Development of Property-Transfer Models for 

Estimating the Hydraulic Properties of Deep 

Sediments at the Idaho National Engineering and 

Environmental Laboratory, Idaho

Scientific Investigations Report 2005–5114

Prepared in cooperation with the U.S. Department of Energy

U.S. Department of the Interior

U.S. Geological Survey

DOE/ID-22196
Cover: Graph showing example of water-retention (q(y)) curve showing components 

of the curve-fit model developed by Rossi and Nimmo (1994). Photographs of core 

specimens used to develop water content and matric pressure relations. Photographs 

courtesy of Kari A. Winfield, U.S. Geological Survey.
Development of Property-Transfer Models 

for Estimating the Hydraulic Properties 

of Deep Sediments at the Idaho National 

Engineering and Environmental 

Laboratory, Idaho

By Kari A. Winfield

Scientific Investigations Report 2005-5114

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

Gale A. Norton, Secretary

U.S. Geological Survey

P. Patrick Leahy, Acting Director

U.S. Geological Survey, Reston, Virginia: 2005

For sale by U.S. Geological Survey, Information Services

Box 25286, Denver Federal Center

Denver, CO 80225

For more information about the USGS and its products:

Telephone: 1-888-ASK-USGS

World Wide Web: http://www.usgs.gov/

Any use of trade, product, or firm names in this publication is for descriptive purposes only and does not imply 

endorsement by the U.S. Government.

Suggested citation:

Winfield, K.A., 2005, Development of property-transfer models for estimating the hydraulic properties of deep 

sediments at the Idaho National Engineering and Environmental Laboratory, Idaho: U.S. Geological Survey Scientific 

Investigations Report 2005-5114, 49 p. 
iii

Contents

Abstract ................................................................................................... 1

Introduction................................................................................................ 1

Site Background .................................................................................... 2

Geohydrologic Setting.............................................................................. 6

Previous Investigations ........................................................................... 6

INEEL Hydraulic Property Measurements ................................................ 6

Overview of Property-Transfer Models ................................................... 7

Purpose and Scope ................................................................................. 8

Property-Transfer Model Calibration and Approach ................................................ 8

Data Sets and Measurement Techniques ...................................................... 10

Water-Retention Curve-Fit Parameters ......................................................... 11

Particle-Size Parameters........................................................................... 12

Multiple Linear Regression ........................................................................ 14

Property-Transfer Model Analyses ..................................................................... 16

Evaluation of Calibration Data ..................................................................... 17

Effect of Repacked Core Samples ......................................................... 17

Errors in Fitted, Calculated, or Measured Parameters.................................... 20

Linearity Between Explanatory and Response Variables .................................... 25

Collinearity Between Explanatory Variables ................................................... 29

Water-Retention Property-Transfer Model ...................................................... 31

Saturated Water Content..................................................................... 31

Scaling Parameter for Matric Pressure ................................................... 34

Curve-Shape Parameter ..................................................................... 36

Saturated Hydraulic Conductivity Property-Transfer Model ................................. 36

Model Discrimination .............................................................................. 38

Property-Transfer Model Application at the INEEL ............................................. 42

Summary and Conclusions .............................................................................. 45

References Cited.......................................................................................... 46
iv

Figures

Figure 1. Map showing location of the Idaho National Engineering Environmental

Laboratory and selected facilities, Idaho ................................................ 3

Figure 2. Maps showing boreholes in the Radioactive Waste Management Complex,

Idaho National Engineering and Environmental Laboratory, Idaho, where

measurements on sedimentary interbed core samples were used to develop

hydraulic property-transfer models ...................................................... 4

Figure 3. Maps showing boreholes in the Vadose Zone Research Park, Idaho

National Engineering and Environmental Laboratory, Idaho, where

measurements on sedimentary interbed core samples were used to develop

hydraulic property-transfer models ...................................................... 5

Figure 4. Diagram outlining steps used to develop multiple linear-regression models

for water-retention parameters............................................................ 9

Figure 5. Graph showing example of water-retention (q(y)) curve showing

components of the curve-fit model developed by Rossi and Nimmo (1994) ......... 11

Figure 6. Graph showing example particle-size distribution showing three parameter

sets (textural-class percentages, geometric statistics, and graphical

statistics) used to develop property-transfer models for water retention and

saturated hydraulic conductivity ......................................................... 13

Figure 7. Diagrams showing textural classification of core samples (based on the

system of the U.S. Department of Agriculture) used to develop a

property-transfer model for estimating water retention (q(y))........................ 21

Figure 8. Diagrams showing textural classification of core samples (based on the

system of the U.S. Department of Agriculture) used to develop a

property-transfer model for estimating saturated hydraulic conductivity (Ksat) ... 25

Figure 9. Residual plots showing studentized residuals and predicted hydraulic

property values for multiple-linear regression model C (where the median

particle diameter, uniformity coefficient, and bulk density were used as the

explanatory variables) ..................................................................... 32

Figure 10. Graphs showing predicted and observed hydraulic property values for

multiple-linear regression model C (where the median particle diameter,

uniformity coefficient, and bulk density were used as the explanatory

variables) .................................................................................... 33

Figure 11. Graphs showing estimated water-retention (q(y)) curves for two silt loam

core samples based on three multiple linear-regression models with

different particle-size parameter sets as input .......................................... 39

Figure 12. Boxplot showing comparison of root-mean-square-error (RMSE)

distributions for four combinations of multiple linear-regression models with

different particle-size parameter sets as input .......................................... 40

Figure 13. Boxplots showing root-mean-square-error distributions for the multiple

linear-regression model with median particle diameter, uniformity

coefficient, and bulk density as input (model combination C-C-C) .................. 41

Figure 14. Graphs showing relation between water-retention parameters of the

Rossi-Nimmo (1994) junction model and the van Genuchten (1980) model ......... 44
Tables

Table 1. Sedimentary interbed core samples with hydraulic and bulk-physical

property measurements at the Idaho National Engineering and

Environmental Laboratory (INEEL), Idaho ................................................ 7

Table 2. Statistics for hydraulic data used to develop property-transfer models, listed

by data source .............................................................................. 18

Table 3. Statistics for bulk physical-property data used to develop property-transfer

models, listed by data source ............................................................ 19

Table 4. Core samples removed from multiple linear-regression analyses due to

missing data ................................................................................. 20

Table 5. Core samples removed from multiple linear-regression analyses due to

errors in fitted, calculated, or measured parameters ................................. 23

Table 6. Correlation matrix between water-retention parameters and bulk-physical

properties, without transformation of selected variables .............................. 26

Table 7. Correlation matrix between saturated hydraulic conductivity and

bulk-physical properties, without transformation of selected variables ............ 27

Table 8. Correlation matrix between water-retention parameters and bulk-physical

properties, with transformation of selected variables ................................. 28

Table 9. Correlation matrix between saturated hydraulic conductivity and

bulk-physical properties, with transformation of selected variables.................. 29

Table 10. Variance inflation factors for explanatory variables used to develop

property-transfer models for water retention and saturated hydraulic

conductivity ................................................................................. 30

Table 11. Multiple linear-regression results comparing alternative representations of

particle-size distribution for water-retention parameters and saturated

hydraulic conductivity ..................................................................... 31

Table 12. Multiple linear-regression model coefficients, confidence limits, standard

errors, and partial t-values for the water-retention parameter, saturated

water content .............................................................................. 34

Table 13. Multiple linear-regression model coefficients, confidence limits, standard

errors, and partial t-values for the water-retention scaling parameter for

matric pressure .............................................................................. 35

Table 14. Multiple linear-regression model coefficients, confidence limits, standard

errors, and partial t-values for the water-retention curve-shape parameter ...... 36

Table 15. Multiple linear-regression model coefficients, confidence limits, standard

errors, and partial t-values for saturated hydraulic conductivity ..................... 37
vi

Conversion Factors, Datums, and Acronyms

Conversion Factors

Multiply By To obtain

centimeter (cm) 0.3937 inch

centimeter of water (cm-water) 0.01419 pound per square inch 

centimeter per second (cm/s) 0.03281 foot per second

cubic centimeter (cm3) 0.06102 cubic inch 

gram (g) 0.03527 ounce, avoirdupois

gram per cubic centimeter (g/cm3) 62.4220 pound per cubic foot 

kilometer (km) 0.6214 mile

meter (m) 3.281 foot 

millimeter (mm) 0.03937 inch

square kilometer (km2) 247.1 acre

square kilometer (km2) 0.3861 square mile

Temperature in degrees Celsius (°C) may be converted to degrees Fahrenheit (°F) as follows:

 °F=(1.8×°C)+32

Datums

Vertical coordinate information is referenced to the National Geodetic Vertical Datum of 1929 

(NGVD of 1929).

Horizontal coordinate information is referenced to the North American Datum of 1983 (NAD83).

Altitude, as used in this report, refers to distance above the vertical datum.

Acronyms

ICPP Idaho Chemical Processing Plant

INEEL Idaho National Engineering and Environmental Laboratory

INTEC Idaho Nuclear Technology and Engineering Center

PPCC probability-plot correlation coefficient

PTM property-transfer model

RMSE root-mean-square error

RWMC Radioactive Waste Management Complex

SDA Subsurface Disposal Area

SRP Snake River Plain

TAN Test Area North

TRA Test Reactor Area

TSA Transuranic Storage Area

VIF variance inflation factor

VZRP Vadose Zone Research Park

USDA U.S. Department of Agriculture

USGS U.S. Geological Survey
Abstract

Because characterizing the unsaturated hydraulic 

properties of sediments over large areas or depths is costly 

and time consuming, development of models that predict 

these properties from more easily measured bulk-physical 

properties is desirable. At the Idaho National Engineering 

and Environmental Laboratory, the unsaturated zone is 

composed of thick basalt flow sequences interbedded with 

thinner sedimentary layers. Determining the unsaturated 

hydraulic properties of sedimentary layers is one step in 

understanding water flow and solute transport processes 

through this complex unsaturated system. Multiple linear 

regression was used to construct simple property-transfer 

models for estimating the water-retention curve and saturated 

hydraulic conductivity of deep sediments at the Idaho National 

Engineering and Environmental Laboratory. The regression 

models were developed from 109 core sample subsets with 

laboratory measurements of hydraulic and bulk-physical 

properties. The core samples were collected at depths of 

9 to 175 meters at two facilities within the southwestern 

portion of the Idaho National Engineering and Environmental 

Laboratory—the Radioactive Waste Management Complex, 

and the Vadose Zone Research Park southwest of the Idaho 

Nuclear Technology and Engineering Center. Four regression 

models were developed using bulk-physical property 

measurements (bulk density, particle density, and particle size) 

as the potential explanatory variables. Three representations 

of the particle-size distribution were compared: (1) texturalclass percentages (gravel, sand, silt, and clay), (2) geometric 

statistics (mean and standard deviation), and (3) graphical 

statistics (median and uniformity coefficient). The four 

response variables, estimated from linear combinations of 

the bulk-physical properties, included saturated hydraulic 

conductivity and three parameters that define the waterretention curve. 

For each core sample, values of each water-retention 

parameter were estimated from the appropriate regression 

equation and used to calculate an estimated water-retention 

curve. The degree to which the estimated curve approximated 

the measured curve was quantified using a goodness-of-fit 

indicator, the root-mean-square error. Comparison of the rootmean-square-error distributions for each alternative particlesize model showed that the estimated water-retention curves 

were insensitive to the way the particle-size distribution was 

represented. Bulk density, the median particle diameter, and 

the uniformity coefficient were chosen as input parameters 

for the final models. The property-transfer models developed 

in this study allow easy determination of hydraulic properties 

without need for their direct measurement. Additionally, 

the models provide the basis for development of theoretical 

models that rely on physical relationships between the poresize distribution and the bulk-physical properties of the media. 

With this adaptation, the property-transfer models should have 

greater application throughout the Idaho National Engineering 

and Environmental Laboratory and other geographic locations. 

Introduction

Flow processes within the highly stratified unsaturated 

zone at Idaho National Engineering and Environmental 

Laboratory (INEEL) are not well understood. The thick basalt 

layers are fractured to massive, contain rubble zones, and 

are interbedded with thinner sedimentary layers. All these 

features potentially cause preferential contaminant and water 

flow through the unsaturated zone. To better understand these 

processes, the role that sedimentary interbeds and surficial 

sediments play in unsaturated flow needs to be determined; 

this typically is accomplished using flow simulations. Flow 

models require input of basic unsaturated hydraulic properties, 

such as water-retention and unsaturated hydraulic conductivity 

curves, which are representative of the modeled media. The 

water-retention curve (q(y)) is a measure of how the water 

content (q) varies with the negative water (or matric) pressure 

(y) of a particular soil or sedimentary material. Unsaturated 

hydraulic conductivity (K), or the rate at which water is 

transmitted through a sedimentary material, varies with q or 

y. Saturated hydraulic conductivity (Ksat) is the flow rate at 

the maximum degree of saturation (qsat). Knowledge of Ksat is 

needed to estimate K(q) from established models. Ksat often is 

measured in place of K(q) because Ksat is easier to determine 

Development of Property-Transfer Models for Estimating 

the Hydraulic Properties of Deep Sediments at the Idaho 

National Engineering and Environmental Laboratory, Idaho

by Kari A. Winfield
than K(q), and because it can be used along with q(y) to 

estimate K(q) using parametric models (for example, Mualem, 

1976). Field tests can provide information about the average 

unsaturated hydraulic properties over a given region at depth, 

but when studying sedimentary interbeds this task becomes 

more challenging because of the overlying and underlying 

fractured basalt units. Alternatively, minimally disturbed 

core samples can be collected by drilling deep boreholes, and 

their hydraulic properties can be determined in a laboratory; 

however, this process is costly and time consuming. Drilling 

can require months to complete and obtaining undisturbed 

core samples can be problematic, especially in coarse-textured 

materials. Laboratory measurement of unsaturated hydraulic 

properties requires specialized equipment and expertise, in 

addition to requiring several weeks to complete measurements 

for a single core sample. These difficulties are compounded 

at the INEEL by the thick vadose zone (up to 200 m), the 

complex stratigraphy, and the large site area.

An alternative to measuring the unsaturated hydraulic 

properties of core samples in the laboratory is to develop 

a property-transfer model (PTM) that can predict these 

properties from more easily measured bulk-physical 

properties, such as bulk density (rbulk), particle density (rpart), 

and particle-size distribution. General purpose models of this 

type exist, but they are inadequate for sediments at the INEEL 

because they were not developed from measurements specific 

to the site. Hydraulic PTMs have not been developed for the 

sedimentary interbeds at the INEEL. These types of models 

will be a cost effective way to estimate unsaturated hydraulic 

properties for use in flow models, to predict contaminant travel 

times and flow paths, and to provide greater understanding of 

unsaturated flow processes in the complex unsaturated zone at 

the INEEL. Property-transfer modeling is a useful approach 

to estimating hydraulic properties, even when the estimates 

have a large margin of error, because hydraulic-property 

measurements cannot be made for every point in space. 

Often when core samples are not obtained during 

drilling, borehole cuttings are collected that can be analyzed 

for particle size. The majority of laboratory data available at 

the INEEL consists of particle-size distributions and other 

bulk-physical-property measurements obtained from borehole 

cuttings. The U.S. Geological Survey (USGS), Bechtel BWXT 

Idaho, LLC, previous contractors, and other institutions have 

investigated hydrologic processes within the vadose zone of 

the INEEL for several decades; however, relatively little work 

has been done in the laboratory to characterize the hydraulic 

properties of sedimentary interbeds. Unsaturated hydraulic 

property estimates by PTMs can save considerable time and 

effort by reducing the need to measure unsaturated hydraulic 

properties in the laboratory. Additionally, estimation of 

hydraulic properties will enable a broader understanding of the 

distribution of hydraulic and bulk-physical properties at the 

INEEL, which is essential in developing models of groundwater flow and contaminant transport. 

Site Background

The INEEL was established in 1949 under the U.S. 

Department of Energy (formerly, the U.S. Atomic Energy 

Commission) for nuclear energy research. The site hosts 

several facilities (fig. 1) at least four of which have generated, 

stored, or disposed of radioactive, organic, and inorganic 

wastes. These include the (1) Radioactive Waste Management 

Complex (RWMC), (2) Idaho Nuclear Technology and 

Engineering Center (INTEC), formerly known as the Idaho 

Chemical Processing Plant (ICPP), (3) Test Reactor Area 

(TRA), and (4) Test Area North (TAN). Because of the 

site’s history of chemical-waste production and groundwater resource contamination, vadose-zone investigations 

are used to determine or forecast critical pathways for 

contaminant migration to the Snake River Plain (SRP) aquifer, 

and to understand flow and transport processes within the 

geologically complex subsurface. Because hydraulic property 

measurements are available only on core samples collected 

from the Vadose Zone Research Park (VZRP) and RWMC, 

this report focuses on these two locations within the INEEL. 

The RWMC (figs. 1 and 2), which covers approximately 

0.6 km2, was established in 1952 as an area for controlled 

management of solid radioactive waste produced on site. 

Facilities include the Subsurface Disposal Area (SDA) for 

shallow burial of mixed low-level wastes and the Transuranic 

Storage Area (TSA) for above-ground waste storage 

containing elements with atomic numbers equal to or higher 

than uranium. Mixed transuranic and low-level wastes were 

buried in pits and trenches at the SDA from 1952 to 1970. 

Since 1970, transuranic wastes have been stored in aboveground containers on asphalt pads at the TSA, and low-level 

waste burial has continued at the SDA (Anderson and Lewis, 

1989). 

The INTEC is in the southwestern part of the INEEL 

covering an area of about 0.8 km2 (figs. 1 and 2). The 

INTEC was established in the 1950’s for reprocessing spent 

nuclear fuel from government reactors. The facility houses 

an underground tank farm for storing high-level wastes, and 

above-ground storage areas for spent nuclear wastes (Bechtel 

BWXT, LLC, 2003). Between 1952 and 1984, radiochemical 

and chemical wastes produced on site were disposed of in 

a 183-m-deep injection well, which penetrated the SRP 

aquifer. Wastes produced after 1984 were disposed of in two 

percolation ponds (fig. 3) directly south of the facility (Cecil 

and others, 1991; Bartholomay and others, 1997; Bartholomay, 

1998). Contaminants detected in perched water bodies beneath 

these ponds led to their discontinued use in 1995. In 2001, new 

INTEC percolation ponds were constructed within the 3-km2

VZRP (fig. 3), about 9.5 km southwest of the original ponds. 

 Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
Figure 1. Location of the Idaho National Engineering Environmental Laboratory and selected facilities, Idaho.

IDSIR20055114_fig01

Big

Lost

River

Big Lost

River Sinks

Little Lost

River

Birch

Creek

Mud Lake 

Camas Creek

SR

22

SR

33

SR

22

SR

33

SR

22

SR

33

SR

33

US

26

US

26

US

20

US

26

US

20

SR

28

CR

A1

US

93

US

20 US

26

US

20

US

93

SR

28

SR

28

0 5

0

10

5 10

20

20 MILES

30 KILOMETERS

44°

113°15' 113° 45' 30' 112°15'

30'

43°15'

44°15'

45'

Howe

Arco

Atomic City

Terreton

IDAHO

Boise

Pocatello

Eastern Snake

River Plain

Twin Falls 

Idaho Falls 

Idaho National Engineering and

Environmental Laboratory

Base from U.S. Geological Survey, 1:100,000, 

Arco quadrangle, 1989, 

Blackfoot quadrangle, 1997, 

Borah Peak quadrangle, 1989, 

Circular Butte quadrangle, 1980, 

Craters of the Moon quadrangle, 1978, 

and Dubois quadrangle, 1983 

Lost River Range 

Lemhi Range 

East Butte 

Middle Butte 

Big Southern Butte 

Vadose Zone 

Research Park 

Test Reactor Area 

Idaho Nuclear Technology 

and Engineering Center 

Central Facilities Area

Experimental

Breeder Reactor-1

Argonne National

Laboratory-West

Test Area North

Bitterroot Range 

Naval Reactors

Facility

Radioactive Waste

Management Complex

Idaho National Engineering and

Environmental Laboratory 

Figure 3 

Figure 2 

Introduction 
Figure 2. Boreholes in the Radioactive Waste Management Complex, Idaho National Engineering 

and Environmental Laboratory, Idaho, where measurements on sedimentary interbed core samples 

were used to develop hydraulic property-transfer models.

