material designation is Soil or 

Weathered Rock.

d The 1961 data consists of the values shown on FSAR [TVA, 2009] Figure 2.4.12-204. The September 2005 

data consists of the values shown on FSAR Figure 2.4.12-210. Average Soil and Average Bedrock are 

based on data from the FSAR Table 2.4.12-204 as described in Section 1.2.5. These four categorizations 

are used to group wells for viewing calibration results, but all wells are used in the calibration regardless of 

category. Wells 6, 7, 8, 9, and 10 from the 1961 data set and well W32 from the September 2005 data set 

are outside of the model domain and are thus not used in the calibration process.

e In the model, MW-1202a was moved 13.5 ft to the northeast so that this point would lie in an active grid cell 

in the post-construction simulations. The modified coordinates are (629146, 1532579).

APPENDIX C

Spreadsheets for Calculating Model Selection Criteria and ModelAveraged Quantities

C

-1

SSWR 4244.350 Num. Data 71 ln|(Cov)| AIC AICc BIC KIC ln|F| -14.6625

Std. Error 8.019 Num. Param 5 -11.185 302.437 303.750 311.751 292.798 -9.763 <- from reported covariance

-32.170 313.783 11.222 <- from covariance with parameters in natural units

Paramete Final Value Lower Upper Log10 Trans ln(Final VaStd.Dev.hk_11 14.358 1.74479 118.152 1 2.6643 14.4605 2.11E-01 1.90E-01 2.02E-01 1.47E-01 2.04E-01

hk_12 1.60E-01 5.62E-03 4.57E+00 1 -1.8310 0.4592 1.90E-01 5.33E-01 1.17E-01 -0.1864 1.86E-01

hk_13 0.154633 1.30E-02 1.835 1 -1.8667 0.2107 2.02E-01 1.17E-01 2.90E-01 7.63E-02 2.25E-01

hk_14 3.10E-03 1.21E-05 0.793635 1 -5.7749 0.0876 1.47E-01 -1.86E-01 7.63E-02 1.458 1.09E-01

rch_34 3.88E-04 4.73E-05 3.19E-03 1 -7.8542 0.0004 2.04E-01 1.86E-01 2.25E-01 1.09E-01 2.10E-01

Name Group Measured Modeled Residual Weight Weight*Measured Weight*Modelled Weight*Residual Measurement_sd Natural_weight Modeled Measured

hed1 head 609 616.0303 -7.0303 0.391993 238.7237 241.4796 -2.755828 18.81828 5.31E-02 616.0303 609

hed2 head 615 617.1609 -2.1609 0.391993 241.0757 241.9228 -0.8470577 18.81828 5.31E-02 617.1609 615

hed3 head 617 618.9467 -1.9467 0.391993 241.8597 242.6228 -0.7630928 18.81828 5.31E-02 618.9467 617

hed4 head 605 598.1456 6.8544 0.391993 237.1558 234.4689 2.686877 18.81828 5.31E-02 598.1456 605

hed5 head 619 609.6112 9.3888 0.391993 242.6437 238.9633 3.680344 18.81828 5.31E-02 609.6112 619

hed6 head 620 599.213 20.787 0.391993 243.0357 234.8873 8.148358 18.81828 5.31E-02 599.213 620

hed7 head 609 6.09E+02 0.3021 0.391993 238.7237 238.6053 0.1184211 18.81828 5.31E-02 608.6979 609

hed8 head 595 598.1434 -3.1434 0.391993 233.2358 234.468 -1.232191 18.81828 5.31E-02 598.1434 595

hed9 head 591 601.8558 -10.8558 0.391993 231.6679 235.9233 -4.255398 18.81828 5.31E-02 601.8558 591

hed10 head 593.87 598.076 -4.206 0.979982 581.9819 586.1037 -4.121804 7.527318 0.1328495 598.076 593.87

hed11 head 596.04 602.7067 -6.6667 0.979982 584.1085 590.6417 -6.533246 7.527318 0.1328495 602.7067 596.04

hed12 head 607.39 609.9702 -2.5802 0.979982 595.2313 597.7598 -2.52855 7.527318 0.1328495 609.9702 607.39

hed13 head 597.87 607.9201 -10.0501 0.979982 585.9018 595.7508 -9.848917 7.527318 0.1328495 607.9201 597.87

hed14 head 595.78 623.8251 -28.0451 0.979982 583.8537 611.3374 -27.48369 7.527318 0.1328495 623.8251 595.78

hed15 head 595.26 596.7755 -1.5155 0.979982 583.3441 584.8292 -1.485163 7.527318 0.1328495 596.7755 595.26

hed16 head 597.26 601.8979 -4.6379 0.979982 585.304 589.8491 -4.545059 7.527318 0.1328495 601.8979 597.26

hed17 head 605.06 606.9639 -1.9039 0.979982 592.9479 594.8137 -1.865788 7.527318 0.1328495 606.9639 605.06

hed18 head 611.42 612.8198 -1.3998 0.979982 599.1806 600.5524 -1.371779 7.527318 0.1328495 612.8198 611.42

hed19 head 598.34 601.7953 -3.4553 0.979982 586.3624 589.7486 -3.386132 7.527318 0.1328495 601.7953 598.34

hed20 head 625.85 622.3165 3.5335 0.979982 613.3217 609.859 3.462766 7.527318 0.1328495 622.3165 625.85

hed21 head 594.58 594.9848 -0.4048 0.979982 582.6777 583.0744 -0.3966967 7.527318 0.1328495 594.9848 594.58

hed22 head 613.8 615.386 -1.586 0.979982 601.513 603.0672 -1.554251 7.527318 0.1328495 615.386 613.8

