NUREG/CR-6805 

A Comprehensive Strategy 

of Hydrogeologic Modeling 

and Uncertainty Analysis 

for Nuclear Facilities and Sites 

University of Arizona

U.S. Nuclear Regulatory Commission 

Office of Nuclear Regulatory Research

Washington, DC 20555-0001

i 
NUREG/CR-6805

A Comprehensive Strategy 

of Hydrogeologic Modeling 

and Uncertainty Analysis 

for Nuclear Facilities and Sites

Manuscript Completed: December 2002

Date Published: July 2003

Prepared by

S.P. Neuman and P.J. Wierenga 

University of Arizona

Tucson, AZ 85721

T.J. Nicholson, NRC Project Manager 

Prepared for 

Division of Systems Analysis and Regulatory Effectiveness 

Office of Nuclear Regulatory Research 

U. S. Nuclear Regulatory Commission 

Washington, DC 20555-0001

NRC Job Code W6790 
A-iii 

ABSTRACT 

This report describes a strategy that 

embodies a systematic and comprehensive 

approach to hydrogeologic 

conceptualization, model development and 

predictive uncertainty analysis. The strategy 

is comprehensive in that it considers all 

stages of model building and accounts 

jointly for uncertainties that arise at each of 

them. The stages include regional and site 

characterization, hydrogeologic 

conceptualization, development of 

conceptual-mathematical model structure, 

parameter estimation on the basis of 

monitored system behavior, and assessment 

of predictive uncertainty. In addition to 

parameter uncertainty, the strategy concerns 

itself with uncertainties arising from 

incomplete definitions of (a) the conceptual 

framework that determines model structure, 

(b) spatial and temporal variations in 

hydrologic variables that are either not fully 

captured by the available data or not fully 

resolved by the model, and (c) the scaling 

behavior of hydrogeologic variables. The 

strategy is generic but designed to be of 

practical use to NRC licensing staff in their 

review of decommissioning plans, and 

performance assessment of high-level and 

low-level radioactive waste disposal sites as 

well as uranium recovery facilities. An 

important component of the strategy is a 

systematic sequence of logical questions, 

guidelines and criteria with analytical 

methods appropriate for NRC review and 

performance evaluation. 
iv 
v 

CONTENTS 

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

EXECUTIVE SUMMARY .........................................................................................................xv 

FOREWORD.............................................................................................................................. xxi 

ACKNOWLEDGMENTS .......................................................................................................xxiii 

1 INTRODUCTION ...................................................................................................................1

1.1 Background .....................................................................................................................1 

1.2 State-of-the-Art Summary .............................................................................................2 

1.2.1 Preliminary Assessment........................................................................................2

1.2.2 Key Conclusions from Preliminary State-of-the-Art Assessment .........................2

1.2.3 Uniqueness of Sites versus Non-Uniqueness of Models ....................................3

1.2.4 Desired Level of Model Complexity......................................................................4

1.2.5 Traditional Approach to Model Uncertainty Analysis..........................................5

1.2.6 Accounting for Conceptual Model Uncertainty....................................................6

1.2.7 Multiobjective Tradeoffs between Model Fit and Complexity..............................9

1.2.8 Combined Measures of Model Fit and Complexity ............................................10

1.2.9 Effects of Scale on Hydrologic Predictability.....................................................10

1.2.10 Prediction Uncertainty Due to Random Fluctuations in Hydrologic Variables...11

1.3 Outline of Proposed Strategy.......................................................................................13

1.3.1 Comprehensive Approach to Modeling and Uncertainty Analysis.....................13 

1.3.2 NRC Licensing Context.......................................................................................14 

1.3.3 Hydrogeologic Conceptualization and Data ......................................................14 

1.3.4 Iterative Approach ..............................................................................................14 

1.3.5 When to Stop Collecting Data and Refining the Model?....................................15 

1.3.6 Assimilating Regional and Site Characterization Data......................................16 

1.3.7 Testing and Calibration against Monitoring Data .............................................16 

1.3.8 Predictive Uncertainty of Deterministic Models ................................................17 

1.3.9 Assessing the Need for More Data......................................................................18 

1.3.10 Accounting for Unresolved Variations in Space-Time .......................................18 

1.3.11 Stochastic Modeling............................................................................................19 

2 NATURE OF HYDROGEOLOGIC SITE MODELS .......................................................21 

2.1 Site-Specific Nature of Hydrogeologic Models...........................................................21

2.2 What is a Hydrogeologic Model?.................................................................................21 

2.3 What is a Conceptual Model of Site Hydrogeology? .................................................22 

2.4 What is a Conceptual-Mathematical Model of Site Hydrogeology?........................32 

2.5 What is a Hydrogeologic Model?.................................................................................34 

2.6 Non-Uniqueness of Hydrogeologic Models.................................................................35 

2.7 Accounting for Hydrogeologic Complexity ................................................................36 

2.7.1 Effects of Heterogeneity and Scale .....................................................................36 

2.7.2 "Simple" Models of Complex Systems.................................................................36
vi 

CONTENTS (continued) 

3 CONTEXTUAL FRAMEWORK.........................................................................................39

3.1 Purpose of Hydrogeologic Models Within NRC Regulatory Context......................39

3.2 Questions that Hydrogeologic Models Must Address................................................39 

3.3 Performance Measures.................................................................................................40 

3.4 Hydrogeologic Context of NRC Performance Measures ..........................................42 

3.4.1 Hydrogeologic Aspects of Performance Criteria ...............................................42 

3.4.2 Implications vis-à-vis Hydrogeologic Conceptualization and Modeling ...........42 

4 ASSEMBLY AND ORGANIZATION OF HYDROGEOLOGIC KNOWLEDGE BASE..45

4.1 What Constitutes a Hydrogeologic Knowledge Base?...............................................45 

4.2 Why Does One Need Site Characterization and Monitoring Data?.........................45 

4.3 Sources of Hydrogeologic Information .......................................................................47 

4.3.1 Example: The Arizona Water Information Directory .........................................47 

4.3.2 Example: Maricopa Agricultural Center............................................................48 

4.3.3 Example: EDR Environmental Data Resources, Inc. .........................................48 

4.4 Data Management and Expert Software Systems......................................................48 

4.4.1 Example: GMS....................................................................................................49 

4.4.2 Example: FRAMES .............................................................................................51 

4.4.3 Example: SEDSS.................................................................................................52 

4.4.4 Example: FIELDS...............................................................................................53 

4.4.5 Example: SADA ..................................................................................................53 

4.4.6 Example: LANL EES Integrated GIS and Model-Based Data Management System .54 

4.4.7 Example: LANL EES Geologic Model of LANL Site and EspaZola Basin.........55 

5 QUALITATIVE CONCEPTUALIZATION AND SCREENING OF HYPOTHESES..57 

5.1 What is a Qualitative Conceptual Model of Site Hydrogeology?.............................58 

5.2 Why Interpret Site Hydrogeology in Three Dimensions?.........................................58 

5.3 How to Formulate Hydrogeologic Hypotheses?.........................................................59 

5.4 Why Formulate Multiple Conceptual Site Models? ..................................................62 

5.5 How to Formulate Alternative Conceptual Models?.................................................62 

5.6 How to Assess and Rank Alternative Conceptual Models? ......................................62 

5.6.1 Example: Proposed High-Level Nuclear Waste Site ..........................................63 

6 MATHEMATICAL CONCEPTUALIZATION AND 

QUANTITATIVE EXPLORATION OF HYPOTHESES.................................................73

6.1 What is a Conceptual-Mathematical Model of Site Hydrogeology?........................73 

6.2 How to Formulate a Conceptual-Mathematical Model for a Site? ..........................75 

6.2.1 What to Specify? .................................................................................................75 

6.2.2 How to Determine a Suitable Level of Model Complexity?................................75 

6.2.3 How to Simplify Models?....................................................................................81 

6.2.4 How to Factor Uncertainty Into a Conceptual-Mathematical Model? ..............83 

6.2.5 How To Interpret Traditional Deterministic Models Stochastically? ................88
vii 

CONTENTS (continued) 

6.3 How to Formulate and Explore Alternative Conceptual-Mathematical Models?..89 

6.3.1 Where to Start? ...................................................................................................89 

6.3.2 How to Proceed?.................................................................................................90 

7 RENDERING A MODEL COMPATIBLE WITH DATA................................................93 

7.1 Why Worry About Compatibility With Data? ..........................................................93 

7.2 How to Achieve Compatibility in Scale and Magnitude?..........................................93 

7.3 How to Infer Prior Input Parameters and Statistics? ...............................................96 

7.3.1 What to Infer from Site Characterization Data and How?.................................96 

7.3.2 How to Incorporate Soft Data?...........................................................................98 

7.3.3 How to Use Generic and Subjective Probabilities? ...........................................99 

7.4 How to Calibrate Traditional Deterministic Models?.............................................100 

7.4.1 Where to Start for what Purpose? ....................................................................100 

7.4.2 How to Calibrate?.............................................................................................100 

7.4.3 How to Select Model Fit Criteria?....................................................................102 

7.4.4 What to Do with Insufficient Data? ..................................................................104 

7.5 How to Calibrate Stochastic Models? .......................................................................106 

7.5.1 How to Calibrate High-Resolution Monte Carlo Models?...............................106 

7.5.2 How to Calibrate Moment Equation Models?..................................................107 

7.6 How to Gauge Quality of Model Fit? ........................................................................108 

7.6.1 Example: Whiteshell Research Area.................................................................109 

7.7 How to Quantify Posterior Parameter Uncertainty? ..............................................114 

8 QUANTITATIVE COMPARISON AND RANKING OF ALTERNATIVE MODELS....117

8.1 How to Establish Comparative Measures of Model Quality and Complexity? ....117 

8.2 How to Compare and Select Alternative Models Based on Quality 

and Complexity? 118 

8.2.1 Iterative Model Testing and Selection ..............................................................118 

8.2.2 Multiobjective Approach...................................................................................118 

8.2.3 Combined Measures of Model Fit and Complexity ..........................................119 

8.2.4 Example: Chalk River Monzonitic Block..........................................................120 

8.2.5 Example: Los Monegros Basin, Spain..............................................................122 

8.2.6 Example: Apache Leap Research Site ..............................................................122 

9 RENDERING PREDICTIONS AND ASSESSING PREDICTIVE UNCERTAINTY 

USING MULTIPLE MODELS ..........................................................................................125

9.1 How to Assess Predictive Uncertainty for a Single Deterministic Model?............125 

9.1.1 Monte Carlo Method.........................................................................................125 

9.1.2 Linearization.....................................................................................................126 

9.2 Why Consider Stochastic Models?............................................................................126 

9.3 How to Assess Predictive Uncertainty for a Single Stochastic Model?..................127 

9.3.1 High-Resolution Monte Carlo Method .............................................................127 

9.3.2 Moment Equation Method.................................................................................127 
viii 

CONTENTS (continued) 

9.4 How to Render Joint Predictions with a Set of Alternative Deterministic Models? ....127 

9.4.1 How to Render Joint Predictions?....................................................................128 

9.4.2 How to Assess Joint Predictive Uncertainty?...................................................128 

9.5 How to Evaluate Hydrogeologic Performance?.......................................................129 

9.6 Conditional Nature of Hydrogeologic Predictions...................................................129 

10 MODEL CONFIRMATION AND ASSESSMENT OF DATA NEEDS ........................131 

10.1 How to Confirm Calibrated Models?........................................................................131 

10.1.1 Example: Chalk River Monzonitic Block..........................................................132 

10.2 How to Assess Additional Data Needs.......................................................................132 

10.2.1 Role of Sensitivity Analyses ..............................................................................132 

10.2.2 How to Decide When To Stop Collecting Data? ..............................................132 

11 CASE STUDIES...................................................................................................................133 

11.1 Maricopa Agricultural Center...................................................................................133 

11.1.1 Introduction.....................................................................................................133 

11.1.2 Brief Description of Experiments 1–3.............................................................134 

11.1.3 Summary of Relevant Public Information .......................................................135 

11.1.4 Conceptual-Mathematical Modeling Based On Public Data..........................139 

11.1.5 Variogram Analysis of Soil Composition At The MAC Site ............................140 

11.1.6 Postulation of Layered Structure ....................................................................142 

11.1.7 Estimation of Hydraulic Parameters Using Pedotransfer Models .................144 

11.1.8 Bayesian Updating of Hydraulic Parameter Estimates..................................149 

11.1.9 Variogram Analysis of Hydraulic Parameter Estimates.................................152 

11.1.10 Forward Flow Modeling Based on Public Data.............................................153 

11.1.11 Forward Flow Modeling Based on Site Data .................................................156

11.1.12 Inverse Flow Modeling Based on Site Data....................................................162

11.1.13 Confirmation of Inverse Modeling Results......................................................171

11.1.14 Inverse Modeling of Solute Transport.............................................................171

11.1.15 Conclusions.....................................................................................................171

11.2 Apache Leap Research Site........................................................................................172 

11.2.1 Introduction.......................................................................................................172 

11.2.2 Experimental Setup ...........................................................................................175 

11.2.3 Findings............................................................................................................175 

11.3 Fanay-Augères.............................................................................................................179 

11.3.1 Introduction.......................................................................................................179 

11.3.2 Indicator analysis and simulation of hydraulic conductivities ..........................185 

11.3.3 Geostatistical analysis and simulation of flow ....................................................188 

11.3.4 Simulation of transport .....................................................................................193 

11.3.5 Conclusions..............................................................................................................196 
ix 

CONTENTS (continued) 

12 GUIDELINES FOR REVIEW OF HYDROGEOLOGIC MODELING AND 

UNCERTAINTY ANALYSES ...........................................................................................197 

12.1 Introduction.................................................................................................................197 

12.2 Summary of Guidelines..............................................................................................197 

12.2.1 Define Hydrogeologic Context........................................................................199 

12.2.2 Collect a Broad Range of Regional and Site Data..........................................201 

12.2.3 Postulate Alternative Conceptual Models.......................................................202 

12.2.4 Avoid Conflict with Data.................................................................................205 

12.2.5 Rank Conceptual Models.................................................................................205 

12.2.6 Develop Conceptual-Mathematical Models....................................................206 

12.2.7 Consider Simplifying the Model......................................................................207 

12.2.8 Select Computational Code or Algorithm .......................................................211 

12.2.9 Define Prior Statistics.....................................................................................209 

12.2.10 Perform Preliminary Test of Conceptual-Mathematical Model .....................210 

12.2.11 Calibrate Model ..............................................................................................212 

12.2.12 Confirm Model ................................................................................................213 

12.2.13 Compare Calibrated Models...........................................................................217 

12.2.14 Render Predictions Using Individual Models.................................................218 

12.2.15 Render Joint Predictions by All Models..........................................................219 

12.2.16 Identify Additional Data Needs.......................................................................222 

12.2.17 Repeat Iterative Process with New Data.........................................................222 

13 REFERENCES.....................................................................................................................223 

Appendices 

A PERFORMANCE MEASURES AND HYDROLOGY ISSUES PERTAINING 

TO GROUNDWATER FLOW AND TRANSPORT MODELS OF LICENSED 

NUCLEAR SITES .............................................................................................................. A-1 

B VERTICAL AVERAGING OF WATER TABLE EQUATION.....................................B-1 

C AVERAGING OF GROUNDWATER FLOW IN PROBABILITY SPACE ................C-1 

D MIXING, DILUTION, AND DISPERSION IN HYDROGEOLOGIC ENVIRONMENTS. D-1 

E MAXIMUM LIKELIHOOD BAYESIAN MODEL AVERAGING...............................E-1 
x 

Figures 

2-1 Conceptual diagram and flow chart of exposure pathways in RESRAD..........................23 

2-2 Conceptual model as depicted by EPA, DOE, and NRC guide to model selection..........24 

2-3 Input data and processes considered in performance assessment of geosphere transport 

by Gautschi........................................................................................................................24 

2-4 Processes affecting performance of an engineered barriers system ..................................25 

2-5a Translation of geologic information into conceptual model for numerical modeling: 

Floridan aquifer system, W to E cross section for central Florida ....................................25 

2-5b Translation of geologic information into conceptual model for numerical modeling: 

Snake River Plain aquifer system, southwest to northeast cross section ..........................26 

2-5c Translation of geologic information into conceptual model for numerical modeling: 

Sand and gravel aquifer in Pensacola, Florida ..................................................................27 

2-5d Translation of geologic information into conceptual model for numerical modeling: 

Glacial-drift river-valley aquifer in Rhode Island.............................................................27 

2-6 Theoretical flow pattern and flow systems according to Tóth ..........................................28 

2-7 Hydrogeologic block diagram of arid zone hydrogeology................................................29 

2-8 Map of groundwater flow and basin divides in the Death Valley area .............................29 

2-9 Conceptual model of crystalline basement in Switzerland with features relevant to flow, 

and corresponding concept of its characterization ............................................................30 

2-10 Procedure for joint conceptualization and characterization of envirochemical 

and groundwater systems according to Kolm and van der Heijde ....................................31 

2-11 Mathematical formulation of hypotheses ..........................................................................33 

2-12 Different conceptual-mathematical models associated with a given qualitative conceptual 

model of crystalline basement in Northern Switzerland, depending on scale...................34 

2-13 Simplified water pathways and well contamination processes in the RESRAD model....38 

3-1 Contextual framework of modeling...................................................................................41 

4-1 Content of knowledge base ...............................................................................................46 

5-1 Qualitative conceptual-mathematical modeling................................................................57 

5-2 Qualitative conceptualization and screening of hypotheses..............................................60 

5-3 Local and regional water levels in the area .......................................................................63 

5-4 Details of conditions in borehole G-2 ...............................................................................66 

5-5 Isopach map of Calico Hills and flow conditions in borehole G-2 ...................................67 

5-6 Temperature profiles and water level recovery in borehole G-2.......................................67 

5-7 Water levels versus topographic elevations ......................................................................68 

5-8 Drain (A) and spillway (B) models with section showing postulated buried graben........69 

5-9 Residual gravity contours and heat flow distribution in unsaturated zone........................70 

5-10 Heat flow variation with thickness of unsaturated zone and distribution of isotherms 

at water table......................................................................................................................71 

5-11 Aeromagnetic survey map.................................................................................................71

6-1 Mathematical conceptualization........................................................................................74 

6-2 Example of hydrogeologic schematization for purposes of mathematical modeling........76 

6-3 Example of hydrogeologic schematization for purposes of mathematical modeling........76 

6-4 Geology (a), regional flow conceptualization (b) and simulation (c) ...............................77 

6-5 Regional groundwater flow conceptualization..................................................................78 

6-6 Vertical flow nets ..............................................................................................................78 
Figures (continued) 

6-7 Horizontal flow net............................................................................................................79 

6-8 Dispersion affected by heterogeneities on various scales .................................................80 

6-9 Alternative conceptual models and their relative permeability characteristics.................82 

7-1 Variation of log-likelihood criterion, sum of squared head residuals, and sum of squared 

log-transmissivity estimation errors with λ ...................................................................104 

7-2 Residual versus parameter plausibility criteria................................................................105 

7-3 Whiteshell Research Area ...............................................................................................108 

7-4 Selected logs of boreholes WD3 (left) and WG4 (right).................................................109 

7-5 Landsat lineaments in WRA and outline of model finite element grid with embedded 

fracture zones...................................................................................................................110 

7-6 Equivalent fresh water head profiles in boreholes WD3 (upper left), WG4 (upper right), 

WA1 (lower left), and WG1 (lower right).......................................................................111 

7-7 URL borehole leaching test setup and results .................................................................112 

7-8 Flow chart of AECL model calibration procedure ..........................................................113 

7-9 Comparison between computed and recorded equivalent fresh water head profiles ......114 

8-1 Block diagrams (a) and section (b) showing fracture zones and boreholes at Chalk River ....121 

8-2 Four alternative parameter zonation patterns used in Chalk River model ......................121 

8-3 Gray scale images of kriged log air permeabilities and their variances along a vertical

cross-section at the ALRS using two geostatistical models............................................123 

10-1 Drawdown versus log time in response to pumping from well FS-11-2.........................131 

11-1 Location of monitoring equipment..................................................................................134 

11-2 Well log of D-04-04 20 BCD near the MAC .................................................................136 

11-3 An East-West cross-section near the MAC.....................................................................137 

11-4 Elevation contours (in feet) of water level in the UAU aquifer in Nov. 1988 (left) and 

Nov. 1993 (right).............................................................................................................137 

11-5 Monthly irrigation (a), reference evapotranspiration ET0 (b), and precipitation (c) 

in MAC area ....................................................................................................................138 

11-6 Omni-directional sample variograms and fitted spherical models for percent sand, silt, 

and clay at depths 0 to 30 cm ..........................................................................................141 

11-7 Vertical sample variograms and fitted models of percent sand, silt, and clay from the 

surface to 15 m depth ......................................................................................................143 

11-8 Neutron count ratio and soil class variations with depth at borehole 402.......................143 

11-9 Local stratigraphy based on soil and neutron data ..........................................................144 

11-10 Distribution of hydraulic parameters for sand based on Meyer et al. .............................146 

11-11 Hydraulic parameters at the MAC using Rawls-Brakensiek regressions with site-specific 

soil component and bulk density data at shallow depths of less than 5 m ......................147 

11-12 Hydraulic parameters at the MAC using Rosetta software with site-specific soil 

component (all the depths) and bulk density data (above 5 m).......................................148 

11-13 Bayesian updates of hydraulic parameters for sand using Rosetta estimates..................150 

11-14 Omni-directional lateral (left, at depth 0–30 cm) and vertical (right) sample variograms

and fitted models for log( Ks )..........................................................................................152 

11-15 1-D forward simulation of infiltration during experiment 3 at borehole 422 using mean 

hydraulic parameters from ROSETTA............................................................................154 

xi 
Figures (continued) 

11-16 1-D forward simulation of infiltration during experiment 3 at borehole 422 

using mean hydraulic parameters from RAWLS ............................................................155 

11-17 1-D forward simulation of infiltration during experiment 3 at borehole 422 

using mean hydraulic parameters from CARSEL...........................................................156 

11-18 1-D simulation of infiltration experiment 3 at borehole 402 using mean hydraulic 

parameter values from CARSEL at various depths.........................................................158 

11-19 1-D simulation of infiltration experiment 3 at borehole 402 using mean hydraulic 

parameter values from ROSETTA at various depths......................................................159 

11-20 1-D simulation of infiltration experiment 3 at borehole 402 using Bayesian updates of 

saturated hydraulic conductivity, van Genuchten's α and n based on Rosetta estimates....160 

11-21 2-D forward simulation along N-S uniformly layered transect using mean hydraulic 

parameter values from CARSEL at various depths.........................................................161 

11-22 1-D simulation of infiltration experiment 3 in borehole 402 using inverse estimates 

of saturated hydraulic conductivity and van Genuchten's α and n ................................164 

11-23 2-D simulation of infiltration experiment 3 along western N-S transect 

(boreholes 402, 422, 442) using inverse estimates of saturated hydraulic conductivity 

and van Genuchten's α and n. Uniform soil layers........................................................165 

11-24 2-D simulation of infiltration experiment 3 along western N-S transect 

(boreholes 402, 422, 442) using inverse estimates of saturated hydraulic conductivity 

and van Genuchten's α and n. Non-uniform layers .......................................................166 

11-25 Histogram of differences between observed and simulated water contents along western 

N-S transect (boreholes 402, 422, 442) following inversion. Non-uniform layers.........167 

11-26 2-D simulation of experiment 1 along the N-S transect (boreholes 402, 422, 442) 

using layered non-uniform conceptual model and hydraulic parameters obtained 

from experiment 3 ...........................................................................................................169

11-27 Simulated (curves) and measured (dots) bromide concentrations at the south monitoring 

island obtained using inverse estimates of dispersivity, immobile or excluded water 

content, and mass transfer coefficient .............................................................................170 

11-28 Experimental site with plots, instrumentation, and surface exposure of fractures..........177 

11-29 Cumulative outflow from each subplot pond ..................................................................177 

11-30 Plot 1 neutron probe count ratios versus depth ...............................................................178 

11-31 Plot 1 soil water potentials versus depth .........................................................................178 

11-32 Plot 1 bromide concentrations versus time at 3 and 5 m depths .....................................178 

11-33 Layout of drift and boreholes..........................................................................................180 

11-34 Measured inflow rates into drift and periods of tracer tests............................................180 

11-35 Tracer release intervals and monitoring locations in drift...............................................181 

11-36 3-D network of circular disks with 1-D flow and transport paths...................................181 

11-37 Normal probability plot of natural log-hydraulic conductivity from 2.5 m packer tests...182 

11-38 Scatter plot of fracture density (number per unit length of boreholes) versus hydraulic 

conductivity from 2.5 m scale packer tests .....................................................................182 

11-39 Projection of observed piezometric heads (in meters) onto transverse model plane ......184 

11-40 Boundary conditions used for simulating tracer injections.............................................184 

11-41 Omni-directional indicator variogram models of Y = ln K normalized with respect 

to sill of variogram model corresponding to indicator no. 3 ...........................................187 

xii 
xiii 

Figures (continued) 

11-42 Omni-directional sample and spherical model semivariogram for indicator no. 3 .........187 

11-43 Kriged estimate of probability that hydraulic conductivity exceeds the fifth cutoff.......188 

11-44 Two random realizations of hydraulic conductivity honoring the data...........................189 

11-45 Cumulative distribution of hydraulic conductivity estimates in Figure 11-44................189 

11-46 Sample (dots) and spherical model (solid curve) omni-directional variogram 

of pressure head at stage 1 with quadratic polynomial drift............................................190 

11-47 Kriged pressure head at stage 1 (left) and standard deviations of corresponding kriging 

errors (right) ....................................................................................................................190 

11-48 Flow model......................................................................................................................191 

11-49 Conditional simulation no. 48 of pressure head (left) and total head (right) at stage 1 ..191 

11-50 Histograms of simulated drift inflow rates at stages 1 (left) and 2 (right) ......................191 

11-51 Measured and simulated flow rate into drift....................................................................193 

11-52 Scattergrams of simulated (using conductivity realization no. 48) versus measured heads 

at stages 1 (left) and 2 (right) ..........................................................................................194 

11-53 Computed versus measured concentrations normalized by maximum for tests 6F3 

(upper left), 4F3 (upper right), 3F3 (lower left), and 7F2 (lower right)..........................195 

11-54 Simulated versus measured peak arrival times................................................................196 

Tables

5-1 Altitudes (and thicknesses) of hydrogeologic units in deep boreholes at the mountain ..68 

8-1 Model fit and discrimination criteria for three geostatistical models applied 

to log air permeabilities from the ALRS .........................................................................122 

11-1 Distribution of Soil Samples with Depth ........................................................................140 

11-2 Omni-directional variogram models fitted to soil composition data at various depths...142 

11-3 Comparison of hydraulic parameter estimates using two different pedotransfer models.....149 

11-4 Prior (based on CARSEL database) and posterior (following Bayesian updating) 

mean hydraulic parameter estimates and their standard deviations by soil class............151 

11-5 Omni-directional variogram models fitted to hydraulic parameter estimates 

at various depths..............................................................................................................153 

11-6 Results of forward simulations using 1-D layered models..............................................162 

11-7 Results of forward simulations using 2-D uniformly layered models.............................162 

11-8 Model quality criteria for four 1-D and two 2-D models associated with N-S transect..168 

11-9 Model quality criteria for three 1-D and two 2-D models associated with E-W transect.....168 

11-10 Indicator cutoffs of hydraulic conductivity data and percent of data not exceeding 

each cutoff .......................................................................................................................187 

11-11 Statistics of simulated inflow rates into drift (in l/min) .....................................................191 

11-12 Statistics of pressure head residuals (differences between simulated and kriged values 

in meters) .........................................................................................................................192 

11-13 Statistics of pressure head residuals (differences between simulated and measured values 

in meters) .........................................................................................................................192 

11-14 Simulated and measured tracer arrival times...................................................................195 
xiv 
xv 

EXECUTIVE SUMMARY 

This report describes a strategy that 

embodies a systematic and comprehensive 

approach to hydrogeologic 

conceptualization, model development and 

predictive uncertainty analysis. The strategy 

is comprehensive in that it considers all 

stages of model building and accounts 

jointly for uncertainties that arise at each of 

them. The stages include regional and site 

characterization, hydrogeologic 

conceptualization, development of 

conceptual-mathematical model structure, 

parameter estimation on the basis of 

monitored system behavior, and assessment 

of predictive uncertainty. In addition to 

parameter uncertainty, the strategy concerns 

itself with uncertainties arising from 

incomplete definitions of (a) the conceptual 

framework that determines model structure, 

(b) spatial and temporal variations in 

hydrologic variables that are either not fully 

captured by the available data or not fully 

resolved by the model, and (c) the scaling 

behavior of hydrogeologic variables. 

The strategy is generic but designed to be of 

practical use to NRC licensing staff in their 

review of decommissioning plans and 

performance assessment of high-level and 

low-level radioactive waste disposal sites as 

well as uranium recovery facilities. For this 

purpose, the strategy is cast in the context of 

a framework that is useful to NRC staff 

review and performance evaluation needs. 

The context is defined in terms of 

corresponding performance measures (as 

identified by the NRC staff in Appendix A), 

hydrogeologic analyses that are needed to 

assess them, the desired reliability of such 

assessments, and the expenditure (in time, 

effort and money) that is allowed to achieve 

it. Examples from various case studies are 

included to help illustrate some aspects of 

these analyses. An important component of 

the strategy is a systematic sequence of 

logical questions, guidelines and criteria 

with analytical methods useful for NRC staff 

review and performance evaluation. 

The strategy encourages exploration of 

varied conceptual frameworks and 

assumptions at all stages of hydrogeologic 

model development through a 

comprehensive evaluation of a broad range 

of regional and site data, their translation 

into coherent and internally consistent 

conceptual-mathematical models, and 

computation and visualization based on 

these data and models. Included among 

these frameworks and assumptions are 

various model simplification and abstraction 

schemes. The strategy recognizes that site 

characterization and monitoring data are 

expensive and difficult to collect, leading to 

a ubiquitous scarcity of hard site 

information. It is therefore critically 

important to assess the role that such data 

play in rendering the hydrogeologic 

performance analysis credible. The strategy 

stresses the role of characterization and 

monitoring data in helping one identify and 

test alternative conceptual models, make 

rational choices among them, gauge and 

reduce model bias and uncertainty through 

proper model selection and calibration, 

assess the reliability of model predictions, 

and confirm the assessment through 

independent peer review as well as at least 

some degree of direct verification. 

The strategy encourages an iterative 

approach to modeling, whereby a 

preliminary conceptual-mathematical model 

is gradually altered and/or refined until one 

or more likely alternatives have been 

identified and analyzed. The idea is to start 

with a conceptual-mathematical model that 

is as simple as may appear warranted by the 
xvi 

context of the problem and the available 

data. This model is then tested for 

qualitative consistency with the available 

data, made quantitatively compatible with 

the data through calibration (parameter 

estimation via an inverse solution), and 

subjected to an assessment of its predictive 

reliability (in terms of potential bias and 

uncertainty). Next, the model structure is 

altered and/or refined (by modifying its 

hydrogeologic makeup or features; 

dimensionality; scale of resolution in spacetime; type of driving forces or events 

including sources, initial and boundary 

conditions; and governing equations or 

processes), the model is retested and 

compared with the former. 

The iterative procedure is repeated until 

there is good reason to conclude that any 

additional alterations or refinements of the 

models would lead at best to minor 

improvements in their predictive capability. 

Models whose predictive capability is 

deemed low in comparison to other are 

discarded and the rest retained for further 

analysis. The retained models are analyzed 

jointly to assess their aggregate predictive 

bias and uncertainty. If the latter are deemed 

acceptable for the stated purpose of the 

analysis, the iterative process is halted and 

the models declared reliable and credible 

within a clearly stated conditional (on the 

available information and selected set of 

models) margin of error. The conditional 

nature of this error estimate is taken to imply 

that it constitutes a lower bound, and that the 

actual predictive error may (and most 

probably is) larger to an unknown degree.

If the models are deemed unacceptable, a 

sensitivity analysis is performed to help 

identify the type and quantity of additional 

site data that might materially enhance their 

reliability and credibility. A decision is then 

made, based on the potential benefit (in 

terms of bias and uncertainty reduction) and 

cost (in terms of time, effort and money) of 

such data, whether or not to collect them and 

how. If and when new data of significant 

weight are obtained, the iterative process is 

repeated till its cost-benefit ratio reaches a 

value that is not considered worth 

exceeding. 

The strategy recognizes that it is often 

possible to postulate hydrogeologic 

conceptual models or hypotheses for a site 

on the basis of publicly available geologic 

and geographic information about its 

surroundings. Additional conceptualization 

can be done on the basis of generic data 

about similar regions and the properties of 

similar materials elsewhere. Several such 

regional and generic sources of information 

are identified and discussed in this report. 

Yet each site is unique and so virtually 

guaranteed to reveal additional features, 

properties and behaviors when characterized 

in some detail locally. Hence the strategy 

considers local characterization essential for 

the postulation of acceptably robust 

conceptual hydrogeologic models for a site. 

Regional and site characterization data tend 

to represent a wide range of measurement 

scales, not all of which are compatible with 

the intended scale of hydrogeologic model 

resolution. The strategy promotes 

recognition of this important issue and an 

effort to render the scale of measurement 

compatible with the scale of model 

resolution. This can be done by either 

rescaling the data to fit the scale of model 

resolution (which often entails averaging or 

upscaling over computational grid cells) or 

adapting model resolution to fit the scale of 

measurement (which often entails adapting 

the size of grid cells to the size of the data 

support). Recent advances in scaling theory 

and practice are briefly reviewed to promote 
xvii 

awareness of fundamental issues associated 

with the scaling of hydrogeologic variables. 

The strategy encourages the analyst to 

associate model parameters, derived from 

regional and site characterization data, with 

statistical measures of their uncertainty. 

Such prior statistics provide the only way to 

assess model predictive uncertainty when 

suitable monitoring data are not available. 

Methods are discussed and an example 

given to derive such statistics from generic 

data sources on the basis of similarity in soil 

or rock textural features; to update these 

statistics on the basis of site-specific data 

using a Bayesian approach; and to 

interpolate, extrapolate and/or average 

parameters across the site geostatistically on 

the basis of measured values at discrete 

points in space. 

While hydrogeologic characterization of a 

site and its surroundings makes 

conceptualization possible, it does not 

provide the means to test conceptual models 

or compare them with other alternatives. For 

this, it is necessary to have monitoring data 

that constitute observations of actual 

hydrologic behavior at and around the site. 

Only with such data can one evaluate the 

ability of models to mimic real system 

behavior (qualitatively at the conceptual 

level, quantitatively at the conceptualmathematical level), improve their ability to 

do so through calibration against the 

monitoring data, determine their optimum 

degree of refinement or complexity, and 

compare them with each other (qualitatively 

and quantitatively). The strategy discusses 

ways to accomplish these tasks in the 

context of traditional deterministic 

groundwater flow and transport models. 

The strategy encourages the analyst to 

employ statistical measures of model 

performance (or fit between simulated and 

observed hydrologic behaviors) where 

possible; to determine feasible tradeoffs 

between model fit and complexity by relying 

on multiobjective approaches; and to rank 

models based on (a) statistical indicators of 

the quality of fit they produce (as obtained 

from an analysis of residuals), and (b) 

likelihood-based model discrimination 

criteria (which penalize models for using an 

excessive number of free parameters to 

achieve a given quality of fit). Models that 

show relatively poor fits and rank low on the 

list can be subjectively eliminated from 

further consideration at this stage. 

An important byproduct of model 

calibration is information about parameter 

uncertainty. When reliable and relevant (in 

terms of type and scale) prior statistics about 

the parameters are available from site 

characterization, the strategy encourages 

their use as input into the calibration 

process. Regardless of whether or not 

suitable prior statistics are available and 

employed for model calibration, the latter 

should produce posterior statistics for the 

optimum parameter estimates. The strategy 

considers the posterior parameter estimates, 

and their error statistics, more suitable for 

the analysis of model predictive uncertainty 

than the prior estimates and their statistics. 

This is so because calibration improves the 

ability of the model to reproduce simulated 

hydrologic behavior, and a calibrated model 

is therefore deemed more reliable and 

credible as a predictive tool than an 

uncalibrated model. The same holds true for 

the associated parameter estimation 

statistics. 

The strategy considers two alternative 

methods to assess the predictive uncertainty 

of a deterministic groundwater model under 

assumed future scenarios. The first method 

relies on Monte Carlo simulation of these 

scenarios using either prior or posterior 
xviii 

parameter estimates and statistics. The 

second method establishes approximate 

error bounds, or confidence limits, for such 

scenarios by linearization, using the same 

parameter estimates and statistics. The first 

approach is more accurate but costly in 

terms of computer time and effort. In both 

cases, the strategy favors the use of posterior 

estimates over that of priors on the 

understanding that the former are less biased 

and uncertain than the latter. 

To render optimum predictions by means of 

all calibrated models that have been retained 

as constituting potentially viable 

alternatives, and to assess their joint 

predictive uncertainty, the strategy relies on 

a newly developed Maximum Likelihood 

Bayesian Model Averaging approach, 

MLBMA. The final outcome is conditional 

on the choice of models and data. It 

accounts jointly for uncertainties in the 

conceptual-mathematical model structure 

and its parameters. 

If suitable monitoring data are not available, 

then there is neither an objective nor a 

quantitative basis for the comparison of 

potentially viable conceptual-mathematical 

models. To quantify the predictive 

uncertainty of each such model, one can 

either weigh all models equally or associate 

them with subjective weights based on the 

professional judgment of the analyst or a 

group of experts. 

To help evaluate what if any additional data 

might be worth collecting so as to materially 

reduce model uncertainty (by further 

constraining the range of alternative 

structures and parameters), the strategy 

suggests conducting a sensitivity analysis to 

indicate what system behavior appears to be 

most sensitive to which parameters at what 

locations. The next step is to consider 

performing additional site characterization 

where existing parameter estimates are least 

certain and the model is relatively 

insensitive to their values, and monitoring 

system behavior where it is most sensitive to 

model parameters while prediction errors 

appear to be relatively large and 

consequential. 

Calibration of a deterministic groundwater 

model yields effective parameter estimates, 

which compensate to some degree for lack 

of knowledge about variability on spacetime scales that are not resolved by the data 

or the model. This is less true for transport 

than for flow, because transport processes 

due to unresolved variations in permeability 

and porosity cannot be validly represented 

by traditional effective parameters (such as a 

constant dispersivity) except in special cases 

or as a crude approximation. The reliance on 

effective parameters is also less valid for 

strongly heterogeneous media with 

preferential flow paths such as some 

fractured rocks and soils that contain 

relatively large and elongated openings 

created by burrowing animals or plant roots. 

When the deterministic effective parameter 

approach fails, the most common and 

powerful alternative is to employ stochastic 

concepts and models. 

A stochastic approach to groundwater 

modeling requires that the spatial variability 

of input parameters, such as permeability 

and unsaturated soil properties, be 

characterized with the aid of geostatistical 

methods. A key component of geostatistical 

characterization is the assessment of spatial 

covariance structure (in terms of a 

covariance function or variogram). The 

strategy promotes this approach and the 

report illustrates it by examples. In the 

absence of site characterization data that are 

amenable to geostatistical analysis, one can 

treat the structural parameters (of the 

variogram or covariance function) as free 
xix 

parameters to be estimated by model 

calibration. This too is illustrated by 

example in the report. 

The strategy supports two general 

approaches to predict stochastic 

groundwater flow and transport on the basis 

of geostatistical input: High-resolution 

numerical Monte Carlo simulation and 

direct (deterministic) prediction of mean 

behavior. The latter approach relies on 

stochastically derived deterministic 

ensemble mean flow and transport equations 

in which the dependent variables represent 

not actual system states (such as head, 

concentration or flux) but rather their 

(ensemble) mean values or statistical 

expectations. These mean or expected values 

represent optimum unbiased predictors of 

the unknown, actual system states. Similar 

values are obtained upon averaging the 

results of numerous Monte Carlo 

simulations. Both the Monte Carlo and the 

mean equation approaches allow the 

reduction of predictive uncertainty by 

conditioning the predictions on actual 

measurements of the input parameters at 

discrete locations in space. 

The variance-covariance of high-resolution 

(conditional) Monte Carlo simulations 

serves to quantify predictive uncertainty. An 

alternative is to compute this variancecovariance directly (deterministically) by 

means of corresponding moment equations. 

Both options are supported by the strategy. 

Conditioning the stochastic flow and 

transport equations not only on measured 

input variables (derived from site 

characterization) but also on monitored 

system behavior tends to improve their 

reliability. This is accomplished through 

novel inverse procedures, which form part 

of the proposed strategy. 

In virtually all other respects, the strategy is 

the same for stochastic as for deterministic 

models of groundwater flow and transport.
xx 
xxi 

FOREWORD 

This technical report was prepared by 

researchers at the University of Arizona 

(UAZ) under a contract (NRC-04-97-056) 

with NRC’s Office of Nuclear Regulatory 

Research. The research objective was to 

develop a methodology with supporting 

technical bases and guidance for selecting 

and evaluating appropriate ground-water 

flow and transport models for use in 

simulating performance of nuclear waste 

disposal facilities and possible remediation 

actions at decommissioning sites. The 

technical focus was on identifying and 

quantifying sources of uncertainty in 

ground-water models as related to specific 

performance measures defined by NRC 

regulations and guidance. These 

performance measures include welldischarge concentrations, expected values or 

potential maximum values of point 

concentrations, and dose from drinking 

water. The focus was not on computer code 

development, but rather on how to evaluate 

and compare ground-water models 

beginning with selection of alternative 

conceptual models and ending in 

performance assessment of the nuclear 

facility and site. Use of this research to 

support regulatory decisions will contribute 

to the NRC Strategic Plan’s performance 

goal of making NRC activities and decisions 

more effective, efficient and realistic. 

This report provides a systematic strategy 

for evaluating site-specific ground-water 

models and their attendant uncertainties. 

Specifically the strategy identifies tools and 

analysis methods for interpreting hydrologic 

and environmental data important to 

formulating alternative conceptual models 

and in estimating their parameter values and 

modeling assumptions. The report presents 

approaches for quantitative comparison and 

ranking of alternative models with field 

examples. It also addresses fundamental 

questions related to assessing model 

predictive uncertainty. Finally, the report 

considers important questions on model 

confirmation and assessment of data needs 

(e.g., when shall data collection end?). 

Many of these technical questions and the 

related narratives were developed through 

discussions between the UAZ researchers 

and the NRC licensing staff who are seeking 

ways to reduce the regulatory burden 

through more realistic modeling. 

This report serves as a technology transfer 

document to the NRC staff for their 

consultative use in reviewing ground-water 

models. The implementation of the strategy 

and its research findings provide a 

framework for more realistic modeling. 

Earlier drafts of this report were used in 

training workshops presented at NRC 

Headquarters on August 14-15, 2001 and 

August 1-2, 2002. The report was also 

circulated to cooperating federal agencies 

working under a Memorandum of 

Understanding on research into multi-media 

environmental models. The report was peerreviewed by Pacific Northwest National 

Laboratory investigators who developed a 

companion hydrologic parameter 

uncertainty methodology. 

Cheryl A. Trottier, Chief 

Radiation Protection, Environmental Risk 

and Waste Management Branch 

Division of Systems Analysis and 

Regulatory Effectiveness 

Office of Nuclear Regulatory Research 
xxii 
xxiii 

ACKNOWLEDGMENTS 

This work was supported by the U.S. 

Nuclear Regulatory Commission (NRC) 

under contract NRC-04-97-056. Special 

credit is due to Thomas J. Nicholson and 

Ralph Cady of the NRC for having 

conceived and championed this research. 

The authors acknowledge with thanks 

contributions by the following individuals: 

Mr. Tim McCartin of the U.S. NRC for his 

recent update of Appendix A; Professor 

Ghislain de Marsily of the University of 

Paris, France, for having provided data 

concerning the Fanay-Augères experimental 

site; Dr. Weihsueh Chiu of the U.S. 

Environmental Protection Agency for 

having brought to our attention the extensive 

work on BMA conducted by the “Seattle 

school” of statistics at the University of 

Washington, and for his helpful comments 

on Appendix E; Drs. Philip D. Meyer and 

Mark L. Rockhold of Pacific Northwest 

National Laboratory for their formal review 

of a draft version of this manuscript. 
xxiv 
1 

1 INTRODUCTION 

1.1 Background

This report describes a comprehensive 

strategy for constructing hydrogeologic flow 

and transport models and assessing their 

predictive uncertainty. The emphasis is on 

hydrogeologic conceptualization and its 

effect on the reliability of predictions 

relevant to decommissioning reviews and 

performance assessment (PA) of high-level 

(HLW) and low-level (LLW) radioactive 

waste disposal sites, as well as uranium 

recovery facilities. 

The strategy is the outcome of a study 

commissioned in 1997 by the U.S. Nuclear 

Regulatory Commission (NRC). Its purpose 

is to address a concern on the part of the 

NRC staff that conceptual models of site 

hydrogeology, and factors (physicochemical processes, parameters, forcing 

functions representing sources and 

initial/boundary conditions) which control 

subsurface flow and transport, constitute a 

major source of uncertainty in assessing the 

expected performance of such sites. The 

stated objective of the study was to develop 

a strategy, with supporting technical bases 

and guidance, for selecting and evaluating 

appropriate groundwater flow and transport 

models and assessing uncertainties of related 

performance measures. 

Some of the questions addressed in the 

report have been the subject of two roundtable discussions held at a Workshop on 

Ground-Water Modeling Related to Dose 

Assessments on June 23–24, 1999, at NRC 

Headquarters in Rockville, Maryland. 

The first round-table discussion, devoted to 

Conceptual Model Uncertainties, focused on 

the following questions: What is a 

conceptual model? Through what process of 

information gathering and analysis does one 

develop and identify a conceptual model? 

What are typical examples of conceptual 

models used in groundwater flow and 

contaminant transport analyses? How 

important are the assumptions dealing with 

parameter representation of heterogeneities, 

dimensionality and scaling factors? What 

role do conceptual models have in 

performance assessments? How does one 

identify alternative conceptual models for 

a specific site and specified performance 

measure(s)? How does one quantify 

differences in the competing conceptual 

models? Are these differences significant 

to the performance measure of interest 

(e.g., well-discharge concentrations, point 

concentrations of expected values or 

potential maximum values, drinking-waterrelated dose? How does one quantify these 

performance-measure uncertainties for site 

specific hydrogeologic systems? How 

important is the issue of model confirmation 

for conceptual models? 

The second round-table discussion dealt 

with Parameter Uncertainties. It asked: 

How are conceptual model and parameter 

uncertainties related? What statistical 

methods are appropriate for evaluating 

parameter uncertainty? How can the 

following topics be handled in a parameter 

uncertainty strategy: (a) representation 

heterogeneities; (b) scaling of representative 

parameter values; (c) assumptions of 

dimensionality, transient versus steady state 

flow and transport, and initial and boundary 

conditions; and (d) model confirmation 

using specific examples such as the 

INTRAVAL Project test cases, the 

decommissioning reviews, and in situ leach 

uranium extraction license applications? 

What reasonable expectations can be met 

in quantifying parameter uncertainty in the 

next 2 years? 
2 

•

•

•

The strategy described in this report 

addresses and answers many of the above 

questions. 

1.2 State-of-the-Art Summary

1.2.1 Preliminary Assessment 

Prior to developing the strategy described in 

this report, a study was conducted to identify 

information needs and objectives for the 

evaluation of hydrogeologic flow and 

transport models relevant to the PA of HLW 

and LLW radioactive waste disposal sites, 

decommissioning reviews, and uranium 

recovery facilities. The study included an 

assessment of hydrogeologic issues common 

to performance analyses of nuclear facilities 

and sites; a review and summary of 

approaches currently used to identify 

conceptual, mathematical and computational 

frameworks for the hydrogeologic flow and 

transport modeling of such facilities and 

sites; a discussion of uncertainties that may 

arise in the context of such framework 

formulations; a review and summary of 

approaches currently used to assess these 

uncertainties; and a preliminary 

identification of ways to extend and/or 

modify existing approaches so as to achieve 

the objectives of the contract. The review 

included technical literature that focuses on 

relations among site characterization, 

monitoring and modeling strategies related 

to constructing and applying groundwater 

flow and transport models, and their 

application to site specific information and 

databases. 

1.2.2 Key Conclusions from Preliminary 

State-of-the-Art Assessment 

The preliminary state-of-the-art assessment 

led to the following interim conclusions: 

• Analyses of groundwater flow and 

transport typically rely on a single 

conceptual model of site hydrogeology. 

This exposes such analyses to Type I 

model errors, which arise when one 

rejects (in this case by omission) valid 

alternative models. 

Critiques of groundwater flow and 

transport analyses, and legal challenges 

to such analyses, typically focus on the 

validity of the underlying conceptual 

model. By adopting (not rejecting) an 

invalid conceptual framework, one 

commits a Type II model error. If 

exposed to be severe, it may damage 

one's professional credibility and result 

in the loss of a legal contest. 

Analyses of model uncertainty based on 

a single hydrogeologic concept are prone 

to statistical bias (by committing a Type 

II error through reliance on an invalid 

model) and underestimation of 

uncertainty (by committing a Type I 

error through under-sampling of the 

relevant model space). 

The bias and uncertainty that result from 

reliance on an inadequate conceptual 

model are typically much larger than 

those introduced through an inadequate 

choice of model parameter values. Yet 

most uncertainty analyses of flow and 

transport ignore the former and focus 

exclusively on the latter. This often leads 

to overconfidence in the predictive 

capabilities of the model, for which there 

is no justification. 
3 

•

•

•

Each site is unique and there is no valid 

way to represent its hydrogeology by 

means of a generic conceptual model or 

set of parameters, regardless of purpose. 

Whereas simplicity may at times be 

desirable and justified, there is no 

guarantee that simple models of 

hydrogeology would necessarily yield 

conservative assessment of system 

performance; the opposite was found to 

be true in virtually all cases examined by 

the report. 

There is no established literature on 

ways to construct alternative conceptual 

models of site hydrogeology or methods 

to assess related uncertainties in 

groundwater flow and transport 

analyses. 

1.2.3 Uniqueness of Sites versus 

Non-Uniqueness of Models 

The idea that each site is unique and there is 

no valid way to represent its hydrology by 

means of a generic conceptual model or set 

of parameters has found eloquent support in 

a recent article be Beven (2000). According 

to him, 350 years of scientific hydrology 

have established some general hydrologic 

principles at all scales from single pore to 

continental river basins. Why then are 

hydrologic predictions, which are so 

important in water resources management 

and pollution control, so uncertain? Why 

does it appear so difficult to use our 

understanding, as embodied in the form of 

theories and models, in practice? The 

primary reason is that practicing 

hydrologists cannot deal in generalities but 

must deal with specific instances of sites 

that are unique in their characteristics over 

periods that are unique in their atmospheric 

and anthropogenic forcing. 

According to Beven, the latter has not been 

considered as a problem in the past. The 

attitude has been that if the modeling 

concepts were sufficiently accurate then it 

would be possible to represent the 

uniqueness of individual sites by specific 

values of model parameters. In principle this 

might be possible; in practice it proves to be 

untenable because of the impossibility of 

fully describing the system, and particularly 

the heterogeneity of its subsurface 

characteristics. 

In Beven's view, experience with predictive 

models in a wide variety of hydrological 

contexts suggests that, given the data 

available in even intensively studied 

locations, there is generally a very wide 

range of models and parameter sets within 

models that will yield acceptable 

simulations. Beven and Freer (2001) have 

attributed to Hornberger and Speer (1981) 

the notion that this is not simply a problem 

of identifying a correct or optimal model 

given limited data. Beven (1993) has called 

this the equifinality problem to stress that 

this is a generic problem in modeling 

complex hydrologic systems, and not simply 

a problem of identifying the "true" model 

structure or parameter set. According to 

Beven (2000), the equifinality problem 

arises from the limitations of current model 

structures in representing heterogeneous 

surface and subsurface flow systems, from 

limitations of measurement techniques and 

scales in defining system characteristics and 

the initial and boundary conditions for a 

model, and from the uniqueness of 

individual sites. 

Beven (2000) opines that in recent years 

there has been a move towards more and 

more detailed studies of flow in rivers, soils 

and aquifers, using more and more 

sophisticated instrumentation. The result has 
4 

generally been to demonstrate complexity of 

flow pathways, due to heterogeneity on 

different scales and interactions between the 

geometry of the flow domain and the 

prevailing hydraulic gradients and flow 

pathways (including both "dead zones" and 

"preferential" flows). It has proven difficult 

to use such information in applications of 

predictive models which are generally 

required to be used at scales larger than the 

detailed measurements. 

The problem is much less one of 

understanding than of uniqueness of place. It 

is not so much that the complexity of the 

measurements is not accessible to adequate 

interpretation (or even in some cases to 

being adequately modeled). The problem is 

that the measurements are specific to the 

study site with little or no guarantee that 

they can be considered representative even 

of sites that are hydrologically similar (in 

climate, physiography, ecology and 

geology). But to do detailed measurements 

throughout an area of interest is both 

impractical and unfeasibly expensive. The 

unique characteristics of a site of interest are 

therefore inherently unknowable. All that 

can be done is to constrain the model 

representations of the site to those that are 

acceptably realistic, usually in the sense of 

being consistent with the data. 

Given limited measurements on a 

hydrologic system, a uniquely correct model 

would not be identifiable even if one had a 

perfect model available. The current 

generation of model structures, while useful, 

may not be sufficiently accurate 

representations of the complexities of the 

flow pathways that are easy to perceive and 

understand but difficult to represent in 

mathematical form without making grossly 

simplifying assumptions. This implies that 

there may also then be many different model 

structures that might be consistent with the 

available observational data. Likewise, it 

will always be impossible to have sufficient 

information to identify a uniquely correct set 

of model parameters for any given model 

structure. We should therefore expect

equifinality of model structures and 

parameter sets in the representation of 

unique places and hydrologic systems. 

In Beven’s view, existing models are useful 

only because of a form of circular reasoning 

called model calibration. Hydrologists are 

prepared to accept that a model is adequate 

if it gives reasonable predictions of the 

available observations. Because of 

uniqueness of place, one does not expect 

that a model will provide reasonable 

predictions with only a priori estimates of 

parameter values (why should it?). Some 

form of calibration is therefore necessary. 

Such models may be more useful in 

allowing a decision to be made than in 

providing accurate predictions. 

1.2.4 Desired Level of Model Complexity 

The degree of complexity that a 

groundwater model should incorporate was 

the subject of a special session, 

"Groundwater models: How much 

complexity is warranted?" at the 1998 

Spring Meeting of the American 

Geophysical Union (Hunt and Zheng, 1999). 

The central questions addressed at the 

session were: We know that the natural 

world is characterized by complex and 

interrelated processes. Yet do we need to 

explicitly incorporate these intricacies and to 

perform the tasks we are charged with? In 

this era of expanding computer power and 

development of sophisticated preprocessors 

and postprocessors, are bigger and better 

machines making better models? 
5 

A common concern expressed at the special 

AGU session was that the extensive data 

required to construct a complex groundwater 

model may result in less time spent 

understanding the system and more time 

spent constructing and managing data input 

and output. It was suggested that careful 

quality assurance and quality control are 

required when the input data are large, and 

that peer review of modeling and data 

acquisition activities may be important in 

the regulatory context. On one hand, the 

opinion was voiced that contaminant 

transport models do not necessarily enjoy 

greater success as predictive tools with the 

inclusion of increasing levels of complexity; 

one should take models for what they are, 

namely, powerful heuristic tools with 

limited predictive capabilities. On the other 

hand, it was recognized that to capture 

certain flow and transport phenomena, 

models of appropriate complexity may be 

required. The degree of complexity should 

thus be in part a function of its intended use. 

Fractal scaling and stochastic methods were 

recognized as capable of capturing flow and 

transport behaviors in complex 

hydrogeologic systems by means of 

relatively simple models. 

1.2.5 Traditional Approach to Model 

Uncertainty Analysis 

The traditional approach to hydrologic 

model uncertainty analysis has been to 

postulate a deterministic model structure and 

treat its parameters as being imperfectly 

known. To quantify this imperfect 

knowledge, one must postulate a prior 

parameter uncertainty model. 

In cases where a statistically significant set 

of measured, site specific parameter values 

is available, one could postulate a Type A 

probabilistic model of prior parameter 

uncertainty based on statistics derived from 

these data. Ideally, the data would include 

measurements of the parameters proper on 

spatial and temporal scales compatible with 

the scale of resolution to be adopted in the 

intended model. The data would also include 

probabilistic information about errors of 

measurement and test interpretation that 

suffer from a known amount (ideally zero) 

of statistical bias. 

In cases where no such data are available in 

statistically significant quantities, one has 

the option of postulating a Type B model of 

prior parameter uncertainty on the basis of 

subjective probabilities. Such a model 

should always be suspected of suffering 

from an unknown amount of statistical and 

personal bias. Statistical bias is introduced 

due to lack of site specific information about 

mean values of the parameters in question. 

A personal bias tends to manifest itself in 

the form of assigned uncertainty measures 

(most importantly bias and error variance) 

that are either too small or too large. The 

first is a manifestation of over-confidence in 

the model parameters, the second of unduly 

low confidence in their values. 

Intermediate between Type A and Type B 

parameter uncertainty models is the case 

where indirect information about the 

parameters is available, from which relevant 

prior statistics can be derived formally. Such 

information may include (a) off-site 

measurements of the parameters proper 

(quite often on scales other than those 

corresponding to the intended scale of model 

resolution) and/or (b) surrogate 

measurements on site that are known to 

correlate with the parameters of interest (for 

example, porosities or geophysical 

signatures that correlate in known ways with 

permeabilities, water contents or fracture 

densities). Statistics derived from off-site 
6 

data must be considered potentially biased 

(due to a lack of site-specific information 

about mean parameter values and 

incompatibility of geology and scale). The 

associated variance may be too small or too 

large, depending on the quantity and quality 

of such data. Statistics derived from 

surrogate data may suffer from poorly 

defined correlations and incompatibility of 

scale. 

The traditional approach to reduce 

parameter bias and uncertainty has been to 

calibrate the model against monitored 

observations of hydrologic system behavior. 

The process is also known as parameter 

identification or estimation, inverse solution 

or history matching. Its goal is to render the 

model compatible with available monitoring 

data and thereby increase the chance that it 

would provide reliable predictions of 

hydrologic behavior in the future. 

The last thirty years have seen major 

advances in the development of theories and 

algorithms for the estimation of 

deterministic model parameters. Many 

(though not all) of these theories and 

algorithms are "statistical" in that they 

include analyses of parameter estimation 

uncertainty. Such analyses typically accept, 

but do not necessarily require, information 

about prior parameter statistics as input. The 

output includes posterior statistics of 

parameter estimation errors, which are 

generally less biased and smaller than the 

prior estimation errors. A recent summary 

and comparison of various statistical inverse 

methods for groundwater flow models has 

been published by Zimmerman et al. (1998). 

Public domain or commercially available 

inverse codes include those developed by 

Poeter and Hill (1998), Doherty et al.

(1994), Finsterle (1999a-c) and Simunek et 

al. (1999). A detailed set of guidelines for 

the effective calibration of deterministic 

groundwater flow models has been prepared 

by Hill (1998). 

Bias and uncertainty in model parameters 

translate into bias and uncertainty in 

hydrologic model predictions. To assess the 

latter a priori, it is necessary to "propagate" 

the former through the model under assumed 

future scenarios. This is in contrast to a 

posteriori assessment on the basis of socalled post audits, in which the model 

predictions are compared with monitored 

future behavior of the hydrologic system. If 

the post facto comparison is deemed 

successful, the model is considered 

"confirmed" or "validated" with reference to 

the available monitoring data. 

The most common way to propagate input 

errors through an otherwise deterministic 

model is by means of Monte Carlo 

simulations. This is done by generating 

multiple, equally likely sets of randomized 

inputs; computing deterministically a set of 

corresponding model outputs for each; and 

analyzing the resultant multiple, equally 

likely random output sets statistically. This 

approach is common in the performance 

assessment of nuclear facilities and sites, in 

which prior input statistics of Type B are 

often used due to insufficiency of site 

specific parameter and monitoring data. 

Another approach is to associate the model 

predictions with approximate error bounds, 

or confidence limits, computed on the basis 

of linear regression theory applied to the 

(typically nonlinear) groundwater inverse 

model (Poeter and Hill, 1998; Hill, 1998). 

1.2.6 Accounting for Conceptual Model 

Uncertainty 

Carrera and Neuman (1986a-b) and Samper 

and Neuman (1989a-b) have noted that an 

inadequate model structure 
7 

(conceptualization) is far more detrimental 

to its predictive ability than is a suboptimal 

set of model parameters. This helps explain 

why the National Research Council (2000), 

in a recent study of Research Needs in 

Subsurface Science for the Environmental 

Management Program of the U.S. 

Department of Energy, has listed as second 

among four recommended research 

emphases the development of tools and 

methodologies for conceptual modeling with 

emphasis on heterogeneity, scale and 

uncertainty bounds on the basis of field 

experimental data. 

Recently, a panel was convened by the 

National Research Council (2001) to 

describe the process through which 

conceptual models of flow and transport in 

the fractured vadose zone are developed, 

tested, refined and reviewed. The panel 

concluded that development of the 

conceptual model is the most important part 

of the modeling process. The conceptual 

model is the foundation of the quantitative 

mathematical representation of the field site 

(i.e., the mathematical model), which in turn 

is the basis for the computer code used for 

simulation. Reasonable alternative 

conceptualizations and hypotheses should be 

developed and evaluated. In some cases, the 

early part of a study might involve multiple 

conceptual models until alternatives are 

eliminated by field results. 

According to the panel, it is important to 

recognize that model predictions require 

assumptions about future events or 

scenarios, and are subject to uncertainty. 

Meaningful quantification of uncertainty 

should be considered an integral part of any 

modeling endeavor, as it establishes 

confidence bands on predictions given the 

current state of knowledge about the system. 

A suite of predictions for a range of different 

assumptions and future scenarios is more 

useful than a single prediction. 

In fact, we have seen earlier that hydrologic 

models are prone to equifinality (Beven, 

2000) in both their structures and parameter 

sets. This means that there is uncertainty not 

only about the parameter values that should 

enter into a given model (as characterized by 

its structure), but also about the very 

structure (conceptual and mathematical) of 

the model that should represent the 

hydrologic system of interest. The 

traditional approach to model uncertainty 

analysis, which considers only a single 

deterministic model structure, fails to 

adequately sample the complete space of 

plausible hydrologic models. As such, it is 

prone to modeling bias and underestimation 

of model uncertainty. 

An example of how one could account 

quantitatively for structural model 

uncertainties was given by James and 

Oldenburg (1997). They investigated the 

uncertainty of simulated TCE 

concentrations, at the point of potential 

human exposure, due to uncertainty in the 

parameters (permeability, porosity, 

diffusivity, solubility, adsorption) and 

variations in the conceptual-mathematical 

model (injection rate of TCE source; initial 

TCE source saturation; regional 

groundwater flow; heterogeneity of 

permeability). The authors used the threedimensional code T2VOC to simulate threephase (gas, aqueous, NAPL), threecomponent (air, water, VOC) nonisothermal 

flow based on an actual site with a 25 m 

thick vadose zone and a saturated zone. To 

assess parameter uncertainty associated with 

a given model, they used the inverse code 

ITOUGH2. Their final step was to assess the 

range of outcomes that one obtains with the 

entire set of alternative conceptual-
8 

mathematical models. James and Oldenburg 

found that uncertainties in their model 

outcomes span orders of magnitude, and that 

both parameter and model uncertainty 

contribute significantly to this wide range of 

outcomes. They concluded that "risk 

assessment and remediation selection ... is 

meaningful only if analysis includes 

quantitative estimates of ... uncertainty" in 

both the parameters and the conceptualmathematical models. 

A similar approach has been advocated more 

recently by Samper and Molinero (2000). 

The authors consider the main uncertainties 

in predicting groundwater flow and transport 

to be those associated with the selection of 

future scenarios, choice of model structure 

and assignment of model parameters. The 

authors consider parameter uncertainty to be 

minor in comparison to structural (i.e. 

conceptual) model errors. They suggest to 

evaluate model predictive uncertainty by 

calibrating a number of conceptualmathematical models against available 

monitoring data, to retain those calibrated 

models that can adequately reproduce past 

observations, to assess the predictive 

uncertainty of each model due to the 

uncertainty of its parameters, to treat the 

predictive uncertainty of each model as 

being equally likely, and to produce a single 

combined range of predictive uncertainties. 

Rather than relying on model calibration and 

treating the outcomes of different structural 

models as being equally likely, Beven and 

Binley (1992) have proposed a strategy to 

which they refer as GLUE (Generalized 

Likelihood Uncertainty Estimation). The 

strategy calls for the identification of several 

alternative structural models and the 

postulation of a prior probabilistic model of 

parameter uncertainty for each. Each 

structural model, coupled with its 

corresponding parameter uncertainty model, 

is used to generate Monte Carlo realizations 

of past hydrologic behaviors and to compare 

the results with monitored system behavior 

during the same period. Likelihood 

measures are defined to gauge the degree of 

correspondence between each simulated and 

observed record of system behavior. If a 

likelihood measure falls below a 

subjectively defined "rejection criterion," 

the corresponding combination of model 

structure and parameter set are discarded. 

Those combinations which pass this test are 

retained to provide predictions of system 

behavior under selected future scenarios. 

Each prediction is weighted by a 

corresponding normalized likelihood 

measure (so as to render the sum of all 

likelihood measures equal to one), to 

produce a likelihood-weighted cumulative 

distribution of all available predictions. 

As of 1992, the GLUE strategy has been 

applied to a variety of hydrologic problems 

including rainfall-runoff modeling, solute 

transport in rivers and groundwater, flood 

frequency and flood inundation analyses, 

critical load modeling, nitrogen budget 

calculations and the analysis of distributed 

land surface to atmosphere vapor and heat 

fluxes. For recent discussions of GLUE and 

its applications the reader is referred to 

Beven (2000) and Beven and Freer (2001). 

A Bayesian approach to the quantification of 

errors in a single groundwater model was 

recently proposed by Gaganis and Smith 

(2001). Like GLUE, it relies on Monte Carlo 

simulations without model calibration and 

on subjective criteria of “model 

correctness.” 

It must be understood that the set of 

predictions one produces with any given 

choice of alternative structural models and 
9 

parameter sets, by whatever method, is 

conditional on the choice of models and the 

data used to support them. As such, these 

predictions do not represent all possibilities 

but only a limited range of such possibilities, 

associated with these models and data. Any 

change in the latter would generally lead to a 

different assessment of predictive model 

uncertainty. There thus appears to be no way 

to assess the uncertainty of hydrologic 

predictions in an absolute sense, only in a 

conditional or relative sense. 

1.2.7 Multiobjective Tradeoffs between 

Model Fit and Complexity 

The National Research Council (2001) panel 

on conceptual models of flow and transport 

in the fractured vadose zone noted that a 

conceptual model is by necessity a 

simplification of the real system. However, 

the degree of simplification must be 

commensurate with the problem being 

addressed. Yet neither the panel nor the 

broader literature offer any systematic 

guidelines about how to determine the 

optimum degree of model simplification for 

a given purpose. 

One important measure of model complexity 

is the number of free parameters (degrees of 

freedom) that it contains. This is the number 

of parameters that are free to be adjusted 

during calibration so as to achieve an 

optimum model fit with monitored system 

behavior. A model with a given set of free 

parameters, which describe a spatially 

varying quantity such as permeability or 

porosity, might be more or less complex 

depending on their mode and degree of 

variation in space. Several such measures of 

complexity have been introduced by 

Emsellem and de Marsily (1971), Neuman 

(1973), Distefano and Rath (1975) and 

Neuman and De Marsily (1976). 

The approach taken by these authors was to 

calibrate a series of hydrologic models with 

gradually increasing degrees of model 

complexity. Each calibration yields a 

measure of model fit that corresponds to a 

particular measure of model complexity. By 

plotting the measure of fit versus the 

measure of complexity, one usually finds the 

former to improve (decrease) as the model 

becomes more complex. The improvement 

is rapid when the model is relatively simple 

and slow when the model is relatively 

complex. Clearly, the standard hydrologic 

objectives of achieving a good fit between 

simulated and observed system behavior, 

and keeping the model simple, are in 

conflict. Instead of one optimum solution to 

this multiobjective or Pareto optimization 

problem, there is a range of feasible (or 

Pareto) solutions among which one must 

choose the optimum tradeoff subjectively 

(Neuman, 1973). The optimum tradeoff is 

achieved at a point beyond which any 

further increase in model complexity would 

bring about only a relatively insignificant 

improvement in model fit. A similar idea has 

been espoused more recently by Sun and 

Yeh (1985) and summarized by Sun (1994) 

in a book on Inverse Problems in 

Groundwater Modeling. 

It is possible to postulate multiple, noncommensurate model fit criteria (Gupta et 

al., 1998) and model complexity measures 

(Neuman, 1973) within a multiobjective 

framework. Typically, models lack the 

ability to simulate all relevant aspects of 

hydrologic behavior with an equal degree of 

fidelity. For example, a subsurface transport 

model may not be able to simulate 

accurately observed contaminant 

concentration versus time records at several 

monitoring locations simultaneously without 

becoming overly complex and cumbersome. 

Or the model may be unable to reproduce 
10 

equally well the peak arrival time, 

magnitude, rate of ascent and rate of 

recession of a contaminant breakthrough 

record. One then has the option of defining 

multiple criteria of model fit, each 

representing a different aspect of desired 

model performance. Multiobjective 

programming techniques are available (e.g., 

Duckstein et al., 1991) that should then 

make it possible for the modeler to 

investigate the range of feasible solutions to 

the problem and solve it by determining 

(subjectively) an acceptable tradeoff 

between all of its conflicting and 

incommensurate objectives. A 

computationally efficient and 

mathematically rigorous method to generate 

binary graphs of tradeoff between each pair 

of conflicting objectives, and to determine 

an optimum tradeoff between all of them 

interactively, was developed by Neuman and 

Krzystofowicz (1977). 

1.2.8 Combined Measures of Model Fit 

and Complexity 

Several combined measures of model fit and 

complexity have been developed in the 

context of time series analysis by Akaike 

(1974, 1977), Hannan (1980) and Kashyap 

(1982). In recent years, such measures have 

been used with increasing frequency to rank 

and discriminate between alternative 

geostatistical (Samper and Neuman, 1989ab; Chen et al., 2000) and dynamic (Carrera 

and Neuman, 1986a-b; Finsterle and 

Faybishenko, 1999) groundwater models. 

The model discrimination criteria combine a 

log-likelihood function, which gauges the 

quality of model fit to monitoring data, with 

a term proportional to the number of free 

model parameters. By ranking the models in 

ascending order of their associated 

discrimination criteria, one favors the least 

complex among those that perform equally 

well in terms of model fit. Among models of 

equal complexity, those that fit the 

monitoring data better are favored. 