ID20055114_fig02

INEEL BOUNDARY

SPREADING AREAS 

(SEE INSET A)

RADIOACTIVE WASTE 

MANAGEMENT COMPLEX

SOUTH-MON-A-009

SOUTH-SCI-V-015

UZ98-2

EXPLANATION

UZ98-2 Borehole and identifier

INSET A 

Big

Lost

River

0 .5 1 2 3 KILOMETERS 

0 .5 1 2 MILES 

0

0

500 1000 METERS

1000 2000 3000 4000 FEET

USGS 118

D-10

D-15

D-02

88-01D

INSET A

SOUTH-SCI-V-011

SOUTH-SCI-V-012

SOUTH-SCI-V-013

SOUTH-SCI-V-014

SOUTH-SCI-V-016

RWMC-SCI-V-158

RWMC-SCI-V-159

RWMC-SCI-V-160

SOUTH-SCI-V-018

RWMC-SCI-V-154

RWMC-SCI-V-156

RWMC-SCI-V-157

RWMC-SCI-V-153

SUBSURFACE DISPOSAL AREA

TRANSURANIC

STORAGE AREA

RWMC-SCI-V-155

Base from U.S. Geological Survey, 1:24,000, 

Arco Hills SE quadrangle, 1972, 

and Big Southern Butte quadrangle, 1972

113°05' 113°02'30” 113°

43°30'

43°28'

113°03' 113°02'16”

43°30'

43°29'47”

Big Lost

River Sinks

Big Lost River

Vadose Zone Research Park

Radioactive Waste

Management Complex

Idaho Nuclear Technology 

and Engineering Center 

Test Reactors Area IDAHO NATIONAL ENGINEERING AND

ENVIRONMENTAL LABORATORY

 Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
Figure 3. Boreholes in the Vadose Zone Research Park, Idaho National Engineering and Environmental Laboratory, Idaho, 

where measurements on sedimentary interbed core samples were used to develop hydraulic property-transfer models.

ID20055114_fig03

EXPLANATION

PAVED ROADS

DIRT ROADS

STREAMS

USGS GAGING STATION

ICPP-SCI-V-189 BOREHOLE AND IDENTIFIER

TEST REACTOR

AREA

IDAHO

NUCLEAR

TECHNOLOGY

AND

ENGINEERING

CENTER

Lincoln Boulevard

VADOSE ZONE

RESEARCH PARK

(SEE INSET A)

Former

percolation

ponds

Current percolation

pond area

Big Lost

River Sinks

Big Lost River

Vadose Zone Research Park

Radioactive Waste

Management Complex

Idaho Nuclear Technology

and Engineering Center

Test Reactors Area IDAHO NATIONAL ENGINEERING AND

ENVIRONMENTAL LABORATORY

Big

Lost River

INSET A

0 100 200 300

0 250 500 750 1000 FEET

METERS

Big

Lost River

112°59'

43°35'

43°33'

34'

58' 57' 56' 112°55'

43°33'45”

43°33'15”

30”

112°58'30” 30” 112°58'

0

0

500 1000 1500 2000 KILOMETERS 

2000 4000 6000 8000 FEET 

Base from U.S. Geological Survey, 1:24,000, 

Circular Butte 3 SW quadrangle, 1973

ICPP-SCI-V-213

ICPP-SCI-V-189

ICPP-SCI-V-214

ICPP-SCI-V-198

ICPP-SCI-V-205

ICPP-SCI-V-204

ICPP-SCI-V-215

Introduction 
Geohydrologic Setting

The INEEL occupies about 2,300 km2 within the eastern 

SRP, a northern extension of the Basin and Range Province. 

The eastern SRP is a northeast trending basin, approximately 

320 km long and 80 to 110 km wide, which slopes gently to 

the southwest and is bordered by northwest trending mountain 

ranges. 

The eastern SRP is underlain by interbedded volcanic 

and sedimentary units extending up to 3,000 m below the 

surface. The sedimentary layers represent quiet intervals 

between volcanic eruptions and are of eolian, fluvial, and 

lacustrine origin with large amounts of sand, silt, and clay. 

Volcanic units, composed primarily of basalt flows, welded 

ash flows, and rhyolite, may be vesicular to massive with 

either horizontal or vertical fracture patterns. Near the 

INTEC facility, boreholes drilled to 200-m depths penetrate a 

sequence of 23 basalt-flow groups and 15 to 20 sedimentary 

interbeds (Anderson, 1991). The surficial sediments near the 

INTEC consist of gravelly alluvium, range from 2- to 20-m 

thick, and are thickest to the northwest (Anderson and others, 

1996). Beneath the RWMC, 10 basalt-flow groups and 7 major 

sedimentary interbeds have been identified from boreholes 

drilled up to 220-m depth. Surficial sediments are composed 

primarily of silt and clay, and range in thickness from 0 to 7 m 

(Rightmire and Lewis, 1987; Anderson and Lewis, 1989). 

The climate of the eastern SRP is semi-arid with an 

average annual precipitation of 22 cm. Parts of the SRP 

aquifer underlie the INEEL. The depth to the water table 

ranges from 60 m in the northern part of the INEEL to about 

200 m toward the south (Barraclough and others, 1981; 

Liszewski and Mann, 1992). The depth to the water table is 

about 145 m below the INTEC and about 180 m below the 

RWMC. The predominant direction of ground-water flow is 

from northeast to southwest. Recharge to the aquifer primarily 

is from irrigation water diversions from streams, precipitation 

and snowmelt, underflow from tributary-valley streams, 

and seepage from surface water bodies (Hackett and others, 

1986). Within the INEEL boundaries, the Big Lost River 

(fig. 1) is an ephemeral stream, which flows from southwest 

to northeast about 3 km north of the RWMC (fig. 2) and 

less than 1 km north of the INTEC (fig. 3). Because of the 

proximity of the Big Lost River to waste disposal and storage 

facilities, a diversion dam was constructed in 1958 upstream 

of the SDA to reduce the threat of flooding (Barraclough and 

others, 1967). During high flow periods, flow is diverted to 

topographic depressions (referred to as spreading areas) less 

than 2 km west of the RWMC (fig. 2). 

Previous Investigations

This section describes studies in which hydraulic property 

measurements specific to the INEEL site were measured and 

gives an overview of PTMs. Few studies have been done in 

which the hydraulic properties of sedimentary interbeds have 

been determined on core samples. Only two locations within 

the INEEL have been characterized, the RWMC and the 

VZRP. Two classes of PTMs exist in the literature, those based 

on empirical relationships and those based on theoretical 

ones. Within the empirical model category, multiple linear 

regression often is used as a tool for model development. 

INEEL Hydraulic Property Measurements

Most studies of contaminant transport and measurement 

of hydraulic properties at the INEEL have been done near the 

RWMC (fig. 1), where potential migration of shallow, buried 

hazardous waste through the unsaturated zone poses concern 

for water quality in the SRP aquifer. McElroy and Hubbell 

(1990) presented measurements of q(y), particle size, and 

rbulk on intact sedimentary core samples collected from eight 

boreholes near the RWMC. Core samples from sedimentary 

interbeds were collected from five of these boreholes (USGS 

118, 88-01D, D-02, D-10, and D-15; shown in fig. 2). Deep 

core-sample measurements of q(y) and K(q), in addition 

to bulk-physical properties, were presented by Perkins and 

Nimmo (2000) for borehole UZ98-2, about 1.5 km southwest 

of the SDA (fig. 2). Measurements from an additional 

16 boreholes near the SDA also have been completed on 

sediment core samples collected from depths of 29.45 to 

78.02 m below land surface (S. Magnuson, written commun., 

2002; fig. 2 and table 1). Perkins (2003) and Winfield (2003) 

presented unsaturated hydraulic- and bulk-physical-property 

measurements for seven boreholes near the current percolation 

pond area for the INTEC facility at the VZRP (fig. 3). 

Several studies have been done to characterize the 

unsaturated hydraulic properties of surficial sediments, defined 

as the sediment and soil overlying the shallowest basalt 

layer (Barraclough and others, 1976; McElroy and Hubbell, 

1990; Borghese, 1991; Shakofsky, 1995; Nimmo and others, 

1999). Measurements on surficial soils and sediments are not 

considered in the models presented in this report because their 

hydraulic properties are affected by structural features arising 

from plant growth, compaction by anthropogenic sources, 

animal burrows, and shrink-swell phenomena. Nimmo and 

others (2004), table 1, present a more complete summary of 

available INEEL measurements.

 Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
Borehole identification No. Borehole

location

Depth

interval

(m)

Number and type of core sample measurements

Saturated 

hydraulic 

conductivity 

(Ksat)

Unsaturated 

hydraulic 

conductivity as 

a function of 

water content 

(K(q))

Water 

retention 

(q(y))

Bulk 

density 

(rbulk)

Particle 

density 

(rpart)

Particle-size 

distribution

Magnuson, Swen, Bechtel BWXT Idaho, LLC (written commun., 2002)

RWMC-SCI-V-153, RWMC-SCI-V-154, 

RWMC-SCI-V-155, RWMC-SCI-V-156, 

RWMC-SCI-V-157, RWMC-SCI-V-158, 

RWMC-SCI-V-159, RWMC-SCI-V-160,

SOUTH-MON-A-009, SOUTH-SCI-V-011, 

SOUTH-SCI-V-012, SOUTH-SCI-V-013, 

SOUTH-SCI-V-014, SOUTH-SCI-V-015, 

SOUTH-SCI-V-016, SOUTH-SCI-V-018 

RWMC 29.45–78.02 41 0 51 51 51 51

McElroy and Hubbell (1990)

88-01D, D-02, D-10, D-15, USGS 118 RWMC 9.60–17.48 16 0 16 16 16 116

Perkins and Nimmo (2000)

UZ98-2 RWMC 42.98–50.30 18 18 18 18 18 18

Perkins (2003)

ICPP-SCI-V-215 VZRP 45.54–59.92 12 12 14 14 14 14

Winfield (2003)

ICPP-SCI-V-189, ICPP-SCI-V-198,

ICPP-SCI-V-204, ICPP-SCI-V-205,

ICPP-SCI-V-213, ICPP-SCI-V-214

VZRP 36.59–56.33 10 10 10 10 10 10

Number of samples, n 97 40 109 109 109 1109

1Partial particle-size data were reported for sample USGS 118 at 173.48 to 174.39 m from McElroy and Hubbell (1990).

Table 1. Sedimentary interbed core samples with hydraulic and bulk-physical property measurements at the Idaho National Engineering and 

Environmental Laboratory (INEEL), Idaho.

[Water retention is defined as water content (q) as a function of matric pressure (y). Abbreviations: m, meter; RWMC, Radioactive Waste Management 

Complex; VZRP, Vadose Zone Research Park]

Overview of Property-Transfer Models

PTMs can be classified into two groups: (1) empirical 

models, which rely on statistical methods to determine 

patterns in the data, and (2) quasi-physical models, which are 

based on theoretical or physical relationships between pore 

sizes and particle or aggregate sizes (Arya and Paris, 1981; 

Haverkamp and Parlange, 1986; Nimmo, 1997; Haverkamp 

and Reggiani, 2002). Methods for predicting q(y) are 

described here, however, these methods also can be applied to 

prediction of K(q). 

Empirical models typically involve multiple linearregression analyses (although some use a neural network 

procedure) and can be further subdivided based on the 

specific approach used to estimate unsaturated hydraulicproperty curves. One approach involves fitting a parametric 

q(y) function (or set of functions) to the q(y) measurements, 

and developing separate regression equations for each 

q(y) parameter (Campbell, 1985; Saxton and others, 

1986; Wösten and van Genuchten, 1988; Vereecken and 

others, 1989; Campbell and Shiozawa, 1992; Schaap and 

others, 1998, 2001). Another approach involves developing 

unique equations for q at the values of y determined during 

measurement of q(y) (Gupta and Larson, 1979; Rawls and 

Brakensiek, 1982; Puckett and others, 1985; Mecke and 

others, 2002). A third approach uses at least one measured 

value of q(y), in addition to bulk-physical properties, as input 

(Gregson and others, 1987; Schaap and others, 1998). 

Introduction 
Several studies also have predicted Ksat using multiple 

linear regression (Puckett and others, 1985; Campbell and 

Shiozawa, 1992; Jabro, 1992; Mecke and others, 2002) 

or neural networks (Schaap and others, 1998, 2001). 

By empirical modeling, several equations are required 

to completely predict q(y) because q varies uniquely in 

relation to y. Because Ksat relates only to one q value, qsat, 

a single equation can be used to directly relate Ksat to a set 

of bulk-physical properties using statistical techniques. The 

remainder of the K(q) curve can be predicted using any of 

the aforementioned techniques, providing measurements of 

K(q) are available. Alternatively, estimates of Ksat and q(y)

obtained by PTMs can be used to predict K(q) using Mualem’s 

(1976) capillary bundle model.

Most PTMs available in the literature were developed for 

specific applications, such as for widespread use (Schaap and 

others, 1998, 2001) or for a specific site or soil type (Jabro, 

1992; Puckett and others, 1985; Mecke and others, 2002). 

PTMs developed for use in agricultural studies typically are 

calibrated using repacked samples where gravels have been 

removed. Mecke and others (2002) reaffirmed the conclusion 

of Puckett and others (1985) and Williams and others (1992) 

that a predictive model should be developed on materials of 

similar mineralogy and genesis. PTMs developed for a specific 

site cannot describe the diverse range of soil textures and 

structures found at other sites. Factors that may cause PTMs 

to differ between sites include water chemistry, mineralogy 

of sediment, liquid-solid interactions, colloidal transport and 

potential blockage of pores, development of cements in pores, 

and sediment structure (macropores, grain shape, orientation, 

roundness, and packing arrangement).

Most PTMs available in the literature have been 

developed for surficial soils, often with characteristics 

particular to a given site and soil type, such as Lower Coastal 

Plain ultisols (Puckett and others, 1985) or glacially-derived 

podzols (Mecke and others, 2002). The development of 

reliable PTMs for deeper sediments (below the zone of soil 

development) is needed, especially as the USGS and other 

agencies move toward characterizing larger portions of 

the subsurface for studies of ground-water recharge. Other 

PTMs calibrated using large databases of soil properties are 

inappropriate for the INEEL because the databases often 

contain too diverse a mixture of samples with conflicting 

characteristics, such as repacked samples included with intact 

samples, or samples with pedologic structure included with 

single-grained soils.

Purpose and Scope

This report describes the development of PTMs for 

predicting q(y) and Ksat of the sedimentary interbeds at the 

INEEL. The PTMs are constrained by available data to two 

locations within the INEEL, the RWMC and the VZRP. 

Because surficial sediments possess different physical 

characteristics than the sedimentary interbeds, these data 

were not used to develop the PTMs in this study. Multiple 

linear-regression techniques were used to estimate hydraulic 

properties from bulk-physical properties based on 109 sets of 

core-sample measurements compiled from previous studies 

(S. Magnuson, written commun., 2002; McElroy and Hubbell, 

1990; Perkins and Nimmo, 2000; Perkins, 2003; Winfield, 

2003; see table 1). Three regression equations were developed 

for parameters defining the q(y) curve (qsat and two parameters 

from the Rossi-Nimmo (1994) junction model). The q(y)

parameters and Ksat were used as the dependent variables in 

the regression models. Parameters describing the particlesize distribution, rbulk, and rpart values were used as potential 

independent variables. This report describes the available 

data, data processing, multiple linear-regression assumptions 

and approach, and regression equations developed for each 

hydraulic parameter. Three representations of particle-size data 

were compared to determine the optimal regression variate 

to use in the final PTMs. Selection of the best PTM was 

based on ease of calculating particle-size parameters, model 

behavior (in terms of meeting the assumptions of multiple 

linear regression), model fit, and potential for developing 

a theoretical model. The results of this report contribute to 

the characterization of the unsaturated zone at the INEEL 

by providing a tool that can be used to predict hydraulic 

properties required as input to unsaturated-flow simulations. 

Property-Transfer Model Calibration 

and Approach

To develop PTMs for the INEEL sedimentary interbeds, 

multiple linear-regression equations were used to estimate 

hydraulic properties (measurements of qsat and Ksat, and curvefit parameters for q(y)) from linear combinations of bulkphysical properties (rbulk, rpart, and particle-size parameters). 

To estimate the entire q(y) curve from saturation to oven 

dryness, the Rossi-Nimmo (1994) junction model was used 

to characterize the q(y) measurements. Multiple linearregression equations then were developed for each of the 

three parameters defining the Rossi-Nimmo (1994) junction 

model. Three alternative models were compared that used 

different representations of the particle-size distribution as 

input, along with rbulk and rpart: (1) model A included texturalclass percentages, (2) model B included geometric statistical 

parameters, and (3) model C included graphical statistical 

parameters. The basic steps of the multiple linear-regression 

procedure used in this study are outlined in figure 4 for q(y). 

The same steps were followed for Ksat, except no curve-fitting 

was involved. 

 Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
Figure 4. Steps used to develop multiple linear-regression models for water-retention parameters.

ID20055114_fig04

RAW DATA DATA

PROCESSING

REGRESSION

MODEL INPUT

REGRESSION

MODEL OUTPUT

Water-retention

measurements

Rossi-Nimmo

(1994) junction 

model, curve fit

Least-squares

multiple linear 

regression

Coefficients for 

each y and set of 

x variables, 

goodness-of-fit

values

HYDRAULIC PROPERTIES

(RESPONSE VARIABLES, Y)

EACH Y 

VARIABLE

9 separate 

equations

Saturated water 

content (Qsat)

Scaling parameter for

matric pressure (Yo)

OR

OR

Particle-size

measurements

MODEL B:

Geometric statistics

 (mean and standard 

deviation)

Frequency of 

particles per bin 

MODEL C: 

Graphical statistics 

(median and uniformity 

coefficient)

MODEL A: 

USDA textural-class 

percentages

Bulk-density and 

particle-density

measurements

BULK PHYSICAL PROPERTIES 

(EXPLANATORY VARIABLES, X)

Bulk-density and 

particle- density 

measurements

Cumulative

frequency

distribution

EACH SET 

OF X 

VARIABLES

no processing

OR

OR

AND

Curve-shape

parameter (L)

Property-Transfer Model Calibration and Approach 
Data Sets and Measurement Techniques

The hydraulic- and bulk-physical-property data sets 

used to develop the q(y) and Ksat models for the INEEL 

sedimentary interbeds, including the number and type of coresample measurements, are summarized in table 1. These data 

represent the potential number of core samples for developing 

the q(y) and Ksat PTMs; smaller subsets were used to develop 

the final regression models after evaluating the data for errors 

and missing values. To develop the multiple linear-regression 

equations, complete data sets of hydraulic (q(y) or Ksat) 

and bulk-physical (rbulk, rpart, and particle-size distribution) 

properties were needed for each core sample. Core samples 

with missing values of either the hydraulic or bulk-physical 

properties were removed from the regression analyses. The 

particle-size distributions needed to have sufficient resolution, 

including points in the clay-sized (less than 0.002 mm) 

fraction, to compute geometric particle-size statistics, texturalclass percentages, or particle sizes at particular cumulative 

percentiles (10th, 50th, and 60th). Because hydraulic property 

measurements are only available for the RWMC and VZRP, 

the PTMs presented in this report can only be generalized 

to interbeds in the southwestern portion of the INEEL. 

Surficial sediments were excluded from the PTM development 

because of likely physical differences from the interbed 

sediments caused by soil development processes, freeze-thaw 

phenomena, or animal or microbial activity. Five data sets 

were used in model development: 

1. RWMC, 51 core samples of Bechtel BWXT Idaho, 

LLC (S. Magnuson, written commun., 2002), 

2. RWMC, 16 core samples of McElroy and Hubbell 

(1990), 

3. RWMC, 18 core samples of Perkins and Nimmo 

(2000), 

4. VZRP, 14 samples of Perkins (2003), and 

5. VZRP, 10 samples of Winfield (2003).

Boreholes where core samples were collected are shown 

in figure 2 for the RWMC and in figure 3 for the VZRP. The 

McElroy and Hubbell (1990) data were measured by Daniel 

B. Stephens and Associates, Inc., New Mexico. The data of 

Magnuson (written commun., 2002) were measured by the 

Southwest Research Institute, Texas. The data of Perkins 

and Nimmo (2000), Perkins (2003), and Winfield (2003) 

were measured by the USGS. Partial particle-size data were 

reported for one McElroy and Hubbell (1990) sample from 

borehole USGS 118 at 173.48 to 174.39 m. Ten samples from 

Magnuson (written commun., 2002) and two samples from 

Perkins (2003) did not have Ksat measurements. 

Techniques used to determine unsaturated hydraulic 

properties of the 109 core samples differed among studies. 