Reported Parameter Estimation Covariance

95% Confidence Limits

640635630625620615610605600595590

590 600 610 620 630 640

Measured

Modeled

Figure C-1. Spreadsheet for model evaluation and calculating model selection criteria
C

-2

Model Num. Data Num. Param. WSSR AIC AICc BIC KIC ln|F| AIC AICc BIC KIC

H1R1D1 71 2 8010.42 341.53 341.89 344.06 349.90 9.51 0.00% 0.00% 0.00% 0.00%

H2R1D1 71 3 6026.49 323.33 323.93 328.12 328.31 5.71 0.00% 0.00% 0.01% 0.01%

H3R1D1 71 4 5590.22 319.99 320.92 327.04 323.02 3.33 0.01% 0.01% 0.03% 0.20%

H4R1D1 71 5 4244.35 302.44 303.75 311.75 313.78 11.22 32.64% 39.59% 52.83% 20.25%

H1R2D1 71 5 5137.42 316.00 317.31 325.31 354.44 38.32 0.04% 0.05% 0.06% 0.00%

H2R2D1 71 6 4932.50 315.11 316.88 326.68 344.02 28.37 0.06% 0.06% 0.03% 0.00%

H3R2D1 71 7 4021.04 302.60 304.92 316.44 337.15 33.58 30.09% 22.03% 5.07% 0.00%

H4R2D1 71 8 4176.33 307.29 310.24 323.39 349.18 40.49 2.88% 1.54% 0.16% 0.00%

H1R1D2 71 2 7767.93 339.35 339.71 341.88 347.91 9.71 0.00% 0.00% 0.00% 0.00%

H2R1D2 71 3 6056.45 323.68 324.29 328.47 321.71 -1.25 0.00% 0.00% 0.01% 0.38%

H3R1D2 71 4 5725.10 321.69 322.61 328.74 321.30 -0.08 0.00% 0.00% 0.01% 0.47%

H4R1D2 71 5 4276.54 302.97 304.29 312.29 311.07 7.97 24.96% 30.28% 40.40% 78.68%

H1R2D2 71 5 5227.29 317.23 318.54 326.54 354.05 36.70 0.02% 0.02% 0.03% 0.00%

H2R2D2 71 6 4941.69 315.24 317.02 326.81 337.98 22.20 0.05% 0.05% 0.03% 0.00%

H3R2D2 71 7 4184.85 305.43 307.76 319.27 337.38 30.98 7.29% 5.34% 1.23% 0.00%

H4R2D2 71 8 4222.75 308.07 311.03 324.18 349.85 40.38 1.95% 1.04% 0.11% 0.00%

denom = sum(exp(-0.5*(IC-ICmin))) {array formula} 3.063546147 2.525996 1.89298 1.27092

Model Probabilities

Figure C-2. Spreadsheet for calculating model probabilities
C

-3

Post-construction groundwater head uncertainty -Standard Deviations of Predicted values from UCODEThree head predictions at the east, south, and west corners of each reactor block

0.085034014 Head Measurement Error Std Dev.

Post-construction models

0.681908836 s_a 0.674019 s_a 0.68449 s_a 0.68761 s_a

1.996564419 95% Critical value 1.99773 95% Critical value 1.996564 95% Critical value 1.99773 95% Critical value

Location Predicted Std Dev Conf.Int. Pred.Int. Location Predicted Std Dev Conf.Int. Pred.Int. Location Predicted Std Dev Conf.Int. Pred.Int. Location Predicted Std Dev Conf.Int. Pred.Int.

Unit3-01 601.67 0.92 603.51 603.96 Unit3-01 601.95 1.34 604.63 604.95 Unit3-01 601.80 0.89 603.57 604.04 Unit3-01 602.13 1.38 604.89 605.21

Unit3-02 604.11 1.07 606.25 606.64 Unit3-02 606.13 2.02 610.15 610.37 Unit3-02 604.03 1.00 606.04 606.46 Unit3-02 606.05 1.90 609.84 610.08

Unit3-03 603.15 0.83 604.81 605.30 Unit3-03 604.66 1.60 607.85 608.12 Unit3-03 602.99 0.77 604.53 605.05 Unit3-03 604.34 1.50 607.33 607.63

Unit4-01 610.10 0.70 611.49 612.05 Unit4-01 610.87 1.50 613.87 614.16 Unit4-01 610.12 0.78 611.68 612.19 Unit4-01 610.37 1.44 613.25 613.56

Unit4-02 611.29 1.17 613.63 613.99 Unit4-02 613.81 1.48 616.77 617.06 Unit4-02 611.38 1.30 613.98 614.32 Unit4-02 613.99 1.43 616.84 617.15

Unit4-03 609.83 0.58 610.99 611.62 Unit4-03 613.05 1.25 615.54 615.88 Unit4-03 609.72 0.74 611.19 611.73 Unit4-03 612.88 1.23 615.35 615.70

Model Probabilities Unit4-02AICc KIC Pred.Val Param. AIC mean KIC mean

H4R1D1 39.59% 20.25% 611.29 1.17 1.39 1.17

H3R2D1 22.03% 0.00% 613.81 1.48 2.31 2.86

H4R1D2 30.28% 78.68% 611.38 1.30 1.46 1.30H3R2D2 5.34% 0.00% 613.99 1.43 2.42 2.99ModAvgMean 612.04 611.36

ModAvgSD 1.68 1.28

ConfInt 615.39 613.91

Error 3.355226 2.5556016

Std. Dev.

Param+Model

H4R1D1 H3R2D1 H4R1D2 H3R2D2

Linear Upper Limits Linear Upper Limits Linear Upper Limits Linear Upper Limits

Figure C-3. Spreadsheet for calculating model-averaged predictions and predictive uncertainty





NUREG/CR

-7211 Application of a Hydrological Uncertainty Methodology 

to Nuclear Reactor Site Evaluations