The model discrimination criterion of 

Kashyap contains yet another term that 

gauges the information content of the 

available data. It thus allows considering 

models of growing complexity as the data 

base improves in quantity and quality. 

Stated otherwise, the criterion recognizes 

that when the data base is limited and/or of 

poor quality, one has little justification for 

selecting an elaborate model with numerous 

parameters. Instead, one should then prefer a 

simpler model with fewer parameters, which 

nevertheless reflects adequately the 

underlying hydrogeologic structure of the 

system, and the corresponding flow and 

transport regime. Kashyap's criterion favors 

that model which, among all alternatives 

considered, is least probable (or likely, in an 

average sense) to be incorrect. Stated 

otherwise, the criterion minimizes the 

average probability of selecting the wrong 

model among a set of alternatives. 

1.2.9 Effects of Scale on Hydrologic 

Predictability 

In September 2000, a workshop entitled 

Predictability and Limits to Prediction in 

Hydrologic Systems was held at the 

National Center for Atmospheric Research 

in Boulder, Colorado, under the auspices of 

the National Research Council. The 

workshop concluded that (Entekhabi, 2000) 

hydrologic systems contain heterogeneous 

geological, topographic, and ecological 

features that vary on multiple scales. This 

multiplicity of scales has a pervasive effect 

on hydrologic characterization and 

prediction. In general, the effective response 

of systems at larger scales is not completely 

determined by scaling local processes. 
11 

In hydrologic science, heterogeneity is a rule 

and it cannot necessarily be fully captured 

by randomization of parameters (essentially 

an after-thought). Interactions among microscale features often lead to effects that are 

not completely represented in macro-scale 

predictions based on effective parameters 

for micro-scale models. 

Examples include enhanced surface flux due 

to land-breeze circulations over 

heterogeneous patches, regional recharge 

and discharge patterns over complex terrain. 

There are processes and conditions when the 

effective parameter approach to scaling may 

be feasible. In the remaining circumstances, 

the macro-scale and micro-scale predictive 

relations for hydrologic processes may have 

different functional forms and dependencies. 

Furthermore, there may be organizing 

principles at work that result in simple 

procedures for statistically relating variables 

across a wide range of scales in the 

hydrologic system. We add that such 

principles have been noted and procedures 

developed (with applications to the 

permeability and dispersivity of geologic 

media) by Neuman (1990, 1994), Di 

Federico and Neuman (1997; see also 

Neuman and Di Federico, 1998) and Di 

Federico et al. (1999). 

According to Entekhabi (2000), it has been 

traditional to assume that spatially and 

temporally averaged hydrologic quantities 

are less prone to predictive uncertainty than 

quantities that are localized in space-time. 

While this may be true for systems that are 

statistically homogeneous and stationary, 

such systems are the exception rather than 

the rule. Spatial and temporal averages are 

not necessarily more predictable as 

traditionally believed if the averaging covers 

a scale that contains a strong transition or 

change in behavior (analogous to bifurcation 

in nonlinear dynamic systems). 

1.2.10 Prediction Uncertainty Due to 

Random Fluctuations in 

Hydrologic Variables 

Hydrogeologic environments consist of 

natural soils and rocks that exhibit both 

systematic and random spatial variations in 

hydraulic and transport properties on a 

multiplicity of scales. Groundwater flow and 

solute transport take place under the action 

of forces whose exact nature, magnitude and 

space-time distribution are generally 

uncertain. Traditional deterministic methods 

of analyses capture at best the larger-scale, 

systematic components of these variations 

and distributions. They however fail to 

resolve smaller scale variations or account 

for their uncertain nature. Yet capturing 

hydrologic complexity on a wide range of 

scales, and quantifying the associated 

uncertainty, is proving to be of paramount 

importance for a wide array of applications 

such as the characterization and remediation 

of contaminated sites, risk analysis and 

monitoring design for sites subject to 

potential future contamination, and 

performance assessment of nuclear facilities 

and sites. 

A recent study of groundwater and soil 

cleanup by the National Research Council 

(1999) recognizes that the subsurface is 

usually highly heterogeneous and 

characterizing this variability is extremely 

difficult. This heterogeneity and difficulty in 

characterization complicate the design of 

subsurface cleanup systems because 

predicting system performance under such 

uncertain conditions is difficult. An accurate 

knowledge of geological heterogeneities is 

vital for evaluating the hydrogeologic limits 

on subsurface contaminant remediation. 
12 

In a recent editorial titled "It's the 

Heterogeneity!", the Editor of the most 

widely read groundwater journal (Wood, 

2000) opines that the natural system is so 

complex that it will be many years before 

one can effectively deal with heterogeneity 

on societally important scales. 

The complex nature of the subsurface and of 

the conditions that control groundwater flow 

and transport within it throw into question 

the reliability of traditional deterministic 

methods of hydrogeologic analysis. The 

emphasis is therefore shifting from 

deterministic to probabilistic methods that 

are better suited for these needs. The trend 

has become to describe the spatial variability 

and scaling of hydrogeologic medium 

properties geostatistically, and to analyze 

subsurface fluid flow and solute transport 

stochastically. This trend has been 

documented in a number of recent books 

including those by Dagan and Neuman 

(1997) and Zhang (2001). 

The most common method of stochastic 

analysis is high-resolution computational 

Monte Carlo simulation that produces a 

large number of equally likely results. These 

nonunique results are summarized in terms 

of statistically averaged quantities, their 

variance-covariance, and perhaps higher 

moments of the corresponding sample 

probability distributions. Results that honor 

measured values of medium properties are 

said to be conditioned on these data. Upon 

conditioning the simulations on measured 

values of parameters in space, one obtains 

(among others) conditional mean flow and 

transport variables that constitute optimum 

unbiased predictors of these unknown 

random quantities. One also obtains 

conditional second moments (variancecovariance) that provide a measure of the 

associated prediction errors. To condition 

the predictions on system monitoring data, 

one must either discard random simulations 

that do not reproduce the observations, or 

employ an inverse procedure of the kind 

developed for this purpose by Sahuiquillo et 

al. (1992), G\mez-Hern<ndez et al. (1997; 

2000) and Capilla et al. (1997). 

Monte Carlo analysis requires knowing the 

multivariate probability distribution of 

relevant hydrogeologic properties, which is 

difficult to infer from commonly available 

data. To achieve a high space-time 

resolution of relevant stochastic phenomena, 

it requires the use of large space-time grids 

with very small discretization intervals. To 

yield sample statistics that converge to their 

theoretical (ensemble) counterparts requires 

numerous repetitions (realizations). The net 

result is a large amount of computational 

time and storage, which are considered 

uneconomical for many practical 

applications. 

This has given impetus to the development 

of alternative stochastic methods that allow 

one to compute the conditional mean, 

variance and covariance of groundwater 

flow and transport variables directly, 

without Monte Carlo simulation. This is 

done on the basis of conditional moment 

equations that may be nonlocal (integrodifferential) and contain parameters that 

depend on more than one point in space 

and/or time. Conditioning additionally 

renders the parameters of such equations 

dependent not only on medium properties 

but also on the information that one has 

about these properties (scale, location, 

quantity and quality of data). Hence the 

parameters of conditional moment equations 

are nonunique. Exact conditional moment 

equations have been developed for steady 

state (Neuman and Orr, 1993; Neuman et 

al., 1996) and transient (Tartakovsky and 
13 

Neuman, 1998, 1999) groundwater flow in 

saturated porous media, steady state flow in 

a certain class of unsaturated soils 

(Tartakovsky et al., 1999; Lu et al., 2002), 

and advective (Neuman, 1993) or advectivedispersive (Zhang and Neuman, 1996) 

transport of a nonreactive solute. 

Though the underlying stochastic flow and 

transport regimes are taken to obey Darcy's 

law and Fick's analogy, these relations are 

generally not obeyed by the conditional 

mean flow and transport regimes except in 

special cases or as localized approximations. 

Such localized approximations yield 

familiar-looking differential equations 

which, however, acquire a non-traditional 

meaning in that their parameters (hydraulic 

conductivity, seepage velocity, dispersivity) 

and state variables (hydraulic head, 

concentration, fluid and solute fluxes) are 

information-dependent and thus nonunique 

(Guadagnini and Neuman, 1999a-b, 2001; 

Ye et al., 2002). Whereas nonlocal moment 

equations contain information about 

predictive uncertainty, localized 

(differential) equations generally do not. 

To render conditional moment equations 

workable, one must adopt some form of 

closure approximation, such as perturbation 

(Guadagnini and Neuman, 1999a-b, 2001; 

Ye et al., 2002; Lu et al., 2002) or an 

assumption of Gaussianity (Neuman et al., 

1999; Amir and Neuman, 2001; Wang et al., 

2002). Though perturbative closure 

approximations are not guaranteed to work 

for strongly heterogeneous media, they have 

proven to work remarkably well in twodimensional finite element analyses of 

superimposed mean uniform and convergent 

steady state (Guadagnini and Neuman, 

1999a-b) and transient (Ye et al., 2002) 

flows in saturated porous media and steadystate flow in unsaturated media (Lu et al., 2002). 

In the aforementioned moment analyses, 

conditioning was limited to prior 

measurements of parameters (mainly 

permeability) at discrete points in space. To 

condition the predictions on state variables 

such as hydraulic head and concentration, it 

is necessary to develop corresponding 

inverse algorithms of the kind recently 

described by Hernandez et al. (2002). 

1.3 Outline of Proposed Strategy 

1.3.1 Comprehensive Approach to 

Modeling and Uncertainty Analysis 

This report describes a strategy that 

embodies a systematic and comprehensive 

approach to hydrogeologic 

conceptualization, model development and 

predictive uncertainty analysis. The strategy 

is comprehensive in that considers all stages 

of model building and accounts jointly for 

uncertainties that arise at each of them. The 

stages include regional and site 

characterization, hydrogeologic 

conceptualization, development of 

conceptual-mathematical model structure, 

parameter estimation on the basis of 

monitored system behavior, and assessment 

of predictive uncertainty. In addition to 

parameter uncertainty, the strategy concerns 

itself with uncertainties arising from 

incomplete definitions of (a) the conceptual 

framework that determines model structure, 

(b) spatial and temporal variations in 

hydrologic variables that are either not fully 

captured by the available data or not fully 

resolved by the model, and (c) the scaling 

behavior of hydrogeologic variables. 

The strategy addresses these issues within a 

unified and coherent theoretical framework. 

It does so by relying on both traditional and 

nontraditional ideas and techniques. Some 

nontraditional ideas and techniques are 
14 

taken from the recent literature and some are 

entirely new. Most are amenable to 

immediate implementation with the aid of 

existing software. Though there is at present 

no publicly available software for the 

solution of stochastic flow and transport 

problems by means of the moment equation 

method (which is part of the strategy), such 

software is expected to become available 

within a few years. In the meantime, such 

problems can be tackled effectively with 

existing software by means of the Monte 

Carlo option provided in the strategy. 

1.3.2 NRC Licensing Context 

 

The strategy is generic but designed to be of 

practical use to NRC staff in their review of 

decommissioning plans, and performance 

assessment of high-level and low-level 

radioactive waste disposal sites as well as 

uranium recovery facilities. For this 

purpose, the strategy is cast in the context of 

a framework that is useful to NRC staff 

review and performance evaluation needs. 

The regulatory context is defined in terms of 

corresponding performance measures (as 

identified by the NRC staff in Appendix A), 

hydrogeologic analyses that are needed to 

assess them, the desired reliability of such 

assessments, and the expenditure (in time, 

effort and money) that is allowed to achieve 

it. Examples from various case studies are 

included to help illustrate some aspects of 

these analyses. 

An important component of the strategy is a 

systematic sequence of logical questions, 

guidelines and criteria with analytical 

methods that may be useful to the NRC staff 

in their review and performance evaluation. 

Though the contract does not call for the 

development or demonstration of practical 

data processing and modeling tools or 

software, such tools and software are 

identified where possible. To the extent 

feasible, the strategy is supported by field 

experimental data that cover a range of 

technical problems facing the NRC staff. 

1.3.3 Hydrogeologic Conceptualization 

and Data 

The strategy encourages exploration of 

varied conceptual frameworks and 

assumptions at all stages of hydrogeologic 

model development through a 

comprehensive evaluation of a broad range 

of regional and site data, their translation 

into coherent and internally consistent 

conceptual-mathematical models, and 

computation and visualization based on 

these data and models. Included among 

these frameworks and assumptions are 

various model simplification and abstraction 

schemes. The strategy recognizes that site 

characterization and monitoring data are 

expensive and difficult to collect, leading to 

a ubiquitous scarcity of hard site 

information. It is therefore critically 

important to assess the role that such data 

play in rendering the hydrogeologic 

performance analysis credible. The strategy 

stresses the role of characterization and 

monitoring data in helping one identify and 

test alternative conceptual models, make 

rational choices among them, gauge and 

reduce model bias and uncertainty through 

proper model selection and calibration, 

render and assess the reliability of model 

predictions, and confirm the assessment 

through independent peer review as well as 

at least some degree of direct verification. 

1.3.4 Iterative Approach 

The strategy encourages an iterative 

approach to modeling, whereby a 

preliminary conceptual-mathematical model 

is gradually altered and refined until one or 
15 

more likely alternatives have been identified 

and analyzed. The idea is to start with a 

conceptual-mathematical model that is as 

simple as may appear warranted by the 

context of the problem and the available 

data. This model is then tested for 

qualitative consistency with the available 

data, made quantitatively compatible with 

the data through calibration, and subjected 

to an assessment of its predictive reliability 

(in terms of potential bias and uncertainty). 

Next, the model structure is altered and/or 

refined (by modifying its hydrogeologic 

makeup or features; dimensionality; scale of 

resolution in space-time; type of driving 

forces or events including sources, initial 

and boundary conditions; and governing 

equations or processes), the model is 

retested and compared with the former. 

The iterative procedure is repeated till there 

is good reason to conclude that any 

additional alterations or refinements of the 

models would lead at best to minor 

improvements in their predictive capability. 

Models whose predictive capability is 

deemed low in comparison to other are 

discarded and the rest retained for further 

analysis. The retained models are used to 

render a joint prediction. The corresponding 

prediction error is quantified by means of a 

joint variance. If the joint prediction and 

associated error variance are deemed 

acceptable for the stated purpose of the 

analysis, the iterative process is halted and 

the models declared reliable and credible 

within a clearly stated conditional (on the 

available information and selected set of 

models) margin of error. The conditional 

nature of this error estimate is taken to imply 

that it constitutes a lower bound, and that the 

actual predictive error may (and most 

probably is) larger to an unknown degree.

If the models are deemed unacceptable, a 

sensitivity analysis is performed to help 

identify the type and quantity of additional 

site data that might materially enhance their 

reliability and credibility. A decision is then 

made, based on the potential benefit (in 

terms of bias and uncertainty reduction) and 

cost (in terms of time, effort and money) of 

such data, whether or not to collect them and 

how. If and when new data of significant 

weight are obtained, the iterative process is 

repeated till its cost-benefit ratio reaches a 

value that is not considered worth 

exceeding. 

1.3.5 When to Stop Collecting Data and 

Refining the Model? 

It is worth repeating that the questions (1) 

what degree of model refinement is enough 

for a given regulatory purpose, and (2) how 

much and what kind of data are enough for 

the task, are recast here in terms of 

economic and policy questions, which are 

the responsibility of managers and decisionmakers to answer. The strategy is designed 

to help the analyst address the questions by 

how much would further model refinements, 

and/or additional site characterization and 

monitoring, improve the prediction of 

performance measures derived from the 

hydrogeologic analysis. In other words, the 

strategy addresses the worth of model 

refinement and data collection in terms of 

their contributions to the potential 

enhancement of model reliability and 

credibility, not in terms of their marginal 

cost-benefit. Its aim is to allow managers 

and decision-makers to make informed 

decisions about the time, manpower and 

budget that they deem worth allocating to 

these activities in light of such enhancement 

under existing administrative, budgetary and 

policy constraints. 
16 

1.3.6 Assimilating Regional and Site 

Characterization Data 

The strategy recognizes that it is often 

possible to postulate hydrogeologic 

conceptual models or hypotheses for a site 

on the basis of publicly available geologic 

and geographic information about its 

surroundings. Additional conceptualization 

can be done on the basis of generic data 

about similar regions and the properties of 

similar materials elsewhere. Several such 

regional and generic sources of information 

are identified and discussed in this report. 

Yet each site is unique and so virtually 

guaranteed to reveal additional features, 

properties and behaviors when characterized 

in some detail locally. Hence the strategy 

considers local characterization essential for 

the postulation of acceptably robust 

conceptual hydrogeologic models for a site. 

Regional and site characterization data tend 

to represent a wide range of measurement 

scales, not all of which are compatible with 

the intended scale of hydrogeologic model 

resolution. The strategy promotes 

recognition of this important issue and an 

effort to render the scale of measurement 

compatible with the scale of model 

resolution. This can be done by either 

rescaling the data to fit the scale of model 

resolution (which often entails averaging or 

upscaling over computational grid cells) or 

adapting model resolution to fit the scale of 

measurement (which often entails adapting 

the size of grid cells to the size of the data 

support). Recent advances in scaling theory 

and practice are briefly reviewed to promote 

awareness of fundamental issues associated 

with the scaling of hydrogeologic variables. 

The strategy encourages the analyst to 

associate model parameters, derived from 

regional and site characterization data, with 

statistical measures of their uncertainty. 

Such prior statistics provide the only way to 

assess model predictive uncertainty when 

suitable monitoring data are not available. 

Methods are discussed and an example 

given to derive such statistics from generic 

data sources on the basis of similarity in soil 

or rock textural features; to update these 

statistics on the basis of site-specific data 

using a Bayesian approach; and to 

interpolate, extrapolate and/or average 

parameters across the site geostatistically on 

the basis of measured values at discrete 

points in space. 

1.3.7 Testing and Calibration against 

Monitoring Data 

While hydrogeologic characterization of a 

site and its surroundings makes 

conceptualization possible, it does not 

provide the means to test conceptual models 

or compare them with other alternatives. For 

this, it is necessary to have monitoring data 

that constitute observations of actual 

hydrologic behavior at and around the site. 

Only with such data can one evaluate the 

ability of models to mimic real system 

behavior (qualitatively at the conceptual 

level, quantitatively at the conceptualmathematical level), improve their ability to 

do so through calibration against the 

monitoring data, determine their optimum 

degree of refinement or complexity, and 

compare them with each other (qualitatively 

and quantitatively). The proposed strategy 

introduces ways to accomplish these tasks in 

the context of traditional deterministic 

groundwater flow and transport models. 

The strategy encourages the analyst to 

employ statistical measures of model 

performance (or fit between simulated and 

observed hydrologic behaviors) where 

possible; to determine feasible tradeoffs 
17 

between model fit and complexity by relying 

on multiobjective approaches; and to rank 

models based on (a) statistical indicators of 

the quality of fit they produce (as obtained 

from an analysis of residuals), and (b) 

likelihood-based model discrimination 

criteria (which penalize models for using an 

excessive number of free parameters to 

achieve a given quality of fit). Models that 

show relatively poor fits and rank low on the 

list can be subjectively eliminated from 

further consideration at this stage. 

An important byproduct of model 

calibration is information about parameter 

uncertainty. When reliable and relevant (in 

terms of type and scale) prior statistics about 

the parameters are available from site 

characterization, the strategy encourages 

their use as input into the calibration 

process. Regardless of whether or not 

suitable prior statistics are available and 

employed for model calibration, the latter 

should produce posterior statistics for the 

optimum parameter estimates. The strategy 

considers the posterior parameter estimates, 

and their error statistics, more suitable for 

the analysis of model predictive uncertainty 

than the prior estimates and their statistics. 

This is so because calibration generally 

improves the ability of the model to 

reproduce simulated hydrologic behavior, 

and a calibrated model is therefore deemed 

more reliable and credible as a predictive 

tool than an uncalibrated model. The same 

holds true for the associated parameter 

estimation statistics. 

1.3.8 Predictive Uncertainty of 

Deterministic Models 

The strategy considers two alternative 

methods to assess the predictive uncertainty 

of a deterministic groundwater model under 

assumed future scenarios. The first method 

relies on Monte Carlo simulation of these 

scenarios using prior or posterior parameter 

estimates and statistics. The second method 

establishes approximate error bounds, or 

confidence limits, for such scenarios by 

linearization, using the same parameter 

estimates and statistics. The first approach is 

more accurate but costly in terms of 

computer time and effort. In both cases, the 

strategy favors the use of posterior estimates 

over that of priors on the understanding that 

the former are less biased and uncertain than 

the latter. 

To render a joint prediction and assess the 

corresponding joint uncertainty based on all 

calibrated models that have been retained as 

constituting potentially viable alternatives, 

the strategy introduces a new approach. The 

approach is based on the idea of Maximum 

Likelihood Bayesian Model Averaging 

(MLBMA) recently proposed by Neuman 

(2002). MLBMA relies on probabilistic 

maximum likelihood (ML) concepts to (a) 

calibrate each model against observed 

space-time variations in system states 

(pressure, water content, concentration), 

considering prior information about relevant 

soil properties; (b) eliminate models that 

cannot be so calibrated with acceptable 

fidelity; (c) predict future system behavior 

or performance measures (travel times, 

concentrations, mass rates), and assess 

corresponding predictive uncertainty, using 

each model; and (d) average the results 

using posterior model probabilities as 

weights. MLBMA supports the principle of 

parsimony in that among models having 

similar predictive capabilities, it favors those 

having fewer parameters and are thus 

simpler. It renders an earlier Bayesian 

Model Averaging (BMA) approach (Hoeting 

et al., 1999) computationally feasible by 

basing it on a maximum likelihood 

approximation due to Kashyap (1982) and 
18 

the maximum likelihood parameter 

estimation method of Carrera and Neuman 

(1986a). Its final outcome is a conditional 

assessment of predictive uncertainty that 

accounts jointly for uncertainties in the 

conceptual-mathematical model structure 

and its parameters. 

MLBA differs in important ways from 

earlier approaches. It differs fundamentally 

from the GLUE approach of Beven and 

Binley (1992; see also Beven, 2000, and 

Beven and Freer, 2001), as well as from the 

Bayesian approach of Gaganis and Smith 

(2001), in that it (a) relies on posterior (postcalibration) rather than prior statistics, (b) 

avoids the need for time-consuming Monte 

Carlo simulations, and (c) obviates the need 

for subjective model acceptance thresholds 

or criteria. MLBA differs from the approach 

of James and Oldenburg (1997) and Samper 

and Molinero (2000) in that it does not 

generally treat calibrated models as being 

equally likely. 

It is important to reemphasize that the set of 

predictions one produces with any given 

choice of alternative structural models and 

parameter sets, by whatever method, is 

conditional on the choice of models and the 

data used to support them. As such, these 

predictions do not represent all possibilities 

but only a limited range of such possibilities, 

associated with these models and data. Any 

change in the latter would generally lead to a 

different assessment of predictive model 

uncertainty. There thus appears to be no way 

to assess the uncertainty of hydrologic 

predictions in an absolute sense, only in a 

conditional or relative sense. 

If suitable monitoring data are not available, 

then there is neither an objective nor a 

quantitative basis for the comparison of 

potentially viable conceptual-mathematical 

models. Likewise, there are no posterior but 

only prior parameter estimates and statistics 

to deal with. To quantify the predictive 

uncertainty of each such model, one again 

has the option of relying on Monte Carlo or 

a linearized analysis based on the available 

prior statistics. To aggregate these results, 

one can either weigh all models equally or 

associate them with subjective weights 

based on the professional judgment of the 

analyst or a group of experts. 

1.3.9 Assessing the Need for More Data 

To help evaluate what if any additional data 

might be worth collecting so as to materially 

reduce model uncertainty (by further 

constraining the range of alternative 

structures and parameters), the strategy 

suggests conducting a sensitivity analysis to 

indicate what system behavior appears to be 

most sensitive to which parameters at what 

locations. The next step is to consider 

performing additional site characterization 

where existing parameter estimates are least 

certain and the model is relatively 

insensitive to their values, and monitoring 

system behavior where it is most sensitive to 

model parameters while prediction errors 

appear to be relatively large and 

consequential. 

1.3.10 Accounting for Unresolved 

Variations in Space-Time 

Calibration of a deterministic groundwater 

model yields effective parameter estimates, 

which compensate to some degree for lack 

of knowledge about variability on spacetime scales that are not resolved by the data 

or the model. This is less true for transport 

than for flow, because transport processes 

due to unresolved variations in permeability 

and porosity cannot be validly represented 

by traditional effective parameters (such as a 
19 

constant dispersivity) except in special cases 

or as a crude approximation. The reliance on 

effective parameters is also less valid for 

strongly heterogeneous media with 

preferential flow paths such as some 

fractured rocks and soils that contain 

relatively large and elongated openings 

created by burrowing animals or plant roots. 

When the deterministic effective parameter 

approach fails, the most common and 

powerful alternative is to employ stochastic 

concepts and models. 

A stochastic approach to groundwater 

modeling requires that the spatial variability 

of input parameters, such as permeability 

and unsaturated soil properties, be 

characterized with the aid of geostatistical 

methods. A key component of geostatistical 

characterization is the assessment of spatial 

covariance structure (in terms of a 

covariance or variogram function). The 

strategy promotes this approach and the 

report illustrates it by examples. In the 

absence of site characterization data that are 

amenable to geostatistical analysis, one can 

treat the structural parameters (of the 

covariance or variogram function) as free 

parameters to be estimated by model 

calibration. This too is illustrated by 

example in the report. 

1.3.11 Stochastic Modeling 

The strategy supports two general 

approaches to predict stochastic 

groundwater flow and transport on the basis 

of geostatistical input: High-resolution 

numerical Monte Carlo simulation and 

direct (deterministic) prediction of mean 

behavior. The latter approach relies on 

stochastically derived deterministic 

ensemble mean flow and transport equations 

in which the dependent variables represent 

not actual system states (such as head, 

concentration or flux) but rather their 

(ensemble) mean values or statistical 

expectations. These mean or expected values 

represent optimum unbiased predictors of 

the unknown, actual system states. Similar 

values are obtained upon averaging the 

results of numerous Monte Carlo 

simulations. Both the Monte Carlo and the 

mean equation approaches allow the 

reduction of predictive uncertainty by 

conditioning the predictions on actual 

measurements of the input parameters at 

discrete locations in space. 

The variance-covariance of high-resolution 

(conditional) Monte Carlo simulations 

serves to quantify predictive uncertainty. An 

alternative is to compute this variancecovariance directly (deterministically) by 

means of corresponding moment equations. 

Both options are supported by the strategy. 

Conditioning the stochastic flow and 

transport equations not only on measured 

input variables (derived from site 

characterization) but also on monitored 

system behavior tends to improve their 

reliability. This is accomplished through 

novel inverse procedures, which form part 

of the proposed strategy. 

In virtually all other respects, the strategy is 

similar in principle (though not in all details) 

for stochastic and deterministic models of 

groundwater flow and transport.
20 
21 

2 NATURE OF HYDROGEOLOGIC SITE MODELS 

Prior to developing a model, one must 

define what one means by the term. 

2.1 Site-Specific Nature of 

Hydrogeologic Models 

Hydrogeologic models are by nature sitespecific. Though there is an established (and 

evolving) set of general hydrogeologic 

principles that apply to many sites on many 

scales, they are insufficient to either 

describe (conceptualize) or quantify (model) 

the hydrogeology of a particular site over a 

given range of scales. Because each site is 

unique, general principles must always be 

supplemented by regional and site-specific 

data to be useful for conceptualizing and 

modeling subsurface flow and transport at a 

site, regardless of purpose. 

General hydrogeologic principles may serve 

to develop generic methods and tools of 

hydrogeologic analysis that are applicable, 

in principle, to many sites on varied scales. 

Among numerous examples one may cite 

software tools for reconstruction and 

visualization of site hydrostratigraphy and 

structure, statistical and geostatistical 

analysis of site data, interpretation of 

pumping tests and simulation of flow and 

transport processes. Whereas numerical 

simulation codes are often referred to as 

"models," we avoid such usage of the term 

on the understanding that they are tools 

rather than models. To transform simulation 

codes into models, one must apply them to 

particular hydrogeologic circumstances that 

represent either a hypothetical or a real 

setting. Hydrogeologic models of 

hypothetical settings are research and 

demonstration tools that may be useful for 

the exploration and exposition of generic 

principles and hypotheses. They have also 

been incorporated in multi-media codes 

sometimes used for dose and risk 

assessment. Though they may serve a useful 

purpose, hydrogeologic models of 

hypothetical settings must not be mistaken 

for valid models of particular sites. To 

render them such, one must ground these 

models in real data which represent specific 

geographic locations, hydrogeologic 

conditions and natural or anthropogenic 

forcing that prevails at these locations. 

The strategy described in this report deals 

with site-specific rather than hypothetical 

models of subsurface flow and transport 

within a regulatory context that is relevant to 

NRC review and performance evaluation 

needs. 

2.2 What is a Hydrogeologic Model? 

For purposes of the strategy developed in 

this report, a hydrogeologic model is a 

conceptual and mathematical construct that 

serves to analyze, qualitatively and 

quantitatively, subsurface flow and transport 

at a site in a way that is useful for NRC 

review and performance evaluation. It 

consists of a description of the site and 

circumstances being modeled, a set of 

regional and site data to support this 

description, a mathematical framework that 

incorporates both the description and the 

data, and a mathematical (analytical or 

numerical) evaluation of system behavior 

and performance. The evaluation typically 

consists of simulating (reconstructing or 

predicting) and analyzing space-time 

variations in quantities such as hydraulic 

head or pressure, solute concentration, fluid 

and solute flux and velocity, solute travel 

time and associated performance measures. 

A hydrogeologic model thus consists of a 
22 

conceptual and a mathematical component. 

Whereas much has been written about the 

mathematical component of hydrogeologic 

models, relatively little attention has been 

devoted to the conceptual component. In 

most mathematical models of subsurface 

flow and transport, the conceptual 

framework is tacitly assumed to be given, 

accurate and unique. All three premises are 

challenged by the strategy in this report, 

which focuses in a major way on issues 

associated not only with mathematical but 

also with conceptual aspects of 

hydrogeologic model building and 

evaluation. For this, we need to define what 

is meant by conceptual and mathematical 

modeling, and how do the two interrelate? 

This is done next. 

2.3 What is a Conceptual Model of Site 

Hydrogeology?

We stated that a hydrogeologic model 

consists of a description of the site and 

circumstances being modeled, a set of 

regional and site data to support this 

description, a mathematical framework that 

incorporates both the description and the 

data, and a mathematical (analytical or 

numerical) evaluation of system behavior 

and performance. A conceptual model 

embodies the descriptive component of the 

model, which may be both qualitative and 

quantitative. 

According to an exchange of letters between 

Paul Roman, Alfred A. Brooks, and Lorenzo 

de la Torre in Physics Today (1998), 

physical (and hydrogeologic) reality exists 

objectively but is not fully or directly 

accessible to us. Instead, it reveals itself to 

us via sensory impressions and experiences 

(through observations and experiments) in 

signatures or bits. To assimilate these 

separate pieces of information we describe, 

contemplate, correlate, connect, systematize, 

interpret and integrate them into a body of 

knowledge. On the basis of this knowledge 

we form mental constructs, or contextual 

structures, which we call conceptual models 

of reality. 

Conceptual descriptions of hydrogeologic 

systems may take many forms. According to 

Lein (1997, p. 85), conceptual models take 

the forms of charts, pictures and diagrams 

depicting system arrangements and flows. 

Examples relevant to NRC staff 

performance assessment use include a 

conceptual diagram and a flow chart of 

exposure pathways considered in the 

RESRAD (Yu, 1993) dose assessment 

model (Figure 2-1), a similar flow chart 

(Figure 2-2) from a technical guide to 

groundwater model selection issued jointly 

by three agencies (EPA, DOE, NRC, 1994); 

a flow chart of input data and processes 

considered in the performance assessment of 

geosphere transport (Figure 2-3) by 

Gautschi (1995), and a flow chart of 

processes that may impact the performance 

of an engineered barriers system at a 

potential HLW repository in Nevada (Figure 

2-4). 

A definition more directly relevant to 

hydrogeologic modeling has been given by 

Anderson and Woessner (1992). According 

to them (ibid, p. 28), a conceptual model is a 

pictorial representation of the groundwater 

flow system, frequently in the form of a 

block diagram or a cross section (Figure 2-

5). Its development includes (ibid, p. 6) 

identification of hydrostratigraphic units and 

system boundaries; assembly of field data 

including information on the water balance 

and data needed to assign values to aquifer 

parameters and hydrologic stresses; and a 

field visit to help keep the modeler tied into 
23 

reality and exert a positive influence on his 

subjective modeling decisions. The purpose 

of a conceptual model is (ibid, p. 28) to 

simplify the field problem and organize the 

associated field data so that the system can 

be analyzed more readily by means of a 

numerical model; in their view, the nature of 

the conceptual model determines the 

dimensions of this model and the design of 

the grid. 

Figure 2-1. Conceptual diagram and flow chart of exposure pathways in RESRAD 

(after Yu, 1993). 
24 

Figure 2-2. Conceptual model 

as depicted by EPA, DOE and NRC (1994) guide to model selection. 

Figure 2-3. Input data and processes considered in performance assessment of geosphere transport

by Gautschi (1995). With permission, NAGRA. 
25 

Figure 2-4. Processes affecting performance of an engineered barriers system 

(from USGS presentation to Expert Elicitation Panel, 1997). 

Figure 2-5. Translation of geologic information into conceptual model for numerical modeling: 

(a) Floridan aquifer system, W to E cross section for central Florida 

(after Anderson and Woessner, 1992). With permission, Academic Press. 
26 

Figure 2-5 (Continued). Translation of geologic information into conceptual model for 

numerical modeling: (b) Snake River Plain aquifer system, southwest to northeast cross section; 

(after Anderson and Woessner, 1992). With permission, Academic Press. 
27 

Figure 2-5 (Continued). Translation of geologic information into conceptual model for 

numerical modeling: (c ) Sand and gravel aquifer in Pensacola, Florida; 

(d) Glacial-drift river-valley aquifer in Rhode Island 

(after Anderson and Woessner, 1992). With permission, Academic Press. 
28 

Figure 2-6. Theoretical flow pattern and flow systems according to Tóth (1963). 

With permission, AGU. 

One of the earliest and best known pictorial 

hydrogeologic concept representations is the 

classic depiction, in Figure 2-6, of 

streamlines defining hydrogeologic systems 

and subsystems due to Tóth (1963). Other 

examples of pictorial concept 

representations include a hydrogeologic 

block diagram of arid zone hydrogeology in 

the southwestern United States (Figure 2-7), 

a map of groundwater flow and basin 

divides in the Death Valley area (Figure 2-

8), and a conceptual model of crystalline 

basement with features relevant to flow, and 

its exploration, in Switzerland (Figure 2-9). 

Meyer and Gee (1999) consider a conceptual 

model to be a hypothesis that describes the 

main features of site geology, hydrology, 

geochemistry, and relationships between 

geologic structure and fluid flow and 

contaminant transport. In their view,

mathematical modeling is a process of 

testing this hypothesis. A conceptual model 

is a pictorial, qualitative description of the 

groundwater system in terms of 

hydrogeologic units, system boundaries 

including time-varying inputs and outputs, 

and hydraulic as well as transport properties 

including their spatial variability. A 

mathematical model constitutes a 

quantitative representation of such a 

conceptual model.
29 

Figure 2-7. Hydrogeologic block diagram of arid zone hydrogeology (after Eakin et al., 1976). 

Figure 2-8. Map of groundwater flow and basin divides in the Death Valley area 

(after D'Agnese et al., 1996). 
30 

 

Figure 2-9. Conceptual model of crystalline basement in Switzerland with features relevant to flow, 

and corresponding concept of its characterization (after Gautschi, 1995). With permission, NAGRA.

Kolm and van der Heijde (1996) describe an 

integrated, stepwise method for the 

qualitative conceptualization and 

quantitative characterization of natural and 

anthropogenic subsurface envirochemical 

systems. They define the latter as 

hydrogeologic and hydrochemical systems 

that contain chemical species of concern to 

environmental management. According to 

the authors, a conceptual model of an 

envirochemical system includes a qualitative 

assessment of how chemicals enter, move 

through or are retained in and leave the 

system. It also includes a qualitative 

description of the source, transport, fate and 

distribution of each targeted chemical or, in 

the case of unknown sources, a hypothesis 

concerning source locations and strengths 

based on conceptualized transport and fate 

processes and known distribution of 

chemicals. The conceptual model is 

described and visualized using crosssections and plan view illustrations. The 

authors do not present examples of 

conceptual models but instead illustrate, by 

means of flow charts (Figure 2-10), how 

their suggested procedure for the joint 

conceptualization and characterization of 

envirochemical and groundwater systems 

would operate.
31 

Figure 2-10. Procedure for joint conceptualization and characterization of envirochemical 

and groundwater systems according to Kolm and van der Heijde (1996). 

With permission, IAHS press. 

It is of interest to note that conceptual 

models do not appear in the otherwise 

extensive index of a book by the National 

Research Council (1990) devoted to 

scientific and regulatory applications of 

groundwater models. However, they are the 

focus of a more recent study by a National 

Academy panel (NRC, 2001) of conceptual 

flow and transport models in the fractured 

vadose zone. The panel defines a conceptual 

model as an evolving hypothesis identifying 

the important features, processes and events 

controlling fluid flow and contaminant 

transport of consequence at a specific field 

site in the context of a recognized problem. 

A conceptual model is qualitative and 

expressed by ideas, words and figures. It is a 

hypothesis because it must be tested for 

internal consistency and for its ability to 

represent the real system in a meaningful 

way. The hypothesis evolves (is revised and 

refined) during testing and as new 

information is gathered. Another study by a 

National Academy panel (NRC, 2000) 

considers conceptual modeling to include 

the representation of hydrogeologic 

heterogeneity and scale on the basis of field 

experimental data. 
32 

2.4 What is a Conceptual-Mathematical 

Model of Site Hydrogeology? 

We stated that a hydrogeologic model 

consists of a description of the site and 

circumstances being modeled, a set of 

regional and site data to support this 

description, a mathematical framework that 

incorporates both the description and the 

data, and a mathematical (analytical or 

numerical) evaluation of system behavior 

and performance. A conceptualmathematical model embodies the 

descriptive component of the model, cast in 

the form of mathematical equations suitable 

for system evaluation. The equations 

represent the symbolic framework, or 

structure, of the conceptual-mathematical 

model. Ideally, they provide a mathematical 

description of all physico-chemical (and 

other) processes that are considered relevant 

to flow and transport at the site on a given 

range of space-time scales subject to welldefined forcing. Defining the scales of the 

system includes specifying its spatial and 

temporal dimensions. Forcing includes 

sources, initial and boundary conditions. 

Suitability for system evaluation means that 

the symbolic conceptual-mathematical 

framework, or structure, is cast in a form 

that lends itself to computation and 

subsequent presentation (numerical, 

graphical) and analysis (deterministic, 

statistical) of the results. 

Mathematical hydrogeologic models are 

described succinctly in Figure 2-11. Their 

essence is the formulation of conceptual 

models of site hydrogeology and 

flow/transport dynamics in mathematical 

language. This includes mathematical 

definition and description of relevant spacetime frames, dimensions and scales, system 

topology and geometry, interactions (called 

processes) between kinematic (flux, 

velocity) and dynamic (force, stress, energy) 

quantities, forcing terms (initial and 

boundary conditions, sources), state 

equations (functional relationships between 

state variables), and corresponding 

parameters (their functional representations). 

It leads to a system of mathematical 

statements, relationships and equations 

(algebraic, differential, integro-differential, 

Boolean, deterministic, statistical, 

stochastic) that ultimately allow one to 

interpret and explain existing observations, 

and predict future conditions, events and 

observations related to site hydrogeology 

and corresponding flow/transport dynamics. 

These statements, relationships and 

equations constitute a mathematical 

conceptualization of the system, not merely 

a calculational tool. They complement the 

qualitative conceptual picture and, together 

with it, form a conceptual-mathematical 

framework or model for the hydrogeologic 

system. Whether such a model is analytical 

or numerical (written in the language of 

calculus or algebra) is merely a technical, 

not a fundamental, distinction. 

For us this implies that one cannot 

legitimately disassociate hydrogeologic 

interpretations, or conceptual models, from 

corresponding mathematical and 

computational models for purposes of 

quantitative environmental impact and 

performance assessments of a given nuclear 

facility or site; these models are intimately 

linked and define a single conceptualmathematical framework for quantitative 

site analyses. An example of how a given 

qualitative conceptual model may be 

associated with different conceptualmathematical models, depending on scale, is 

given by Figure 2-12 for the crystalline 

basement of Northern Switzerland. 
33 

Figure 2-11. Mathematical formulation of hypotheses. 
34 

Figure 2-12. Different conceptual-mathematical models associated with a given qualitative 

conceptual model of crystalline basement in Northern Switzerland, depending on scale 

(after Gautschi, 1995). With permission, NAGRA. 

We mentioned that a groundwater code is a 

tool and not a model. However, selecting a 

particular code implies identifying specific 

processes that may govern flow and 

transport at a site, their symbolic 

mathematical representation, and their 

numerical approximation. Selecting the 

space-time dimensions and size of a 

computational grid determines the 

dimensions and scale of the system being 

modeled. Choosing the sizes of space-time 

discretization intervals defines the scales at 

which flow and transport processes are 

resolved. Specifying the location and type of 

sources, initial and boundary conditions 

identifies the forcing. Both material 

properties and forcing terms are associated 

with parameters that must eventually be 

assigned numerical values. Choosing the 

modes of their representation 

(parameterization) defines the resolution 

scales of these parameters (and thus of 

material properties and forcing terms) 

in space-time. Once this has been 

accomplished in a way that is supported by 

all relevant regional and site data, the code 

has been transformed from a mere tool to a 

bona fide conceptual-mathematical model of 

site hydrogeology. 

2.5 What is a Hydrogeologic Model? 

A conceptual-mathematical model describes 

subsurface flow and transport in symbolic 

mathematical and algorithmic language. As 

such, it constitutes the mathematical 

framework, or structure, of a hydrogeologic 

model for a site. The framework contains 

input parameters that represent material 

properties and forcing terms (sources, initial 

and boundary conditions). Assigning 

numerical values to these parameters

transforms the symbolic framework into a 

working mathematical model of the 

hydrogeologic system. This hydrogeologic 

model can then be run to simulate 
35 

(reconstruct or predict) and analyze spacetime variations in quantities such as 

hydraulic head or pressure, solute 

concentration, fluid and solute flux and 

velocity, solute travel time and associated 

performance measures. 

A hydrogeologic model thus consists of a 

conceptual-mathematical structure (which 

embodies a qualitative mathematical 

description of the system) and a set of 

parameter values (which render the model 

quantitative). 

2.6 Non-Uniqueness of Hydrogeologic 

Models

As pointed out by Roman, Brooks and de la 

Torre (Physics Today, 1998), physical 

reality exists objectively but is not fully or 

directly accessible to us. Instead, it reveals 

itself to us via sensory impressions and 

experiences (through observations and 

experiments) in signatures or bits. To 

assimilate these separate pieces of 

information we describe, contemplate, 

correlate, connect, systematize, interpret and 

integrate them into a body of knowledge. On 

the basis of this knowledge we form mental 

constructs, or contextual structures, which 

we call conceptual models of reality. It has 

been recognized by Einstein that such 

constructs, or structures, cannot be extracted 

from experience but must be freely invented. 

It therefore follows that observed reality 

may lend itself not just to one but to 

multiple conceptualizations. 

The same applies to hydrogeologic systems. 

The formulation of a conceptual model is 

inherently subjective in that it relies on 

limited available site observations and data, 

as well as experience and insights developed 

through work on similar sites and/or related 

problems; hydrogeologic conceptualization 

is susceptible to biases arising from the 

disciplinary background and experience of 

the analyst, and/or by different perceptions 

of the problem as influenced by external 

social and political factors (NRC, 2001). 

Contrary to engineering systems which are 

generally closed, relatively simple and welldefined, hydrogeologic systems are open, 

complex and only partially defined. The 

open nature of hydrogeologic systems 

means that they are not amenable to fully 

controlled experimentation; as they are 

additionally complex, their description must 

remain forever incomplete and imprecise. 

Corresponding conceptual-mathematical 

models are therefore fallible scientific 

constructs, not credible engineering tools; 

they undergo a never ending process of 

modification, rejection and/or replacement 

as new scientific evidence and/or reasoning 

emerge. In other words, the conceptualmathematical structure of a hydrogeologic 

site model is nonunique. 

The latter has not been considered as a 

problem in the past (Beven, 2000). The 

attitude has been that if the modeling 

concepts were sufficiently accurate then it 

would be possible to represent the site 

uniquely by specific values of model 

parameters. In principle this might be 

possible; in practice it proves to be 

untenable because of the impossibility of 

fully describing the system, and particularly 

the heterogeneity of its subsurface 

characteristics. 

Experience with models in a wide variety of 

hydrologic settings and contexts indicates 

that, regardless of the amount and quality of 

regional and site data available at a given 

site, it is generally neither possible nor 

justifiable to describe all relevant aspects of 

site behavior and performance by a unique 
36 

model. Instead, there is generally a wide 

range of model structures and parameter 

values that appear to "work" for a given site 

and purpose. In other words, neither the 

structure nor the parameter values of a 

hydrologic site model are unique. This 

"equifinality" problem (Beven, 1993, 2000) 

is generic and not simply one of identifying 

a system's "true" model structure or 

parameter values. In fact, a "true" model for 

a hydrologic system does not exist. It is 

therefore best (NRC, 2001) to consider a 

broad range of reasonable alternative 

hypotheses and base the model on a variety 

of different types of data. 

2.7 Accounting for Hydrogeologic 

Complexity 

2.7.1 Effects of Heterogeneity and Scale 

Hydrologic systems are inherently 

heterogeneous on a multiplicity of scales 

(Entekhabi, 2000). The nonuniqueness of 

hydrologic models, applied to unique sites, 

arises from the limitations of current model 

structures in representing heterogeneous 

surface and subsurface flow systems, from 

limitations of measurement techniques and 

scales in defining system characteristics and 

the initial and boundary conditions for a 

model, and from the uniqueness of 

individual sites (Beven, 2000). 

In recent years there has been a move 

towards more and more detailed studies of 

flow in rivers, soils and aquifers, using more 

and more sophisticated instrumentation 

(Beven, 2000). The result has generally been 

to demonstrate complexity of flow 

pathways, due to heterogeneity on different 

scales and interactions between the 

geometry of the flow domain and the 

prevailing hydraulic gradients and flow 

pathways (including both "dead zones" and 

"preferential" flows). It has proven difficult 

to use such information in applications of 

predictive models, which are generally 

required to be used at scales larger than the 

detailed measurements. 

The problem is not so much that the 

complexity of the measurements is not 

accessible to adequate interpretation or even, 

in some cases, to being adequately modeled 

(Beven, 2000). The problem is that detailed 

characterization of a site is technically 

impractical and unfeasibly expensive. Even 

if this was practical and feasible, it would 

not necessarily lead to an improved 

representation of the system on larger scales. 

This is so because the effective behavior of a 

hydrologic system on larger scales is not 

completely determined by the scale-up of 

local processes; interactions among smallerscale features and processes often lead to 

effects that are not completely captured by 

effective parameters in larger-scale models 

(Entekhabi, 2000). 

There are processes and conditions when the 

effective parameter approach to scaling may 

be feasible. In the remaining circumstances, 

the macro-scale and micro-scale predictive 

relations for hydrologic processes may have 

different functional forms and dependencies 

(Entekhabi, 2000). 

2.7.2 "Simple" Models of Complex Systems 

The unique heterogeneous and scaledependent characteristics of a site are 

inherently unknowable and not given to 

accurate or unique representation by a 

model. All one can do is to represent system 

complexities in simplified form and 

constrain the model so it provides an 

acceptably realistic representation of the 

system, as manifested by its consistency 

with available site data. 
37 

Demonstrating consistency with site data is 

important because a model is a valid 

reflection of reality only in the sense, and to 

the extent, that it yields qualitative and 

quantitative explanations and predictions of 

real experiences (Physics Today, 1998). 

Only if a hydrogeologic model is 

comparable with site characterization and 

monitoring data would it reflect actual and 

anticipated conditions at a site. Models may 

legitimately consider conditions and/or data 

that lie outside the observed range, but they 

must nevertheless remain compatible with 

available site data when applied within their 

range. Models which cannot be so applied 

may constitute appealing logical constructs 

which are however speculative and not 

subject to verification, confirmation or 

validation. 

The level of hydrogeologic complexity that 

one incorporates in a model should be 

commensurate with its purpose (NRC, 

2001). Some flow and transport phenomena 

may not be modeled with desired fidelity 

without accounting for system complexities 

that control them. Yet the details of these 

complexities may be deemed less relevant to 

the problem than their effect on flow and 

transport, and to characterize and model 

them may be considered too difficult and 

costly. One should then seek a model that 

captures the effects of these complexities 

implicitly, without reproducing them in 

detail explicitly. 

A widely practiced approach has been to 

simplify complex hydrogeologic system for 

modeling purposes in an ad hoc and 

subjective manner (as illustrated for a 

popular dose assessment code in Figure 2-

13). It is generally not clear that this 

approach captures adequately all aspects of 

site complexity that have a significant 

impact on the problem being tackled. It is 

also not clear that such models have the 

space-time resolution, which may be 

required to render them comparable and 

compatible with site data. Without this, the 

models remain interesting logical constructs 

which, however, may not be valid 

representations of actual site conditions and 

may thus yield unreliable predictions. 

To insure that relevant aspects of 

hydrogeologic complexity are reflected in 

modeled system behavior and performance, 

and that the model is compatible with site 

data, it is important that the process of 

simplification be done systematically and 

objectively. This can be done by filtering out 

undesirable details through formal averaging 

of the governing equations in space-time or 

in probability space, in a way which retains 

and renders their influence on the model 

implicit. 

Averaging three- or two-dimensional 

equations across one spatial dimension 

renders them two- or one-dimensional, 

respectively. Averaging transient equations 

over time may (but need not) render them 

representative of a steady state. Averaging 

can also be done over subdomains of the site 

being modeled, and over multiple time 

intervals. In each case, the averaging results 

in governing flow and transport equations 

that contain upscaled quantities. If the 

space-time scales of these quantities differ 

from those of the available site data, then 

either the model or the data must be rescaled 

to render them compatible and comparable 

with each other. 
38 

 

Figure 2-13. Simplified water pathways and well contamination processes in the RESRAD model 

(after Yu, 1993). 

It has been traditional to assume that 

spatially and temporally averaged 

hydrologic quantities are less prone to 

predictive uncertainty than quantities that 

are localized in space-time. While this may 

be true for systems that are statistically 

homogeneous and stationary, such systems 

are the exception rather than the rule. Spatial 

and temporal averages are not necessarily 

more predictable as traditionally believed if 

the averaging covers a scale that contains a 

strong transition or change in behavior 

(Entekhabi, 2000). 

This and other scale-related problems are 

generally avoided if the averaging is done in 

probability space. Such "ensemble" 

averaging leads to stochastic equations that 

contain statistical moments of hydrogeologic 

variables (considered random), most 

commonly the mean and variancecovariance. The mean is a predictor of 
39 

system behavior or performance, and the 

variance-covariance is a measure of 

predictive error. Both are smoother (vary 

more slowly in space-time) than their 

random counterparts and, in this sense, 

render the model relatively "simple." 

Despite their smoothness, both moments are 

defined on the same space-time scales as are 

the random hydrogeologic variables on 

which they are based. Stochastic models 

thus achieve smoothness and simplicity 

without any need to average or upscale in 

space-time. As they are typically 

conditioned on site measurements (i.e., they 

honor the data), stochastic models are 

compatible with these measurements both in 

scale and magnitude. Yet another advantage 

of the stochastic method over space-time 

averaging is that it yields measures of 

predictive uncertainty. Stochastic 

approaches are increasingly recognized as 

offering a way to deal with complex, scaledependent heterogeneous systems by means 

of relatively simple models (Hunt and 

Zheng, 1999).
40 

•

•

•

•

•

•

3 CONTEXTUAL FRAMEWORK 

The first step in model development is to 

define its context and purpose. 

3.1 Purpose of Hydrogeologic Models 

Within NRC Regulatory Context 

Within a NRC regulatory context, the 

purpose of hydrogeologic site models is to 

help one analyze, qualitatively and 

quantitatively, subsurface flow and transport 

at a site in a way that is useful for NRC staff 

review of decommissioning plans and 

performance of high-level and low-level 

radioactive waste disposal sites and uranium 

recovery facilities. For this purpose, the 

strategy developed in this report is cast in 

the context of a framework that is useful to 

NRC staff review and performance 

evaluation needs. The framework is defined 

in terms of performance measures identified 

by the NRC licensing staff (see Appendix A). 

3.2 Questions that Hydrogeologic 

Models Must Address 

The contextual framework defines key 

questions to which groundwater flow and 

transport models are expected to provide 

answers; it helps narrow down the problem 

as well as the range and type of potential 

model applications. The questions are stated 

in Figure 3-1 in the following order: 

For what purposes will the model be 

used? The answer includes defining the 

nature and magnitude of an existing 

and/or potential problem, location of the 

problem area, its causes, potential shortand long-term solutions and/or remedies, 

their anticipated consequences and 

costs-benefits, issues needing resolution, 

and criteria by which a solution and/or 

remedy will be selected. 

To what hydrogeologic system will the 

models be applied? The answer includes 

defining the hydrogeologic environment 

that is or may potentially be affected by 

the problem and/or its solution or 

remedy, the corresponding site and 

surrounding environs of (potential) 

concern, and the regional hydrogeologic 

setting. 

Under what circumstances and 

scenarios will the modeled 

hydrogeologic system operate? This 

requires defining predevelopment, 

current and potential undisturbed and 

disturbed site and regional conditions, 

and corresponding natural as well as 

anthropogenic influences, under which 

flow and transport would be modeled. 

What measures will be adopted to assess 

performance of the hydrogeologic 

system? This requires specifying 

measures and/or criteria related to site 

and regional hydrogeology that would be 

used to identify issues, potential 

solutions and/or remedies, their 

cost-efficiency and ability to meet 

regulatory requirements. 

What aspects of site and regional 

hydrogeology and flow/transport 

dynamics are expected to impact these 

performance measures, and how? The 

answer involves defining key elements 

of the hydrogeologic system, key flow 

and transport mechanisms, key natural 

and anthropogenic influences, and 

space-time scales that may potentially 

impact these measures, as well as the 

manners in which such impacts might 

occur. 

With what reliability, certainty and 

accuracy does one need to predict 

performance measures? How important

a role will such measures play in the 
41 

decision process? What is the worth

(in time/effort/resources) of assessing 

performance measures for the site at 

specified levels of reliability, certainty, 

and accuracy? These are the most 

basic questions that must be addressed 

before any site investigations and 

modeling efforts are initiated, as the 

answer would have a major impact on 

the time, effort and resources that could 

validly be expended in pursuit of 

performance assessment for the site. 

3.3 Performance Measures 

The strategy in this report focuses on 

groundwater flow and transport primarily in 

the sense, and to the extent, that they impact 

performance measures. Such measures are 

typically articulated within the broader 

context of a system in which groundwater is 

only one among several components. 

However, our strategy focuses solely on 

groundwater aspects of these broader 

criteria. These have been identified by the 

NRC staff in Appendix A as a “Note on 

Performance Measures and Hydrology 

Issues Pertaining to Groundwater Flow and 

Transport Models of Licensed Nuclear 

Sites.” Appendix A provides information 

about performance measures based on 

regulatory standards, pertinent hydrology 

and groundwater issues that are related to 

performance, and uncertainty commonly 

encountered in groundwater flow and 

transport modeling. The NRC staff note 

focuses on low-level waste, 

decommissioning, high-level waste and 

uranium recovery sites. It emphasizes that 

the end point of the analysis is the 

performance measure for the licensed 

facility obtained through a performance 

assessment. As outlined in Appendix A, the 

hydrologic measures and issues need only be 

explored and dealt with to the extent 

required by performance assessment, taking 

into account the significant uncertainties and 

the hazard involved, and that some 

uncertainties may be satisfactorily addressed 

through bounding analyses. 

Our interpretation of Appendix A is that the 

following groundwater-related performance 

measures need to be assessed by means of 

hydrogeologic flow and transport models. 

Performance Measures for Low-Level 

Waste Disposal Sites: 

• Concentrations of radioactive materials 

at receptor locations contributing to 

annual dose to an average member of a 

critical group. 

• Concentrations in well water at site 

boundary resulting in the highest 

composite dose. 

• Cumulative releases to the environment 

to be As Low As Reasonably Achievable 

(ALARA). 

Performance Measures for 

Decommissioning Sites: 

• Residual radioactivity resulting in Total 

Effective Dose Equivalent (TEDE) to an 

average member of a critical group at 

receptor locations, or human access 

points, including drinking water. 

• Radionuclide concentrations that 

contribute toward the dose received by 

an individual who uses well water on 

site, or by an individual off-site who 

uses water from a well at the site 

boundary. 

• Cumulative releases to the environment 

ALARA.
42 

Figure 3-1. Contextual framework of modeling. 

Performance Measure for High-Level 

Disposal at Yucca Mountain: 

• Radionuclide concentrations that 

contribute to the expected dose or TEDE 

received by an average member of a 

critical group 20 km down gradient of 

the proposed repository. 

Performance Measures for Uranium 

Recovery and Tailing Sites: 

• Concentrations of specified chemicals 

and radionuclides. 

At DOE mill tailing sites under Title I 

• The model is to provide a technical basis 

for selecting a restoration strategy and 

for determining risk to humans and the 

environment. 

At non-DOE mill tailing sites under Title II 

• Maximum concentrations of selected

constituents and chemical parameters. 

• The model is aid in guiding and 

supporting 

9 Detailed groundwater 

characterization, 

9 Specific criteria for monitoring and 

restoration. 

At In-Situ Leach (ISL) uranium extraction 

sites 

• Monitoring to assure restoration of ore 

zone and affected aquifers, based on 

background levels and EPA standards. 

• Restoration goals for each constituent on 

a well-by-well or well-field average 

basis. 
43 

• Primary restoration goal of returning the 

ore zone and aquifers to their 

preoperational (baseline) levels, which 

could rely on statistical measures such as 

average parameters, range of baseline 

concentrations, and 99% confidence 

intervals. 

• Secondary restoration standards of 

returning the ore zone and aquifers to 

a pre-ISL class of use such as drinking 

water, livestock, agricultural or limited use. 

• If a parameter cannot be restored to a 

secondary standard, demonstrate that it 

would not 

9 threaten public health and safety or 

the environment, 

9 significantly degrade water use. 

3.4 Hydrogeologic Context of NRC 

Performance Measures 

3.4.1 Hydrogeologic Aspects of 

Performance Criteria 

The above performance criteria establish the

contextual framework within which the 

strategy in this report is developed. 

From a hydrogeologic point of view, the 

main differences between the various 

performance measures relate to the spatial 

and temporal scales of the problem; the type 

of environment (whether saturated or 

unsaturated, confined or unconfined, porous 

or fractured); the driving mechanisms 

(whether infiltration or pumping, recharge or 

discharge); the space-time resolution needed 

to accurately assess performance measures; 

the practicality and cost of attaining such a 

resolution; the potential effect of a mismatch 

between required and attained resolution on 

the accuracy of the computed performance 

measures; and the level of inaccuracy and 

uncertainty that performance assessment can 

tolerate. 

Common to all the above performance 

measures is that they involve complex, 

three-dimensional, heterogeneous 

hydrogeologic environments; threedimensional, generally transient 

groundwater flow regimes; open flow 

systems with indeterminate driving (forcing) 

terms (initial conditions, recharge and 

discharge, other source terms and boundary 

conditions); severely limited access to, and 

possibility to explore, the interior of the 

groundwater flow system; the uncertain 

nature and distribution of contaminants and 

their sources both at and below the soil 

surface, on site and off site; the lack of 

adequate science and technology to render 

reliable long-term predictions of pollutant 

migration and dilution with groundwater; 

and the difficulty in verifying such 

predictions by direct measurement. 

3.4.2 Implications vis-à-vis 

Hydrogeologic Conceptualization 

and Modeling 

The commonalties just identified among the 

various performance criteria imply that all of 

them require developing a good 

understanding, and reliable models, of 

subsurface flow and transport. One cannot 

do so without recognizing and considering 

the full complexity of a site before 

attempting to represent it by means of a 

simplified conceptual-mathematical model. 

This is true regardless of the specific 

performance criteria one needs to address. 

The process of model simplification or 

abstraction must be based on a prior 

decision by the hydrogeologist and 

performance assessment analyst on the 

degree of reliability, certainty and accuracy 

with which given performance measures 

need to be predicted, their importance in the 

decision process, and the amount of time, 

effort and resources that assessing them at 
44 

the specified levels of reliability, certainty 

and accuracy would justify in each specific 

case. These questions are of a regulatory 

nature and cannot be addresses by 

hydrologists without a case-by-case dialogue 

with regulators. As such, they are not 

addressed by the strategy in this report. 

Instead, the strategy stipulates that these 

fundamental questions must be addressed 

and resolved by the NRC licensing staff 

before any site investigation and modeling 

are initiated. This is so because the answers 

impact in a major way the degree to which 

one justifies the use of a simplified 

conceptual-mathematical model for what is 

ubiquitously a complex hydrogeologic 

system with three-dimensional, transient 

groundwater flow and contaminant 

transport. 

Therefore, it is very important that the 

hydrogeologist first articulated these 

processes in their naturally complex setting 

as best understood in light of available site 

data and the state of prevailing 

hydrogeologic knowledge prior to 

postulating groundwater flow and transport 

models for performance assessment of a 

given site. Even if performance assessment 

is ultimately conducted with the aid of 

highly simplified conceptual-mathematical 

models of groundwater flow and transport, 

the strategy deems it essential for the 

credibility of the assessment that these 

models derive objectively (and if possible 

formally) from a more complete description 

of site hydrogeology, based on clearly 

reasoned and properly defended arguments. 

Rather than taking the attitude that a limited 

performance assessment goal justifies a 

limited view of hydrogeology, the strategy 

adopts the attitude that a comprehensive 

description of hydrogeology is required to 

properly adapt a hydrogeologic model to 

such a limited goal. This manifests itself as a 

key criterion and guiding principle for the 

construction of conceptual-mathematical 

models in the strategy.
45 
46 

4 ASSEMBLY AND ORGANIZATION OF HYDROGEOLOGIC KNOWLEDGE BASE 

Having determined the context and purpose 

of a model, one must next assemble and 

organize a corresponding knowledge base. 

4.1 What Constitutes a Hydrogeologic 

Knowledge Base? 

As illustrated diagrammatically in Figure 4-

1, a hydrogeologic knowledge base includes 

all relevant (to the contextual framework) 

qualitative and quantitative data, 

observations, concepts, theories and models 

which pertain to past, current and 

anticipated groundwater flow and transport 

conditions and system behavior at and 

around a site, of both a generic and a sitespecific nature. It includes information 

concerning the accessibility and reliability 

of data and knowledge. Key data categories 

include site and regional physiography, 

topography, climate, meteorology, soils, 

vegetation, land use, geomorphology, 

geology, geophysics, surface and subsurface 

hydrology, inorganic and organic 

hydrochemistry, radiochemistry, natural and 

anthropogenic isotopes, remotely sensed 

data, etc. Of special relevance to 

hydrogeologic model development are 

regional and site data that allow one to 

define the distribution of hydrostratigraphic 

units on a variety of scales; their geologic 

structure; rock and soil types; their textural, 

physical, flow and transport properties; fluid 

types; their state of saturation, pressure, 

temperature and density; chemical 

constituents and isotopes; major 

contaminants in soil, rock and groundwater; 

and their sources. 

Key knowledge categories include available 

concepts, methods and tools to integrate, 

manage, access, visualize, analyze, interpret 

and process such data in a manner suitable 

for flow and transport analyses under 

uncertainty, to perform such analyses, and to 

assess the corresponding ambiguities and 

uncertainties at various levels of 

hydrogeologic, conceptual and mathematical 

complexity in various contextually relevant 

hydrogeologic environments. 

4.2 Why Does One Need Site 

Characterization and Monitoring 

Data? 

Hydrogeologic modeling must not take 

place in a vacuum. All sites are unique and 

so cannot be validly represented by a 

generic model. Instead, hydrogeologic 

models must be solidly grounded in a broad 

array of regional and site data to validly 

represent particular locales. 

Site characterization data form the 

foundation on which one postulates one or 

more conceptual-mathematical models for 

an area and assigns initial values to their 

input parameters (i.e., material properties 

and forcing terms such as sources, initial and 

boundary conditions). To test and compare 

these models among themselves 

qualitatively and/or quantitatively, one also 

needs monitoring data that constitute 

observations of actual hydrologic behavior 

at and around the site. Only with such data 

can one evaluate the ability of models to 

mimic real system behavior (qualitatively at 

the conceptual level, quantitatively at the 

conceptual-mathematical level), improve 

their ability to do so through calibration 

against the monitoring data, determine their 

optimum degree of refinement or 

complexity, rank them and assess their 

cumulative impact on predictive uncertainty, 

and compare them with each other.
47 

Figure 4-1. Content of knowledge base. 
48 

The strategy in this report recognizes that 

site characterization and monitoring data are 

expensive and difficult to collect, leading to 

a ubiquitous scarcity of hard site 

information. It is therefore critically 

important to assess the role that such data 

play in rendering the hydrogeologic 

performance analysis credible. The strategy 

stresses the role of characterization and 

monitoring data in helping one identify and 

test alternative conceptual models, make 

rational choices among them, gauge and 

reduce model bias and uncertainty through 

proper model selection and calibration, 

assess the reliability of model predictions, 

and confirm the assessment through 

independent peer review as well as at least 

some degree of direct verification. 

4.3 Sources of Hydrogeologic 

Information 

Sources of site-specific information include 

technical reports, maps, charts and data files 

from local, state and federal agencies and 

institutions, including universities and their 

research arms; corresponding sites on the 

Internet; and visits to the site and its 

surroundings. Sources of generic 

information include textbooks, monographs, 

reference books and compendia on 

hydrogeology; soil physics; the dynamics of 

flow and transport in porous and fractured 

media; testing, sampling and monitoring of 

soils and aquifers; groundwater modeling; 

isotope hydrology; and hydrogeochemistry. 

Additional sources of generic information 

include research and review papers and 

reports pertaining to specific aspects of the 

above topics. 

4.3.1 Example: The Arizona Water 

Information Directory

An excellent example of how one can 

identify sources of information relevant to 

hydrogeologic analysis and modeling is 

provided by the Arizona Water Information 

Directory (Tellman, 2001). This guide to 

locating agencies, organizations, and 

university specialists with information about 

water in Arizona is based in part on a 1998 

publication titled "Where to Get Free (or 

Almost Free) Information About Water in 

Arizona" by Tellman. It documents an 

extensive and broad range of available 

resources for water information in Arizona, 

many of which (primarily federal agencies) 

may also serve as sources of information for 

other parts of the United States. The 

publication is free upon request from the 

University of Arizona Water Resources 

Research Center, whose web site 

(www.ag.arizona.edu/azwater/) also 

supports complex search queries and is 

updated periodically as new information 

becomes available. 

Examples of data sources related to geology 

include U.S. Geological Survey (USGS) 

(www.usgs.gov), Arizona Geological 

Survey (www.azgs.state.az.us) and 

University of Arizona Institute for the Study 

of Planet Earth (www.ispe.arizona.edu). 

Data concerning aquifers can be obtained 

from the USGS, Arizona Department of 

Water Resources (www.water.az.gov), U.S. 

Environmental Protection Agency (U.S. 

EPA) (www.epa.gov), and U.S. Water 

Conservation Laboratory 

(www.uswcl.ars.ag.gov). Well data are 

available from the Arizona Department of 

Water Resources, Arizona Small Utilities 

Association (www.asua.org), and Salt River 

Project (www.srpnet.com). Information 

about soils is available from the Arizona 

Department of Agriculture 

(www.agriculture.state.az.us) and 

Cooperative Extension 

(www.ag.arizona.edu/extension).

Data related to climate and weather can be 

obtained from the U.S. Natural Resources 
49 

Conservation Service 

(www.az.nrcs.usda.gov), U.S. Water 

Conservation Laboratory, Southwest 

Watershed Research Center 

(www.tucson.arg.ag.gov), Arizona State 

University Office of Climatology 

(www.geography.asu.edu/climatolog), and 

University of Arizona Institute for the Study 

of Planet Earth. Evaporation and 

evapotranspiration data are obtainable from 

the Arizona Meteorological Network 

(AZMET) (www.ag.arizona.edu/azmet) and 

Southwest Watershed Research Center. 

Streamflow data are available from the 

USGS, U.S. Forest Service (www.fs.fed.us) 

and Salt River Project. Numerous additional 

sources of information are listed by various 

categories. 

4.3.2 Example: Maricopa Agricultural 

Center 

In the process of developing this strategy, 

we have investigated experimentally 

groundwater flow and transport in 

unsaturated and saturated soils at the 

Maricopa Agricultural Center near Phoenix, 

Arizona. To support this investigation, we 

relied on several publicly available sources 

of information. These included reference 

maps downloaded from the University of 

Arizona library 

(www.srnr.arizona.edu/nbs/gap/nbiidata.

html), meteorological data from the AZMET 

Network, distribution of soils and irrigated 

land from the Arizona Land Resource 

Information System (ALRIS) 

(www.land.state.az.us/alris/htmls/

data2.html), as well as irrigation rates, 

distribution of wells, well logs, well 

construction data, pumping rates, and 

groundwater levels from archived records of 

the Arizona Department of Water Resources 

and/or the U.S. Geological Survey. 

4.3.3 Example: EDR Environmental 

Data Resources, Inc.

EDR Environmental Data Resources, Inc. 

(www.edrnet.com), is a good commercial 

source of environmental information. EDR 

offers current and historical environmental 

risk management information, industry 

publications and market research, 

newsletters, a daily business news service, 

training workshops, and state-of-the-art 

online services including interactive 

mapping. 

The service allows one to search a variety of 

government records for information about 

specified target areas within a given radius. 

Examples include topographic maps and 

gradients from the U.S. Geological Survey; 

flood zone data from the Federal Emergency 

Management Agency; information about 

wetlands from the U.S. Fish and Wildlife 

Service; groundwater flow direction, 

velocity and depth to water table from 

published sources; geologic age and rock 

stratigraphic units from the USGS; soil maps 

and soil survey data from the U.S. 

Department of Agriculture Soil 

Conservation Service; locations of water 

wells and springs from the USGS, U.S. EPA 

and state agencies; as well as other 

environmental data. 