All q(y) measurements were determined on minimally 

disturbed samples except for those of Magnuson (written 

commun., 2002). These samples were sieved to remove 

the gravel (> 2 mm) fraction and then repacked. For the 

Magnuson (written commun., 2002) data, five q points were 

determined per sample, at y values of -200, -400, -600, -800, 

and -1,000 cm-water, using the pressure-plate technique. 

For the McElroy and Hubbell (1990) data, eight points on 

the q(y) curve were determined per sample, spanning the 

range of y from about -50 to -15,000 cm-water. Points in 

the wet range (> -400 cm-water) were determined by the 

hanging column method, and drier points were determined 

by the pressure-plate technique. The q(y) measurements of 

Perkins and Nimmo (2000), Perkins (2003), and Winfield 

(2003) were completed as part of the steady-state centrifuge 

method (Nimmo and others, 1994; Conca and Wright, 1998; 

Nimmo and others, 2002), which was used to determine 

K(q). Tensiometers were used to determine equilibrium y at 

different q values. When samples were too dry to use with 

a tensiometer, the filter paper method (Fawcett and CollisGeorge, 1967; Greacen and others, 1987) was used. For the 

Perkins and Nimmo (2000) samples, additional dry-range q(y)

points were determined for the range of y between -105 and 

-107 cm-water using a chilled mirror hygrometer (Winfield, 

written commun., 2001). The number of q(y) points for these 

studies varied from 7 to 18, with an average of 14 points for 

Perkins and Nimmo (2000), 10 points for Perkins (2003), and 

9 points for Winfield (2003).

Degree of saturation was either measured directly or 

inferred from other properties. For the Perkins and Nimmo 

(2000), Perkins (2003), and Winfield (2003) studies, saturation 

was achieved by submerging the core samples in a wetting 

solution up to the sample height. To determine qsat values, 

the samples were weighed after measurement of Ksat and the 

weights were converted to volumetric q values using rbulk. 

For the McElroy and Hubbell samples, qsat was determined 

prior to beginning q(y) and Ksat measurements, after 

vacuum saturating the samples. qsat was not measured for the 

Magnuson samples. In this report, the maximum water content 

for the Magnuson samples was assumed to equal the total 

porosity determined from the repacked sample rbulk. 

For all studies, Ksat was measured on intact core samples. 

Ksat values for the McElroy and Hubbell (1990) samples 

were determined using either the falling-head or the constanthead method. Perkins and Nimmo (2000), Perkins (2003), 

and Winfield (2003) used either the traditional falling-head 

method, or the falling-head method adapted for use with a 

centrifuge (Nimmo and Mello, 1991; Nimmo and others, 

2002) for low Ksat materials. Ksat values from Magnuson 

(written commun., 2002) were measured using the constanthead method. 

10 Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
Particle-size distributions were determined by sieve 

analysis, hydrometer, or laser light scattering. McElroy and 

Hubbell (1990) used sieve analysis for particle sizes greater 

than 0.075 mm, and the hydrometer method for sizes less than 

0.075 mm. The number of points analyzed per sample was 20, 

with the exception of four samples that only had seven points 

greater than 0.075 mm. For one of these samples (USGS 118 

at 173.48 to 174.39 m) the diameter at the 10th percentile (d10), 

the geometric statistics, and textural-class percentages could 

not be calculated. For the Magnuson (written commun., 2002) 

samples, sieve analysis was used for particle sizes greater than 

0.075 mm, and the hydrometer method was used for sizes less 

than 0.075 mm, with 20 points determined per sample. Perkins 

and Nimmo (2000), Perkins (2003), and Winfield (2003) 

used sieve analysis for particles greater than 2 mm. A laser 

diffraction method was used for particle sizes from 

4  ́ 10-5 mm to 2 mm, with 116 points analyzed. 

Total porosity was calculated from measurements of rpart

and rbulk. Values of rpart were determined on small subsamples 

from the original core samples. The pycnometer method of 

Blake and Hartge (1986) was used to measure rpart on the core 

samples from Perkins and Nimmo (2000), Perkins (2003), 

and Winfield (2003). The entire particle-size distribution 

was analyzed for these samples. For the core samples of 

Magnuson (written commun., 2002) and McElroy and Hubbell 

(1990), rpart was determined using the standard pycnometer 

method D854-83 (now an archived method) of the American 

Society for Testing and Materials (1989). For the McElroy 

and Hubbell (1990) samples, material less than 4.75 mm was 

analyzed, while material less than 2 mm was analyzed for the 

Magnuson (written commun., 2002) samples. For the data set 

of Perkins and Nimmo (2000), the average rpart value of 

2.65 g/cm3 (measured for 4 core samples) was used for all 

18 core samples. Values of rbulk were determined on intact 

core samples for all data sets used in model calibration. 

Total porosity was calculated by 1 – rbulk/rpart. Although total 

porosity values were not used in developing the regression 

models, total porosity was used as a baseline for comparing 

what the core sample degree of saturation should be. Because 

total porosity is a measure of all void space in a sample, 

including any dead-end pores and the internal porosity 

of mineral grains, it should be greater than the measured 

qsat value (which can be considered a measure of effective 

porosity).

Water-Retention Curve-Fit Parameters

Because q varies nonlinearly as a function of y, one or 

more functions involving multiple parameters are needed to 

describe the entire range of q. One approach to developing a 

PTM for q(y) involves developing multiple linear-regression 

equations for several q values associated with particular 

y values. Difficulties applying this method to INEEL 

measurements included that (1) the number of measured q(y) 

points varied between data sets, and (2) q values were not 

obtained at the same y values for all samples due to different 

methods used to measure q(y). An alternate way to develop 

a PTM for q(y) is to fit a curve to the measured q(y) points 

for each core sample. This approach yields parameters that 

uniquely define each curve. Multiple linear regression is then 

used to develop equations relating each of these parameters to 

a set of bulk-physical properties.

The Rossi-Nimmo (1994) junction model (fig. 5) was 

chosen to fit the q(y) measurements because this model 

is more physically realistic over the entire range of q from 

saturation to oven dryness than other parametric models 

(Brooks and Corey, 1964; van Genuchten, 1980) that use 

residual water content (qr) as an optimized parameter; qr

has no physical meaning. According to capillary theory the 

largest pores are associated with y values near zero and drain 

first, followed by drainage of successively smaller pores 

as q approaches qr. The asymptotic approach to qr means 

that the number of small pores approaches infinity, which 

is physically unrealistic. The q(y) curve represented by the 

Rossi-Nimmo (1994) junction model goes to zero q at a fixed 

value of y calculated for the conditions of oven dryness (yd). 

Another advantage to using the junction model is that it can be 

combined with the capillary-bundle model of Mualem (1976) 

to estimate K(q) (Fayer and others, 1992; Rossi and Nimmo, 

1994; Andraski, 1996; Andraski and Jacobson, 2000).

WATER CONTENT (Q)

MATRIC PRESSURE (y)

yj Junction point for 2. and 3.

EXPLANATION

1. Parabolic function, yi £ y £ 0

2. Power law function, yj £ y £ yi

3. Logarithmic function, yd £ y £ yj

qsat Saturated water content

yo Scaling parameter for matric pressure

l Curve-shape parameter

yi Junction point for 1. and 2.

yd Oven-dry matric pressure

qsat

0 -107

0

yo

yd

yj

yi

l

1.

2.

3.

Figure 5. Example of water-retention (q(y)) curve showing components 

of the curve-fit model developed by Rossi and Nimmo (1994). 

Property-Transfer Model Calibration and Approach 11
The Rossi-Nimmo (1994) junction model requires three 

parameters to describe the entire q(y) curve, therefore, three 

separate multiple linear-regression equations are needed. By 

contrast, models that estimate q at particular y values usually 

require 7 to 12 separate regression equations. These equations 

typically range from y = 0 at saturation to approximately 

-15,000 cm-water. For the abundance of silt-loam-textured 

samples found at the INEEL, -15,000 cm-water corresponds to 

saturations that vary from 23 to 88 percent, a range that does 

not adequately describe the dry end of the q(y) curve. Using 

a q(y) model that extends from 100 to 0 percent saturation 

avoids this problem. Additionally, developing multiple linearregression equations for the q(y) model parameters saves the 

extra step of having to fit the estimated q(y) points in order to 

use these in unsaturated-flow simulations.

The Rossi-Nimmo (1994) junction model consists of 

three functions joined at two points (fig. 5): 

1. Parabolic function for the wet range of y,

2. Power law function (Brooks and Corey, 1964) for the 

middle range of y, and 

3. Logarithmic function for the dry range of y. 

This model has two independent parameters: (1) the 

scaling factor for y (yo), and (2) the curve-shape parameter 

(l). Sometimes, yo is associated with y at which air first 

enters a porous material during desaturation (referred to 

as “air-entry pressure”), but, actually, air begins displacing 

water in the largest pores at a higher (less negative) y than 

yo as evidenced by the departure of q from saturation earlier 

than yo on the q(y) curve (fig. 5). In this study, the units of 

yo are expressed in centimeters of water (cm-water). The 

curve-shape parameter l indicates the relative steepness of the 

middle portion of the q(y) curve, described by the power-law 

function. Larger l values cause the drainage portion of the 

q(y) curve to appear steeper. Unlike the model of Brooks and 

Corey, which holds q fixed between y = 0 and the “air-entry 

pressure”, the junction model produces a smooth curve near 

saturation, represented by a parabolic function, that allows the 

pore-size distribution (the first derivative of the q(y) curve) to 

be represented more realistically. 

The parabolic function applies for yi £ y £ 0, and is 

represented by:

q

q

y

sat yo

= -c

æ

è

ç

ç

çç

ö

ø

÷

÷

÷

÷ 1

2

, (1)

where 

qsat is expressed volumetrically and 

c is a dimensionless constant calculated from an 

analytical function involving the parameter l, 

which also is dimensionless. 

The power law function applies for yj £ y £ yi, and is 

represented by:

q

q

y

y

l

sat

= o æ

è

ç

ç

ç

ö

ø

÷

÷

÷÷ , (2)

The logarithmic function applies for yd £ y £ yj, and is 

represented by:

q

q

y

sat y = d æ

è

ç

ç

ç

ö

ø

÷

÷

÷÷ Aln

, (3)

where A is the slope of the dry-range part of the q(y) curve 

on a semilog scale, calculated from a combination of l, yo, 

and yd. The junction points (yi and yj) are calculated from 

analytical functions of yo and l.

Optimized values of yo and l for each core sample were 

determined using a nonlinear regression program based on 

the modified Gauss-Newton least-squares method found in 

the Statistics Toolbox version 3 of MATLAB 6, Release 12 

(The Mathworks, Inc., Natick, MA). qsat can be treated as an 

optimized parameter in the Rossi-Nimmo (1994) junction 

model, however, its value was known by direct measurement, 

and it was treated as a fixed parameter. yd also was treated as 

a fixed parameter because its value can be calculated from the 

Kelvin equation, using average laboratory temperature during 

oven drying of samples and the relative humidity of water 

at average oven and room temperatures (Ross and others, 

1991; Andraski, 1996). Under laboratory conditions of 50 

percent relative humidity and 22°C, and an oven temperature 

of 105°C, yd has a value near -8  ́ 106 cm-water. For 

convenience, a yd value of -1  ́ 107 cm-water was used in the 

model fits for all core samples.

Particle-Size Parameters

To apply multiple linear-regression techniques, discrete 

measurements of the explanatory variables used to predict the 

hydraulic parameters (qsat, yo, l, and Ksat) are needed. These 

explanatory variables include rbulk, rpart, and a representation of 

particle-size data. Because particle-size measurements spanned 

different size ranges and varied in resolution between data 

sets, the first step in representing particle-size data involved 

calculating cumulative particle-size distributions on a percentfiner-than basis. The cumulative particle-size distribution, like 

the q(y) curve, is nonlinear. Three alternative representations 

of the particle-size distribution were chosen for comparison 

in developing the PTMs (fig. 6): (1) textural classes (percent 

gravel, sand, silt, and clay), (2) geometric statistics (geometric 

mean particle diameter (Mg) and geometric standard deviation 

(sg)), and (3) graphical statistics (median particle diameter 

(d50) and uniformity coefficient (Cu)). 

12  Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
Figure 6. Example particle-size distribution showing three parameter sets (textural-class percentages, geometric 

statistics, and graphical statistics) used to develop property-transfer models for water retention and saturated 

hydraulic conductivity. 

ID20055114_fig06

CLAY

(< 0.002 MM) 

GRAVEL

(> 2 MM)

SILT

(0.002 -0.05 MM)

SAND

(0.05 - 

2 MM)

PERCENTAGE OF PARTICLES

FINER THAN INDICATED SIZE

PARTICLE DIAMETER, IN MILLIMETERS

B. Cumulative particle-size distribution showing the frequency of 

particles finer than a given particle diameter.

PARTICLE DIAMETER, IN MILLIMETERS

FREQUENCY, IN PERECENT

A. Differential frequency distribution showing the frequency of 

particles at a given particle diameter. 

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

10-3 10-2 10-1 100 101 10-4

0

10

20

30

40

50

60

70

80

90

100

10-4 10-3 10-2 10-1 100 101

d60

d50

d10

Mg

d50

sg

Cu = d60/d10

Mg Geometric mean particle diameter

sg Geometric standard deviation

d10 Diameter at the 10th percentile

d50 Median particle diameter

Measured data

Normal distribution

EXPLANATION

Shaded regions indicate 

particle-size limits defining 

the USDA textural classes 

(gravel, sand, silt, and clay).

d60 Diameter at the 60th percentile

Cu Uniformity coefficient

Property-Transfer Model Calibration and Approach 13
Textural-class percentages commonly are recorded in 

lithologic logs or are easily measured in the field or laboratory, 

requiring only a few measurements of particle size. Texturalclass percentages also have been used to develop simple PTMs 

for other sites or applications (Gupta and Larson, 1979; Cosby 

and others, 1984; Puckett and others, 1985; Wösten and van 

Genuchten, 1988; Schaap and others, 1998, 2001; Schaap, 

1999). For this reason, textural-class percentages were chosen 

as one way of representing the particle-size data. Weight 

percentages were calculated from the cumulative particlesize distributions using the definitions of gravel, sand, silt, 

and clay from the U.S. Department of Agriculture’s (USDA) 

soil classification system (Soil Survey Staff, 1975; fig. 6B). 

Linear interpolation between points on the cumulative particlesize distribution was required to determine the frequency of 

particles falling between the particle-size limits defined by this 

soil classification system.

Because particle-size distributions of sediments typically 

follow a lognormal distribution (Krumbein, 1938; Pettijohn, 

1975), descriptive statistical formulas, using the method of 

moments (Beyer, 1991, p. 470-471), were used to calculate Mg

and sg. Mg is a summary measure of the most abundant particle 

diameter in a distribution. For a distribution with skewness, or 

excess particles in one-half the distribution compared to the 

other, Mg is weighted by the asymmetry away from the mean 

of the normal distribution in the direction of excess particles 

(fig. 6A). Mg and sg together define a particle-size distribution 

having a single mode. Mg was calculated by the formula:

log

( )log ( )

( )

10

10

1

1

M

f d d

f d

g

i ci ci

i

k

i ci

i

( ) = k

[ ] =

=

å

å , (4)

where 

k is the number of bins, 

dci is the geometric center of the i

th bin, and 

f

i

(dci) is the frequency of sizes occurring within i

th bin 

assigned to dci. 

sg was calculated by:

log

log log

10

2 10 10

2

1 sg

i ci ci g

i

k

f d d M

( ) é

ë

ê ù

û

ú =

( ) ( )- ( ) é

ë

ê ù

û { ú } =

å

f d i ci

i

k

( ) =

å1 . (5)

The third representation of particle-size data uses 

graphical statistics, where particle diameters are determined at 

specific cumulative frequencies (fig. 6B). The midpoint of the 

cumulative particle-size distribution is the particle diameter 

at the 50th percentile, or d50. One disadvantage of using d50

is that it excludes the influence of distribution extremes, so 

skewed distributions are not well represented (Folk, 1980). 

A measure of the spread in particle sizes, referred to as the 

uniformity coefficient (Cu), is calculated as the diameter at 

the 60th percentile (d60) divided by the diameter at the 10th

percentile (d10). When the cumulative particle-size distribution 

is plotted on a log-log scale, these percentiles commonly lie in 

the part of the distribution that can be fit with a straight line, 

and hence, Cu describes the slope (Fetter, 1994). A small Cu

reflects a steep curve, or a narrow range of particle sizes. The 

diameter at any given percentile can be determined graphically 

by drawing a horizontal line from the percentile of interest, 

usually plotted on the y-axis, until it intersects the particle-size 

distribution. A vertical line, perpendicular to the horizontal 

line, is drawn until it meets the x-axis, where the particle 

diameter is then read. Because a large number of samples 

needed to be processed in this study, a computer program 

was used to calculate d10, d50, and d60 by linearly interpolating 

between the two points on the cumulative particle-size 

distribution nearest the percentile of interest. 

Multiple Linear Regression

Multiple linear-regression analyses were done using 

custom programs written with Matlab (The MathWorks, Inc., 

Version 6, Release 12) and the Statistics Toolbox utility. The 

basic steps of the multiple linear-regression procedure used 

in this study are outlined in figure 4. Hydraulic property data 

included measurements of qsat, Ksat, and curve-fit parameters 

(yo and l) for the q(y) measurements. Bulk-physical 

properties included rbulk, rpart, and particle-size parameters 

(textural-class percentages; Mg and sg; d50 and Cu). Hydraulic 

properties (referred to as response variables) were predicted 

from a linear combination of the bulk-physical properties, 

referred to as explanatory variables. 

The basic functional form for multiple linear 

regression is:

y b j ix e j n i

p

= +i j j = =

å0

, ,1 2,...

, (6)

where 

n is number of observations (note, in this 

application, n refers to the number of core 

samples with complete sets of measured 

hydraulic and bulk-physical properties), 

yj is observed value of the response variable for the 

j

th core sample, 

p is number of x (explanatory) variables including 

an intercept term, 

bi is a coefficient estimated by regression and 

associated with xi, and 

ej is prediction error term, or residual. 

14  Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
To calculate the regression equations, a complete set of 

hydraulic and bulk-physical-property measurements was 

necessary for each core sample; otherwise, the sample was 

removed from the analysis. The right-hand side of the equation 

that includes regression coefficients and explanatory variables 

is referred to as the regression variate. An intercept term (i = 

0) must be included by setting x0 = 1 for each core sample. 

For each core sample, ej is calculated by subtracting the 

estimated value of the response variable from the observed 

value. Positive residuals indicate that the response variable is 

underestimated by the regression variate. The least-squares 

procedure yields the best estimates for bi by minimizing the 

objective function: 

S b y b x j i ij

i

p

j

n

( ) = - æ

è

ç

ç

ç

ö

ø

÷

÷

÷÷ = =

å å 1 0

2

. (7)

The strength of the linear relation between a set 

of explanatory variables and a given response variable 

(goodness-of-fit) was assessed using the adjusted coefficient 

of determination (R2

adj), which accounts for the number of 

observations and the number of explanatory variables. R2

adj 

is directly related to the coefficient of determination (R2), 

defined as:

R

SS

SS SS

reg

res reg

2 = + , (8)

where 

SSres is the sum of squared residuals (equivalent to 

equation 7 using the known values of bi) and 

SSreg is the regression sum of squares. 

SSreg is calculated by:

SS b x

y

n reg i ij

j

j

n

i

p

j

n

= -

æ

è

ç

ç

ç

ç

ç

ç

ç

ç

çç

ö

ø

÷

÷

÷

÷

÷

÷

÷

÷

÷

÷

÷

=

= =

å

å å 1

1 0

2

, (9)

Then R2

adj is calculated by:

R n

n p adj R 2 2 1 1 = - 1 -- - ( )( )( )

. (10)

R2

adj measures the ability of the explanatory variables in the 

regression variate to explain variation in the response variable. 

A value of R2

adj equal to 1 indicates that the explanatory 

variables are able to explain all variation in the response 

variable. For comparing the ability of one regression variate 

(A) to explain variation in y over another regression variate 

(B), the percent change in R2

adj is calculated by 100(R2adj

(A)

- R2

adj 

(B)).

The root-mean-square error (RMSE), also referred to as 

the standard error of the estimate, is a measure of the variation 

in the predicted values. It can be used to differentiate between 

competing models of a single response variable. RMSE (in 

units of the response variable) is calculated as:

RMSE

y y

n p

j j

j

n

=

-

-

=

å( ˆ )

( )

2

1

, (11)

where yˆj is the predicted value of the response variable 

(equivalent to the second term in equation 7). Smaller values 

of RMSE indicate that the explanatory variables of a particular 

regression variate are more successful in explaining y than 

those in competing models.