4.4 Data Management and Expert 

Software Systems 

Ideally, the knowledge base would be 

contained in an integrated, interactive, 

flexible, robust and user-friendly data 

management and expert software system 

which allows easy access to each member of 

a multidisciplinary team to simultaneously 

view, edit, and interpret available and/or 

hypothetical site data and results through all 

phases of a project; with applications 

organized by discipline, each with a variety 
50 

of modules, which work together and share a 

common data base; and with seamless 

access to data as well as to flow and 

transport analytic and interpretive tools of 

varied types and levels of complexity. 

4.4.1 Example: GMS 

The Groundwater Modeling System (GMS) 

(www.scisoft-gms.com) is a sophisticated 

package for groundwater modeling, 

marketed by the Scientific Software Group 

(www.scisoftware.com) and other 

commercial entities. It provides tools for 

various phases of the modeling process 

including site characterization, model 

development, post-processing, calibration, 

and visualization. GMS supports 

triangulated irregular networks (TINs), 

solids, borehole data, two- and threedimensional geostatistics, and both finite 

element and finite difference models in two 

and three spatial dimensions. Currently 

supported groundwater flow and transport 

codes include MODFLOW, MODPATH, 

MT3D, RT3D, FEMWATER, SEEP2D, 

SEAM3D, PEST, UCODE and UTCHEM. 

Due to its modular nature, GMS can be 

customized to include selected modules and 

interfaces. 

GMS comes in identical PC and UNIX 

versions. It allows one to directly import 

data from, or export data to, ARC/INFO, 

ArcCAD, and ArcView. GMS makes it 

relatively easy for the user to quickly 

develop a conceptual model and a 

corresponding numerical model for the area 

being studied. For example, a TIFF or JPEG 

image of an aerial photo/scanned-in map, or 

an AutoCAD or MicroStation DXF drawing 

of the site can be displayed as background 

images allowing the user to define points, 

polylines, and polygons to represent 

spatially associated modeling data. 

Boundary conditions and parameter values 

can be directly assigned to these graphical 

entities. Points can define well pumping data 

or point sources for contaminants; polylines 

can define rivers, drains, or model 

boundaries; and polygons can define areal 

data such as lakes, differing recharge zones 

or hydraulic conductivities. Once the 

conceptual model has been defined, GMS 

will construct a grid, automatically refined 

around wells with the cells outside the 

model boundary already deactivated. The 

defined modeling data is then superimposed 

onto the grid with the appropriate 

parameters. For example, conductances 

assigned to polylines such as drains and 

rivers are automatically computed according 

to the length of the polyline segment within 

each cell. At this stage, the model is 

completely defined and no cell editing is 

required. If the user decides to change the 

conceptual model (move a boundary, add 

additional wells, etc.), these changes can be 

made quickly. Drawing tools are also 

provided, which allow the user to draw text, 

lines, polylines, arrows, rectangles, etc., in 

order to add annotation to the graphical 

representation of the model. 

GMS allows the user to construct TINs and 

solid models, and display borehole data. 

TINs are formed by connecting a set of x-yz points (either scattered, gridded, or from 

boreholes) with edges to form a network of 

triangles. TINs can be used to represent the 

surface of a geologic stratum and can be 

displayed in oblique view with hidden 

surfaces removed. Three-dimensional 

models of stratigraphy can be constructed 

using solids and cut to create cross sections 

anywhere on the grid. The solid model can 

be shaded to generate realistic images. 

GMS allows one to manage borehole data 

for site characterization. A borehole can 

contain either stratigraphy data or sample 

data or both. Stratigraphy data are used to 
51 

represent soil layers that are encountered in 

a soil boring. The soil layers are represented 

using contacts and segments. A segment 

represents a soil layer, and a contact is the 

interface between two segments. Contacts 

and segments can be used to construct TINs, 

solids and 3D finite-element meshes. 

Sample data represent data obtained by 

continuous sampling along the length of the 

hole. Cone penetrometer data and down-hole 

geophysical data are examples of sample 

data. Sample data are stored in data sets that 

can be manipulated as other data sets in 

GMS. For example, sample data from a cone 

penetrometer test may include data sets for 

tip resistance, sleeve resistance, and friction 

ratio. Sample data can be converted to 

scatter points, which can be interpolated to a 

three-dimensional grid, or mesh from which 

isosurfaces and color-shaded contours can 

be generated. Sample data can also be used 

to infer soil stratigraphy. 

With GMS, one can interpolate from groups 

of scattered data in two or three dimensions 

to other objects (meshes, grids, TINs). Twodimensional interpolation is used (for 

example) to generate transmissivities for a 

layer of a three-dimensional grid. Threedimensional interpolation is used (for 

example) to assign initial conditions (head 

and/or concentrations) to each node in a 

three-dimensional grid. Interpolation 

schemes supported by GMS include linear 

(2D), inverse distance weighting (2-3D), 

Clough-Tocher piece-wise cubic patch based 

on finite-elements (2D), natural neighbor

based on Thiessen polygons (2D) or 

polyhedrals (3D), ordinary and universal 

kriging using the GSLIB software package 

with graphical (anisotropic) variogram 

editing, and log interpolation. One can also 

interpolate from groups of 3D scatter points 

to any of the other data types (meshes, grids, 

TINs). 

The GMS user interface is divided into ten 

separate modules. A module is provided for 

each of the basic data types supported by the 

system. These include a Triangulated 

Irregular Network (TIN) Module, Borehole 

Module, Solid Module, 2D Mesh Module, 

2D Grid Module, 2D Scatter Point Module, 

3D Mesh Module, 3D Grid Module, 3D 

Scatter Point Module, and Map Module. 

Switching from one module to another can 

be done instantaneously to facilitate the 

simultaneous use of several data types when 

necessary. To provide a consistent interface 

for all the modules, GMS associates generic 

scalar or vector data set values with any 

object. Each data set can be either steady 

state or transient. TINs, meshes, grids, and 

scatter point sets all have an associated list 

of scalar data sets and a list of vector data 

sets. Boreholes have a list of scalar data sets. 

Each set has a single vector or scalar value 

for each node, cell, borehole sample point, 

or scatter point. Data sets can be used to 

represent various quantities such as total 

heads computed by a groundwater model or 

starting heads used as initial conditions for 

input to a transient groundwater model. Data 

sets can be imported from a file or created 

by interpolation from a group of scattered 

points. The data can be altered through 

various mathematical operations and 

contoured or displayed as isosurfaces. 

A variety of tools are available in GMS for 

panning, zooming, and rotating 3D objects. 

Editing and model interaction can take place 

in any view. Visualization options range 

from 2D contour plots to 3D isosurfaces. A 

groundwater model can be displayed in plan 

view or 3D oblique view and rotated 

interactively. Cross sections and fence 

diagrams may be cut arbitrarily anywhere in 

the model. Hidden surface removal and 

color and light source shading can be used to 

generate high-quality images. Contours and 

color fringes can be used to display the 
52 

variation of input data or computed results. 

Cross sections and isosurfaces can be 

interactively generated from 3D meshes, 

grids and solids, allowing the user to quickly 

visualize the 3D model. Both steady state 

and transient solutions can be displayed in 

animation using vector, isosurface, color 

fringe or contour animation. For example, 

animation of a transient solution allows the 

user to observe how head, drawdown, 

velocity and contaminant concentration vary 

with time. In addition, GMS can also sweep 

an isosurface through the 3D model. The 

minimum and maximum isosurface values 

are determined from the model and the 

program will then linearly interpolate and 

display multiple isosurfaces in rapid 

succession. This allows the user to quickly 

understand the spatial variation of a 

contaminant plume. Film loops are saved in 

Microsoft Video for Windows (*.avi) format 

and can be played back outside of GMS 

using almost any multimedia player or 

presentation package such as Microsoft 

PowerPoint. 

4.4.2 Example: FRAMES 

A Framework for Risk Analysis in 

Multimedia Environmental Systems 

(FRAMES) is being developed by Pacific 

Northwest National Laboratory 

(www.pnl.gov/hightechcomm/license/

programs/frames.pdf). FRAMES is an openarchitecture, object-oriented system that 

provides an environmental database. It aids 

decision-makers and environmental users in 

constructing a Conceptual Site Model that is 

real-world based. FRAMES provides a 

platform to link the users' preferred codes 

with others that are required to perform 

environmental assessment, analyze the 

results and display them graphically. 

FRAMES contains sockets for a collection 

of computer models that simulate elements 

of a source, fate and transport, exposure, and 

risk-assessment system. Currently available 

sockets include 

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

Contaminant Source and Release to 

Environment 

Overland 

Vadose Zone 

Saturated Zone 

Atmospheric 

Surface Water 

Exposure 

Dose 

Human Health Impacts 

Sensitivity/Uncertainty 

Future planned sockets will include 

Ecological Impacts 

GIS 

Remediation Technology 

Cost Analysis 

Data Quality Objective 

Life-Cycle Management 

Conceptual Site Module 

FRAMES provides file specifications that 

describe how all site information is stored 

within the framework and passed between 

modules. These file specifications are not 

associated with the model-specific 

information, only with the transfer of 

information to the outer framework or 

another modules. 

With a Windows-based plug-and-play user 

interface, the user builds a Conceptual Site 

Model using media icons to represent the 

flow of contamination through the 

environment. Then, the user selects the 

simulation models to be used for the 

analysis. FRAMES allows the user to 

produce multiple unique analyses with one 

software tool. 
53 

FRAMES’ modularization produces several 

types of time varying outputs including 

• Contaminant mass remaining at the source 

• Contaminant fluxes from the source 

• Atmospheric concentrations and soil 

deposition 

• Intake or dose 

• Hazard quotient or risk 

The platform allows one to 

• Evaluate protectiveness for hazardous 

waste sites (U.S. EPA) 

• Support D&D license termination rule 

(U.S. NRC) 

• Link models related to both human health 

and ecological impacts from past and 

current practices at government 

installations (U.S. Department of Defense) 

• Provide flexible environmental modeling 

tools for assessing human health and 

environmental compounds for a waste site, 

program, installation, and complex wide 

scale (U.S. DOE and its contractors) 

• Assess environmental issues and set policy 

(international companies and government 

agencies) 

For further information contact: 

4.4.3 Example: SEDSS 

The U.S. DOE, Office of Science and 

Technology, through the Mixed Waste 

Landfill Integrated Demonstration program, 

has funded the development of the Sandia 

Environmental Decision Support System 

(SEDSS) (www.sandia.gov/Subsurface/

factshts/ert/sedss.pdf). The latter is a 

decision support strategy and automated 

decision support tool for aiding 

environmental decision support makers in 

selecting appropriate characterization and 

remediation schemes. It is a concise, widely 

applicable process or framework for 

formulating, addressing and solving 

environmental problems. SEDSS is designed 

to extend the application of risk-based 

performance assessment methodologies to 

environmental restoration activities. It 

explicitly accommodates uncertainty while 

integrating risk analysis with data collection. 

SEDSS provides access to site data through 

a GIS; a set of analysis tools necessary to 

qualitatively and quantitatively evaluate a 

site based on available information; and a 

platform for graphically displaying and 

documenting the results of these analyses. 

To address uncertainty, SEDSS uses 

probabilistic methods such as Monte Carlo 

simulation and geostatistical analysis. 

SEDSS includes a Conceptual Model 

Manager (CMM) that assists users in 

defining and documenting the environmental 

problem. A series of interactive windows 

allows the user to step through the problem 

to identify assumptions about the site 

environment as well as sources of 

contamination, transport pathways, and 

human and environmental exposure routes. 

The user's response to the CMM queries 

provides the framework for the quantitative 

evaluation of risk. 

Some examples of decisions that SEDSS 

may help answer are the following: How 

should resources (cost and time) be 

prioritized based on estimates of risk? Is a 

site safe? If the site is unsafe, what remedial 

action or containment is necessary and 

optimal? When is remediation complete and 

how can it be defended as adequate? Is a 

potential new waste-facility-design safe (i.e., 

can a permit application be defended)? Is a 

monitoring network adequate? Eventually, a 

user of SEDSS will be: 

• Guided in establishing performance 

objectives for a given site; 
54 

•

•

•

•

•

Provided with immediate access to both 

data stored in numerical or graphical 

forms; 

Guided in the development of a 

conceptual model of site conditions, 

including the types and condition of the 

waste, the pathways through which the 

waste migrates, the mechanisms by 

which human receptors can be exposed 

to the waste, and the human health risks 

associated with that exposure; 

Assisted in setting up and executing 

numerical analysis to evaluate the user's 

concept of contaminant movement, 

exposure, and risks that accommodate 

the users existing uncertainty in site 

characteristics; 

Provided with the ability to determine 

which additional data would be 

important in altering or substantiating a 

current decision on risk or remediation 

approach, and tools to evaluate the 

cost/benefit of acquiring that data; and 

Guided in on-site, real-time collection of 

additional data. 

A prototype of SEDSS exists for a Sun 

workstation or server running the Unix 

operating system. Its capabilities include the 

first four of the above steps, providing the 

ability to determine risk from radioactive or 

hazardous contaminants moving via the 

ground-water pathway. The prototype 

software will optimize the location of a 

monitoring well network for detection of 

hazardous contaminants under U.S. EPA 

RCRA regulations, and perform risk 

analysis for the groundwater pathway. 

4.4.4 Example: FIELDS 

U.S. EPA Region 5 is developing a Fully 

Integrated Environmental Location Decision 

Support (FIELDS) System (www.epa.gov/

region5fields/static/pages/mission.htm) to 

help identify, assess, communicate and solve 

priority environmental problems in specific 

geographic areas. FIELDS began as an 

effort to more effectively solve 

contaminated sediment problems in and 

around the Great Lakes. The FIELDS team 

now supports and has applied their 

technology tools to numerous sediment, soil 

and groundwater sites in U.S. EPA Region 5 

and beyond. The system can be applied to 

environmental problem on various scales 

involving soils, surface water, ground water, 

air and the ecology at diverse site facilities, 

watersheds and urban areas. FIELDS 

employs Geographic Information Systems 

(GIS), Global Positioning System (GPS), 

database, analysis and imaging software 

combined in a modular fashion. It supports 

the EQuIS database by Earthsoft 

(www.earthsoft.com) which holds chemistry 

data from wells, bore holes, groundwater, 

surface water and soil samples. It also has a 

geology database, which can contain 

information on well development. 

4.4.5 Example: SADA 

The University of Tennessee at Knoxville is 

collaborating with Oak Ridge National 

Laboratory in the development of a Spatial 

Analysis and Decision Assistance (SADA) 

System (www.sis.utk.edu/cis/sada/) under 

funding by the U.S. EPA and Department of 

Energy (U.S. DOE). SADA is an evolving 

freeware system that incorporates tools from 

environmental assessment fields into a 

unified problem solving environment. The 

tools include integrated modules for 

visualization, geo-spatial analysis, statistical 

analysis, human health risk assessment, 

cost-benefit analysis, sampling design, and 

decision analysis. The capabilities of SADA 

can be used independently or collectively to 

address site specific concerns when 

characterizing a contaminated site, assessing 

risk, determining the location of future 
55 

samples, and designing remedial action. The 

U.S. EPA FIELDS team has funded the 

development of an ecological risk module 

for SADA. 

SADA provides a number of methods for 

the exploration of environmental data that is 

categorized by depth during remedial 

investigations (generally soil and 

groundwater). Data exploration tools include 

two- and three- dimensional data 

visualization options. Three-dimensional 

information is presented as multiple slices 

(layers) or by volume. The volume approach 

allows visualization of all depths at once. 

SADA accepts map layers from GIS and 

allows one to select a sub-region of the site 

for geo-spatial and risk analyses. Geo-spatial 

analysis tools include methods for assessing 

spatial correlation among data, modeling 

spatial correlation, and producing 

concentration, risk, probability, variance, 

and cleanup maps. Spatial data can be 

interpolated via ordinary kriging, indicator 

kriging, inverse distance or nearest neighbor 

methods. 

SADA can produce site-specific cost-benefit 

curves that demonstrate the specific 

relationship between a given remedial 

cleanup goal and the corresponding cost. 

This cleanup goal can be a concentration 

value or a particular human health risk 

scenario. It also provides different strategies 

to determine future sample locations, 

depending on the choice of geo-spatial 

interpolator. The estimate rank approach 

identifies unsampled locations that are 

modeled to have high concentration levels 

relative to the existing data. It can be useful 

for verifying the extent of hotspot regions 

and is available for any of the interpolation 

schemes. It does not account, however, for 

data variability. Consequently, it may place 

sampling points at locations that are high in 

concentration values but are relatively well 

characterized. The variance rank approach 

fills new samples into unsampled locations 

that have high estimation variances. Since it 

gives no weight to the magnitude of 

concentrations, samples may appear where 

data are sparse but where corresponding 

concentrations are very low relative to the 

decision rule. This approach is available 

only with ordinary kriging. The percentile 

rank approach considers both the magnitude 

and variance so as to avoid sampling wellcharacterized hot spots or sparse areas with 

very low detected or non-detected values. 

The uncertainty rank approach places new 

samples in areas where there is the greatest 

uncertainty about exceeding a cleanup goal. 

It helps delineate the boundaries of an area 

of concern. Finally, the secondary constraint 

approach allows the user to specify a 

minimum distance between any new sample 

locations and any previously sampled data. 

 

4.4.6 Example: LANL EES Integrated GIS 

and Model-Based Data Management 

System

The Earth and Environmental Sciences 

(EES) Division at Los Alamos National 

Laboratory (LANL) is integrating numerical 

vadose zone modeling results with GIS into 

a single platform to store, retrieve, query 

and display data (EES Division, 2001). The 

GIS system is comprised of a Raid Storage 

device, an Oracle database, a SDE (Spatial 

Data Engine) for querying, and Arc/View 

GIS to view the data. Traditional GIS data 

such as contours, historical building 

footprints, and known source locations are 

stored as points, lines and polygons with 

attributes. Vadose zone flow and transport 

modeling results obtained with the threedimensional FEHM code are stored as points 

with attributes, such as temperature or 

pressure at a given node. The system is able 

to store, query, visualize and compare both 

numerical model results and geographic data 
56 

in an efficient manner. This has proven to be 

helpful in creating both conceptual and 

numerical models of subsurface flow and 

transport at the site. 

4.4.7 Example: LANL EES Geologic 

Model of LANL Site and EspaZola 

Basin

The EES Division at LANL is developing 

(personal communication, 2001) geologic 

models for the Environmental Restoration 

Project and Groundwater Protection 

Program on two scales: Laboratory (138 

square miles) and EspaZola Basin (2500 

square miles). The models, assembled from 

geologic maps, borehole data, highresolution total station mapping, and 

interpretative cross-sections, are managed 

using Oracle. GIS Arc/Info is used to 

assemble the various data sources and to 

generate gridded surfaces representing 

geologic unit boundaries. These surfaces are 

assembled into three-dimensional geologic 

models using Stratamodel. 

At present, the models are built into 30 

geologic units, from approximately 50,000 

data records. The geology ranges from 

Precambrian basement to recent ash-flow 

tuffs to localized basalt flows. The models 

embed all of the spatial relations of the 

actual geology and can be used to generate 

geologic maps, cross-sections, etc. They will 

be the framework for numerical models of 

hydrologic and contaminant transport 

processes in the area. The Laboratory scale 

model will serve as a framework for detailed 

studies of the fate of contaminants such as in 

the Los Alamos Canyon region. At the 

EspaZola Basin scale, the model will help 

interpret transport pathways through the 

regional groundwater system.
57 
58 

5 QUALITATIVE CONCEPTUALIZATION AND SCREENING OF HYPOTHESES 

Having assembled and organized a 

hydrogeologic knowledge base for a site, 

one should be ready to describe in 

qualitative terms the hydrogeologic makeup 

of the site, the prevailing flow and transport 

regimes, and the way in which these regimes 

are expected to evolve under future 

scenarios of interest. As each site is defined 

by a geographic area and a depth dimension, 

its qualitative hydrogeologic description 

should capture all three of these spatial 

dimensions. Several alternative descriptions 

should be considered, examined in light of 

the available information and eliminated if

in conflict with key data.

Figure 5-1. Qualitative conceptual-mathematical modeling. 
59 

5.1 What is a Qualitative Conceptual 

Model of Site Hydrogeology? 

The qualitative aspect of conceptual 

hydrogeologic modeling is encapsulated in 

Figure 5-1. A conceptual hydrogeologic 

model is a mental construct or hypothesis 

accompanied by verbal, pictorial, 

diagrammatic and/or tabular interpretations 

and representations of site hydrogeologic 

conditions as well as corresponding 

flow/transport dynamics. It is customarily 

presented in the form of written text 

accompanied by pictures, charts and 

diagrams (including maps, block-diagrams, 

cross-sections, panel diagrams, 

vertical/horizontal profiles) of system 

components, arrangements and relationships 

(flows) known collectively as structure. A 

conceptual model identifies relevant 

hydrogeologic units (soils, aquifers, 

aquitards, aquicludes, bedrock) and features 

(faults, intrusions, fractures), their makeup 

(mineralogy, petrography, texture, 

cementation, porosity, permeability, related 

petrophysical and geophysical properties), 

geometry (horizontal and vertical 

dimensions, boundary and internal 

configurations), system states under 

undisturbed and disturbed conditions (types 

of fluids that permeate the soils and rocks, 

their pressure, saturation, density and 

temperature, dissolved solutes and their 

concentrations), flow and transport 

dynamics and kinematics (active flow and 

transport processes; their driving 

mechanisms; fluid and solute fluxes and 

velocities; fluid, solute and energy 

balances), hydrochemistry and isotopes. 

5.2 Why Interpret Site Hydrogeology in 

Three Dimensions? 

A hydrogeologic knowledge base typically 

includes information obtained from 

boreholes, soil and geologic outcrops, 

underground openings and geophysical 

surveys. This adds a depth dimension to 

geographic information about the site and its 

surroundings which is largely twodimensional. A hydrogeologic knowledge 

base is thus inherently three-dimensional, 

and should allow one to describe the system 

in three-dimensions. This is important 

because hydrogeology is inherently 

three-dimensional, complex, and manifests 

itself on a multiplicity of scales. As such, it 

cannot be meaningfully captured in a single 

two-dimensional map or cross-section, nor 

can it be described adequately by a 

one-dimensional profile such as a 

stratigraphic column or a lithologic log, that 

show only selected details on a given scale. 

Instead, hydrogeologic data must be 

interpreted in three-dimensions and in some 

detail on a range of scales (from regional 

down to thin sections). There virtually 

always is at least some regional and site 

information that should allow one to do so; 

seldom can lack of data be validly quoted as 

a reason for doing less. This is true 

regardless of how complex or simple a set of 

models one ultimately employs for the 

assessment of site performance and the 

analysis of corresponding uncertainty. 

Not only the hydrogeologic makeup of a site 

but also the corresponding flow and 

transport regimes can and must be explored 

and described qualitatively in threedimensions on a range of scales. Seldom can 

an inability or lack of need to do so be 

validly quoted as justification for doing less. 

This is true regardless of how narrow is the 

contextual framework, how limited is the 

available knowledge (data) base, or how 

restricted are the available (financial, 

technological, computational) resources. 

Doing otherwise may result in 

mischaracterization of the system and 

misrepresentation of flow and transport in 

the mathematical model. 
60 

5.3 How to Formulate Hydrogeologic 

Hypotheses? 

A systematic approach to the qualitative 

conceptualization and screening of 

hydrogeologic hypotheses is depicted in 

Figure 5-2. It requires as a prerequisite the 

availability or acquisition of expertise in the 

qualitative and quantitative interpretation of 

hydrogeologic field data. With such 

expertise, and a healthy dose of 

hydrogeologic insight, one may proceed to 

conceptualize hydrogeologic units and 

features on the regional, site and subsite 

scales in terms of their hydrostratigraphy, 

lithology, vertical and horizontal boundaries, 

structural features such as folds, faults, 

offsets and intrusions, textural features such 

as grain size, cementation, microstructure 

and fracturing, flow and transport properties 

such as porosity, permeability, dispersivity 

and sorption coefficients, related 

pedological and petrophysical properties 

(spatial, directional and textural) as well as 

distributions and variations (heterogeneity 

and anisotropy) on a range of scales, both 

between and within hydrogeologic units and 

relevant structural and textural features. 

One must assemble or develop descriptions 

of geologic outcrops, well logs, air and 

satellite images, maps and cross-sections, 

panel diagrams, quantitative records of 

measured variables, and qualitative 

descriptions of observed phenomena. It is 

useful, but not necessary, to embed these in 

tools for computer management of 

comprehensive hydrogeologic data of the 

kind discussed in Chapter 4. 

The next step is to describe the space-time 

distributions of fluid types (water, air, 

nonaqueous phase liquids) and 

corresponding states such as saturation, 

pressure, temperature and density, followed 

by significant solutes and their 

concentrations, as well as the delineation of 

saturated, vadose and perched zones, all on a 

range of scales between and within 

hydrogeologic, structural and textural units 

and features. This in turn allows one to 

describe active and anticipated flow and 

transport phenomena such as advection (of 

solutes or particulates), convection (of heat), 

diffusion, dispersion and sorption, their 

modes (discrete features; single, dual or 

multiple continua) and scales of 

manifestation (regional, site, subsite, longterm, short-term) boundary and internal 

mechanisms, forces and sources that drive 

them (infiltration, evapotranspiration, 

recharge, discharge, pumping), as well as 

their relative intensities and significance. 

The next step is to delineate in a qualitative 

but internally consistent and coherent 

manner contours of equal hydraulic potential 

(head), pressure, saturation, density, 

temperature and solute concentration 

together with corresponding flowlines 

(streamlines, pathlines, streaklines), vectors 

of fluid and solute flux and velocity, and 

isochrones (of groundwater residence time 

and solute travel time). The potential for the 

development of fast flow paths, on scales 

smaller than is represented by these contours 

and vectors (focused and episodic 

infiltration, preferential wetting, highpermeability channels, instability of fluid 

fronts, fingering), must be articulated at this 

stage. One should also assess the overall 

balance of fluids, solutes and energy within 

the system. The task can be facilitated by 

means of descriptions, maps, cross-sections 

and panel diagrams concerning site geology 

and hydrogeology; records of precipitation, 

irrigation, evapotranspiration, runoff, river 

stage and discharge, wetlands; well and 

spring hydrographs; hand-drawn horizontal 

and vertical flow nets; simple water balance 

formulae; and published case studies of 

hydrologically similar sites. 
61 

Figure 5-2. Qualitative conceptualization and screening of hypotheses.
62 

To conceptualize contaminant transport on 

regional, site and subsite scales, one must 

describe the space-time distribution of major 

contaminants in the soil, vadose zone and 

groundwater on these scales. The description 

must include the space-time distributions of 

contaminant sources; mechanisms and rates 

of source contaminant mobilization and 

leaching; active transport phenomena such 

as average and fast advection, diffusion, 

dispersion, radioactive or biochemical 

decay, sorption, colloid transport; spacetime distribution of migration, spreading, 

and dilution patterns; and overall mass and 

ionic balance for key contaminants. 

Among tools that may be suitable for this 

task we list records, maps, cross-sections 

and panel diagrams of existing 

contamination, sources and groundwater 

flow patterns; published discussions and 

case studies of mechanisms that cause 

contaminant release, migration, spreading, 

and dilution along average and fast flow 

paths; simple formulae to assess 

corresponding rates and parameters; handdrawn horizontal and vertical average and 

fast migration pathlines, velocity arrows, 

and isochrones of travel time; and simple 

formulae of mass and ionic balance. 

A hydrogeologic conceptualization is not 

complete without a description of 

hydrogeochemistry and isotope hydrology 

on regional, site and subsite scales in threedimensions. This includes the space-time 

distributions of major hydrochemical 

constituents, environmental isotopes and 

their sources above and below the water 

table; space-time distribution of 

groundwater ages; implications concerning 

flow between and within hydrogeologic, 

structural and textural units and features 

(including infiltration, evapotranspiration, 

recharge, discharge, directions and rates); 

and implications concerning transport 

(directions and velocities, possible 

compartmentalization, isolation and mixing 

of groundwater bodies, chemical reactions, 

and water-rock interactions). 

Possible tools include maps, cross-sections, 

panel and compositional diagrams of 

constituents, isotopes and groundwater flow 

patterns; published discussions and case 

studies of mechanisms causing spatialtemporal variations in water chemistry, 

isotopes and ages; simple formulae to assess 

corresponding rates and parameters; handdrawn horizontal and vertical, average and 

fast migration pathlines, velocity arrows and 

isochrones; and simple mixing and chemical 

balance formulae. 

It often helps to conceptualize temperature 

and heat flow on regional, site and subsite 

scales in three-dimensions. This may include 

the space-time distributions of temperature 

and heat flow above and below the water 

table, and their implications concerning flow 

between and within hydrogeologic, 

structural and textural units and features 

(including infiltration, evapotranspiration, 

recharge, discharge, directions and rates). 

Among tools that may facilitate such 

conceptualization are maps and crosssections that include measured values and 

contours of temperature and heat flow; 

published heat conductivity and capacity 

values for soils, rocks and moisture 

conditions similar to those found at the site; 

published discussions and case studies of 

mechanisms causing spatial and temporal 

variations in water temperature and heat 

flow; and simple formulae that associate 

heat flow with fluid convection. 
63 

5.4 Why Formulate Multiple 

Conceptual Site Models? 

Hydrogeologic systems are open and 

complex and the corresponding knowledge 

base is invariably incomplete and imprecise. 

Therefore, such systems almost always lend 

themselves to multiple conceptualizations 

and the postulation of several alternative 

hypotheses. It is therefore important to 

explore varied conceptual frameworks and 

assumptions through a comprehensive 

evaluation of a broad range of regional and 

site data, their translation into coherent and 

internally consistent conceptual models or 

hypotheses, and an in-depth examination of 

these hypotheses in light of the available 

knowledge base. The more experts with a 

wider range of earth and environmental 

specialties are given access to the 

knowledge base, the larger and more varied 

are the alternative site descriptions they may 

identify. 

The conceptualization is not complete 

without a clear articulation of ambiguities 

and uncertainties associated with each 

alternative description and interpretation 

(conceptual model or hypothesis) of site 

hydrogeology. 

5.5 How to Formulate Alternative 

Conceptual Models?

To develop alternative conceptual models 

for a site, one should consider (among 

others) alternative representations of spacetime scales; number and type of 

hydrogeologic units such as layers and 

structures such as faults; flow and transport 

properties (their values and statistics, spatial 

distribution and geostatistics, internal 

heterogeneity, anisotropy); location and type 

of system boundaries; space-time 

distribution of fluids and their states 

(pressure, density, saturation, temperature); 

space-time distribution of saturated, vadose 

and perched zones; space-time distribution 

of driving forces (infiltration, recharge, 

discharge, initial system states, boundary 

conditions); space-time distribution of flow 

patterns; existence and nature of fast flow 

paths; overall water balance; space-time 

distribution of contaminants; space-time 

distribution of contaminant sources; 

mechanisms and rates of source contaminant 

mobilization and leaching; controlling 

transport phenomena; migration, spreading 

and dilution patterns of contaminants; 

contaminant mass balance; space-time 

distribution of groundwater ages; space-time 

relationships between major chemical 

constituents, isotopes, temperatures and heat 

flows; and their implications regarding flow 

and transport on regional, site and subsite 

scales in three dimensions and time. 

The alternative conceptualizations should be 

firmly grounded in the available knowledge 

base. Each alternative conceptualization 

should be supported by key data. 

Conceptualizations that are contradicted by 

key data should be avoided. A major attempt 

should be made to articulate conceptual 

models that contain a minimum number of 

inconsistencies, anomalies and ambiguities 

with the lowest possible amount of 

remaining uncertainty about the site and the 

corresponding flow and transport regimes. 

5.6 How to Assess and Rank Alternative 

Conceptual Models? 

Once a number of alternative hydrogeologic 

conceptualizations have been articulated, 

they must be systematically examined, 

compared, screened and ranked according to 

acceptance criteria that include logical 

consistency and coherence, and the extent to 

which they are supported or contradicted by 

available observations and data. Among 

otherwise equal conceptual models, we 
64 

favor the least complex based on the 

principle of parsimony. Models that do not 

meet reasonable acceptance criteria of 

internal consistency, coherence and 

correspondence with the available data 

should be eliminated from further 

consideration at this stage of the analysis. 

5.6.1 Example: Proposed High-Level 

Nuclear Waste Site 

As an example of how one may qualitatively 

formulate alternative conceptual 

hydrogeologic models, assess them, select 

the best among them, and confirm or refute 

some of them through additional site 

exploration we consider the case of a large 

apparent hydraulic gradient (LAHG) 

identified on the basis of water level 

measurements in boreholes on the north side 

of Yucca Mountain, Nevada. The site is 

being considered as a potential repository of 

high-level nuclear waste (Figure 5-3). 

The material that follows is taken from 

Neuman (1997b) and represents his personal 

view.

Figure 5-3. Local and regional water levels in the area (after Fridrich et al., 1994). 

The LAHG is defined by only two boreholes 

on its north (G-2, WT-6) and three boreholes 

on its south (G-1, H-1, WT-16). There are 

no data to confirm that water level highs 

recorded in G-2 and WT-6 persist further to 

the north, northwest or northeast. The 

LAHG is on the order of (200 m - 280 

m)/2,000 m = 0.01 - 0.14 in contrast to a 

moderate gradient of 0.02 - 0.04 across the 

Solitario Canyon Fault, and small gradient 
65 

of 0.0001 - 0.0004 south and east of the 

former. Regionally, steep hydraulic 

gradients tend to be associated with known 

geologic or topographic features such as 

edges of thick confining units, faults with 

major offsets, caldera boundaries and 

mountain range fronts. A large gradient on 

the west side of Yucca Flat coincides with 

the edge of the Eleana Formation (Figure 5-

3). The LAHG under consideration is unique 

in that it does not correspond to any obvious 

geologic or topographic feature. 

Several conceptual models have been 

proposed for the LAHG. They attribute the 

LAHG to: 

1. A perched system such that water levels 

in boreholes G-2 and WT-6 on the north 

side of the LAHG reflect the upper 

volcanic aquifer, those on the south 

reflect the lower volcanic aquifer, and 

downward vertical leakage takes place 

from the former to the latter through the 

intervening Calico Hills aquitard; 

2. A semiperched system which is similar 

but considers the Calico Hills aquitard to 

be saturated; 

3. Topographic control on the regional and 

local water table; 

4. A drain model according to which the 

LAHG coincides with the effective 

northern limit of the deep carbonate 

aquifer and a fault buried under the 

Calico Hills Formation, marking the 

northern boundary of a buried graben; 

water drains through the fault downward 

from the volcanic system into the 

underlying carbonate aquifer, then 

returns in part by upwelling along the 

Solitario Canyon, Bow Ridge and 

Paintbrush Faults; 

5. A spillway model in which a buried fault 

delineates the effective northern 

boundary of the lower volcanic aquifer, 

causing water to spill from the upper 

into the lower volcanic aquifer; 

6. A model according to which a buried 

fault acts as a barrier to flow from north 

to south due to the juxtaposition of lowand high-permeability layers and/or the 

presence of low-permeability gouge 

material within the fault; 

7. A reduction in permeability due to 

increased rock alteration and decreased 

fracturing to the north; 

8. Presence of the Eleana formation which 

causes the thinning of overlying 

volcanics and acts to reduce flow rates 

from north to south; 

9. Neotectonic phenomena such as 

rotational extension and/or increased 

stress to the north causing elevated water 

levels in this region. 

For purposes of performance assessment, it 

is important to adopt an appropriate 

interpretive model for the LAHG so as to 

demonstrate an understanding of site 

hydrogeology and reduce uncertainty in both 

the qualitative (conceptual) and quantitative 

(computational) representations of flow and 

transport in the area. A reliable conceptual 

framework for the LAHG is especially 

important to correctly define inflow 

boundary conditions for the site 

groundwater flow model and to answer 

questions such as: Is inflow into the area 

from the north significant and, if so, where 

does it occur and at what rates? Is flow from 

the north diverted into Crater Flat or along 

Fortymile Wash? If so, how and to what 

extent? 

In selecting among alternative conceptual 

models for the LAHG, Neuman favored 

those which are supported by the largest, 

most reliable and relevant set of 

observations and experimental data. Among 

all alternative models that conform to a 

given set of observations and experimental 
66 

data, he favors the least complex (this is 

known as the principle of parsimony). 

Based on these principles, Neuman 

tentatively favored the perched system 

model because it is conceptually straight 

forward and, as shown below, is supported 

by numerous direct and indirect 

observations and/or data while contradicting 

none. The same is not true about any other 

model with the possible exception of the 

semiperched concept which is easy to 

accommodate jointly with the perched 

interpretation (more on this later). A single 

additional borehole, drilled strategically into 

the LAHG area and logged as well as 

sampled appropriately, should in his view 

suffice to confirm or deny that the LAHG is 

an artifact of perched conditions, as Neuman 

proposed. In the absence of new data from 

such a borehole (at the time of Neuman's 

writing), he tentatively associated a 

probability of 0.95 with the perched system 

model, 0.4 with the semiperched model, and 

0.1 with all remaining conceptual models for 

the LAHG. 

The following illustrates how observations 

and data (Figures 5-4 – 5-6, Table 5-1) 

support, more-or-less in the order listed, a 

perched interpretation of the LAHG: (1) 

Recorded water levels in G-2 and WT-6 are 

near the contact between the Topopah 

Spring basal vitrophyre and the underlying 

Calico Hills Formation; perched conditions 

are known to exist near this contact in other 

wells (UZ-14, NRG-7A, SD-7, SD9); (2) 

recorded water levels in G-2 and WT-6 

(1,020 - 1,030 m) are not anomalous when 

compared to perched levels (from north to 

south) in UZ-14 (960 m), NRG-7A (860 m), 

SD-9 (890 m) and SD-7 (860 m); (3) the 

upper volcanic confining unit is much 

thicker in G-2 (326 m) and WT-6 than in G1 (156 m), H-1 (135 m) and WT-16; these 

wells define the LAHG; (4) geophysical logs 

suggest that rock saturation along G-2 is at 

and/or slightly below unity at altitudes 

above 730 m which coincide with the Calico 

Hills aquitard, but at unity in the underlying 

lower volcanic aquifer; water levels in G-1 

(750 m ), H-1 (731 m) and G-4 (731 m) 

further south are just below the aquiferaquitard contact (27 m, 5 m and 2 m, 

respectively); thus, the top of the saturated 

zone in G-2 is not anomalous when 

compared to G-1, H-1 and G-4; (5) water 

levels in G-2 declined by 12 m between 

1981 and 1994, while those in WT-6 rose by 

4 m; (6) thermal gradients in G-2 decreased 

gradually with time; (7) pumping in April 

1996 resulted in an asymptotic residual 

drawdown of about 0.5 m in G-2 by 

December 17, 1996; (8) wet walls and 

dripping was observed in the air-filled part 

of this borehole above the Topopah Spring 

basal vitrophyre; and (9) pulsed-heat flow 

meter logs have indicated downward flow in 

the water-filled part of the borehole, 

suggesting leakage into the lower volcanic 

aquifer. It is important to point out that 

other observations employed to support 

alternative models neither support nor 

contradict the perched system concept. In 

particular, since the perched system is 

relatively shallow, it should not be expected 

to explain phenomena and/or geophysical 

anomalies which are associated with deeper 

parts of the Yucca Mountain geologic 

environment. 

Since the Calico Hills aquitard appears to be 

either at or just below full saturation in the 

area of the LAHG, a minute addition of 

water to it would render this unit fully 

saturated. It is therefore very likely that 

conditions in the LAHG area vary 

temporally, and spatially, between perched 

and semiperched. If full saturation occurs 

only intermittently within this unit (as 

current data suggest), then one can expect 

flow within it to take place vertically 
67 

downward under a near-unit hydraulic 

gradient and the hydraulic conductivity of 

the aquitard to be near its maximum 

saturated value. If, on the other hand, 

semiperched conditions were allowed to 

persist, flow within the aquitard would 

develop horizontal components and the 

LAHG would dissipate to form a milder 

lateral hydraulic gradient. Since the 

observed lateral gradient is not mild but 

steep, the semiperched model does not offer 

a viable standalone interpretation for LAHG.

 Figure 5-4. Details of conditions in borehole G-2 

(presented to Expert Elicitation Panel by Czarnecki, 1997). 

None of the remaining conceptual models 

for the LAHG are consistent with key 

observations and data used earlier to support 

the perched system concept. In particular, 

these models are inconsistent with the G-2 

geophysical log which suggests partial 

saturation within the Calico Hills Formation; 

with the strong correspondence between 

recorded water levels in G-2, and confirmed 

perched conditions along the contact 

between the Calico Hills Formation and the 

Topopah Spring basal vitrophyre in other 

wells; and with the striking correspondence 

between the geophysically indicated top of 

the saturated zone in G-2 and water table 

elevations recorded in other wells. The idea 

that topography controls the LAHG is 

additionally contradicted by the lack of any 

correlation between recorded water levels 

and topographic elevations of wells across 

the LAHG (Figure 5-7); topographic control 

appears to be evident only on a much larger, 

regional scale and may help explain the 

general increase in water levels from their 

approximate values of 730 - 780 m in much 

of the Mountain area to about 1,200 m in the 

upper Fortymile Wash and 1,400 m at 

Pahute Mesa.
68 

 

Figure 5-5. Isopach map of Calico Hills (after Fridrich et al., 1994) 

and flow conditions in borehole G-2 (presented to Expert Elicitation Panel by Czarnecki, 1997). 

 

Figure 5-6. Temperature profiles and water level recovery in borehole G-2 

(presented to Expert Elicitation Panel by Czarnecki, 1997). 
69 

Table 5-1. Altitudes (and thicknesses) of hydrogeologic units in deep boreholes at the mountain 

(after Luckey et al., 1996; modified by Neuman, 1997b). 

Figure 5-7. Water levels versus topographic elevations (Neuman, 1997b). 
70 

 Figure 5-8. Drain (A) and spillway (B) models with section showing postulated buried graben 

(after Fridrich et al., 1994). 

The drain model (Figure 5-8A) relies on a 

conjectured fault and graben (Figure 5-8) for 

which there is no direct evidence, although it 

is consistent with a measured gravity 

anomaly (Figure 5-9) and the thickening of 

the Crater Flat Group between G-2 and G-1. 

It also postulates a complex and improbable 

flow system which requires that the drain 

(fault), and the entire volcanic system north 

of its inlet, be hydraulically isolated from

the same system south of the inlet; what else 

would prevent water in the volcanic system

from bypassing the inlet of the drain and 

thereby causing the LAHG to dissipate? The 

model further attributes the low heat flow 

"anomaly" in the unsaturated zone (Figure 

5-9) to cooling of the deep carbonate aquifer 

by water draining into it from above, which 

does not explain the observed strong 

negative correlation between heat flow in 

the unsaturated zone and the thickness of 

this zone (Figure 5-10), suggesting that the 

proposed mechanism may not be the cause 

of the anomaly. Finally, the model attributes 

relatively high water table temperatures 

along the Solitario Canyon, Bow Ridge, and 

Paintbrush Faults (Figure 5-10) to upwelling 

of relatively warm water from the carbonate 

aquifer through these faults, which however 

does not require postulating a drain.
71 

 

Figure 5-9. Residual gravity contours and heat flow distribution in unsaturated zone 

(after Fridrich et al., 1994; modified by Neuman, 1997b). 

The spillway model (Figure 5-8B) likewise 

relies on a conjectured fault and graben for 

which there is no direct evidence but which 

is consistent with a measured gravity 

anomaly and the thickening of the Crater 

Flat Group between G-2 and G-1. It further 

proposes an effective termination of the 

Crater Flat Group as an aquifer north of the 

LAHG without offering any direct evidence 

in support of such a termination. There also 

is no direct evidence for the conjectured 

reduction in permeability from south to 

north due to enhanced alteration and reduced 

fracturing, or for the proposed presence of 

the Eleana Formation north of the mountain; 

though such presence is consistent with an 

aeromagnetic high (Figure 5-11) attributed 

to magnetite-bearing argillites, it is 

consistent neither with the above mentioned 

gravity anomaly nor with the stratigraphy 

observed in G-2; the Eleana Formation is 

not present in UE-25p1. The proposed 

neotectonic models require postulating 

complex and unproven tectonophysical 

effects on hydrogeologic conditions and 

flow.
72 

 

Figure 5-10. Heat flow variation with thickness of unsaturated zone (after Sass et al., 1988) 

and distribution of isotherms at water table (after Fridrich et al., 1994). 

Figure 5-11. Aeromagnetic survey map (after Fridrich et al., 1994).
73 

Members of the Expert Elicitation Panel 

were not unanimous about whether the large 

hydraulic gradient is important to 

understanding the amount of inflow to 

Yucca Mountain from the north, and 

whether it is associated with a perched 

system or is connected to the regional 

saturated system. They, however, all agreed 

that if the hydraulic high is caused by a 

perched zone, then the underlying 

unsaturated zone is near saturation. They 

also agreed that a carefully drilled well 

should go a long way toward resolving the 

issue. 

In 1998, a new well (USW WT-24) was 

completed just southeast of well G-2. 

During drilling, a water-bearing fracture was 

encountered at about 760 m below land 

surface in the lower part of the Calico Hills 

unit, and water rose in the borehole to an 

altitude of about 840 m above sea level 

(Tucci, 2001). According to Tucci, “drilling 

continued for another 104 m below the 

water-bearing fracture without any 

significant change in the water level ... 

Because the potentiometric level persisted as 

the borehole was considerably deepened, 

and because the water level remained 

relatively stable after completion of the well, 

the 840 m level is assumed to represent the 

regional potentiometric level and not a 

perched level." 

We note that there appears to be nothing in 

the new information from well WT-24 to 

indicate that the Calico Hills unit is 

saturated between elevations 840 m (the 

water level recorded within the underlying 

upper volcanic aquifer in WT-24) and 1,020 

- 1,030 m (the water levels recorded in the 

nearby wells G-2 and WT-6). It therefore 

remains very likely that conditions in the 

LAHG area vary temporally, and spatially, 

between perched and semi-perched as 

proposed by Neuman (1997b). 
74 

6 MATHEMATICAL CONCEPTUALIZATION AND QUANTITATIVE

EXPLORATION OF HYPOTHESES

Once alternative qualitative descriptions of 

site hydrogeology have been articulated and 

examined, the next step is to formulate 

corresponding conceptual-mathematical 

models of flow and transport at the site 

under existing and anticipated conditions. 

These models can then be used to explore 

and screen quantitatively alternative 

hypotheses regarding the hydrogeologic 

makeup and behavior of the site. 

6.1 What is a Conceptual-Mathematical 

Model of Site Hydrogeology?

As stated earlier, a conceptual-mathematical 

model is obtained upon formulating a 

qualitative conceptual model in 

mathematical language. Its purpose is to 

help define and describe the hydrogeologic 

system in terms of space-time dimensions; 

topology; geometry; interactions (called 

processes) between kinematic (mass, 

concentration, flux, velocity) and dynamic 

(energy, force, stress) quantities; parameters 

and forcing terms (sources, initial and 

boundary conditions). Whether such a model 

is analytical or numerical (written in the 

language of calculus or algebra) is merely a 

technical, not a fundamental, distinction. 

The model ultimately allows one to explain 

and interpret existing observations, and to 

predict new observations, quantitatively. 

A conceptual-mathematical model embodies 

the descriptive component of a 

hydrogeologic model, cast in the form of 

mathematical equations suitable for system 

evaluation. The equations represent the 

symbolic framework, or structure, of the 

conceptual-mathematical model. Ideally, 

they provide a mathematical description of 

all physico-chemical (and other) processes 

that are considered relevant to flow and 

transport at the site on a given range of 

space-time scales subject to well-defined 

forcing. Defining the scales of the system 

includes specifying its spatial and temporal 

dimensions. Forcing includes sources, initial 

and boundary conditions. Suitability for 

system evaluation means that the symbolic 

conceptual-mathematical framework, or 

structure, is cast in a form that lends itself to 

computation and subsequent presentation 

(numerical, graphical) and analysis 

(deterministic, statistical) of the results. 

One cannot legitimately disassociate 

hydrogeologic interpretations, or conceptual 

models, from corresponding mathematical 

and computational models for purposes of 

quantitative environmental impact and 

performance assessments of a given nuclear 

facility or site; these models are intimately 

linked and define a single conceptualmathematical framework for quantitative 

site analyses. 

Conceptual-mathematical modeling may, 

but need not, be accompanied by the 

selection of a corresponding computational 

code (or several codes). We mentioned 

earlier that a code is a tool rather than a 

model. However, selecting a particular code 

implies identifying specific processes that 

may govern flow and transport at a site, their 

symbolic mathematical representation, and 

their numerical approximation. Selecting the 

space-time dimensions and size of a 

computational grid determines the 

dimensions and scale of the system being 

modeled. Choosing the sizes of space-time 

discretization intervals defines the scales at 

which flow and transport processes are 

resolved. Specifying the location and type of 

sources, initial and boundary conditions 

identifies the forcings. Both material 
75 

properties and forcing terms are associated 

with parameters that must eventually be 

assigned numerical values. Choosing the 

modes of their representation 

(parameterization) defines the resolution 

scales of these parameters (and thus of 

material properties and forcing terms) in 

space-time. Once this has been 

accomplished in a way that is supported by 

all relevant regional and site data, the code 

has been transformed from a mere tool to a 

bona fide conceptual-mathematical model of 

site hydrogeology.

Figure 6-1. Mathematical conceptualization. 
76 

6.2 How to Formulate a ConceptualMathematical Model for a Site? 

6.2.1 What to Specify? 

Mathematical conceptualization and the 

quantitative exploration and screening of 

hypotheses are encapsulated in Figure 6-1. 

A prerequisite is the availability, or 

acquisition, of expertise in the theoretical 

and mathematical interpretation and analysis 

of hydrogeologic field data. For each 

alternative conceptual model identified 

during the qualitative stage of the process, 

one needs to define the hydrogeologic units 

and features (such as folds, faults, offsets 

and intrusions) that are to be explicitly 

modeled; their three-dimensional topology 

and geometry in terms of location, shape, 

size and relationship to other units and 

features (extent of external and internal 

boundaries and structural elements); 

equations that govern flow and transport 

phenomena included in the model; 

corresponding equations of state, in the form 

of functional relationships between 

parameters and state variables in the model; 

spatial variability of parameters that enter 

into the governing and state equations, 

within each hydrogeologic unit and feature; 

spatial variability of initial states in each 

unit and feature; boundary equations for 

flow and transport; and the space-time 

distribution of boundary and source 

parameters and values. 

The quantification of system states includes 

functional and quantitative representations 

of fluid saturation, pressure, temperature, 

density, solute mass and concentration and 

regional as well as perched water tables on a 

range of scales (between and within 

hydrogeologic, structural and textural units 

and features). Governing and boundary 

equations include mathematical definitions 

and descriptions of active and anticipated 

flow and transport phenomena such as 

advection, convection, diffusion, dispersion 

and sorption for selected modes (discrete 

features; single, dual or multiple continua) 

and scales of manifestation, in the interior of 

units and features as well as on their 

boundaries and interfaces, including source 

terms, parameters, functional relationships 

between these quantities, and their spacetime distributions. Examples of this process 

are given in Figures 6-2 – 6-3. 

6.2.2 How to Determine a Suitable Level 

of Model Complexity? 

The purpose of conceptual-mathematical 

models is to help quantify alternative 

hypotheses regarding the hydrogeologic 

makeup and behavior of a site. Since such 

hypotheses are always cast in three spatial 

dimensions, ideally so should be the 

corresponding conceptual-mathematical 

models. Both model types must allow for the 

hydrogeologic regime to evolve in time. 

In reality, it is much more difficult and time 

consuming to set up and run mathematical 

models in three than in two spatial 

dimensions. Working in two spatial 

dimensions is often feasible because 

eliminating the third dimension usually 

entails a far lesser phenomenological change 

than reducing the dimensionality of the 

problem from two to one. Modeling and 

visualization of complex flow and transport 

phenomena is much easier in two than in 

three dimensions, but not much harder than 

in one dimension. Hence the strategy in this 

report supports two-dimensional flow and 

transport analyses (whether mathematicalanalytical or computational-numerical) for 

exploratory purposes. Such simulations and 

analyses are illustrated in Figures 6-4 – 6-7. 

The strategy supports a similar approach for 

performance assessment purposes in cases 

where the effect of the third dimension is 
77 

demonstrably minor. However, the strategy 

discourages one-dimensional analyses 

unless a very strong and convincing 

hydrogeologic argument is made in their 

favor. This is so because flow and transport 

behaviors predicted by one- and multidimensional models often differ from each 

other in fundamental ways. 

Figure 6-2. Example of hydrogeologic schematization for purposes of mathematical modeling 

(after NRC, 1990). With permission, National Academies Press. 

Figure 6-3. Example of hydrogeologic schematization for purposes of mathematical modeling 

(after Anderson and Woessner, 1992). With permission, Academic Press. 
78 

Figure 6-4. Geology (a), regional flow conceptualization (b) and simulation (c) 

(after Domenico and Schwartz, 1990). 
79 

Figure 6-5. Regional groundwater flow conceptualization 

(after NRC, 1990). With permission, National Academies Press. 

Figure 6-6. Vertical flow nets (after Freeze, 1969). With permission, AGU. 
80 

Figure 6-7. Horizontal flow net (after Bennet and Meyer, 1952). 

The spatial and temporal scales at which 

flow and transport phenomena are modeled 

depend in part on the contextual framework 

(area, depth and time-frame of concern; 

space-time scales on which performance 

measures are defined). These phenomena are 

affected to a large extent, but not 

exclusively, by hydrogeologic complexities, 

heterogeneities and driving-mechanisms that 

manifest themselves on similar scales. It is 

therefore important that conceptualmathematical modeling start by identifying 

these "site-scale" features, factors and/or 

phenomena and by incorporating them 

directly as explicit elements in the model. 

Models that do not incorporate such 

elements explicitly, but account for them

implicitly, are considered here to be 

simplified or abstracted; methods to develop 

them are discussed in the next section. 
81 

Models that fail to account for site scales 

complexities, either explicitly or implicitly, 

are oversimplified or would generally be too 

crude to provide a reliable description of site 

hydrogeology for most purposes. 

Site-scale hydrogeologic complexities, 

heterogeneities and driving-mechanisms

may be influenced and/or controlled by 

larger- or regional-scale features and factors 

such as regional recharge, discharge, and 

flow mechanisms and patterns on various 

time scales. A model must account for these 

large-scale influences and controls, as well 

as for associated uncertainties, through the 

appropriate assignment of initial conditions, 

boundary conditions, and source terms such 

as those that describe infiltration, recharge, 

discharge and leakage across aquitards. 

Site-scale hydrogeologic complexities, 

heterogeneities and driving mechanisms

may be significantly influenced by smalleror subsite-scale features and factors such as 

space-time irregularities and fluctuations in 

external and internal boundary and source 

shapes or conditions; internal 

heterogeneities within site-scale 

hydrogeologic units, faults, dikes or other 

features; as well as smaller-scale units, 

faults, dikes, fractures or preferential flow 

channels. Determining what is the nature 

and extent of these influences is an integral 

part of developing a conceptualmathematical model for a site. If features 

and factors that manifest themselves on 

subsite scales are deemed important for the 

modeling of site-scale phenomena (as in the 

case of dispersion in Figure 6-8), one must 

account for them directly (explicitly, by 

embedding such features and factors 

discretely in the site model) or indirectly 

(implicitly, by formally integrating these 

features and factors into the site model 

equations and parameters, such as 

dispersivity) in the model. 

Figure 6-8. Dispersion affected by heterogeneities on various scales 

(after Domenico and Schwartz, 1990). 
82 

Embedding discrete small-scale features and 

factors in a model renders it relatively 

complex. Integrating small-scale features 

and factors into the model equations and 

parameters renders it less complex than 

embedding. However, to compensate for 

loss of information, the equations change 

form and phenomenology (from Stokes to 

Darcy, isotropic to anisotropic, juxtaposed 

to overlapping dual or multiple continua, 

differential local to integro-differential 

nonlocal) and acquire new 

phenomenological parameters (permeability, 

dispersivity, integral kernels) that differ in 

nature and magnitude (from scalar to tensor, 

local to nonlocal, well-defined to scaledependent) from the original parameters. 

Only in special cases can the original form 

and phenomenology be recovered, and even 

then the parameters usually change (due to 

upscaling). It is not presently clear which of 

these two approaches is better suited for 

their intended task. The strategy in this 

report considers both options. Methods of 

embedding are discussed in the next section. 

Ignoring the influence of regional- or 

subsite-scale features or factors on site-scale 

flow or transport without due justification, 

or failing to insure that the treatment of all 

scales is self-consistent without 

demonstrating that some of them are not 

relevant to the problem at hand, constitute 

oversimplifications which may cast doubt on 

the reliability of the model. A large number 

of both steady-state and transient two- and 

three-dimensional analytical solutions and 

computational as well as visualization codes 

are available and easily accessible to general 

users. Many of them are well suited for the 

development, exploration, screening and 

selection of alternative conceptualmathematical models of site hydrogeology. 

Among these are the Groundwater Modeling 

System (GMS) mentioned in Chapter 4, 

which includes the forward and inverse 

groundwater flow and transport codes 

MODFLOW, MODPATH, MT3D, RT3D, 

FEMWATER, SEEP2D, SEAM3D, PEST, 

UCODE and UTCHEM; FEFLOW, an 

interactive graphics-based finite element 

simulation system for groundwater flow, 

contaminant and heat transport in two- and 

three-dimensions; etc. 

6.2.3 How to Simplify Models? 

Narrowly defined contextual or regulatory 

criteria, limited data or resources, and a 

quest for simplicity or transparency may 

motivate the adoption of hydrogeologic flow 

and transport models that are less than 

three-dimensional, ignore time, and include 

few details on limited scales. While such 

motivation for simplification and abstraction 

of hydrogeology and flow or transport 

dynamics may sometimes be justified on 

practical grounds, it does not in itself turn 

simplified and/or abstracted models into 

scientifically valid tools of performance 

assessment. Only a formal demonstration 

that such models capture the essential 

features and capabilities of their more 

complex and complete counterparts, and that 

they thereby provide comparable 

performance assessments or conservative 

bounds thereof, might justify their use for 

regulatory purposes. As already stated, 

to ensure that relevant aspects of 

hydrogeologic complexity are reflected in 

modeled system behavior and performance, 

and that the model can be rendered 

compatible with site data, it is important that 

the process of simplification be done 

systematically and objectively. This can be 

done by filtering out undesirable details 

through formal averaging of the governing 

equations in space-time or in probability 

space, in a way which retains and renders 

their influence on the model implicit. This 

was called embedding in the previous 

section. 
83 

Averaging three- or two-dimensional 

equations across one spatial dimension 

renders them two- or one-dimensional, 

respectively. An example of this procedure 

is presented in Appendix B. Averaging 

transient equations over time may (but need 

not) render them representative of a steady 

state. Averaging can also be done over 

subdomains of the site being modeled, and 

over multiple time intervals. In each case, 

the averaging results in governing flow and 

transport equations that contain upscaled 

quantities (such as the relative permeability 

curves in Figure 6-9). If the space-time 

scales of these quantities differ from those 

of the available site data, then either the 

model or the data must be rescaled to render 

them compatible and comparable with each 

other. 

Figure 6-9. Alternative conceptual models and their relative permeability characteristics 

(after Altman et al., 1996). 
84 

It has been traditional to assume that 

spatially and temporally averaged 

hydrologic quantities are less prone to 

predictive uncertainty than quantities that 

are localized in space-time. While this may 

be true for systems that are statistically 

homogeneous and stationary, such systems 

are the exception rather than the rule. Spatial 

and temporal averages are not necessarily 

more predictable as traditionally believed if 

the averaging covers a scale that contains a 

strong transition or change in behavior 

(Entekhabi, 2000). 

This and other scale-related problems are 

generally avoided if the averaging is done in 

probability space. Mathematical illustrations 

of averaging in probability space are 

provided in Appendices C and D. Such 

"ensemble" averaging leads to stochastic 

equations that contain statistical moments of 

hydrogeologic variables (considered 

random), most commonly the mean and 

variance-covariance. The mean is a predictor 

of system behavior or performance, and the 

variance-covariance is a measure of 

predictive error. Both are smoother (vary 

more slowly in space-time) than their 

random counterparts and, in this sense, 

render the model relatively "simple." 

Despite their smoothness, both moments are 

defined on the same space-time scales as are 

the random hydrogeologic variables on 

which they are based. Stochastic models 

thus achieve smoothness and simplicity 

without any need to average or upscale in 

space-time. As they are typically 

conditioned on site measurements (i.e., they 

honor the data), stochastic models are 

compatible with these measurements both in 

scale and magnitude. Yet another advantage 

of the stochastic method over space-time 

averaging is that it yields measures of 

predictive uncertainty. Stochastic 

approaches are increasingly recognized as 

offering a way to deal with complex, scaledependent heterogeneous systems by means 

of relatively simple models (Hunt and 

Zheng, 1999). 

Whenever possible, simplified and/or 

abstracted models should be deduced 

systematically and objectively from, be 

comparable with, and be judged and 

validated against their more comprehensive 

(complex and complete) 

conceptual-mathematical counterparts. The 

strategy strongly suggests postulating and 

evaluating the latter (at least on paper, not 

necessarily implementing them on the 

computer) before attempting to develop and 

validate the former. 

6.2.4 How to Factor Uncertainty Into a 

Conceptual-Mathematical Model?

Information about site- and regional-scale 

hydrogeology is typically sparse and 

uncertain. To account for this uncertainty, it 

is necessary to treat hydrogeologic 

quantities probabilistically (as random 

variables) or stochastically (as correlated 

random fields or processes). 

The traditional approach to hydrologic 

model uncertainty analysis has been to 

postulate a deterministic conceptualmathematical model structure and treat its 

parameters as being imperfectly known. The 

present strategy considers this approach as 

one of two options. 

The other option is to postulate a stochastic 

rather than a deterministic conceptualmathematical model structure. It considers 

that hydrogeologic environments consist of 

natural soils and rocks, which exhibit both 

systematic and random spatial variations in 

hydraulic and transport properties on a 

multiplicity of scales. Groundwater flow and 

solute transport take place under the action 

of forces whose exact nature, magnitude and 
85 

space-time distribution is generally 

uncertain. Traditional deterministic methods 

of analyses capture at best the larger-scale, 

systematic components of these variations 

and distributions. They however fail to 

resolve smaller scale variations or account 

for their uncertain nature. Yet capturing 

hydrologic complexity on a wide range of 

scales, and quantifying the associated 

uncertainty, is proving to be of paramount 

importance for a wide array of applications 

such as the characterization and remediation 

of contaminated sites, risk analysis and 

monitoring design for sites subject to 

potential future contamination, and 

performance assessment of nuclear facilities 

and sites. 

Under the stochastic option, one describes 

the spatial variability and scaling of 

hydrogeologic medium properties 

geostatistically, and analyzes subsurface 

fluid flow and solute transport 

stochastically. A key component of 

geostatistical characterization is the 

assessment of spatial covariance structure 

(in terms of a covariance or variogram 

function). In the absence of site 

characterization data that are amenable to 

geostatistical analysis, one can treat the 

structural parameters (of the covariance or 

variogram function) as free parameters to be 

estimated by model calibration. 

The strategy supports two general 

approaches to predict stochastic 

groundwater flow and transport on the basis 

of geostatistical input: High-resolution 

numerical Monte Carlo simulation and 

direct (deterministic) prediction of mean 

behavior. In the more widely known Monte 

Carlo approach, one generates a large 

number of equally likely parameter input 

fields, feeds them into a standard 

deterministic flow and transport simulator 

that however uses a very fine computational 

grid, and produces a large number of model 

outputs representing equally likely system 

behaviors. These nonunique outputs are 

summarized in terms of statistically 

averaged quantities, their variancecovariance, and perhaps higher moments of 

the corresponding sample probability 

distributions. Results that honor measured 

values of medium properties are said to be 

conditioned on these data. Upon 

conditioning the simulations on measured 

values of parameters in space, one obtains 

(among others) conditional mean flow and 

transport variables that constitute optimum 

unbiased predictors of these unknown 

random quantities. One also obtains 

conditional second moments (variancecovariance) that provide a measure of the 

associated prediction errors. To condition 

the predictions not only on measured input 

variables (derived from site characterization) 

but also on monitored system behavior tends 

to improve their reliability. This is 

accomplished by either discarding random 

simulations that do not reproduce the 

observations, or employing a suitable 

inverse procedure. 

The more novel and less widely known 

direct approach relies on stochastically 

derived deterministic ensemble mean flow 

and transport equations in which the 

dependent variables represent not actual 

system states (such as head, concentration or 

flux) but rather their (ensemble) mean 

values or statistical expectations. These 

mean or expected values represent optimum 

unbiased predictors of the unknown, actual 

system states (similar to those obtained upon 

averaging the results of numerous Monte 

Carlo simulations). To condition the 

predictions on system monitoring data, one 

must employ a special inverse procedure. 
86 

The variance-covariance of high-resolution 

(conditional) Monte Carlo simulations 

serves to quantify predictive uncertainty. 

Another option is to compute this variancecovariance directly (deterministically) by 

means of corresponding moment equations. 

Various software packages are available to 

help one implement the geostatistical and 

Monte Carlo options in two and three spatial 

dimensions. For up-to-date information 

about software and literature concerning the 

statistics and geostatistics of spatial data the 

reader is referred to AI-GEOSTATS 

(www.ai-geostats.org), a central server for 

GIS and spatial statistics run by the 

University of Lausanne in Switzerland. A 

number of publicly or commercially 

available software packages are listed 

below. 

GMS: This system was described in 

Chapter 4. 

UNCERT: UNCERT is a public domain 

uncertainty analysis and geostatistical 

software package for groundwater flow 

and contaminant transport modeling, 

available from the Department of 

Geology and Geological Engineering at 

the Colorado School of Mines

(uncert.mines.edu). It includes a number 

of software modules that allow one to 1) 

input raw field data or data from a preexisting database, 2) analyze the data 

using classical statistics, 3) evaluate 

trends, 4) evaluate the data using 

geostatistical techniques such as 

semivariogram analysis, various kriging 

techniques (simple, ordinary, indicator, 

and Bayesian), and stochastic 

simulation. When the data are analyzed, 

or prepared from other sources, 

graphical tools are available to view the 

results in two-, two-and-a-half-, and 

three-dimensions. Once the spatial 

variation of materials has been 

determined, tools are available to 

automatically generate finite-difference 

grids for groundwater flow and 

contaminant transport codes such as 

MODFLOW andMT3D, 5) run these 

models, and 6) evaluate the results of 

individual runs, as well as the composite 

results of multiple model simulations. 

MODFLOW and MT3D are widely used 

by hydrogeologists to simulate 

groundwater flow and solute transport, 

respectively, in three-dimensional 

saturated porous media. 

UNCERT is written primarily in C 

language and is compatible with Unix 

and Linux computational platforms. 

Code options include logarithmic 

transformation of data; generation of 

histograms, probability plots, 

variograms, cross-variograms, 

covariance functions, indicator 

variograms and covariance functions; 

jackknifing; manual or automatic 

modeling of standard spatial correlation 

functions; estimation via trend surface 

analysis as well as ordinary, simple and 

indicator kriging; mathematical 

manipulation of numerical series 

associated with two-dimensional and 

three-dimensional grids; and Monte 

Carlo simulation of random fields 

conditioned on both hard and soft data. 

In addition to a manual interface, 

UNCERT connects with GIS packages 

such as Arc/Info and Grass, and offers a 

graphical interface for MODFLOW and 

MT3D. 

The philosophy behind UNCERT is that 

there are a multitude of possible 

interpretations of the subsurface which 

honor any given set of data. To evaluate 

the alternatives manually would take 
87 

considerable time and still only a small 

portion of the possibilities could be 

evaluated. This is true even when the 

subsurface configuration is relatively 

simple. In order to evaluate this inherent 

uncertainty, computers can be used to 

create multiple alternative realizations of 

the subsurface. The process can be 

forced to honor hard data (well logs, 

etc.) by using indicator kriging 

techniques, and incorporate more 

uncertain data (soft data - data with a 

range of uncertainty, e.g., seismic 

information, geophysical well logs, 

expert opinion) through Bayesian 

kriging. By automating this process, 

much of the uncertainty can be 

characterized with comparatively little 

time invested by the hydrogeologist. 

Once multiple realizations are created, 

groundwater flow and contaminant 

transport models can be executed to 

compare modeled and field conditions. 

When a model response clearly doesn't 

match field observations, this possible 

subsurface configuration can be 

disregarded; of the remaining 

realizations (invalidating 90% of the 

realizations might not be unreasonable) 

that appear reasonable, the distribution 

of contaminants may be evaluated, for 

the time already modeled, or for future 

conditions. Based on the results of flow 

and transport modeling in these 

remaining configurations, the probable 

locations of contamination may be 

identified. Also, the probable 

effectiveness of remediation facilities 

designed to contain the contamination 

can be evaluated. A computer can 

evaluate only a limited number of 

realizations, but the number is so large, 

relative to that which can be 

accomplished manually, that a 

representative assessment of the 

reasonable alternatives will be realized. 

The modeler can start with hard and/or 

soft field data and be guided through 

statistical analysis of the data, generation 

of multiple realizations of each data 

property, development of model grids, 

kriging of data properties into model 

grids, generation of input files for flow 

and contaminant transport models, 

execution of models, and visualization of 

model results. 

Gstat: Gstat is a program for the modeling, 

prediction and simulation of 

geostatistical data in one, two or three 

dimensions. It is made available free of 

charge by the Department of Geography 

at Utrecht University in the Netherlands 

(www.gstat.org) and works on various 

platforms including Linus and win32. 

Geostatistical modelling comprises 

calculation of sample variograms and 

cross variograms (or covariograms) and 

fitting models to them. Sample (co-) 

variograms are calculated from ordinary, 

weighted or generalized least squares 

residuals. Nested models are fitted to 

sample (co-)variograms using weighted 

least squares, and during a fit each single 

parameter can be fixed. Restricted 

maximum likelihood estimation of 

partial sills and interactive variogram 

modeling are possible. 

Gstat provides estimation and prediction 

using a model that is the sum of a trend 

modeled as a linear function of 

polynomials of the coordinates or of 

user-defined base functions, and an 

independent or dependent, 

geostatistically modeled residual. This 

allows simple, ordinary and universal 

kriging; simple, ordinary and universal 

cokriging; standardized cokriging; 

kriging with external drift, block kriging 

and trend kriging; as well as 
88 

uncorrelated, ordinary or weighted least 

squares regression prediction. Gstat 

allows univariate or multivariate 

conditional or unconditional multiGaussian sequential simulation of point 

values or block averages, or (multi-) 

indicator sequential simulation. 

Gslib: The Gslib code is included on CD 

with the book of Deutsch and Journel 

(1998). It is written in Fortran language 

for DOS and Unix operating systems. 