Statistical tests are used to make inferences about how 

representative the results of a regression analysis, based on 

a particular sample, are of the population. In this context, 

sample refers to the group of observations (hydraulic and 

bulk-physical properties). Significance testing involves stating 

a hypothesis, specifying a significance level (a) for the test 

(probability of finding a hypothesis true when, in fact, it is 

not), and determining whether the hypothesis is true or false. 

The hypothesis is rejected if the value of the test statistic 

exceeds a certain critical value, calculated using the a-level, 

number of observations, and number of explanatory variables. 

In this study, the significance level (a) was specified at 0.05. 

For the significance tests to be valid and to yield accurate 

results, the assumption of normally-distributed residuals must 

be met. 

For the t-test, the hypothesis states that the value of a 

particular regression coefficient is equal to zero. Partial tvalues were calculated by dividing the value of the coefficient 

by its standard error. The standard error denotes the expected 

variation of a coefficient across multiple samples of data 

from a given population. Smaller standard error values imply 

more reliable predictions (smaller confidence intervals for the 

coefficient). To determine whether a regression coefficient 

is significantly different from zero, the partial t-value is 

compared to the critical t-value, which is determined from the 

Student’s t-distribution for the number of observations and alevel specified. If the magnitude of the partial t-value exceeds 

the critical value, then the coefficient is significantly different 

from zero, indicating that the explanatory variable is useful in 

explaining the variation in the response variable. 

For purposes of predicting the response variable from a 

set of explanatory variables, hypothesis testing, and estimating 

confidence intervals, the assumptions of multiple linear 

regression are that 

1. Residuals are normally distributed, 

2. Response variable displays equal levels of variance 

across the range of values for the explanatory 

variables (homoscedasticity), 

Property-Transfer Model Calibration and Approach 15
3. Relation between the response variable and each 

explanatory variable is linear, and 

4. No correlated errors or systematic effects exist in the 

data. 

The assumption that the residuals are normally distributed 

must be met for valid hypothesis testing or calculation of 

confidence intervals. 

To determine whether the residuals were normally 

distributed, the residuals were plotted on a normal probability 

plot and the probability-plot correlation coefficient (PPCC) 

was calculated. PPCC measures the departure of the residual 

distribution from a straight line, which represents the normal 

distribution on a probability plot. A normal distribution would 

have a PPCC value of 1.0 and skewness of zero. Skewness 

was used as another means of assessing normality. Skewness 

measures the degree of asymmetry in a distribution, with 

negative skewness indicating a distribution with an excess 

of smaller values compared to the mean of the normal 

distribution. Skewness should be near zero for a residual 

distribution to be considered normal, and the PPCC value 

should be near or greater than the critical value for the number 

of observations comprising the distribution (Looney and 

Gulledge, 1985; Helsel and Hirsch, 1992). 

Residual plots were used to identify heteroscedastic 

trends (unequal variance in either the response variable or 

one of the explanatory variables) that would violate a basic 

assumption of multiple linear regression. A residual plot 

is constructed by plotting the residuals from a particular 

regression model against the predicted values of the response 

variable. More commonly the residual is studentized. This 

refers to a special form of standardization where the j

th

observation is omitted. Standardization is used to create a 

common scale for comparing the residuals from different 

models. Each residual is standardized by dividing its value 

by the standard deviation of residuals for all observations 

from a particular regression analysis. After standardization, 

the residual distribution has a mean value of 0 and a standard 

deviation of 1 (Hair and others, 1998). For n greater than or 

equal to 50, the studentized residual distribution approximates 

the Student’s t-distribution. Residuals whose absolute values 

exceed the critical t-value (typically near 2) are considered 

outliers. Heteroscedasticity exists if the pattern of residuals 

appears nonrandom, with either a nonlinear trend, clustering 

of points in a particular region, or both. Heteroscedasticity can 

be corrected, although sometimes not entirely, by transforming 

one or more explanatory variables or the response variable.

Linearity between response and explanatory variables 

was assessed by examining scatter plots and variable 

distributions, and by calculating Pearson’s correlation 

coefficient (r). Scatter plots of each response variable and each 

explanatory variable, in addition to Pearson’s r, were used to 

identify univariate trends in the data. Pearson’s r measures 

the correlation between two variables. It can vary from -1 to 

+1, with 0 indicating no correlation and ±1 indicating perfect 

correlation. 

While Pearson’s r is used to identify univariate trends in 

the data and to identify explanatory variables that exhibit high 

covariance, the variance inflation factor (VIF) is used to test 

for multicollinearity resulting from the combined effect of two 

or more variables in the regression variate. VIF indicates the 

effect of the other explanatory variables on the explanatory 

variable of interest; each explanatory variable (xi) is in turn 

regressed against the remaining explanatory variables to 

calculate Ri

2 (as defined in equation 8), where VIF = 1/(1-

Ri

2

). Large values of VIF, typically greater than 10, indicate 

a high degree of collinearity or multicollinearity among the 

explanatory variables examined (Hair and others, 1998). 

The existence of collinearity between explanatory 

variables invalidates hypothesis tests, for example if 

interpretation of estimated coefficients (bi) is important. 

However, if prediction is the main application of the 

regression models, then collinearity is acceptable as long 

as the prediction is made within the range of data values 

used in model development (Helsel and Hirsch, 1992). 

Multicollinearity makes it difficult to separate the effects 

among explanatory variables on the regression model, for 

example when stepwise procedures are used to iteratively 

select variables. Other effects of multicollinearity include 

a limited size to R2 (because the predictive power of each 

explanatory variable is shared or overlaps with other 

variables), incorrect magnitudes or signs of the coefficients, 

and instability in the coefficient values during the regression 

(Helsel and Hirsch, 1992; Hair and others, 1998).

Another assumption of multiple linear regression is that 

the predicted values of the response variable are not related 

to each other, or that the values are not sequenced by any 

particular explanatory variable. Sequencing often occurs when 

time- or event-based variables are included in the regression 

analysis. Plotting residuals against a given explanatory 

variable should show a random pattern if the predicted values 

are independent. A clustered or nonlinear pattern in this 

type of plot indicates error dependence, or that an essential 

explanatory variable has been omitted from the regression 

variate (Hair and others, 1998). Hydraulic and bulk-physical 

properties of core samples used to develop multiple linearregression equations in this study should be independent of 

seasonal fluctuations in moisture content and timing of core 

collection, therefore, the error terms for each regression model 

were considered independent without further testing of this 

assumption.

Property-Transfer Model Analyses

Multiple linear-regression techniques were used to 

develop two separate PTMs, one for q(y) and the other for 

Ksat. The q(y) PTM involved a separate regression equation 

for each of the three parameters (qsat, yo, and l) that define 

the Rossi-Nimmo (1994) junction model. For each of the four 

16  Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
hydraulic properties (qsat, yo

, l, and Ksat), three models were 

examined to identify the best set of particle-size parameters to 

use as explanatory variables. Model A refers to the regression 

variate that includes rbulk, rpart, and textural-class percentages 

(gravel, sand, silt, and clay). Model B includes rbulk, rpart, Mg, 

and sg as explanatory variables. Model C includes rbulk, rpart, 

d50, and Cu. 

The minimum, maximum, average, and standard 

deviation of the hydraulic properties (qsat, yo, l, and Ksat), 

(table 2), and bulk-physical properties (rbulk, rpart, and various 

particle-size parameters), (table 3), are summarized for each 

data source. Although y is a negative pressure measured 

relative to atmospheric pressure, absolute y values were 

used to develop the regression equations for yo. Textural 

classes (gravel, sand, silt, and clay) are reported as weight 

percentages, or the weight of material falling within a given 

range of particle diameters divided by the total weight of the 

material analyzed. Particle-size limits for each textural class 

are defined according to the U.S. Department of Agriculture’s 

soil classification system (Soil Survey Staff, 1975). Ninety-one 

out of 108 samples were classified as sands or loams, with 38 

samples classified as silt loams. The texture of sample USGS 

118 at 173.48 to 174.39 m from McElroy and Hubbell (1990) 

could not be determined due to incomplete particle-size data. 

The Magnuson (written commun., 2002) data set included the 

finest-textured samples, with 17 samples classified as clay 

loams, silty clay loams, silty clays, and clays. Because gravelsized particles (> 2 mm) were removed prior to measuring 

q(y) of the Magnuson (written commun., 2002) samples, the 

particle-size distributions of this data set were adjusted to zero 

percent gravel before calculating the particle-size parameters 

used to develop the q(y) PTMs. In table 3, two summaries 

of the particle-size parameters are reported for Magnuson 

(written commun., 2002), one for use in developing the Ksat

PTM and the other for developing the q(y) PTM. 

Evaluation of Calibration Data

During the multiple linear-regression analyses, data were 

thoroughly examined for typographical errors, measurement 

errors, or differences between groups of core samples (arising 

from differences in measurement technique, collection 

method, or sample handling). In some cases, core samples 

were removed from the regression analyses due to missing or 

incomplete data. 

Effect of Repacked Core Samples

For the Magnuson (written commun., 2002) data set, q(y)

measurements were completed on disturbed samples (after 

removing gravel and repacking). For the other data sets used 

in this study, q(y) curves were measured on intact samples. 

Although qsat was not measured on the Magnuson (written 

commun., 2002) samples, the repacking effect was observed 

by comparing the total porosity of the repacked samples (taken 

as the qsat values for these samples) with the total porosity 

calculated from the intact samples used for measurement of 

Ksat. Measured qsat values should be less than or equal to the 

values of total porosity calculated for intact core samples. For 

the Magnuson (written commun., 2002) samples, total porosity 

of the repacked samples exceeded total porosity of the intact 

samples by 25 percent or more for 37 out of 51 samples. In 

contrast, only 3 out of 58 of the intact samples from the other 

data sets had measured qsat values that exceeded total porosity 

by 25 percent. Because the interbed samples are deep and 

consolidated, repacking “fluffs” the material, creating a larger 

mean pore size compared to an intact sample. As a result, the 

repacked rbulk will be smaller than the rbulk of the intact core 

sample. The repacked total porosity is then larger than that of 

the intact sample.

“Fluffing” the material by repacking also should affect 

yo and l values because the sample mean pore size has been 

increased. From capillary theory, y is inversely related to 

the pore size at which drainage occurs. An increase in mean 

pore size as a result of repacking suggests that the absolute 

value of yo would be smaller than for an identical, intact 

sample, with drainage occurring “earlier” on the q(y) curve. 

A more uniform pore-size distribution is expected for a 

repacked sample because the material has been homogenized, 

destroying both large- and small-sized pores in the process. 

This would be reflected in the q(y) curve by a steeper drainage 

slope or a larger l value. 

The Mann-Whitney rank-sum test (Zar, 1996) was used 

to determine if the medians of the distributions of qsat, yo, 

and l for repacked (S. Magnuson, written commun., 2002) 

and intact (McElroy and Hubbell, 1990; Perkins and Nimmo, 

2000; Perkins, 2003; Winfield, 2003) samples were equal. For 

qsat, the calculated test statistic, Zc, was 3.191, compared to 

the critical two-tailed value of 1.645, using the t-distribution 

approximation to a normal distribution with infinite degrees 

of freedom and an a-level of 0.10. For yo, Zc was 2.930, and 

for l, Zc was 1.473. The rank-sum test indicated a statistical 

difference between repacked and intact samples for qsat and 

yo. Aside from the way samples were pretreated before 

experimental analyses, this difference could be due to the high 

clay content of the Magnuson samples (table 3) compared 

to the other data sets. However, the test was repeated after 

excluding 17 samples classified as clay loams, silty clay 

loams, silty clays, and clays from the Magnuson data set. 

Results showed that the medians of the qsat and yo distributions 

between intact and repacked samples were again dissimilar, 

suggesting that repacking has a stronger effect on the 

distributions of qsat and yo than percent clay does. For l, the 

distributions of repacked and intact samples were found to be 

dissimilar after excluding the samples with high clay content, 

indicating that repacking also affects l.

Property-Transfer Model Analyses 17
Statistic Saturated water content,

qsat (cm3/cm3)

Scaling parameter for 

matric pressure, 

yo (-cm-water)

Curve-shape 

parameter, 

l

Saturated hydraulic 

conductivity, 

Ksat (cm/s)

Magnuson, written commun. (2002)1

Minimum 0.4427 0.06 0.0820 1.14×10-7

Maximum .7333 241.77 .6568 8.41×10-2

Average .5098 45.73 .2380 4.03×10-3

Standard deviation .0482 54.63 .1292 1.32×10-2

Number of samples 51 51 51 41

McElroy and Hubbell (1990)

Minimum 0.3481 7.97 0.0853 1.13×10-8

Maximum .6290 3,499.98 .7082 6.73×10-3

Average .5191 295.56 .2340 1.33×10-3

Standard deviation .0814 831.84 .1836 2.20×10-3

Number of samples 16 16 16 16

Perkins and Nimmo (2000)

Minimum 0.3473 3.13 0.1066 2.35×10-7

Maximum .4497 556.67 1.1850 3.90×10-3

Average .4083 150.55 .3056 8.73×10-4

Standard deviation .0326 179.37 .2607 1.62×10-3

Number of samples 18 18 18 18

Perkins (2003)

Minimum 0.4516 2.12 0.0765 1.66×10-7

Maximum .5773 399.93 .5312 1.42×10-3

Average .5075 67.79 .1742 2.78×10-4

Standard deviation .0371 106.36 .1249 3.81×10-3

Number of samples 14 14 14 12

Winfield (2003)

Minimum 0.3585 16.65 0.0950 2.07×10-7

Maximum .5661 1,207.78 .2593 8.31×10-4

Average .4796 430.15 .1681 8.88×10-5

Standard deviation .0531 334.98 .0543 5.34×10-3

Number of samples 10 10 10 10

Total number of samples, n 109 109 109 97

1Water-retention measurements were completed on samples that were repacked after removing gravel (particle sizes greater than 2 millimeters).

Table 2. Statistics for hydraulic data used to develop property-transfer models, listed by data source.

[Abbreviations: cm3/cm3, cubic centimeter per cubic centimeter; cm-water, centimeter of water; cm/s, centimeter per second]

18  Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
Statistic

Bulk 

density, 

rbulk

(g/cm3)

Particle 

density,

rpart

(g/cm3)

Gravel

(percent)

Sand

(percent)

Silt

(percent)

Clay

(percent)

Geometric 

mean particle 

diameter,

Mg

(mm)

Geometric 

particle-size 

standard 

deviation,

sg

Median 

particle 

diameter, d50

(mm)

Uniformity 

coefficient, 

Cu

Magnuson, S., written commun. (2002)1

Minimum 1.06 2.04 0.00 0.89 3.33 1.76 0.0013 2.76 0.0009 2.94

Maximum 1.97 2.66 74.70 91.05 70.07 63.81 2.4070 71.43 5.5197 2,254.05

Average 1.58 2.42 11.79 34.77 33.23 20.20 .2280 10.70 .4035 85.85

Standard deviation .21 .13 18.04 25.14 18.95 18.35 .5154 11.16 1.1436 311.74

Number of samples 51 51 51 51 51 51 51 51 51 51

Magnuson, S., written commun. (2002), adjusted2

Minimum 1.06 2.04 0.00 0.89 3.92 2.17 0.0012 0.00 0.0007 2.86

Maximum 1.97 2.66 .00 92.92 70.07 81.40 .2427 14.76 .6163 257.62

Average 1.58 2.42 .00 41.92 36.25 21.83 .0568 6.32 .0825 30.78

Standard Deviation .21 .13 .00 29.25 17.87 19.18 .0678 2.84 .1216 42.50

Number of samples 51 51 51 51 51 51 51 51 51 51

McElroy and Hubbell (1990)

Minimum 1.18 2.52 0.00 21.83 2.84 0.12 0.0133 1.91 0.0115 2.59

Maximum 1.94 2.89 .26 96.91 67.36 15.75 .2363 7.78 .2345 79.44

Average 1.44 2.63 .03 61.37 32.12 6.50 .0708 4.27 .0820 20.45

Standard deviation .20 .08 .07 24.11 21.10 5.08 .0720 1.59 .0699 24.01

Number of samples 16 16 15 15 15 15 15 15 15 15

Perkins and Nimmo (2000)

Minimum 1.22 2.65 0.00 0.04 2.37 0.14 0.0065 2.14 0.0082 2.77

Maximum 1.63 2.65 6.81 97.25 82.88 17.83 .4267 8.02 .4395 32.95

Average 1.42 32.65 .82 34.73 53.31 9.97 .0881 4.22 .0995 13.61

Standard deviation .09 .00 1.92 33.03 29.62 6.04 .1328 1.37 .1437 7.34

Number of samples 18 18 18 18 18 18 18 18 18 18

Perkins (2003)

Minimum 1.11 2.60 0.00 11.86 7.76 1.29 0.0094 2.71 0.0108 3.30

Maximum 1.46 2.75 .02 90.97 74.13 13.99 .1992 9.42 .3970 118.17

Average 1.31 2.63 .00 59.20 34.54 6.28 .0856 5.75 .1361 32.70

Standard deviation .10 .04 .00 24.97 20.95 4.20 .0579 2.08 .1145 33.33

Number of samples 14 14 14 14 14 14 14 14 14 14

Winfield (2003)

Minimum 1.18 2.74 0.00 0.00 61.43 9.91 0.0039 3.01 0.0046 6.98

Maximum 1.63 2.84 .00 25.20 80.80 25.08 .0160 4.67 .0183 16.40

Average 1.41 2.78 .00 9.00 73.59 17.41 .0083 3.87 .0097 10.95

Standard deviation .12 .03 .00 8.85 5.45 4.44 .0036 .55 .0042 2.83

Number of samples 10 10 10 10 10 10 10 10 10 10

Total number of 

samples, n

109 109 108 108 108 108 108 108 108 108

1 Core sample properties were used to develop the saturated hydraulic conductivity property-transfer model.

2 Adjusted samples: water-retention measurements were completed on samples that were repacked after removing gravel (particle sizes greater than 2 mm). 

Particle-size distributions were normalized to zero gravel content prior to calculating parameters. These core sample properties were used to develop the waterretention property-transfer model.

3 The average particle density of 2.65 g/cm3

, for measurements on 4 core samples, was used for all 18 samples.

Table 3. Statistics for bulk physical-property data used to develop property-transfer models, listed by data source.

[The uniformity coefficient is defined as the particle diameter at the 60th percentile divided by the diameter at the 10th percentile, calculated from the cumulative 

particle-size distribution on a percent-finer-than basis. Abbreviations: g/cm3, gram per cubic centimeter; mm, millimeter]

Property-Transfer Model Analyses 19
Including the repacked samples in preliminary regression 

analyses had a pronounced effect on the results compared to 

analyses that excluded these samples. The goodness-of-fit 

values, indicated by R2

adj, decreased by 30.9 to 31.2 percent 

for models A, B, and C for qsat, by 22.8 to 39.6 percent for yo, 

and by 4.2 to 34 percent for l. Note that for the Magnuson 

data set, Mg, sg, d50, and Cu were determined on particle-size 

distributions reconstituted without gravel (to match the way 

the q(y) measurements were completed), although rbulk values 

included gravel. To adjust the measured rbulk values to zero 

percent gravel, the rbulk of the gravel fraction would have to 

be known (Andraski, 1996). As a result of these findings, 

the Magnuson data were excluded from the development of 

the q(y) PTMs. However, this data set was included in the 

development of the Ksat PTM because Ksat and rbulk were 

measured on intact core samples.

Errors in Fitted, Calculated, or Measured 

Parameters

Complete core sample measurement sets were needed to 

do the regression analyses for qsat, yo, l, and Ksat,. Because 

some core samples had missing values, the number of core 

samples available for developing the PTMs was reduced. 

One sample from McElroy and Hubbell (1990), USGS 118 

at 173.48 to 174.39 m, had only partial particle-size data. 

Because the particle-size parameters for this sample could not 

be calculated, this sample was excluded from all regression 

analyses, reducing the number of samples for developing the 

q(y) PTMs from 109 to 108 (table 4). Ksat measurements 

were not completed for 10 samples from Magnuson (written 

commun., 2002) and two samples from Perkins (2003). The 

total number of samples for developing the Ksat PTM was 

reduced from 109 to 96, accounting for the sample with 

missing particle-size data and the 12 samples with missing Ksat

values (table 4). 

Borehole 

identification 

No.

Depth 

interval 

(m)

Water retention (q(y)) parameters Saturated hydraulic 

conductivity, 

log(Ksat) Saturated water content, 

qsat

Scaling parameter for 

matric pressure, 

log(yo)

Curve-shape parameter, 

log(l)

Model A Model B Model C Model A Model B Model C Model A Model B Model C Model A Model B Model C

Magnuson, S., written commun. (2002)

RWMC-SCI-V-153 32.74–33.54 H H H

RWMC-SCI-V-154 33.99–34.30 H H H

RWMC-SCI-V-157 71.04–71.19 H H H

RWMC-SCI-V-159 72.56–72.71 H H H

RWMC-SCI-V-160 73.84–73.99 H H H

SOUTH-SCI-V-012 35.98–36.13 H H H

SOUTH-SCI-V-013 71.98–72.13 H H H

SOUTH-SCI-V-014 35.18–35.37 H H H

36.40–36.59 H H H

SOUTH-SCI-V-018 74.21–74.39 H H H

McElroy and Hubbell (1990)

USGS 118 173.48–174.39 B B B B B B B B B B B B

Perkins (2003)

ICPP-SCI-V-215 45.53–45.59 H H H

45.59–45.65 H H H

Total number of samples 109 109 109 109 109 109 109 109 109 109 109 109

Number of samples with missing 

data

1 1 1 1 1 1 1 1 1 13 13 13

Adjusted total number of samples, n 108 108 108 108 108 108 108 108 108 96 96 96

Table 4. Core samples removed from multiple linear-regression analyses due to missing data.

[Model A, regression model with textural class percentages and bulk density as explanatory variables; Model B, regression model with geometric mean particle 

diameter, geometric standard deviation, and bulk density as explanatory variables; Model C, regression model with median particle diameter, uniformity 

coefficient, and bulk density as explanatory variables. Abbreviations: m, meter; B, Core sample removed due to error in one or more bulk physical-properties, 

typically a particle-size parameter; H, Core sample removed due to error in the hydraulic property value indicated]

20 Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
Figure 7. Textural classification of core samples (based on the system 

of the U.S. Department of Agriculture) used to develop a property-transfer 

model for estimating water retention (q(y)). 

Because errors in hydraulic and bulk-physical properties 

can cause the regression models to be strongly influenced 

by a few points, the values of the explanatory and response 

variables were carefully examined. Errors can be due to 

sample collection, handling, measurement technique, and 

measurement resolution. Errors in sample collection and 

handling can adversely affect the rbulk of a core sample, 

which needs to be representative of a minimally disturbed 

sample. Sample disturbance also affects the hydraulic 

properties, particularly those most sensitive to the structure 

of the medium, such as qsat or Ksat. The resolution of the q(y)

measurements can affect the quality of the Rossi-Nimmo 

(1994) junction curve fits and cause errors in the optimized 

curve-fit parameters, yo and l. The textural, geometric, and 

graphical parameters calculated from particle-size distributions 

also are sensitive to the resolution and range of measurements. 

Measurements of rbulk and rpart were examined for errors 

by dividing the values into each of the nine USDA textures 

represented by the data (fig. 7A). rbulk is a complex property 

affected by multiple factors, such as depth (or amount of 

overburden pressure), texture, particle arrangement, and 

aggregation or macropores. The influences of these factors, 

in addition to the possible effects of sample disturbance 

during core collection and subsequent handling, on rbulk

are difficult to separate. Because natural variation in rbulk is 

expected, no values of rbulk seemed unusually high or low 

for a given texture. Values of rpart ranged from 2.04 to 2.89 

g/cm3 (table 3). Minerals commonly found in the sedimentary 

interbeds beneath the RWMC include quartz, plagioclase, 

orthoclase, pyroxene, and calcite (Rightmire and Lewis, 

1987; Bartholomay, 1990; Reed and Bartholomay, 1994). The 

rpart values of these minerals range from 2.57 to 2.76 g/cm3. 

Compared to these values, the rpart values of the other data 

sets, and the rpart values of similarly textured samples, one 

value of rpart equal to 2.04 g/cm3 (for sample SOUTH-SCIV-015 at 38.81 to 39.02 m, classified as a clay) seemed too 

low; therefore, this sample was excluded from the regression 

analyses. 

Several samples were excluded from the q(y) and Ksat 

PTMs because their particle-size distributions did not extend 

to small enough diameters (or cumulative frequencies) to 

accurately calculate statistical and textural-class parameters. 

The resolution of the particle-size distribution also will affect 

the values of most particle-size parameters; however, this 

issue is not addressed in this report. To adequately define the 

distribution shape, it is important to have as many points at 

the particle-size distribution’s fine end as at the coarse end. 

Definition at the distribution’s fine end is needed to interpolate 

d10 (for calculating Cu) and the frequency at 0.002 mm (for 

determining percent clay and percent silt). If the cumulative 

frequency at the smallest measured particle size is greater than 

10 percent, then d10 can have significant error because this 

part of the cumulative distribution’s slope is unknown. 

ID20055114_fig07

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

CLAY

SANDY CLAY

SANDY

CLAY LOAM

SANDY LOAM

LOAMY SAND

SAND

CLAY

LOAM

SILTY

CLAY

SILTY CLAY

LOAM

SILT LOAM LOAM

SILT

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

CLAY

SANDY CLAY

SANDY

CLAY LOAM

SANDY LOAM

LOAMY SAND

SAND

CLAY

LOAM

SILTY

CLAY

SILTY CLAY

LOAM

SILT LOAM

LOAM

SILT

A. All potential core samples, except those 

with missing data (n = 108).

PERCENTAGE OF SAND

PERCENTAGE OF CLAY

PERCENTAGE OF SAND

B. Core samples used in model development, 

shown with the maximum number removed 

due to errors and missing data (n = 51). 

PERCENTAGE OF CLAY

RWMC: Magnuson 

(2002, written commun.)

RWMC: McElroy and 

Hubbell (1990)

RWMC: Perkins and Nimmo (2000)

VZRP: Perkins (2003)

VZRP: Winfield (2003)

EXPLANATION

DATA SOURCE

VZRP, Vadose Zone Research Park

RWMC, Radioactive Waste 

 Management Complex

Property-Transfer Model Analyses  21
If a significant percentage of particles is finer than the smallest 

measured particle diameter, then the width of the last bin at 

the distribution’s fine end, defined as the interval between 

the smallest measured diameter and zero, will be large, 

weighting Mg toward larger values and sg toward smaller 

values. The cumulative frequency at which deviations in Mg 

and sg became unacceptable was near the 10th percentile. 

Therefore, samples that did not have measurements on the 

cumulative particle-size distribution that extended to at least 

the 10th percentile were excluded from the multiple linearregression analyses. Samples that did not have particle-size 

measurements extending to at least 0.002 mm were excluded 

because percent clay and percent silt could not be determined. 

Four McElroy and Hubbell (1990) samples (USGS 118 at 

173.48 to 174.39 m, D-02 at 70.91 to 71.41 m, D-02 at 71.41 

to 71.71 m, and D-15 at 33.11 to 33.84 m) were excluded from 

model A due to lack of particle-size data in the clay range 

(down to 0.002 mm). For models B and C, three samples 

from this data set (USGS 118 at 173.48 to 174.39 m, D-15 

at 33.11 to 33.84 m, and D-15 at 69.94 to 71.16 m) were 

excluded because the size of the last particle-size bin was too 

large, causing errors in the calculated values of Mg, sg, and 

Cu. For the Magnuson data set, all particle-size measurements 

extended to small enough diameters to allow determination of 

percent silt and percent clay. However, 26 out of 51 samples 

were excluded from models B and C (table 5) due to poor 

definition at the fine end of the particle-size distribution, 

which affected the values of Mg, sg, and Cu. These 26 samples 

were classified as silt loams, clay loams, silty clay loams, silty 

clays, and clays. Among the Ksat models, these samples are the 

finest in texture compared to the remaining samples (fig. 8A); 

as a result, model A included more fine-textured samples than 

models B and C. 

Values of each hydraulic property (qsat, yo, l, and Ksat) 

were evaluated by examining their distributions within each of 

the nine USDA textures. A few anomalous values of Ksat and 

yo were identified this way. One core sample, SOUTH-SCIV-011 at 32.29 to 32.44 m, from the Magnuson data set had a 

measured Ksat value of 1.24 x 10-3 cm/s, which is too large for 

a sample containing 41 percent clay. This sample was removed 

from the final regression analyses for Ksat. Cracks in the 

sample or gaps between the sample and sample holder could 

have contributed to this large Ksat measurement. Swelling 

of a sample containing a large amount of clay would tend to 

lower Ksat. The maximum yo value of all 58 samples used for 

calibrating the q(y) models (sample D-15 at 33.11 to 33.84 m 

from the McElroy and Hubbell (1990) data set) was 3,499 cm 

for a sandy sample. Because this yo value is much too large 

for a sample of this texture, sample D-15 at 33.11 to 33.84 m 

was removed from the yo regression models. Values of yo are 

affected by the texture (particle-size distribution) and structure 

(particle arrangement, macropores) of the porous media, the 

influences of which are difficult to separate. For fine-textured 

samples and samples with a wide range of particle sizes, yo

will tend to be large. For sample D-15 at 33.11 to 33.84 m, 

the optimization of yo may have been weighted by a few 

erroneous points in the wet range of q. A mismatch of reported 

particle-size distribution with measured q(y) curve (McElroy 

and Hubbell (1990) listed the texture of this sample as ‘clay’) 

also could be the reason for this large yo value. 

To evaluate the qsat measurements, values for each core 

sample were compared to total porosity values calculated from 

rbulk and rpart. Values of qsat were converted to percent saturation 

by dividing by total porosity and multiplying by 100. Values of 

percent saturation exceeding 125 percent were removed from 

the regression analyses for qsat. This conservative threshold 

was selected to keep most samples in the analyses, but exclude 

samples with unreasonably high values of saturation compared 

to total porosity. Three core samples from the McElroy and 

Hubbell (1990) data set (from boreholes 88-01D, D-10, and 

USGS 118) had qsat values that met this criterion. Because the 

rbulk values seemed reasonable for these three samples, ranging 

from 1.29 to 1.57 g/cm3, the high qsat values could be due to 

sample disturbance (although the Ksat values of these samples 

are reasonable) or to saturation in excess of total porosity. As 

mentioned above, qsat was not measured for the Magnuson 

(written commun., 2002) core samples. For this reason, and 

due to repacking effects on the Rossi-Nimmo (1994) junction 

model parameters, all 51 of the Magnuson (written commun., 

2002) samples were excluded from the q(y) PTMs. 

The l parameter approximates the slope of the drainage 

portion of the q(y) curve on a semilog scale, with larger values 

typically associated with coarse-textured materials or materials 

with a narrow range of particle sizes. The maximum value of 

l was 1.19 for a sandy sample (UZ98-2 at 43.09 to 43.21 m 

from Perkins and Nimmo (2000)) and the minimum was 0.077 

for a sandy loam sample (ICPP-SCI-V-215 at 46.10 to 46.20 m 

from Perkins (2003)). Because no upper limit on l exists, 

there is no reason to omit the 1.19 value from the regression 

analyses. The minimum value is low for a sandy loam material 

but this could be due to the way the particles are arranged 

in the sample, or the particular proportions of sand, silt, and 

clay. The values of l and yo are affected by the number and 

quality of measurements defining the q(y) curve. For sample 

ICPP-SCI-V-215 at 46.10 to 46.20 m, most of the measured 

points are clustered between 20 and 400 cm-water. Lack of 

q(y) points in the dry range makes it difficult for the RossiNimmo (1994) junction model to find an inflection point that 

would cause the curve fit to have a steeper drainage slope 

(corresponding to a larger l value). However, no core samples 

were excluded from the multiple linear-regression analyses for 

having anomalous l values.

22  Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
Table 5. Core samples removed from multiple linear-regression analyses due to errors in fitted, calculated, or measured parameters.

[Model A, regression model with textural class percentages and bulk density as explanatory variables; Model B, regression model with geometric mean particle 

diameter, geometric standard deviation, and bulk density as explanatory variables; Model C, regression model with median particle diameter, uniformity 

coefficient, and bulk density as explanatory variables. Abbreviations: m, meter; B, core sample removed due to error in one or more of its bulk physicalproperties, typically a particle-size parameter; H, core sample removed due to error in the hydraulic property value indicated]

Borehole 

identification 

no.

Depth 

interval 

(m)

Water retention (q(y)) parameters Saturated hydraulic 

conductivity, 

log(Ksat) Saturated water content, 

qsat

Scaling parameter for 

matric pressure, 

log(yo)

Curve-shape parameter, 

log(l)

Model A Model B Model C Model A Model B Model C Model A Model B Model C Model A Model B Model C

Magnuson, written commun. (2002)

RWMC-SCI-V-153 30.79–31.59 H H H H H H H H H

31.98–32.16 H B, H B, H H B, H B, H H B, H B, H B B

32.74–33.54 H B, H B, H H B, H B, H H B, H B, H

RWMC-SCI-V-154 30.34–31.55 H H H H H H H H H

32.16–32.32 H B, H B, H H B, H B, H H B, H B, H B B

33.99–34.30 H B, H B, H H B, H B, H H B, H B, H

RWMC-SCI-V-155 67.99–68.14 H H H H H H H H H

RWMC-SCI-V-156 30.95–31.25 H H H H H H H H H

RWMC-SCI-V-157 69.51–69.66 H B, H B, H H B, H B, H H B, H B, H B B

71.04–71.19 H B, H B, H H B, H B, H H B, H B, H

RWMC-SCI-V-158 29.73–29.94 H H H H H H H H H

RWMC-SCI-V-159 68.17–68.35 H B, H B, H H B, H B, H H B, H B, H B B

69.82–70.27 H B, H B, H H B, H B, H H B, H B, H B B

72.56–72.71 H B, H B, H H B, H B, H H B, H B, H

72.71–72.87 H B, H B, H H B, H B, H H B, H B, H B B

RWMC-SCI-V-160 68.29–68.45 H H H H H H H H H

72.29–72.74 H H H H H H H H H

73.84–73.99 H B, H B, H H B, H B, H H B, H B, H

SOUTH-MON-A-009 40.64–40.82 H H H H H H H H H

43.29–43.75 H H H H H H H H H

46.01–46.19 H H H H H H H H H

SOUTH-SCI-V-011 29.88–31.89 H H H H H H H H H

32.29–32.44 H B, H B, H H B, H B, H H B, H B, H H B, H B, H

SOUTH-SCI-V-011 33.54–34.15 H B, H B, H H B, H B, H H B, H B, H B B

71.04–71.22 H H H H H H H H H

SOUTH-SCI-V-012 33.54–33.96 H H H H H H H H H

35.98–36.13 H B, H B, H H B, H B, H H B, H B, H

37.68–38.11 H H H H H H H H H

SOUTH-SCI-V-013 29.45–29.88 H H H H H H H H H

69.70–71.55 H H H H H H H H H

71.04–71.22 H H H H H H H H H

71.98–72.13 H B, H B, H H B, H B, H H B, H B, H

SOUTH-SCI-V-014 31.40–31.65 H H H H H H H H H

35.18–35.37 H B, H B, H H B, H B, H H B, H B, H

36.40–36.59 H B, H B, H H B, H B, H H B, H B, H

73.78–73.96 H H H H H H H H H

74.21–74.39 H B, H B, H H B, H B, H H B, H B, H B B

SOUTH-SCI-V-015 35.67–35.88 H H H H H H H H H

36.89–37.10 H B, H B, H H B, H B, H H B, H B, H B B

38.81–39.02 H B, H B, H H B, H B, H H B, H B, H B B B

70.73–70.95 H H H H H H H H H

75.09–75.30 H B, H B, H H B, H B, H H B, H B, H B B

77.53–77.74 H B, H B, H H B, H B, H H B, H B, H B B

Property-Transfer Model Analyses  23
Core samples that were removed from models A, B, and 

C for qsat, yo, l, and Ksat due to measurement errors, errors in 

fitted parameters, or errors in calculated parameters are listed 

in table 5. Samples removed due to errors in a measured or 

fitted hydraulic parameter are indicated with an ‘H’, while 

samples removed due to errors in a bulk property are indicated 

with a ‘B’. The adjusted total number of samples reflects the 

number of samples removed after subtracting from the original 

109 samples. For the q(y) PTM, textural classification of core 

samples is shown in figure 7A (before excluding samples, 

n = 108; one sample had missing particle-size data and its 

texture could not be determined), and in figure 7B (after 

removing the maximum number of samples with errors and 

missing data, n = 51). For the Ksat PTM, textural classification 

of core samples is shown in figure 8A (before excluding 

samples with errors, n = 96; 13 samples had missing data) and 

in figure 8B (after removing the maximum number of samples 

with errors and missing data, n = 78). 

Table 5. Core samples removed from multiple linear-regression analyses due to errors in fitted, calculated, or measured parameters.—Continued

[Model A, regression model with textural class percentages and bulk density as explanatory variables; Model B, regression model with geometric mean particle 

diameter, geometric standard deviation, and bulk density as explanatory variables; Model C, regression model with median particle diameter, uniformity 

coefficient, and bulk density as explanatory variables. Abbreviations: m, meter; B, core sample removed due to error in one or more of its bulk physicalproperties, typically a particle-size parameter; H, core sample removed due to error in the hydraulic property value indicated]

Borehole 

identification 

no.

Depth 

interval 

(m)

Water retention (q(y)) parameters Saturated hydraulic 

conductivity, 

log(Ksat) Saturated water content, 

qsat

Scaling parameter for 

matric pressure, 

log(yo)

Curve-shape parameter, 

log(l)

Model A Model B Model C Model A Model B Model C Model A Model B Model C Model A Model B Model C

Magnuson, written commun. (2002)—Continued

SOUTH-SCI-V-016 37.96–38.11 H B, H B, H H B, H B, H H B, H B, H B B

41.55–41.77 H B, H B, H H B, H B, H H B, H B, H B B

45.21–45.43 H B, H B, H H B, H B, H H B, H B, H B B

74.70–74.85 H H H H H H H H H

SOUTH-SCI-V-016 75.76–75.91 H H H H H H H H H

SOUTH-SCI-V-018 68.69–68.87 H H H H H H H H H

71.04–71.25 H H H H H H H H H

74.21–74.39 H B, H B, H H B, H B, H H B, H B, H

McElroy and Hubbell (1990)

88-01D 170.15–71.13 H H H

D-02 70.91–71.41 B B B B

71.41–71.71 B B B B

D-10 110.58–10.98 H H H

D-15 33.11–33.84 B B B B, H B, H B, H B B B B B B

69.94–71.16 B B B B B B B B

USGS 118 68.60–70.12 H H H

Perkins (2003)

ICPP-SCI-V-215 45.53–45.59 H H H

45.59–45.65 H H H

Total number of samples (excluding 

missing)

108 108 108 108 108 108 108 108 108 96 96 96

Number of samples removed due to 

errors

57 56 56 54 53 53 54 53 53 5 18 18

Adjusted total number of samples, n 51 52 52 54 55 55 54 55 55 91 78 78

1Two core samples were analyzed from the depth interval indicated. Samples indicated by McElroy and Hubbell (1990) as duplicates were removed.

24  Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
Linearity Between Explanatory and Response 

Variables

Linearity between explanatory and response variables 

is an important assumption in applying the multiple 

linear-regression approach. Multiple linear regression is a 

correlation-based method, where measuring the strength of 

relations between variables assumes linearity. The strength 

of these relations will be underestimated if nonlinearity is 

present. The most direct way of correcting nonlinearity is by 

transforming one or both of the variables (Hair and others, 

1998). Data transformations can improve the correlation 

between variables and correct violations of multivariate 

assumptions. Linearity was evaluated by examining scatter 

plots, using Pearson’s r, and examining residual plots from 

preliminary regression analyses. Nonlinear trends identified 

in the scatter plots and residual plots were corrected by 

transforming the explanatory variable, the response variable, 

or both.

By examining scatter plots (not shown), and from the 

Pearson’s r values (tables 6 and 7), several explanatory 

variables (Mg, sg, d50, Cu, and percent gravel) appeared to 

vary nonlinearly with the response variables (qsat, yo, l, and 

Ksat). Based on the curvature of the relations between the 

explanatory and response variables, transformations such 

as the square root, inverse, or logarithm were selected as 

candidates for correcting the nonlinearity (Hair and others, 

1998). Because the values of Mg, d50, and Cu spanned 

approximately two orders of magnitude for the q(y) models, 

and more than three orders of magnitude for the Ksat models, 

a logarithmic transformation was chosen to establish linearity. 

The spread in particle sizes is described by both sg and Cu. 