Code options include the generation of 

histograms, probability plots, two- and 

three-dimensional variograms and crossvariograms; point or block estimation by 

ordinary, simple and universal kriging, 

kriging with an external drift, cokriging, 

nonlinear kriging, indicator principal 

component kriging, and soft kriging; 

cross-validation and jackknifing; and 

conditional Monte Carlo simulation by 

indicator principal components, LUdecomposition, simulated annealing, 

sequential Gaussian simulation, 

sequential indicator simulation of a 

categorical or continuous variable, and 

the turning bands method. Output is in 

postscript format but can be displayed in 

a graphical raster format by means of the 

UPFILE and 3Plot codes that are 

available without charge. 

FSS: This set of tools for DOS, Unix and 

Linux platforms is available from FSS 

International Europe, a consulting firm 

based in Valencia, Spain 

(www.fssintl.com). It uses the Gslib data 

format to perform univariate and 

bivariate statistical analyses of data; to 

generate variograms, covariance and 

correlation functions, H-scattergram and 

variogram plots; to estimate random 

field values by ordinary and simple 

kriging with global or local mean, 

cokriging, collocated cokriging, 

universal kriging, indicator ordinary and 

simple kriging, and multi-Gaussian 

kriging; and to conduct Monte Carlo 

simulations by sequential Gaussian and 

indicator methods. 

The direct stochastic modeling option is 

supported by a menu of exact conditional 

moment equations developed for steady 

state (Neuman and Orr, 1993; Neuman et 

al., 1996) and transient (Tartakovsky and 

Neuman, 1998, 1999) groundwater flow in 

saturated porous media, steady state flow in 

a certain class of unsaturated soils 

(Tartakovsky et al., 1999; Lu et al., 2002), 

and advective (Neuman, 1993) or advectivedispersive (Zhang and Neuman, 1996) 

transport of a nonreactive solute. 

Even though the underlying stochastic flow 

and transport regimes are taken to obey 

Darcy's law and Fick's analogy, these 

relations are generally not obeyed by the 

conditional mean flow and transport regimes 

except in special cases or as localized 

approximations. Such localized 

approximations yield familiar-looking 

differential equations which, however, 

acquire a non-traditional meaning in that 

their parameters (hydraulic conductivity, 

seepage velocity, dispersivity) and state 

variables (hydraulic head, concentration, 

fluid and solute fluxes) are informationdependent and thus nonunique (Guadagnini 

and Neuman, 1999a-b, 2001; Ye et al., 

2002). Whereas nonlocal moment equations 

contain information about predictive 

uncertainty, localized (differential) 

equations generally do not. 

There are presently no publicly available 

computer codes to implement the 

conditional moment approach, though some 

prototype codes exist and other are being 

developed. To implement the approach, one 
89 

must start by adopting a suitable closure 

approximation such as perturbation 

(Guadagnini and Neuman, 1999a-b, 2001; 

Ye et al., 2002; Lu et al., 2002) or an 

assumption of Gaussianity (Neuman et al., 

1999; Amir and Neuman, 2001; Wang et al., 

2002). Though perturbative closure 

approximations are not guaranteed to work 

for strongly heterogeneous media, they have 

proven to work remarkably well in twodimensional finite element analyses of 

superimposed mean uniform and convergent 

steady state (Guadagnini and Neuman, 

1999a-b) and transient (Ye et al., 2002) 

flows in saturated porous media and for 

similar type steady state flow in unsaturated 

media (Lu et al., 2002). A different 

perturbative approach to conditional 

moment analysis and its various applications 

can be found in the recent book of Zhang 

(2001). 

6.2.5 How To Interpret Traditional 

Deterministic Models 

Stochastically? 

A comparison of stochastic moment 

equations of flow and transport with 

traditional deterministic equation shows that 

the latter can be interpreted stochastically. 

This is seen upon "localizing" conditional 

mean equations that are otherwise nonlocal 

(integro-differential) in a way which renders 

them approximately differential 

(Appendices C and D). Localized 

conditional mean flow and transport 

equations can be solved with the aid of 

standard flow and transport codes or 

modifications thereof. While this is an 

important advantage, a major disadvantage 

of localized equations is that they do not 

yield information about predictive 

uncertainty. 

The prevailing tradition has been to model 

flow and transport by means of standard 

(local) deterministic equations. In 

Appendices C and D we examine this 

tradition from a stochastic viewpoint and 

comment on its validity and meaning. More 

specifically, we ask what is the meaning of 

hydrogeologic variables that enter into 

traditional deterministic groundwater flow 

and transport models, when the latter are 

applied to randomly heterogeneous media? 

What is the significance of parameters 

obtained by calibrating a deterministic flow 

model against randomly varying data? What 

do corresponding parameter estimation 

errors imply about predictive uncertainty? 

We show, in a rigorous manner, that 

applying traditional deterministic flow and 

transport models to randomly heterogeneous 

media is valid at best as an approximation, 

due to localization. The following are some 

important implications of interpreting 

traditional deterministic flow and transport 

models stochastically: (1) their parameters 

are inherently nonunique, regardless of how 

well or poorly posed an associated inverse 

problem may be; (2) these parameters differ 

from their optimum (geostatistical) estimates 

as obtained on the basis of field 

measurements; and (3) uncertainties in these 

parameters reflect uncertainties in predicted 

(conditional) mean system behavior, not 

uncertainties associated with actual system 

behavior. 

It follows that analyzing the effect of 

parameter uncertainty on system behavior 

and performance by means of traditional 

deterministic models generally 

underestimates (often severely) behavior and 

performance uncertainty. Therefore, the 

strategy in this report supports the use of 

traditional deterministic models only to the 

extent that they are interpreted and 

implemented in their proper stochastic 

context, as illustrated in Appendices C 

and D. 
90 

6.3 How to Formulate and Explore 

Alternative ConceptualMathematical Models? 

6.3.1 Where to Start? 

The qualitative stage of conceptualization 

should lead to several alternative hypotheses 

concerning the hydrogeologic makeup of a 

site, the prevailing flow and transport 

regimes, and the way in which these regimes 

are expected to evolve under future 

scenarios of interest. With each of these, one 

must associate at least one mathematical 

structure to form a corresponding 

conceptual-mathematical model. This 

structure should initially include a direct 

representation of site-scale hydrogeologic 

complexities, heterogeneities and 

driving-mechanisms in the form of explicit 

model elements. It should account directly 

for larger- or regional-scale features and 

factors through the appropriate assignment 

of initial conditions, boundary conditions, 

and source terms such as those that describe 

infiltration, recharge, discharge and leakage 

across aquitards. If features and factors that 

manifest themselves on subsite scales are 

deemed important for the modeling of sitescale phenomena, one must account for 

them explicitly or implicitly at this initial 

stage of conceptual-mathematical model 

building. This may result either in a 

deterministic or a stochastic model structure. 

Ignoring the influence of regional-, site- or 

subsite-scale features or factors at this initial 

stage, without having demonstrated that they 

are not relevant to the problem at hand, 

constitutes an oversimplification which 

renders the model a priori suspect of being 

invalid and unreliable. This is true even if 

the model can be made to reproduce 

observed system behavior, because there is 

reason to suspect that it would be unable to 

predict reliably behaviors under future 

scenarios, which differ in significant ways 

from that under which the observations have 

been made. 

The initial conceptual-mathematical model 

should identify the sources and nature of 

uncertainties that may impact its predictive 

power. This includes uncertainties in the 

very structure of the model, as well as 

uncertainties in its input parameters 

(material properties and forcings). Both 

deterministic and stochastic models are 

expected to have uncertain structural 

elements and parameter values. 

Having associated a deterministic and/or 

stochastic mathematical structure with each 

hypothesis that had been postulated for a 

site, the next step is to explore them 

quantitatively and graphically. The aim of 

this initial exploration is qualitative: To 

identify the potential of each conceptualmathematical model to explain observed 

system behavior and predict it under future 

scenarios. For this, it is useful to run the 

model with a plausible range of input 

parameters and study its output in light of 

the available site information. This may 

reveal ambiguities and uncertainties that 

may prompt a reexamination of the data 

and/or a revision of the model (which may, 

but need not, render the model more 

complex). It may also reveal differences 

between the various conceptualmathematical models that may help 

eliminate some of them and rank the rest in 

the order of their apparent ability to explain 

and predict system behavior. 

The initial exploration of alternative 

conceptual-mathematical models can often 

be done in the two-dimensional horizontal 

and/or vertical planes. This is so because 

two-dimensional analyses differ from their 

three-dimensional counterparts to a much 

lesser extent than do one-dimensional 
91 

analyses, yet are much less demanding than 

three-dimensional analyses. Twodimensional scoping analyses should yield 

contour and/or color representations of 

hydrogeologic parameter distributions; 

system states (hydraulic head, pressure, 

saturation, density, temperature, solute 

concentration); flowlines (streamlines, 

pathlines, streaklines); and vectors of fluid 

and solute fluxes and velocities (Figures 6-2 

– 6-7). If the conceptual-mathematical 

model involves transients, the analyses 

should consider variations of hydrogeologic 

variables with time. Some three-dimensional 

exploratory analyses may also be possible at 

this stage, including the preliminary 

quantification of fluid, solute and energy 

balances. Three-dimensional analyses may 

also be required if the two-dimensional 

results are insufficient to provide insight into 

relevant processes or fail to resolve key 

ambiguities and uncertainties. 

6.3.2 How to Proceed? 

Once an initial set of conceptualmathematical models have been formulated 

and explored in a preliminary fashion, once 

has the option of postulating and exploring 

less complex alternatives to each. There is 

no reason to contemplate more complex 

alternatives at this stage of the modeling 

process. Model simplification may entail a 

reduction in dimensionality (from three 

spatial dimensions to two or one; from 

transient to steady state), model size 

(smaller area and/or reduced depth); or 

details of various features, events and 

processes (fewer layers, faults or fracture 

zones; a less detailed representation of 

internal heterogeneity; single rather than 

dual continuum or discrete representation of 

a fractured rock; fewer discrete rainfall or 

infiltration events; a less detailed delineation 

of contaminant sources; constant rather than 

scale-dependent dispersion; equilibrium 

rather than kinetic sorption). Various types 

and levels of simplification can be 

entertained, leading to a number of 

simplified model structures for each initial 

(and more complex) conceptualmathematical model. 

As stated earlier, the strategy discourages 

the use of one-dimensional models unless a 

very strong and convincing hydrogeologic 

argument is made in their favor. This is so 

because system behavior and performance 

predicted by a one-dimensional model tend 

to be fundamentally different from those 

predicted by two- and three-dimensional 

models. 

Some of the details lost in the simplification 

process may have an important impact on 

the predictive capabilities of the model and 

some may have a lesser impact or none at 

all. This should be established by comparing 

each simplified model with its more 

complex initial version vis-a-vis their 

potentials to explain observed system 

behavior and predict it under anticipated 

future scenarios. For this, it is useful to run 

both models with a comparable range of 

plausible input parameters and compare 

their outputs in light of the available site 

information. To do so requires that any 

differences in scale between the input and 

output variables of the two models be 

resolved so as to render the models mutually 

compatible and comparable. Inability to do 

so in a scientifically defensible way should 

disqualify the simplified model. Other 

factors that should disqualify a simplified 

model are a much reduced ability (in 

comparison to the initial model) to explain 

observed system behavior that is considered 

important for the problem at hand, or an 

inability to predict future system behavior 

and/or performance that are qualitatively 

and quantitatively similar to those predicted 

by the initial model. 
92 

To render a simplified model compatible 

and comparable with its more complex 

(initial) alternative, it helps to derive it 

formally and objectively from the latter. 

This can be done through averaging or 

filtering in space, time and/or probability 

space. Appendix B shows in detail how the 

three-dimensional equations that govern 

flow in a water-table (unconfined) aquifer 

are formally averaged over the vertical, 

based on the well-known Dupuit 

assumptions, so as to yield the 

corresponding Boussinesq equation of 

horizontal flow. This formal analysis reveals 

with clarity under what conditions is one 

justified simplifying unconfined flow by 

restricting it to the two-dimensional 

horizontal plane. Appendix C explains and 

illustrates how the steady state stochastic 

groundwater flow equation, and 

corresponding hydrogeologic variables, are 

formally averaged over an ensemble of 

random hydraulic conductivity, head and 

flux realizations. Appendix D does the same 

for transport. 

Yet another important example is provided 

by Domenico and Robbins (1984) who 

demonstrate formally that characterizing the 

two- or three-dimensional spread of a solute 

with a one-dimensional transport model 

requires an apparent dispersivity, which 

increases as a function of distance. This is so 

even when the dimensionally correct (twoor three-dimensional) transport model is 

associated with a constant dispersivity. It 

follows that replacing the latter model by a 

one-dimensional version with constant 

dispersivity is an error, which does not 

necessarily lead to conservative results. 

A fifth example is provided by the 

theoretical and field study of Neuman and 

Witherspoon (1972) on multiple (leaky) 

aquifer systems. A pumping test conducted 

by these authors near Oxnard, California, 

has shown that when drawdowns in the 

pumped Oxnard aquifer are interpreted by 

means of the well-known and widely-used 

Theis model, the aquifer transmissivities 

appear to increase systematically with 

distance from the pumping well. The authors 

demonstrate theoretically that this apparent 

aquifer heterogeneity is, at least in part, an 

artifact of having simplified a complex 

multiaquifer flow regime by means of a 

model, which disregards interactions 

between the Oxnard aquifer and surrounding 

hydrogeologic units. Other examples 

abound, which collectively demonstrate that 

simplifying hydrogeologic flow and 

transport models on the basis of intuition or 

convenience, rather than on the basis of a 

more comprehensive formal analysis, may 

lead to serious errors of omission and 

commission. 
93 
94 

7 RENDERING A MODEL COMPATIBLE WITH DATA 

Once an acceptable set of conceptualmathematical models and corresponding 

simplifications have been identified, the next 

step is to render them compatible with 

available site characterization and 

monitoring data. Models that cannot be so 

rendered are unsuitable for quantitative 

representation of site hydrogeology and 

must be eliminated from further 

consideration. 

7.1 Why Worry About Compatibility 

With Data? 

A fundamental premise of this strategy is 

that a model must be operational, i.e., that it 

must be based on quantities which are either 

measurable, or can be derived from 

measurements, on well-defined space-time 

scales. The quantities we speak of include 

all parameters, initial and forcing terms 

(including sources and boundary 

conditions), and state variables that enter 

into the model (such as pressure and 

concentration). For example, if the flow 

model is based on Darcy's law, and this law 

is deemed relevant on a field length scale of 

1 m (equivalently, area scale of 1 m2

 or 

volume scale of 1 m3

), then the Darcy flux, 

permeability, and pressure or its gradient, 

must be amenable to direct or indirect 

measurement or evaluation on this same 

scale. The latter does not imply that it must 

be practical to perform such measurements 

within every cubic meter of the rock, only 

that it must be possible to do so in a 

sufficiently large sample of similar rock 

volumes to yield a statistically significant 

sample. Only then can a model be 

meaningfully tested against experiment 

and/or observations. 

A model that includes nonmeasurable 

quantities, or quantities that cannot be 

evaluated on the basis of measurements, can 

neither be applied directly to real data nor 

confirmed experimentally. Such a model is 

therefore non-operational and can neither be 

tested nor confirmed. 

7.2 How to Achieve Compatibility in 

Scale and Magnitude? 

To render a model compatible with site data, 

one starts by asking whether the scales of 

parameterization and resolution in the model 

are consistent with the support 

(measurement) scales of relevant data (site 

characterization data that define model 

structure and parameters, and site 

monitoring data that record observed state 

variables) and their statistics. If the answer 

is negative, one needs to either rescale the 

data and their statistics to fit the model, or to 

rescale the model to fit the available (as well 

as anticipated) data and their statistics. 

In many cases, the support scale of 

hydrogeologic data is smaller than the scale 

of parameterization and resolution of 

corresponding hydrogeologic models. In 

recent years, there has been a concerted 

effort on the part of many researchers to 

develop methods for the scaling up of 

hydrogeologic variables, most notably 

permeability. As explained by Neuman 

(1997a), the search has focused in large part 

on methods of upscaling that ascribe 

equivalent parameters to the grid blocks of 

numerical flow and transport models on the 

basis of smaller-scale random or nonrandom 

parameter values. One approach has been to 

postulate more-or-less ad hoc rules for 

upscaling based on numerically determined 

criteria of equivalence. Another approach 

has been to develop upscaling rules 

analytically through volume or stochastic 

averaging of smaller-scale values. Both sets 
95 

of rules are still evolving and it may be 

premature to recommend specific methods 

of upscaling for the purposes of this 

strategy. Instead, we mention two examples. 

Neuman and Depner (1988) have shown 

how one can upscale hydraulic conductivity 

data from single-hole hydraulic injection 

tests in fractured crystalline rocks at the 

Oracle site near Tucson, Arizona, as well as 

their spatial statistics. Their approach is 

based on a stochastically derived steady 

state formula due to Gelhar and Axness 

(1983) that applies to statistically 

homogeneous and infinite continua (a 

similar formula for finite three-dimensional 

blocks, under either steady state or transient 

conditions, has been developed by 

Tartakovsky and Neuman, 1998). The 

authors compare their upscaled values 

successfully with larger-scale hydraulic 

conductivities from cross-hole injection tests 

at the site. 

The above approach does not work at the 

Apache Leap Research Site (ALRS) in 

Arizona, where air permeability is not 

statistically homogeneous but exhibits the 

properties of a random fractal field (Chen et 

al., 2000). A method to rescale the 

permeability of such a field has recently 

been proposed by Di Federico et al. (1999) 

and used by Hyun et al. (2002) to interpret a 

pronounced permeability scale effect at the 

ALRS. 

A rule of thumb that often works reasonably 

well in practice is to upscale permeability by 

considering the geometric mean of smallerscale values. This is known to be exact 

(Neuman and Orr, 1993) for a statistically 

homogeneous Gaussian log permeability 

field in an infinite planar domain, but is less 

accurate for three-dimensional or finite 

domains (and entirely unsuitable for onedimensional domains, for which the 

harmonic mean is more appropriate). 

Despite this potential inaccuracy, the 

geometric mean is often used in conjunction 

with block kriging for the purpose of 

ascribing equivalent permeabilities to finite 

subdomains of a site on the basis of smaller 

scale measurements. 

Rubin et al. (1999) used a stochastic 

approach to develop analytical expressions 

for a grid-block effective dispersivity tensor 

due to subgrid-scale random variations in 

permeability. Their expressions can be 

conditioned on measurements of 

permeability. 

Efendiev et al. (2000) developed a strategy 

for incorporating subgrid effects in coarsescale numerical models of flow in 

heterogeneous media by upscaling a 

deterministic fine-grid permeability 

description and then solving the flow 

equation over the coarse grid to obtain 

coarse-scale velocities. A coarse-grid 

saturation equation is formed through a 

volume average of the fine-scale equations 

and includes terms involving both the 

average and fluctuating components of the 

velocity field. The terms involving the 

fluctuating components are subgrid effects 

that appear as length- and time-dependent 

dispersivities. A simplified model for the 

coarse-scale dispersivity, in terms of these 

subgrid velocity fluctuations, was proposed 

and a numerical scheme based on it was 

implemented in two dimensions. 

Some practitioners set the ratio between the 

longitudinal dispersivity assigned to a 

numerical grid block, and the length scale of 

this block, proportional to 10. This is based 

on a generalized scaling rule proposed by 

Neuman (1990) as a way of explaining an 

observed increase in apparent longitudinal 

dispersivity with the scale of observation in 

a large number of tracer studies, in diverse 
96 

hydrogeologic environments under varied 

circumstances, worldwide. It is not clear that 

the same scaling rule applies locally to 

specific sites under particular conditions of 

flow and transport. 

Numerical methods to upscale unsaturated 

soil hydraulic properties have been 

discussed by Bagtzoglou et al. (1994) and 

Desbarats (1995, 1999). Rockhold et al.

(1999) use geostatistical indicator simulation 

techniques for spatial interpolation of fieldmeasured water contents and porosities, and 

a conditional simulation method based on 

similar media scaling for estimating 

hydraulic properties from a set of scalemean parameters and the initial water 

content and porosity distributions. They then 

apply an upscaling algorithm to determine 

effective grid-scale parameters in threedimensions. 

As pointed out by Neuman (1997a), one 

major difficulty with upscaling is that it 

postulates local relationships between 

upscaled driving forces and fluxes (Darcy's 

and Fick's laws) when in fact these 

relationships are generally nonlocal. 

Another conceptual difficulty with upscaling 

is that it requires the a priori definition of a 

numerical grid and resolution scale. We add 

that improper upscaling may rendered the 

prior parameter estimates biased and their 

uncertainty mischaracterized. 

Upscaling can often be avoided if one 

adopts a stochastic approach to groundwater 

modeling. One choice under the stochastic 

option is to use a high-resolution conditional 

Monte Carlo method. In this method, the 

grid is fine in comparison to the spatial 

correlation scale of the hydrogeologic input 

variables, most importantly the permeability. 

Ideally, one would want the length scale of 

each grid cell to be of the same order as that 

of the input variable support scale. In 

practice, it is usually enough to set the 

length scale of each grid cell equal to or less 

than one fifth the spatial correlation scale of 

the input variable. Experience has shown 

that with such a grid, one is often able to 

reproduce random spatial fluctuations in the 

input variable with high fidelity. As the 

correlation scales of output variables (such 

as head, concentration and flux) are 

generally larger, this also allows resolving 

the latter with equal or higher fidelity. 

Another choice under the stochastic option 

is to employ conditional moment equations. 

Here scale-related problems are generally 

avoided because the input as well as the 

output variables represent ensemble 

averages in probability space rather than 

volume and/or time averages in real space. 

Though they are smoother (vary more 

slowly in space-time) than their random 

counterparts, these variables are 

nevertheless defined on the same space-time 

scales. Hence stochastic moment equations 

achieve smoothness and simplicity without 

any need to average or upscale 

hydrogeologic variables in space-time. By 

being smoother than their random 

counterparts, ensemble moments can in 

principle be resolved on grids that are 

coarser than those required for highresolution Monte Carlo simulation. 

Stochastic Monte Carlo and moment 

equation models honor site characterization 

data by being conditioned upon them. They 

are thus a priori compatible with site data 

both in scale and magnitude. Their output 

can be made similarly compatible with site 

monitoring data by conditioning the models 

a posteriori on observed behavior through 

appropriate inverse procedures. By doing so, 

the models may still honor the site 

characterization data and so remain 

conditional on both data types. 
97 

Site characterization data may be used to 

derive prior input parameters for a 

deterministic model through appropriate 

upscaling and estimation procedures. In this 

limited sense, the deterministic model can 

be made to honor the site characterization 

data and so being conditional upon them as 

well as compatible with them a priori in 

both scale and magnitude. To render the 

same model compatible with (conditional 

on) site monitoring data, it is almost always 

necessary to calibrate them a posteriori

against such data. This invariably modifies 

the model input parameters to generate 

posterior estimates, which often differ 

significantly from their prior counterparts. It 

is therefore generally not possible to render 

a deterministic model compatible in 

magnitude with both site characterization 

and monitoring data at the same time and to 

the same degree. The best one can do is to 

seek an optimum tradeoff between model fit 

(a measure of compatibility with the 

monitoring data) and parameter plausibility 

(a measure of compatibility with the 

characterization data) either by viewing the 

problem in a multiobjective context as 

proposed by Neuman (1973), or by 

incorporating the priors into a statistical 

statement of the inverse problem as 

proposed by Carrera and Neuman (1986a-b). 

Another (we think less desirable) possibility 

is to calibrate the model without considering 

prior parameter estimates (and, by 

implication, site characterization data on 

which such estimates are based) at all, as 

recommended by Hill (1998). More on these 

and other inverse modeling options later. 

7.3 How to Infer Prior Input 

Parameters and Statistics? 

7.3.1 What to Infer from Site 

Characterization Data and How? 

Since hydrogeologic medium properties and 

forcing functions are always uncertain, so 

are the model input variables. In other 

words, the input parameters of groundwater 

flow and transport models are inherently 

uncertain regardless of whether the model is 

deterministic or stochastic. The difference is 

that whereas in deterministic models the 

input parameters are viewed as imperfectly 

known deterministic quantities, in stochastic 

models they are viewed as correlated 

random fields or processes that may be 

perfectly or imperfectly known at discrete 

sampling locations in space-time. In both 

cases, imperfect knowledge is characterized 

by random errors that may, but often are not, 

considered to be mutually correlated. This 

renders the input parameters of deterministic 

models correlated or uncorrelated random 

variables, and those of stochastic models 

correlated random fields or processes 

conditional on either exact or random 

measurements, which may themselves be 

correlated or uncorrelated among 

themselves. Whereas the second moment of 

uncorrelated random variables is a diagonal 

matrix of their respective variances, that of 

correlated random variables is a full square 

symmetric and positive definite matrix of 

their respective covariances. 
98 

Estimating input parameters for a model 

(deterministic or stochastic) on the basis of 

incomplete and/or uncertain data is 

equivalent to inferring their mean values 

from these data. If the mean values are 

based on site characterization data, they 

represent prior parameter estimates. If they 

are based on site monitoring data, they 

represent posterior parameter estimates. In 

this section we consider only prior inference 

of model input parameters. 

If the inferred parameter estimates vary in 

space and/or time in a way which reflects 

similar variations in the underlying site 

characterization data, they are said to be 

conditional on these data. If the inferred 

mean values do not reflect such spatial 

and/or temporal variability, they are said to 

be unconditional. Unconditional inference 

may take place when the data are too few or 

too clustered to allow defining their 

variability in space-time. 

Assume that a set of site specific 

measurements are available, which represent 

the same hydrogeologic variable (say 

permeability or porosity) as a corresponding 

set of model input parameters. Such 

measurements are considered to be "hard." 

Ideally, the data would include probabilistic 

information about errors of measurement 

and test interpretation that suffer from a 

known amount (ideally zero) of statistical 

bias. If one has a statistically significant set 

of such data, one should be able to estimate 

a prior set of model input parameters on 

their basis. One should also be able to 

postulate a probabilistic model of prior 

parameter uncertainty based on statistics 

derived from these data. Such a probabilistic 

model is known to be of Type A. 

A key measure of parameter uncertainty is 

the second statistical moment, or variancecovariance, of their estimation errors. Since 

input variables into both deterministic and 

stochastic computational models are 

specified in terms of a discrete set of input 

parameters, the corresponding variancecovariance forms a matrix. In the 

conditional case, the off-diagonal covariance 

terms of the matrix may reflect spatial 

and/or temporal correlation between 

parameters of a given type (say permeability 

or porosity) as well as cross-correlations 

between parameters of different types (say 

permeability and porosity). In the 

unconditional case, there is no inferred 

space-time variability and the covariance 

terms represent at most cross-correlations 

between different types of parameters. 

Prior parameter estimates can be obtained 

from clustered data by means of common 

statistical methods, and from spatially 

distributed data by means of standard 

geostatistical techniques. This is true 

regardless of whether the parameters are 

intended for a deterministic or a stochastic 

model. The main difference is that in the 

deterministic case, there may be a need to 

upscale the parameters whereas in the 

stochastic case, this may not be necessary. 

The simplest and most practical form of 

geostatistical inference with upscaling is 

block kriging. While this may not always be 

the most accurate and sophisticated way to 

proceed, it is quite adequate for many 

purposes. This is especially true in situations 

where enough site monitoring data are 

available to later modify the parameter 

estimates through model calibration. 

To characterize prior estimation uncertainty 

one should, as a minimum, infer from the 

data a variance-covariance matrix of prior 

estimation errors. Only in rare circumstances 

would there be enough data to permit 

inferring from them higher statistical 

moments of these errors. 
99 

Standard geostatistical analysis consists of 

identifying the spatial autocovariance 

structure of each variable being analyzed (in 

terms of an autocovariance or variogram 

function) and providing a smooth (kriged) 

conditional estimate of this variable on a 

two- or three-dimensional grid, as well as 

the associated estimation (kriging) variance 

at each grid point. A more advanced analysis 

may also yield an autocovariance matrix for 

the estimate across the grid. Another level of 

sophistication may be achieved by 

identifying cross-covariance functions or 

cross-variograms for two or more variables, 

estimating them simultaneously by 

cokriging, and computing their co- and 

cross-covariances across the grid. Both 

kriging and cokriging may be used to 

estimate average values of the variables over 

finite blocks or subdomains of a two- or 

three-dimensional grid. 

7.3.2 How to Incorporate Soft Data? 

If there are insufficient hard data of a given 

hydrogeologic variable to conduct a 

meaningful statistical or geostatistical 

analysis, then the use of soft (qualitative) 

data coupled with indicator geostatistical 

analysis are recommended. This yields an 

uncertainty model for the prior parameters 

that is intermediate between Type A and 

Type B (defined in the next section). Soft or 

indirect information about the parameters 

may include (a) off-site measurements of the 

parameters proper (quite often on scales 

other than those corresponding to the 

intended scale of model resolution) and/or 

(b) surrogate measurements on site that are 

known to correlate with the parameters of 

interest (for example, porosities or 

geophysical signatures that correlate in 

known ways with permeabilities, water 

contents or fracture densities). Statistics 

derived from off-site data must be 

considered potentially biased (due to a lack 

of site-specific information about mean 

parameter values and incompatibility of 

geology and scale). The associated variance 

may be too small or too large, depending on 

the quantity and quality of such data. 

Statistics derived from surrogate data may 

suffer from poorly defined correlations and 

incompatibility of scale. 

Indicator geostatistics yields indicator 

variograms for various classes of the 

variable, a smooth (kriged) estimate of this 

variable on a two- or three-dimensional grid, 

and the probability that the variable is larger 

(or smaller) than specified at each grid point. 

Most geostatistical software packages 

include an indicator option and allow one to 

generate random realizations of the variable 

on a grid by means of indicator Monte Carlo 

simulation. 

An example of soft data use is that of 

pedotransfer functions, which allow one to 

estimate soil hydraulic characteristics on the 

basis of soil textural data. One such software 

package is Rosetta (www.ussl.ars.usda.gov), 

developed by Schaap at the U.S. Salinity 

Laboratory in Riverside, California. The 

Rosetta code predicts van Genuchten (1980) 

water retention parameters and saturated 

hydraulic conductivity on the basis of soil 

textural class, textural distribution, bulk 

density and one or two water retention data 

points. It follows a hierarchical approach 

that is based on five models, depending on 

the available data. The simplest model 

consists of a lookup table of average 

hydraulic parameters for each textural class 

such as sand, silty loam, clay loam, etc. The 

other models rely on neural network 

analyses (Schaap et al., 1999) of over 2,000 

soil samples for water retention, over 1,000 

samples for saturated hydraulic conductivity 

(Schaap and Leij, 1998), and over 200 

samples for unsaturated hydraulic 

conductivity (Schaap and Leij, 2000). 
100 

The samples represent a large number of 

agricultural and non-agricultural soils in 

temperate climate zones of the northern 

hemisphere, most from the U.S. and some 

from Europe. The authors warn the user that 

the application of Rosetta to other climate 

zones, in which pedogenic processes are 

different, could lead to inaccurate 

predictions. Included in the output are 

uncertainty estimates of the predicted 

hydraulic parameters, which can be used to 

assess the reliability of the predictions. 

We have used Rosetta in our analysis of data 

from the Maricopa Agricultural Center in 

Arizona as illustrated later in this report. 

Rossini et al. (1994) and Rovellini et al.

(1998) utilize the concept of facies as the 

basis for petroleum reservoir 

characterization. They treat facies as the 

elementary units of the reservoir, determine 

them through a log-based cluster analysis, 

and associate them with petrophysical 

properties from cores. In the case study of 

Rossini et al., the reservoir shows extreme 

variability, comprising all the transitional 

lithologies from sand to dolomite. Due to the 

limited continuity of its lithologic facies, 

their spatial distribution is difficult to 

identify. The authors started by identifying 

deterministically three hydraulically 

separated layers in a vertical cross-section, 

distinguished in part by their diverse original 

gas-oil and oil-water contacts. After 

developing a petrophysical criterion to 

distinguish between sandy and dolomitic 

facies, they established a frequency 

distribution of porosities, and a correlation 

between log permeabilities and porosities, 

for each facies on the basis of core data. 

Likewise, they developed horizontal and 

vertical indicator semivariograms of 

porosity for each facies. Next, the authors 

divided the three layers into 1,645,000 cells 

measuring 50 m horizontally and 0.5 m 

vertically. They then generated ten equally 

likely random images of the facies across 

this grid by conditional stochastic indicator 

simulation, and assigned random porosities 

and permeabilities to grid blocks within each 

facies. To simulate flow through each of the 

ten generated reservoirs, the authors 

superimposed a coarse grid over the original 

fine grid and assigned an upscaled porosity 

and permeability to each coarse grid block. 

Rovellini et al. (1998) also used a stochastic 

model to distribute facies and related 

petrophysical parameters (porosity, 

permeability and irreducible water 

saturation) within the volume of a reservoir. 

They then generated several equally 

probable realizations of the reservoir and ran 

it through a flow simulator. The realization 

providing the best history match was 

selected for production forecasts. 

7.3.3 How to Use Generic and 

Subjective Probabilities? 

If the available hard and soft data are not 

amenable to geostatistical analysis (due to 

insufficient information about their spatial 

location, an inappropriate spatial pattern, 

insufficient number and/or poor quality of 

data), an alternative is to rely on generic 

and/or subjective probabilities and statistics. 

Doing so is equivalent to postulating a Type 

B statistical model of prior parameter 

uncertainty. Such a model should always be 

suspected of suffering from an unknown 

amount of statistical and personal bias. 

Statistical bias is introduced due to lack of 

site-specific information about mean values 

of the parameters in question. A personal 

bias tends to manifest itself in the form of 

assigned uncertainty measures (most 

importantly bias and error variance) that are 

either too small or too large. The first is a 

manifestation of over-confidence in the 

model parameters, the second of unduly low 

confidence in their values. 
101 

Generic probabilities for unsaturated 

hydraulic properties associated with various 

soil textures are given by Carsel and Parish 

(1988) and Meyer et al. (1997). An example 

of generic probability distributions of soil 

hydraulic properties for sand by the latter 

authors is given in Figure 11-10. The same 

authors also describe a Bayesian procedure 

that makes it possible to update these 

probabilities, and reduce the corresponding 

uncertainty in hydraulic parameters, on the 

basis of site-specific data. 

We have used both sets of generic 

probabilities, and the Bayesian updating 

procedure of Meyer et al. (1997), in our 

analysis of data from the Maricopa 

Agricultural Center in Arizona as illustrated 

later in this report. 

7.4 How to Calibrate Traditional 

Deterministic Models?

7.4.1 Where to Start for what Purpose? 

The traditional approach to hydrogeologic 

modeling has been to postulate a 

deterministic model structure and treat its 

parameters as being imperfectly known. One 

would then derive prior estimates of these 

parameters from site characterization data (if 

available) and modify them so as to achieve 

an acceptable fit between model outputs and 

available monitoring data (if such exist). 

The process is known as model calibration, 

parameter estimation, history matching or 

inverse modeling. The parameters that yield 

the best match between observed and 

recorded system behavior form posterior 

estimates. It is common to consider the latter 

to be more suitable for predictive purposes 

than prior parameter estimates. 

Although this is not always done, it is 

generally advisable to start the model 

calibration process by postulating a prior

parameter uncertainty model. We saw how 

to do this and mentioned that it may lead to 

a Type A probability model when prior 

parameter uncertainty is characterized on the 

basis of a statistically significant set of site 

specific measurements that represent the 

parameter; a Type B model when the 

uncertainty is characterized by generic 

and/or subjective probabilities; and an 

intermediate type model when parameter 

uncertainty is inferred from indirect 

information about the parameters. We noted 

that a Type B model should always be 

suspected of an unknown amount of 

statistical and personal bias. A model that is 

intermediate between Type A and Type B 

must also be considered potentially biased, 

with variances that may be too small or too 

large, poorly defined correlations, and 

incompatibility of scale. 

We noted earlier that improper upscaling 

may render the prior parameter estimates 

biased and their uncertainty 

mischaracterized, regardless of what type 

probability model one adopts. It is generally 

believed that model calibration tends to 

reduce both the bias and the uncertainty in 

prior parameter estimates by insuring that it 

reproduces adequately observed system 

behavior. This should render a calibrated 

model more reliable as a predictor than an 

uncalibrated model. Appendix C points out 

that calibrating a traditional deterministic 

flow model against measured values of head 

and flux is tantamount to conditioning it on 

such measurements. As the model hydraulic 

parameters are by nature conditional, the 

very act of adding measured heads and 

fluxes to the database alter their values. This 

inherent nonuniqueness persists regardless 

of whether the inverse problem is well- or 

ill-posed. It explains why model parameter 

estimates tend to change every time one 

redefines the underlying database. 
102 

7.4.2 How to Calibrate? 

The last thirty years have seen major 

advances in the development of theories and 

algorithms for the estimation of 

deterministic model parameters. Many 

(though not all) of these theories and 

algorithms are "statistical" in that they 

include analyses of parameter estimation 

uncertainty. Such analyses typically accept, 

but do not necessarily require, information 

about prior parameter statistics as input. The 

output includes posterior statistics of 

parameter estimation errors, which tend to 

be less biased and smaller than the prior 

estimation errors. A recent summary and 

comparison of various statistical inverse 

methods for groundwater flow models has 

been published by Zimmerman et al. (1998). 

A detailed set of guidelines for the effective 

calibration of deterministic groundwater 

flow models has been prepared by Hill 

(1998). Among inverse codes that can be 

used to estimate the parameters of standard 

deterministic flow and transport models we 

mention the following: 

UCODE: This is a public domain universal 

inverse modeling tool developed by 

Poeter and Hill (1998) 

(www.mines.edu/igwmc/freeware/ucode/) 

at the Department of Geology and 

Geological Engineering of the Colorado 

School of Mines in Golden, and the U.S. 

Geological Survey in Lakewood, 

Colorado, respectively. The code is 

available for DOS, Windows and Unix 

operating systems. The code is universal 

in that it works with any model that has 

suitable numerical (ASCII or text only) 

input and output files. The code 

generates sensitivity coefficients and 

statistics for parameter estimates. It is 

included as an option in GMS (described 

earlier). 

PEST: This is a universal parameter 

estimation software developed by 

Doherty et al. (1994). The code is 

marketed by the International GroundWater Modeling Center 

(www.mines.edu/igwmc/) at the 

Colorado School of Mines in Golden, 

the Scientific Software Group 

(www.scisoftware.com) and other 

vendors. It uses text input-output files 

and provides estimation statistics 

including confidence intervals, 

covariance and correlation matrices 

between the parameters. A parallel 

version of the code can run 

simultaneously on several processors or 

on multiple machines across a PC 

network. PEST is an optional component 

of GMS. 

PEST has been used successfully by 

Vesselinov and Neuman (2001) and 

Vesselinov et al. (2001a-b) to interpret 

multi-rate single-hole and cross-hole 

pneumatic injection tests at the Apache 

Leap Research Site near Superior, 

Arizona. Our inverse interpretation of 

the cross-hole data yields tomographic 

images of air permeability and porosity 

variations in three-dimensional space. 

The latter represent geostatistical 

(kriged) parameter estimates based on 

pressure monitoring data collected 

during each cross-hole test. We coupled 

PEST with a three-dimensional finite 

volume flow simulator, FEHM 

(Zyvoloski et al., 1997), an automatic 

mesh generator, X3D (Trease et al., 

1996), and the geostatistical program 

GSTAT (Pebesma and Wesseling, 1998) 

mentioned earlier. Our computations 

were conducted in parallel on 32 

processors of the SGI Origin 2000 

supercomputer at The University of 

Arizona in Tucson. 
103 

ITOUGH2: ITOUGH2 (inverse TOUGH2) 

is an inverse code developed by Finsterle 

(1999a-c) specifically for the TOUGH2 

model, which simulates multiphase, 

multicomponent, non-isothermal flows 

in multidimensional fractured-porous 

media. The inverse code is available for 

a fee from the Earth Sciences Division 

of Lawrence Berkeley National 

Laboratory at esd.lbl.gov/ITOUGH2/. 

Any TOUGH2 input parameter can be 

estimated based on any observation for 

which a corresponding TOUGH2 output 

can be calculated. ITOUGH2 conducts 

sensitivity analyses and provides 

statistical information about estimation 

residuals, uncertainties, and the ability to 

discriminate among model alternatives. 

It also allows one to propagate parameter 

uncertainties so as to quantify prediction 

errors. Our group has used ITOUGH2 to 

help interpret infiltration and tracer 

experimental data from the Maricopa 

Agricultural Center, as discussed 

elsewhere in this report. 

HYDRUS-2D: This is a Windows-based 

software package developed by Simunek 

et al. (1999) at the U.S. Salinity 

Laboratory in Riverside, California 

(www.ussl.ars.usda.gov). The program 

simulates water, heat and solute 

movement in variably saturated media in 

a vertical plane, a horizontal plane, or a 

three-dimensional region with a vertical 

axis of symmetry. The package includes 

a mesh generator, a graphic interface, 

and an automatic parameter estimation 

module for the estimation of selected 

soil hydraulic and/or solute transport 

parameters. It is available for a fee from 

the International Ground-Water 

Modeling Center at the Colorado School 

of Mines in Golden 

(www.mines.edu/igwmc/). 

7.4.3 How to Select Model Fit Criteria? 

Differences between values of model 

simulated system states (most commonly 

hydraulic head, pressure or concentration) at 

discrete points in space-time, and observed 

values of these same state variables, are 

termed "residuals" in model calibration 

parlance. The residuals are considered to 

represent errors that are distributed 

randomly about the simulated model output. 

An underlying (often tacit) assumption is 

that the conceptual-mathematical model is 

exact and associated with an unknown set of 

"true" parameters. If these parameters were 

known, the model (structure plus 

parameters) would be exact. To the extent 

that it would produce nonzero residuals, this 

would be entirely due to errors in the 

monitoring record. These data errors (or 

their logarithmic transform, as is usually 

done with permeability and transmissivity) 

are typically taken to have zero mean and be 

normally distributed. Hence ideally (in the 

theoretical event that the true parameters 

were known), the residuals would exhibit a 

multivariate Gaussian distribution with zero 

mean (i.e., they would be unbiased) and a 

variance-covariance identical to that of the 

monitoring data. 

Model calibration is seen as the process of 

estimating the model true parameters 

without bias and as closely as possible. This 

is accomplished by defining an appropriate 

calibration criterion (or objective function) 

in terms of the residuals, and "optimizing" 

the parameters in a way which comes closest 

to satisfying this criterion (achieving the 

objective). The extent to which the criterion 

is satisfied (the objective achieved) becomes 

a measure of model fit. Theoretically, the 

optimized parameters are associated with 

estimation errors that have zero mean and 

minimum variance. 
104 

The most commonly used measure of model 

fit is the weighted sum of squared residuals. 

This is meaningful in the above statistical 

sense provided the monitoring data errors 

are mutually uncorrelated and each weight is 

inversely proportional to the corresponding 

error variance. Model calibration then 

reduces to a weighted least squares fit of the 

model to the data (or regression of the 

parameters on the data). Since groundwater 

model parameters are usually related to state 

variables in a nonlinear fashion, the 

weighted least squares fit is nonlinear. This 

is why model calibration must usually be 

done iteratively. 

If all the weights (and variances) are equal, 

the model fit criterion can be normalized to 

form a simple sum of squared residuals. 

Model calibration then reduces to an 

ordinary nonlinear least squares fitting 

process (or nonlinear regression). 

A less common but still widely used 

measure of model fit is the generalized sum 

of squared residuals. Whereas in the 

previous cases the weights formed a 

diagonal matrix proportional to the inverse 

of a diagonal matrix of observational error 

variances, here the weights form a full 

matrix proportional to the inverse of a 

(usually full) variance-covariance matrix. 

This is statistically meaningful in the 

previous sense when the errors are mutually 

correlated. Calibration now becomes a 

generalized nonlinear least squares or 

regression process. 

If some of the statistical parameters that 

define the observational errors are unknown, 

they may sometimes be estimated jointly 

with the other model parameters by the 

maximum likelihood method. This entails a 

likelihood function, which is the likelihood 

of the parameters given a set of (conditional 

on) observational data. The latter is the 

probability density of the (error corrupted 

and so random) data given (conditional on) 

the parameters. The objective would be to 

maximize the likelihood function. 

Since the data (or their log transform) are 

assumed to be normally distributed, the 

likelihood function is multivariate Gaussian. 

As such, it includes an exponential term 

whose negative logarithm is equal to half the 

weighted (by the inverse covariance matrix) 

sum of theoretical square residuals. 

Parameters are estimated by minimizing the 

negative logarithm of the likelihood 

function. In the special case where all 

statistical parameters are known, the 

negative log likelihood function reduces to 

the standard generalized least squares 

criterion. 

A maximum likelihood approach to model 

calibration, which incorporates information 

about prior parameter statistics into the 

statement of the inverse problem, was 

proposed by Carrera and Neuman (1986a-b). 

It yields a negative log likelihood criterion 

that includes two weighted square residual 

terms instead of one. The first is the usual 

generalized sum of squared differences 

between simulated and observed state 

variables. The second is a generalized sum 

of squared differences between posterior and 

prior parameter estimates. The 

corresponding weight matrix is proportional 

to the inverse covariance matrix of prior 

parameter estimation errors. The constant of 

proportionality, λ, is treated as a free 

statistical parameter that may be estimated 

jointly with the remaining parameters by 

maximum likelihood. The authors have done 

so by calibrating the model for various 

values of λ, plotting the negative log 

likelihood (S) against λ, and finding 

graphically the value of λ that minimizes S 

(Figure 7-1). Allowing λ to be initially 
105 

unspecified means that neither the 

covariance matrix of the observational data 

nor that of the prior parameters need be fully 

specified; it is enough to specify each of 

them up to its own constant of 

multiplication. A method to estimate these 

constants of multiplication on the basis of λ

has been described by the authors (see also 

Carrera et al., 1997). 

Including prior information in the 

calibration criterion allows Carrera and 

Neuman (1986a-b) to condition the 

parameter estimates not only on site 

monitoring (observational) data but also on 

site characterization data from which prior 

parameter estimates are usually derived. 

When both sets of data are considered to be 

statistically meaningful, the posterior 

parameter estimates are compatible with a 

wider array of measurements than they 

would be otherwise and are therefore better 

constrained (potentially rendering the model 

a better predictor).

Figure 7-1. Variation of log-likelihood criterion, sum of squared head residuals, 

and sum of squared log-transmissivity estimation errors with λ 

(after Carrera and Neuman, 1986b). With permission, AGU. 

7.4.4 What to Do with Insufficient Data? 

When either set of data is too small, 

clustered or otherwise unsuitable for a 

meaningful assessment of prior statistics, it 

may still be possible to come up with prior 

parameter estimates and weigh them

subjectively relative to each other, based on 

their perceived reliability. Similar weights 

may be assigned to site monitoring data. 

One can then adopt a calibration criterion 

J + λP equal to a weighted sum (J) of 

squared residuals (differences between 

simulated and observed state variables), plus 
106 

the product of λ with a weighted sum (P) of 

squared differences between posterior and 

prior parameter estimates. One can then 

calibrate the model for various choices of λ

and plot the weighted sum of squared 

residuals versus 1/λ to see how the latter 

affects the former. As shown by Neuman 

(1973), one would typically find that the 

sum of squared residuals is largest when 1/λ

= 0 and decreases more-or-less 

monotonically to an asymptote as 1/λ

increases (Figure 7-2). This is clear 

considering that 1/λ = 0 corresponds 

toλ = ∞ , which is equivalent to giving 

infinite weight to the prior parameter

estimates and forcing the posterior estimates 

to coincide with them. As 1/λ is allowed to 

increase, the weight placed on the prior 

parameters decreases and their constraining 

effect on the posterior estimates gradually 

diminishes. Eventually 1/λ becomes large 

enough to virtually eliminate any effect of 

the prior on the posterior estimates. The 

calibration is now unconstrained by site 

characterization data and relies entirely on 

site monitoring data. 

Figure 7-2. Residual versus parameter plausibility criteria 

(after Neuman, 1973). With permission, AGU. 

Rather than one calibration criterion one 

now has two, the weighted sum of squared 

residuals (J) and the weighted sum of 

squared differences between posterior and 

prior parameter estimates (P). These two 

criteria are mutually incompatible in that to 

satisfy one, it is necessary to sacrifice the 

other (to achieve the objective of 

minimizing one, the other must be allowed 

to remain arbitrarily large, as seen in Figure 

7-2). In most cases, the optimum solution to 

this multiobjective or Pareto problem is not 
107 

to minimize either objective but to find an 

acceptable tradeoff between them on the 

basis of subjective value judgment. The idea 

is discussed in detail by Neuman (1973). 

In the absence of sufficient or reliable site 

monitoring data, the model must rely 

entirely on prior parameter estimates and is 

only as good as are the latter. The model 

remains uncalibrated, untested and 

unconfirmed. 

7.5 How to Calibrate Stochastic Models?

7.5.1 How to Calibrate High-Resolution 

Monte Carlo Models?

High-resolution Monte Carlo modeling 

entails generating multiple random 

realizations of flow and transport parameters 

on a fine grid, solving standard deterministic 

flow and transport equations with these 

parameters on the same grid, averaging the 

results and analyzing them statistically. 

Conditional Monte Carlo simulations honor 

measured values of the parameters at 

discrete points in space. They however do 

not guarantee that the simulations produce 

results which correspond to actual records of 

system behavior. To render Monte Carlo 

simulations compatible with observational 

data, the approach has been to simply 

discard realizations that do not conform to 

such data. This is the approach that underlies 

the GLUE (Generalized Likelihood 

Uncertainty Estimation) strategy of Beven 

and Binley (1992; see also Beven, 2000) and 

the Bayesian approach of Gaganis and Smith 

(2001). The problem with it is that it usually 

requires a tremendously large number of 

simulations without ever guaranteeing full 

compatibility with the observations. 

Sahuiquillo et al. (1992), G\mez-Hern<ndez 

et al. (1997) and Capilla et al. (1997) have 

developed an inverse method that allows one 

to condition individual Monte Carlo 

realizations not only on measured 

hydrogeologic input parameters (notably 

transmissivity) but also on observed system 

states (notably hydraulic head). In the 

version described by G\mez-Hern<ndez et 

al. (1997), first a random transmissivity field 

is generated conditional only on 

measurements of the same (via pumping 

tests). Next, the transmissivity field (and 

possibly the boundary conditions) is 

modified iteratively, without destroying the 

spatial pattern of its variability, until head 

simulations based upon it come close to 

honoring the observational data, by yielding 

a sufficiently small sum of weighted squared 

residuals. The transmissivity is 

parameterized geostatistically in terms of 

discrete values associated with actual and 

fictitious "master" (analogous to de 

Marsily's "pilot", 1978) points of 

measurement. Values associated with real 

measurements are taken to represent known 

transmissivities. Values associated with pilot 

points are treated as hydrogeologic 

parameters to be estimated by the inverse 

algorithm. Both sets of values are projected 

onto a computational grid via kriging to 

yield parameter estimates for the entire 

model. An option is provided whereby 

values associated with real measurement 

points can also be modified during the 

parameter estimation process within their 

respective uncertainty ranges. 

To allow variations in boundary conditions, 

head and flux are also parameterized in 

terms of values at a few master locations on 

the boundary, which are estimated jointly 

with the transmissivities. In the first 

iteration, the flow equation is linearized to 

allow direct solution of the optimization 

problem by quadratic programming. In 

subsequent iterations, the equation is solved 

in its nonlinear form and optimization 

proceeds using a gradient method. 
108 

Capilla et al. (1997) have applied the 

method to two-dimensional steady state flow 

using a finite difference numerical scheme. 

In their synthetic examples, two or three 

iterations have sufficed to yield excellent 

results even for cases where natural log 

transmissivity variance is as high as 10. 

G\mez-Hern<ndez et al. (1997) stress that 

their aim is not to obtain "the optimal 

transmissivity field" but any transmissivity 

field that satisfies the conditions of honoring 

both transmissivity and head data while 

displaying a plausible pattern of spatial 

variability. It is clear that their optimal field 

is only one among many equally likely 

random fields that satisfy the same 

conditions. 

In principle, one should be able to generate a 

large number of random fields that satisfy 

the above conditions by combining the 

proposed inverse method with the Monte 

Carlo approach. Though this might require 

considerable computer time and resources, it 

would most probably be much more 

efficient than merely discarding standard 

conditional Monte Carlo simulations that do 

not honor the observational data. 

7.5.2 How to Calibrate Moment 

Equation Models?

Inverse algorithms based on the moment 

equation approach are presently under 

development by our group. The algorithms 

yield directly (without Monte Carlo 

simulation) optimized unbiased predictors of 

groundwater flow and transport variables for 

randomly heterogeneous hydrogeologic 

environments, under the action of uncertain 

source and boundary terms. They also yield 

the variance-covariance of associated 

estimation and prediction errors. The 

algorithms do so while accounting explicitly 

for the multiscale (e.g., fractal) nature of 

hydrogeologic heterogeneity. They allow 

optimum use of field information through 

joint conditioning on measured values of 

hydraulic parameters, hydraulic heads and 

solute concentrations. 

An inverse algorithm recently developed by 

us for steady state flow in saturated media 

(Hernandez et al., 2002) is based on leading 

finite element approximations of exact first 

and second conditional moment equations. It 

parameterizes log permeability 

geostatistically in terms of discrete values 

associated with actual and fictitious "pilot" 

(de Marsily, 1978) points of measurement. 

Values associated with real measurements 

are taken to represent prior log permeability 

estimates. If and when enough such values 

are available, they are projected onto a 

computational grid via kriging to yield prior 

parameter estimates for the model. Values 

associated with pilot points are treated as 

unknown parameters to be estimated by the 

inverse algorithm. An option is provided 

whereby values associated with real 

measurement points can also be modified 

during the parameter estimation process. 

This is useful when they are uncertain 

and/or suspected of statistical bias. 

Free statistical parameters may be 

introduced such as the variance or 

correlation scale of the underlying random 

field and/or its kriging statistics. These are 

then estimated jointly with the 

hydrogeologic parameters by using the 

nonlinear maximum likelihood approach of 

Carrera and Neuman (1986a-b). The 

negative log likelihood function is 

minimized by means of the Gauss-Newton 

nonlinear optimization method (Carrera et 

al., 1997). 
109 

7.6 How to Gauge Quality of Model Fit?

To gauge the extent to which calibration has 

helped improve the quality of model fit, it is 

helpful to compare values of the calibration 

criterion prior to and after calibration. The 

first is computed with prior parameter

estimates, and the second with posterior 

estimates. A significant improvement would 

manifest itself in a large relative move 

toward the objective (say of maximizing a 

likelihood function or minimizing a negative 

log likelihood criterion). 

Other useful ways to gauge the quality of 

model fit, before and after calibration, 

include comparing contours, time records, 

peaks and trends exhibited by observed and 

simulated state variables. Systematic 

differences between observed and simulated 

behaviors are often an indication of 

systematic errors (biases) in the underlying 

model structure and/or parameters. 

Calibration should help eliminate systematic 

errors in the parameters. It follows that the 

persistence of such errors in a properly 

calibrated model may be taken as an 

indication of bias in model structure 

(provided the observational data are 

considered to be reliable). This may call for 

a reevaluation and revision or abandonment 

of the underlying conceptual-mathematical 

model. 

It is also very important to perform an 

analysis of residuals with the aim of 

checking how closely they satisfy the 

assumptions on which the statistical inverse 

approach is predicated. We recall that these 

assumptions include multivariate Gaussian 

distribution, zero mean, and a variancecovariance similar to that of the 

observational data. Several statistical tests 

and graphical procedures to perform such 

checks are discussed by Hill (1998).

The following example illustrates how 

examining the quality of calibrated model fit 

may help one disqualify the central 

hydrogeologic hypothesis that underpins the 

model. 

Figure 7-3. Whiteshell Research Area (after Ophori et al., 1996).
110 

7.6.1 Example: Whiteshell Research 

Area

The Whiteshell Research Area (WRA) near 

Pinawa, Manitoba (Figure 7-3), and the 

underground research laboratory (URL) 

contained within it, have been used as a 

generic field site to investigate the potential 

suitability of crystalline rocks in the 

Canadian shield to host a repository for 

spent nuclear fuel. 

As part of his service on the Scientific 

Review Group (SRG) for the Canadian 

concept of geologic spent fuel disposal, 

Neuman (Scientific Review Group, 1996) 

has examined a recently revised conceptual 

and mathematical model that has been 

proposed for the area by Atomic Energy for 

Canada Limited (AECL). This model rests 

on a number of fundamental premises, 

which state that 

1. Large intact bodies of sparsely fractured 

rock (SFR) are found at the site; 

2. Uniformly low permeabilities and 

hydraulic gradients in the SFRs render 

transport through them predominantly 

diffusive; 

3. There is a consistent decrease in 

permeability with depth in all rock units 

at the site, including fracture zones (FZs) 

of relatively high permeability; 

4. Vertical and horizontal hydraulic 

communication between permeable rock 

units, including FZs, is limited; 

5. Ambient flow in all rock units takes 

place at steady state. 

The review revealed that none of these 

simplifying assumptions are supported, and 

most appear to be contradicted, by the 

available data. A reinterpretation of these 

data by Neuman has led him, and the SRG 

(1996), to a fundamentally different 

conceptual-mathematical model of site 

hydrogeology that is considerably more 

complex.

 

Figure 7-4. Selected logs of boreholes WD3 (left) and WG4 (right) (after Stevenson et al., 1996). 
111 

Figure 7-5. Landsat lineaments in WRA and outline of model finite element grid 

with embedded fracture zones (after Ophori et al., 1996). 
112 

 

 

Figure 7-6. Equivalent fresh water head profiles in boreholes WD3 (upper left), 

WG4 (upper right), WA1 (lower left), and WG1 (lower right) 

(after Stevenson et al., 1995, and personal communication). 

The SRG review has found the available 

data to show and suggest the following: 

1. Boreholes intersect intervals with up to 

100+ m of SFR (Figure 7-4), which have 

been interpreted in the original model as 

thicknesses of laterally extensive SFR 

bodies. There is however compelling 

evidence (below) that numerous 

permeable fracture zones (FZ) and 

moderately fractured rock (MFR) bodies 

form a hydraulically interconnected 

network which envelopes less permeable 

SFR bodies on many if not all sides. 

2. Whereas MFR dominates at shallow 

depths, it is sometimes found down to 

depths of 1,000 m (Figure 7-4). 

3. 500+ lineaments and FZs have been 

identified on the surface, but relatively 

few have been encountered in boreholes 

or included in the model (Figures 7-4–7-5). 

4. Only small differences have been found 

between equivalent fresh water head 

values in high-permeability zones at all 

depths across the region (Figure 7-6). 

This suggests that such zones form an

interconnected regional network which 

allows rapid hydraulic communication 

and equilibration in all directions, 

thereby allowing flow within the 

network to be at a near-steady state. 

5. Heads in high-permeability zones vary 

more in a horizontal direction (between 

boreholes) than in a vertical direction 

(within boreholes; Figure 7-6). This 

suggests that flow in the regional highpermeability network is predominantly 

horizontal. 

6. Steep local hydraulic gradients prevail 

between high-permeability zones and

SFR interiors (Figure 7-6). This suggests 
113 

that embedded within the regional highpermeability network are lowpermeability SFR blocks from which 

water leaks into the surrounding network 

under steep gradients. Indeed, field 

salinity experiments (below) confirm 

such advective leakage. A plausible 

interpretation is that past heads were 

higher than those of today, possibly due 

to glacial loading. As the corresponding 

load relaxed, heads in the highpermeability network dissipated rapidly 

toward a new dynamic equilibrium 

(steady state) while heads within the 

low-permeability SFR blocks are still 

dissipating under a transient regime. 

Salts have been flushed from shallower 

parts of the network, but not yet fully 

from the SFR blocks. 

7. The latter explains why salinity 

increases regionally with depth, and is 

locally higher within SFR blocks than 

within more permeable adjacent rock 

units. 

8. Gascoyne et al. (1996) drilled three 

boreholes, up to 100 m in length, into the 

SFR at a depth 420 m (Figure 7-7). They 

packed off six sections which contained 

no apparent fractures, filled them with 

deionized water without pressurizing, 

and observed a continuous increase in 

salinity throughout the period 1992-

1996. Core leaching tests and 

calculations indicated that ionic 

diffusion alone would give a salinity 

increase orders of magnitude lower than 

those actually observed. Gascoyne et al.

therefore concluded that solute transport 

within the SFR, under sufficiently large 

hydraulic gradients, is predominantly 

advective rather than diffusive. As very 

steep hydraulic gradients between SFR 

interiors and adjacent high-permeability 

rocks have been recorded throughout the 

WRA (Figure 7-7), there appears to be 

no justification for the simplifying 

assumption that radionuclide transport 

through SFR blocks would be 

predominantly diffusive. This 

assumption is evidently nonconservative.
114 

 

Figure 7-7. URL borehole leaching test setup and results (after Gascoyne et al., 1996). 

The numerical model used for performance 

assessment additionally assumed a 

downward trend in SFR permeability for 

which there is no evidence; disregarded the 

documented three-order variability in SFR 

permeabilities at depths below 200 m;

underrated MFR permeabilities by many 

orders of magnitude at depths below 100 m;

neglected spatial variability in FZ 

permeabilities; and set the latter 1-3 orders 

lower than is their mean at depths below 400 

m. The groundwater flow and transport 

model was calibrated in the manner depicted 

in Figure 7-8. Though quantitatively the 

agreement between computed and observed 

equivalent fresh water heads was said by the 

modelers to be satisfactory, qualitatively the 

agreement is poor. This is clearly seen in 

Figure 7-9 where computed heads increase

near-monotonically with depth while 

measures heads are much lower in highpermeability rock units than in lowpermeability zones. The most probable 

reason for this mismatch is a conceptualmathematical framework that fails to capture 

important complexities of hydrogeology at 

the URL site.
115 

Figure 7-8. Flow chart of AECL model calibration procedure (after Ophori et al., 1996). 
116 

Figure 7-9. Comparison between computed and recorded equivalent fresh water head profiles 

(after Ophori et al., 1996). Modified by Neuman (Scientific Review Group, 1996). 

7.7 How to Quantify Posterior 

Parameter Uncertainty?

The statistical approach to deterministic 

model calibration allows one to quantify the 

uncertainty associated with posterior 

parameter estimates. One measure of 

posterior parameter estimation errors is the 

Cramer-Rao lower bound on their 

covariance matrix. This lower bound is 

given by the inverse Fisher information 

matrix (Carrera and Neuman, 1996a-b; 

Carrera et al., 1997). Each term in the Fisher 

information matrix is one half the ensemble 

average of the second derivative of the 

negative log likelihood criterion with respect 

to a pair of parameters. It thus measures the 

average rate at which model sensitivity to 

one parameter is affected by changes in 

another parameter. From it, one can obtain a 

corresponding lower bound on the 

correlation matrix of the estimates. Ideally, 

the parameter estimates should be 

uncorrelated. 

The Fisher information matrix is usually 

approximated by linearizing the relationship 

between system states and parameters when 

the latter are optimal. It is then expressed as 

a function of the calibration weights (inverse 

covariance matrices of observed and prior 

input data) and a Jacobian matrix, whose 

terms represent sensitivities of system states 

to the parameters. It is thus clear that an 

analysis of parameter uncertainty includes in 

it a sensitivity analysis of the model and its 

output. 

A large variance associated with a given 

parameter indicates a high level of 

uncertainty in its estimate. A high degree of 

correlation between estimates implies that
117 

they are linearly related in a statistical sense. 

This means that they cannot be estimated 

individually with the available data, and 

should either be lumped into one parameter 

or estimated anew when more data become 

available. 

Linearization implies that if the prior 

(observation and parameter) errors are 

multivariate Gaussian (as one generally 

assumes), so are the posterior parameter (or 

log parameter) estimation errors. This allows 

one to associate them with linear confidence 

intervals, which can be viewed as 

corresponding error bounds. 

More accurate measures of uncertainty, 

which compensate to some extent for 

nonlinearity, are discussed by Vecchia and 

Cooley (1987). 

We illustrate these and related concepts later 

in the report in connection with our inverse 

analysis of experimental data from the 

Maricopa Agricultural Center. Another 

illustration related to inverse analysis of 

pneumatic pressure data from the Apache 

Leap Research Site is provided by 

Vesselinov et al. (2001a-b). 
118 
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8 QUANTITATIVE COMPARISON AND RANKING OF ALTERNATIVE MODELS 

Alternative conceptual-mathematical models 

that have been made compatible with the 

available database must now be compared 

among themselves to see how they rank on a 

relative scale of quality and complexity. 

Models deemed to rank low are iteratively 

reevaluated, revised or discarded. Those 

retained are ranked again on the basis of 

likelihood based model discrimination 

criteria and optionally eliminated from 

further consideration at this final stage of 

model testing and selection. 

8.1 How to Establish Comparative 

Measures of Model Quality and 

Complexity? 

Alternative conceptual-mathematical models 

that have been successfully calibrated can be 

compared on the basis of quality criteria 

such as model fit and posterior parameter 

uncertainty. Examples include: 

Likelihood or negative log likelihood of 

posterior parameter estimates, given the 

available data; 

Inverse variance-covariance weighted 

sums of squared residuals, which 

measure the overall lack of fit between 

model generated and observed system 

states; 

Various statistics of the residuals, which 

measure how closely they satisfy the 

assumptions of Gaussianity, 

unbiasedness, and similarity of their 

variance-covariance to that of the 

observational data; 

Consistent measures of systematic 

differences between contours, time 

records, peaks and trends exhibited by 

observed and simulated state variables; 

Invariant properties of the Cramer-Rao 

lower bound on the posterior covariance 

matrix of parameter estimation errors 

such as— 

Trace, also known as A-optimality 

criterion (which measures the 

cumulative variance of all estimation 

errors); 

Determinant, also known as Doptimality criterion (which provides 

an overall measure of the estimation 

error variance-covariance matrix); 

Eigenvectors (the components of 

which represent relative 

contributions by the various 

parameter estimates, implying that 

parameters associated with a single 

eigenvector have uncorrelated 

estimation errors, and those 

associated with multiple 

eigenvectors have cross-correlated 

estimation errors, rendering them 

less amenable to discrimination); and 

Eigenvalues (parameters associated 

with eigenvectors that have small 

eigenvalues being less uncertain than 

those associated with eigenvectors 

that have large eigenvalues); the 

maximum absolute eigenvalue is 

known as E-optimality criterion; 

Linear confidence intervals; 

Invariant properties of the Jacobian 

sensitivity matrix (of system states to the 

parameters); etc. 

Models that have not been formally 

compared against monitoring data have 

indeterminate quality in terms of their ability 

to reproduce real system behavior. 

Calibrated models rank higher than 

uncalibrated models in terms of parameter 

quality criteria because (posterior) estimates 

tend to be less biased and uncertain than 

prior estimates, on which uncalibrated 

models are based. 
120 

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If all models remain uncalibrated, their main 

comparative quality criteria are those that 

pertain to prior parameter uncertainty (if 

monitoring data are available against which 

model predictions can be compared, it is 

advisable to further calibrate the models 

against such data). These include 

The variance-covariance of prior 

parameter estimation errors; 

Various invariant properties thereof; and 

Corresponding linear confidence 

intervals. 

All models can be compared on the basis of 

model complexity criteria such as 

Dimensionality; 

Number of simulated processes; 

Degree of nonlinearity; 

Number of parameters; and 

Number or size of grid cells. 

8.2 How to Compare and Select 

Alternative Models Based on 

Quality and Complexity? 

8.2.1 Iterative Model Testing and 

Selection 

Alternative conceptual-mathematical models 

that have been made compatible with the 

available database must be compared among 

themselves to see how they rank on a 

relative scale of quality and complexity. 

Models deemed to rank low are iteratively 

reevaluated, revised or eliminated from 

further consideration. 

In particular, models that do not appear to 

meet acceptable criteria of quality (by 

exhibiting excessive uncertainty or bias) or 

complexity (by being either too refined or 

too crude) are reevaluated, revised or 

discarded. This process of testing, 

comparison, revision and screening is 

repeated iteratively till no further 

improvements appear necessary or feasible. 

A set of models is identified which represent 

acceptable tradeoffs between quality and 

complexity. The retained models are gauged 

and ranked in terms of likelihood based 

model discrimination criteria, which account 

for both model fit and complexity. Models 

that rank low on the list can be eliminated 

from further consideration at this final stage 

of model testing and selection. 

Examples of comparison and selection 

between models based on various criteria are 

included in our analysis of data from the 

Maricopa Agricultural Center discussed later 

in this report. 

8.2.2 Multiobjective Approach 

To help identify a set of models which 

represent acceptable tradeoffs between 

quality and complexity, it is helpful to 

employ a multiobjective approach. 

Consider a single hydrogeologic conceptual 

model for a site, expressed in terms of 

several conceptual-mathematical models 

that differ in only one complexity measure, 

such as their number of input parameters (I). 

Suppose that the models have been 

calibrated without the benefit of prior 

parameter estimates and one decides to 

gauge their quality in terms of a single 

criterion, such as negative log likelihood (S). 

Upon plotting S versus I, one typically finds 

that S decreases as I increases, in a manner 

similar to Figure 7-2. In other words, model 

quality improves as the model becomes 

more complex, and deteriorates as the model 

becomes simpler. If the objective is to have 

a high-quality but simple model, then there 

is no obvious way to achieve this objective. 

Instead, one must decide subjectively how 

much quality one is willing to sacrifice for 
121 

the sake of simplicity (or vice versa), and 

based on this select from the graph a range 

of S and I values that represent an acceptable 

tradeoff between the two. This "feasible" 

range constitutes a "Pareto" solution to the 

dual-objective optimization (or Pareto) 

problem. 

It is often advisable to gauge model quality 

in terms of more than a single criterion. For 

example, the calibration objectives for a 

transport model may include minimizing 

three criteria of model fit: negative log 

likelihood (S), a measure (T) of how well the 

model reproduces the observed peak arrival 

time of a plume at a monitoring point, and a 

measure (P) of how accurately it reproduces 

the magnitude of the peak. The objectives 

may also include minimizing two model 

complexity criteria: Its dimensionality (D) 

and number of parameters (I). The overall 

objective thus consists of minimizing 5 

incommensurate criteria simultaneously. Not 

all of them, however, can be achieved 

without sacrificing some other objectives. 

The only solution is to find an acceptable 

tradeoff between them. 

To do so, one must consider a range of 

models with different values of D (say 3, 2 

and 1 corresponding to three-, two- and onedimensional models, respectively) and I (say 

100, 10 and 1 to represent highly 

heterogeneous, less heterogeneous and 

uniform media, respectively). One must then 

calibrate these models and determine their 

corresponding values of the model fit 

parameters S, T and P. Next, one must 

determine an optimum tradeoff between 

these various model choices. 