Although sg varied by less than an order of magnitude for the 

q(y) and Ksat models, a logarithmic transformation also was 

applied to sg to provide a better comparison with log(Cu) in the 

comparison of competing regression models. Percent gravel 

varied by less than one order of magnitude. By definition, the 

textural-class percentages are constrained to vary between 

zero and 100 percent; the other particle-size parameters have 

no such lower and upper limits. Several core samples had zero 

gravel content, causing nonlinear patterns in the scatter plots. 

Transformation of percent gravel using the logarithm, inverse, 

or square root was not possible due to the zero values. 

After transforming Mg, sg, d50, and Cu, preliminary 

regression analyses were done and residuals for models 

A, B, and C were examined for each response variable. If 

residuals displayed heteroscedasticity, then the response 

variable was transformed. This procedure typically corrected 

all nonlinear effects in the scatter plots and in the residual 

plots. The response variables Ksat, yo, and l showed nonlinear 

relations with the explanatory variables. The logarithmic, Figure 8. Textural classification of core samples (based on the system 

of the U.S. Department of Agriculture) used to develop a property-transfer 

model for estimating saturated hydraulic conductivity (Ksat).

ID20055114_fig08

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

CLAY

SANDY CLAY

SANDY

CLAY LOAM

SANDY LOAM

LOAMY SAND

SAND

CLAY

LOAM

SILTY

CLAY

SILTY CLAY

LOAM

SILT LOAM

LOAM

SILT

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

CLAY

SANDY CLAY

SANDY

CLAY LOAM

SANDY LOAM

LOAMY SAND

SAND

CLAY

LOAM

SILTY

CLAY

SILTY CLAY

LOAM

SILT LOAM

LOAM

SILT

RWMC: Magnuson 

(2002, written commun.)

RWMC: McElroy and 

Hubbell (1990)

RWMC: Perkins and Nimmo (2000)

VZRP: Perkins (2003)

VZRP: Winfield (2003)

EXPLANATION

DATA SOURCE

VZRP, Vadose Zone Research Park

RWMC, Radioactive Waste 

 Management Complex

A. All potential core samples, except those 

with missing data (n = 96).

PERCENTAGE OF SAND

PERCENTAGE OF CLAY

B. Core samples used in model development, 

shown with the maximum number removed due to 

errors and missing data (n = 78). 

PERCENTAGE OF CLAY

PERCENTAGE OF SAND

Property-Transfer Model Analyses  25
square-root, and inverse transformations were tried based 

on the shape of the relation between Ksat, yo, and l and the 

explanatory variables observed in the scatter plots (Hair 

and others, 1998). Because transformation of the response 

variable changes the model units, transformed models 

cannot be compared directly using the R2

adj, RMSE, or other 

measures of model fit (Helsel and Hirsch, 1992). Instead the 

residuals were examined to determine the best transformation 

of Ksat, yo, and l to implement. Ksat varied over five orders 

of magnitude, yo varied over three, and l varied over one. 

The transformation providing the best residual behavior for 

Ksat and yo was the logarithmic transformation. After the 

logarithmic transformation was applied, values of Ksat and yo

spanned less than an order of magnitude and the linearity of 

Bulk 

density, 

rbulk

(g/cm3)

Particle 

density, 

rpart

(g/cm3)

Gravel

(percent)

Sand

(percent)

Silt

(percent)

Clay

(percent)

Geometric 

mean 

particle 

diameter, 

Mg (mm)

Geometric 

particle-size 

standard 

deviation, sg

Median 

particle 

diameter, 

d50 (mm)

Uniformity 

coefficient, 

Cu

Saturated water

content, qsat (cm3/cm3)

-0.568 -0.046 -0.247 0.283 -0.250 -0.256 0.016 -0.036 0.074 0.047

Scaling parameter for matric 

pressure, yo (-cm-water)

0.346 0.112 -0.097 -0.092 0.087 0.136 -0.091 -0.292 -0.134 -0.178

Curve-shape parameter, l -0.202 -0.057 0.221 0.490 -0.484 -0.505 0.721 -0.538 0.572 -0.364

Bulk density, rbulk (g/cm3) 1 -0.058 0.063 -0.268 0.233 0.334 -0.320 -0.034 -0.399 -0.125

Particle density, rpart (g/cm3) 1 -0.039 -0.460 0.435 0.479 -0.221 -0.289 -0.256 -0.310

Gravel (percent) 1 0.201 -0.276 -0.271

Sand (percent) 1 -0.992 -0.901

Silt (percent) 1 0.858

Clay (percent) 1

Geometric mean particle 

diameter, Mg (mm)

1 -0.229

Geometric particle-size 

standard deviation, sg

1

Median particle diameter, 

d50 (mm)

1 0.217

Uniformity coefficient, Cu 1

Table 6. Correlation matrix between water-retention parameters and bulk-physical properties, without transformation of selected variables.

[The statistical relation between parameters listed in the header and first column is indicated by Pearson’s correlation coefficient (r). Values of r can vary from 

-1 to +1, indicating parameters are either perfectly negatively or positively correlated. Values near zero indicate no correlation between parameters. Calculations 

were made using a subset of the original 109 core samples that excluded the samples with missing values and the Magnuson (2002) data, for a total of 57 

samples. Abbreviations: g/cm3, gram per cubic centimeter; mm, millimeter; cm3/cm3, cubic centemeter per cubic centimeter; cm-water, centimeter of water]

Ksat and yo with the explanatory variables greatly improved. 

For l, the inverse and logarithmic transformations produced 

the best linear patterns in the scatter plots, but nonlinearity 

was not entirely corrected. The studentized residual patterns 

were similar for the logarithmic and inverse transformations, 

typically with slight heteroscedasticity existing for smaller 

predicted l values. Applying the logarithmic transformation 

to l improved the linearity with the explanatory variables, 

corrected most of the heteroscedasticity in the residual plots, 

and provided the most normally distributed residuals. An 

added benefit of using the logarithm of l is that predicted 

values cannot be negative, which is a restriction on l in the 

Rossi-Nimmo (1994) junction model.

26  Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
The effect of transforming the variables is reflected in 

Pearson’s r values calculated before and after transformation 

by comparing tables 6 and 8 for the q(y) models and tables 

7 and 9 for the Ksat models. To simplify the discussion of 

correlation between variable pairs, the r values reported 

between each of the q(y) parameters and the explanatory 

variables were based on 57 core samples (excluding the 51 

Magnuson (written commun., 2002) samples and the sample 

with missing particle-size data from McElroy and Hubbell 

(1990)). The r values between Ksat and the explanatory 

variables were based on 96 core samples (excluding the 13 

samples with missing particle-size or Ksat values). Similar 

r values were obtained using the final number of samples 

included in models A, B, and C after removing erroneous 

values. After applying a logarithmic transformation to log(yo), 

the correlations with most textural-class parameters greatly 

improved. Prior to transformation, the r values between yo and 

percent sand, percent silt, and percent clay were -0.092, 0.087, 

0.136, respectively (table 6), reflecting the nonlinear relations 

observed in the scatter plots. The r values between log(yo) 

and percent sand, percent silt, and percent clay improved 

after transformation of yo (r = -0.645, 0.629, and 0.653, 

respectively, (table 8)). Similar improvements in r values were 

observed after applying the logarithmic transformation to other 

response variables (Ksat and l) and some of the explanatory 

variables (Mg, sg, d50, Cu). 

Bulk

density, 

rbulk

(g/cm3)

Particle 

density, 

rpart

(g/cm3)

Gravel

(percent)

Sand

(percent)

Silt

(percent)

Clay

(percent)

Geometric 

mean 

particle 

diameter, 

Mg (mm)

Geometric 

particle-size 

standard 

deviation, 

sg

Median 

particle 

diameter, 

d50 (mm)

Uniformity 

coefficient, 

Cu

Saturated hydraulic 

conductivity, Ksat

(cm/s)

-0.138 -0.032 0.326 0.132 -0.261 -0.148 0.250 0.127 0.210 0.077

Bulk density, rbulk (g/cm3) 1 -0.199 0.282 -0.311 0.013 0.379 0.216 0.177 0.231 0.039

Particle density, rpart

(g/cm3)

1 -0.160 -0.003 0.305 -0.408 -0.025 -0.328 -0.011 -0.155

Gravel (percent) 1 -0.045 -0.394 -0.213

Sand (percent) 1 -0.819 -0.662

Silt (percent) 1 0.381

Clay (percent) 1

Geometric mean particle 

diameter, Mg (mm)

1 0.178

Geometric particle-size 

standard deviation, sg

1

Median particle diameter, 

d50 (mm)

1 0.107

Uniformity coefficient, 

Cu

1

Table 7. Correlation matrix between saturated hydraulic conductivity and bulk-physical properties, without transformation of selected variables.

[The statistical relation between parameters listed in the header and first column is indicated by Pearson’s correlation coefficient (r). Values of r can vary from 

-1 to +1, indicating parameters are either perfectly negatively or positively correlated. Values near zero indicate no correlation between parameters. Calculations 

were made using a subset of the original 109 core samples that excluded the samples with missing values, for a total of 96 samples. Abbreviations: g/cm3, gram 

per cubic centimeter; mm, millimeter; cm/s, centimeter per second]

Property-Transfer Model Analyses  27
Poor correlations between some variable pairs indicated 

either that the explanatory variables had no relation with 

the response variable or that the linearity assumption was 

poorly met after applying transformations. All response 

variables were poorly correlated with percent gravel because 

most core samples had zero gravel content. As a result, the 

distribution for percent gravel was asymmetric, with large 

positive skewness, causing points on the scatter plots to be 

clustered near zero. rpart was poorly correlated with qsat (r = 

-0.046), log(l) (r = 0.017), and log(Ksat) (r = -0.204), with 

only a slightly better correlation with log(yo) (r = 0.315). 

By examining the scatter plots, it was determined that the 

relation between rpart and the response variables was weak. 

This was an early indication that rpart would not help explain 

the variation in qsat, log(l), and log(Ksat) during the regression 

analyses. The implications for poor correlations between some 

variable pairs are that the coefficients may become unstable 

or may not be found significantly different from zero when 

in fact they are (incorrect significance testing). Regression 

models with poor correlations between variable pairs may be 

less desirable to use than models with explanatory variables 

better able to explain the variation in the response variable 

of interest. Assumptions of multiple linear regression are, on 

rare occasions, perfectly met. In this study, the explanatory 

variables for models A, B, and C were pre-selected to simplify 

comparison of different particle-size parameter sets; therefore, 

some variable pairs with low correlations were included in 

the regression analyses. Explanatory variables that are poorly 

correlated with the response variables will not likely contribute 

to explaining variation in response variables. Goodness-offit values likely would increase or decrease only slightly 

by including these explanatory variables in the regression 

variates. 

Bulk

density,

rbulk

(g/cm3)

Particle 

density,

rpart 

(g/cm3)

Gravel

(percent)

Sand

(percent)

Silt

(percent)

Clay

(percent)

log(Geometric mean 

particle 

diameter, 

Mg (mm))

log(Geometric 

particle-size 

standard 

deviation, 

sg)

log(Median 

particle 

diameter, 

d50 (mm))

log(Uniformity 

coefficient, 

Cu)

Saturated water content, qsat

(cm3/cm3)

-0.568 -0.046 -0.247 0.283 -0.250 -0.256 0.173 -0.038 0.175 -0.068

log(Scaling parameter for 

matric pressure, yo

(-cm-water))

0.565 0.315 -0.196 -0.645 0.629 0.653 -0.699 -0.183 -0.708 -0.095

log(Curve-shape parameter, 

l)

-0.168 0.017 0.213 0.420 -0.406 -0.481 0.461 -0.684 0.408 -0.662

Bulk density, ρbulk (g/cm3) 1 -0.058 0.063 -0.268 0.233 0.334 -0.317 -0.007 -0.323 0.044

Particle density, rpart (g/cm3) 1 -0.039 -0.460 0.435 0.479 -0.437 -0.224 -0.450 -0.223

Gravel (percent) 1 0.201 -0.276 -0.271

Sand (percent) 1 -0.992 -0.901

Silt (percent) 1 0.858

Clay (percent) 1

log(Geometric mean particle 

diameter, Mg (mm))

1 -0.140

log(Geometric particle-size 

standard deviation, sg)

1

log(Median particle 

diameter, d50 (mm))

1 -0.060

log(Uniformity coefficient, 

Cu)

1

Table 8. Correlation matrix between water-retention parameters and bulk-physical properties, with transformation of selected variables.

[The statistical relation between parameters listed in the header and first column is indicated by Pearson’s correlation coefficient (r). Values of r can vary from 

-1 to +1, indicating parameters are either perfectly negatively or positively correlated. Values near zero indicate no correlation between parameters. Calculations 

were made using a subset of the original 109 core samples that excluded the samples with missing values and the Magnuson (2002) data, for a total of 57 

samples. Abbreviations: g/cm3, gram per cubic centimeter; mm, millimeter; cm3/cm3, cubic centimeter per cubic centimeter; cm-water, centimeter of water]

28  Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
Collinearity Between Explanatory Variables

Because collinearity affects the size of R2, and can 

produce errors in coefficient magnitudes or signs, or 

instabilities in coefficient values, assessing collinear relations 

among explanatory variables is needed to understand the 

results of the multiple linear regression analyses. Collinearity 

was evaluated by using scatter plots, Pearson’s r, and VIF 

values. Scatter plots between explanatory variables in model 

A indicated that percent sand was collinear with both percent 

silt and percent clay, and that percent silt was collinear with 

percent clay. This is true because any given textural-class 

percentage can be calculated by subtracting the sum of the 

remaining variables from 100 percent. For the q(y) PTM, 

Pearson’s r between percent sand and percent silt was 

-0.992; Pearson’s r between percent sand and percent clay 

was -0.901; and Pearson’s r between percent silt and percent 

clay was 0.858 (table 8). For similar reasons, the degree of 

correlation between percent sand and percent silt and between 

percent sand and percent clay was also high for the Ksat PTM 

(table 9). The correlation of percent gravel with the other 

textural-class percentages is smaller because a large number 

of core samples had zero gravel content. For the q(y) PTM 

and the Ksat PTM, Pearson’s r values indicated a poor degree 

of correlation among the explanatory variables in models B 

(rbulk, Mg, and sg) and C (rbulk, d50, and Cu). VIF indicates the 

degree of multicollinearity in a model, which can come from 

the combined effect of two or more explanatory variables in 

the regression variate. Calculations of VIF for the q(y) PTM 

were based on 57 core samples, excluding the 51 Magnuson 

(written commun., 2002) samples and the sample with missing 

particle-size data from McElroy and Hubbell (1990). For Ksat, 

96 samples were used to calculate VIF values, after excluding 

the 13 samples with missing particle-size or Ksat values. For 

model A and the q(y) PTM, VIF values ranged from 286 

to 38,600 for percent gravel, sand, silt, and clay (table 10). 

Bulk 

density,

rbulk

(g/cm3)

Particle 

density,

rpart

(g/cm3)

Gravel

(percent)

Sand

(percent)

Silt

(percent)

Clay

(percent)

log(Geometric mean 

particle 

diameter, 

Mg (mm))

log(Geometric particlesize standard 

deviation, 

sg)

log(Median 

particle 

diameter, 

d50 (mm))

log(Uniformity 

coefficient, 

Cu)

log(Saturated hydraulic 

conductivity, Ksat

(cm/s))

-0.035 -0.204 0.369 0.655 -0.735 -0.476 0.728 0.270 0.710 0.197

Bulk density, rbulk

(g/cm3)

1 -0.199 0 .282 -0.311 0.013 0.379 -0.131 0.199 -0.151 0.165

Particle density, rpart

(g/cm3)

1 -0.160 -0.003 0.305 -0.408 -0.010 -0.347 0.063 -0.162

Gravel (percent) 1 -0.045 -0.394 -0.213

Sand (percent) 1 -0.819 -0.662

Silt (percent) 1 0.381

Clay (percent) 1

log(Geometric mean 

particle diameter, Mg

(mm))

1 0.316

log(Geometric particlesize standard 

deviation, sg)

1

log(Median particle 

diameter, d50 (mm))

1 0.279

log(Uniformity 

coefficient, Cu)

1

Table 9. Correlation matrix between saturated hydraulic conductivity and bulk-physical properties, with transformation of selected variables.

[The statistical relation between parameters listed in the header and first column is indicated by Pearson’s correlation coefficient (r). Values of r can vary from 

-1 to +1, indicating parameters are either perfectly negatively or positively correlated. Values near zero indicate no correlation between parameters. Calculations 

were made using a subset of the original 109 core samples that excluded the samples with missing values, for a total of 96 samples. Abbreviations: g/cm3, gram 

per cubic centimeter; mm, millimeter; cm/s, centimeter per second]

Property-Transfer Model Analyses  29
For the Ksat model, VIF values ranged from 124 to 605 for 

percent gravel, sand, silt, and clay. For VIF much greater than 

10, results indicate a high degree of multicollinearity among 

the textural-class parameters. For models B and C, the VIF 

values ranged from 1.09 to 1.59 for explanatory variables in 

the q(y) PTM and from 1.07 to 1.33 for those in the Ksat PTM 

(table 10), indicating a lack of multicollinearity among the 

explanatory variables. Because rbulk does not relate directly 

to particle size, lack of multicollinearity is indicated by the 

low VIF values between rbulk and the explanatory variables of 

models A, B, and C (ranging from 1.19 to 1.40 for the q(y)

PTM and from 1.10 to 1.40 for the Ksat PTM). The VIF values 

for rpart also were low because rpart has no relation with particle 

size. 

Collinearity is acceptable as long as the prediction 

is made within the range of data values used in model 

development (Helsel and Hirsch, 1992). However, because 

the PTMs presented in this report will potentially be used 

to predict a wide range of sediment textures at the INEEL, 

beyond the range of textures in the calibration data, 

collinearity between textural-class percentages was deemed an 

undesirable aspect of model A. The sediment texture for the 

q(y) models ranged from sands to silt loams (fig. 7B), which 

does not include materials with higher clay content found 

at the INEEL site. Solutions for correcting multicollinearity 

between variables in model A, but not used in this report, 

include centering the data, eliminating variables, collecting 

additional data, or performing a more sophisticated type of 

regression, such as ridge, Bayesian, or weighted least squares 

(Helsel and Hirsch, 1992; Hair and others, 1998). 

Explanatory variable

Water-retention (q(y)) models Saturated hydraulic conductivity (Ksat) 

models

Coefficient of 

determination 

(R2)

Variance inflation 

factor 

(VIF)

Coefficient of 

determination 

(R2)

Variance inflation 

factor 

(VIF)

Model A

Bulk density, rbulk 0.284 1.40 0.287 1.40

Particle density, rpart .304 1.44 .436 1.77

Gravel .997 286 .993 134

Sand 1.000 38,600 .998 605

Silt 1.000 27,400 .998 420

Clay 1.000 1,580 .992 124

Model B

Bulk density, rbulk 0.165 1.20 0.095 1.11

Particle density, rpart .324 1.48 .145 1.17

log(Geometric mean particle diameter, Mg) .372 1.59 .146 1.17

log(Geometric particle-size standard deviation, sg) .137 1.16 .250 1.33

Model C

Bulk density, rbulk 0.157 1.19 0.092 1.10

Particle density, rpart .308 1.45 .063 1.07

log(Median particle, diameter, d50) .348 1.53 .124 1.14

log(Uniformity coefficient, Cu) .084 1.09 .142 1.17

Table 10. Variance inflation factors for explanatory variables used to develop property-transfer models for water retention and saturated hydraulic 

conductivity. 

[VIF, variance inflation factor, measures multicollinearity between explanatory variables in a particular regression model. To calculate VIF, each explanatory 

variable is regressed against the remaining explanatory variables in the model, and the goodness-of-fit, indicated by the coefficient of determination (R2), is 

calculated. VIF equals 1/ (1-R2). VIF less than 10 indicates little or no multicollinearity between the explanatory variable of interest and the rest of the variables 

used in the regression variate. VIF greater than 10 indicates the variable of interest is correlated with one or more of the other variables. Calculations were made 

using a subset of the original 109 core samples that excluded the samples with missing values, for a total of 57 samples for the water-retention models and a total 

of 96 samples for the saturated hydraulic conductivity models.]

30 Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
Water-Retention Property-Transfer Model

Because q varies nonlinearly with y, the Rossi-Nimmo 

(1994) junction model was used to characterize the q(y)

curve. The Rossi-Nimmo (1994) junction model requires three 

parameters (qsat, yo, and l) to describe the entire range of y; 

therefore, three separate multiple linear-regression equations 

are needed. 