Consider yet another case where each of 

several hydrogeologic conceptual models for 

a site is represented by a number of 

conceptual-mathematical models, the latter 

differing from each other in one or more 

measures of complexity (number of 

parameters, dimensionality, etc.). Upon 

considering one or more measures of quality 

for these models, one again faces a 

multiobjective problem that may entail 

many more potential choices than those 

obtained in the previous case. 

Pareto problems with more than two 

objectives are difficult to solve by 

inspection, as we saw is possible in the dualobjective case. The alternative is to employ 

multiobjective programming techniques. 

Several methods have been developed for 

the articulation and solution of 

multiobjective decision problems which 

include the ELECTRE technique, 

compromise programming, multiattribute 

utility theory, surrogate worth trade-off, 

cooperative game theory, metagame 

analysis, Q-analysis, multiobjective simplex, 

and other (Duckstein et al., 1991). The idea 

is to make it possible for the analyst to 

investigate a range of feasible solutions to 

the problem and solve it by determining 

(subjectively) an acceptable tradeoff 

between all of its conflicting and 

incommensurate objectives. A multiobjective programming method developed 

by Neuman and Krzystofowicz (1977) is 

especially well suited for this purpose. It 

provides a computationally efficient way to 

generate binary graphs of tradeoff between 

pairs of conflicting objectives, and to 

determine an optimum tradeoff between all 

objectives interactively, in an iterative 

manner that is guaranteed to converge under 

commonly encountered conditions. 

8.2.3 Combined Measures of Model Fit 

and Complexity 

The multiobjective approach helps one 

identify a set of models that represent 

acceptable tradeoffs between quality and 

complexity. To further discriminate between 
122 

these models, it is useful to gauge and rank 

them in terms of likelihood based model 

discrimination criteria, which account for 

both model fit and complexity. This applies 

only to calibrated models because 

uncalibrated models have unknown 

likelihood. 

Consider several models that differ in their 

number of input parameters (I). The quality 

of their model fit is measured in terms of 

negative log likelihood (S). This common 

but special case has been studied intensively 

in connection with models of random time 

series. In particular, several criteria have 

been developed by Akaike (1974, 1977), 

Hannan (1980) and Kashyap (1982) to help 

one discriminate between such models. In 

recent years, the same criteria have been 

used with increasing frequency to 

discriminate between alternative 

geostatistical (Samper and Neuman, 1989ab; Chen et al., 2000) and dynamic (Carrera 

and Neuman, 1986a-b; Finsterle and 

Faybishenko, 1999) groundwater models. 

The model discrimination criteria are based 

on a combination of the model fit criterion 

(S) and the complexity criterion (I). By 

ranking the models in ascending order of 

their associated discrimination criteria, one 

favors the least complex among those that 

perform equally well in terms of model fit. 

Among models of equal complexity, those 

that fit the monitoring data better are 

favored. 

The model discrimination criterion of 

Kashyap contains yet another term that 

gauges the information content of the 

available data. It thus allows one to consider 

models of growing complexity as the data 

base improves in quantity and quality. 

Stated otherwise, the criterion recognizes 

that when the data base is limited and/or of 

poor quality, one has little justification for 

selecting an elaborate model with numerous 

parameters. Instead, one should then prefer a 

simpler model with fewer parameters, which 

nevertheless reflects adequately the 

underlying hydrogeologic structure of the 

system, and the corresponding flow and 

transport regime. Kashyap's criterion favors 

that model which, among all alternatives 

considered, is least probable (or likely, in an 

average sense) to be incorrect. Stated 

otherwise, the criterion minimizes the 

average probability of selecting the wrong 

model among a set of alternatives. 

Other model discrimination criteria can also 

be used, such as those employed by Hills 

and Wierenga (1994) in connection with 

INTRAVAL studies of experiments at the 

Las Cruces Trench Site in New Mexico. 

The model testing and selection process 

ends with the final ranking of models on the 

basis of likelihood based discrimination 

criteria, and the optional elimination of low 

ranking models from further consideration. 

We do so for alternative models of the 

Maricopa experimental site, as illustrated 

later in the report. 

8.2.4 Example: Chalk River Monzonitic 

Block 

The above quantitative method of 

discriminating between conceptualmathematical models has been applied 

successfully by Carrera et al. (1990) to the 

inverse modeling of transient flow during 

pumping tests conducted in fractured 

crystalline rock at Chalk River National 

Laboratories in Canada (Figure 8-1). They 

used formal model discrimination criteria to 

rank four different zonation patterns of 

hydraulic parameters, labeled Models 1 - 4 

in Figure 8-2, to find that Model 4 is favored 

and indeed agrees extremely well with 

measurements.
123 

Figure 8-1. Block diagrams (a) and section (b) showing fracture zones and boreholes 

at Chalk River (after Carrera et al., 1990). With permission, Swets & Zeitlinger Publishers. 

Figure 8-2. Four alternative parameter zonation patterns used in Chalk River model 

(after Carrera et al., 1990). With permission, Swets & Zeitlinger Publishers. 
124 

8.2.5 Example: Los Monegros Basin, 

Spain

Samper-Calvete and Garcia-Vera (1999) 

applied similar criteria to the inverse

modeling of groundwater flow in the 

semiarid basin of Los Monegros in Spain. 

They compared isotropic and anisotropic 

representations of two aquifers in the basin, 

and found that anisotropy is very important 

in one of them but not so much in the other. 

8.2.6 Example: Apache Leap Research 

Site

Chen et al. (2000) compared three different 

conceptual-geostatistical models of spatial 

variability for single-hole natural log air 

permeability data collected on a nominal 

scale of 1 m at the Apache Leap Research 

Site in Arizona: (1) a fractal power-law 

variogram model, (2) an exponential 

variogram model with linear drift, 

and (3) an exponential variogram model 

with quadratic drift. Upon applying the 

maximum likelihood cross-validation 

method of Samper and Neuman (1989a-b) to 

these data, they found that whereas all three 

models yield comparable kriged estimates of 

log permeability in three dimensions, they 

yield very different estimation variances. 

This is illustrated along a vertical crosssection for models 1 and 3 in Figure 8-3. 

The negative log likelihood model fit 

criterion, NLL, in Table 8-1 is very similar 

for all three models, implying that all three 

fit the data equally well. Yet all four model 

discrimination criteria, AIC, MAIC, HIC 

and KIC (due to Kashyap), consistently rank 

model 1 first and model 3 last. The reason is 

that, even though all three models fit the 

data reasonably well, model 1 is the most 

parsimonious and model 3 the least 

parsimonious among the three, hence the 

former is the choice of preference. 

Table 8-1. Model fit and discrimination criteria for three geostatistical models 

applied to log air permeabilities from the ALRS. 
125 

 

Figure 8-3. Gray scale images of kriged log air permeabilities and their variances 

along a vertical cross-section at the ALRS using two geostatistical models.
126 
127 

9 RENDERING PREDICTIONS AND ASSESSING PREDICTIVE UNCERTAINTY 

USING MULTIPLE MODELS 

Alternative models that have been tested, 

compared, ranked and retained for further 

consideration are considered to constitute 

potentially valid simulators and predictors of 

site hydrogeology. There is no valid basis to 

prefer one of these models over another, and 

one must therefore use them in tandem. This 

means applying all of them to any given 

scenario and averaging their results to form 

a single "best" prediction. The prediction is 

best only in a relative and not in an absolute 

sense: It is conditional on the available 

models and data. This implies that better 

predictions could potentially be produced 

with other (as yet unidentified) models or 

data. The next task is to render a joint 

prediction by means of all the retained 

models and to identify their joint conditional 

predictive uncertainty. 

9.1 How to Assess Predictive 

Uncertainty for a Single 

Deterministic Model?

9.1.1 Monte Carlo Method

Monte Carlo simulation is by far the most 

common method of assessing the predictive 

uncertainty of a model. The method is 

conceptually straight forward and has the 

advantage of applying to a very broad range 

of both linear and nonlinear flow and 

transport models. Given information about 

the statistical properties of the model input 

parameters (including those that represent 

forcing terms) or their log transformed 

values, one generates numerous equally 

likely realizations of the parameters (or log 

parameters, as in the case of permeability 

and transmissivity). If the model has been 

calibrated, the input statistics correspond to 

the posterior parameter estimates. If the 

model is uncalibrated, they correspond to 

the prior estimates. In both cases, the 

estimates are generally associated with a full 

variance-covariance matrix. Upon assuming 

that the estimation errors are multivariate 

Gaussian, one has all the information one 

needs to produce random realizations of the 

parameters. 

One way to produce random realizations of 

a multivariate Gaussian vector (of 

parameters) is by Cholesky decomposition 

of its variance-covariance matrix. This LUfactorization method, introduced by Clifton 

and Neuman (1982), is well suited for full 

matrices and highly efficient when the latter 

are not too large. Its efficiency stems from 

the fact that only one decomposition is 

required for an unlimited number of 

realizations. 

Once a parameter realization has been 

generated, one runs the model under a given 

scenario to generate a corresponding random 

prediction of system states. Upon averaging 

the predictions over all realizations, one 

obtains a sample mean prediction. As the 

number of such Monte Carlo runs increases, 

one hopes that the sample mean converges 

to a theoretical ensemble mean. While such 

convergence can neither be insured nor 

verified in most cases, it is important to at 

least plot a representative measure of the 

sample mean versus the number of runs to 

verify that it has reached a stable value. The 

stable sample mean is then taken to 

constitute the best prediction the model can 

produce. 

In addition to computing the sample mean, 

one typically also computes the sample 

variance of the predictions and generates a 

corresponding frequency histogram. Both 

converge more slowly to their theoretical 
128 

ensemble equivalents than does the sample 

mean. It is therefore very important to verify 

that both have stabilized with the number of 

runs. Since most groundwater models are 

nonlinear, one need not expect the histogram

of the predictions to resemble either a 

normal or a log normal distribution. 

9.1.2 Linearization

Linearizing the flow and transport models 

allows one to establish approximate error 

bounds, or confidence limits, on model 

predictions on the basis of linear regression 

theory. Linearization expresses each

observed value of a state variable as a linear 

combination of unknown "true" parameters 

plus a random error, ε . As the model is 

considered exact and its parameters true, the 

latter implies that ε represents an error of 

observation. For regression (calibration) 

purposes, ε is replaced by a corresponding 

residual and an optimum (posterior) estimate 

of the unknown parameters is obtained by 

minimizing a suitably weighted sum of 

squared residuals. This sum thus becomes a 

measure of the error (variance) with which 

observed state variables are represented by 

the calibrated model. The variance is

generally larger than that of ε because the 

model is based on estimated rather than true 

parameters. 

The weighted sum of squared residuals does 

not, however, measure the error with which 

state variables that have not yet been 

observed (or have not been considered for 

calibration) would be predicted by the 

model. This is so because future predictions 

would be done with parameter estimates that 

are themselves corrupted by an estimation 

error. Linearization expresses each predicted 

value of a state variable as a linear 

combination of unknown parameters plus a 

random error, ε . Each unknown parameter 

is the sum of a known posterior parameter 

estimate and an associated estimation error. 

Hence the error in predicted system states is 

the sum of two errors: one due to replacing 

the true parameters by their posterior 

(calibrated) estimates (proportional to the 

weighted sum of residuals), and the other 

due to rendering a prediction with the aid of

uncertain posterior parameters (the 

uncertainty being characterized by the 

estimation variance-covariance of the 

parameters).

Hill (1998) explains how one can establish 

linearized confidence limits, or error 

bounds, for predictions rendered by models 

calibrated using nonlinear least squares with 

a diagonal weight matrix. 

9.2 Why Consider Stochastic Models? 

Appendix C explains that assessing the 

uncertainty of standard deterministic model 

parameter estimates may allow one to

quantify the uncertainty in computed 

conditional mean system state predictors 

(mean head, concentration, flux etc.). This, 

however, says nothing about how actual 

(random and unknown) system states 

fluctuate about their predictors; such

information is provided only by stochastic 

models that treat hydraulic parameters as 

random fields. 

Appendix C concludes that traditional 

methods of assessing uncertainty in the 

output of a calibrated deterministic model, 

of the kind just described, fail to account for 

the stochastic component of prediction error. 

To do so, one must adopt a stochastic model. 
129 

9.3 How to Assess Predictive 

Uncertainty for a Single Stochastic 

Model?

9.3.1 High-Resolution Monte Carlo 

Method

High-resolution Monte Carlo simulation is 

by far the most common method to assess 

predictive uncertainty stochastically. It 

entails generating multiple random 

realizations of flow and transport parameters 

on a fine grid, solving standard deterministic 

flow and transport equations with these 

parameters on the same grid under scenarios 

of interest, averaging the results and 

analyzing them statistically. If based on 

prior parameter estimates, the Monte Carlo 

results honor measured values of the 

parameters at discrete points in space. If 

based additionally on the inverse method of 

Sahuiquillo et al. (1992) and G\mezHern<ndez et al. (1997), they also honor 

observed values of state variables. 

In all other respects, the analysis of 

predictive uncertainty by the stochastic 

Monte Carlo method is similar to that 

described previously in connection with 

deterministic models. 

9.3.2 Moment Equation Method

We mentioned earlier that forward and 

inverse algorithms based on the moment 

equation approach are presently under 

development by our group. The algorithms 

yield directly (without Monte Carlo 

simulation) optimized unbiased predictors of 

groundwater flow and transport variables for 

randomly heterogeneous hydrogeologic 

environments, under the action of uncertain 

source and boundary terms. They also yield 

the variance-covariance of associated 

estimation and prediction errors. The 

algorithms do so while accounting explicitly 

for the multiscale (e.g., fractal) nature of 

hydrogeologic heterogeneity. They allow 

optimum use of field information through 

joint conditioning on measured values of 

hydraulic parameters, hydraulic heads and 

solute concentrations. To see how this works 

for steady state flow in saturated media the 

reader is referred to Hernandez et al. (2002). 

9.4 How to Render Joint Predictions 

with a Set of Alternative 

Deterministic Models?

Alternative models that have been tested, 

compared, ranked and retained for further 

consideration are considered to constitute 

potentially valid simulators and predictors of 

site hydrogeology. There is no valid basis to 

prefer one of these models over another, and 

one must therefore use them in tandem. This 

raises the question how to render an 

optimum joint prediction using multiple 

models, and how to assess their joint 

predictive uncertainty? 

We propose to do so by adopting a novel 

Maximum Likelihood Bayesian Model 

Averaging (MLBMA) approach recently 

described by Neuman (2002). His paper is 

included in Appendix E. As mentioned 

earlier, the approach differs in fundamental 

ways from previous methods. It relies on 

probabilistic maximum likelihood (ML) 

concepts to (a) calibrate each model against 

observed space-time variations in system 

states (pressure, water content, 

concentration), considering prior 

information about relevant soil properties; 

(b) eliminate models that cannot be so 

calibrated with acceptable fidelity; 

(c) predict future system behavior or 

performance measures (travel times, 

concentrations, mass rates), and assess 

corresponding predictive uncertainty, using 

each model; and (d) average the results 

using posterior model probabilities as 
130 

)

weights. MLBMA supports the principle of 

parsimony in that among models that have 

similar predictive capabilities, it favors those 

having fewer parameters and being therefore 

simpler. 

9.4.1 How to Render Joint Predictions? 

Let M = ( 1,..., M MK be the set of all 

retained models and let be a quantity one 

wants to predict. The optimum joint 

prediction of by means of all the models, 

given a discrete set of data D, is given by its 

conditional ensemble mean (expectation) 

∆

∆

E ⎡∆⎣ ⎤ D⎦ . According to MLBMA, the latter 

can be approximated by (Appendix E) 

( )

1

ˆ , ,

K

l l l

l

E E M p M =

∆ ∆⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ∑ ⎣ ⎦ D  θ D D

where ˆ , , E Ml l ⎡∆

⎣ ⎤⎦ θ D is the conditional 

mean associated with model Ml and its 

maximum likelihood parameter estimates 

ˆθ l , and p M( l D) is the posterior 

(conditional) probability of this model. The 

term ˆ , , E Ml l ⎡∆

⎣ ⎤⎦ θ D represents simply a 

prediction of ∆ generated by model Ml

after it has been calibrated against data D

using the maximum likelihood method. The 

posterior model probability p M( l D) can 

be calculated as a byproduct of this 

calibration using equation (6) in Appendix 

E. The joint prediction E ⎡∆⎣ ⎤ D⎦ is thus seen 

to be a weighted average of predictions 

obtained using individual calibrated models 

where the weights are the posterior (post 

calibration) probabilities of these models. 

9.4.2 How to Assess Joint Predictive 

Uncertainty? 

The joint posterior probability of predictions 

∆ is given approximately by 

( ) ( ) ( )

1

ˆ,θ ,

K

k k k

k

p p M p M =∆ ∆ D  ∑ D D

where ( ) ˆ , , k k p M∆ θ D is the posterior 

probability of ∆ associated with model Mk

after it has been calibrated against data D

using the maximum likelihood method. The 

joint predictive probability is thus seen to be 

a weighted average of predictive 

probabilities associated with individual 

calibrated models where the weights are the 

posterior (post calibration) probabilities of 

these models. 

The joint posterior variance of predictions is 

given approximately by 

Var⎡∆ ⎤ ⎣ D⎦ 

{ }

2

1

ˆ ˆ , , , ,

K

k k k k

l

Var M E M =

⎡∆ +⎤ ⎡∆ ⎤ ∑ ⎣ ⎦ ⎣ ⎦ θ θ D D

( ) 2

l ⋅ − p M E ⎡∆ ⎤ D ⎣ D⎦

Methods to evaluate ˆθ k by calibrating a 

deterministic model Mk against 

hydrogeologic data D, which may include 

prior information about the parameters, are 

described by Carrera and Neuman (1986a-b) 

and Carrera et al. (1997). The same can be 

done with a stochastic model based on 

moment equations in a manner patterned 

after Hernandez et al. (2002). These 

methods also yield an approximate 

covariance matrix for the estimation errors 

of ˆθ k . Upon considering the parameter 

estimation errors of a calibrated 

deterministic model Mk to be Gaussian or 

log Gaussian, one easily determines 
131 

( ˆ , , k k p M∆ θ D) by Monte Carlo simulation 

of through random perturbation of the 

parameters. The simulation also yields 

corresponding approximations 

∆

ˆ , , E Mk k

⎡ ⎤ ∆

⎣ ⎦ θ D of , E Mk ⎡∆ ⎤ ⎣ ⎦ D , and 

ˆ , , Var Mk k

⎡ ⎤ ∆

⎣ ⎦ θ D of , Var Mk ⎡∆ ⎤ ⎣ ⎦ D . 

If Mk is a stochastic model based on 

moment equations, it can yield 

ˆ , , E Mk k ⎡ ⎤ ∆

⎣ ⎦ θ D and ˆ , , Var Mk k ⎡ ⎤ ∆⎣ ⎦ θ D

directly without Monte Carlo simulation 

(Hernandez et al. 2002). Such a model does 

not, however, yield ( ) ˆ , , k k p M∆ θ D . 

9.5 How to Evaluate Hydrogeologic 

Performance? 

Hydrogeologic performance of a site, under 

a given scenario, is evaluated in terms of

corresponding performance measures. These 

in turn are computed on the basis of system

states (such as travel time and concentration)

that are predicted using groundwater flow 

and transport models. This implies that the 

same models and methods that one uses to 

predict system states, and to assess their 

predictive uncertainty, can also be used to 

predict performance measures and their 

uncertainty. No special methodology is 

needed for this purpose.

9.6 Conditional Nature of 

Hydrogeologic Predictions 

We recall that predictions of hydrogeologic 

system behavior and performance are 

conditional on the available models and 

data. Many other predictions are potentially 

(and most probably) possible with other 

models and data that are equally valid and 

telling, perhaps more so. The uncertainty 

measures one associates with these 

predictions are likewise conditional. They 

therefore provide at best a lower bound on 

actual predictive uncertainty, which may 

(and most probably is) larger to an unknown 

degree.
132 
133 

10 MODEL CONFIRMATION AND ASSESSMENT OF DATA NEEDS 

Once a set of predictive models have been 

developed and their predictive uncertainty 

assessed, one should try to confirm the 

models through comparison with 

observational data not previously used for 

calibration. This provides a suitable 

opportunity to evaluate the adequacy of the 

available database and make a decision 

whether or not to expand it, and how. 

10.1 How to Confirm Calibrated Models? 

In rare situations where enough 

observational data are available to allow 

using only some of them for model 

calibration, it may be possible to use the 

remaining data for purposes of model 

verification or confirmation. For this to be 

valid, the data used for confirmation must 

represent a different mode of system

behavior that those used for calibration. 

Otherwise, the confirmation data are merely 

an extension of the calibration data and 

being able to reproduce them does not 

constitute a meaningful test of model 

predictive capability. As both the model and 

the data are generally uncertain, model 

confirmation must be cast in a suitable 

probabilistic and/or stochastic framework.

Figure 10-1. Drawdown versus log time in response to pumping from well FS-11-2. 

Continuous lines represent predictions and their 95% confidence intervals; dots represent 

measurements (after Carrera et al., 1990). With permission, Swets & Zeitlinger Publishers. 
134 

10.1.1 Example: Chalk River Monzonitic 

Block

In section 8.2.4 we discussed briefly inverse 

modeling by Carrera et al. (1990) of 

transient flow during pumping tests 

conducted in fractured crystalline rock at 

Chalk River National Laboratories in 

Canada (Figure 8-1). The authors used 

formal model discrimination criteria to rank 

four different zonation patterns of hydraulic 

parameters, labeled Models 1 - 4 in Figure 

8-2, to find that Model 4 is favored and 

indeed agrees extremely well with 

measurements taken while water was 

withdrawn from well FS-10. They then 

checked the extent to which the same model 

reproduces drawdowns during another 

pumping test when water is withdrawn from 

well FS-11-2. As shown in Figure 10-1, the 

predicted drawdown (dots) is within the 

95% prediction confidence interval in some 

of the monitoring intervals but not in others. 

10.2 How to Assess Additional Data 

Needs

10.2.1 Role of Sensitivity Analyses 

To help evaluate what if any additional data 

might be worth collecting so as to materially 

reduce model uncertainty (by further 

constraining the range of alternative 

structures and parameters), one may conduct 

a sensitivity analysis to indicate what system 

behavior appears to be most sensitive to 

which parameters at what locations. The 

next step is to consider performing 

additional site characterization where 

existing parameter estimates are least certain 

and the model is relatively insensitive to 

their values, and monitoring system 

behavior where it is most sensitive to model 

parameters while prediction errors appear to 

be relatively large and consequential. This 

may indicate the type and quantity of 

additional site data that might materially 

enhance model reliability and credibility. 

10.2.2 How to Decide When To Stop 

Collecting Data? 

The question how much and what kind of 

data are enough for model development and 

evaluation is one of economics and policy, 

not of hydrogeologic analysis or modeling. 

It is therefore the responsibility of managers 

and decision-makers to answer, not 

hydrogeologists or modelers. The strategy in 

this report is designed to help one address 

the question by how much would additional 

site characterization and monitoring improve 

the prediction of performance measures 

derived from the hydrogeologic analysis. In 

other words, the strategy addresses the 

worth of data in terms of their contributions 

to the potential enhancement of model 

reliability and credibility, not in terms of 

their marginal cost-benefit. Its aim is to 

allow managers and decision-makers to 

make informed decisions about the time, 

manpower and budget that they deem worth 

allocating to these activities in light of such 

enhancement under existing administrative, 

budgetary and policy constraints.
135 

11 CASE STUDIES 

The case studies in this chapter illustrate 

selected components of the proposed 

strategy. These include the use of public, 

generic and site data in the 

conceptualization, characterization, and 

modeling of flow and transport, 

geostatistical analysis of hard and soft data, 

model calibration, assessment of prior and 

posterior parameter uncertainty, and model 

discrimination. 

11.1 Maricopa Agricultural Center

This case study is based on the doctoral 

dissertation of Wang (2002). 

11.1.1 Introduction 

Field infiltration and tracer experiments 

were conducted at the Maricopa Agricultural 

Center (MAC) near Phoenix, Arizona 

(Young et al., 1999; Yao et al., 2003). We 

(Wang, 2002) have conducted a study to 

address the questions 

• How accurately can site hydrogeology, 

with emphasis on the vadose zone, be 

conceptualized on the basis of public 

data and generic databases? 

• How reliably can flow and transport 

within the vadose zone at a site be 

reproduced by means of simple models, 

and how simple can such models be? 

• What methods can be used to quantify 

related uncertainties? 

• What is the role of site data in bringing 

about a reduction in uncertainty? 

To address these questions, we employ 

various methods to postulate, compare and 

rank alternative conceptual-mathematical 

models of unsaturated flow and transport 

during selected infiltration and tracer 

experiments at the MAC site. Our models 

include one- and two-dimensional flow and 

transport in a uniform soil, a soil consisting 

of uniform layers, and a stratified soil 

having laterally varying properties. 

Characterization of the soil as a uniform 

medium with a relatively deep regional 

water table is based on information obtained 

from public sources. Characterization of the 

soil as a layered medium with a relatively 

shallow water table is based on site data. 

At the time of this study, only a handful of 

soil hydraulic properties were available for 

the MAC site. We use various ways to 

estimate these properties indirectly on the 

basis of soil pedologic data. We start by 

ascribing uniform soil hydraulic properties 

to each layer on the basis of soil type using 

mean values of three generic databases. We 

then ascribe variable soil hydraulic 

properties to individual soil samples based 

on soil type and bulk density using 

regression and neural network pedotransfer 

models. Upon treating these variable 

hydraulic property estimates as 

measurements, we incorporate them in the 

original databases to obtain Bayesian 

updates of their mean values and variances 

of the corresponding estimates. Next, we 

conduct a geostatistical analysis of soil 

pedologic and hydraulic properties which 

provides support for a layered conceptual 

model with relatively large-scale lateral 

variability in each layer. 

In the next stage of our analysis, we use the 

above conceptual-mathematical models and 

hydraulic parameter estimates to compare 

simulated and observed water contents 

during one of the infiltration experiments at 

the MAC. We show that in order to 

reproduce observed behavior, it is necessary 

to further modify the hydraulic parameter 

estimates through inverse modeling. We 
136 

compare and rank the various conceptualmathematical models and parameter 

estimates using likelihood-based model 

discrimination criteria and confirm our 

choice of best model by successfully 

simulating flow during an earlier infiltration 

experiment. Finally, we adopt onedimensional inverse estimates of soil 

hydraulic parameters based on one 

infiltration experiment for one-dimensional 

solute transport modeling of bromide tracer 

data collected during an earlier experiment. 

An advection-dispersion model with linear 

mass transfer to and from a zone of 

immobile water, or anion exclusion, is 

employed in forward and inverse modes. 

Our study illustrates how a combination of 

methods and data sets can be used 

sequentially and in tandem to improve one's 

understanding of unsaturated flow and 

transport conditions at a site. 

50 m

50 m

0 5 10 15 20 25 30 35 40 45 50 55 60

0

5

10

15

20

25

30

35

40

45

50

55

60

Solute samplers

Shallow N. Probe

boreholes

Monitoring Islands

Deep N. Probe 

boreholes

Figure 11-1. Location of monitoring equipment (Scale in meters). 

11.1.2 Brief Description of Experiments 1–3 

Three infiltration and tracer experiments 

were conducted at the MAC by Young et al.

(1999). During these experiments, water was 

uniformly applied to a 50 50 area 

using a drip irrigation system at a controlled 

rate. The area was covered by a 60 

× 2m

× 60 

thick Hypalon® pond liner to minimize 

evaporation. Monitoring took place along a 

trench, within two islands, and in boreholes 

(Figure 11-1). The backfilled trench was 65 

m long, 1.5 m deep and contained various 

instrument clusters at 10 m intervals. The 

2

m
137 

islands consisted of highway culverts, 1.5 m 

in diameter, placed vertically in the soil 

down to 3 m depth. They were instrumented 

to measure soil water tension, soil water 

content and tracer concentration. Borehole 

monitoring equipment consisted of neutron 

probe access tubes, tensiometers and 

solution samplers. Nine neutron probe 

boreholes were drilled to 15 m depth and 

several others to 3 m depth. Solution 

samplers near the nine deep boreholes were 

installed at depths of 3, 5, and 10 m. 

Solution samplers in the monitoring islands 

were installed at depths from 0.5 to 3.0 m at 

0.5 m intervals. 

We focus on the first and last of the three 

experiments. Experiment 1 lasted 93 days 

starting April 28 and ending July 30, 1997. 

Water was applied at an average rate of 1.85 

cm/day to the field for 24 days, with a 

bromide tracer added for the first 15 days at 

a mean concentration of 31.6 ppm. The 

water application period was followed by a 

redistribution period of 69 days. Experiment 

3 lasted more than 200 days; in this study, 

we use data from the first 56 days starting 

April 24 and ending June 19, 2001. Water 

was applied at an average rate of 2.66 

cm/day for 28 days and redistribution 

measured for the following 28 days. In 

experiment 3, monitoring was limited to the 

nine deep boreholes, yet data were collected 

more frequently, using fewer field 

monitoring devices (neutron probes and 

deep tensiometers). 

11.1.3 Summary of Relevant Public 

Information 

To develop a conceptual hydrogeologic 

model for the MAC, we started by collecting 

public data on geology, soils, aquifers, 

driller's logs, well logs, climate and weather, 

evaporation and evapotranspiration as well 

as related information from monographs and 

research papers. These data were obtained 

from various sources including United 

States Geological Survey (USGS), Arizona 

Department of Water Resources (ADWR), 

Arizona Department of Agriculture, Arizona 

Land Department, Arizona Land Resource 

Information System (ALRIS), and Arizona 

Meteorological Network (AZMET). They 

led to the following conceptual model of the 

experimental site. 

The site lies within the upper fill of the 

Maricopa-Stanfield alluvial basin (Andersen 

et al., 1992). The upper fill, which has the 

hydrostratigraphic designation of upper 

alluvial unit (UAU), consists mainly of 

gravel, sand and silt. It has a relatively 

uniform thickness of between 200 and 300 

feet, becoming thinner as one nears 

mountain fronts. The unit was deposited by 

ancestral surface rivers during the final 

stages of basin development. It comprises 

alluvial channel, terrace, floodplain and 

alluvial fan deposits. 

Historically, the unconfined UAU has been 

the most productive aquifer of the basin. 

However, groundwater pumping has greatly 

reduced its saturated thickness. Currently, 

groundwater levels are recovering 

throughout the basin. The rise in water 

levels results from an overall reduction in 

groundwater withdrawals, recharge 

following floods, and the introduction and 

widespread use of Central Arizona Project 

(CAP) water. 

The State Soil Geographic Database 

(STATSGO), developed by the Natural 

Resource Conservation Service and 

accessible through the ALRIS website, 

includes a soil map for the State of Arizona. 

According to this map, soils in the MAC 

area consist of sandy loam. 
138 

Abundant well logs provide public 

information about the geology of the MAC 

area. Figure 11-2 shows a schematic log for 

a well about 200 meters southwest of the 

MAC. Figure 11-3 depicts our attempt to 

correlate several well logs along a west-east 

section near the site. Data about well 

locations, pumping rates and historical water 

levels are available for more than 1000 wells 

across the basin from the ADWR. Figure 

11-4 shows water table elevations within the 

UAU aquifer across part of the basin in 

November 1988 (left) and 1993 (right), 

respectively. For reasons mentioned earlier, 

water levels in the MAC area have increased 

by about 50 feet between 1988 and 1993.

Figure 11-2. Well log of D-04-04 20 BCD near the MAC (depth and thickness in feet). 
139 

Figure 11-3. An East-West cross-section near the MAC 

(depth and thickness in feet; see legend in Figure 2). 

 

Figure 11-4. Elevation contours (in feet) of water level in the UAU aquifer 

in Nov. 1988 (left) and Nov. 1993 (right). 
140 

Dec-93 Dec-94 Dec-95 Dec-96 Dec-97 Dec-98

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Pre

cipitatio

n (

m)

Year-month

0.00

0.05

0.10

0.15

0.20

0.25

0.30

ET

0 (

m) 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

(c)

(b)

(a)

irrigation(

m)

Figure 11-5. Monthly irrigation (a), reference evapotranspiration ET0 (b), and precipitation (c) 

in MAC area. 

Meteorological data from the AZMET 

include air temperature, relative humidity, 

solar radiation, precipitation, soil 

temperatures and wind velocity. Information 

about agricultural irrigation is available 

from the MAC Administration and ADWR. 

Figure 11-5 shows how monthly irrigation, 

reference evapotranspiration (ET0) and 

precipitation varied at the MAC during 

1993-1998. ET0 is evapotranspiration from
141 

3–6 inches tall cool season grass that 

completely covers the ground and is

supplied with adequate water. Average 

annual irrigation during 1989 - 1998 over an 

area of 430 acres was 1.09 m, average 

annual reference evapotranspiration during 

1990 - 1998 was 1.98 m, and average annual 

precipitation during 1988 - 1998 was 0.18 

m. Information about the distribution of 

irrigated land is available from the ALRIS 

website. 

11.1.4 Conceptual-Mathematical Modeling 

Based on Public Data 

The available public data suggest the 

following conceptual hydrogeologic model 

for the MAC experimental site: 

• Soils at the experimental site consist of 

sandy loam down to a depth of about 16 

m. 

• The UAU aquifer is unconfined with a 

regional water table locally at a depth of 

22 m.

• At the site, groundwater within the UAU 

flows from northeast to southwest under 

a gradient of about 0.3 %. 

• Local recharge of the UAU is due to 

irrigation and precipitation; floods are a 

source of recharge elsewhere in the 

basin. At the covered experimental site, 

recharge is entirely due to applied 

irrigation. 

• Local discharge from the UAU is due to 

pumping. 

• Loss of water from the unsaturated zone 

is due to evapotranspiration, which is 

suppressed at the covered experimental 

site. 

To simulate flow in the unsaturated zone, we 

adopt the following constitutive 

relationships between soil hydraulic 

variables (Van Genuchten, 1980) 

( )

1

1

m

r

n

s r h

θ θ

θ θ α

⎡ ⎤ − = ⎢ ⎥ − ⎢ + ⎥ ⎣ ⎦

( ) ( )

( )

2 1/ 1 1

0.5(1 1/ )

1 1

1

n n n

s n n

h h

K K

h

α α

α

− −

−

⎡ ⎤ − + ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ = ⎡ ⎤ + ⎣ ⎦

or equivalently 

2

1/ 2 1/

1 1

m m

r r

s

s r s r

K K θ θ θ θ

θ θ θ θ

⎧ ⎫ ⎡ ⎤ ⎛ ⎞ − − ⎪ ⎛ ⎞ ⎪ = − ⎜ ⎟ ⎨ ⎢ ⎥ −⎜ ⎟ ⎬ ⎝ ⎠ − − ⎪ ⎣ ⎦ ⎢ ⎥ ⎝ ⎠ ⎪ ⎩ ⎭

where θ is volumetric water content; θ r and 

θ s are residual and saturated water content, 

respectively; h is capillary pressure head; 

α and are van Genuchten parameters 

with 

n

m =1 1− / n ; K is unsaturated 

hydraulic conductivity; and Ks is saturated 

hydraulic conductivity. 

The available public information does not 

include soil hydraulic properties for the site. 

We start by estimating soil hydraulic 

parameters indirectly on the basis of soil 

class. Several generic databases have been 

developed for this purpose including 

RAWLS (Rawls et al., 1982), ROSETTA 

(Schaap and Leij, 1998) and CARSEL 

(Carsel and Parrish, 1988). 

The RAWLS database includes 5401 soil 

samples from across the United States. 

Rawls et al. (1982) published a table of 

mean parameter estimates and their standard 

deviations for 11 USDA soil texture classes, 

based on the constitutive model of Brooks 

and Corey (1964). The parameters include a 

pore size distribution index,λ , and the airentry pressure head, . These are related to 

parameters of the van Genuchten (1980) 

constitutive model through 

hb

λ = n −1 and 

1/ hb = α . 
142 

The ROSETTA database is pooled from part 

of the AHUJA (Schaap and Leij, 1998), 

UNSODA (Leij et al., 1996) and RAWLS 

databases. It contains water retention 

parameters for 2134 soil samples and Ks for 

1306 samples. Schaap and Leij (1998) 

tabulate mean values of these parameters for 

12 USDA soil classes. 

The CARSEL database contains 15,737 soil 

textural samples collected by the Natural 

Resources Conservation Service (formerly 

Soil Conservation Service) from 42 of the 

United States. The database does not contain 

measured hydraulic parameters. Based on a 

regression model due to Rawls and 

Brakensiek (1985) coupled with Monte 

Carlo simulations, Carsel and Parrish (1988) 

derived from the same database probability 

distributions for saturated volumetric water 

content θ s , residual volumetric water 

content θ r , saturated hydraulic 

conductivity Ks , and van Genuchten’s 

parameters α and n for twelve USDA soil 

textural classes. Meyer et al. (1997) 

extended their results to include probability 

distributions for the Brooks-Corey 

parameters λ and , effective porosity, 

field capacity, wilting point, and available 

water content. 

hb

The distribution of samples among soil 

classes in RAWLS, ROSETTA and 

CARSEL is quite uneven. Whereas the sand 

portion of samples in RAWLS and 

ROSETTA is much larger than in CARSEL, 

the sandy loam portion in RAWLS is

smaller and there are fewer fine-textured 

samples in RAWLS and ROSETTA than in 

CARSEL. We apply all three databases to 

the MAC and assess the extent to which they

are supported by site data. 

11.1.5 Variogram Analysis of Soil 

Composition at the MAC Site 

Pedologic data at the MAC site include soil 

composition down to a depth of 15 m and 

bulk density down to 5 m (Young et al., 

1999). Soil samples were collected from the 

trench, monitoring islands and boreholes at 

depth intervals of about 0.3 m. Most 

samples are concentrated in the upper 1.8 m

(Table 11-1) and we therefore focus our 

variogram analysis on this shallow depth 

interval. Samples below 1.8 m originate 

mainly from boreholes. Histograms and 

statistics of soil composition at the site can 

be found in Wang (2002).

Table 11-1. Distribution of Soil Samples with Depth. 

Sampling Depth Number of Samples 

0-30 cm 70 

31-60 cm 33 

61-90 cm 46 

91-120 cm 39 

121-150 cm 88 

151-180 cm 74 

181-1545 cm 198 
143 

Figure 11-6. Omni-directional sample variograms and fitted spherical models 

for percent sand, silt, and clay at depths 0 to 30 cm. 

Figure 11-6 shows omni-directional sample 

variograms and fitted spherical models for 

percent sand, silt and clay at depths 0 to 30 

cm. Similar variograms for underlying 30-

cm depth intervals, down to 1.8 m, can be 

found in Wang (2002). Though some appear 

to fit a linear or Gaussian model, most fit 

spherical models with ranges of 20 - 25 

meters (Table 11-2). This implies that soil 

composition should be expected to vary 

laterally on scales exceeding 20–25 m. 

Figure 11-7 shows a vertical sample 

variograms and fitted spherical models of

percent sand, silt and clay for depths 0–15 m. 

They reveal that the vertical correlation 

scale is about 2 m. In other words, soil 

composition should be expected to vary 

vertically on scales exceeding 2 m.
144 

Table 11-2. Omni-directional variogram models fitted to soil composition data at various depths. 

Depth (cm) Component Model Nugget Sill 

Range 

(m) Slope 

0-30 Sand+gravel Spherical 47.0 113.0 24.0 

 Silt Spherical 13.0 33.0 21.0 

 Clay Spherical 20.0 35.0 22.0 

31-60 Sand+gravel Spherical 2.0 27.0 25.0 

 Silt Spherical 3.5 23.5 20.0 

 Clay Spherical 6.0 35.0 23.0 

61-90 Sand+gravel Spherical 13.0 163.0 20.0 

 Silt Spherical 5.0 56.0 21.0 

 Clay Spherical 9.5 39.5 21.0 

91-120 Sand+gravel Linear 40.0 1.5 

 Silt Linear 15.0 0.9 

 Clay Spherical 7.0 20.0 24.0 

121-150 Sand+gravel Linear 40.0 0.2 

 Silt Linear 25.5 0.3 

 Clay Spherical 6.5 13.0 22.0 

151-180 Sand+gravel Spherical 24.0 44.0 35.0 

 Silt Gaussian 15.8 29.8 26.0 

 Clay Spherical 5.7 13.0 33.0 

11.1.6 Postulation of Layered Structure 

Our variogram analysis of soil composition 

supports a conceptual model whereby the 

soil consists of horizontal layers that are 

about 2 m thick and laterally heterogeneous 

on scales exceeding 20 – 25 m. A layered 

structure of similar average thickness is 

supported by neutron count ratios which 

correlate with soil compositional data as 

illustrated for borehole 402 by Figure 11-8; 

see Wang (2002) for similar correlations in 

other boreholes. Neutron readings were 

taken in 9 boreholes down to a depth of 14 

meters at 0.25 m intervals. The two sets of 

data suggest a subdivision of the soil profile 

down to a depth of 20 m into 10 layers 

consisting of four different soil types (sandy 

loam, gravelly loam sand, sand and sandy 

clay loam). Figure 11-9 is a panel diagram 

showing our interpretation of site 

stratigraphy at the MAC based on neutron 

probe and soil compositional data in the 9 

boreholes. In preparing this diagram, we 

have given more weight to the neutron than 

to the soil sample data. The neutron data 

have revealed a perched water table at a 

depth of about 13 m across the site. 
145 

Figure 11-7. Vertical sample variograms and fitted models of percent sand, silt, and clay 

from the surface to 15 m depth. 

Figure 11-8. Neutron count ratio and soil class variations with depth at borehole 402. 
146 

Figure 11-9. Local stratigraphy based on soil and neutron data 

(scale in meters; see legend in Figure 11-9). 

11.1.7 Estimation of Hydraulic Parameters

Using Pedotransfer Models 

Various methods (called pedotransfer 

functions; Bouma and van Lanen, 1997) 

have been developed to translate soil 

pedologic data into hydraulic parameters. 

One approach is to translate soil class into 

mean hydraulic parameter values or 

corresponding probability distributions, 

based on generic databases, as we have done 

earlier. Another approach is to estimate soil 

hydraulic properties based on soil physical 

properties such as soil composition, bulk 

density, and organic matter content by 

regression or neural network models. This 

provides continuously varying soil hydraulic 

properties across the entire triangle of 

possible soil compositions. We use 

regression equations due to Rawls and 
147 

Brakensiek (1985) and a the Rosetta neural 

network software developed by Schaap et al.

(1998) to estimate the hydraulic properties 

of individual samples across the MAC based 

on their particle size distribution and bulk 

density. 

Meyer et al. (1997) provide probability 

distributions for the hydraulic parameters of 

various soil classes based on the CARSEL 

database. They use the KolmogorovSmirnov test of the null hypothesis that 

a sample is drawn from a population having 

a particular probability distribution. Kenkel 

(1989) points out that this test is 

inappropriate when statistical parameters 

such as mean and variance are estimated 

from sample data. We therefore use the chisquare test to modify somewhat their 

probability distributions for hydraulic 

parameters of soil classes found at the MAC. 

Our approach consists of (a) postulating the 

null hypothesis that the data are drawn 

from one of several theoretical probability 

distributions (we consider normal, 

lognormal and beta); (b) assigning the 

sample mean and variance to the selected 

theoretical distribution; (c) drawing random

“observations” out of the sample distribution 

and grouping them into K bins; (d) 

calculating the Chi-square statistic 

Ho

Ho

( )2

2

1

K

i i

i i

o e

e

χ =

− = ∑ (4)

where 1 2 , ,....., K o o o are the frequencies of 

observations in the K bins and 1 2 , ,...., K e e e

are corresponding theoretical frequencies;

(e) minimizing 2 χ with respect to the mean 

and variance (we do so using the LevenbergMarquardt algorithm); (f) repeating the 

process for all distributions; and (g) 

associating the data with that distribution 

which yields the smallest value of 2 χ . We 

found in many cases that using sample mean 

and variance, the Kolmogorov-Smirnov test 

would not reject at a given level of 

significance 

H0

α (0.05) for either the normal 

or lognormal distributions. However, using 

the above procedure we were always able to 

identify a single best-fit distribution among 

the three considered. 

Figure 11-10 shows histograms and fitted 

distributions (solid curves) of hydraulic 

parameters for sand; histograms and fitted 

distributions for other soil components can 

be found in Wang (2002). The dashed curve 

of van Genuchten parameter n is adopted 

from the distribution of Meyer et al. (1997) 

based on a Kolmogorov-Smirnov test. 

Distributions based on the latter tests are 

associated with 2 χ values that tend to be an 

order of magnitude larger than those 

obtained by our procedure. 

The regression model of Rawls-Brakensiek 

requires saturated water content as input. 

The latter is equal to porosity, which we 

infer from bulk density according to θs = 1- 

ρb /ρs where ρb is bulk density and ρs is the 

density of the solids. Unfortunately, bulk 

density was determined only for shallow 

samples at depths of less than 5 m. Figure 

11-11 shows corresponding estimates of 

saturated and residual water content, α , n 

and Ks at various depths. Saturated water 

content tends to decrease with depth due to a 

corresponding increase in bulk density. 

To estimate hydraulic parameters at all 

depths we use the Rosetta neural network 

model. Results are shown in Figure 11-12. 

The statistics of these estimates differ from

those obtained by regression (Table 11-3).
148 

0 .2 5 0 .3 0 0 .3 5 0 .4 0 0 .4 5 0 .5 0 0 .5 5 0 .6 0

0

2 0

4 0

6 0

 S a tu ra te d w a te r c o n te n t

Frequency

n o rm a l

c h i^2 = 0.316

m ean = 0.4 3

s td = 0 .1208

0 .0 2 0 .0 3 0 .0 4 0 .0 5 0 .0 6 0 .0 7 0 .0 8 0 .0 9

0

1 0

2 0

3 0

4 0

 R e s id u a l w a te r c o n tent

 Frequency

M o del: L ogN o rm a l

L N (-3 .141 9 ,0 .224)

C h i^2 = 0.4 8 745

m e an = 0.04429 

s td = 0 .0 1

0 .0 8 0 .1 0 0 .1 2 0 .1 4 0 .1 6 0 .1 8 0 .2 0 0 .2 2

0

1 0

2 0

3 0

4 0

 Frequency

v a n G enu c h te n 's alp h a (1 /c m )

norm a l

ch i^2 = 1 .2829

m ean = 0.1 4 8 9

s td = 0.053

2 .0 2 .2 2 .4 2 .6 2 .8 3 .0 3 .2 3 .4 3 .6

0

2 0

4 0

6 0

M o d e l: Lo g N o rm a l

L N (0 .9678, 0.1 )

C h i^2 = 0.133 4

m e a n = 2.6 418 

s td = 0 .2 6 7

 Frequency

 v an G e n u c h ten's n

0 .5 0 0 .5 5 0 .6 0 0 .6 5 0 .7 0

0

2 0

4 0

 Lam b da (1 -1 /n )

 Frequency

M o del: L ogN o rm a l

L N (-0 .4 7 28, 0 .06)

C h i^2 = 4.1 2838

m e an = 0.6243

s td = 0.0 3749

0 .0 0 0 0 .0 0 5 0 .0 1 0 0 .0 1 5 0 .0 2 0

0

2 0

4 0 M odel: B e ta

C h i^2 = 0.78013

m ean= 8.22E -3

s td = 4.39E -3

 Frequency

 Ks (c m /s )

Figure 11-10. Distribution of hydraulic parameters for sand based on Meyer et al. (1997). 
149 

0 100 200 300 400 500

0.30

0.32

0.34

0.36

0.38

0.40

0.42

0.44

0.46

0.48

Saturated water content

Depth (cm)

0 100 200 300 400 500

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

Residual water content

Depth (cm)

0 100 200 300 400 500

0.00

0.05

0.10

0.15

0.20

0.25

Van Genuchten's alpha (1/cm)

Depth (cm)

0 100 200 300 400 500

1.25

1.30

1.35

1.40

1.45

1.50

1.55

1.60

Van Genuuchten's n

Depth (cm)

0 100 200 300 400 500

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

 Ks (cm/s)

Depth (cm)

Figure 11-11. Hydraulic parameters at the MAC using Rawls-Brakensiek (1985) regressions 

with site-specific soil component and bulk density data at shallow depths of less than 5 m.
150 

-200 0 200 400 600 800 1000 1200 1400 1600

0.30

0.32

0.34

0.36

0.38

0.40

0.42

0.44

0.46

Saturated water content 

 Depth (cm)

-200 0 200 400 600 800 1000 1200 1400 1600

0.03

0.04

0.05

0.06

0.07

Residual water content

Depth (cm)

-200 0 200 400 600 800 1000 1200 1400 1600

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

0.055

Van Genuchten's alpha (1/cm)

 Depth (cm)

-200 0 200 400 600 800 1000 1200 1400 1600

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Van Genuchten's n

 Depth (cm)

-200 0 200 400 600 800 1000 1200 1400 1600

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0.012

 Ks (cm/s)

Depth (cm)

Figure 11-12. Hydraulic parameters at the MAC using Rosetta software 

with site-specific soil component (all the depths) and bulk density data (above 5 m). 
151 

Table 11-3. Comparison of hydraulic parameter estimates 

using two different pedotransfer models.

Estimates Using Parameter Mean St.D. Min Max 

θr (cm3 / cm3) 4.98E-02 6.67E-03 2.99E-02 7.05E-02

θr (cm3 / cm3) 3.80E-01 1.97E-02 3.11E-01 4.40E-01

α (1/cm) 3.21E-02 5.81E-03 1.21E-02 5.03E-02 

n 1.84E+00 6.58E-01 1.31E+00 4.20E+00

Rosetta 

Neural Network 

 

 

 Ks (cm/s) 1.69E-03 2.49E-03 0.00E+00 1.15E-02

θr (cm3 / cm3) 7.47E-02 1.25E-02 3.67E-02 1.02E-01

θr (cm3 / cm3) 4.15E-01 2.78E-02 3.19E-01 4.75E-01

α (1/cm) 1.26E-01 3.83E-02 3.42E-02 2.73E-01 

n 1.39E+00 5.07E-02 1.26E+00 1.56E+00

Rawls-Brakensiek 

Regression 

 

 

 Ks (cm/s) 2.79E-03 2.32E-03 9.68E-05 1.62E-02

We note in particular that whereas van 

Genuchten parameters n and α estimated by 

Rosetta differ from corresponding estimates 

using Rawls-Brakensiek regression, most 

other parameter estimates are comparable. 

The mean of Rosetta n estimates is much 

larger that of Rawls-Brakensiek estimates, 

the opposite being true for the mean of α

estimates. The mean estimates of from 

Rosetta and 

n

α from regression are 

comparable to mean values of corresponding 

CARSEL distributions for sandy loam, 

loamy sand and sand, the three dominant 

soil classes at the MAC. 

11.1.8 Bayesian Updating of Hydraulic 

Parameter Estimates 

Generic probability distributions of 

hydraulic parameters based solely on soil 

class can be updated by treating our 

pedotransfer estimates of these parameters 

as "measurements." In particular, we 

incorporate these measurements in the prior 

CARSEL database to obtain posterior 

estimates of the population mean values and 

the variance of these estimates. We do so by 

using a slightly modified version of a 

Bayesian updating code developed for this 

purpose by Meyer et al. (1997). 

Updated mean hydraulic parameter 

distributions for sand obtained using 

“measurements” based on the Rosetta neural 

network model are depicted in Figure 11-13; 

corresponding distributions for this and 

other soil components, using either Rosetta 

or Rawls-Brakensiek regression, can be 

found in Wang (2002). The Bayesian 

approach of Meyer et al. (1997) equates the 

variance of prior mean values with that of 

soil hydraulic parameters in the CARSEL 

database. In this sense, Figures 11-10 can be 

viewed as representing the prior distribution 

of mean values. As expected, a comparison 

of Figures 11-10 and 11-13 reveals that 

Bayesian updating has brought about a 

change in the mean (reduction in bias) and a 

drastic reduction in its estimation variance. 

The same happens in the case of all soil 

classes at the MAC, regardless of what 

pedotransfer model one uses to obtain the 

“measurements” (Table 11-4).
152 

0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0

20

40

Frequency

Saturated water content

normal

mean=0.3837

std=0.00237

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0

20

40

Residual water content

Frequency

normal

mean=0.048

std=8.161E-4

0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0

20

40

60

van Genuchten's alpha (1/cm)

 Frequency

Model: LogNormal

LN(-3.236, 0.0735)

Chi^2 = 0.40872

mean=0.0394

std=8.42E-6

2.0 2.4 2.8 3.2 3.6 0

10

20

30

 Frequency

 van Genuchten's n

Model: LogNormal

Chi^2 = 0.45493

log_mean=1.116

log_std=0.03815

0.50 0.55 0.60 0.65 0.70 0

10

20

30

lambda(1-1/n)

 Frequency

Model: LogNormal

LN(-0.3914, 0.0182)

Chi^2 = 0.47496

mean=0.6762

std=1.51E-4

0.000 0.005 0.010 0.015 0.020 0

20

40

60

 Ks (cm/s)

 Frequency

Model: LogNormal

LN(-4.9351, 0.09621)

Chi^2 = 1.33165

mean=0.007223

std=4.85E-7

Figure 11-13. Bayesian updates of hydraulic parameters for sand using Rosetta estimates. 
153 

Table 11-4. Prior (based on CARSEL database) and posterior (following Bayesian updating) 

mean hydraulic parameter estimates and their standard deviations by soil class. 

Soil Type θ s St. D. θ r St. D. α St. D. n St. D. Ks St. D. 

unit 1/cm cm/s

Sandy loam 

Meyer et al 0.410 9.00E02 0.065 1.70E02 

7.57E02 3.68E02 1.890 1.55E01 

1.17E03 1.37E03 

Updated by 

RawlsBrakensiek 

regression 

0.422 1.30E03 0.077 6.80E04 

1.24E01 2.39E03 1.380 1.13E05 

1.97E03 9.30E09 

Updated by 

Rosetta 0.385 1.00E03 0.052 3.51E04 

2.90E02 1.47E07 1.556 1.20E04 

5.61E04 5.71E10 

Gravel Loamy 

Sand 

Meyer et al 0.410 9.00E02 0.057 1.40E02 

1.26E01 4.04E02 2.270 2.70E01 

1.40E04 3.17E03 

Updated by 

RawlsBrakensiek 

regression 

0.391 7.70E03 0.062 3.00E03 

1.33E01 1.00E02 1.459 1.20E04 

3.45E03 1.42E10 

Updated by 

Rosetta 0.365 3.24E03 0.049 1.00E03 

4.00E02 1.17E05 1.882 1.15E02 

1.80E03 9.98E08 

Sand 

Meyer et al 0.430 6.00E02 0.047 1.00E02 

1.47E01 2.55E02 2.670 2.67E01 

8.22E03 4.39E03 

Updated by 

Rosetta 0.384 2.37E02 0.048 8.16E04 

3.94E02 8.42E06 3.055 1.35E02 

7.22E03 4.85E07 
154 

0 5 10 15 20 25

Lag Distance (m)

0

0.05

0.1

0.15

0.2

0.25

0.3

Variogram

012 345

Lag Distance (m)

0

0.05

0.1

0.15

0.2

0.25

0.3

Variogram

a

b

Figure 11-14. Omni-directional lateral (left, at depth 0–30 cm) and vertical (right) 

sample variograms and fitted models for log( Ks ). 

11.1.9 Variogram Analysis of Hydraulic 

Parameter Estimates 

We conduct a variogram analysis of 

hydraulic parameter estimates obtained 

using the Rosetta neural network software. 

Figure 11-14a shows omni-directional 

sample variograms and fitted spherical 

models for log Ks at depths of 0–30 cm. 

Similar variograms are obtained down to a 

depth of 1.8 m for log Ks and saturated 

water content (θ s ) (Wang, 2002). Most of 

them fit spherical models and one fits a 

Gaussian model with horizontal range 

values between 20 and 36 m; some fit linear 

models that do not possess finite correlation 

scales (Table 11-5). 

Vertical sample variogram and fitted 

spherical model of log Ks are depicted in 

Figure 11-14b. Similar results are obtained 

for saturated water content (θ s ). Both have a 

vertical range on the order of 1–2 m.

Our variograms of hydraulic parameter 

estimates closely corresponds to those of 

soil composition. This is not surprising

considering that the former were estimated 

in part on the basis of the latter. Together 

with neutron probe data they support a 

conceptual model whereby the soil consists 

of horizontal layers that are about 2 m thick 

and laterally heterogeneous on scales of 20–

30 m.
155 

Table 11-5. Omni-directional variogram models fitted to hydraulic parameter estimates 

at various depths. 

Parameter Depth (cm) Model Nugget Sill Range (m) Slope 

31-60 Spherical 1.70E-05 2.70E-05 25

61-90 Linear 3.60E-05 3.2E-07

91-120 Spherical 0.00E+00 3.40E-05 30

121-150 Linear 3.84E-05 1.09E-07

θ s

151-180 Gaussian 9.00E-06 2.30E-05 22

0-30 Spherical 0.025 0.18 20

31-60 Spherical 0.009 0.119 20

61-90 Spherical 0.013 0.168 21

91-120 Linear 0.04 0.0015

121-150 Spherical 0.029 0.055 24

Ks

151-180 Spherical 0.036 0.064 36

11.1.10 Forward Flow Modeling Based on 

Public Data

As pointed out earlier, publicly available 

data suggest that soils at the experimental 

site consist of sandy loam down to a depth 

of about 16 m with a regional water table 

located at a depth of 22 m. In this paper we 

examine the extent to which such public 

information allows reproducing observed 

behavior during field infiltration experiment 

3 at the MAC. We do so by postulating 

uniform vertical flow through the shallow 

subsurface, which we take to consist of 

uniform sandy loam. To characterize the 

sandy loam hydraulically, we ascribe to it 

mean parameter values taken from the three 

generic databases RAWLS (Rawls et al., 

1982), ROSETTA (Schaap and Leij, 1998) 

and CARSEL (Carsel and Parrish, 1988; 

Meyer et al., 1997). 

Volumetric water contents are obtained from

neutron probe readings using a calibration 

curve developed by Young (1999). The 

calibration is based on measured water 

contents from core samples and neutron 

probe readings at depths of 3.0 meters or 

less. We adopt water content measurements 

in borehole 422 prior to experiment 3 as 

initial water contents and set the water table 

as a constant head boundary. A constant flux 

of 2.66 cm/day is maintained at the soil 

surface during the first 28 days of the 

experiment. During the remainder of the 

experiment, the soil surface constitutes a noflow boundary. We use the TOUGH2 code 

of Pruess et al. (1999) to simulate 

unsaturated flow. 

Figures 11-15 – 11-18 compare measured 

and simulated water contents in borehole 

422 using mean hydraulic parameters from

ROSETTA, RAWLS and CARSEL, 

respectively. Only the latter yield acceptable 

matches with observations at depths of less 

than 4.0 m.

We conclude that 

• Relying entirely on publicly available 

data leads to a poor reproduction of 

observed infiltration at the Maricopa 

site. 

• Among the three generic databases 

examined, the best results are obtained 

with the CARSEL set.
156 

Time (days)

water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

3.00 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

4.00 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.50 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

7.00 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.00 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

9.75 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

10.00 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.75 m

Figure 11-15. 1-D forward simulation of infiltration during experiment 3 at borehole 422 

using mean hydraulic parameters from ROSETTA. 
157 

Time (days)

water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.75 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

3.00 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

4.00 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.50 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

7.00 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.00 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

9.75 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

10.00 m

Figure 11-16. 1-D forward simulation of infiltration during experiment 3 at borehole 422 

using mean hydraulic parameters from RAWLS. 
158 

Time (days)

water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.75 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

3.00 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

4.00 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.50 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

7.00 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.00 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

9.75 m

Time (days)

Water content

-20 0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

10.00 m

Figure 11-17. 1-D forward simulation of infiltration during experiment 3 at borehole 422 

using mean hydraulic parameters from CARSEL. 

11.1.11 Forward Flow Modeling Based on 

Site Data 

Whereas public data imply that soils at the 

MAC consist of sandy loam down to a depth 

of about 16 m, site data indicate that the 

upper 20 m of soil consist of 10 layers 

composed of four different soil types (sandy 

loam, gravelly loam sand, sand and sandy 

clay loam), which are laterally 

heterogeneous on scales exceeding 20 – 30 

m (see paper 1, this issue). Whereas publicly 

available data identify a regional water table 

that is locally about 22 m deep, site data 

indicate the additional presence of a perched 

water table at a depth of about 13 m across 
159 

the site. We conduct a series of one- and 

two-dimensional forward numerical 

simulations of flow during infiltration 

experiment 3 based on this site-specific 

conceptual model. Our computational 

domain consists of 9 - 11 layers in line with 

Figures 11-9 – 11-10. In one-dimensional 

simulations, the water table at depth 13 m 

constitutes a constant head boundary. In 

two-dimensional simulations, either a 

constant head boundary is prescribed at a 

depth of 13 m or a no-flow boundary at a 

depth of 20 m. The latter condition is 

motivated by the fact that soils below 16 m 

consist of low-permeability sandy clay loam 

or clay. Neutron-probe data suggest that the 

water table at the eastern and northern 

boundaries is higher by about 0.2 m than at 

the western and southern boundaries. This is 

reflected in the head boundaries prescribed 

on the sides of the two-dimensional domain. 

The latter extends 110 m horizontally. A 

constant flux of 2.66 cm/day is maintained 

at the soil surface during the first 28 days of 

the experiment within the irrigated plot, and 

a flux of zero outside this plot. During the 

remainder of the experiment, the soil surface 

constitutes a no-flow boundary. 

Figure 11-18 shows the results of a onedimensional forward simulation in borehole 

402 using mean hydraulic parameters from 

CARSEL. The computed response (curves) 

captures in a very crude way the observed 

behavior (dots). However, there are large 

differences between measured and computed 

water contents and wetting-front arrival times. 

This suggests that the parameter estimates 

would have to be improved by model 

calibration against the observed water 

contents. 

Figure 11-19 shows what happens when we 

adopt mean hydraulic parameters from 

ROSETTA. The results are much poorer 

than those in Figure 11-18. The same 

happens when we adopt mean hydraulic 

parameter values from RAWLS. 

Figure 11-20 depicts results obtained using 

Bayesian updates of ROSETTA parameter 

estimates. Though there is some 

improvement in comparison to results 

obtained without updating, the results are 

still poorer than those obtained using mean 

hydraulic parameters from CARSEL. The 

same happens when we use updated 

parameter estimates from RAWLS. 

These and similar simulations corresponding 

to borehole 422, summarized in Table 11-6, 

indicate that mean hydraulic parameters 

from CARSEL provide best fit simulations 

to measured water content data in all cases. 

Bayesian updates improve the accuracy of 

generic ROSETTA and RAWLS hydraulic 

parameters, but not significantly. 

Figure 11-21 compares two-dimensional 

forward simulations and measurements 

along a N-S uniformly layered transect 

passing through boreholes 402, 422 and 442 

(see Figure 11-10) using mean hydraulic 

parameter values from CARSEL. The 

quality of the results varies with depth, with 

simulated wetting front arrival times lagging 

by up to ten days behind those measured. 

Results obtained using mean parameter 

estimates from ROSETTA or RAWLS, as 

well as those obtained using Bayesian 

updates of these parameters, are much less 

satisfactory. The same is true for a central EW transect through boreholes 422, 423 and 

425 (Figure 11-10). A summary of all these 

runs in Table 11-7 indicates that hydraulic 

parameter estimates from CARSEL are 

generally superior to those from ROSETTA, 

RAWLS, or their Bayesian updates. 

However, none of the forward simulations 

are entirely satisfactory and there is an 

obvious need to calibrate the models against 

observed system behavior.
160 

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.5 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2.75 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

3.0 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

7.0 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.0 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.25 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.5 m

Figure 11-18. 1-D simulation of infiltration experiment 3 at borehole 402 using mean hydraulic parameter 

values from CARSEL at various depths. 
161 

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.5 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2.75 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

3.0 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

7.0 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.0 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.25 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.5 m

Figure 11-19. 1-D simulation of infiltration experiment 3 at borehole 402 using mean hydraulic parameter 

values from ROSETTA at various depths. 
162 

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.5 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2.75 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

3.0 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

7.0 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.0 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.25 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.5 m

Figure 11-20. 1-D simulation of infiltration experiment 3 at borehole 402 using Bayesian updates of

saturated hydraulic conductivity, van Genuchten's α and n based on Rosetta estimates. 
163 

Time (days)

water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.5 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.75 m 422

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.5 m 442

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2.75 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

3.0 m 422

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2.75 m 442

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 422

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 442

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.0 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.0 m 422

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

10.0 m 442

Figure 11-21. 2-D forward simulation along N-S uniformly layered transect using mean 

hydraulic parameter values from CARSEL at various depths. 
164 

Table 11-6. Results of forward simulations using 1-D layered models. 

Borehole Parameter source Sum of weighted squared residuals

402 CARSEL 7.28E+02 

402 ROSETTA 3.62E+03 

402 RAWLS 3.35E+04 

402 Bayesian updates based on Rosetta neural network 2.05E+03 

402 Bayesian updates based on Rawls-Brakensiek regression 2.59E+04 

422 CARSEL 1.53E+03 

422 ROSETTA 3.74E+03 

422 RAWLS 3.66E+03 

422 Bayesian updates based on Rosetta neural network 2.30E+03 

422 Bayesian updates based on Rawls-Brakensiek regression 3.21E+03 

Table 11-7. Results of forward simulations using 2-D uniformly layered models. 

Transect Parameter source 

Sum of weighted 

squared residuals 

N-S CARSEL 2.65E+03 

N-S ROSETTA 5.35E+03 

N-S RAWLS 5.53E+03 

 N-S Bayesian updates from Rosetta 3.17E+03 

 N-S Bayesian updates from Rawls-Brakensiek regression 4.06E+03 

 E-W CARSEL 2.01E+03 

E-W ROSETTA 6.14E+03 

 E-W RAWLS 6.73E+03 

 E-W Bayesian updates from Rosetta 3.46E+03 

 E-W Bayesian updates from Rawls-Brakensiek regression 4.72E+03 

11.1.12 Inverse Flow Modeling Based on 

Site Data 

We use the inverse code ITOUGH2 

(Finsterle, 1999 a-b) to calibrate our oneand two-dimensional flow models against 

observed water contents at the MAC. The 

inverse code estimates hydraulic parameters 

by minimizing a negative log likelihood 

criterion equal to the weighted sum of 

squared water content and parameter 

residuals. Water content residuals are 

differences between simulated and observed 

water contents, the square of each being 

weighted by the inverse variance of the 

corresponding observation error. Parameter 

residuals are differences between posterior 

(inverse) and prior (input) parameter 

estimates, the square of each being weighted 

by the inverse variance of the corresponding 

prior estimation error. 

Previously we described various ways to 

obtain prior hydraulic parameter estimates 

and error variances for the MAC site. 

Parameter estimates obtained on the basis of 

Bayesian updating have very small 

variances and therefore very large weights. 
165 

The latter allow only minimal departure of 

the posterior parameters from their initial 

(input) values, which is not enough to yield 

a significant improvement in model fit 

(reduction in the weighted sum of squared 

water content residuals). As unsaturated 

flow equations are highly nonlinear and soil 

hydraulic parameter estimates are correlated 

with each other, it is difficult to identify soil 

hydraulic properties uniquely using 

inversion. The more accurate are the input 

parameters, the higher is the prospect of 

obtaining meaningful inverse estimates. 

Since the CARSEL database has proven to 

yield best results in forward simulations, we 

adopt the mean and variance of hydraulic 

parameters from Meyer et al. (1997) as input 

into the inverse code. 

Calibrating a one-dimensional model 

consisting of a single uniform layer against 

water content data in various boreholes 

during infiltration experiment 3 brings about 

only a minor improvement over the 

uncalibrated model. This model is clearly 

inferior to the multilayer models we 

consider below. 

As the sandy loam at depths 0 - 2 m has a 

different bulk density than deeper sandy 

loam layers, we estimate its hydraulic 

properties separately. This yields a total of 

four materials for each sequence of layers: 

Sandy loam in the top layer, sandy loam in 

deeper layers, gravel loamy sand and sand. 

Sensitivity analysis about the prior 

parameter estimates suggests (see Wang, 

2002, for details) that it should be possible 

to estimate independently the saturated 

hydraulic conductivity Ks and van 

Genuchten’s n and α for each of these 

materials, and this is what we do. 

Figure 11-22 shows matches between 

simulated (curves) and measured (dots) 

water contents in borehole 402 during 

infiltration experiment 3 following 

inversion. Inverse modeling is seen to have 

improved these matches significantly as 

compared to the forward modeling results in 

Figure 11-18. It has also brought about a 

significant change in the estimate (mean) 

and reduction in the estimation error

(variance) of each parameter. The same

happens when we calibrate our onedimensional model against water content 

data from boreholes 422, 442, 423 and 425. 

Figure 11-23 shows the results of twodimensional inverse modeling along the 

western N-S transect by considering the soil 

to consist of horizontally uniform layers. 

The matches are seen to be much better than 

those obtained prior to inversion in Figure 

11-21. While some simulation results fit the 

data well, other are systematically too low or 

too high. For example, water content in the 

top sandy loam layer in borehole 402 is 

systematically under-predicted, whereas in 

sandy loam and sand layers at depths 6 - 10 

m in boreholes 422 and 442 it is 

systematically over-predicted. Figure 11-23 

suggests indirectly that computed wettingfront arrival times in deeper sections of 

borehole 442 lag considerably behind the 

measured ones. Similar results are obtained 

for the central E-W transect. We conclude 

that it may be necessary to account for 

lateral variations in layer properties. 

Variogram analysis has shown that the 

dominant horizontal correlation scale of soil 

hydraulic parameters at the MAC is 20 - 25 

m. We therefore subdivide the transect into 

3 horizontal segments, one per borehole. We

ascribe to each segment initial parameter 

values equal to those previously obtained 

from corresponding one-dimensional inverse 

modeling results. This yields a total of 36 

parameters per transect. 
166 

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.5 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2.75 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

3.0 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

7.0 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.0 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.25 m

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.5 m

Figure 11-22. 1-D simulation of infiltration experiment 3 in borehole 402 using inverse estimates of

saturated hydraulic conductivity and van Genuchten's α and n. 
167 

Time (days)

water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.5 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.75 m 422

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.5 m 442

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2.75 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

3.0 m 422

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2.75 m 442

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 422

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 442

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.0 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.0 m 422

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

10.0 m 442

Figure 11-23. 2-D simulation of infiltration experiment 3 along western N-S transect 

(boreholes 402, 422, 442) using inverse estimates of saturated hydraulic conductivity 

and van Genuchten's α and n. Uniform soil layers. 
168 

Time (days)

water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.5 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.75 m 422

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.5 m 442

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2.75 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

3.0 m 422

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2.75 m 442

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 422

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 442

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.0 m 402

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.0 m 422

Time (days)

Water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

10.0 m 442

Figure 11-24. 2-D simulation of infiltration experiment 3 along western N-S transect 

(boreholes 402, 422, 442) using inverse estimates of saturated hydraulic conductivity 

and van Genuchten's α and n. Non-uniform layers. 
169 

-0.04 -0.02 0.00 0.02 0.04

0

50

100

 Frequency

Normal distribution

mean = 0.00137

St.D = 0.00693

Figure 11-25. Histogram of differences between observed and simulated water contents 

along western N-S transect (boreholes 402, 422, 442) following inversion. Non-uniform layers. 