Saturated Water Content

Regression results for qsat were analyzed for overall 

goodness-of-fit, normality of residuals, homoscedasticity 

of residuals, and significance of coefficients. A summary of 

regression results for qsat, including values of R2

adj, RMSE, 

PPCC, and residual-distribution skewness, for models A, 

B, and C are presented in table 11. The regression analyses 

were completed after removing as many as 57 core samples 

identified as erroneous from the original total of 109 (see 

table 5). For samples included in models A, B, and C, qsat

ranged from 0.3473 to 0.6151. Goodness-of-fit values (R2

adj) 

were comparable for each model, ranging from 0.491 to 

0.568, with model A having the highest value. The residuals 

were normally distributed for all models, as indicated by 

probability plots, PPCC values, and skewness values. PPCC 

values ranged from 0.989 to 0.992 compared to the critical 

value of 0.981. Residual distributions showed slight negative 

skewness, with values ranging from -0.05 to -0.20. Residual 

plots did not show heteroscedastic or nonlinear patterns (see 

example in fig. 9A for model C). Predicted qsat values are 

plotted against observed values in figure 10A for model C. The 

qsat values for the samples from Perkins and Nimmo (2000) 

were consistently overestimated, falling above the 1:1 line. 

Most similarly textured samples had higher qsat values than the 

Perkins and Nimmo (2000) samples, which may be a cause for 

qsat overestimation. 

Model Explanatory variables R2

adj RMSE PPCC for residual 

distribution

Critical 

PPCC

Skewness 

of residual 

distribution

Number of 

observations, n

Saturated water content, qsat (cm3/cm3)

A rbulk, gravel, sand, silt, clay 0.568 0.043 0.989 0.981 -0.20 51

B rbulk, log(Mg

), log(sg) .491 .047 .992 .981 -.06 52

C rbulk, log(d50), log(Cu

) .491 .047 .992 .981 -.05 52

Scaling parameter for matric pressure, yo (-cm-water)

A rbulk, gravel, sand, silt, clay 0.667 0.406 0.993 0.982 -0.04 54

B rbulk, log(Mg

), log(sg) .727 .367 .986 .982 -.33 55

C rbulk, log(d50), log(Cu

) .693 .389 .987 .982 -.32 55

Curve-shape parameter, l

A rbulk, gravel, sand, silt, clay 0.192 0.230 0.990 0.982 0.23 54

B rbulk, log(Mg

), log(sg) .667 .157 .993 .982 -.16 55

C rbulk, log(d50), log(Cu

) .607 .170 .989 .982 -.05 55

Saturated hydraulic conductivity, Ksat (cm/s)

A rbulk, gravel, sand, silt, clay 0.632 0.918 0.994 0.988 -0.19 91

B rbulk, log(Mg

), log(sg) .606 .982 .991 .987 -.34 78

C ρbulk, log(d50), log(Cu

) .589 1.003 .989 .987 -.36 78

Table 11. Multiple linear-regression results comparing alternative representations of particle-size distribution for water-retention parameters and 

saturated hydraulic conductivity. 

[Hydraulic parameters used as response variables in regression models included saturated water content, the scaling parameter for matric pressure, the waterretention curve-shape parameter, and saturated hydraulic conductivity. Three models (A, B, and C) with alternative forms of the particle-size distribution as 

input were compared for each hydraulic parameter. R2

adj indicates the goodness-of-fit for the model, accounting for the number of explanatory variables and the 

number of observations; values close to 1 indicate that the variables successfully explain the variation in the hydraulic parameter. RMSE, in units of the hydraulic 

parameter, measures the variation in the predicted values of each hydraulic parameter, similar to the standard deviation of a variable around its mean. PPCC 

indicates the normality of a distribution by comparing it to the normal distribution (a straight line) on a probability plot. If PPCC is greater than the critical PPCC 

value, then the data are considered normally distributed. Abbreviations: R2

adj, adjusted coefficient of determination; RMSE, root-mean-square-error; PPCC, 

probability plot correlation coefficient; ρbulk. bulk density; Mg, geometric mean particle diameter; σg, geometric particle-size standard deviation; d50, median 

particle diameter; Cu, uniformity coefficient; cm3/cm3, cubic centimeter per cubic centimeter; cm-water, centimeter of water; cm/s, centimeter per second]

Property-Transfer Model Analyses  31
Figure 9. Studentized residuals and predicted hydraulic property values for multiple-linear regression model C (where the median 

particle diameter, uniformity coefficient, and bulk density were used as the explanatory variables). 

32  Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho

ID20055114_fig09

10-2 10-1 100

-2

-1

0

1

2

0.3 0.4 0.5 0.6 0.7

-2

-1

0

1

2

100 101 102 103

-2

-1

0

1

2

10-8 10-6 10-4 100

-2

-1

0

1

2

10-2

RWMC: Magnuson (2002, written commun.)

RWMC: McElroy and Hubbell (1990)

RWMC: Perkins and Nimmo (2000)

VZRP: Perkins (2003)

VZRP: Winfield (2003)

EXPLANATION

DATA SOURCE

VZRP, Vadose Zone Research Park

RWMC, Radioactive Waste Management Complex

PREDICTED SATURATED HYDRAULIC CONDUCTIVITY, 

IN CENTIMETERS PER SECOND

STUDENTIZED RESIDUAL

PREDICTED SATURATED WATER CONTENT

D. Saturated hydraulic conductivity (KSAT)

PREDICTED SCALING PARAMETER FOR 

MATRIC PRESSURE, IN CENTIMETERS OF WATER

STUDENTIZED RESIDUAL

B. Scaling parameter for matric pressure (yO)

PREDICTED CURVE-SHAPE PARAMETER

A. Saturated water content (qSAT)

STUDENTIZED RESIDUAL

STUDENTIZED RESIDUAL

C. Curve-shape parameter (l)
Figure 10. Predicted and observed hydraulic property values for multiple-linear regression model C (where the median particle 

diameter, uniformity coefficient, and bulk density were used as the explanatory variables). 

ID20055114_fig10

MEASURED CURVE-SHAPE PARAMETER

PREIDICTED CURVE-SHAPE PARAMETER

C. Curve-shape parameter (l)

MEASURED SATURATED WATER CONTENT

PREDICTED SATURATED WATER CONTENT

A. Saturated water content (qSAT)

MEASURED SCALING PARAMETER FOR MATRIC 

PRESSURE, IN CENTIMETERS OF WATER

PREDICTED SCALING PARAMETER FOR MATRIC 

PRESSURE, IN CENTIMETERS OF WATER

B. Scaling parameter for matric pressure (yO) D. Saturated hydraulic conductivity (KSAT)

MEASURED SATURATED HYDRAULIC 

CONDUCTIVITY, IN CENTIMETERS PER SECOND

PREDICTED SATURATED HYDRAULIC 

CONDUCTIVITY, IN CENTIMETERS PER SECOND

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

10-2 10-1 100 101

10-2

10-1

100

101

100 101 102 103 104

100

101

102

103

104

10-8 10-6 10-4 10-2 100

10-8

10-6

10-4

10-2

100

RWMC: Magnuson (2002, written commun.)

RWMC: McElroy and Hubbell (1990)

RWMC: Perkins and Nimmo (2000)

VZRP: Perkins (2003)

VZRP: Winfield (2003)

EXPLANATION

DATA SOURCE

VZRP, Vadose Zone Research Park

RWMC, Radioactive Waste Management Complex

Property-Transfer Model Analyses  33
Because the residual behavior was equivalent among 

models A, B, and C, the best model for qsat was chosen as 

model A, which had the highest R2

adj and lowest RMSE. 

However, model A was able to explain only about 8 percent 

more of the variation in qsat, as indicated by the values of R2

adj, 

than the other two models. The null hypothesis test states 

that coefficients are significantly different from zero if their 

t-value magnitudes exceed the critical t-value for the specified 

a-level and number of observations in the model. The partial 

t-value magnitudes for Mg, sg, d50, and Cu were less than the 

critical t-value of 2.011 (table 12). These variables did not 

help explain the variation in qsat, which is why the goodnessof-fit values for models B and C were nearly identical. For 

models A, B, and C, the partial t-value magnitudes for rbulk

ranged from -6.975 to -6.960, greatly exceeding the critical tvalue of about 2.010 at a = 0.05. rbulk was the most significant 

variable for explaining the variation in qsat. This finding also 

was supported by the values of Pearson’s r calculated between 

each explanatory variable and qsat, where rbulk was the only 

explanatory variable that had at least a moderate correlation 

with qsat (r = -0.568; table 8). Textural-class percentages in 

model A are affected by multicollinearity, and are not highly 

correlated with qsat (table 8). The geometric and graphical 

particle-size variables in models B and C do not correlate 

highly with qsat, which is a likely reason why their coefficients 

were not significantly different from zero. 

Because qsat is related to total porosity, which can be 

calculated from both rbulk and rpart, the effect of including 

rpart in models A, B, and C was evaluated. The R2

adj value 

decreased by 0.4 percent and 0.5 percent for models B and 

C, respectively, and did not change for model A. From the 

scatter plots and Pearson’s r (table 8), rpart was not strongly 

correlated with qsat. Additionally, rpart was poorly correlated 

with yo, l, and Ksat (tables 7 and 9). Therefore, rpart was 

excluded from the regression analyses of all response 

variables. 

Scaling Parameter for Matric Pressure

The regression results for yo are listed in tables 11 and 

13. The regression analyses were based on a subset of the 

original 109 core samples, with the final number of samples 

ranging from 54 to 55. Values of yo ranged from 2.12 cmwater for a sandy loam to 1,207.78 cm-water for a silt loam. 

Explanatory variable Coefficient Lower 

boundary

Upper 

boundary

Standard 

error

Partial

t-value

Critical

t-value

Model A

Y-intercept -5.0123 -12.6245 2.6001 3.780 -1.326 2.014

Bulk density, rbulk -.3974 -.5121 -.2826 .057 -6.973

Gravel .0604 -.0161 .1369 .038 1.590

Sand .0595 -.0170 .1359 .038 1.567

Silt .0641 -.0129 .1411 .038 1.677

Clay .1353 -.0502 .3209 .092 1.469

Model B

Y-intercept 1.0029 0.8453 1.1605 0.078 12.796 2.011

Bulk density, rbulk -.4002 -.5156 -.2848 .057 -6.975

log(Geometric mean particle diameter, Mg) -.0130 -.0379 .0119 .012 -1.049

log(Geometric particle-size standard 

deviation, sg)

-.0022 -.0872 .0827 .042 -.053

Model C

Y-intercept 1.0063 0.8531 1.1595 0.076 13.207 2.011

Bulk density, rbulk -.3998 -.5153 -.2843 .057 -6.960

log(Median particle diameter, d50) -.0123 -.0362 .0115 .012 -1.039

log(Uniformity coefficient, Cu) -.0029 -.0372 .0314 .017 -.171

Table 12. Multiple linear-regression model coefficients, confidence limits, standard errors, and partial t-values for the water-retention parameter, 

saturated water content.

[Three regression models (A, B, and C) with alternative forms of the particle-size distribution as input were compared. Regression coefficients and the upper 

and lower 95-percent confidence limits are listed for each explanatory variable in a given model. Standard error denotes the expected variation of a coefficient 

across multiple samples of the data from a given population. Smaller standard error values imply more reliable predictions (smaller confidence intervals for the 

coefficient). Partial t-value, used to test whether a coefficient is significantly different from zero, is calculated by dividing the coefficient by its standard error. 

If the magnitude of the partial t-value is greater than the critical t-value (calculated from the Student’s t-distribution at a significance level of 0.05 and for the 

number of observations in the model), then the coefficient is significantly different from zero]

34  Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
The R2

adj values were high for all models, ranging between 

0.667 and 0.727. Model B had the highest R2

adj (0.727) and 

smallest RMSE (0.367). The residuals for model A were the 

most normally distributed. Although the PPCC for model B 

was greater than the critical value (0.986 > 0.982), the residual 

distribution for model B was more negatively skewed than 

that of model A or C (with skewness of -0.33 for model B 

compared to -0.04 for model A and -0.32 of model C). For 

models B and C, the studentized residuals were homoscedastic 

(fig. 9B for model C). Observations from different data sets 

were not preferentially underestimated or overestimated, as 

shown by the example for model C in figure 10B. 

The coefficients for each model were examined for 

significance from zero using the partial t-test. If the partial 

t-value magnitude exceeded the critical t-value calculated from 

Student’s t-distribution at a = 0.05, then the coefficient was 

significantly different from zero. As with qsat, only rbulk was 

significant in explaining the variation in log(yo) for model 

A (table 13). The correlation between rbulk and log(yo) was 

moderate, with r = 0.565 (table 8). However, compared with 

the qsat results for model A, the textural parameters—percent 

sand, silt, and clay—are at least moderately correlated with 

log(yo), with values of Pearson’s r of -0.645, 0.629, and 0.653, 

respectively (table 8). The coefficients for these parameters 

did not pass the partial t-test, indicating the influence of 

multicollinearity on model A. The coefficients for rbulk, Mg, and 

sg were significantly different than zero because the partial 

t-value magnitudes were greater than the critical t-value of 

2.008. Log(yo) was moderately correlated with log(Mg) (r = 

-0.699) and poorly correlated with log(sg) (r = -0.183). For 

model C, the coefficients for the intercept term, rbulk, and d50

were significantly different from zero. rbulk and d50 also showed 

the highest univariate correlations with log(yo) for model 

C. Overall, model B had the greatest number of explanatory 

variables that would be helpful in predicting log(yo). 

Based on residual behavior and goodness-of-fit values, 

model B was chosen as the best model for predicting log(yo). 

The residuals of model B were the most normally distributed 

according to the PPCC and did not display unequal variances 

when plotted against predicted values of log(yo). Model B had 

the highest goodness-of-fit values, explaining 6 percent more 

of the variation in log(yo) than model A and 3 percent more 

Explanatory variable Coefficient Lower

boundary

Upper

boundary

Standard

error

Partial

t-value

Critical

t-value

Model A

Y-intercept 51.4054 -17.4564 120.2670 34.249 1.501 2.011

Bulk density, rbulk 1.8613 .8042 2.9183 .526 3.540

Gravel -.5303 -1.2235 .1628 .345 -1.538

Sand -.5199 -1.2124 .1727 .344 -1.509

Silt -.4977 -1.1964 .2011 .348 -1.432

Clay -1.2932 -2.9783 .3920 .838 -1.543

Model B

Y-intercept -1.1183 -2.3143 0.0777 0.596 -1.877 2.008

Bulk density, rbulk 1.4866 .6178 2.3554 .433 3.435

log(Geometric mean particle diameter, Mg) -.9035 -1.0956 -.7115 .096 -9.445

log(Geometric particle-size standard 

deviation, sg)

-.8969 -1.5605 -.2332 .331 -2.713

Model C

Y-intercept -1.4080 -2.6350 -0.1810 0.611 -2.304 2.008

Bulk density, rbulk 1.5344 .6139 2.4549 .459 3.346

log(Median particle diameter, d50) -.8394 -1.0330 -.6458 .096 -8.703

log(Uniformity coefficient, Cu) -.1510 -.4347 .1328 .141 -1.068

Table 13. Multiple linear-regression model coefficients, confidence limits, standard errors, and partial t-values for the water-retention scaling parameter 

for matric pressure.

[Three regression models (A, B, and C) with alternative forms of the particle-size distribution as input were compared. Regression coefficients and the upper 

and lower 95-percent confidence limits are listed for each explanatory variable in a given model. Standard error denotes the expected variation of a coefficient 

across multiple samples of the data from a given population. Smaller standard error values imply more reliable predictions (smaller confidence intervals for the 

coefficient). Partial t-value, used to test whether a coefficient is significantly different from zero, is calculated by dividing the coefficient by its standard error. 

If the magnitude of the partial t-value is greater than the critical t-value (calculated from the Student’s t-distribution at a significance level of 0.05 and for the 

number of observations in the model), then the coefficient is significantly different from zero]

Property-Transfer Model Analyses  35
than model C. All coefficients for model B were significantly 

different from zero, indicating that all explanatory variables 

were important for describing the variation in log(yo).

Curve-Shape Parameter

Multiple linear-regression analyses for log(l) are 

summarized in tables 11 and 14 for models A, B, and C. After 

removing samples with erroneous or missing values (table 

5), the final number of samples used in model A was 54, and 

the number used in models B and C was 55. Model B had 

the highest R2

adj and smallest RMSE. The residuals also were 

the most normally distributed for this model, with a PPCC 

value of 0.993 compared to the critical PPCC value of 0.982, 

and a skewness value of -0.16. The residual plot for model B 

showed the least heteroscedasticity. Residual plots for models 

A and C showed some clustering of points at smaller values 

of predicted log(l) (fig. 9C), and the residual plot for model 

A showed slight nonlinearity. Most of the l values for the 

Perkins and Nimmo (2000) data set were underestimated 

(fig. 10C for model C). The coefficients for log(Mg), log(sg), 

log(d50), and log(Cu) were significantly different from zero for 

models B and C; whereas for model A, only the coefficient 

for percent silt was significantly different from zero. From 

Pearson’s r values (table 8), log(sg) and log(Cu) were most 

highly correlated with log(l). Based on the comparison of 

regression results for models A, B, and C, model B was 

selected as the best model for predicting log(l). 

Saturated Hydraulic Conductivity PropertyTransfer Model

The Ksat PTM was developed using the same core 

samples as the q(y) PTM in addition to the 51 samples from 

Magnuson (written commun., 2002), for 96 potential samples 

(excluding the samples with missing Ksat and particle-size 

measurements). Twelve samples had missing Ksat values (10 

samples from Magnuson (written commun., 2002) and two 

samples from Perkins (2003)) and one sample had partial 

particle-size data (USGS 118 at 173.48 to 174.39 m from 

McElroy and Hubbell (1990)); these samples were treated 

Explanatory variable Coefficient Lower

boundary

Upper

boundary

Standard

error

Partial

t-value

Critical

t-value

Model A

Y-intercept 38.0186 -0.9624 76.9996 19.387 1.961 2.011

Bulk density, rbulk .1387 -.4596 .7371 .298 .466

Gravel -.3881 -.7805 .0043 .195 -1.989

Sand -.3867 -.7788 .0053 .195 -1.983

Silt -.4099 -.8055 -.0144 .197 -2.084

Clay -.9143 -1.8683 .0396 .474 -1.927

Model B

Y-intercept 0.2489 -0.2615 0.7594 0.254 0.979 2.008

Bulk density, rbulk .0799 -.2909 .4507 .185 .433

log(Geometric mean particle diameter, Mg) .2029 .1209 .2848 .041 4.970

log(Geometric particle-size standard 

deviation, sg)

-1.2227 -1.5059 -.9394 .141 -8.666

Model C

Y-intercept -0.0411 -0.5777 0.4955 0.267 -0.154 2.008

Bulk density, rbulk .0974 -.3051 .4999 .201 .486

log(Median particle diameter, d50) .1925 .1079 .2772 .042 4.565

log(Uniformity coefficient, Cu) -.4910 -.6151 -.3669 .062 -7.944

Table 14. Multiple linear-regression model coefficients, confidence limits, standard errors, and partial t-values for the water-retention curve-shape 

parameter.

[Three regression models (A, B, and C) with alternative forms of the particle-size distribution as input were compared. Regression coefficients and the upper 

and lower 95-percent confidence limits are listed for each explanatory variable in a given model. Standard error denotes the expected variation of a coefficient 

across multiple samples of the data from a given population. Smaller standard error values imply more reliable predictions (smaller confidence intervals for the 

coefficient). Partial t-value, used to test whether a coefficient is significantly different from zero, is calculated by dividing the coefficient by its standard error. 

If the magnitude of the partial t-value is greater than the critical t-value (calculated from the Student’s t-distribution at a significance level of 0.05 and for the 

number of observations in the model), then the coefficient is significantly different from zero]

36  Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
as missing data and subtracted from the original set of 109 

samples. Ksat values of the 96 samples ranged from 1.13  ́

10-8 to 8.41  ́ 10-2 cm/s (table 2), with textures ranging from 

clays to sands (fig. 8A). Thirty six of the 96 samples were 

classified as silt loams, with the second greatest number of 

samples (19) classified as sandy loams. As discussed above, 

five additional core samples were removed from model A due 

to either inadequacies in measured particle-size distributions 

or errors in measured Ksat and rpart values. An additional 18 

samples were removed from models B and C for similar 

reasons. The total number of core samples used to develop 

model A was 91, with 78 samples used to develop models B 

and C. The range of textures for models B and C is shown on 

the textural triangle in figure 8B.

Regression results for each Ksat model are summarized in 

tables 11 and 15. Model A had the highest R2

adj value (0.632), 

but no coefficients were significantly different from zero due 

to the multicollinearity between the textural-class parameters. 