Figure 11-24 compares simulated and 

observed water contents using inverse 

parameter estimates along the western N-S 

transect. The fit is seen to be good in all 

cases. A histogram of residuals (Figure 11-

25) suggests that they are close to normal 

with a near-zero mean and small standard 

deviation. At a confidence level of 95%, 

only 14 out of the 300 residuals are 

identified as outliers. The parameter 

estimates differ only slightly from their onedimensional counterparts. Similar results are 

obtained for the central E-W transect.

Tables 11-8 and 11-9 compare the various 

one- and two-dimensional models along the 

western N-S and central E-W transects, 

respectively, following inversion. The 

quality of model fit is compared on the basis 

of a D-optimality criterion equal to the 

determinant of the covariance matrix of

parameter estimation errors, an A-optimality 

criterion equal to the trace of this matrix, an 

E-optimality criterion equal to the largest 

absolute eigenvalue of the same matrix 

(Steinberg and Hunter, 1984), the negative 

log likelihood criterion that is minimized 

during the inverse process, and the 

logarithm of the corresponding likelihood 

function (which is being maximized). These 

model fit criteria can be used to compare the 

quality of different models that have similar 

structure and number of parameters (such as 

our N-S and E-W two-dimensional models 

with uniform layers), but not models that 

have different structures or numbers of 

parameters (such as our two-dimensional 

models with uniform and non-uniform 
170 

layers). To validly compare the quality of all 

models in Tables 11-8 and 11-9, we employ 

likelihood-based model discrimination 

criteria due to Akaike (1974) and Kashyap 

(1982) as done previously by Carrera and 

Neuman (1986a-c). The smaller (or more 

negative) are these criteria, the better is the 

model. The model discrimination criteria 

consistently identify the uniform onedimensional model as being the worst 

among those considered and the twodimensional non-uniformly layered model as 

being the best.

Table 11-8. Model quality criteria for four 1-D and two 2-D models associated with N-S transect. 

1-D models 2-D N-S models Model Quality 

Criteria 422 uniform 422 402 442 Uniform 

layers 

Non-uniform 

layers 

D-optimality 4.79E-11 5.77E-68 5.65E-58 1.15E-58 5.83E-57 5.50E-179 

A-optimality 9.21E-03 2.00E-04 4.80E-04 8.95E-04 1.96E-03 1.11E-03 

E-optimality 9.00E-03 1.30E-04 2.21E-04 3.78E-04 8.49E-04 2.35E-04 

Negative log 

likelihood 1.28E+03 2.34E+02 1.99E+02 2.29E+02 8.20E+02 2.70E+02 

Log likelihood -3.45E+01 6.28E+02 6.09E+02 6.96E+02 6.11E+02 1.00E+03 

Akaike 7.51E+01 -1.23E+03 -1.19E+03 -1.37E+03 -1.20E+03 -1.93E+03 

Kashyap 1.02E+02 -1.06E+03 -1.05E+03 -1.22E+03 -1.04E+03 -1.45E+03 

Table 11-9. Model quality criteria for three 1-D and two 2-D models associated with E-W transect. 

1-D Models 2-D E-W Models 

Model Quality Criteria 

422 423 425 Uniform 

layers 

Non-uniform 

layers 

D-optimality 5.77E-68 3.16E-74 4.06E-56 7.55E-62 1.44E-194 

A-optimality 2.00E-04 9.79E-05 1.10E-03 1.04E-03 3.55E-04 

E-optimality 1.30E-04 5.29E-05 6.19E-04 5.27E-04 8.80E-05 

Negative log likelihood 2.34E+02 2.45E+02 2.61E+02 8.37E+02 4.54E+02 

Log likelihood 6.28E+02 7.00E+02 6.99E+02 6.32E+02 9.55E+02 

Akaike -1.23E+03 -1.38E+03 -1.37E+03 -1.24E+03 -1.84E+03 

Kashyap -1.06E+03 -1.19E+03 -1.23E+03 -1.07E+03 -1.31E+03 
171 

Time (days)

water content

0 20 40 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 402

Time (days)

Water content

0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.5 m 402

Time (days)

Water content

0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.75 m 422

Time (days)

Water content

0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.5 m 442

Time (days)

Water content

0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2.75 m 402

Time (days)

Water content

0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

3.0 m 422

Time (days)

Water content

0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2.75 m 442

Time (days)

Water content

0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 402

Time (days)

Water content

0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 422

Time (days)

Water content

0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6.5 m 442

Time (days)

Water content

0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.0 m 402

Time (days)

Water content

0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

8.0 m 422

Time (days)

Water content

0 20 40 60 80 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

10.0 m 442

Figure 11-26. 2-D simulation of experiment 1 along the N-S transect (boreholes 402, 422, 442) 

using layered non-uniform conceptual model and hydraulic parameters obtained from experiment 3. 
172 

T im e [D a ys]

Concentration (ppm)

0 5 0 1 0 0 0

1 0

2 0

3 0

4 0

5 0

6 0

0 .5 m

T im e [D a ys]

Concentration (ppm)

0 5 0 100 0

1 0

2 0

3 0

4 0

5 0

6 0

1 .0 m

T im e [D a ys]

Concentration (ppm)

0 5 0 1 0 0 0

1 0

2 0

3 0

4 0

5 0

6 0

1 .5 m

T im e [D a ys]

Concentration (ppm)

0 5 0 100 0

1 0

2 0

3 0

4 0

5 0

6 0

2 .0 m

T im e [D a ys]

Concentration (ppm)

0 5 0 1 0 0 0

1 0

2 0

3 0

4 0

5 0

6 0

2 .5 m

T im e [D a ys]

Concentration (ppm)

0 5 0 100 0

1 0

2 0

3 0

4 0

5 0

6 0

3 .0 m

Figure 11-27. Simulated (curves) and measured (dots) bromide concentrations at the south monitoring 

island obtained using inverse estimates of dispersivity, immobile or excluded water content, 

and mass transfer coefficient. 
173 

11.1.13 Confirmation of Inverse Modeling 

Results 

Our conceptual model and inverse hydraulic 

parameter estimates are based on data 

collected during infiltration experiment 3. 

We use them to simulate water contents at 

the MAC site during experiment 1. 

Figure 11-26 shows a two-dimensional 

simulation of infiltration experiment 1 along 

the western N-S transect using the nonuniform layered model and inverse 

hydraulic parameters, which are obtained by 

calibrating against water contents observed 

during experiment 3. The good matches 

between simulated and observed water 

contents for experiment 1 constitute a 

confirmation of the calibrated model. 

Similar results were obtained for the central 

E-W transect. 

11.1.14 Inverse Modeling of Solute 

Transport 

One-dimensional inverse estimates of soil 

hydraulic parameters based on infiltration 

experiment 3 were adopted for onedimensional solute transport modeling of 

bromide tracer data collected during 

experiment 1. We model transport as an 

advection-dispersion process with linear 

mass transfer to and from a zone of 

immobile water or, equivalently, anion 

exclusion. 

Figure 11-27 shows simulated (solid curve) 

and measured (dots) bromide concentrations 

at the south monitoring island obtained 

using inverse estimates of dispersivity, 

immobile or excluded water content, and 

mass transfer coefficient. Varying the latter 

has no effect on our simulations. 

11.1.15 Conclusions 

Our work at the MAC site leads to the 

following conclusions: 

• Publicly available data suggest that soils 

at the experimental MAC site consist of 

sandy loam down to a depth of about 16 

m. Site data indicate instead that the soil 

profile down to a depth of 20 m consists 

of 10 layers composed of four different 

soil types (sandy loam, gravelly loam 

sand, sand and sandy clay loam), which 

are laterally heterogeneous on scales 

exceeding 20 – 30 m. 

• Publicly available data identify a 

regional water table that is locally about 

22 m deep. Site data indicate the 

additional presence of a perched water 

table at a depth of about 13 m across the 

site. 

• Generic databases make it possible to 

associate each soil type at the MAC with 

prior probability distributions of relevant 

hydraulic parameters. The distributions 

vary with the database. 

• Regression and neural network 

pedotransfer models allow translating 

information about the composition and 

density of discrete soil samples at the 

MAC into estimates of hydraulic 

parameters. The statistics of these 

estimates vary with the model and 

generic database on which it is based. 

• Generic probability distributions of 

hydraulic parameters based solely on 

soil class can be updated by treating our 

pedotransfer estimates of these 

parameters as "measurements." 

Bayesian updating has brought about a 

change in the mean (reduction in bias) 

and a drastic reduction in its estimation 

variance. 

• Our conclusion about the layered and 

laterally heterogeneous structure of soils 

at the MAC is supported in large part by 
174 

variograms of soil compositional data 

and hydraulic parameter estimates 

obtained using pedotransfer models. 

• The same conclusion is supported by 

neutron probe readings, which correlate 

with soil compositional data and appear 

to be more consistent across the site than 

are these data. 

• Relying entirely on publicly available 

data leads to a poor reproduction of 

observed infiltration at the MAC site. 

• Mean hydraulic parameter estimates 

based on the generic CARSEL database, 

published by Meyer et al. (1997) on the 

basis of data assembled by Carsel and 

Parrish (1988), allow a much better 

reproduction of observed water contents 

at the MAC than do estimates based on 

the RAWLS (Rawls et al., 1982) or 

ROSETTA (Schaap and Leij, 1998) 

databases. Bayesian updating of mean 

hydraulic parameters based on RAWLS 

and ROSETTA leads only to a marginal 

improvement in their ability to 

reproduce observed water contents at the 

site. 

• Regardless of our choice of database, 

pedotransfer function or flow model we 

were unable to reproduce observed water 

contents at the site without calibrating 

the flow model against such 

observations. We used the mean and 

variance of hydraulic parameters 

associated with the CARSEL database as 

inputs into our inverse flow models. 

• Calibrating a one-dimensional model 

consisting of a single uniform layer 

against water content data in various 

boreholes during infiltration experiment 

3 brings about only a minor 

improvement over the uncalibrated 

model. This model is clearly inferior to 

multilayer models we consider for the 

site. 

• Inverse modeling brings about a 

significant improvement in our ability to 

reproduce observed water contents at the 

MAC using multilayer models regardless 

of whether the latter are one- or twodimensional with uniform or laterally 

heterogeneous layers. Inverse modeling 

is accompanied by a significant change 

in the estimate (mean) and reduction in 

the estimation error (variance) of each 

hydraulic parameter. 

• Likelihood-based model discrimination 

criteria consistently identify the uniform 

one-dimensional model as being the 

worst among those considered and the 

two-dimensional model with laterally 

heterogeneous layers as being the best. 

• The latter model, calibrated against 

water content data observed during 

infiltration experiment 3, reproduces 

with fidelity water content data observed 

during infiltration experiment 1. 

• One-dimensional inverse estimates of 

soil hydraulic parameters based on 

infiltration experiment 3 were adopted 

for one-dimensional solute transport 

modeling of bromide tracer data 

collected during experiment 1. An 

advection-dispersion model with linear 

mass transfer to and from a zone of 

immobile water, or anion exclusion, was 

employed in forward and inverse modes. 

Following calibration, the model 

reproduced reasonably well measured 

bromide breakthroughs at the monitoring 

island during experiment 1. 

11.2 Apache Leap Research Site

11.2.1 Introduction 

Issues associated with the site 

characterization of fractured rock terrains, 

the analysis of fluid flow and contaminant 

transport in such terrains and the efficient 
175 

handling of contaminated sites are typically 

very difficult to resolve. A major source of 

this difficulty is the complex nature of the 

subsurface "plumbing system” of pores and 

fractures through which flow and transport 

in rocks take place. This is especially true 

for fractured rocks that are only partially 

saturated. 

One relatively convenient way to 

characterize such rocks is by means of 

single-hole and cross-hole pneumatic 

injection tests. Considerable experience with 

such testing of unsaturated fractured tuffs 

has been accumulated in recent years by the 

University of Arizona (U of A) at the 

Apache Leap Research Site (ALRS) near 

Superior, Arizona, under the auspices of the 

U.S. NRC (Chen et al., 2000; Guzman et al., 

1996; Illman et al., 1998; Illman and 

Neuman, 2000, 2001; Neuman et al., 2001; 

Vesselinov et al., 2001a-b; Vesselinov and 

Neuman, 2001: Hyun et al., 2002). This 

research has established the following 

conceptual framework for the ALRS: 

• It is possible to model airflow through 

unsaturated fractured tuffs at the ALRS 

by treating water as if it was immobile, 

and the rock as if it was a porous 

continuum. 

• During a pneumatic injection test, air 

moves primarily through fractures most 

of which contained relatively little water, 

and the test therefore yields 

permeabilities and porosities which 

reflect closely the intrinsic properties of 

the surrounding fractures. This is so 

because capillary forces tend to draw 

water from fractures into porous (matrix) 

blocks of rock, leaving the fractures 

saturated primarily with air, and making 

it difficult for air to flow through matrix 

blocks. 

• Pneumatic permeabilities increase 

systematically with applied pressure, as 

air appears to displace water under twophase flow. In a few single-hole tests, 

where the injection intervals were 

intersected by widely open fractures, air 

permeabilities decrease with applied 

pressure due to inertial effects. Twophase flow and inertial phenomena 

decay rapidly with distance from the 

injection interval. Enhanced permeability 

due to slip flow (the Klinkenberg effect) 

appeares to be of little relevance to the 

interpretation of single-hole or crosshole air injection tests at the ALRS. 

• Flow in the vicinity of most relatively 

short pneumatic test intervals is threedimensional regardless of the number or 

orientation of fractures in the surrounding 

rock. This implies that such flow is 

controlled by a single continuum, 

representative of a three-dimensional 

network of interconnected fractures, 

rather than by discrete planar features. 

Only in a small number of single-hole 

test intervals, known to be intersected by 

widely open fractures, do such features 

dominated flow. Some pressure records 

indicate radial flow during early and 

intermediate times, but none do so fully 

at late times. 

• It is generally not possible to distinguish 

between the permeabilities of individual 

fractures, and the bulk permeability of 

the fractured rock in the immediate 

vicinity of a test interval, by means of 

pneumatic injection tests. Hence there is 

little justification for attempting to 

model flow through individual fractures 

at the site. The explicit modeling of 

discrete features appears to be justified 

only when one can distinguish clearly 

between layers, faults, fracture zones, or 

major individual fractures on scales not 

much smaller than the domain of interest. 
176 

• Air permeabilities obtained from singlehole tests are poorly correlated with 

fracture densities, as is known to be the 

case for hydraulic conductivities at many 

water-saturated fractured rock sites 

worldwide (Neuman, 1987). This 

provides further support for Neuman’s 

conclusion that the permeability of 

fractured rocks cannot be reliably 

predicted from information about fracture 

geometry (density, trace lengths, 

orientations, apertures and their roughness) 

but must be determined directly by means 

of hydraulic and/or pneumatic tests. 

• Core and single-hole measurements, 

conducted over short segments of a 

borehole, provide information only 

about a small volume of rock in the 

immediate vicinity of each measurement 

interval. They tend to vary erratically in 

space in a manner that renders the rock 

randomly heterogeneous and anisotropic. 

• Local-scale air permeabilities from 

single-hole tests vary by orders of 

magnitude between test intervals across 

the site; their spatial variability is much 

more pronounced than their dependence 

on applied pressure. 

• It is possible to interpolate some of the 

core and single-hole measurements at 

the ALRS between boreholes by means 

of geostatistical methods, which view 

the corresponding variables as correlated 

random fields defined over a continuum. 

This is especially true about air 

permeability, porosity, fracture density, 

water content, and the van Genuchten 

water retention parameter α, for each of 

which there are enough measurements to 

constitute a workable geostatistical 

sample. It supports the application of 

stochastic continuum flow and transport 

theories and models to unsaturated 

fractured porous tuffs at the ALRS on 

scales of one meter or more. 

• Air-permeability is well characterized by 

a power variogram, which is 

representative of a random fractal field 

with multiple scales of spatial 

correlation. 

• Cross-hole pneumatic injection test data 

from individual monitoring intervals are 

amenable to analysis by type-curve and 

numerical inverse models, which treat 

the rock as a uniform and isotropic 

fractured porous continuum. Analyses of 

pressure data from individual monitoring 

intervals by the two methods provide 

information about pneumatic 

connections between injection and 

monitoring intervals, corresponding 

directional air permeabilities, and airfilled porosities. All of these quantities 

vary considerably from one monitoring 

interval to another in a given cross-hole 

test on scales ranging from a few meters 

to several tens of meters. Thus, even 

though the analyses treat the rock as if it 

was pneumatically uniform and 

isotropic, they ultimately yield 

information about the spatial and 

directional dependence of pneumatic 

connectivity, permeability and porosity 

across the site. 

• The pneumatic permeabilities and 

porosities of unsaturated fractured tuffs 

at the ALRS vary strongly with location 

and scale of measurement. The scale 

effect is most probably due to the 

presence in the rock of various size 

fractures that are interconnected on a 

variety of scales. 

• As there is consistency between singlehole and cross-hole test results, the 

pronounced permeability scale effect at 

the ALRS is unrelated to the method of 

testing. As there is consistency between 

results obtained by means of diverse 

steady-state and transient, analytical and 

numerical methods of test interpretation, 
177 

the scale effect is unrelated to the 

method of interpretation. As neither the 

single-hole nor the cross-hole test results 

have been affected by any skin effect of 

consequence, the scale effect is unrelated 

to phenomena associated with borehole 

drilling and completion. The observed 

permeability scale effect at the ALRS 

appears to be real. 

• The question was asked to what extent 

can one interpret the observed 

permeability scale effect at the ALRS by 

means of a stochastic scaling theory due 

to Di Federico and Neuman (1997) and 

Di Federico et al. (1999), which views 

log permeability as a truncated random 

fractal. There is considerable uncertainty 

about the magnitude of the observed 

scale effect at the ALRS, making it 

difficult to answer this question 

conclusively. However, there is 

sufficient correspondence between the 

site data and the theory to suggest that 

the latter may indeed provide a viable (if 

not complete) explanation of the 

observed scale effect. 

These findings amount to a conceptual 

model of the pneumatic characteristics of, 

and airflow through, unsaturated fractured 

tuffs at the ALRS. It however leaves many 

fundamental questions unanswered, 

implying that the present conceptual model 

is incomplete and uncertain. Among 

conceptual issues that require resolution are 

the following: 

• To what extent do conceptualizing the 

air permeability and air-filled porosity of 

unsaturated fractured tuffs at the ALRS 

as multiscale random fields, defined over 

a continuum, apply to saturated and 

unsaturated hydraulic conductivities and 

porosities? 

• To what extent do bulk air permeabilities 

and air-filled porosities represent 

hydraulic conductivities and porosities 

of the fractured rock? 

• How can unsaturated hydraulic rock 

properties be determined under field 

conditions? 

• What roles do fractures play in 

controlling these properties? 

• Would a (stochastic or deterministic) 

continuum representation of these 

properties be appropriate? 

• How variable are these properties in 

space? Could their variability be 

described geostatistically, or would there 

be a need to consider discrete features 

such as fractures or channels in the 

conceptual model of medium 

heterogeneity? 

• What role does preferential flow play 

under saturated and unsaturated 

conditions? Can preferential flow 

channels or unstable fingers be seen, 

identified and characterized in the field? 

• Is there a need to identify fractures and 

preferential-flow-channels discretely? If 

so, is this feasible, and how? 

• What role does dispersion play in the 

movement of tracers with water? Can it 

be predicted by means of continuum 

stochastic concepts? 

• What role does matrix-fracture 

interaction play during saturated and 

unsaturated water flow and solute 

transport in fractured porous rocks? Are 

dual porosity, dual permeability and/or 

matrix diffusion active during such flow 

and transport? 

• Can these interaction phenomena be 

detected in the field, and how? 

These fundamental questions of conceptual 

understanding cannot be answered without 

conducting appropriate experiments with 

water and dissolved tracers at the ALRS. 
178 

To address them, a long-term ponding 

infiltration and tracer experiment was 

conducted at the site by Yao et al. (2002). 

Here we provide a brief synopsis of the 

experiment and our findings. 

11.2.2 Experimental Setup 

A 9 x 9 meter plot was divided into nine 3 x 

3 meter subplots and instrumented as shown 

in Figure 11-28. An air compressor was used 

to expose the surface of the rock and major 

fractures. Ponding started on November 12, 

1999 and bromide tracer was added on May 

30, 2000. Ponding ceased on August 28, 

2001. The advance of the water front was 

measured with tensiometers and a neutron 

probe within each subplot. Solute movement 

was determined based on solution samples 

collected from suction lysimeters at various 

locations and depths.

Figure 11-29 shows cumulative outflow 

from the pond over each subplot. Examples 

of neutron count and soil potential variations 

with depth are given for subplot 1 in Figures 

11-30 and 11-31, respectively. Neutron 

counts were also available for vertical and 

slanted deep boreholes in and around the 

experimental plot which are not indicated in 

Figure 11-28. Figure 11-32 shows bromide 

concentrations versus time at 3 and 5m 

depth beneath pond 1. 

11.2.3 Findings 

The long-term ponding infiltration and 

tracer experiment at the ALRS has shown 

that arrivals of water fronts were clearly 

detectable from the tensiometer data but not 

as clearly from the neutron probe data. 

Higher infiltration rates were found in plots 

with apparent surface fractures than in other 

plots. After 650 days of flooding with 

bromide labeled water, 14 of the 45 suction 

lysimeters have indicated the arrival of 

surface applied bromide, but only 3 have 

indicated early arrival. Concentration spikes 

of bromide were detected in two lysimeters 

under plot 8 (at 2 and 3 m) and in one 

lysimeter under plot 9 (0.5 m). Though there 

was a clear increase in matric potential 

under all plots at depths of 0.5 and 1.0 m, 

there was no measurable bromide at 6 of the 

9 lysimeters at these depths. High apparent 

dispersivity values were calculated from 

bromide breakthrough data at eight sampling 

points using a one-dimensional model with 

uniform velocity. The relatively high 

infiltration rates in some plots, early arrivals 

of bromide spikes at deep sampling points 

without showing up at shallow points, and 

high apparent dispersivity values at some 

points suggest the presence of a complex 

system of preferential flow paths at the site. 

Though fracture data beneath the plots are 

lacking, it stands to reason that these 

preferential flow paths are associated with 

fractures. 
179 

Figure 11-28. Experimental site with plots, instrumentation, and surface exposure of fractures. 

Figure 11-29. Cumulative outflow from each subplot pond. 
180 

Figure 11-30. Plot 1 neutron probe count ratios versus depth. 

Figure 11-31. Plot 1 soil water potentials versus depth. 

Figure 11-32. Plot 1 bromide concentrations versus time at 3 and 5 m depths. 
181 

11.3 Fanay-Augères

This case study is based on a paper by Ando 

et al. (2002). 

11.3.1 Introduction 

Cacas et al. (1990a-b) described flow 

experiments in a uranium mine at FanayAugères near Limoges, France. The 

experiments were conducted in a 100 m 

horizontal drift at a depth of 150 m within 

highly fractured granite. Ten 50 m long 

boreholes were drilled radially from the drift 

in three vertical planes at distances of 25.9 m 

and 23.7 m from each other (Figure 11-33). 

Planes 1 and 3 contain three boreholes each 

and plane 2 contains four boreholes. Fractures 

were mapped in the drift and observed in

borehole cores. Water was injected into 180 

packed-off chambers of length 2.5 m and 

diameter 7.6 cm at 10 MPa. Similar injection

tests were conducted in 50 chambers of length 

10 m and 10 chambers of length 50 m. Head

was monitored in 68 chambers of length 5 m

for over 400 days starting February 2, 1985 

and ending March 3, 1986. They reveal the 

presence of a near-horizontal water table 

(zero-pressure isobar) a short distance above 

the drift. Below it, the rock is fully saturated 

with water. 

Inflow into the drift was measured 

continually before and after partial 

desaturation due to borehole drilling. It 

exhibits two stages of near steady state with 

an intervening transient flow period (Figure 

11-34). Two tracer tests were conducted by 

injecting different tracers into 5 m long 

chambers in boreholes F2 and F3 (Figure 

11-35). Chambers are identified by symbols 

in which the first character refers to chamber 

number (increasing toward the drift, as 

illustrated in Figure 11-35) and the rest 

identify the borehole (for example, 4F3 is 

chamber 4 in borehole F3). During the first 

test that commenced on September 19, 1985, 

two tracers were injected into 7F2 and one 

into each of 6F2, 4F2, 1F2 and 4F3. During 

the second test that commenced on 

December 19, 1985, one tracer was injected 

into each of 3F2, 6F3, 3F3 and 1F3. 

Measurements included tracer 

concentrations at location "A" (a plastic 

sheet glued to the roof of the gallery near its 

intersection with borehole F2 at 50 m as 

measured from the north end of the drift); 

chambers 1 - 7 of borehole F4; rhodamine 

WT, injected into 4F3 at 50 m, along the 

drift intervals 43.5 m - 66.5 m on October 

13 and 70.0 m - 99.0 m on October 30, 

1985; location "B" in a 25m long drain 

along the gallery extending from the 

injection plane of boreholes F1 - F4 to the 

plane of boreholes F5 - F7; and location "C" 

along a 25 m - 50 m segment of the gallery 

downstream of the injection plane (Figure 

11-35). 

A stochastic discrete fracture network model

was developed for the site by Cacas et al.

(1990a-b). In their model, fractures were

represented by circular disks with randomly 

prescribed radia and orientations based on 

statistics inferred from in situ geometric 

observations (Figurer 11-36). Flow in each 

disk was one-dimensional, purportedly to 

represent channels. The probability 

distribution of flow rates through fractures 

across the network was assumed to be lognormal, in analogy to the distribution inferred 

from steady state flow rates (and hydraulic 

conductivities) measured during 2.5 m scale 

constant-pressure packer tests. This inference 

ignored about 1/3 of the test values, which 

fell below the detection limit of

10 K 6 10− = × m/s (Figure 11-37). In 

postulating the analogy, no notice was taken 

of the fact that correlation between the 

intensity of fracturing (as represented by 

fracture density, measured by number of 
182 

fractures per unit length of borehole) and 

hydraulic conductivities from the 2.5 m

scale tests is highly tenuous (Figure 11-38). 

A similar lack of clear correlation between 

fracturing and permeability has been noted 

by us at other sites (Neuman, 1997; Chen et 

al., 2000), throwing into question the 

validity of models that relate the latter to the 

former.

Figure 11-33. Layout of drift and boreholes (after Cacas et al., 1990a). With permission, AGU. 

Figure 11-34. Measured inflow rates into drift and periods of tracer tests. 
183 

Figure 11-35. Tracer release intervals and monitoring locations in drift 

(after Cacas et al., 1990b). With permission, AGU. 

Figure 11-36. 3-D network of circular disks with 1-D flow and transport paths 

(after Cacas et al., 1990a). With permission, AGU. 
184 

Figure 11-37. Normal probability plot of natural log-hydraulic conductivity from 2.5 m packer tests. 

Figure 11-38. Scatter plot of fracture density (number per unit length of boreholes) 

versus hydraulic conductivity from 2.5 m scale packer tests. 

Cacas et al. (1990a) generated 17 random

networks of fractures within 10 m scale 

cubes. No spatial correlation between the 

fractures was considered even though the 

2.5 m scale hydraulic conductivities (to 

which the fracture statistics were deemed 

related) are strongly auto-correlated, as we 

show later. By computing steady state flow 

across each cube under an externally 

imposed hydraulic gradient (and noting, as 

should have been anticipated by 

construction, that cube orientation has little 

effect on the results) the authors had 

obtained 17 scalar equivalent hydraulic 

conductivity values that they considered 

representative of 10-m scale rock volumes. 

Ke
185 

The values were treated as a sample 

from a statistically homogeneous and isotropic 

multivariate log-normal random field across 

the experimental site. An approximate threedimensional stochastic continuum formula 

was then used to calculate an effective scalar 

hydraulic conductivity 

Ke

( ) 2 8 1 / 6 1.6 10 K K eff g σ −  +  ×

dimensional networks of one-dimensional 

do

s

ly to 

n B

om

ir

acas et al. (1990b) stated that it would not

 of

d

r 

than 10 m.

 m/s for the

rock mass on scales much larger than 10 m. 

Next, a circular finite element model was 

constructed on a plane transverse to the drift. 

Pressure head in the drift was set equal to 

zero and head was prescribed along 7 

concentric circular boundaries with radii 

increasing from 5.5 m to 47.5 m, based on 

borehole measurements projected onto the 

model plane (Figure 11-39). For each of the 

7 boundaries, the two-dimensional model 

was calibrated against measured flow into 

the drift (it is not clear how well did the 

calibrated models reproduce heads projected 

onto the interior of the model plane, a 

question we address later in the context of 

our own model). The calibrations yielded 7 

equivalent uniform scalar hydraulic 

conductivity estimates that tended to 

decrease with domain size. Even though the 

estimates had not fully stabilized at the 

maximum radius of 47.5 m, their smallest 

value of m/s was taken to 

represent the large-scale hydraulic 

conductivity of the rock. That this estimate 

is within the range m/s of 17 

values generated by means of the threedimensional discrete network model, and is 

close to the stochastic 

average m/s of these values, 

was considered by Cacas et al. (1990a) to 

constitute a validation of their fracture 

network model with respect to flow. 

8 1.8 10− ×

8 1.5 2.3 10− − ×

Ke

8 1.6 10 Keff−  ×

To simulate tracer transport, Cacas et al.

(1990b) generated 20 random threechannels interconnected at 10,000 nodes. 

The network covered part of the domain, 

which was subjected to prescribed head 

boundary conditions (Figure 11-40) that

not appear to be entirely consistent with 

those observed (Figure 11-39). Tracer wa

advected through each network by particle 

tracking. Local-scale dispersion was 

simulated by routing particles random

one of several downstream channels. Largescale dispersion was simulated by drawing 

random particle residence times in channels

from a normal distribution with variance 

proportional to an (apparently arbitrary) 

dispersivity of 0.8 m. The mean residence

time was made proportional to a retardation

coefficient that was calibrated against peak 

arrival times at location A for tracers

injected into 7F2 and 6F2, and locatio

for tracers injected into 6F3 and 4F3. As 

injection into 3F3 at a distance of 35 m fr

the drift would have required including 

25,000 fractures in the model, it was not

simulated. Measured and simulated mass

recovery fractions did not match well and

the paper includes no comparison between

simulated and observed tracer breakthrough

curves. Their finding that the model was 

able to reproduce observed breakthrough 

durations with some fidelity was taken by

Cacas et al. to constitute a validation of the

fracture network model with respect to 

solute transport. 

C

be realistic to interpret the tracer experiments

at Fanay-Augères without accounting for 

discrete channels in the model. In the view

Cacas et al. (1990a), continuum models are

not capable of adequately interpreting small 

scale measurements; at best, they may be use

at some distance where only average behavior 

is required. Judging by their papers, Cacas 

et al. considered the continuum approach to

be invalid at Fanay-Augères on scales smalle
186 

Figure 11-39. Projection of observed piezometric heads (in meters) onto transverse model plane 

(after Cacas et al., 1990a). With permission, AGU. 

Figure 11-40. Boundary conditions used for simulating tracer injections 

e obtained data relating to the flow and 

e

t

and transport at Fanay-Augères by viewing 

proposed by

(after Cacas et al., 1990b). With permission, AGU. 

W

transport experiments at Fanay-Augères 

courtesy of Professor G. de Marsily of th

University of Paris in France. In the presen

paper, we utilize these data to model flow 

the fractured rock as a stochastic continuum

on the nominal 2.5 m support 

(measurement) scale of available packer test 

data. This approach, originally
187 

s 

ds to 

le 

ort in 

situ 

t 

ow 

ied 

t al.

 Hyun et 

or transport are affected by a few 

ominant features (fracture zones, faults or 

e 

d 

ch may 

.

re 

ne 

c 

e 

eat 

owing is an abbreviation and 

nthesis of material contained in the M.S. 

95). 

cator Analysis and Simulation 

of Hydraulic Conductivities 

We start b

onductivities (K) derived at Fanay-Augères 

le intervals 

ned 

n 

und areas 

Neuman (1987, 1988), is based on the 

recognition that (a) it is expensive and 

difficult to characterize the geometry, 

hydraulic and transport properties of 

individual fractures on such small scale

across a site, (b) fracture geometry ten

be a poor indicator of how such small sca

hydraulic and transport properties vary 

across a site, and therefore (c) it is neither 

feasible nor necessary to collect detailed

information about individual fractures on 

such small scales for the purpose of 

analyzing flow and transport on similar or 

larger scales. Instead, flow and transp

many fractured rock environments are 

amenable to analysis by continuum models 

that account adequately for medium 

heterogeneity. This includes the stochastic 

continuum approach, which treats in 

measurements of hydraulic conductivity 

(and possibly other rock properties) as a 

correlated random field and obviates the 

need for either detailed information abou

fracture geometry or assumptions about h

individual fractures control flow and 

transport. The concept, and methods of 

analysis based upon it, have been appl

successfully to fractured granites by 

Neuman and Depner (1988) and to 

unsaturated fractured tuffs by Chen e

(2000), Vesselinov et al. (2001) and

al. (2002). 

When flow 

d

dykes) on relatively large scales, one has the 

option of embedding them as discrete slabs 

within the model and treating the internal 

properties of each feature as spatially autocorrelated random fields (Neuman, 1997). 

Though a fault zone has been identified at 

the Fanay-Augères experimental site (Figur

1), neither Cacas et al. (1990a-b) nor we 

have found it necessary to include this fault 

in our models. In fact, we show that it is 

possible to reproduce a larger selection of 

experimental results than those considere

by Cacas et al. merely upon modeling the 

rock as a statistically homogeneous 

continuum in two dimensions. Our results 

demonstrate that a continuum approa

be well suited for the analysis of flow and 

transport in fractured rock. This does not 

constitute a validation of the continuum 

approach, just as the results of Cacas et al

fall short of validating the discrete fractu

approach. Instead, the two sets of results 

illustrate jointly the well established 

principle that an open system, especially o

as complex as fractured hydrogeologi

environments tend to be, cannot be 

described uniquely on the basis of spars

data and need not be described in gr

detail to capture its salient behavior by a 

model. 

The foll

sy

theses of Kostner (1993) and Ando (19

Readers interested in details beyond those 

presented here are encouraged to consult 

these theses. 

11.3.2 Indi

y subjecting hydraulic 

c

from packer tests in 2.5 m boreho

to indicator geostatistical analysis. Such 

analysis treats the data as a sample from a 

spatially auto-correlated random field defi

on a continuum. We use indicator rather tha

standard geostatistical analysis to account for 

the spatial distribution of hydraulic 

conductivities below the detection limit 

(Figure 11-37), which may have 

considerable impact on the movement of

tracers by forcing it to migrate aro

of low permeability. 
188 

The first step in the analysis is to transform

K ( ) x into a binary indicator function 

( ; ) c I x K equal to 1 if ( ) K K≤ c x and 0

, where Kc ed

utoff) va and x is a vec

coordinates. We subdivide the available 

range of K data into 7 classes defined by

indicator cutoffs, listed in Table 11-10 

together with the percent of K data that

lie at or below each cutoff. The first cutoff is

the detection limit and the third is the 

median of the data. Indicator analysis 

requires no distributional assumptions

K. Instead the expectation of ( ) ; c

if

K

indicator (c tor of

6

that

about

( ) K > c x is a specifi

lue

I x K , 

conditional on a set of measur

( ) 1 2 , ,... K K KN

T K = where K is a vect

is an estimate of the 

conditional cumulative probability 

distribution of K ( ) x , 

( )

ements 

or and

T denotes transpose,

E I⎡ x;K ⎡ ( ) T

c K P K Kc ⎤ = ≤ ⎤ ⎣ ⎦ ⎣ ⎦ x T K . 

igure 11-37 shows that if one ignores

, the

cumulat

rst 

d

he next step is to infer from the indicators of

ll 6 

te

mnis noted by Cacas et al. (1990a), the spatial

stify 

all the

e

dicator kriging yields a best linear 

F

values of Y K = ln at the detection limit

remaining ive distribution is nearnormal with slight positive skewness. The 

indicator approach makes it possible to 

extrapolate this distribution below the fi

and above the sixth cutoffs with the aid of 

suitable parabolic and hyperbolic models, 

respectively. The augmented distribution 

yields a variable probability of encountering 

a hydraulic conductivity below the detection 

limit and a finite probability of encountering 

one above the highest value measured. It 

increases slowly above the sixth cutoff an

thus preserves the statistical properties of the

data including their slight positive skewness. 

T

transformed data sample and theoretical 

(model) indicator variograms. Variogram

models for all the cutoffs fit isotropic

spherical models with ranges (measure

spatial auto-correlation) that vary from 18 m

to 23 m. Variogram models for cutoffs 1 – 5 

have sills (total variances) varying form 0.16

to 0.25 and nuggets (variances of white 

background noise) varying from 0.10 to 

0.16. The variogram of cutoff 6 has much

smaller sill (0.05) and nugget (0.02) values

which, however, are based on a relatively 

small sample (Table 11-10) and are 

therefore less reliable. Normalizing a

variograms with respect to the sill of the 

median variogram shows that they are qui

similar (Figure 11-41). As the sill has no 

effect on indicator kriging weights, we 

calculate them on the basis of a single o

directional variogram corresponding to the 

median cutoff (Figure 11-42). 

A

distribution of near steady state heads in the 

three parallel planes that contain the

boreholes are sufficiently similar to ju

modeling flow into the drift in two 

dimensions. We do so by projecting

available K and head data onto a single plan

transverse to the drift. 

In

unbiased estimate of the conditional 

probability ( ) P K Kc ⎡ ≤ ⎤ ⎣ ⎦ x T K for ea

cutoff (G\m Srivastava

1990). We estimate the latter on a 44

ch

ez-Hern<ndez and ,

× 47

grid of 2 m × 2 m cells in a plane tran vers

to the drift using an algorithm found in 

Deutsch and Journel (1998). For exampl

Figure 11-43 shows how our estimate of the

probability that hydraulic conductivity 

exceeds the fifth cutoff varies across the

plane. One can use the results to map out

estimates of the conditional mean of K or 

other statistical moments such as median o

mode.

s e

e,

r
189 

Figure 11-41. Omni-directional indicator variogram models of Y K = ln 

normalized with respect to sill of variogram model corresponding to indicator no. 3. 

Table 11-10. Indicator cutoffs of hydraulic conductivity data 

and percent of data not exceeding each cutoff. 

Figure 11-42. Omni-directional sample and spherical model semivariogram for indicator no. 3. 
190 

Figure 11-43. Kriged estimate of probability that hydraulic conductivity exceeds the fifth cutoff. 

Finally, we use conditional Monte Carlo 

simulation to generate 100 stochastic 

realizations of the hydraulic conductivity 

field on the transverse plane. We do so by 

means of a sequential algorithm (Deutsch 

and Journel, 1998) that honors hard 

indicator data, hard indicator constraints, the 

augmented probability distribution of the 

data, and the indicator variogram of the 

median cutoff. The simulations are 

performed on a rectangular grid of 34× 36 

cells measuring 2.5 m × 2.5 m

corresponding to the nominal support scale 

of the packer test data. Both the mean and 

the variance of the generated fields stabilize 

after only about eighty realizations. Two of

the realizations are depicted in Figure 11-44. 

Whereas the sampled data do not fall below 

the detection limit, many of the simulated 

values do. Their histograms and probability 

plots (illustrated for the above two 

realizations in Figure 11-45) show bimodal 

distributions and a slight positive skewness 

as do the sampled data. The only notable 

difference between the original and 

simulated histograms is the wider spread 

(tail) of the latter below the detection limit. 

The generated hydraulic conductivity data 

under the detection limit are log-normally 

distributed.

11.3.3 Geostatistical Analysis and 

Simulation of Flow 

To simulate flow and transport at FanayAugères we limit ourselves to the saturated 

zone. We work with pressure heads rather

than hydraulic heads to allow easy 

identification of the water table as a zeropressure isobar. Near steady state pressure 

head data collected during the first and 

second stages of the experiment, prior to and 

following a transitional nonsteady state 

period (Figure 11-34), are represented well 
191 

by omni-directional residual semivariograms 

with second-order polynomial drifts in the 

transverse (vertical) plane (depicted for 

stage 1 in Figure 11-46). We use them to 

estimate pressure head on the above grid of 

34× 36, 2.5 m 2.5 m cells by kriging, and 

to identify the position of the water table at 

each stage (as shown for stage 1 in Figure 

11-47). Kriging estimation errors 

(represented by their standard deviations) 

are seen to increase with distance from the 

× borehole monitoring intervals.

 

Figure 11-44. Two random realizations of hydraulic conductivity honoring the data. 

 

Figure 11-45. Cumulative distribution of hydraulic conductivity estimates in Figure 11-44. 
192 

Figure 11-46. Sample (dots) and spherical model (solid curve) omni-directional variogram 

of pressure head at stage 1 with quadratic polynomial drift. 

 

Figure 11-47. Kriged pressure head at stage 1 (left) 

and standard deviations of corresponding kriging errors (right). 

Next we prescribe zero pressure head along 

the water table and the drift, and nonzero 

estimated pressure heads along the remaining 

boundaries of the saturated zone (Figure 11-

48). We then simulate flow through this zone 

in each of the 100 generated permeability 

fields at each stage using the finite element 

code SUTRA (Voss 1984, 1990). Pressure

and total heads corresponding to one of these

simulations (no. 48 at stage 1) are depicted in 

Figure 11-49. 

Figure 11-50 shows histograms of simulated 

inflow rates into the 100 meter long drift at 

stages 1 and 2. Corresponding statistics are

given in Table 11-11. Cumulative frequency 

plots (not shown) suggest that the rates are

close to being log-normally distributed with

geometric mean values of 2.09 l/min and 1.56 

l/min, respectively. The measured inflow 

rates of 2.42 l/min (September 1985) and 1.48 

l/min (December 1985) during the two stages

correspond closely to the most frequently 

simulated values. Simulations using 

geometric mean hydraulic conductivities over 

all 100 realizations at each stage (which vary

from grid cell to grid cell) yield inflow rates 

of 2.14 l/min and 1.35 l/min, respectively. 

These are very close to the measured rates.
193 

Figure 11-48. Flow model. 

Figure 11-49. Conditional simulation no. 48 of pressure head (left) and total head (right) 

at stage 1. 

 

Figure 11-50. Histograms of simulated drift inflow rates at stages 1 (left) and 2 (right). 

∇ indicates measured value. 
194 

Table 11-11. Statistics of simulated inflow rates into drift (in l/min).

Stage-1 Stage-2 

Mean of 100 realizations 10.09 5.97 

Maximum of 100 realizations 34.49 23.18 

Minimum of 100 realizations 1.57 0.73 

Median of 100 realizations 8.25 4.90 

Mode of 100 realizations 2 to 8 1 to 5

Using geometric mean hydraulic conductivities 2.14 1.35 

Measured 2.42 1.48 

Tables 11-12 lists statistics of differences 

between simulated and kriged pressure head 

values at stages 1 and 2 for five hydraulic 

conductivity realizations (nos. 15, 41, 48, 72, 

95). Table 11-13 lists corresponding statistics 

of differences between simulated and 

measured pressure heads. The five 

simulations yield inflow rates into the drift, 

and pressure head residuals, that are closest to 

those actually measured. Among these five 

simulations, no. 48 yields the best fit to 

measured drift inflow rates at both stages 

(Figure 11-51). Corresponding scattergrams 

of simulated versus measured head residuals 

in Figure 11-52 show an acceptable fit (we do 

not know how well the model of Cacas et al. 

(1990a) reproduces observed heads in the 

interior of the flow domain). We therefore 

adopt conductivity realization no. 48 for the 

purpose of simulating transport.

Table 11-12. Statistics of pressure head residuals 

(differences between simulated and kriged values in meters). 

Realizatio

n 

#15 #41 #48 #72 #95 

Stage 1 2 1 2 1 2 1 2 1 2 

Mean 0.77 0.95 1.19 1.26 1.30 1.29 1.251

. 

1.28 1.19 1.19

Variance 2.39 1.78 3.09 2.42 2.72 2.13 1.89 1.58 3.17 2.25

Max 3.73 3.80 5.07 5.45 5.09 4.32 5.587 4.77 4.54 4.05

Min -4.90 -3.34 -4.91 -3.81 -4.70 -3.49 -2.52 -2.23 -4.28 -2.64

Table 11-13. Statistics of pressure head residuals 

(differences between simulated and measured values in meters). 

Realizatio

n 

#15 #18 #48 #72 #95 

Stage 1 2 1 2 1 2 1 2 1 2 

Mean 1.00 1.30 1.00 1.27 1.00 1.38 1.55 1.71 1.60 1.68

Variance 4.70 2.95 6.38 2.09 6.83 1.96 1.55 1.72 5.13 3.44

Max 3.70 3.65 4.67 5.40 4.77 4.12 4.30 3.78 4.48 4.18

Min -6.52 -4.62 -9.07 -6.82 -6.89 -4.82 -2.96 -1.72 -5.26 -3.52
195 

Figure 11-51. Measured and simulated flow rate into drift. 

11.3.4 Simulation of Transport 

For the simulation of transport we select a 

single realization (no. 48) that yields the best

reproduction of inflow rates into the drift and 

a reasonable reproduction of measured heads 

at both stages of the analysis. We focus on

four tracer tests during which injection took 

place into chambers 3F3, 4F3, 6F3 and 7F2. 

According to Cacas et al. (1990b), these 

four tests are the most reliable and offer the 

most complete database among all tracer 

tests at the site. Since their model was 

limited to a relatively small rock volume, 

Cacas et al. were unable to simulate test 

3F3, analyzing instead test 6F2. Our model 

has no such limitation (mainly because we 

do not consider individual fractures, and to a 

lesser extent because we limit our analysis 

to two dimensions) and we therefore analyze 

test 3F3. We add that injection during test 

6F2 took place above the water table, a fact 

not explicitly considered by Cacas et al.

The periods during which breakthrough of 

the four tracers injected into 3F3, 4F3, 6F3 

and 7F2 were monitored are indicated by 

bars in Figure 2. Injection took place for 30 

minutes at the start of each monitoring 

period. To mimic the transient stage during 

tracer tests 4F3 and 7F2, we interpolate 

linearly over time between the preceding 

steady state head configuration of stage 1 

and the subsequent steady state head 

configuration of stage 2. 

We use SUTRA and the two-dimensional 

grid in Figure11-48 to simulate each of the 

four tracer tests. Neglecting the third 

dimension in our model means that we do 

not allow the tracer to advect or disperse 

perpendicular to our planar flow domain. 

For example, the distance from 7F2 to 

observation site B is 5m in the two 

dimensional section, but in reality varies 

from 5m to 25.5m with distance along B, 

implying that transport at the site is threedimensional. We however achieve 

sufficiently accurate results for our purposes 

with a two-dimensional model. 

Maloszewski and Zuber (1993) estimated 

porosity indirectly to range between 

 and . We estimate -4 1.25×10 -3 1.10×10
196 

effective porosity and longitudinal as well as 

transverse dispersivities in three steps:

1. Select dispersivities and modify 

effective porosity to fit peak arrival 

times (all tracers were conservative).

2. Modify longitudinal and transverse 

dispersivities to fit observed shapes of 

breakthrough curves. 

3. Repeat steps 1 and 2 until a satisfactory 

fit to the data is obtained.

Figure 11-53 compares computed (curves) 

and measured (solid squares) breakthrough 

curves, normalized with respect to 

maximum concentration, for tracer tests 

6F3, 4F3, 3F3 and 7F2, respectively. 

Concentration data for 4F3 and 3F3 were 

derived by conversion from fluorescence 

data. The corresponding best-fit effective 

porosities are , , 

and , all but the first of which are 

within the range proposed by Maloszewski 

and Zuber (1993). The large effective 

porosity in 6F3 suggests that transport may 

be taking place through a preferential path 

of high permeability, such as a channel or 

wide fracture. Longitudinal dispersivity is 

17.5 m and transverse dispersivity is one 

tenth this value in all four cases. We

attribute the high dispersivities in part to 

mixing within the drift and to neglect of the 

third dimension in our model. 

-3 8.0×10 -4 8.0×10 -4 6.0×10

-4 4.0×10

Cacas et al. (1990b) did not provide 

simulated breakthrough curves. We

therefore compare our simulated peak 

arrival times with those measured (Figure 

11-54) and computed by Cacas et al. (Table 

11-14). It appears that our two-dimensional 

stochastic continuum model reproduces 

observed behavior at least as well as does 

the three-dimensional stochastic discrete 

network model of these authors, if not 

better.

 

Figure 11-52. Scattergrams of simulated (using conductivity realization no. 48) 

versus measured heads at stages 1 (left) and 2 (right). 
197 

 

 

Figure 11-53. Computed versus measured concentrations normalized by maximum 

for tests 6F3 (upper left), 4F3 (upper right), 3F3 (lower left), and 7F2 (lower right). 

Por = effective porosity, Alfl = longitudinal dispersivity, Alft = transverse dispersivity. 
198 

Table 11-14. Simulated and measured tracer arrival times. 

Tracer 

Number 

Simulated peak arrival 

time (hr) 

Measured peak arrival

time (hr) 

Peak arrival time from 

Cacas et al. (hr) 

6F3 748 757 1 to 900 

4F3 179 199 27 to 14900

3F3 236 274 out of computation 

Figure 11-54. Simulated versus measured peak arrival times. 

It might be possible to improve model fit 

further by conditioning a particular random 

conductivity field not only on measured 

conductivities as we have done but also on 

measured drift inflow rates and heads 

(G\mez-Hern<ndez et al., 1997, 2000) as

well as tracer concentration data (G\mezHern<ndez et al., 2002) via an automated

inverse procedure. We deem this

unnecessary because the much simpler 

approach we use yields simulations of flow 

and transport that are sufficiently accurate 

for our purpose of juxtaposing the stochastic 

discrete network and continuum approaches. 

11.3.5 Conclusions 

1. Our work confirms the suitability of

stochastic continuum modeling for the 

analysis of flow and transport in 

fractured crystalline rocks at FanayAugères in France. Claims in the 

literature that only discrete fracture 

network models are suitable for this 

purpose are unfounded. 

2. Stochastic continuum conceptualization 

avoids the need for detailed information 

about fracture geometry and assumptions 

about the nature of flow within 

individual fractures. Embedding such 

information and assumptions in a 

discrete fracture network model has 

proven not to have any advantage over 
199 

the stochastic continuum approach in the 

case considered here. 

Our results do not constitute a validation of 

the continuum approach just as those of 

Cacas et al. (1990a-b) fall short of validating 

the discrete fracture approach. Instead, the 

two sets of results illustrate jointly the well 

established principle that an open system, 

especially one as complex as fractured 

hydrogeologic environments tend to be, 

cannot be described uniquely on the basis of 

sparse data and need not be described in 

great detail to capture its salient behavior by 

a model.
200 

12 GUIDELINES FOR REVIEW OF HYDROGEOLOGIC MODELING 

AND UNCERTAINTY ANALYSES

12.1 Introduction

The following guidelines for review of hydrogeologic modeling and uncertainty analyses are 

intended to help insure that documents reviewed conform in a broad way to the methodology 

described in this report. This means that such documents should reflect a systematic and 

comprehensive approach to hydrogeologic conceptualization, model development and predictive 

uncertainty analysis. The guidelines cover all stages of model building and uncertainty analysis. 

Model building stages include regional and site characterization, hydrogeologic 

conceptualization, development of conceptual-mathematical model structure, parameter 

estimation on the basis of monitored system behavior, and assessment of predictive uncertainty. 

Uncertainty analysis includes assessment of estimation and predictive errors arising from the 

conceptual framework that determines model structure, the definition of model parameters, 

spatial and temporal variations in hydrologic variables that are either not fully captured by the 

available data or not fully resolved by the model, and the scaling behavior of hydrogeologic 

variables. 

The guidelines are generic but designed to be of practical use to NRC staff in their review of 

decommissioning plans and performance assessment of high-level and low-level radioactive 

waste disposal sites as well as uranium recovery facilities. For this purpose, the guidelines are 

cast in the context of a framework that is relevant to NRC staff review and performance 

evaluation needs. The context is defined in terms of corresponding performance measures, 

hydrogeologic analyses that are needed to assess them, the desired reliability of such 

assessments, and the expenditure (in time, effort and money) that is allowed to achieve it. 

12.2 Summary of Guidelines

The guidelines are summarized below and explained further in subsequent sections of this 

chapter. The guidelines are arranged in an iterative sequence of steps that correspond to our 

proposed strategy of model building and uncertainty assessment. 

• Define the hydrogeologic context of the problem. 

• Collect a broad range of regional and site data from public and private sources that are 

relevant to the hydrogeologic context of the problem. Provide a detailed rationale for your 

choice of data. 

• Interpret the data in a way that leads to several plausible conceptual models (hypotheses 

and assumptions) of regional and site hydrogeology in three spatial dimensions. Provide a 

detailed rationale for each interpretation and explain why no other hydrogeologic 

interpretations of the data would be plausible. 

• Verify in detail that none of the selected conceptual models are in conflict with any of the 

data; discard those that are. 
201 

• Rank the conceptual models of regional and site hydrogeology in the order of their 

plausibility in light of the data, regardless of context. Provide a detailed rationale for your 

ranking and discard models that appear much less plausible than those retained. 

• Starting with the most plausible conceptual model and continuing down the ranks: 

▫ Develop a complete, coherent and internally consistent conceptual-mathematical model 

of three-dimensional transient flow and transport that is as simple as warranted by the 

context of the problem and the data. Provide a detailed rationale for your schematization 

of regional and/or site hydrogeology; choice of governing, state and constitutive 

equations; type and space-time distribution of forcing terms (sources, initial and 

boundary conditions); parameterization (functional form and number of parameters); and 

all other conceptual-mathematical aspects of the model. Discard models that do not 

lend themselves to such mathematical description. 

▫ If mathematical modeling in three spatial dimensions and/or in time is considered 

unnecessary or computationally infeasible, provide a detailed explanation of how the 

three-dimensional transient conceptual-mathematical model reduces to a twodimensional and/or steady state model. In particular, demonstrate (preferably through a 

mathematical formalism and/or computationally) how the lower-dimensional model 

captures all relevant aspects of the original three-dimensional transient model. Avoid 

one-dimensional models at this stage. 

▫ Select a suitable computational code or algorithm to represent the conceptualmathematical model. 

▫ Associate objective or subjective prior statistics with all input parameters and forcing 

terms that are considered to be uncertain. Provide a detailed rationale for considering any 

of these inputs to be known with certainty. 

▫ Test the conceptual-mathematical model for qualitative consistency with the available 

data through computational exploration and visualization in at least two spatial 

dimensions, within a plausible range of input parameters and forcing terms. If 

inconsistent, repeat the previous steps in modified form (by altering and/or refining the 

hydrogeologic scheme and mathematical description of flow and/or transport) or discard 

the underlying conceptual model and select an alternative. 

▫ Use an inverse (preferably Maximum Likelihood) approach to calibrate the 

computational model against observational (monitoring) data. This should yield 

posterior estimates of the model parameters, statistical measures of their estimation 

errors and associated model quality criteria. If the model cannot be properly calibrated 

against reliable data, repeat the previous steps in modified form (by altering and/or 

refining the hydrogeologic scheme and mathematical description of flow and/or 

transport) or discard the underlying conceptual model and select an alternative. 

▫ If feasible, confirm aspects of the model by predicting system behavior under 

conditions other than those used for calibration and comparing with observed system 

behavior. Quantify model predictive uncertainty and check the extent to which the 

observations lie within corresponding uncertainty bounds. If not, repeat the previous 

steps in modified form (by altering and/or refining the hydrogeologic scheme and 

mathematical description of flow and/or transport) or discard the underlying conceptual 

model and select an alternative. 
202 

•

•

•

• Compare all retained calibrated models on the basis of suitable model quality criteria. 

Discard models that are clearly inferior to others. 

• Use each retained calibrated model to predict system behavior as well as relevant 

hydrogeologic performance measures under various scenarios of relevance to the problem, 

and quantify the corresponding predictive uncertainty. 

• Use Maximum Likelihood Bayesian Model Averaging to render joint predictions by all 

the retained calibrated models, and to assess their joint predictive uncertainty. 

Acknowledge the conditional nature of the predictions and their corresponding uncertainty 

measures. 

• If none of the models are deemed acceptable, perform a sensitivity analysis to help identify 

the type and quantity of additional site data that might materially enhance their reliability 

and credibility. Decide based on the potential benefit (in terms of bias and uncertainty 

reduction) and cost (in terms of time, effort and money) of such data whether or not to collect 

them and how. 

• If and when new data of significant weight are obtained, repeat the iterative process till its 

cost-benefit ratio reaches a value that is not considered worth exceeding. 

The following provides additional detail concerning each guideline. 

12.2.1 Define Hydrogeologic Context 

The first step in model development is to define its context and purpose. Within the context of 

NRC staff criteria (see Appendix A), the purpose of hydrogeologic site models is to help one 

analyze, qualitatively and quantitatively, subsurface flow and transport at a site in a way that is 

useful for review of decommissioning plans and performance of high-level and low-level 

radioactive waste disposal sites and uranium recovery facilities. For this purpose, the strategy 

developed in this report is cast in the context of a framework that is useful to NRC staff review 

and performance evaluation needs. The framework is defined in terms of performance measures 

identified by the NRC staff (see Appendix A). 

The contextual framework defines key questions to which groundwater flow and transport 

models are expected to provide answers; it helps narrow down the problem as well as the range 

and type of potential model applications. Each reviewed document should provide clear answers 

to the following questions: 

For what purposes will the model be used? The answer should include defining the nature 

and magnitude of an existing and/or potential problem, location of the problem area, its 

causes, potential short- and long-term solutions and/or remedies, their anticipated 

consequences and costs-benefits, issues needing resolution, and criteria by which a solution 

and/or remedy will be selected. 

To what hydrogeologic system will the models be applied? The answer should include 

defining the hydrogeologic environment that is or may potentially be affected by the problem 

and/or its solution or remedy, the corresponding site and surrounding environs of (potential) 

concern, and the regional hydrogeologic setting. 

Under what circumstances and scenarios will the modeled hydrogeologic system operate? 
203 

•

•

•

This requires defining predevelopment, current and potential undisturbed and disturbed site 

and regional conditions, and corresponding natural as well as anthropogenic influences, 

under which flow and transport would be modeled. 

What measures will be adopted to assess performance of the hydrogeologic system? This 

requires specifying measures and/or criteria related to site and regional hydrogeology that 

would be used to identify issues, potential solutions and/or remedies, their cost-efficiency 

and ability to meet regulatory requirements. 

What aspects of site and regional hydrogeology and flow/transport dynamics are expected to 

impact these performance measures, and how? The answer involves defining key elements 

of the hydrogeologic system, key flow and transport mechanisms, key natural and 

anthropogenic influences, and space-time scales that may potentially impact these measures, 

as well as the manners in which such impacts might occur. 

With what reliability, certainty and accuracy does one need to predict performance 

measures? How important a role will such measures play in the decision process? What is 

the worth (in time/effort/resources) of assessing performance measures for the site at 

specified levels of reliability/certainty/accuracy? These are the most basic questions that 

must be addressed before any site investigations and modeling efforts are initiated, as the 

answer would have a major impact on the time, effort and resources that could validly be 

expended in pursuit of performance assessment for the site. 

Performance measures are typically articulated within the broader context of a system in which 

groundwater is only one among several components. However, our strategy focuses solely on 

groundwater aspects of these broader criteria. Groundwater-related performance measures that 

need to be assessed by means of hydrogeologic flow and transport models are listed in Chapter 3. 

All of them require developing a good understanding, and reliable models, of subsurface flow 

and transport. One cannot do so without recognizing and considering the full complexity of a site 

before attempting to represent it by means of a simplified conceptual-mathematical model. This 

is true regardless of the specific performance criteria one needs to address. 

The process of model simplification or abstraction must consider the degree of reliability, 

certainty and accuracy with which given performance measures need to be predicted, their 

importance in the decision process, and the amount of time, effort and resources that assessing 

them at the specified levels of reliability, certainty and accuracy would justify in each specific 

case. These questions involve regulatory issues which might be best addressed by hydrologists 

through case-by-case dialogues with regulators. As such, they are not addressed by the strategy 

in this report. Instead, the strategy stipulates that these fundamental questions should be 

addressed and resolved through discussions with the NRC staff before any site investigation and 

modeling are initiated. This is so because the answers impact in a major way the degree to which 

one justifies the use of a simplified conceptual-mathematical model for what is ubiquitously a 

complex hydrogeologic system with three-dimensional, transient groundwater flow and 

contaminant transport. 

Therefore, it is very important the hydrogeologists first articulate these processes in their 

naturally complex setting as best understood in light of available site data and the state of 

prevailing hydrogeologic knowledge prior to postulating a groundwater flow and transport for 
204 

performance at a given site. Even if performance assessment is ultimately conducted with the aid 

of highly simplified conceptual-mathematical models of groundwater flow and transport, the 

strategy deems it essential for the credibility of the assessment that these models derive 

objectively (and if possible formally) from a more complete description of site hydrogeology, 

based on clearly reasoned and properly defended arguments. 

Rather than taking the attitude that a limited performance assessment goal justifies a limited view 

of hydrogeology, we propose that a comprehensive description of hydrogeology is required to 

properly adapt a hydrogeologic model to such a limited goal. 

12.2.2 Collect a Broad Range of Regional and Site Data 

Collect a broad range of regional and site data from public and private sources that are relevant 

to the hydrogeologic context of the problem. Provide a detailed rationale for your choice of data. 

The guideline is based on the recognition that it is often possible to postulate hydrogeologic 

conceptual models or hypotheses for a site on the basis of a broad range of publicly available 

geologic and geographic information about its surroundings. Additional conceptualization can be 

done on the basis of generic data about similar regions and the properties of similar materials 

elsewhere. Several such regional and generic sources of information are identified and discussed 

in this report. Yet each site is unique and so virtually guaranteed to reveal additional features, 

properties and behaviors when characterized in some detail locally. Hence the strategy considers 

local characterization essential for the postulation of acceptably robust conceptual hydrogeologic 

models for a site. The broader is the available database, the more robust is the conceptualization. 

Key data categories include site and regional physiography, topography, climate, meteorology, 

soils, vegetation, land use, geomorphology, geology, geophysics, surface and subsurface 

hydrology, inorganic and organic hydrochemistry, radiochemistry, natural and anthropogenic 

isotopes, remotely sensed data, etc. Of special relevance to hydrogeologic model development 

are regional and site data that allow one to define the distribution of hydrostratigraphic units on a 

variety of scales; their geologic structure; rock and soil types; their textural, physical, flow and 

transport properties; fluid types; their state of saturation, pressure, temperature and density; 

chemical constituents and isotopes; major contaminants in soil, rock and groundwater; and their 

sources. 

Site characterization data form the foundation on which one postulates one or more conceptualmathematical models for an area and assigns initial values to their input parameters (i.e., material 

properties and forcing terms such as sources, initial and boundary conditions). To test and 

compare these models among themselves qualitatively and/or quantitatively, one also needs 

monitoring data that constitute observations of actual hydrologic behavior at and around the site. 

Only with such data can one evaluate the ability of models to mimic real system behavior 

(qualitatively at the conceptual level, quantitatively at the conceptual-mathematical level), 

improve their ability to do so through calibration against the monitoring data, determine their 

optimum degree of refinement or complexity, rank them and assess their cumulative impact on 

predictive uncertainty, and compare them with each other. 
205 

Site characterization and monitoring data are expensive and difficult to collect, leading to a 

ubiquitous scarcity of hard site information. It is therefore critically important to assess the role 

that such data play in rendering the hydrogeologic performance analysis credible. One cannot 

overemphasize the role of characterization and monitoring data in helping one identify and test 

alternative conceptual models, make rational choices among them, gauge and reduce model bias 

and uncertainty through proper model selection and calibration, assess the reliability of model 

predictions, and confirm the assessment through independent peer review as well as at least some 

degree of direct verification. 

Regional and site characterization data tend to represent a wide range of measurement scales, not 

all of which are compatible with the intended scale of hydrogeologic model resolution. It is 

important to recognize this important issue explicitly and to make a conscious effort rendering 

the scale of measurement compatible with the scale of model resolution. This can be done by 

either rescaling the data to fit the scale of model resolution (which often entails averaging or 

upscaling over computational grid cells) or adapting model resolution to fit the scale of 

measurement (which often entails adapting the size of grid cells to the size of the data support). 

12.2.3 Postulate Alternative Conceptual Models 

Interpret available data in a way that leads to several plausible conceptual models (hypotheses 

and assumptions) of regional and site hydrogeology in three spatial dimensions. Provide a 

detailed rationale for each interpretation and explain why no other hydrogeologic 

interpretations of the data would be plausible. 

A conceptual hydrogeologic model is a mental construct or hypothesis accompanied by verbal, 

pictorial, diagrammatic and/or tabular interpretations and representations of site hydrogeologic 

conditions as well as corresponding flow and transport dynamics. It should be presented in the 

form of written text accompanied by pictures, charts and diagrams (including maps, blockdiagrams, cross-sections, panel diagrams, vertical/horizontal profiles) of system components, 

arrangements and relationships (flows) known collectively as structure. A conceptual model 

identifies relevant hydrogeologic units (soils, aquifers, aquitards, aquicludes, bedrock) and 

features (faults, intrusions, fractures), their makeup (mineralogy, petrography, texture, 

cementation, porosity, permeability, related petrophysical and geophysical properties), geometry 

(horizontal and vertical dimensions, boundary and internal configurations), system states under 

undisturbed and disturbed conditions (types of fluids that permeate the soils and rocks, their 

pressure, saturation, density and temperature, dissolved solutes and their concentrations), flow 

and transport dynamics and kinematics (active flow and transport processes; their driving 

mechanisms; fluid and solute fluxes and velocities; fluid, solute and energy balances), 

hydrochemistry and isotopes. 

A hydrogeologic knowledge base typically includes information obtained from boreholes, soil 

and geologic outcrops, underground openings and geophysical surveys. This adds a depth 

dimension to geographic information about the site and its surroundings which is largely twodimensional. A hydrogeologic knowledge base is thus inherently three-dimensional in space, and 

should allow one to describe the system in as many dimensions. This is important because 

hydrogeology is inherently three-dimensional (with many quantities depending additionally on 
206 

time), complex, and manifests itself on a multiplicity of scales. As such, it cannot be 

meaningfully captured in a single two-dimensional map or cross-section, nor can it be described 

adequately by a one-dimensional profile such as a stratigraphic column or a lithologic log, that 

show only selected details on a given scale. Instead, hydrogeologic data must be interpreted in 

three-dimensions and in some detail on a range of scales (from regional down to thin sections). 

There virtually always is at least some regional and site information that should allow one to do 

so; seldom can lack of data be validly quoted as a reason for doing less. This is true regardless of 

how complex or simple a set of models one ultimately employs for the assessment of site 

performance and the analysis of corresponding uncertainty. 

A systematic approach to the qualitative conceptualization and screening of hydrogeologic 

hypotheses requires as a prerequisite the availability or acquisition of expertise in the qualitative 

and quantitative interpretation of hydrogeologic field data. With such expertise, and a healthy 

dose of hydrogeologic insight, one may proceed to conceptualize hydrogeologic units and 

features on the regional, site and subsite scales in terms of their hydrostratigraphy, lithology, 

vertical and horizontal boundaries, structural features such as folds, faults, offsets and 

intrusions, textural features such as grain size, cementation, microstructure and fracturing, flow 

and transport properties such as porosity, permeability, dispersivity and sorption coefficients, 

related pedological and petrophysical properties (spatial, directional and textural) as well as their 

distributions and variations (heterogeneity and anisotropy) on a range of scales, both between 

and within hydrogeologic units and relevant structural and textural features. 

To help in this task, one must assemble or develop descriptions of geologic outcrops, well logs, 

air and satellite images, maps and cross-sections, panel diagrams, quantitative records of 

measured variables, and qualitative descriptions of observed phenomena. It is useful, but not 

necessary, to embed these in tools for the computer management of comprehensive 

hydrogeologic data of the kind discussed in Chapter 4. 

The next step is to describe the space-time distributions of fluid types (water, air, nonaqueous 

phase liquids) and corresponding states such as saturation, pressure, temperature and density, 

followed by significant solutes and their concentrations, as well as the delineation of saturated, 

vadose and perched zones, all on a range of scales between and within hydrogeologic, structural 

and textural units and features. This in turn allows one to describe active and anticipated flow 

and transport phenomena such as advection (of solutes or particulates), convection (of heat), 

diffusion, dispersion and sorption, their modes (discrete features; single, dual or multiple 

continua) and scales of manifestation (regional, site, subsite, long-term, short-term) boundary 

and internal mechanisms, forces and sources that drive them (infiltration, evapotranspiration, 

recharge, discharge, pumping), as well as their relative intensities and significance. 

The next stage is to delineate in a qualitative but internally consistent and coherent manner 

contours of equal hydraulic potential (head), pressure, saturation, density, temperature and solute 

concentration together with corresponding flowlines (streamlines, pathlines, streaklines), vectors 

of fluid and solute flux and velocity, and isochrones (of groundwater residence time and solute 

travel time). The potential for the development of fast flow paths, on scales smaller than are 

represented by these contours and vectors (focused and episodic infiltration, preferential wetting, 

high-permeability channels, instability of fluid fronts, fingering), must be articulated at this 

stage. One should also assess the overall balance of fluids, solutes and energy within the system. 
207 

To conceptualize contaminant transport on regional, site and subsite scales, one must describe 

the space-time distribution of major contaminants in the soil, vadose zone and groundwater on 

these scales. The description must include the space-time distributions of contaminant sources; 

mechanisms and rates of source contaminant mobilization and leaching; active transport 

phenomena such as average and fast advection, diffusion, dispersion, radioactive or biochemical 

decay, sorption, colloid transport; space-time distribution of migration, spreading, and dilution 

patterns; and overall mass and ionic balance for key contaminants. 

A hydrogeologic conceptualization is not complete without a description of hydrogeochemistry 

and isotope hydrology on regional, site and subsite scales in three-dimensions. This includes the 

space-time distributions of major hydrochemical constituents, environmental isotopes and their 

sources above and below the water table; space-time distribution of groundwater ages; 

implications concerning flow between and within hydrogeologic, structural and textural units and 

features (including infiltration, evapotranspiration, recharge, discharge, directions and rates); and 

implications concerning transport (directions and velocities, possible compartmentalization, 

isolation and mixing of groundwater bodies, chemical reactions, and water-rock interactions). 

It often helps to conceptualize temperature and heat flow on regional, site and subsite scales in 

three-dimensions. This may include the space-time distributions of temperature and heat flow 

above and below the water table, and their implications concerning flow between and within 

hydrogeologic, structural and textural units and features (including infiltration, 

evapotranspiration, recharge, discharge, directions and rates). 

Hydrogeologic systems are open and complex and the corresponding knowledge base is 

invariably incomplete and imprecise. Therefore, such systems almost always lend themselves to 

multiple conceptualizations and the postulation of several alternative hypotheses. It is therefore 

important to explore varied conceptual frameworks and assumptions through a comprehensive 

evaluation of a broad range of regional and site data, their translation into coherent and internally 

consistent conceptual models or hypotheses, and an in-depth examination of these hypotheses in 

light of the available knowledge base. The more experts with a wider range of earth and 

environmental specialties are given access to the knowledge base, the larger and more varied are 

the alternative site descriptions they may identify. 

To develop alternative conceptual models for a site, one should consider (among others) 

alternative representations of space-time scales; number and type of hydrogeologic units such as 

layers and structures such as faults; flow and transport properties (their values and statistics, 

spatial distribution and geostatistics, internal heterogeneity, anisotropy); location and type of 

system boundaries; space-time distribution of fluids and their states (pressure, density, saturation, 

temperature); space-time distribution of saturated, vadose and perched zones; space-time 

distribution of driving forces (infiltration, recharge, discharge, initial system states, boundary 

conditions); space-time distribution of flow patterns; existence and nature of fast flow paths; 

overall water balance; space-time distribution of contaminants; space-time distribution of 

contaminant sources; mechanisms and rates of source contaminant mobilization and leaching; 

controlling transport phenomena; migration, spreading and dilution patterns of contaminants; 

contaminant mass balance; space-time distribution of groundwater ages; space-time relationships 
208 

between major chemical constituents, isotopes, temperatures and heat flows; and their 

implications regarding flow and transport on regional, site and subsite scales in three dimensions 

and time. 

The alternative conceptualizations should be firmly grounded in the available knowledge base. 

Each alternative conceptualization should be supported by key data. 

The conceptualization is not complete without a clear articulation of ambiguities and 

uncertainties associated with each alternative description and interpretation (conceptual model or 

hypothesis) of site hydrogeology. 

12.2.4 Avoid Conflict with Data 

Verify in detail that none of the selected conceptual models are in conflict with any of the data; 

discard those that are. 

Make sure that the selected conceptual models contain a minimum number of inconsistencies, 

anomalies and ambiguities with the lowest possible amount of remaining uncertainty about the 

site and the corresponding flow and transport regimes. 

12.2.5 Rank Conceptual Models 

Rank all alternative conceptual models in the order of their plausibility in light of the data, 

regardless of context. Provide a detailed rationale for the ranking and discard models that 

appear to be much less plausible than those retained. 

Once a number of alternative hydrogeologic conceptualizations have been articulated, they must 

be systematically examined, compared, screened and ranked according to acceptance criteria that 

include logical consistency and coherence, and the extent to which they are supported or 

contradicted by available observations and data. Among otherwise equal conceptual models, we 

favor the least complex based on the principle of parsimony. Models that do not meet reasonable 

acceptance criteria of internal consistency, coherence and correspondence with the available data 

should be eliminated from further consideration at this stage of the analysis. 

12.2.6 Develop Conceptual-Mathematical Models 

For each alternative conceptual model, develop a complete, coherent and internally consistent 

conceptual-mathematical model of three-dimensional transient flow and transport within the 

schematized hydrogeologic system that is as simple as warranted by the context of the problem 

and the data. Provide a detailed rationale for your schematization of regional and/or site 

hydrogeology; choice of governing, state and constitutive equations; type and space-time 

distribution of forcing terms (sources, initial and boundary conditions); parameterization 

(functional form and number of parameters); and all other conceptual-mathematical aspects of 

the model. Discard models that do not lend themselves to such mathematical description. 
209 

A conceptual-mathematical model helps define and describe the hydrogeologic system in terms 

of space-time dimensions; topology; geometry; interactions (called processes) between kinematic 

(mass, concentration, flux, velocity) and dynamic (energy, force, stress) quantities; parameters 

and forcing terms (sources, initial and boundary conditions). Whether such a model is analytical 

or numerical (written in the language of calculus or algebra) is merely a technical, not a 

fundamental, distinction. The model ultimately allows one to explain and interpret existing 

observations, and to predict new observations, quantitatively. 

For each alternative conceptual model identified during the qualitative stage of the process, one 

needs to define the hydrogeologic units and features (such as folds, faults, offsets and intrusions) 

that are to be explicitly modeled; their three-dimensional topology and geometry in terms of 

location, shape, size and relationship to other units and features (extent of external and internal 

boundaries and structural elements); equations that govern flow and transport phenomena 

included in the model; corresponding equations of state, in the form of functional relationships 

between parameters and state variables in the model; spatial variability of parameters that enter 

into the governing and state equations, within each hydrogeologic unit and feature; spatial 

variability of initial states in each unit and feature; boundary equations for flow and transport; 

and the space-time distribution of boundary and source parameters and values. 

The quantification of system states includes functional and quantitative representations of fluid 

saturation, pressure, temperature, density, solute mass and concentration and regional as well as 

perched water tables on a range of scales (between and within hydrogeologic, structural and 

textural units and features). Governing and boundary equations include mathematical definitions 

and descriptions of active and anticipated flow and transport phenomena such as advection, 

convection, diffusion, dispersion and sorption for selected modes (discrete features; single, dual 

or multiple continua) and scales of manifestation, in the interior of units and features as well as 

on their boundaries and interfaces, including source terms, parameters, functional relationships 

between these quantities, and their space-time distributions. 

12.2.7 Consider Simplifying the Model 

If mathematical modeling in three spatial dimensions and/or in time is considered unnecessary 

or computationally infeasible, provide a detailed explanation of how the three-dimensional 

transient conceptual-mathematical model reduces to a two-dimensional and/or steady state 

model. In particular, demonstrate (preferably through a mathematical formalism and/or 

computationally) how the lower-dimensional model captures all relevant aspects of the original 

three-dimensional transient model. Avoid one-dimensional models at this stage. 

The purpose of conceptual-mathematical models is to help quantify alternative hypotheses 

regarding the hydrogeologic makeup and behavior of a site. Since such hypotheses are always 

cast in three spatial dimensions, ideally so should the corresponding conceptual-mathematical 

models. Both model types must allow for the hydrogeologic regime to evolve in time. 
210 

Model simplification may entail a reduction in dimensionality (from three spatial dimensions to 

two or one; from transient to steady state), model size (smaller area and/or reduced depth); or 

details of various features, events and processes (fewer layers, faults or fracture zones; a less 

detailed representation of internal heterogeneity; single rather than dual continuum or discrete 

representation of a fractured rock; fewer discrete rainfall or infiltration events; a less detailed 

delineation of contaminant sources; constant rather than scale-dependent dispersion; equilibrium 

rather than kinetic sorption). Various types and levels of simplification can be entertained, 

leading to a number of simplified model structures for each initial (and more complex) 

conceptual-mathematical model. 

In reality, it is much more difficult and time consuming to set up and run mathematical models in 

three than in two spatial dimensions. Working in two spatial dimensions is often feasible because 

eliminating the third dimension usually entails a far lesser phenomenological change than 

reducing the dimensionality of the problem from two to one. Modeling and visualization of 

complex flow and transport phenomena is much easier in two than in three dimensions, but not 

much harder than in one dimension. Hence the strategy in this report supports two-dimensional 

flow and transport analyses (whether mathematical-analytical or computational-numerical) in 

cases where the effect of the third dimension is demonstrably minor. However, the strategy 

discourages one-dimensional analyses unless a very strong and convincing hydrogeologic 

argument is made in their favor. This is so because flow and transport behaviors predicted by 

one- and multi-dimensional models often differ from each other in a fundamental way. 

The spatial and temporal scales at which flow and transport phenomena are modeled depend in 

part on the contextual framework (area, depth and time-frame of concern; space-time scales on 

which performance measures are defined). These phenomena are affected to a large extent, but 

not exclusively, by hydrogeologic complexities, heterogeneities and driving-mechanisms that 

manifest themselves on similar scales. It is therefore important that conceptual-mathematical 

modeling start by identifying these "site-scale" features, factors and/or phenomena and by 

incorporating them directly as explicit elements in the model. Models that do not incorporate 

such elements explicitly, but account for them implicitly, are considered here to be simplified or 

abstracted. Models that fail to account for site scales complexities, either explicitly or implicitly, 

are oversimplified and would generally be too crude to provide a reliable description of site 

hydrogeology for most purposes. 

Site-scale hydrogeologic complexities, heterogeneities and driving-mechanisms may be 

influenced and/or controlled by larger- or regional-scale features and factors such as regional 

recharge, discharge, and flow mechanisms and patterns on various time scales. A model must 

account for these large-scale influences and controls, as well as for associated uncertainties, 

through the appropriate assignment of initial conditions, boundary conditions, and source terms 

such as those that describe infiltration, recharge, discharge and leakage across aquitards. 

Site-scale hydrogeologic complexities, heterogeneities and driving mechanisms may be 

significantly influenced by smaller- or subsite-scale features and factors such as space-time 

irregularities and fluctuations in external and internal boundary and source shapes or conditions; 

internal heterogeneities within site-scale hydrogeologic units, faults, dikes or other features; as 

well as smaller-scale units, faults, dikes, fractures or preferential flow channels. Determining 
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what is the nature and extent of these influences is an integral part of developing a conceptualmathematical model for a site. If features and factors that manifest themselves on subsite scales 

are deemed important for the modeling of site-scale phenomena, one must account for them 

directly (explicitly, by embedding such features and factors discretely in the site model) or 

indirectly (implicitly, by formally integrating these features and factors into the site model 

equations and parameters) in the model. 

Embedding discrete small-scale features and factors in a model renders it relatively complex. 

Integrating small-scale features and factors into the model equations and parameters renders it 

less complex than embedding. However, to compensate for loss of information, the equations 

change form and phenomenology (from Stokes to Darcy, isotropic to anisotropic, juxtaposed to 

overlapping dual or multiple continua, differential local to integro-differential nonlocal) and 

acquire new phenomenological parameters (permeability, dispersivity, integral kernels) that 

differ in nature and magnitude (from scalar to tensor, local to nonlocal, well-defined to scaledependent) from the original parameters. Only in special cases can the original form and 

phenomenology be recovered, and even then the parameters usually change (due to upscaling). It 

is not presently clear which of these two approaches is better suited for their intended task. The 

strategy in this report considers both options. 

Ignoring the influence of regional- or subsite-scale features or factors on site-scale flow or 

transport without due justification, or failing to insure that the treatment of all scales is 

self-consistent without demonstrating that some of them are not relevant to the problem at hand, 

constitute oversimplifications which may cast doubt on the reliability of the model. 

Narrowly defined contextual or regulatory criteria, limited data or resources, and a quest for 

simplicity or transparency may motivate the adoption of hydrogeologic flow and transport 

models that are less than three-dimensional, ignore time, and include few details on limited 

scales. While such motivation for simplification and abstraction of hydrogeology and flow or 

transport dynamics may sometimes be justified on practical grounds, it does not in itself turn 

simplified and/or abstracted models into scientifically valid tools of performance assessment. 

Only a formal demonstration that such models capture the essential features and capabilities of 

their more complex and complete counterparts, and that they thereby provide comparable 

performance assessments or conservative bounds thereof, might justify their use. 

To insure that relevant aspects of hydrogeologic complexity are reflected in modeled system 

behavior and performance, and that the model can be rendered compatible with site data, it is 

important that the process of simplification be done systematically and objectively. This can be 

done by filtering out undesirable details through formal averaging of the governing equations in 

space-time or in probability space, in a way which retains and renders their influence on the 

model implicit. This is equivalent to embedding. 

Averaging three- or two-dimensional equations across one spatial dimension renders them twoor one-dimensional, respectively. Averaging transient equations over time may (but need not) 

render them representative of a steady state. Averaging can also be done over subdomains of the 

site being modeled, and over multiple time intervals. In each case, the averaging results in 

governing flow and transport equations that contain upscaled quantities. If the space-time scales 
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of these quantities differ from those of the available site data, then either the model or the data 

must be rescaled to render them compatible and comparable with each other. 

Many scale-related problems are avoided if the averaging is done in probability space. Such 

"ensemble" averaging leads to stochastic equations that contain statistical moments of 

hydrogeologic variables (considered random), most commonly the mean and variancecovariance. The mean is a predictor of system behavior or performance, and the variancecovariance is a measure of predictive error. Both are smoother (vary more slowly in space-time) 

than their random counterparts and, in this sense, render the model relatively "simple." Despite 

their smoothness, both moments are defined on the same space-time scales as are the random 

hydrogeologic variables on which they are based. Stochastic models thus achieve smoothness 

and simplicity without any need to average or upscale in space-time. As they are typically 

conditioned on site measurements (i.e., they honor the data), stochastic models are compatible 

with these measurements both in scale and magnitude. Yet another advantage of the stochastic 

method over space-time averaging is that it yields measures of predictive uncertainty. Stochastic 

approaches are increasingly recognized as offering a way to deal with complex, scale-dependent 

heterogeneous systems by means of relatively simple models. 

12.2.8 Select Computational Code or Algorithm 

Select a suitable computational code or algorithm to represent the conceptual-mathematical 

model. 

A computational code is a tool rather than a model. However, selecting a particular code implies 

identifying specific processes that may govern flow and transport at a site, their symbolic 

mathematical representation, and their numerical approximation. Selecting the space-time 

dimensions and size of a computational grid determines the dimensions and scale of the system 

being modeled. Choosing the sizes of space-time discretization intervals defines the scales at 

which flow and transport processes are resolved. Specifying the location and type of sources, 

initial and boundary conditions identifies the forcings. Both material properties and forcing terms 

are associated with parameters that must eventually be assigned numerical values. Choosing the 

modes of their representation (parameterization) defines the resolution scales of these parameters 

(and thus of material properties and forcing terms) in space-time. Once this has been 

accomplished in a way that is supported by all relevant regional and site data, the code has been 

transformed from a mere tool to a bona fide conceptual-mathematical model of site 

hydrogeology. 

12.2.9 Define Prior Statistics 

Associate objective or subjective prior statistics with all input parameters and forcing terms that 

are considered to be uncertain. Provide a detailed rationale for considering any of these inputs 

to be known with certainty. 

Since hydrogeologic medium properties and forcing functions are always uncertain, so are the 

model input variables. In other words, the input parameters of groundwater flow and transport 

models are inherently uncertain regardless of whether the model is deterministic or stochastic. 
213 

The difference is that whereas in deterministic models the input parameters are viewed as 

imperfectly known deterministic quantities, in stochastic models they are viewed as correlated 

random fields or processes that may be perfectly or imperfectly known at discrete sampling 

locations in space-time. In both cases, imperfect knowledge is characterized by random errors 

that may, but often are not, considered to be mutually correlated. This renders the input 

parameters of deterministic models correlated or uncorrelated random variables, and those of 

stochastic models correlated random fields or processes conditional on either exact or random 

measurements, which may themselves be correlated or uncorrelated among themselves. Whereas 

the second moment of uncorrelated random variables is a diagonal matrix of their respective 

variances, that of correlated random variables is a full square symmetric and positive definite 

matrix of their respective covariances. 

Estimating input parameters for a model (deterministic or stochastic) on the basis of incomplete 

and/or uncertain data is equivalent to inferring their mean values from these data. If the mean 

values are based on site characterization data, they represent prior parameter estimates. If they 

are based on site monitoring data, they represent posterior parameter estimates. Here we consider 

only prior inference of model input parameters. 

If the inferred parameter estimates vary in space and/or time in a way which reflects similar 

variations in the underlying site characterization data, they are said to be conditional on these 

data. If the inferred mean values do not reflect such spatial and/or temporal variability, they are 

said to be unconditional. Unconditional inference may take place when the data are too few or 

too clustered to allow defining their variability in space-time. 

Assume that a set of site specific measurements are available, which represent the same 

hydrogeologic variable (say permeability or porosity) as a corresponding set of model input 

parameters. Such measurements are considered to be "hard." Ideally, the data would include 

probabilistic information about errors of measurement and test interpretation that suffer from a 

known amount (ideally zero) of statistical bias. If one has a statistically significant set of such 

data, one should be able to estimate a prior set of model input parameters on their basis. One 

should also be able to postulate a probabilistic model of prior parameter uncertainty based on 

statistics derived from these data. Such a probabilistic model is known to be of Type A. 

A key measure of parameter uncertainty is the second statistical moment, or variance-covariance, 

of their estimation errors. Since input variables into both deterministic and stochastic 

computational models are specified in terms of a discrete set of input parameters, the 

corresponding variance-covariance forms a matrix. In the conditional case, the off-diagonal 

covariance terms of the matrix may reflect spatial and/or temporal correlation between 

parameters of a given type (say permeability or porosity) as well as cross-correlations between 

parameters of different types (say permeability and porosity). In the unconditional case, there is 

no inferred space-time variability and the covariance terms represent at most cross-correlations 

between different types of parameters. 

Prior parameter estimates can be obtained from clustered data by means of common statistical 

methods, and from spatially distributed data by means of standard geostatistical techniques. This 

is true regardless of whether the parameters are intended for a deterministic or a stochastic 
214 

model. The main difference is that in the deterministic case, there may be a need to upscale the 

parameters whereas in the stochastic case, this may not be necessary. The simplest and most 

practical form of geostatistical inference with upscaling is block kriging. While this may not 

always be the most accurate and sophisticated way to proceed, it is quite adequate for many 

purposes. This is especially true in situations where enough site monitoring data are available to 

later modify the parameter estimates through model calibration. 

To characterize prior estimation uncertainty one should, as a minimum, infer from the data a 

variance-covariance matrix of prior estimation errors. Only in rare circumstances would there be 

enough data to permit inferring from them higher statistical moments of these errors. 

Standard geostatistical analysis consists of identifying the spatial autocovariance structure of 

each variable being analyzed (in terms of an autocovariance or variogram function) and 

providing a smooth (kriged) conditional estimate of this variable on a two- or three-dimensional 

grid, as well as the associated estimation (kriging) variance at each grid point. A more advanced 

analysis may also yield an autocovariance matrix for the estimate across the grid. Another level 

of sophistication may be achieved by identifying cross-covariance functions or cross-variograms 

for two or more variables, estimating them simultaneously by cokriging, and computing their coand cross-covariances across the grid. Both kriging and cokriging may be used to estimate 

average values of the variables over finite blocks or subdomains of a two- or three-dimensional 

grid. 

If there are insufficient hard data of a given hydrogeologic variable to conduct a meaningful 

statistical or geostatistical analysis, then the use of “soft” (qualitative) data coupled with 

indicator geostatistical analysis are recommended. This yields an uncertainty model for the prior 

parameters that is intermediate between Type A and Type B (defined below). Soft or indirect 

information about the parameters may include (a) off-site measurements of the parameters proper 

(quite often on scales other than those corresponding to the intended scale of model resolution) 

and/or (b) surrogate measurements on site that are known to correlate with the parameters of 

interest (for example, porosities or geophysical signatures that correlate in known ways with 

permeabilities, water contents or fracture densities). Statistics derived from off-site data must be 

considered potentially biased (due to a lack of site-specific information about mean parameter 

values and incompatibility of geology and scale). The associated variance may be too small or 

too large, depending on the quantity and quality of such data. Statistics derived from surrogate 

data may suffer from poorly defined correlations and incompatibility of scale. 

An example of soft data use is that of pedotransfer functions, which allow one to estimate soil 

hydraulic characteristics on the basis of soil textural data. 

Indicator geostatistics yields indicator variograms for various classes of the variable, a smooth 

(kriged) estimate of this variable on a two- or three-dimensional grid, and the probability that the 

variable is larger (or smaller) than specified at each grid point. Most geostatistical software 

packages include an indicator option and allow one to generate random realizations of the 

variable on a grid by means of indicator Monte Carlo simulation. 

If the available hard and soft data are not amenable to geostatistical analysis (due to insufficient 

information about their spatial location, an inappropriate spatial pattern, insufficient number 
215 

and/or poor quality of data), an alternative is to rely on generic and/or subjective probabilities 

and statistics. Doing so is equivalent to postulating a Type B statistical model of prior parameter 

uncertainty. Such a model should always be suspected of suffering from an unknown amount of 

statistical and personal bias. Statistical bias is introduced due to lack of site-specific information 

about mean values of the parameters in question. A personal bias tends to manifest itself in the 

form of assigned uncertainty measures (most importantly bias and error variance) that are either 

too small or too large. The first is a manifestation of over-confidence in the model parameters, 

the second of unduly low confidence in their values. 

Methods are available to update generic statistics on the basis of site-specific data using a 

Bayesian approach. 

12.2.10 Perform Preliminary Test of Conceptual-Mathematical Model 

Test the conceptual-mathematical model for qualitative consistency with the available data 

through computational exploration and visualization in at least two spatial dimensions, within a 

plausible range of input parameters and forcing terms. If inconsistent, repeat the previous steps 

in modified form (by altering and/or refining the hydrogeologic scheme and mathematical 

description of flow and/or transport) or discard the underlying conceptual model and select an 

alternative. 

Having associated a deterministic and/or stochastic mathematical structure with each hypothesis 

that had been postulated for a site, the next step is to explore them quantitatively and graphically. 

The aim of this initial exploration is qualitative: To identify the potential ability of each 

conceptual-mathematical model to explain observed system behavior and predict it under future 

scenarios. For this, it is useful to run the model with a plausible range of input parameters and 

study its output in light of the available site information. This may reveal ambiguities and 

uncertainties that may prompt a reexamination of the data and/or a revision of the model (which 

may, but need not, render the model more complex). It may also reveal differences between the 

various conceptual-mathematical models that may help eliminate some of them and rank the rest 

in the order of their apparent ability to explain and predict system behavior. 

The initial exploration of alternative conceptual-mathematical models can often be done in the 

two-dimensional horizontal and/or vertical planes. This is so because two-dimensional analyses 

differ from their three-dimensional counterparts to a much lesser extent than do one-dimensional 

analyses, yet are much less demanding than three-dimensional analyses. Two-dimensional 

scoping analyses should yield contour and/or color representations of hydrogeologic parameter 

distributions; system states (hydraulic head, pressure, saturation, density, temperature, solute 

concentration); flowlines (streamlines, pathlines, streaklines); and vectors of fluid and solute 

fluxes and velocities. If the conceptual-mathematical model involves transients, the analyses 

should consider variations of hydrogeologic variables with time. Some three-dimensional 

exploratory analyses may also be possible at this stage, including the preliminary quantification 

of fluid, solute and energy balances. Three-dimensional analyses may also be required if the twodimensional results are insufficient to provide insight into relevant processes or fail to resolve 

key ambiguities and uncertainties. 
216 

Once an initial set of conceptual-mathematical models have been formulated and explored in a 

preliminary fashion, once has the option of postulating and exploring less complex alternatives to 

each. There is no reason to contemplate more complex alternatives at this stage of the modeling 

process. 

12.2.11 Calibrate Model 

Use an inverse (preferably Maximum Likelihood) approach to calibrate the computational model 

against observational (monitoring) data. This should yield posterior estimates of the model 

parameters, statistical measures of their estimation errors and associated model quality criteria. 

If the model cannot be properly calibrated against reliable data, repeat the previous steps in 

modified form (by altering and/or refining the hydrogeologic scheme and mathematical 

description of flow and/or transport) or discard the underlying conceptual model and select an 

alternative.

If monitoring data are available to allow comparing model predictions with observed system 

behavior, there usually is no reason to avoid calibrating the model against these data. The 

strategy strongly suggests that this be done. 

The traditional approach to hydrogeologic modeling has been to postulate a deterministic model 

structure and treat its parameters as being imperfectly known. One would then derive prior 

estimates of these parameters from site characterization data (if available) and modify them so as 

to achieve an acceptable fit between model outputs and available monitoring data (if such exist). 

The process is known as model calibration, parameter estimation, history matching or inverse 

modeling. The parameters that yield the best match between observed and recorded system 

behavior form posterior estimates. It is common to consider the latter to be more suitable for 

predictive purposes than prior parameter estimates. 

Though this is not always done, it is generally advisable to start the model calibration process by 

postulating a prior parameter uncertainty model. We explained how to do this and mentioned that 

it may lead to a Type A probability model when prior parameter uncertainty is characterized on 

the basis of a statistically significant set of site specific measurements that represent the 

parameter; a Type B model when the uncertainty is characterized by generic and/or subjective 

probabilities; and an intermediate type model when parameter uncertainty is inferred from 

indirect information about the parameters. We noted that a Type B model should always be 

suspected of an unknown amount of statistical and personal bias. A model that is intermediate 

between Type A and Type B must also be considered potentially biased, with variances that may 

be too small or too large, poorly defined correlations, and incompatibility of scale. 

Improper upscaling may render the prior parameter estimates biased and their uncertainty 

mischaracterized, regardless of what type probability model one adopts. It is generally believed 

that model calibration tends to reduce both the bias and the uncertainty in prior parameter 

estimates by insuring that it reproduces adequately observed system behavior. This should render 

a calibrated model more reliable as a predictor than an uncalibrated model. 
217 

Calibrating a traditional deterministic flow model against measured values of head and flux is 

tantamount to conditioning it on such measurements. As the model hydraulic parameters are by 

nature conditional, the very act of adding measured heads and fluxes to the database alter their 

values. This inherent nonuniqueness persists regardless of whether the inverse problem is wellposed or ill-determined. It explains why model parameter estimates tend to change each time one 

redefines the underlying database. 

Our guidelines stipulate that one use "statistical" model calibration algorithms which include 

analyses of parameter estimation uncertainty. Such analyses typically accept, but do not 

necessarily require, information about prior parameter statistics as input. The output includes 

posterior statistics of parameter estimation errors, which tend to be less biased and smaller than 

the prior estimation errors. Several public domain or commercial codes are available for this 

purpose (see section 7.4.2). 

Differences between values of model simulated system states (most commonly hydraulic head, 

pressure or concentration) at discrete points in space-time, and observed values of these same 

state variables, are termed "residuals" in model calibration parlance. The residuals are considered 

to represent errors that are distributed randomly about the simulated model output. An 

underlying (often tacit) assumption is that the conceptual-mathematical model is exact and 

associated with an unknown set of "true" parameters. If these parameters were known, the model 

(structure plus parameters) would be exact. To the extent that it would produce nonzero 

residuals, this would be entirely due to errors in the monitoring record. These data errors (or their 

logarithmic transform, as is usually done with permeability and transmissivity) are typically 

taken to have zero mean and be normally distributed. Hence ideally (in the theoretical event that 

the true parameters were known), the residuals would exhibit a multivariate Gaussian 

distribution with zero mean (i.e., they would be unbiased) and a variance-covariance identical to 

that of the monitoring data. 

Model calibration is seen as the process of estimating the model true parameters without bias and 

as closely as possible. This is accomplished by defining an appropriate calibration criterion (or 

objective function) in terms of the residuals, and "optimizing" the parameters in a way which 

comes closest to satisfying this criterion (achieving the objective). The extent to which the 

criterion is satisfied (the objective achieved) becomes a measure of model fit. Theoretically, the 

optimized parameters are associated with estimation errors that have zero mean and minimum 

variance. 

The most commonly used measure of model fit is the weighted sum of squared residuals. This is 

meaningful in the above statistical sense provided the monitoring data errors are mutually 

uncorrelated and each weight is inversely proportional to the corresponding error variance. 

Model calibration then reduces to a weighted least squares fit of the model to the data (or 

regression of the parameters on the data). Since groundwater model parameters are usually 

related to state variables in a nonlinear fashion, the weighted least squares fit is nonlinear. This is 

why model calibration must usually be done iteratively. 

If all the weights (and variances) are equal, the model fit criterion can be normalized to form a 

simple sum of squared residuals. Model calibration then reduces to an ordinary nonlinear least 
218 

squares fitting process (or nonlinear regression). A detailed set of guidelines for the effective 

calibration of such nonlinear regression models has been prepared by Hill (1998). 

A less common but still widely used measure of model fit is the generalized sum of squared 

residuals. Whereas in the previous cases the weights formed a diagonal matrix proportional to 

the inverse of a diagonal matrix of observational error variances, here the weights form a full 

matrix proportional to the inverse of a (usually full) variance-covariance matrix. This is 

statistically meaningful in the previous sense when the errors are mutually correlated. Calibration 

now becomes a generalized nonlinear least squares or regression process. 

If some of the statistical parameters that define the observational errors are unknown, they may 

sometimes be estimated jointly with the other model parameters by the maximum likelihood 

method. This entails a likelihood function, which is the likelihood of the parameters given a set 

of (conditional on) observational data. The latter is the probability density of the (error corrupted 

and so random) data given (conditional on) the parameters. The objective would be to maximize 

the likelihood function. 

Since the data (or their log transform) are assumed to be normally distributed, the likelihood 

function is multivariate Gaussian. As such, it includes an exponential term whose negative 

logarithm is equal to half the weighted (by the inverse covariance matrix) sum of theoretical 

square residuals. Parameters are estimated by minimizing the negative logarithm of the 

likelihood function. In the special case where all statistical parameters are known, the negative 

log likelihood function reduces to the standard generalized least squares criterion. 

The strategy in this report favors a maximum likelihood approach to model calibration, which 

incorporates information about prior parameter statistics into the statement of the inverse 

problem (see section 7.4.3). This approach yields a negative log likelihood criterion that includes 

two weighted square residual terms instead of one. The first is the usual generalized sum of 

squared differences between simulated and observed state variables. The second is a generalized 

sum of squared differences between posterior and prior parameter estimates. The corresponding 

weight matrix is proportional to the inverse covariance matrix of prior parameter estimation 

errors. The constant of proportionality, λ, is treated as a free statistical parameter that may be 

estimated jointly with the remaining parameters by maximum likelihood. This can be done most 

easily by calibrating the model for various values of λ, plotting the negative log likelihood (S) 

against λ, and finding graphically the value of λ that minimizes S. Allowing λ to be initially 

unspecified means that neither the covariance matrix of the observational data nor that of the 

prior parameters need be fully specified; it is enough to specify each of them up to its own 

constant of multiplication. It is then possible to estimate these constants of multiplication on the 

basis of λ. 

Including prior information in the calibration criterion allows one to condition the parameter 

estimates not only on site monitoring (observational) data but also on site characterization data 

from which prior parameter estimates are usually derived. When both sets of data are considered 

to be statistically meaningful, the posterior parameter estimates are compatible with a wider 
219 

P

array of measurements than they would be otherwise and are therefore better constrained 

(potentially rendering the model a better predictor). 

When either set of data is too small, clustered or otherwise unsuitable for a meaningful 

assessment of prior statistics, it may still be possible to come up with prior parameter estimates 

and weigh them subjectively relative to each other, based on their perceived reliability. Similar 

weights may be assigned to site monitoring data. One can then adopt a calibration criterion 

J + λ equal to a weighted sum (J) of squared residuals (differences between simulated and 

observed state variables), plus the product of λ with a weighted sum (P) of squared differences 

between posterior and prior parameter estimates. One can then calibrate the model for various 

choices of λ and plot the weighted sum of squared residuals versus 1/λ to see how the latter 

affects the former. One would typically find that the sum of squared residuals is largest when 1/λ

= 0 and decreases more-or-less monotonically to an asymptote as 1/λ increases. This is clear 

considering that 1/λ = 0 corresponds toλ = ∞ , which is equivalent to giving infinite weight to 

the prior parameter estimates and forcing the posterior estimates to coincide with them. As 1/λ is 

allowed to increase, the weight placed on the prior parameters decreases and their constraining 

effect on the posterior estimates gradually diminishes. Eventually 1/λ becomes large enough to 

virtually eliminate any effect of the prior on the posterior estimates. The calibration is now 

unconstrained by site characterization data and relies entirely on site monitoring data. 

Rather than a single calibration criterion one now has two, the weighted sum of squared residuals 

(J) and the weighted sum of squared differences between posterior and prior parameter estimates 

(P). These two criteria are mutually incompatible in that to satisfy one, it is necessary to sacrifice 

the other. In most cases, the optimum solution to this multiobjective or Pareto problem is not to 

minimize either objective but to find an acceptable tradeoff between them on the basis of 

subjective value judgment (see section 7.4.4). 

Methods to calibrate stochastic models have also been developed and are discussed in the report. 

In the absence of sufficient or reliable site monitoring data, the model must rely entirely on prior 

parameter estimates and is only as good as are the latter. The model remains uncalibrated, 

untested and unconfirmed. 

The statistical approach to deterministic model calibration allows one to quantify the uncertainty 

associated with posterior parameter estimates. One measure of posterior parameter estimation 

errors is the Cramer-Rao lower bound on their covariance matrix. This lower bound is given by 

the inverse Fisher information matrix. Each term in the Fisher information matrix is one half the

ensemble average of the second derivative of the negative log likelihood criterion with respect to 

a pair of parameters. It thus measures the average rate at which model sensitivity to one 

parameter is affected by changes in another parameter. From it, one can obtain a corresponding 

lower bound on the correlation matrix of the estimates. Ideally, the parameter estimates should 

be uncorrelated. 

The Fisher information matrix is usually approximated by linearizing the relationship between 

system states and parameters when the latter are optimal. It is then expressed as a function of the 
220 

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calibration weights (inverse covariance matrices of observed and prior input data) and a Jacobian 

matrix, whose terms represent sensitivities of system states to the parameters. It is thus clear that 

an analysis of parameter uncertainty includes in it a sensitivity analysis of the model and its 

output. 

A large variance associated with a given parameter indicates a high level of uncertainty in its 

estimate. A high degree of correlation between estimates implies that they are linearly related in 

a statistical sense. This means that they cannot be estimated individually with the available data, 

and should either be lumped into one parameter or estimated anew when more data become 

available. 

Linearization implies that if the prior (observation and parameter) errors are multivariate 

Gaussian (as one generally assumes), so are the posterior parameter (or log parameter) estimation 

errors. This allows one to associate them with linear confidence intervals, which can be viewed 

as corresponding error bounds. 

12.2.12 Confirm Model 

If feasible, confirm aspects of the model by predicting system behavior under conditions other 

than those used for calibration and comparing with observed system behavior. Quantify model 

predictive uncertainty and check the extent to which the observations lie within corresponding 

uncertainty bounds. If not, repeat the previous steps in modified form (by altering and/or refining 

the hydrogeologic scheme and mathematical description of flow and/or transport) or discard the 

underlying conceptual model and select an alternative. 

In rare situations where enough observational data are available to allow using only some of 

them for model calibration, it may be possible to use the remaining data for purposes of model 

verification or confirmation. For this to be valid, the data used for confirmation must represent a 

different mode of system behavior that those used for calibration. 

Otherwise, the confirmation data are merely an extension of the calibration data and being able 

to reproduce them does not constitute a meaningful test of model predictive capability. As both 

the model and the data are generally uncertain, model confirmation must be cast in a suitable 

probabilistic and/or stochastic framework. 

12.2.13 Compare Calibrated Models 

Compare all retained calibrated models on the basis of suitable model quality criteria. Discard 

models that are clearly inferior to others. 

Alternative conceptual-mathematical models that have been successfully calibrated can be 

compared on the basis of quality criteria such as model fit and posterior parameter uncertainty. 

Examples include: 

Likelihood or negative log likelihood of posterior parameter estimates, given the available 

data; 
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Inverse variance-covariance weighted sums of squared residuals, which measure the overall 

lack of fit between model generated and observed system states; 

Various statistics of the residuals, which measure how closely they satisfy the assumptions of 

Gaussianity, unbiasedness, and similarity of their variance-covariance to that of the 

observational data; 

Consistent measures of systematic differences between contours, time records, peaks and 

trends exhibited by observed and simulated state variables; 

Invariant properties of the Cramer-Rao lower bound on the posterior covariance matrix of 

parameter estimation errors, such as— 

Trace, also known as A-optimality criterion (which measures the cumulative variance of 

all estimation errors); 

Determinant, also known as D-optimality criterion (which provides an overall measure of 

the estimation error variance-covariance matrix); 

Eigenvectors (the components of which represent relative contributions by the various 

parameter estimates, implying that parameters associated with a single eigenvector have 

uncorrelated estimation errors, and those associated with multiple eigenvectors have 

cross-correlated estimation errors, rendering them less amenable to discrimination); and 

Eigenvalues (parameters associated with eigenvectors that have small eigenvalues being 

less uncertain than those associated with eigenvectors that have large eigenvalues); the 

maximum absolute eigenvalue is known as E-optimality criterion; 

Linear confidence intervals; 

Invariant properties of the Jacobian sensitivity matrix (of system states to the parameters); 

etc. 

Models that have not been compared with monitoring data have indeterminate quality in that 

nothing is known about their ability to reproduce real system behavior. Calibrated models rank 

higher than uncalibrated models in terms of parameter quality criteria because (posterior) 

estimates tend to be less biased and uncertain than prior estimates, on which uncalibrated models 

are based. 

If all models remain uncalibrated, their only comparative quality criteria are those that pertain to 

prior parameter uncertainty. 

These include— 

The variance-covariance of prior parameter estimation errors; 

Various invariant properties thereof; and 

Corresponding linear confidence intervals. 

All models can be compared on the basis of model complexity criteria such as— 

Dimensionality; 

Number of simulated processes; 

Degree of nonlinearity; 

Number of parameters; and 

Number or size of grid cells. 
222 

To help identify a set of models which represent acceptable tradeoffs between quality and 

complexity, it is helpful to employ a multiobjective approach as described in section 8.2.2. To 

further discriminate between these models, it is useful to gauge and rank them in terms of 

likelihood based model discrimination criteria, which account for both model fit and complexity. 

This applies only to calibrated models because uncalibrated models are associated with zero 

likelihood. Several such criteria are discussed in section 8.2.3. The model testing and selection 

process ends with the final ranking of models on the basis of likelihood based discrimination 

criteria, and the optional elimination of low ranking models from further consideration. 

12.2.14 Render Predictions Using Individual Models 

Use each retained calibrated model to predict system behavior as well as relevant hydrogeologic 

performance measures under various scenarios of relevance to the problem, and quantify the 

corresponding predictive uncertainty. 

Monte Carlo is by far the most common method of assessing the predictive uncertainty of a 

model. The method is conceptually straight forward and has the advantage of applying to a very 

broad range of both linear and nonlinear flow and transport models. Given information about the 

statistical properties of the model input parameters (including those that represent forcing terms) 

or their log transformed values, one generates numerous equally likely realizations of the 

parameters (or log parameters, as in the case of permeability and transmissivity). If the model 

has been calibrated, the input statistics correspond to the posterior parameter estimates. If the 

model is uncalibrated, they correspond to the prior estimates. In both cases, the estimates are 

generally associated with a full variance-covariance matrix. Upon assuming that the estimation 

errors are multivariate Gaussian, one has all the information one needs to produce random 

realizations of the parameters. 

One way to produce random realizations of a multivariate Gaussian vector (of parameters) is by 

Cholesky decomposition of its variance-covariance matrix. This method is well suited for full 

matrices and highly efficient when the latter are not too large. Its efficiency stems from the fact 

that only one decomposition is required for an unlimited number of realizations. 

Once a parameter realization has been generated, one runs the model under a given scenario to 

generate a corresponding random prediction of system states. Upon averaging the predictions 

over all realizations, one obtains a sample mean prediction. As the number of such Monte Carlo 

runs increases, one hopes that the sample mean converges to a theoretical ensemble mean. While 

such convergence can neither be insured nor verified in most cases, it is important to at least plot 

a representative measure of the sample mean versus the number of runs to verify that it has 

reached a stable value. The stable sample mean is then taken to constitute the best prediction the 

model can produce. 

In addition to computing the sample mean, one typically also computes the sample variance of 

the predictions and generates a corresponding frequency histogram. Both converge more slowly 

to their theoretical ensemble equivalents than does the sample mean. It is therefore very 

important to verify that both have stabilized with the number of runs. Since most groundwater 
223 

models are nonlinear, one need not expect the histogram of the predictions to resemble either a 

normal or a log normal distribution. 

Linearizing the flow and transport models allows one to establish approximate error bounds, or 

confidence limits, for model predictions based on linear regression theory, without Monte Carlo 

simulation. Linearization expresses each observed value of a state variable as a linear 

combination of unknown "true" parameters plus a random error(ε ). As the model is considered 

exact and its parameters true, the latter implies that ε represents an error of observation. For 

regression (calibration) purposes, ε is replaced by a corresponding residual and an optimum

(posterior) estimate of the unknown parameters is obtained by minimizing a suitably weighted 

sum of squared residuals. This sum thus becomes a measure of the error (variance) with which 

observed state variables are represented by the calibrated model. The corresponding variance is 

generally larger than that of ε because the model is based on estimated rather than true 

parameters. 

The weighted sum of squared residuals does not, however, measure the error with which state 

variables that have not yet been observed (or have not been considered for calibration) would be 

predicted by the model. This is so because future predictions would be done with parameter 

estimates that are themselves corrupted by an estimation error. Linearization expresses each 

predicted value of a state variable as a linear combination of unknown parameters plus a random

error(ε ). Each unknown parameter is the sum of a known posterior parameter estimate and an 

associated estimation error. Hence the error in predicted system states is the sum of two errors: 

one due to replacing the true parameters by their posterior (calibrated) estimates (proportional to 

the weighted sum of residuals), and the other due to rendering a prediction with the aid of 

uncertain posterior parameters (the uncertainty being characterized by the estimation variancecovariance of the parameters). Hill (1998) explains how one can establish linearized confidence 

limits, or error bounds, for predictions rendered by models calibrated using nonlinear least 

squares with a diagonal weight matrix. 

Assessing uncertainty in the output of a calibrated deterministic model, as just described, fails to 

account for the stochastic component of prediction error. Evaluating the uncertainty of standard 

deterministic model parameter estimates may allow one to quantify the uncertainty in computed 

conditional mean system state predictors (mean head, concentration, flux etc.). This, however, 

says nothing about how actual (random and unknown) system states fluctuate about their 

predictors; such information is provided only by stochastic models that treat hydraulic 

parameters as random fields. 

To do so, one must adopt a stochastic model. High-resolution Monte Carlo simulation is by far 

the most common method to assess predictive uncertainty stochastically. It entails generating 

multiple random realizations of flow and transport parameters on a fine grid, solving standard 

deterministic flow and transport equations with these parameters on the same grid under 

scenarios of interest, averaging the results and analyzing them statistically. If based on prior 

parameter estimates, the Monte Carlo results honor measured values of the parameters at discrete 

points in space. If based additionally on a suitable inverse method, they also honor parameter 

observed values of state variables. 
224 

In all other respects, the analysis of predictive uncertainty by the stochastic Monte Carlo method 

is similar to that described previously in connection with deterministic models. 

It is important to be aware that forward and inverse algorithms based on the moment equation 

approach are presently under development by our group. The algorithms yield directly (without 

Monte Carlo simulation) optimized unbiased predictors of groundwater flow and transport 

variables for randomly heterogeneous hydrogeologic environments, under the action of uncertain 

source and boundary terms. They also yield the variance-covariance of associated estimation and 

prediction errors. The algorithms do so while accounting explicitly for the multiscale (e.g., 

fractal) nature of hydrogeologic heterogeneity. They allow optimum use of field information 

through joint conditioning on measured values of hydraulic parameters, hydraulic heads and 

solute concentrations. We expect these algorithms to be available for general use in the near 

future. 

12.2.15 Render Joint Predictions by All Models 

Use Maximum Likelihood Bayesian Model Averaging to render joint predictions by all the 

retained calibrated models, and to assess their joint predictive uncertainty. Acknowledge the 

conditional nature of the predictions and their corresponding uncertainty measures. 

Alternative models that have been tested, compared, ranked and retained for further 

consideration are considered to constitute potentially valid simulators and predictors of site 

hydrogeology. There is no valid basis to prefer one of these models over another, and one must 

therefore use them in tandem. To render joint predictions using multiple models and assess their 

joint predictive uncertainty, we recommend adopting the Maximum Likelihood Bayesian Model 

Averaging (MLBMA) approach described in section 9.4. This novel approach relies on 

probabilistic maximum likelihood concepts to (a) calibrate each model against observed spacetime variations in system states (pressure, water content, concentration), considering prior 

information about relevant soil properties; (b) eliminate models that cannot be so calibrated with 

acceptable fidelity; (c) predict future system behavior or performance measures (travel times, 

concentrations, mass rates), and assess corresponding predictive uncertainty, using each model; 

and (d) average the results using posterior model probabilities as weights. MLBMA supports the 

principle of parsimony in that among models that have similar predictive capabilities, it favors 

those having fewer parameters and being therefore simpler. Regulatory hydrogeologic performance 

of a site, under a given scenario, is evaluated in terms of corresponding performance measures. 

These in turn are computed on the basis of system states (such as travel time and concentration) 

that are predicted using groundwater flow and transport models. This implies that the same 

models and methods that one uses to predict system states, and to assess their predictive 

uncertainty, can also be used to predict performance measures and their uncertainty. No special 

methodology is needed for this purpose. 

It is important to recognize implicitly that the set of predictions one produces with any given 

choice of alternative structural models and parameter sets, by whatever method, is conditional on 

the choice of models and the data used to support them. As such, these predictions do not 

represent all possibilities but only a limited range of such possibilities, associated with these 
225 

models and data. Any change in the latter would generally lead to a different assessment of 

predictive model uncertainty. There thus appears to be no way to assess the uncertainty of 

hydrologic predictions in an absolute sense, only in a conditional or relative sense. 

12.2.16 Identify Additional Data Needs 

If none of the models are deemed acceptable, perform a sensitivity analysis to help identify the 

type and quantity of additional site data that might materially enhance their reliability and 

credibility. Decide based on the potential benefit (in terms of bias and uncertainty reduction) 

and cost (in terms of time, effort and money) of such data whether or not to collect them and 

how. 

To help evaluate what if any additional data might be worth collecting so as to materially reduce 

model uncertainty (by further constraining the range of alternative model structures and 

parameters), one may conduct a sensitivity analysis to indicate what system behavior appears to 

be most sensitive to which parameters at what locations. The next step is to consider performing 

additional site characterization where existing parameter estimates are least certain and the 

model is relatively insensitive to their values, and monitoring system behavior where it is most 

sensitive to model parameters while prediction errors appear to be relatively large and 

consequential. This may indicate the type and quantity of additional site data that might 

materially enhance model reliability and credibility. 

The question how much and what kind of data are enough for model development and evaluation 

is one of economics and policy, not of hydrogeologic analysis or modeling. It is therefore the 

responsibility of managers and decision-makers to answer, not hydrogeologists or modelers. The 

strategy in this report is designed to help one address the question by how much would additional 

site characterization and monitoring improve the prediction of performance measures derived 

from the hydrogeologic analysis. In other words, the strategy addresses the worth of data in 

terms of their contributions to the potential enhancement of model reliability and credibility, not 

in terms of their marginal cost-benefit. Its aim is to allow managers and decision-makers to make 

informed decisions about the time, manpower and budget that they deem worth allocating to 

these activities in light of such enhancement under existing administrative, budgetary and policy 

constraints. 

12.2.17 Repeat Iterative Process with New Data 

If and when new data of significant weight are obtained, repeat the iterative process till its costbenefit ratio reaches a value that is not considered worth exceeding. 

This final guideline requires no further elaboration.
226 

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APPENDIX A 

PERFORMANCE MEASURES AND HYDROLOGY ISSUES 

PERTAINING TO GROUNDWATER FLOW AND TRANSPORT MODELS 

OF LICENSED NUCLEAR SITES
A-1 

APPENDIX A: PERFORMANCE MEASURES AND HYDROLOGY ISSUES 

PERTAINING TO GROUNDWATER FLOW AND TRANSPORT MODELS 

OF LICENSED NUCLEAR SITES 

This note was prepared by the licensing staff 

in the Office of Nuclear Material Safety and 

Safeguards (NMSS) for the benefit of the 

research staff in connection with an ongoing 

generic research that is being conducted for 

NRC’s Office of Nuclear Regulatory 

Research (RES) by the University of 

Arizona. The main purpose of the research is 

to develop strategy for formulation, 

development, and application of 

representative groundwater flow and 

transport models for licensed sites. More 

specifically, this information is provided to 

ensure that the ongoing research efforts 

identify methods and techniques that can be 

used by the licensing staff to better analyze 

uncertainties arid improve the decisionmaking procedures with regard to the 

selection and application of site-specific 

groundwater flow and transport models for 

licensed sites. 

The note provides the following information 

concerning licensed nuclear facilities, 

including processing and waste disposal 

sites: (1) performance measures based on 

regulatory standards; (2) pertinent 

hydrology/groundwater issues that are 

related to performance; and (3) uncertainties 

commonly encountered in groundwater flow 

and transport modeling. 

The performance measures and pertinent 

hydrology issues and uncertainties are 

provided for licensed sites in the following 

NMSS programs: low-level radioactive 

waste, decommissioning, high-level 

radioactive waste, and uranium recovery. 

Although a variety of derivative hydrologic 

measures and hydrologic issues are stated, 

RES staff and their contractors should keep 

in mind that the end point of the analysis is 

the performance measure for the licensed 

facility obtained through a performance 

assessment. The hydrologic measures and 

issues need only be explored and dealt with 

to the extent required to have confidence in 

the technical basis supporting licensing 

decisions, taking into account the significant 

uncertainties and the hazard involved (note: 

some uncertainties may be satisfactorily 

addressed through bounding analyses). 

Low-Level Waste Disposal Sites

The performance measure in the current 

regulations for low-level radioactive waste 

(LLW) disposal sites is provided by a 

standard for protection of the population and 

the environment in 10 CFR 61.41. 

According to this standard, concentrations of 

radioactive material which may be released 

to the general environment in groundwater, 

surface water, air, soil, plants, or animals 

must not result in an annual exceeding an 

equivalent of 25 millirems (mrem) to the 

whole body, 75 mrem to the thyroid, and 25 

mrem to any other organ of any member of 

the public. In addition, the standard indicates 

that reasonable effort should be made to 

maintain releases of radioactivity in 

effluents to the general environment as low 

as reasonably is reasonably achievable 

(ALARA). 

Staff views on performance assessment 

approaches for LLW sites is provided in 

NUREG-1573. As stated in NUREG-1573 

(p. 3-57), the objective of the groundwater 

flow and transport analyses (including 

groundwater models) is to assess 
A-2 

concentrations of radionuclides released in 

the ground water at receptor locations so as 

to assess the potential annual dose. Staff 

recommended in NUREG-1573 that the 

groundwater transport analysis provide 

concentrations in well water at the site 

boundary representative of composite 

concentration of radionuclides for the 

average member of the critical group 

(critical group is defined as the group of 

individuals reasonably expected to receive 

the greatest dose from radioactive releases 

from the disposal facility over time, given 

the circumstances under which the analysis 

would be carried out - NUREG-1537, page 

3-76). 

Analysis of radionuclide concentrations in 

the groundwater at specific sites is carried 

out usually involves addressing such sitespecific hydrologic issues as infiltration 

through the disposal site cover, release of 

radionuclides from the waste, and flow and 

radionuclide transport in the unsaturated 

zone and in the saturated zone to the 

receptor points. 

Uncertainties commonly encountered in the 

hydrologic analysis of LLW disposal sites 

include both data as well as conceptual 

uncertainties. Data uncertainties include the 

hydraulic properties of the cover (mainly 

permeability/hydraulic conductivity, service 

life); the hydraulic properties of the 

formations beneath the site (mainly 

permeability/hydraulic conductivity, 

anisotropy and inhomogeneity, effective 

porosity); retardation properties and 

coefficients; and pH values. Conceptual 

uncertainties may include lateral flow versus 

vertical flow and possible development of 

perched water conditions in the unsaturated 

zone below the disposal facility; matrix 

versus fracture flow; structural controls on 

flow and radionuclide transport; interaquifer flow; and uncertainty related to 

changes in time-dependent variables (such 

as the water levels, concentrations, and pH). 

Data-related uncertainties are sometimes 

addressed by bounding analyses. 

Decommissioning of Licensed Sites

The performance measure in the current 

regulations for decommissioning of licensed 

sites is provided by the standards in 10 CFR 

20.1402 (for license termination including 

unrestricted use of the decommissioned site) 

and 10 CFR 20.1403 (license termination 

under restricted conditions for use of the 

decommissioned site). According to the 

standards in 10 CFR 20.1402, a site will be 

considered acceptable for unrestricted use if 

the residual radioactivity that is 

distinguishable from background radiation 

results in a Total Effective Dose Equivalent 

(TEDE) to an average member of the critical 

group at receptor locations or human access 

points that does not exceed 25 mrem per 

year, including that from groundwater 

sources of drinking water, and the residual 

radioactivity has been reduced to levels that 

are ALARA. 

The standards in 10 CFR 20.1403 provide 

that a site will be considered acceptable for 

license termination under restricted 

conditions by satisfying certain provisions 

specified in the regulations. These include 

provisions pertaining to meeting the 

ALARA provision; legally enforceable 

institutional controls that provide reasonable 

assurance that the TEDE from the residual 

radioactivity will not exceed 25 mrem per 

year; financial assurance to assume and 

carry out any necessary control and 

maintenance of the site; submittal of a 

decommissioning or a license termination 

plan indicating intent to decommission in 

accordance with the regulations in IO CFR 
A-3 

Subparts 30.36(d), 40.42(d), 50.82(a) and 

(b), 70.38(d), or 72.54; and that if the 

institutional controls were no longer in 

effect, there is reasonable assurance that the 

residual activity to the average member of 

the critical group is ALARA, and would not 

exceed either 100 mrem per year, or 500 

mrem per year provided that the licensee: 

(1) demonstrates that further reductions in 

residual radioactivity necessary to comply 

with the 100 mrem per year are not 

technically achievable, would be 

prohibitively expensive, or would result in 

net public or environmental harm; (2) makes 

provisions for durable institutional controls; 

and (3) provides sufficient financial 

assurance to enable a responsible 

government entity or independent third 

party, both to carry out periodic rechecks of 

the site to assure that the institutional 

controls remain in place as necessary to 

provide reasonable assurance that the TEDE 

from the residual radioactivity will not 

exceed 25 mrem per year. 

Guidance for demonstrating compliance 

with the current regulations for 

decommissioning sites is provided in 

NUREG-1549. This guidance does not 

explicitly address how the groundwater 

analysis should be performed. However, 

guidance is provided for assessment of the 

dose for an individual located on site (e.g., 

using water extracted from a well located on 

the site with an intake point directly beneath 

the waste area), an individual located off site 

(e.g., using water extracted from a well 

located at the site boundary), or both. It 

should be pointed out, as reflected in 

NUREG-1549, the NRC recommended dose 

modeling approach is an iterative approach 

that involves a screening analysis initially, 

but eventually includes more site-specific 

analyses as warranted by the site conditions. 

The screening approach recommended in 

another NRC staff document, NUREG-5512 

(vol. 1), has a predefined ground-water 

conceptual model. 

Operational environmental monitoring of 

groundwater, although adequate for its 

intended purpose, may not be adequate for 

site characterization and to support dose 

assessments for decommissioning. As noted 

in NUREG-1727, "NMSS Decommissioning 

Standard Review Plan," Section 4.6, 

"Groundwater," "[T]he information supplied 

by the licensee should be sufficient to allow 

the staff to fully understand the types and 

movement of radioactive material 

contamination in groundwater at the facility, 

as well as the extent of this contamination." 

The actual number, location, and design of 

monitoring wells depend on the size of the 

contaminated area, the type and extent of 

contamination, the background quality, 

hydrogeologic system, and the objectives of 

the monitoring program. 

As part of the guidance consolidation 

efforts, NUREG-1556, Vol. 1-20, staff is 

currently working on the consolidation of 

Decommissioning License Termination 

Guidance. As part of this effort, the 

guidance is being updated to risk inform. 

This guidance consolidation for 

decommissioning is planned to be 

completed within approximately 2 years and 

will be contained in NUREG-1757. 

 

On October 9, 2002, NRC and EPA signed 

an MOU which outlined consultation and 

finality on decommissioning and 

decontamination of contaminated sites. The 

MOU requires NRC to seek EPA’s expertise 

when NRC determines that there is 

radioactive groundwater contamination in 

excess of EPA’s MCLs or for which 

restricted release or the use of alternate 

criteria for license termination is contemplated. 
A-4 

Uncertainties commonly encountered in the 

hydrologic analysis for decommissioning of 

licensed sites include both data as well as 

conceptual uncertainties that are similar to 

those encountered at LLW sites (see LLW 

Disposal Sites above). 

High-Level Waste Disposal Sites

The performance measures for the potential 

high-level waste (HLW) repository at Yucca 

Mountain, Nevada in NRC’s regulations at 

10 CFR Part 63 are based on the EPA 

standards (40 CFR 197) which provide: (1) 

an overall performance measure (annual, 

individual dose limit of 15 mrem TEDE to a 

reasonably, maximally exposed individual); 

(2) a performance measure for a specified 

human intrusion scenario (annual, individual 

dose limit of 15 mrem TEDE to a 

reasonably, maximally exposed individual); 

and (3) separate limits for the protection of 

ground water (5 pCi/l for radium-226 and 

radium-228, 15 pCi/liter for gross alpha 

activity, and 4 mrem/year to the whole body 

or any organ from combined beta and 

photon emitting radionulcides). Compliance 

with these performance measures is 

demonstrated with a performance 

assessment using a compliance location of 

approximately 18 kilometers downgradient 

from the repository footprint and over a 

compliance period of 10,000 years. 

Additionally, the performance assessment is 

limited to certain conditions (e.g., features, 

events, and processes estimated to have less 

than one chance in 10,000 of occurring over 

10,000 years are not to be considered in the 

performance assessment). 

Hydrologic processes and flow and transport 

issues that may need to be considered for the 

Yucca Mountain site include: infiltration 

from the ground surface under present and 

future climates; deep percolation from the 

root zone into the waste emplacement drift 

(an unsaturated, fractured and anisotropic 

zone above the repository); thermal effects 

on the flow regime, hydraulic and transport 

properties of formations in the unsaturated 

zone; flow and transport of radionuclides in 

the unsaturated zone below the repository; 

and flow and transport of radionuclides in 

the saturated zone between the repository 

and the receptor group location. 

Consideration of these complex these issues 

over the 10,000 year regulatory period is 

expected to require the evaluation of 

hydrologic uncertainties through the use of 

groundwater and transport models of Yucca 

Mountain. 

Uranium Recovery/Tailings Disposal Sites 

The performance measure for reviewing 

uranium mill and tailings sites can be 

divided into three areas: Title I dealing with 

DOE-remedial action programs of former 

mill tailings sites; Title II dealing with nonDOE mill tailings sites; and in situ leach

(ISL) uranium solution mining sites. In all 

three areas, concentration limits of specified 

chemical and radionuclide constituents in 

groundwater are determined. 

Title I

For Title I sites, the performance measures 

are covered in 40 CFR 192. Specifically 

Subparts A, B and C of Part 192 provide the 

regulatory requirements for water resources 

protection. Guidance for implementing 

performance of groundwater standards is 

covered in NUREG-1724 entitled “Draft 

Standard Review Plan for the Review of 

DOE Plans for Achieving Regulatory 

Compliance at Sites With Contaminated 

Ground Water Under Title I of the Uranium 

Mill Tailings Radiation Control Act.” 
A-5 

Protection of water resources at Title I sites 

is a process that encompasses two strategies. 

The first strategy is to prevent or contain the 

spread of contaminants in the groundwater, 

surface water, and surrounding lands. The 

second strategy is to mitigate the threat to 

public health an the environment from 

contaminants that have already been 

mobilized. Performance of these strategies 

is maintained through groundwater 

monitoring programs and corrective action. 

Groundwater models are utilized to predict 

the fate and transport of the constituents and 

to design remedial systems. NRC staff 

utilize NUREG-1724 for specific guidance 

criteria in site characterization, groundwater 

monitoring, corrective action of 

groundwater contamination at Title I 

uranium mill tailings sites. 

The hydrogeologic model plays a role in 

groundwater monitoring to demonstrate 

compliance with standards and in detection 

of contamination, site characterization of 

contamination, and corrective action. For 

example, in estimating the risk of a potential 

receptor (i.e. water supply well, stream) 

from groundwater contamination as a result 

of mill activities, models must be 

sufficiently detailed to provide a technical 

basis for regulatory decisions and assure the 

protection of human health and the 

environment. Factors such as contaminant 

velocity in groundwater through the 

geologic media, geochemical conditions, 

and potential preferential pathways must be 

factored into the transport of radionuclides 

and other constituents. 

Title II

Regulations for the performance measures 

and standards for groundwater at Title II 

uranium mill and tailings sites is covered in 

10 CFR Part 40, Appendix A Criterion 5, 7, 

and 13. Guidance for demonstrating 

performance for Title II uranium mill and 

tailings sites can be found in NUREG-1620 

entitled “Draft, Rev. 1, Standard Review 

Plan for the Review of a Reclamation Plan 

for Uranium Mill Tailings Sites Under Title 

II of the Uranium Mill Tailings Radiation 

Control Act.” The strategies with Title II 

sites are similar to Title I. The first strategy 

is to prevent or contain the spread of 

contaminants in the groundwater, surface 

water, and surrounding lands. The second 

strategy is to mitigate the threat to public 

health an the environment from 

contaminants that have already been 

mobilized. Groundwater protection 

standards are either: (1) Commission 

approved background of a constituent in the 

groundwater, (2) respective values given in 

10 CFR Part 40, Appendix A, paragraph 5C 

[maximum concentration limits] if the 

constituent is listed in the table and if the 

background level of the constituent is below 

the value listed, or (3) an alternate 

concentration limit established by the 

Commission. Alternate concentration limits 

may be approved by the Commission if they 

do not present a significant hazard to human 

health or the environment, are as low as 

reasonably achievable, after considering 

practicable corrective actions. Groundwater 

detection programs are used to demonstrate 

compliance. If groundwater protection 

standards are exceeded, a corrective action 

program must be put into operation pursuant 

to 10 CFR Part 40, Appendix A, Criterion 

5D. The objective of the program is to 

return the hazardous constituent 

concentration levels in groundwater to the 

standards. 

In proposals for alternate concentration 

limits, licensee’s often use groundwater fate 

and transport models to demonstrate that a 

concentration at the point of compliance will 
A-6 

not present a significant hazard at the point 

of exposure. The licensee can account for 

degradation of that constituent through the 

geologic media over time and distance. 

Staff criteria for reviewing these models can 

be found in NUREG-1620. Sufficient site 

characterization is paramount in building a 

technically adequate model. Uncertainties 

are encountered in the site characterization 

process and in the assumption that 

conditions, as they were characterized, will 

remain constant for 1000 years 

(performance period). Extended 

groundwater monitoring programs are 

implemented to lower uncertainty and to 

assure continued protection of human health 

and the environment. 

Once a license for a Title II site is 

terminated, pursuant to 10 CFR Part 40.28, 

the site is transferred under general license 

to the Department of Energy, another 

Federal agency designated by the President, 

or a State where the disposal site is located. 

Groundwater monitoring is detailed in the 

Long-Term Surveillance Plan. 

In Situ Leach (ISL) Uranium Extraction 

Guidance for demonstrating compliance for 

in situ leach (ISL) uranium extraction 

license applications is provided in NUREG1569. This guidance explicitly addresses the 

groundwater information and analysis that is 

specified in Regulatory Guide 3.46 

"Standard Format and Content of License 

Applications, Including Environmental 

Report, for In Situ Uranium Solution 

Mining. NUREG-1569 identifies the NRC 

reviewer's proposed activities in reviewing a 

licensee submittal, specifically the areas of 

review, review procedures, acceptance 

criteria, evaluation findings and references. 

The ground-water issues in NUREG-1569 

relate to ground- water quality restoration. 

The monitoring programs needed to assure 

ground-water quality restoration are 

discussed. The acceptance criteria for the 

ground-water quality are established based 

upon the background water quality prior to 

ISL mining. 

NUREG-1569 states that restoration goals 

are established in the application for each of 

the monitored constituents. The applicant 

has the option of determining restoration 

goals for each constituent on a well-by-well 

basis, or on a well field average basis. 

Restoration goals should be established for 

the ore zone and for any overlying or 

underlying aquifer that remains affected by 

ISL solutions. Performance measures for 

ISL sites can be classified into two groups; 

primary restoration goals, and secondary 

restoration goals. For primary restoration 

standards, the primary goal for a restoration 

program is to return the water quality of the 

ore zone and affected aquifers to preoperational (baseline) water quality. It is 

unlikely that after restoration activities the 

ground-water quality will be returned to the 

exact water quality that existed at every 

location in the aquifer before ISL 

operations. Therefore, it is acceptable to use 

standard statistical methods to set the 

primary restoration goal and to determine 

compliance with it. It is also acceptable for 

the applicant to propose that the baseline 

conditions for each chemical species be 

represented by a range of concentrations. 

The reviewer shall ensure that statistical 

methods used to determine such confidence 

intervals are properly applied. 

For secondary restoration standards, since 

the ISL process requires changing the 

chemistry of the ore zone, it is reasonable to 

expect that ISL may cause permanent 

changes in water quality. For this reason, it 

is acceptable for the applicant to propose, as 

a secondary restoration standard, returning 
A-7 

the water quality to its pre-ISL class of use 

(e.g., drinking water, livestock, agricultural, 

or limited use). Applications should state 

that secondary standards will not be applied 

so long as restoration continues to result in 

significant improvement in ground-water 

quality. It is acceptable to the staff if, on a 

constituent-by-constituent basis, secondary 

goals are determined by applying the lower 

of the State or EPA secondary and primary 

drinking water standards. 

If a ground-water parameter could not be 

restored to its secondary goal, an applicant 

could demonstrate to NRC that leaving the 

parameter at the higher concentration would 

not be a threat to public health and safety 

nor the environment, and that, on a 

parameter-by-parameter basis, water use 

would not be significantly degraded. Such 

proposed alternatives must be evaluated as a 

license amendment request only after 

restoration to the primary or secondary 

standard is shown not to be practical. This 

approach is consistent with the ALARA 

philosophy that is used broadly within NRC. 

Uncertainties commonly encountered in the 

hydrologic analysis for Title I, Title II, and 

ISL sites include both data, as well as, 

conceptual model uncertainties. The data 

uncertainties are similar to those 

encountered in ground-water quality 

monitoring programs (e.g., sampling 

methods, well screen location, and 

laboratory analysis). The uncertainties in the 

conceptual models are those encountered in 

the site characterization process, as well as, 

the process for determining model 

assumptions used in estimating input 

parameters, and data analysis of field tests 

(e.g., pump and pilot study tests) and 

compliance monitoring.
A-8 
APPENDIX B 

VERTICAL AVERAGING OF WATER-TABLE EQUATION 
B-1 

APPENDIX B: VERTICAL AVERAGING OF WATER-TABLE EQUATION 

Transient groundwater flow in a saturated 

porous medium is governed by the threedimensional continuity equation 

3 s

h S

t

∂ −∇ ⋅ = ∂ q

subject to Darcy's law where q K = − ∇3h

3 ( ) / , / , / T

∇ = ∂ ∂x ∂ ∂y ∂ ∂z is the threedimensional gradient operator (the 

superscript T indicating transpose), q is flux 

vector, Ss is specific storage, h is hydraulic 

head, t is time, and K is a 3 x 3 hydraulic 

conductivity tensor. The same equation 

applies at any point within a water-table 

aquifer. Let the bottom of this aquifer be at 

elevation , and the water table at 

elevation 

( , o z x y)

ξ ( x, , y t), above some horizontal 

datum. The same datum serves to define the 

three-dimensional head function 

h x( ) , , y z,t . 

Flow in unconfined aquifers is commonly 

described by a two-dimensional (rather than 

three-dimensional) flow equation in the

horizontal plane. Here we derive this twodimensional equation from its more 

comprehensive three-dimensional counterpart 

by formally averaging the above continuity 

expression over the vertical, 

3 0

o

s

z

h S dz

t

ξ ⎛ ⎞ ∂ ⎜ ⎟ ∇ ⋅ =

⎝ ⎠ ∂ ∫ q +

From Leibnitz’s rule for the differentiation 

of integrals it follows that

( ) ( ) o 0

x o

x x x o

z z

q z dz q dz q q z

x x x

ξ ξ ξ ξ ∂ ∂ ∂ ∂ = − +

∂ ∂ ∂ ∫ ∫ ∂x

( ) o 0 z z

h dz h dz h

t t

ξ ξ

t

ξ ξ ∂ ∂ = − ∂ ∂ ∫ ∫ ∂

∂

This allows us to rewrite the verticallyintegrated water balance equation as 

( ) , , ( )|

o

x y x y z

z

q q dz q q

ξ

∇⋅ − =ξ ξ ∫ ⋅∇

( ) , | o

o

z

x y z z o

z

q q q z dz

z

ξ

=

∂

+ ⋅∇ +

∂ ∫

( ) 0 o

s z S h dz h

t t

ξ ξ ξ ⎡ ⎤ ∂ ∂

+ − = ⎢ ⎥ ⎣ ⎦ ∂ ∂ ∫

where now ( ) / , / T

∇ = ∂ ∂x ∂ ∂y is a twodimensional gradient operator in the 

horizontal plane, and Ss is considered to be 

independent of time. We define the 

vertically-averaged head h and horizontal 

flux ( , )

T

q = q q x y as 

1

oz o

h h

z

ξ

ξ = − dz ∫

( ) 1 , o

T

x y z o

q q dz

z

ξ

ξ − ∫ q =

Next, we adopt the well-known Dupuit 

assumption that flow in the aquifer is

virtually horizontal. This assumption 

implies that head does not vary much in the 

vertical direction and therefore 

h h (ξ ) ≈ (zo ) ≈ h . We can therefore write 

the vertically-integrated continuity equation as 

( o x ) ( , | y ) z ∇⋅ − ⎡ ⎤ ξ z q − q =ξ ⋅∇ξ ⎣ ⎦ q 

( , | ) | | o o q q x y z z o z z z z z + ⋅ = ∇z + q = = ξ −q

s o ( ) 0 h S z

t

ξ ∂ − ≈ ∂

where the last term follows from

( ) ( )

o

o

o

z z

z h dz h h dz h

t t t

ξ ξ ξ ξ ξ ∂ ∂ ∂ ∂ − − ≈ − ∂ ∂ ∂ ∫ ∫ ∂t
B-2 

1 ( )

o o

o

z z o

z h dz h dz

t z t

ξ ξ ξ

ξ

∂ ∂ − = − ∂ − ∂ ∫ ∫

( ) ( ) 1

o

o o z o

h z h dz

t z

ξ

ξ ξ ξ

∂ ⎡ ⎤ = − ⎢ ⎥ = − ∂ − ∂ ⎣ ⎦ ∫ z

t

∂

We treat the water table as a free moving 

upper boundary of the saturated flow region, 

at which gauge pressure is zero 

(atmospheric) and therefore head h at the 

water table is equal to the elevation z. From

the Lagrangian viewpoint of a fluid particle 

that moves with the water table, the quantity 

h - z remains fixed at zero. In other words, 

the Lagrangian derivative of h - z is zero, 

( )|

z

D h z

Dt − =ξ

1 sx sy sz

z

hhh h

q q q

t x y z =ξ

⎡ ⎤

⎥

⎦

∂ ∂ ∂ ⎛ ⎞ ∂

≡ + ⎢ + + − ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠ ∂ ⎣

= 0

where qs is the (seepage) velocity of the 

water table. By virtue of the Dupuit 

assumption, we can disregard in this 

expression and replace h in all the remaining 

terms by 

∂h / ∂z

ξ to yield 

sx sy sz 0

z

qqq

t x y ξ

ξ ξ ξ

=

⎛ ⎞ ∂∂∂ ⎜ ⎟ + + − =

⎝ ⎠ ∂ ∂ ∂

Water balance at the water table requires 

that ( ) where r is a vector 

representing recharge from above, and S

y s q r − ⋅n = S q ⋅n

y is 

specific yield or drainable porosity. It is 

common to assume that recharge is strictly 

vertical so that r = (0,0,−I ) ate 

ce at

S q , q S y yqsy and sz I = S q . 

Multiplying t hout

 where I the r

of net vertical infiltration. Water balan

the free boundary then implies that

qx = = qz +

he above equation throug

by S

y sx y

y therefore yields 

y x y ( ) z 0

z

S q q q I

t x y ξ

ξ ξ ξ

=

⎡ ⎤ ∂ ∂ ∂

+ + − + = ⎢ ⎥ ⎣ ⎦ ∂ ∂ ∂

which can be rewritten as

( ) , | | x y z z z y q q q I S

t ξ ξ

ξ ξ = =∂ ⋅∇ − = − ∂

Substituting this into the vertically-averaged 

continuity equation and rearranging yields 

( ) ( , | ) o x o y z z o ξ z q q z = −∇⋅⎡ ⎤ − + ⋅∇ + I ⎣ ⎦ q

| ( ) o z z z s o y

h q S z S

t = ξ ∂

+ ≈ ⎡ − + ⎤ ⎣ ⎦ ∂

So far the vertical averaging has not affected 

the three-dimensional Darcy equation 

q K = − ∇3h . It is important to note that 

setting h ≈ h in accord with the Dupuit 

assumption eliminates the vertical derivative 

of head, but does not eliminate the vertical 

component of flux. For the vertical flux to 

vanish we must assume that K has a 

principal direction parallel to z. Then the 

vertical averaging of yields a 

two-dimensional form of Darcy's law, 

q K = − ∇3h

q = −K H ∇h

where KH is an equivalent hydraulic 

conductivity tensor given by 

1

o

H z o

dz

z

ξ

ξ = − ∫ K K

Note that KH is a 2 x 2 tensor defined in the 

horizontal plane, which does not depend on 

z but may vary with the horizontal 

coordinates x and y. Note further that KH is 

the arithmetic average of K over the 

saturated thickness o ξ − z of the aquifer. As

the saturated thickness may vary with time, 

so can KH, even if K is a constant. The 

variability of KH with saturated thickness is 

commonly ignored, even though it may be 

significant in heterogeneous (and especially 

stratified) aquifers. 
B-3 

If the principal values of K in the horizontal 

plane do not vary strongly with elevation, 

then the Dupuit assumption implies that 

( ) , | o

T

q q x y z=z ≈ q . The vertically-averaged

continuity equation can therefore be written 

as 

S z s o ( ) Sy t

ξ ξ ∂ ≈ − ⎡ + ⎤ ⎣ ⎦ ∂

The quantity (h z − o ) K H or ( is 

commonly referred to as transmissivity. 

o ) H ξ − z K

It is common to set zo ≡ 0 and disregard 

artesian storage. Then one obtains the wellknown Boussinesq equation either in terms 

of average head, 

( ) | o o o −∇ ξ z z I q = ⋅⎡ ⎤ − ⋅∇ + + ⎣ ⎦ q + q z z z

s o ( ) y

h S z S

t

ξ ∂ ≈ − ⎡ ⎤ + ⎣ ⎦ ∂

( ) | H z o z z

h h h I q S

t =

∂ ∇ ⋅ ∇ + + ≈ ∂ Here ( ) K y o ξ − z q is horizontal flow rate 

across the entire saturated thickness of the 

aquifer per unit transverse horizontal 

distance, o q ⋅∇z is a term that accounts for 

variations in the elevation of the aquifer 

bottom, I is net vertical recharge at the water 

table, is rate of leakage across the 

aquifer bottom,

| o qz z=z

( ) Ss o ξ − z is an internal

(artesian) storage coefficient or storativity 

that depends on saturated thickness, and the 

specific yield Sy is storativity due to 

imbibition or drainage at the water as the 

latter rises or falls. 

or in terms of water-table elevation, 

( ) | H z o z z I q S

t y

ξ ξ ξ =∂ ∇ ⋅ ∇ + + ≈ ∂

K

Our formal averaging of flow in an 

unconfined aquifer was accompanied by a 

number of simplifying assumptions. These 

state that 

1. Flow in the aquifer is essentially 

horizontal. 

2. Recharge above the water table is 

essentially vertical. 

3. One principal direction of the threedimensional hydraulic conductivity 

tensor K is vertical. Introducing the vertically-averaged form of 

Darcy's law into the above expression, and 

replacingξ by h , yields the verticallyaveraged flow equation 

4. The horizontal components of K vary 

weakly with elevation (this is needed

only if the aquifer bottom is uneven 

and/or one treats KH as being ( ) independent of saturated thickness). | o h zo H h H h o z z z z I q = ∇ ⋅⎡ ⎤ − ∇ ∇ ⋅∇ + + ⎣ ⎦ K - K

The standard form of the Boussinesq 

( ) equation assumes additionally that s o y

h S h z S

t

∂ ≈ − ⎡ ⎤ + ⎣ ⎦ ∂ 5. The aquifer bottom is horizontal and flat. 

which is seen to be nonlinear in the 

vertically-averaged head h . Alternatively, 

one can replace h byξ to obtain a similar 

equation in terms of water-table elevation, 

6. Artesian storage is negligible. 

( ) | o H H o o z z z ∇ ⋅ ξ ξ − z ∇ ∇ξ z I q = ⎡ ⎤ ⋅∇ + + ⎣ ⎦ K - K

When these assumptions are violated, a twodimensional description of flow in an

unconfined aquifer in the horizontal plane 

may lead to a systematic bias in predicted 

heads and fluxes. 
B-4 
APPENDIX C 

AVERAGING OF GROUNDWATER FLOW IN PROBABILITY SPACE 
C-1 

APPENDIX C: AVERAGING OF GROUNDWATER FLOW IN PROBABILITY SPACE 

A new look at traditional deterministic flow models and their calibration 

in the context of randomly heterogeneous media 

SHLOMO P. NEUMAN

Department of Hydrology and Water Resources, The University of Arizona, Tucson, AZ, USA 

ALBERTO GUADAGNINI

D.I.I.A.R., Politecnico di Milano, Piazza L. da Vinci, 32, 20133 Milano, Italy 

Abstract We examine the tradition of modelling subsurface flow deterministically from a 

stochastic viewpoint. In particular, we show that traditional deterministic flow equations 

do apply to randomly heterogeneous media, albeit in an approximate manner, provided 

they are interpreted in a non-traditional manner. Our paper explains why parameter 

estimates obtained by traditional inverse methods tend to vary as one modifies the 

database. It also makes clear that the traditional Monte Carlo method of assessing 

uncertainty in the output of a calibrated deterministic model generally overestimates the 

predictive capabilities of the model. The only valid way to assess predictive uncertainty is 

by means of a stochastic model. 

INTRODUCTION 

Parameters such as hydraulic conductivity have been traditionally viewed as well-defined local 

quantities that can be assigned unique values at each point in space. Yet in practice they are 

deduced from measurements at selected well locations and depth intervals where their values 

depend on the scale and mode of measurement. Quite often, the support of the measurements is 

uncertain and the data are corrupted by experimental and interpretative errors. Estimating the 

parameters at points where measurements are not available entails an additional random error. It 

is therefore appropriate to think of the subsurface as being randomly heterogeneous and to 

describe fluid flow in geologic media by means of stochastic equations. 

Yet, the tradition has been to model subsurface flow deterministically. In this paper we 

examine this tradition from a stochastic viewpoint and comment on its validity and meaning. 

More specifically, we ask what is the meaning of hydrogeologic variables that enter into 

traditional deterministic groundwater flow models when the latter are applied to randomly 

heterogeneous media? What is the significance of parameters obtained by calibrating a 

deterministic flow model against randomly varying data? What do corresponding parameter 

estimation errors imply about predictive uncertainty? 
C-2 

To address these questions, we consider steady state flow of groundwater in a randomly 

nonuniform domain, Ω. The flux q(x) and the hydraulic head h(x) obey the continuity equation 

and Darcy’s law, subject to appropriate boundary conditions. All parameters and state variables

are defined on a consistent nonzero support volume, ω, which is small in comparison to Ω, but 

sufficiently large for Darcy’s law to be locally valid. It has been shown (Neuman & Orr, 1993; 

Neuman et al., 1996; Guadagnini £ Neuman, 1999a, b; Tartakovsky & Neuman, 1998, 1999) that 

it is theoretically possible, and computationally feasible, to render optimum unbiased predictions 

of h(x) and q(x) under ubiquitously nonuniform and uncertain field conditions by means of their 

first ensemble (statistical) moments (expected or mean values), 〈h(x)〉c and 〈q(x)〉c, conditioned 

on measurements of K(x). The predictors 〈h(x)〉c and 〈q(x)〉c satisfy the equations 

− 〈q(x)〉c + 〈f(x)〉 = 0 x ∈ Ω (1) 

〈q(x)〉c = − 〈K(x)〉c∇〈h(x)〉c + rc(x) rc(x) = − 〈K’(x) ∇h’(x)〉c (2) 

subject to the boundary conditions 

〈h(x)〉c = 〈H(x)〉; x ∈ ΓD − 〈q(x)〉c ⋅ n(x) = 〈Q(x)〉; x ∈ ΓN (3) 

where the subscript c implies “conditional”; primed quantities represent random fluctuations 

about (conditional) mean values; K(x) is a random field of scalar hydraulic conductivities; rc(x) 

is a residual flux; 〈f(x)〉, 〈H(x)〉, 〈Q(x)〉 are prescribed unconditional first moments of the 

statistically independent random source and boundary forcing terms f(x), H(x), Q(x); and n(x) is 

a unit outward normal to Γ = ΓD ∪ ΓN where ΓD and ΓN are Dirichlet and Neumann boundaries, 

respectively. The residual flux rc(x) is given implicitly by (Neuman et al., 1996) 

∫ ∫

Ω Ω

r x = a y x ∇ y y + d y x r y y c c , h d c , c d c y ( ) ( ) ( ) ( ) ( ) (4) 

where the kernels 

x y c ( , ) =< K'( ) K'( )∇ ∇ G( , ) > T a y x x y y x c (5) 

x y c ( , ) =< K'( ) ∇ ∇ G( , ) > T d y x x y x c (6)

form a symmetric and a non-symmetric tensor, respectively. Here G is a random Green’s 

function, or the solution of the random flow equations for the case where f(x) is a point source of 

unit strength at point y, subject to homogeneous boundary conditions H(x) ≡ Q(x) ≡ 0. 
C-3 

 Due to the integro-differential nature of rc(x), our conditional moment equations include 

nonlocal parameters that depend on more than one point in space (hence the equations are

referred to as nonlocal). The traditional concept of an REV (representative elementary volume) is 

neither necessary nor relevant for their validity or application. The corresponding parameters are

nonunique in that they depend not only on medium properties but also on the information one 

has about these properties (scale, location, quantity, and quality of data). It is clear that the flux 

predictor is generally nonlocal and non-Darcian, in that it depends on the residual flux rc(x). 

 Guadagnini & Neuman (1999a, b) have developed corresponding integro-differential 

equations for the conditional variance-covariance of associated prediction errors in head and 

flux, and have shown how to solve both sets of equations by finite elements. Their solution

entails expansion of the otherwise exact nonlocal moment equations in terms of a small 

parameter, σY, which represents a measure of the standard deviation of natural log hydraulic 

conductivity, Y(x) = ln K(x). 

Since the flux predictors rc(x) and 〈q(x)〉c are generally nonlocal and non-Darcian, the 

traditional notion of effective conductivity looses meaning in the context of flow prediction by

means of conditional ensemble mean quantities. In the following section we show how these

quantities can be localised in an approximate manner, so as to render them Darcian. 

LOCALIZATION OF CONDITIONAL MEAN FLUXES

If one treats the conditional mean hydraulic gradient ∇〈h(y)〉c, and residual flux rc(y), in (4) as if

they were locally uniform, one can take them outside the corresponding integrals and express

them as functions of x. One can then express the residual flux in localised, Darcian form as 

rc(x) ≈ − κc(x) ∇〈h(x)〉c (7)

- (8) ∫ ∫

Ω

−

Ω ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

x = − d y x y a y x y c ( ) 1 c ( , ) d c ( , ) d

1

κ

This leads to a familiar looking Darcian form for the mean flow predictor 

〈q(x)〉c ≈ − Kc(x) ∇〈h(x)〉c Kc(x) = 〈K(x)〉c I + κc(x) (9) 

where I is the identity tensor and Kc(x) is a spatially-varying conditional effective hydraulic

conductivity tensor. The latter is either symmetric positive definite or nonsymmetric, depending 

on whether κc(x) is symmetric or nonsymmetric (as ac in (5) is symmetric while dc in (6) is

nonsymmetric, κc is symmetric only when dc ≡ 0). 
C-4 

Since Kc(x) is defined on the support scale ω, it is a local rather than an upscaled effective 

parameter and does not depend in any way on a grid of any kind. It constitutes a biased estimate 

of the actual ω-scale hydraulic conductivity, K(x), which is represented more faithfully by its 

unbiased conditional mean value, 〈K(x)〉c. Nevertheless, it is Kc(x) rather than 〈K(x)〉c that renders 

(more-or-less) unbiased predictions of head and flux, provided that the assumptions of local 

uniformity of ∇〈h〉c and rc are approximately satisfied. 

We see that the localisation of flow in a randomly nonuniform medium, under the influence 

of uncertain source and boundary terms, yields a deterministic system of differential equations, 

which is identical in form to traditional flow equations commonly written for fully deterministic 

systems. We can therefore comment about the widespread practice of applying such traditional 

equations to groundwater flow under ubiquitously nonuniform and uncertain field conditions. 

First, such applications are valid at best as an approximation, since it is clear that localisation of 

the mean flux introduces a modelling (localisation) error. Second, they are valid only provided 

one interprets the traditional equations (and associated analytical or numerical models of 

groundwater flow) in a non-traditional way. Whereas traditionally the hydraulic conductivity in 

deterministic flow equations has been considered to be a material property, we see that it is 

instead a nonunique effective parameter which depends on how closely the underlying property 

K (which enters into the stochastic, but not the deterministic, flow equations) has been defined 

on the basis of available measurements (i.e., on their quantity, quality and scale ω). Whereas 

traditionally the hydraulic head and flux in deterministic flow equations have been considered to 

represent real system states, we see that they are in fact conditional predictors of these states. As 

such, they are data-dependent and therefore nonunique, relatively smooth, and differ generally 

from their true random counterparts. Calibrating a deterministic groundwater flow model against 

measured values of system states is tantamount to conditioning the model on such measurements. 

It is therefore easy to understand why the resulting parameters tend to vary continuously as more 

and more data are incorporated into the (calibration, inverse or parameter estimation) process: 

the very act of expanding the database alters the (data-dependent, and thus inherently nonunique 

effective) parameters which the process is designed to estimate. While nonlocal moment 

equations provide information about predictive uncertainty, localisation does not. 
C-5 

CONDITIONING ON HYDRAULIC CONDUCTIVITY DATA 

Our nonlocal theory assumes that one has at his/her disposal a conditional unbiased estimate, 

〈K(x)〉c, of the randomly varying hydraulic conductivity function, K(x), together with the second 

conditional moment of associated estimation errors of log hydraulic conductivities, CYc(x, y). 

When conditioning is performed on the basis of existing ω-scale measurements of K at a set of 

discrete points, 〈K(x)〉c and CYc(x, y) can be obtained (in principle) by means of geostatistical 

methods (e.g., Deutsch & Journel, 1998). Geostatistical techniques can also be used to generate 

multiple conditional realisations of K(x), which honour the measured data. Solving the stochastic 

flow equations for each of these realisations by conditional Monte Carlo simulation (CMS) 

allows one to calculate ∇〈h(x)〉c and 〈q(x)〉c. Based on these, one can use the localised equation 

(9) to evaluate Kc(x). The latter can also be evaluated directly by considering K(x) to be 

multivariate log-normal and using stochastically derived formulae to approximate Kc(x). For 

example, if one treats K(x) in a two-dimensional domain as if it was log-normal as well as 

statistically homogeneous and isotropic in the neighbourhood of x, and the conditional mean 

flow as if it was locally uniform, one can use existing stochastic theory for flow in an infinite 

domain to deduce that Kc(x) ≈ KG(x)I where KG(x) is the (conditional) geometric mean of K at x. 

In addition to localisation error, this introduces another modelling error due to the analytical 

approximation of Kc. 

 We illustrate below the effect that these two modelling errors have on the flow predictors 

〈h(x)〉c and 〈q(x)〉c in a two-dimensional square domain, under superimposed mean uniform and 

convergent flows, subject to deterministic boundary conditions. For comparison purpose, we 

compute these conditional first moments by three methods: (a) CMS, which we consider to be 

the most reliable and accurate among the three methods; (b) recursive nonlocal finite elements, 

as proposed by Guadagnini & Neuman (1999a, b), which provide a direct solution to the 

stochastic flow problem to second order of approximation in σY; and (c) localisation, by setting 

Kc(x) ≈ KG(x)I. The flow domain measures L1 = L2 = 10 in arbitrary consistent length units. It is 

discretized into 50 x 50 rows and columns of M = 2500 square elements with sides 

∆x1 = ∆x2 = 0.2. Deterministic no-flow boundary conditions are imposed at the bottom along 

x2 = 0, and at the top along x2 = 10. A uniform deterministic head HL = 10 is prescribed on the 

left boundary along x1 = 0, and HR = 0 on the right boundary along x1 = 10. We consider three 

test problems: TP1 without conditioning on hydraulic conductivity data, and TP2 and TP3 that 

involve conditioning on two different sets of K measurements. In all three test problems, the 

underlying (unconditional) log hydraulic conductivity field Y(x) is multivariate Gaussian, 

statistically homogeneous and isotropic with an exponential spatial autocovariance. 
C-6 

-1.50-1.00-0.500.000.501.00

 0.00

0.100.200.300.400.500.60

10 10

(a) (b) 

x2 x2

5 5 

0 

0 5 x1 10 0

0 

5 x1 10

2.00

0.700.800.901.0

1.50

0

 

-

-1.5 0.0 2.0 

(d) 

0.0 0.5 1.0 

10 (c) 10

x2 x2

5 5 

0 

0 5 x1 10 0

0 

5 x1 10

1.50

-1.00-0.500.000.501.001.502.002.50

0.000.100.200.300.400.500.600.700.800.901.00

 

Fig. 1 Images (a) and (c) of 〈Y(x)〉c, (b) and (d) of (x), obtained with NMC=2,000 for TP2 in (a) and (b), TP3 in

(c) and (d). 

2 σY

-1.5 0.0 2.5 0.0 0.5 1.0 

Overall, the flow is uniform in the mean from left to right. We superimpose convergent flow on 

this mean uniform background by placing a well (point sink) at the centre (x1 = 5, x2 = 5) of the

grid, which pumps at a volumetric flow rate of Q = 5 (measured in arbitrary consistent time 

units). 
C-7 

Unconditional random Y(x) fields are generated by means of the simulator SGSIM (Deutsch 

& Journel, 1998), with unit log conductivity variance and autocorrelation scale λ. Figure 1 

shows images of the conditional sample mean, 〈Y(x)〉

2 σY

c, and variance, (x), obtained with NMC

= 2000 Monte Carlo simulations for TP2 and TP3. The “measured” hydraulic conductivities for 

these test problems are extracted from a single unconditional random realisation of Y(x). For 

purposes of flow analysis by CMS, we assign to each element in the grid a constant Y value,

corresponding to the point value generated at the element centre. This is justified considering

that our grid includes a minimum of five such cells per autocorrelation scale. The conditional 

K

2 σY

G(x) field, which plays the role of an effective parameter field for the localised mean flow 

equations, is obtained by averaging all conditionally generated Y realisation for each test 

problem. 

 

10 0.2 

0.0

〈h(x)〉c

〈h(2)

(x)〉c

5 TP 3 

TP 2 

 TP 1 

 TP 2 

 TP 3 TP 1 

Fig. 2 (a) Nonlocal (solid, same as CMS) and localised (dashed) solutions for 〈h(x)〉c along longitudinal section (x2 = 

5) through well; (b) second-order component, 〈h2

(x)〉c, of nonlocal 〈h(x)〉c solution along same section. 

Fig. 3 Nonlocal (solid, same as CMS) and localised (dashed) solutions for longitudinal mean flux along longitudinal 

section (x2 = 4.9). 

0 

0 5 x1 10 0

-1 

5 x1 10

(a) (b) 

10 

TP 1 TP 2 

0 

Lon

gitudinal mean flux 

5 

TP 3 

-5 

0 5 10 0 5 10 0 5 10 
C-8 

Figure 2(a) depicts the conditional mean head, 〈h(x)〉c, along a longitudinal section through 

the well for the three test problems. Values obtained by CMS and the nonlocal approach are 

virtually indistinguishable from each other, hence only nonlocal results are shown. Figure 2(b) 

shows the second-order (in σY) component, 〈h(2)(x)〉c, of the nonlocal 〈h(x)〉c solution along the 

same section for the three test problems. 

Figure 3 compares values of longitudinal mean flux, as obtained by the nonlocal and 

localised methods, along a longitudinal section passing through elements sharing the well node. 

Values obtained by the nonlocal and CMS approaches are virtually indistinguishable from each 

other, hence only nonlocal results are shown. 

Both the nonlocal and localised conditional mean head solutions agree very well with those 

obtained by CMS. The localised results are slightly less accurate, especially at the well in the 

unconditional case, due to errors introduced through localisation and the approximation of Kc(x) 

by KG(x)I. 

The incorporation of second-order terms in our nonlocal solution adds to its accuracy in all 

test problems. Figure 2(a) shows that the second-order head correction is negative at the well in 

the unconditional case (TP1), but remains positive in TP2 and TP3, in both of which the well 

acts as a conditioning point. It is worth noting that the “measured” K at the well is somehow 

larger than the unconditional mean conductivity 〈K(x)〉. Though both the nonlocal and localised 

solutions compare well with CMS results in all test problems, the nonlocal results do so 

uniformly better than the localised ones. 
C-9 

a

ano a

CONDITIONING ON HEAD DATA THROUGH MODEL CALIBRATION 

Kc(x) in our localised equations can, in principle, be estimated by means of standard inverse 

methods. This is analogous to the common groundwater modelling practice of taking the 

standard deterministic flow equations for granted, and estimating the associated hydraulic 

conductivities by model calibration against measured head and (possibly) fluxes data. Such 

calibration is tantamount to conditioning the localised equations not only on measured values of 

hydraulic conductivity, as we did previously, but also on those of head and (possibly) flux. A 

typical goal of inverse modelling in this context would be to obtain a suitably parameterised 

estimate, Kˆ

c (x), of Kc(x) so as to minimise, in some sense, the difference between computed and

actual head values, 〈 h

ˆ (xi)〉c nd h(xi), at a discrete set of measurement points, xi. If measurement

of flux, q(yj), are available at discrete points yj, an equally important goal should be to minimise

the difference between q(yj) and their computed counterparts, 〈 qˆ (xj)〉c = - Kc

ˆ (xj)∇〈 ˆ

h (xj)〉c. Yet

ther go l could be to keep the parameter estimate, Kc

ˆ (x), close in some sense to a prior 

estimate, * Kc (x), of Kc(x). In the examples considered earlier, a suitable prior might be KG(x)I. 

One way to accomplish this is to use the maximum likelihood method of Carrera & Neuman

(1986a,b). This, and other related methods of parameter estimation, have been recently reviewed

and compared by Zimmerman et al. (1998). By satisfying all three goals, one would end up with 

a localised model that is conditioned on a combination of hydraulic conductivity, head and flux 

data. Consider now the error, 

 εh(x) ≡ h(x) − 〈 h (x)〉 ˆ c (10)

introduced upon using such a calibrated model to predict the actual head, h(x), at any point x in 

the domain by means of 〈 h (x)〉 ˆ c. This error can be expressed as

 εh(x) = ε〈h〉(x) + ε〈 ~

h 〉(x) + ε〈 〉 $

h (x) (11) 

where 

ε〈h〉(x) ≡ h(x) − 〈h(x)〉c (12) 

ε〈 〉 ~

h (x) ≡ 〈h(x)〉c − 〈 h

~ (x)〉c (13)

ε〈 〉 $

h (x) ≡ 〈 h

~ (x)〉c − 〈 h (x)〉 ˆ c (14) 
C-10 

Here 〈h(x)〉c is the (generally) unknown true conditional mean head, 〈 h

~ (x)〉c is conditional mean

head corrupted by localisation and other modelling errors, 〈 h (x)〉 ˆ c is mean head corrupted by 

both localisation and parameter estimation errors, ε〈h〉(x) is the error introduced by using the 

conditional mean 〈h(x)〉c to predict the unknown random variable h(x), ε〈 ~

h 〉(x) is a modelling

error due primarily to localisation, and ε〈 〉 $

h (x) is an additional error due to parameter estimation. 

Note that in our previous examples, the latter error was zero, and the modelling error was 

relatively small (except at the well).

Statistical analyses of parameter estimation errors, such as ε $ ( ) $ ( ) K ≡ − K x c K x c where 

is some consistent norm, typically yield a lower bound for the error (c.f., Carrera & Neuman, 

1996a, b). The corresponding error estimate is then commonly used to assess uncertainties in 

model output, by varying its parameters (in our case Kc) randomly about their calibrated values 

(in our case ), and conducting Monte Carlo simulations of corresponding heads and fluxes. 

The approach yields at best information about errors such as ε

Kc

ˆ

〈 〉 $

h (x), which in our case stems 

from the parameter estimation error ε〈 〉 K$ (x). It provides no information about modelling errors 

such as ε〈 ~

h 〉(x), or prediction errors such as either ε〈h〉(x) or εh(x). As such, it underestimates the

prediction error εh(x), the only type of error that ultimately matters, and overestimates the 

predictive capabilities of the model. The reason is that Kc(x) relates conditional mean hydraulic 

gradient to conditional mean flux, and so ε〈 〉 K$ (x) affects only uncertainties associated with the 

evaluation of these conditional mean quantities. It contains no information about how actual 

heads and fluxes fluctuate about their conditional mean predictors; for that, one needs to evaluate 

higher conditional moments of head and flux, with the aid of stochastic methods that treat 

hydraulic parameters (in our case K or Y) as random fields. For a novel way of doing so, the 

reader is referred to Guadagnini & Neuman (1999a, b). Their work clearly demonstrates that the

stochastic component, ε〈h〉(x), of the total prediction error, εh(x), can be very large in strongly 

heterogeneous media. 

CONCLUSIONS 

 1. Applying traditional deterministic flow models to nonuniform media is valid at best as an 

approximation. Whereas traditionally the hydraulic conductivity in such models was considered 

to be a unique material property, in reality it is a nonunique, data-dependent, spatially-varying

effective hydraulic conductivity tensor. 

 2. Whereas traditionally the hydraulic head and flux in deterministic flow equations have 

been considered to represent real system states, they must in fact be viewed at best as conditional 

predictors of these states, which differ from their unknown counterparts by random errors of 

prediction. Since standard deterministic flow models provide no information about such 

prediction errors, they yield solutions of indeterminate quality. 

3. Calibrating a traditional deterministic flow model against measured values of head and flux 

is tantamount to conditioning it on such measurements. As the model hydraulic parameters are 
C-11 

by nature conditional, the very act of adding measured heads and fluxes to the database alters 

their values. This inherent nonuniqueness persists regardless of whether the inverse problem is 

well-posed or ill-posed. 

 4. Assessing the uncertainty of deterministic model parameters may allow one to quantify the 

uncertainty in computed head and flux predictors. This, however, says nothing about how actual 

heads and fluxes fluctuate about their predictors; such information is provided only by stochastic 

models that treat hydraulic parameters as random fields. The traditional Monte Carlo method of 

assessing uncertainty in the output of a calibrated deterministic model accounts neither for 

modelling errors nor for the stochastic component of the prediction error. As the latter error can 

be very large in strongly heterogeneous media, the traditional method may seriously 

overestimate the predictive capabilities of the calibrated groundwater flow model. The same 

holds for transport. 

REFERENCES 

Carrera, J. & Neuman, S. P. (1986a) Estimation of aquifer parameters under transient and steady state conditions, 1, Maximum 

likelihood method incorporating prior information. Wat. Resour. Res. 22 (2), 199-210. 

Carrera, J. & Neuman, S. P. (1986b) Estimation of aquifer parameters under transient and steady state conditions, 2, Uniqueness, 

stability, and solution algorithms. Wat. Resour. Res. 22 (2), 211-227. 

Deutsch, C. V. & Journel, A. G. (1998) GSLIB Geostatistical Software Library and User’s Guide. Oxford University Press, New York. 

Guadagnini, A. & Neuman, S. P. (1999a) Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly 

nonuniform domains, 1, theory and computational approach. Wat. Resour. Res. 35 (10), 2999-3018. 

Guadagnini, A. & Neuman, S. P. (1999b) Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly 

nonuniform domains, 2, computational examples. Wat. Resour. Res. 35 (10), 3019-3039. 

Neuman, S. P., & Orr S. (1993), Prediction of steady state flow in non uniform geologic media by conditional moments: Exact non local 

formalism, effective conductivities, and weak approximations. Wat. Resour. Res. 29 (2), 341-364. 

Neuman, S. P., Tartakovsky, D. M., Wallstrom, T. C. & Winter, C. L. (1996) Correction to “Prediction of steady state flow in 

nonuniform geologic media by conditional moments: Exact nonlocal formalism, effective conductivities, and weak 

approximation”. Wat. Resour. Res. 32(5), 1479-1480. 

Tartakovsky D. M. & Neuman, S. P. (1998) Transient flow in bounded randomly heterogeneous domains 1. Exact conditional moment 

equations and recursive approximations. Wat. Resour. Res. 34 (1), 1-12. 

Tartakovsky, D. M. & Neuman, S. P. (1999) Extension of "Transient flow in bounded randomly heterogeneous domains: 1. Exact 

conditional moment equations and recursive approximations,". Wat. Resour. Res. 35(6), 1921-1925. 

Zimmerman, D. M., de Marsily, G., Gotway, C. A., Marietta, M. G., Axness, C. L., Beauheim, R. L., Bras, R. L., Carrera, J., Dagan, G., 

Davies, P. B., Gallegos, D. P., Galli, A., Gómez-Hernández, J., Grindrod, P., Gutjahr, A. L., Kitanidis, P. K., Lavenue, A. M.,

McLaughlin, D., Neuman, S. P., RamaRao, B. S., Ravenne, C. & Rubin, Y. (1998) A comparison of seven geostatistically based 

inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow, Wat. Resour. Res. 34(6), 

1373-1413. 
C-12 
APPENDIX D 

MIXING, DILUTION, AND DISPERSION IN HYDROGEOLOGIC ENVIRONMENTS 
D-1 

APPENDIX D: MIXING, DILUTION, AND DISPERSION 

IN HYDROGEOLOGIC ENVIRONMENTS

The concepts of mixing, dilution and 

dispersion of dissolved contaminants are 

central to the assessment of groundwater 

pollution hazards. Yet there appears to be 

considerable confusion in the literature 

about the meaning of these concepts and 

their proper representation in contaminant 

transport models. Most notable is a lack of 

appreciation for the intimate link between 

dispersion and hydrogeologic uncertainty. 

The purpose of this section is to help shed 

light on these concepts and issues. A 

mathematical treatment of this topic can be 

found in Neuman (1993), Zhang and 

Neuman (1996) and Guadagnini and 

Neuman (2001). 

 

It is common to assume for purposes of 

environmental impact and performance 

assessments of nuclear facilities and sites 

that when radionuclides exiting the base of 

the unsaturated zone and enter into the 

saturated, they rapidly (in fact 

instantaneously) mix with a body of 

groundwater that is many meters, even tens 

of meters, deep. Additional dilution is 

sometimes assumed to occur by mixing of 

different groundwater sources, either 

naturally along the flow path between the 

source of radionuclides and a user, or by the 

user tapping alternative sources of water as 

in the slotting of a well over different units. 

Some believe dilution to occur by the 

mixing of groundwaters from adjacent 

sub-basins due to water withdrawal in the 

vicinity of their mutual boundary and lateral 

as well as vertical dispersion. The issue of 

mixing, dilution and dispersion in 

hydrogeologic environments is clearly of 

major significance to performance 

assessment. 

Yet, in reality, the only mechanisms by 

which dilution can actually take place in the 

absence of groundwater withdrawal for 

sampling or use are molecular diffusion, 

advective dispersion caused by space-time 

meandering of pathlines and variations in 

velocity along as well as among 

streamtubes, and turbulent eddies which are 

not likely to occur at most nuclear facilities 

and sites. Otherwise, dilution can occur 

when water is drawn into samplers or wells 

from multiple horizons and/or directions, 

and when waters in samplers or wells mix 

by diffusion and turbulence caused by 

shaking, stirring or rapid flow. Density 

effects due to space-time variations in 

solution chemistry, temperature and pressure 

may either enhance or prevent mixing (as in 

buoyant or gravity segregation of fluids 

having different densities). Another factor 

which may contribute to mixing is instability 

of fluid interfaces and resultant fingering. 

 

There appears to be no scientific basis for 

the "stirred tank" model according to which 

contaminated waters from the unsaturated 

zone mix rapidly with pristine waters in the 

saturated zone down to some specified 

"mixing depth." Quite the contrary, small 

vertical dispersivity commonly observed in 

stratified materials will usually keep the 

incoming plume of radionuclides at shallow 

depth except where it is intercepted by 

major vertical fractures or faults within 

which there is significant downward flow. 

One must likewise question the scientific 

basis for the notion that waters from 

neighboring sub-basins mix naturally along 

a flow path; only where flow paths from the 

two sub-basins converge will their waters 

mix by dispersion under natural conditions. 

 
D-2 

Mixing by dispersion may occur, but it is 

important to recognize that dispersion 

cannot be interpreted as dilution except in 

special cases to be described later. On the 

laboratory scale, dispersion coefficients 

compensate for our inability to resolve the 

intricacies of advective transport and 

diffusion in the pore space, about which we 

thus have no direct information. On the field 

scale, dispersion coefficients compensate 

additionally for our inability to resolve the 

intricacies of advective transport and localscale dispersion in a heterogeneous porous 

and/or fractured rock, about which we have 

only partial information. As space-time 

fluctuations in advective velocity are 

generally greater in the field than in the 

laboratory, the amount of information we 

generally lose through inadequate resolution 

of flow in the field tends to be greater than 

that which we lose through inadequate 

resolution of flow in a laboratory sample of 

rock. It follows that the coefficient of 

dispersion, which compensates for this loss 

of information, is generally 

larger on the field scale than on the 

laboratory scale. In fact, longitudinal 

dispersivities appear to increase consistently 

with the scale of field observations, a 

phenomenon we attribute today to a 

corresponding increases in the scale of 

medium heterogeneity, which our 

groundwater flow and transport models fail 

to resolve (Neuman, 1990, 1995; Di 

Federico and Neuman, 1997, 1998a,b; 

Neuman and Di Federico, 1998). Clearly, 

lack of resolution does not necessarily imply 

mixing and dilution; only if we sample on a 

scale comparable to that of unresolved 

heterogeneities should we expect dispersion 

to imply mixing and dilution. Hence 

confusing dispersion with mixing and 

dilution is generally inappropriate. 

 

Given that field-scale dispersion is 

associated with incomplete resolution of 

medium heterogeneities, one can reduce it 

by including in the groundwater flow and 

transport models a greater amount of detail 

about the spatial variability of medium 

properties. Since one never has exhaustive 

information about all small-scale details of 

medium heterogeneity, the only hope one 

has to resolve such details is by means of 

statistical and geostatistical methods. A 

geostatistical description of medium 

heterogeneity introduces uncertainty into 

groundwater flow and transport models 

which renders these models stochastic. 

Hence stochastic approaches provide an 

appropriate framework for the analysis of 

flow and transport in heterogeneous media 

(Dagan and Neuman, 1997). 

 

Briefly, the stochastic view of field-scale 

dispersion proposed here is as follows. 

Given a sufficiently large sample of localscale medium properties at a site, it is often 

possible to generate by conditional Monte 

Carlo simulation random but potentially 

realistic spatial distributions or images 

(called realizations) of these properties 

which are equally likely and honor the data. 

Here local scale means any scale that (a) is 

much smaller than the rock volume under 

investigation and (b) allows measuring 

many if not all requisite flow and transport 

parameters (permeability, porosity, 

dispersivity) and state variables (heads, 

concentrations) by means of standard field 

techniques; the corresponding length scale is 

typically on the order of meters to tens of 

meters. 
D-3 

Given adequate computer resources, it is 

then also possible to perform random but 

equally likely, and potentially realistic, highresolution flow and transport simulations on 

a computational grid with local-scale cells, 

using local-scale medium properties such as 

permeability, porosity and dispersivity. 

Since local-scale dispersivities are much 

smaller than their larger-scale counterparts, 

these simulations indicate much lesser (but 

more realistic) degrees of mixing and 

dilution than would simulations conducted 

on coarser grids with lesser spatial 

resolution. 

Hence for purposes of investigating mixing 

and dilution, we propose that analysts follow 

the above approach of high-resolution, 

statistically-based conditional flow and 

transport simulations. Even a small number 

of corresponding realizations may be more 

telling with regard to dilution and mixing 

than would deterministic or stochastic 

simulations which do not achieve a 

comparable degree of resolution. 

 

In the same context, it is important to 

recognize that local-scale dispersivities 

increase with their scale of definition; a 

crude rule of thumb (which does not 

consider numerical dispersion) is to set the 

longitudinal dispersivity equal to one tenth 

the length of a local-scale grid cell, and the 

transverse dispersivity a fraction thereof 

(one tenth to one third). 

 

If a statistically significant number of 

conditional Monte Carlo flow and transport 

simulations are performed, their results can 

be averaged to yield conditional mean 

values of head, groundwater flux, 

groundwater velocity, solute concentration 

and solute mass flux at each grid point at 

many discrete time steps. These conditional 

mean values are deterministic (sure, certain) 

and vary much more smoothly in space-time 

than do their random (uncertain) 

counterparts (as represented by individual 

realizations). Most importantly, they 

constitute optimum unbiased predictors of 

the actual but unknown local-scale values of 

head, flux, velocity, concentration and mass 

flux at each point in space-time. The 

variance of the conditional Monte Carlo 

realizations provides a measure of the 

uncertainty associated with these 

predictions, which can in turn be translated 

into optimum predictions and uncertainties 

of performance measures. Even a small 

sample of realizations may yield 

meaningful, though probably not accurate, 

insight into the uncertainty associated with 

simulating flow and transport at a site, and 

assessing the corresponding system 

performance. 

 

There is only one proper way to avoid 

running many high-resolution conditional 

Monte Carlo simulations of flow and 

transport: solve numerically, on a coarser 

computational grid than required for such 

simulations, a single set of deterministic 

flow and transport equations which control 

the space-time evolution of smooth 

conditional mean heads, fluxes, velocities, 

concentrations and mass fluxes (it is the 

relative smoothness of these deterministic 

functions which allows, in principle, 

computing them deterministically on a 

relatively coarse grid). Unfortunately, as 

was stated earlier, the corresponding 

conditional mean flow and transport 

equations are not differential but integrodifferential. Fortunately, they can sometimes 

be approximated by differential equations 

which look like the familiar flow and 

transport equations (the same equations one 

would use, among others, for highresolution conditional Monte Carlo 

simulations) but their coefficients and 
D-4 

variables now have different meanings and 

values. In particular, the conditional mean 

advection-dispersion equation now contains 

a conditional mean velocity vector which is 

much smoother, and a dispersion tensor 

which grows with time to become rapidly 

much larger, than their local counterparts. 

The large dispersion compensates for 

resolution lost in smoothing the velocity and 

concentration fields (recall that solving the 

conditional mean equation is analogous to 

averaging the results of many conditional 

Monte Carlo simulations, each one of which 

is associated with nonsmooth velocity and 

concentration fields, which however become 

smooth upon averaging). The more 

information about local-scale medium 

properties one builds into the model (i.e., the 

more strongly one conditions the model on 

local-scale data), the less smooth is the 

corresponding conditional mean velocity (as 

well as concentration) field and the smaller 

is the corresponding dispersion tensor. 

Clearly, building information into a model 

does not affect mixing and dilution, hence it 

is evident that the dispersion tensor does not 

generally reflect these phenomena. 

 

Since there is always uncertainty about 

local-scale medium heterogeneities, a 

deterministic analysis of transport is never 

warranted unless the advection-dispersion 

model is viewed and interpreted in the 

manner just described. This means that the 

computed concentration and mass flux are 

recognized to represent not actual but 

smooth predicted values with which there is 

associated a quantifiable error of prediction 

and smoothing, and that the computed 

concentrations are recognized to potentially 

spread, or disperse, to a much greater extent 

than do their real but unknown counterparts 

(one can view this enhanced spread, or 

dispersion, not as that of real solute mass but 

that of information about the space-time 

distribution of this mass). 

 

Only in special, so-called quasi-ergodic 

situations can such enhanced dispersion be 

interpreted to imply mixing and dilution. 

These situations arise as the mean travel 

distance becomes large enough for a plume 

to encounter (sample) heterogeneities of all 

relevant scales in the longitudinal direction 

of flow, and when the source of 

contamination is wide enough so that the 

plume can sample all such heterogeneities in 

the transverse direction. An extreme 

example of nonergodic transport is that of an 

imaginary solute "particle" of infinitesimal 

volume which never diffuses, disperses or 

dilutes. If the particle has unit mass 

normalized by porosity then its 

concentration, predicted deterministically 

with a relatively large "field-scale" 

dispersion coefficient, merely represents the 

probability of finding this particle in the 

immediate vicinity of any given point in 

space-time. 

 

Despite significant advances in stochastic 

flow and transport theories over the last two 

decades as reflected in the recent book of 

Dagan and Neuman (1997), stochastically 

derived field-scale dispersivities are known 

to overpredict the vertical spread of solutes 

in stratified media under seemingly quasiergodic conditions. 

 

Model abstraction through reduction of 

dimensionality (as when three-dimensional 

transport is represented by a network of onedimensional flow tubes) is associated with a 

loss of spatial resolution which should be 

compensated for by a corresponding 

increase in dispersivities (Domenico and 

Robbins, 1984). Once again, such an 

increase must not be interpreted to imply 

enhanced dilution. 
APPENDIX E 

MAXIMUM LIKELIHOOD BAYESIAN MODEL AVERAGING 
E-1 

APPENDIX E: MAXIMUM LIKELIHOOD BAYESIAN MODEL AVERAGING

Accounting for conceptual model uncertainty 

Via maximum likelihood Bayesian model averaging 

S. P. NEUMAN 

Department of Hydrology and Water Resources, University of Arizona, Tucson, Arizona 85721, USA

e-mail: neuman@hwr.arizona.edu

Abstract Analyses of groundwater flow and transport typically rely on a single 

conceptual model of site hydrogeology. Yet hydrogeologic environments are open and 

complex, rendering them prone to multiple interpretations. Adopting only one of these 

may lead to statistical bias and underestimation of uncertainty. A comprehensive strategy 

for constructing alternative conceptual-mathematical models, selecting the best among 

them, and using them jointly to render optimum predictions under uncertainty is being 

developed by the author. This paper proposes a Maximum Likelihood Bayesian Model 

Averaging approach, MLBMA, to rendering optimum predictions by means of several 

competing models and assessing their joint predictive uncertainty. 

Key words conceptual models; model uncertainty; predictive uncertainty; Bayesian; maximum likelihood 

INTRODUCTION 

Analyses of groundwater flow and transport typically rely on a single conceptual model of site 

hydrogeology. Yet hydrogeologic environments are open and complex, rendering them prone to 

multiple interpretations. This is true regardless of the quantity and quality of available site data. 

Focusing on only one site interpretation may lead to Type I model errors, which arise when one 

rejects (by omission) valid alternative models. It may also result in a Type II model error, which 

arises when one adopts (fails to reject) an invalid conceptual framework. Indeed, critiques of 

hydrogeologic analyses, and legal challenges to such analyses, typically focus on the validity of 

the underlying conceptual model. If severe, these may damage one's professional credibility; 

result in the loss of a legal contest; and lead to adverse environmental, economic and political 

impacts. 

Analyses of model uncertainty based on a single hydrogeologic concept are prone to 

statistical bias (by committing a Type II error through reliance on an invalid model) and 

underestimation of uncertainty (by committing a Type I error through under sampling of the 

relevant model space). The bias and uncertainty that result from reliance on an inadequate 

conceptual model are typically much larger than those introduced through an inadequate choice 

of model parameter values. Yet most uncertainty analyses of flow and transport ignore the 

former and focus exclusively on the latter. This often leads to overconfidence in the predictive 

capabilities of the model, which the available site data do not justify. 
E-2 

It is argued by Beven and Freer (2001) “that, given current levels of understanding and 

measurement technologies, it may be endemic to mechanistic modeling of complex 

environmental systems that there are many different model structures and many different 

parameter sets within a chosen model structure that may be behavioural or acceptable in 

reproducing the observed behaviour of that system.” They attribute to Hornberger and Speer 

(1981) the notion that this is not simply a problem of identifying a correct or optimal model 

given limited data. Instead, this is a generic problem which Beven (1993) calls equifinality and 

attributes to (Beven, 2000) limitations of current model structures in representing heterogeneous 

surface and subsurface flow systems, limitations of measurement techniques and scales in 

defining system characteristics including initial and boundary conditions for a model, and the 

uniqueness of individual sites. He points out that to do detailed measurements throughout a site 

is both impractical and unfeasibly expensive. The unique characteristics of a site are therefore 

inherently unknowable. All that can be done is to constrain the model representations of the site 

to those that are acceptably realistic, usually in the sense of being consistent with the data. 

There is no established literature on ways to construct alternative conceptual models of site 

hydrogeology, select the best among them, use them jointly to render optimum predictions of 

groundwater flow and transport, and assess the uncertainty of such predictions. A comprehensive 

strategy for doing so is being developed by the author. This paper focuses on one aspect of the 

strategy, which addresses the question how to render optimum predictions by means of several 

competing deterministic or stochastic models and how to assess their joint predictive uncertainty. 

PREDICTION AND UNCERTAINTY ANALYSIS BASED ON SINGLE MODEL 

Deterministic model 

The traditional approach to hydrologic prediction and uncertainty analysis has been to postulate a 

deterministic model structure and treat its parameters as being imperfectly known. To quantify 

this imperfect knowledge, one must postulate a prior parameter uncertainty model. When 

sufficient site parameter values are available, one could postulate a Type A probabilistic model 

of prior parameter uncertainty based on statistics derived from these data. When no such data are 

available in statistically significant quantities, one has the option of postulating a Type B model 

of prior parameter uncertainty on the basis of subjective probabilities. Such a model should 

always be suspected of suffering from an unknown amount of statistical and personal bias. 

Intermediate between Type A and Type B parameter uncertainty models is the case where 

indirect information about the parameters is available, from which relevant prior statistics can be 

derived formally. Such information may include off-site parameter measurements and/or 

surrogate data. Statistics derived from off-site data are potentially biased due to a lack of sitespecific information about mean parameter values and incompatibility of geology and scale. The 

associated variance may be too small or too large, depending on the quantity and quality of such 

data. Statistics derived from surrogate data may suffer from poorly defined correlations and 

incompatibility of scale. 
E-3 

 The traditional approach to reduce parameter bias and uncertainty has been to calibrate the 

model against observed system behavior by means of a suitable inverse method. The last thirty 

years have seen major advances in the development of theories and algorithms for the estimation 

of deterministic model parameters. Many (though not all) of these theories and algorithms are 

"statistical" in that they include analyses of parameter estimation uncertainty. Such analyses 

typically accept, but do not necessarily require, information about prior parameter statistics as 

input. The output includes posterior statistics of parameter estimation errors, which are generally 

less biased and smaller than the prior estimation errors. A recent summary and comparison of 

various statistical inverse methods for groundwater flow models has been published by 

Zimmerman et al. (1998). A detailed set of guidelines for the effective calibration of 

deterministic groundwater flow models has been prepared by Hill (1998). 

 The most common way to propagate input errors through an otherwise deterministic model is 

by means of Monte Carlo simulations. This is done by generating multiple, equally likely sets of 

randomized inputs; computing deterministically a set of corresponding model outputs for each; 

and analyzing the resultant multiple, equally likely random output sets statistically. Another 

approach is to associate the predictions with approximate error bounds, or confidence limits, 

computed on the basis of linear regression theory applied to the (typically nonlinear) 

groundwater inverse model (Hill, 1998). 

Stochastic model 

Hydrogeologic medium properties exhibit both systematic and random spatial variations on a 

multiplicity of scales. Traditional deterministic models capture at best the larger-scale, 

systematic components of these variations. They however fail to resolve smaller scale variations 

or account for their uncertain nature. The emphasis is therefore shifting from deterministic to 

probabilistic methods that are better suited for these needs. The trend has become to describe the 

spatial variability and scaling of hydrogeologic medium properties geostatistically, and to 

analyze subsurface fluid flow and solute transport stochastically. This trend has been 

documented in a number of recent books including those by Dagan and Neuman (1997) and 

Zhang (2001). 

The most common method of stochastic analysis is high-resolution computational Monte 

Carlo simulation that produces a large number of equally likely results. These nonunique results 

are summarized in terms of statistically averaged quantities, their variance-covariance, and 

perhaps higher moments of the corresponding sample probability distributions. Results that 

honor measured values of medium properties are said to be conditioned on these data. Upon 

conditioning the simulations on measured values of parameters in space, one obtains (among 

others) conditional mean flow and transport variables that constitute optimum unbiased 

predictors of these unknown random quantities. One also obtains conditional second moments 

(variance-covariance) that provide a measure of the associated prediction errors. To condition the 

predictions on system monitoring data, one must either discard random simulations that do not 

reproduce the observations, or employ an inverse procedure of the kind developed for this 

purpose by G\mez-Hern<ndez et al. (1997). 
E-4 

Monte Carlo analysis requires knowing the multivariate probability distribution of relevant 

hydrogeologic properties, which is difficult to infer from commonly available data. To achieve a 

high space-time resolution of relevant stochastic phenomena, it requires the use of large spacetime grids with very small discretization intervals. To yield sample statistics that converge to 

their theoretical (ensemble) counterparts requires numerous repetitions (realizations). The net 

result is a large amount of computational time and storage, which are considered uneconomical 

for many practical applications. 

This has given impetus to the development of alternative stochastic methods that allow one to 

compute the conditional mean, variance and covariance of groundwater flow and transport 

variables directly, without Monte Carlo simulation. This is done on the basis of moment 

equations conditional on hydraulic parameter measurements as illustrated for example by 

Guadagnini and Neuman (1999) and Ye et al. (2002). Conditioning additionally on observed 

hydraulic head values requires an inverse procedure (Hernandez et al., 2002). 

PREDICTION AND UNCERTAINTY ANALYSIS BASED ON MULTIPLE MODELS 

Previous hydrologic approaches 

Carrera and Neuman (1986b) have noted that an inadequate model structure (conceptualization) 

is far more detrimental to its predictive ability than is a suboptimal set of model parameters. This 

helps explain why the National Research Council (1999) has listed as second among four 

recommended research directions in subsurface science the development of tools and 

methodologies for conceptual modeling with emphasis on heterogeneity, scale and uncertainty 

bounds on the basis of field experimental data. 

 Recently, a panel was convened by the National Research Council (2001) to describe the 

process through which conceptual models of flow and transport in the fractured vadose zone are 

developed, tested, refined and reviewed. The panel concluded that development of the 

conceptual model is the most important part of the modeling process. The conceptual model is 

the foundation of the quantitative mathematical representation of the field site (i.e., the 

mathematical model), which in turn is the basis for the computer code used for simulation. 

Reasonable alternative conceptualizations and hypotheses should be developed and evaluated. In 

some cases, the early part of a study might involve multiple conceptual models until alternatives 

are eliminated by field results. 

 According to the panel, it is important to recognize that model predictions require 

assumptions about future events or scenarios, and are subject to uncertainty. Meaningful 

quantification of uncertainty should be considered an integral part of any modeling endeavor, as 

it establishes confidence bands on predictions given the current state of knowledge about the 

system. A suite of predictions for a range of different assumptions and future scenarios is more 

useful than a single prediction. 
E-5 

 We have noted earlier that there is uncertainty not only about the parameter values that 

should enter into a given model (as characterized by its structure), but also about the very 

structure (conceptual and mathematical) of the model that should represent the hydrologic 

system of interest. The traditional approach to model uncertainty analysis, which considers only 

a single deterministic model structure, fails to adequately sample the complete space of plausible 

hydrologic models. As such, it is prone to modeling bias and underestimation of model 

uncertainty. 

 An example of how one could account quantitatively for structural model uncertainties was 

given by James and Oldenburg (1997). They investigated the uncertainty of simulated TCE 

concentrations, at the point of potential human exposure, due to uncertainty in the parameters 

(permeability, porosity, diffusivity, solubility, adsorption) and variations in the conceptualmathematical model (injection rate of TCE source; initial TCE source saturation; regional 

groundwater flow; heterogeneity of permeability). The authors used the three-dimensional code 

T2VOC to simulate three-phase (gas, aqueous, NAPL), three-component (air, water, VOC) 

nonisothermal flow based on an actual site with a 25 m thick vadose zone and a saturated zone. 

To assess parameter uncertainty associated with a given model, they used the inverse code 

ITOUGH2. Their final step was to assess the range of outcomes that one obtains with the entire 

set of alternative conceptual-mathematical models. James and Oldenburg found that uncertainties 

in their model outcomes span orders of magnitude, and that both parameter and model 

uncertainty contribute significantly to this wide range of outcomes. They concluded that "risk 

assessment and remediation selection ... is meaningful only if analysis includes quantitative 

estimates of ... uncertainty" in both the parameters and the conceptual-mathematical models. 

 A similar approach has been advocated more recently by Samper and Molinero (2000). The 

authors consider the main uncertainties in predicting groundwater flow and transport to be those 

associated with the selection of future scenarios, choice of model structure and assignment of 

model parameters. The authors consider parameter uncertainty to be minor in comparison to 

structural (i.e. conceptual) model errors. They suggest to evaluate model predictive uncertainty 

by calibrating a number of conceptual-mathematical models against available monitoring data, to 

retain those calibrated models that can adequately reproduce past observations, to assess the 

predictive uncertainty of each model due to the uncertainty of its parameters, to treat the 

predictive uncertainty of each model as being equally likely, and to produce a single combined 

range of predictive uncertainties. 

 Rather than relying on model calibration and treating the outcomes of different structural 

models as being equally likely, Beven and Binley (1992) have proposed a strategy to which they 

refer as GLUE (Generalized Likelihood Uncertainty Estimation). The strategy calls for the 

identification of several alternative structural models and the postulation of a prior probabilistic 

model of parameter uncertainty for each. Each structural model, coupled with its corresponding 

parameter uncertainty model, is used to generate Monte Carlo realizations of past hydrologic 

behaviors and to compare the results with monitored system behavior during the same period. 

Likelihood measures are defined to gauge the degree of correspondence between each simulated 

and observed record of system behavior. If a likelihood measure falls below a subjectively 

defined "rejection criterion," the corresponding combination of model structure and parameter 
E-6 

set are discarded. Those combinations which pass this test are retained to provide predictions of 

system behavior under selected future scenarios. Each prediction is weighted by a corresponding 

normalized likelihood measure (so as to render the sum of all likelihood measures equal to one), 

to produce a likelihood-weighted cumulative distribution of all available predictions. For recent 

discussions of GLUE and its applications the reader is referred to Beven (2000) and Beven and 

Freer (2001). 

A Bayesian approach to the quantification of errors in a single groundwater model was 

recently proposed by Gaganis and Smith (2001). Like GLUE, it relies on Monte Carlo 

simulations without model calibration and on subjective criteria of “model correctness.” 

It must be understood that the set of predictions one produces with any given choice of 

alternative structural models and parameter sets, by whatever method, is conditional on the 

choice of models and the data used to support them. As such, these predictions do not represent

all possibilities but only a limited range of such possibilities, associated with these models and

data. Any change in the latter would generally lead to a different assessment of predictive model 

uncertainty. There thus appears to be no way to assess the uncertainty of hydrologic predictions 

in an absolute sense, only in a conditional or relative sense. 

Proposed approach 

Bayesian model averaging 

At the heart of the approach we propose is the concept of Bayesian Model Averaging (BMA), 

described with clarity in a recent tutorial by Hoeting et al. (1999). According to these authors,

“standard statistical practice ignores model uncertainty ... leading to over-confident inferences 

and decisions that are more risky than one thinks they are. ... (BMA) provides a coherent 

mechanism for accounting for this model uncertainty.” They introduce BMA by noting that if ∆

is a quantity one wants to predict, then its posterior distribution given a discrete set of data D is 

( ) ( ) (

1

,

K

k k

k

p p M p M =∆ = D ∑ ∆ D D)

)

(1)

where M = ( 1,..., M MK is the set of all models (or hypotheses) considered, at least one of

which must be valid. In other words, p(∆ D) is the average of the posterior distributions

( , k p M∆ D) under each model, weighted by their posterior model probabilities ( k p M D) . The 

posterior probability for model Mk is given by Bayes’ rule, 

( ) ( ) ( )

( ) ( )

1

k k

k K

l l

l

p M p M

p M

p M p M =

=

∑

D

D

D

(2)
E-7 

where 

( ) k k ( ) , k ( k k ) k p M p = M p M d D D ∫ θ θ θ (3)

is the integrated likelihood of model Mk , θ k is the vector of parameters associated with model 

Mk , p(θk Mk ) is the prior density of θ k under model Mk , p( D θk , Mk ) is the joint likelihood

of model Mk and its parameters θ k , and p (Mk ) is the prior probability that Mk is the correct

model. All probabilities are implicitly conditional on M. 

The posterior mean and variance of ∆ are (Draper, 1995) 

(

1

,

K

l l

l

E E M p M =⎡∆ ⎤ = ⎡∆ ⎤ ⎣ ⎦ ∑ ⎣ ⎦ D D D) (4)

{ } ( ) 2

1

, ,

K

l l l

l

Var Var M E M p M E =⎡∆ ⎤ = ⎡∆ ⎤ + ⎡∆ ⎤ − ⎡∆ ⎤ ⎣ ⎦ ∑ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

2

D D D D D . (5) 

According to Hoeting et al. (1999), there is considerable empirical evidence that averaging 

over all models in this fashion provides better average predictive ability than relying on a single 

model, Mk , conditional on M. However, they list a number of factors that render the application 

of BMA to complex systems (such as those encountered in hydrology) difficult: (a) The number

of potentially feasible models may be exceedingly large, rendering their exhaustive inclusion in

(1) infeasible; (b) integrals of form (3) may be hard to compute; and (c) the specification of prior 

model probabilities ( ) k p M is challenging, having received little attention. 

A practical way to eliminate the first difficulty is to average over a manageable subset of 

models that are supported by the data. The strategy being developed by the author promotes the 

idea of Occam’s window (Madigan and Raftery, 1994) according to which averaging is limited 

to a relatively small set of the most parsimonious models that are most strongly supported by the 

data while remaining hydrologically plausible. 

Maximum likelihood Bayesian model averaging 

To render BMA computationally feasible, we adopt a suggestion by Taplin (1993) that 

( , k p M∆ D) in (1) be approximated by ( ) ˆ , , k k p M∆ θ D where ˆθ k is the maximum likelihood

estimate of θ k based on the likelihood p( D θk , Mk ). Hoeting et al. (1999) note that Draper 

(1995), Raftery et al. (1996) and Volinsky et al. (1997) have shown this to be useful in the BMA 

context. 
E-8 

Methods to evaluate ˆθ k by calibrating a deterministic model Mk against hydrogeologic data 

D, which may include prior information about the parameters, are described by Carrera and 

Neuman (1986a,b) and Carrera et al. (1997). The same can be done with a stochastic model

based on moment equations in a manner similar to that of Hernandez et al. (2002). These 

methods also yield an approximate covariance matrix for the estimation errors of ˆθ k . Upon 

considering the parameter estimation errors of a calibrated deterministic model Mk to be 

Gaussian or log Gaussian, one easily determines ( ) ˆ , , k k p M∆ θ D by Monte Carlo simulation of 

 through random perturbation of the parameters. The simulation also yields corresponding 

approximations 

∆

ˆ , , E Mk k

⎡ ⎤ ∆

⎣ ⎦ θ D of , E Mk ⎡∆ ⎤ ⎣ D⎦ , and ˆ , , Var Mk k

⎡∆ ⎤ ⎣ ⎦ θ D of , Var Mk ⎡∆ ⎤ ⎣ ⎦ D , in 

(4) and (5). If Mk is a stochastic model based on moment equations, it can yield 

ˆ , , E Mk k

⎡ ⎤ ∆

⎣ ⎦ θ D and ˆ , , Var Mk k

⎡∆

⎣ ⎤⎦ θ D directly without Monte Carlo simulation (Hernandez et

al., 2002). 

To eliminate the need for computer intensive integration according to (3), I propose to 

evaluate the weights p M( l D) in (1) and (4) – (5) based on a result due to Kashyap (1982). The 

author considers a set 1,..., M MK of mutually exclusive models so that any set of observational 

data could have originated from only one of them. I interpret this to mean that only one of the 

models is correct even in the event that some yield similar predictions for a given set of data (in 

which event the degeneracy could be resolved by prior information about which model “makes 

most sense,” parsimony, and/or additional data for which the predictions would differ). The

models may be linear or nonlinear, Gaussian or non-Gaussian. Kashyap proves that, under some

fairly standard conditions, 

( ) ( ) ( ) 2 ˆ ˆ ln ln ln , ln ln

2

k

k k k k k k

N p M C p M p M

N

⎛ ⎞ π

= + + + ⎜ ⎟ ⎝ ⎠ D D θ θ

( ) ( ) 1 ˆ ln , 2 − F D k k θ Mk + R N

)

(6)

where ( ), C c k k = p M ( k p M being the prior probability of model Mk and c a constant that can 

be evaluated from

( )

1

1

K

l

l

p M =

∑ D = , (7)
E-9 

Nk is the dimension of θ k , N is the dimension of D, Fk is the normalized (by N) observed (as

opposed to ensemble mean) Fisher information matrix having components 

( ) 2

,

1 ln , k k

k ij

i j

p M

F

N θ θ

∂ = − ∂ ∂D θ

(8)

and NR(N) tends to a constant almost surely as N → ∞ . 

Kashayp (1982) suggests that, in the absence of any contrary information, the models be 

assigned equal prior probabilities, yielding Ck = C = constant for all k. The assumption that all 

models are a priori equally likely is considered by Hoeting et al. (1999) to be a “reasonable 

‘neutral’ choice” when there is insufficient prior reason to prefer one model over another. Draper 

(1999) and George (1999) express concern that if two models are near equivalent as concerns 

predictions, treating them as separate equally likely models amounts to giving double weight to a 

single model of which there are two slightly different versions, thereby “diluting” the predictive

power of BMA. One way to minimize this effect is to eliminate at the outset models that are

deemed potentially inferior. Another is to retain only models that are structurally distinct and 

non-collinear. Otherwise, one should consider reducing (diluting) the prior probabilities assigned 

to models that are deemed closely related. 

Kashyap’s (1982) purpose in developing (6) was to derive an optimum decision rule for

selecting one among several competing models, unrelated to BMA. Since the first term on the 

right hand side is constant and the last is asymptotically zero, Kashyap proposed to select that 

model which minimizes the criterion

( ) ( ) ( ) 2 1 ˆ ˆ ˆ ˆ ln , ln ln ln , 2 2

k

k k k k k k k

N d p M p M M

N

⎛ ⎞ π = − − − ⎜ ⎟ + ⎝ ⎠ D F θ θ D θ k (9) 

Increasing the number of parameters Nk allows ( ) ˆ ln , k k − p D θ M to decrease and to 

increase. When N

ln Nk N

k is large, the rate of decrease does not compensate for the rate of increase and 

 grows while ˆ

k d ( k p M D) in (6) diminishes. This means that a more parsimonious model with 

fewer parameters is preferred by (9) and assigned a higher probability by (6). On the other hand 

( ˆ ln , − p D θ k Mk ) diminishes with N at a rate higher than linear so that as the latter grows, there

may be an advantage to a more complex model with larger Nk. 
E-10 

The last term in (9) reduces the relative emphasis on model fit as the information content of 

the data diminishes. As illustrated by Carrera and Neuman (1986b), it may cause one to prefer a 

simpler model that leads to a poorer fit with the data over a more complex model that fits the 

data better. The term tends to a constant as N becomes large, so that becomes asymptotically 

equivalent to the Bayes information criterion BIC derived by Akaike (1977), Rissanen (1978) 

and Schwarz (1978) on the basis of different considerations. Raftery (1993) proposed adopting 

the asymptotic BIC approximation 

ˆ

k d

( ) ( ) ˆ ln ln , ln

2

k

k k k

N

p D D M p ≈ θ M − N (10)

for BMA (see also Raftery et al. 1996; Volinsky et al. 1997; Hoeting et al. 1999). 

To my knowledge, the nonasymptotic expression (6) has not been previously incorporated 

into BMA. I propose to do so because environmental models seldom satisfy the assumption that 

N is large. To render the use of (6) in BMA computationally feasible, I propose to follow the

approach of Carrera and Neuman (1986a) who incorporate in D both observational data such as 

head and prior estimates of the parameters, and treat the two sets as being mutually uncorrelated

while allowing internal correlations between members of each set. This allows them to 

incorporate ( ˆ ln p θ k Mk ) into the log likelihood function ( ) ˆ ln , k k p D θ M and to compute 

( ˆ , F D k k θ Mk ) in a straightforward manner (also Carrera et al. 1997). 

CONCLUSION

Bayesian model averaging provides an optimal way to combine the predictions of several 

competing conceptual-mathematical models and to assess their joint predictive uncertainty. It can

be made computationally feasible by basing it on a maximum likelihood approximation due to 

Kashyap (1982) and the parameter estimation method of Carrera and Neuman (1986a). 

Acknowledgment This work was supported by the U.S. Nuclear Regulatory Commission under 

contract NRC-04-97-056. Special credit is due to Thomas J. Nicholson and Ralph Cady of the

USNRC for having conceived and championed this research. I am most grateful to Weihsueh 

Chiu of the U.S. Environmental Protection Agency for having brought to my attention the 

extensive work on BMA conducted by the “Seattle school” of statistics at the University of 

Washington, and for his helpful comments on this paper. 
E-11 

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domains, 2, Computational examples. Wat. Resour. Res. 35(10), 3019-3039. 

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BIBLIOGRAPHIC DATA SHEET

(See instructions on the reverse)

NRC FORM 335

(2-89)

NRCM 1102,

3201, 3202

U.S. NUCLEAR REGULATORY COMMISSION 1. REPORT NUMBER

 (Assigned by NRC, Add Vol., Supp., Rev.,

 and Addendum Numbers, if any.)

NUREG/CR-6805

DATE REPORT PUBLISHED

MONTH

July

YEAR

2003

4. FIN OR GRANT NUMBER

N/A

2. TITLE AND SUBTITLE

A Comprehensive Strategy of Hydrogeologic Modeling and Uncertainty Analysis 

for Nuclear Facilities and Sites

5. AUTHOR(S)

S.P. Neuman and P.J. Wierenga

6. TYPE OF REPORT

Technical

7. PERIOD COVERED (Inclusive Dates)

September 1997 - December 2002

8. PERFORMING ORGANIZATION - NAME AND ADDRESS (If NRC, provide Division, Office or Region, U.S. Nuclear Regulatory Commission, and mailing address; if contractor,

University of Arizona

9. SPONSORING ORGANIZATION - NAME AND ADDRESS (If NRC, type "Same as above"; if contractor, provide NRC Division, Office or Region, U.S. Nuclear Regulatory Commission, 

Division of Systems Analysis and Regulatory Effectiveness

provide name and mailing address.)

and mailing address.)

10. SUPPLEMENTARY NOTES

T.J. Nicholson, NRC Project Manager

11. ABSTRACT (200 words or less)

This report describes a strategy that embodies a systematic and comprehensive a pproach to hydrogeologic conceptualization,

model development and predictive uncertainty analysis. The strategy is comprehe nsive in that it considers all stages of model

building and accounts jointly for uncertainties that arise at each of them. The stages include regional and site characterization,

hydrogeologic conceptualization, development of conceptual-mathematical model structure, parameter estimation on the basis 

of monitored system behavior, and assessment of predictive uncertainty. In addi tion to parameter uncertainty, the strategy

concerns itself with uncertainties arising from incomplete definitions of (a) t he conceptual framework that determines model

structure, (b) spatial and temporal variations in hydrologic variables that are either not fully captured by the available data or 

not fully resolved by the model, and (c) the scaling behavior of hydrogeologic variables. The strategy is generic but designed 

to be of practical use to NRC licensing staff in their review of decommissionin g plans, and performance assessment of

high-level and low-level radioactive waste disposal sites as well as uranium re covery facilities. An important component of the

strategy is a systematic sequence of logical questions, guidelines and criteria with analytical methods appropriate for NRC

review and performance evaluation.

12. KEY WORDS/DESCRIPTORS (List words or phrases that will assist researchers in locating the report.)

conceptual model uncertainty

ground-water analysis

hydrogeologic modeling

model development

performance assessment

predictive uncertainty analysis

sensitivity

uncertainty analysis

14. SECURITY CLASSIFICATION

13. AVAILABILITY STATEMENT

unlimited

(This Page)

unclassified

(This Report)

unclassified

15. NUMBER OF PAGES

16. PRICE

NRC FORM 335 (2-89) This form was electronically produced by Elite Federal Forms, Inc.

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