The residuals for Model A were the most normally distributed, 

with a PPCC value of 0.994 compared to the critical value 

of 0.988, and a skewness of -0.19. Model B had the second 

highest R2

adj value (0.606) and ranked second best in terms of 

residual normality (PPCC of 0.991 compared to the critical 

value of 0.987, and skewness of -0.34). Coefficients for the 

intercept term and log(Mg) were significantly different from 

zero. The correlation between log(Ksat) and log(Mg) (r = 0.728; 

table 9) was highest among all the explanatory variables 

in model B. Model C had the lowest R2

adj value (0.589). 

Although the normality of the residuals was found to be good, 

with a PPCC of 0.989 compared to the critical PPCC value 

of 0.987, the residual distribution was slightly bimodal. Only 

the intercept term and log(d50) had coefficients that passed 

the partial t-test for significance from zero. Log(d50) also was 

more highly correlated with log(Ksat) (r = 0.710; table 9) than 

rbulk or log(Cu). Studentized residual variances, plotted against 

the predicted Ksat values, were nearly random and equal for 

all models (fig. 9D for model C), with the exception of a 

few extreme values exceeding the t-value threshold of about 

2.0. Trends observed in residual plots were basically linear. 

Predicted values of Ksat are shown plotted against observed 

values for model C in figure 10D. The Ksat values for the 

Magnuson samples tended to be underestimated, likely due to 

the large number of samples with high clay contents and high 

Explanatory variable Coefficient Lower

boundary

Upper

boundary

Standard

error

Partial

t-value

Critical

t-value

Model A

Y-intercept -14.2998 -29.4376 0.8379 7.614 -1.878 1.988

Bulk density, rbulk .6625 -.5308 1.8557 .600 1.104

Gravel .1123 -.0381 .2627 .076 1.484

Sand .1127 -.0387 .2641 .076 1.480

Silt .0712 -.0788 .2212 .075 .944

Clay .0826 -.0685 .2336 .076 1.087

Model B

Y-intercept -1.5121 -3.4163 0.3920 0.956 -1.582 1.993

Bulk density, rbulk -.0500 -1.2985 1.1985 .627 -.080

log(Geometric mean particle diameter, Mg) 1.8423 1.4862 2.1984 .179 10.308

log(Geometric particle-size standard 

deviation, sg)

-.2906 -1.3644 .7833 .539 -.539

Model C

Y-intercept -1.7690 -3.6640 0.1260 0.951 -1.860 1.993

Bulk density, rbulk .0794 -1.1796 1.3384 .632 .126

log(Median particle diameter, d50) 1.7507 1.4155 2.0858 .168 10.408

log(Uniformity coefficient, Cu) -.3274 -.8314 .1765 .253 -1.295

Table 15. Multiple linear-regression model coefficients, confidence limits, standard errors, and partial t-values for saturated hydraulic conductivity.

[Three regression models (A, B, and C) with alternative forms of the particle-size distribution as input were compared. Regression coefficients and the upper 

and lower 95-percent confidence limits are listed for each explanatory variable in a given model. Standard error denotes the expected variation of a coefficient 

across multiple samples of the data from a given population. Smaller standard error values imply more reliable predictions (smaller confidence intervals for the 

coefficient). Partial t-value, used to test whether a coefficient is significantly different from zero, is calculated by dividing the coefficient by its standard error. 

If the magnitude of the partial t-value is greater than the critical t-value (calculated from the Student’s t-distribution at a significance level of 0.05 and for the 

number of observations in the model), then the coefficient is significantly different from zero]

Property-Transfer Model Analyses  37
Ksat values compared with the rest of the data. Based on the 

high value of R2

adj, and the normality and homoscedasticity 

of the residuals, model A was identified as the best model for 

predicting Ksat. 

Model Discrimination

Several tests were done to keep the use of the q(y) and 

Ksat models practical by limiting explanatory variables to one 

common set of particle-size parameters and determining the 

sensitivity of the estimated q(y) and Ksat values to the way 

the particle-size distribution was represented. For the qsat, yo, 

and l models, R2

adj values were not consistently higher for 

one common set of particle-size parameters. The set of three 

regression models for qsat, yo, and l that produced the highest 

R2

adj values was model combination A-B-B (textural-class 

percentages (A) for qsat, and geometric statistics (B) for yo

and l). These models also showed the best behavior in terms 

of meeting the assumptions of multiple linear regression. 

Model A had the highest R2

adj and displayed the best model 

behavior for Ksat. Because the combined ability of the qsat, 

yo, and l models in estimating the q(y) curve is of more 

interest than obtaining the highest goodness-of-fit values for 

individual parameters, four model combinations were tested 

based on: (1) highest R2

adj values (A-B-B), (2) textural-class 

percentages (A-A-A), (3) geometric statistics (B-B-B), and (4) 

graphical statistics (C-C-C). Model discrimination then was 

completed by (1) calculating the RMSE between the observed 

and predicted q(y) curve for each core sample, (2) comparing 

the RMSE distributions of the four competing model 

combinations, and (3) evaluating the best model combination 

based on ease of model use, best model behavior, highest 

values of R2

adj, and best representation of the particle-size 

distribution to use for development of a theoretical PTM.

For each core sample, predicted values of qsat, yo, and 

l were calculated using the appropriate regression equations 

developed for models A, B, and C. For each of the four model 

combinations tested, the predicted values of qsat, yo, and l

were used as input to the Rossi-Nimmo (1994) junction model 

(eqs. 1 to 3) to produce predicted q(y) curves for each core 

sample. The q(y) curve could not be estimated if any value 

of qsat, yo, or l was missing for a particular core sample. 

For this reason, the number of estimated curves per model 

combination was constrained to the minimum number of core 

samples used to develop the qsat, yo, and l models as a whole. 

For model combination A-B-B, 50 samples were used, and 

for model combination A-A-A, 51 samples were used. For 

model combinations B-B-B and C-C-C, 52 samples were used 

in the comparisons. Because yo and l were log-transformed, 

and the range of measured qsat values was not close to zero, 

predicted values of the hydraulic parameters were always 

positive. Negative values would cause errors in the q(y) curve 

calculated with the Rossi-Nimmo (1994) junction model.

Due to the difficulty in visually assessing the quality of 

predicted q(y) curves for each model combination (which 

would require up to 52 comparisons per combination), R2 and 

RMSE values were used to assess goodness-of-fit between 

observed and predicted q(y) curves. For each core sample, 

the observed q(y) curve was defined using the measured qsat

value and the two parameters (yo and l) obtained by fitting 

the Rossi-Nimmo (1994) junction model (eqs. 1 to 3) to the 

q(y) measurements. The R2 and RMSE values originally were 

computed from q values at the measured y values, however, 

goodness-of-fit values were found to be sensitive to the range 

and number of measured q(y) points. Among core samples, 

the measured q(y) points spanned different ranges of y and 

varied in number. R2, calculated between the observed and 

predicted q values at the measured y values, was highly 

sensitive to the range and number of q(y) points. For some 

core samples, the R2 value was high (0.90 or greater) even 

when the observed and estimated curves visually appeared 

quite different. To place samples on an equal basis for 

comparison, observed and predicted q values were calculated 

at 1,000 logarithmically-spaced y points to define an 

approximately smooth curve, and R2 and RMSE values were 

recalculated. R2 values increased using the finer resolution 

of y points and still falsely indicated goodness-of-fit for 

some samples. Although RMSE differs from R2 because it is 

expressed in units of the dependent variable (q) and does not 

have a fixed scale like R2, which varies from 0 to 1, RMSE was 

chosen as the best goodness-of-fit indicator for comparing the 

four model combinations because R2 did not always indicate 

true goodness-of-fit.

Predicted q(y) curves for each model combination 

are shown in figure 11 for two core samples of silt loam 

texture with contrasting goodness-of-fit values. For model 

combination C-C-C, the RMSE between observed and 

predicted curves for core sample ICPP-SCI-V-214 at 56.24 to 

56.33 m from Winfield (2003) (fig. 11A) was 0.012, indicating 

a good fit, while the RMSE for core sample UZ98-2 at 49.89 

to 49.99 m from Perkins and Nimmo (2000) (fig. 11B) was 

0.043, indicating a poor fit. The overestimation of qsat inflated 

the RMSE value for UZ98-2 at 49.89 to 49.99 m, indicating 

the importance of accurately estimating qsat because it controls 

the relative position of the q(y) curve along the q axis. 

RMSE distributions for each model combination were 

examined to determine the best particle-size parameter set to 

implement based on the distribution with the smallest RMSE

values. In figure 12, the RMSE distributions for each model 

combination are presented as a boxplot. The box represents 

the interquartile range for each R2 distribution, where the 

upper box limit is the 75th percentile and the lower limit is the 

25th percentile. The median of the distribution (50th percentile) 

is drawn as a line across the box. Skewness is indicated by 

the relative position of the median line within the box, with 

a perfect normal distribution indicated by a line centered in 

38  Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
the box. The “whiskers” extend from the box edge to the data 

points that lie within 1.5 times the interquartile range. Any 

points that lie beyond the ends of the whiskers, plotted as 

circles, are outlying points. If there are no outlying points, a 

dot is placed at the end of the left whisker. The range of RMSE

values was similar between each model combination, varying 

from 0.001 to 0.064 (fig. 12). The medians ranged from 0.015 

to 0.021. These results indicated that the estimated q(y)

curves were insensitive to the way the particle-size distribution 

was represented. The distributions for model combinations 

A-B-B and B-B-B were nearly identical, indicating that the 

estimated qsat values for each core sample were insensitive to 

the regression equation used. This result supports the earlier 

observation that the particle-size parameters explain little 

Figure 11. Estimated water-retention (q(y)) curves for two silt loam core samples 

based on three multiple linear-regression models with different particle-size parameter 

sets as input. 

The q(y) curve was represented by the Rossi-Nimmo (1994) junction model.

0.00

0.10

0.20

0.30

0.40

0.50

0.00

0.10

0.20

0.30

0.40

0.50

-107 -106 -10-1 -100 -101 -102 -103 -104 -105

-107 -106 -10-1 -100 -101 -102 -103 -104 -105

Measured points

Curve fit to measured points

Estimated curve:

model combination A-A-A

(textural-class percentages)

Estimated curve:

model combination B-B-B

(geometric statistics)

Estimated curve:

model combination C-C-C

(graphical statistics)

MATRIC PRESSURE, IN CENTIMETERS OF WATER

VOLUMETRIC WATER CONTENT

B. Core sample UZ98-2 at 49.89 to 49.99 meters (Perkins and Nimmo, 2000)

A. Core sample ICPP-SCI-V-214 at 56.24 to 56.33 meters (Perkins, 2003)

Property-Transfer Model Analyses  39
of the variation in qsat, and that rbulk is the most important 

explanatory variable for qsat. Model combination A-A-A 

possessed the largest interquartile range, or spread of RMSE

values, likely because the l model had such a low value of 

R2

adj (0.192). 

The RMSE distribution for model combination C-C-C 

was divided into groups to examine trends based on texture, 

source data, and location. The 52 core samples for this model 

combination were divided into groups based on the five U.S. 

Department of Agriculture textures (sand, loamy sand, sandy 

loam, loam, and silt loam) represented by the data (fig. 13A). 

The primary texture represented by the data was silt loam 

(n = 29), followed by sandy loam (n = 10). Sandy loam had the 

smallest median RMSE value (0.008) among all textures, with 

silt loam having the second smallest median (0.017). Samples 

of silt loam texture were expected to have the smallest RMSE

values because silt loam is the dominant texture represented 

in the models. Removing the three outlying points of the silt 

loam distribution decreased the median only slightly, from 

0.017 to 0.015. The range of RMSE values for sands compared 

to loams was quite similar, indicating lack of model bias 

toward a particular texture. The same RMSE values divided on 

the basis of source data are shown in figure 13B. The RMSE

distribution for the Winfield (2003) data set had the tightest 

range (0.008 to 0.019), with a median value only slightly 

smaller than the other distributions. This range likely resulted 

from all 10 samples having silt loam textures. The McElroy 

and Hubbell (1990) data set displayed the largest range of 

RMSE values (0.001 to 0.054). The qsat values for five samples 

were either greatly over- or underestimated, with absolute 

residual values of 0.07 or more. Over- or underestimation of 

qsat could cause the RMSE values to be inflated due to the large 

difference between estimated and measured values of q near 

saturation. Textures represented by the McElroy and Hubbell 

Figure 12. Comparison of root-mean-square-error (RMSE) distributions for four combinations of multiple 

linear-regression models with different particle-size parameter sets as input. 

RMSE was calculated for individual core samples between water-retention curves estimated from regression 

models and fitted to measured points. The water-retention curve was represented by the Rossi-Nimmo (1994) 

junction model.

ID200551114_fig12

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

HIGHEST ADJUSTED COEFFICIENT 

OF DETERMINATION

MODEL COMBINATION A-B-B

N = 50

TEXTURAL-CLASS PERCENTAGES

MODEL COMBINATION A-A-A

N = 51

GEOMETRIC PARTICLE-SIZE STATISTICS

MODEL COMBINATION B-B-B

N = 52

GRAPHICAL PARTICLE-SIZE STATISTICS

MODEL COMBINATION C-C-C

N = 52

ROOT-MEAN-SQUARE ERROR, IN CUBIC CENTIMETER 

PER CUBIC CENTIMETER

40 Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
data set were the most diverse among the four data sets shown 

in figure 13B, with a mixture of silt loams, loams, sandy 

loams, and loamy sands. This diverse mixture of textures also 

could explain why the RMSE distribution for the McElroy and 

Hubbell data spanned such a large range. 

Comparison of RMSE distributions in figure 12 showed 

that estimated q(y) curves were insensitive to the way the 

particle-size distribution was represented; therefore, other 

criteria were used to select the best particle-size parameter 

set for the final q(y) and Ksat models. These criteria 

included: (1) ease of model use or ease of determining the 

particle-size parameters, (2) best model behavior for each 

hydraulic parameter (qsat, yo, l) in terms of normality and 

homoscedasticity of the residuals, (3) highest degree of model 

fit for each hydraulic parameter, based on R2

adj, and (4) best 

model combination to use for developing a theoretical model 

from results of the statistical analyses. Model combination 

A-B-B involved computing two sets of parameters from 

the particle-size distribution, textural-class percentages and 

geometric statistics. Although textural-class percentages can 

be readily determined, calculation of geometric statistics 

requires good particle-size distribution resolution and points 

that extend to at least a cumulative frequency of 10 percent 

on a percent-finer-than basis. Model combination A-B-B was 

undesirable to use because two sets of particle-size parameters 

needed to be calculated. Although model combination A-B-B 

involved regression models with the highest R2

adj values for 

each q(y) parameter, the RMSE distribution for this model 

combination differed only slightly from distributions for 

model combinations A-A-A, B-B-B, and C-C-C. For these 

reasons, model combination A-B-B was eliminated from 

further consideration.

Textural-class percentages are the easiest particle-size 

parameters to determine, followed by the graphical statistics, 

and then the geometric statistics. The minimum number of 

particle-size points needed to calculate textural classes is 

three, chosen to correspond to the textural-class limits of a 

particular sedimentary or soil classification system. More 

points are typically measured for those methods where precise 

determination of particle sizes corresponding to classification 

limits is difficult (for example, the pipette or hydrometer 

method). Percent gravel, sand, and silt can be calculated 

directly from the cumulative particle-size distribution, 

graphically or by interpolation between measured points. 

Percent clay is then calculated from the sum of the other 

three percentages subtracted from 100 percent. The second 

easiest set of parameters to calculate includes d10, d50, and d60, 

which can be determined graphically from the cumulative 

particle-size distribution or calculated by interpolating 

between measured points. Calculating the graphical statistics 

requires better particle-size distribution resolution than for 

calculating textural-class percentages, primarily because the 

Figure 13. Root-mean-square-error distributions for the multiple 

linear-regression model with median particle diameter, uniformity 

coefficient, and bulk density as input (model combination C-C-C). 

RMSE was calculated for individual core samples between curves 

estimated from the regression models and fitted to the measured 

water-retention points. The water-retention curve was represented 

by the Rossi-Nimmo (1994) junction model.

ID20055115_fig13

PERKINS AND 

NIMMO (2000)

N = 18

MCELROY AND 

HUBBELL (1990)

N = 10

PERKINS (2003)

N = 14

WINFIELD (2003)

N = 10

SILT LOAM

N = 29

LOAM

N = 2

SANDY LOAM

N = 10

LOAMY SAND

N = 4

SAND

N = 7

ROOT-MEAN-SQUARE ERROR, IN CUBIC

CENTIMETER PER CUBIC CENTIMETER

B. Data source for model development

A. Core sample texture

ROOT-MEAN-SQUARE ERROR, IN CUBIC

CENTIMETER PER CUBIC CENTIMETER

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Property-Transfer Model Analyses  41
values of d10, d50, and d60 are not defined by specific particle 

diameters but by cumulative percentiles. Using geometric 

statistics is attractive because particle-size distributions of 

sediments often are lognormally distributed (Krumbein, 1938; 

Pettijohn, 1975). This means that particle-size distribution 

can be characterized using only two parameters, the mean and 

standard deviation. However, calculating these parameters 

requires good resolution of the particle-size distribution. The 

number of particle-size bins reported for samples in this study 

varied from 20 for 67 samples (McElroy and Hubbell, 1990; 

S. Magnuson, written commun., 2002) to 116 for 42 samples 

(Perkins and Nimmo, 2000; Perkins, 2003; and Winfield, 

2003). Geometric statistics can be calculated directly from 

particle-size data using the moment method (which is sensitive 

to bin size), or by fitting a lognormal function to the data 

(which involves access to software with nonlinear regression 

capabilities). Mg and d50 are the two most common measures 

of central tendency used in the earth sciences. Because a 

small number of observations can strongly influence Mg, it 

is not considered a resistant measure of central tendency. 

However, d50 is determined solely by the relative order of the 

observations in the distribution. The resistant analogs to the 

geometric statistical parameters are d50 and Cu (d60/d10), which 

have the advantage of not including effects due to skewness or 

outliers. 

To estimate K(q), Ksat can be combined with q(y), 

represented by the Rossi-Nimmo (1994) or van Genuchten 

(1980) equations, in Mualem’s (1976) equation. For ease 

of application of this step, the Ksat and q(y) PTMs needed 

to have the same input parameters, which would require 

calculating only one set of particle-size parameters. For Ksat, 

model A had the highest R2

adj (0.631), with models B and C 

differing only by 2.5 and 4.2 percent, respectively, from model 

A. Because the goodness-of-fit values were similar for models 

A, B, and C for Ksat, the final Ksat PTM was selected based on 

the best model combination for q(y). 

Besides ease of use, other criteria used to discriminate 

between competing models included the best behavior of 

residuals for each hydraulic parameter (qsat, yo, l), the highest 

R2

adj, and the best parameters to use for development of a 

theoretical PTM. In general, model combination B-B-B ranked 

the highest for all of the criteria for model selection except 

ease of use. Model combination C-C-C ranked second best. 

Values of R2

adj for each hydraulic parameter were, on average, 

higher for model B (table 14) than for the other models. Model 

B also tended to have the most normally distributed and 

homoscedastic residuals for log(l) and qsat, with the normality 

being only slightly degraded compared to model A for log(yo). 

Model combination B-B-B also was the best candidate for use 

in developing a theoretical model because Mg and sg can be 

related to similar parameters of the pore-size distribution on 

a lognormal basis (Kosugi, 1996; Hwang and Powers, 2003). 

Model A involved four explanatory variables that described 

the particle-size distribution, but because these summed to 

100 percent, only three needed to be measured. Model A 

ranked highest for ease of use because textural data typically 

are the minimum data reported when describing particle size. 

However, due to multicollinearity among the textural-class 

parameters, model A was less desirable to use as the final 

PTM than models B or C. The parameters for model C readily 

can be calculated from the cumulative particle-size distribution 

as long as enough points are available to interpolate accurately 

d10, d50, and d60. Accurate calculation of model B particle-size 

parameters depends on particle-size distribution resolution. 

Resolution affects the width of the bin sizes, and wider bins 

create larger errors in Mg and sg. In addition, distribution’s 

fine end needs to be adequately defined (at least down to a 

cumulative frequency of 10 percent), otherwise calculated 

Mg and sg values are greatly affected. Because d50 and Cu are 

closely related to Mg and sg, respectively, and given that model 

C ranked second highest for the model criteria, model C was 

chosen as the final model. The following are the final q(y)

equations using the graphical particle-size statistics and rbulk as 

input:

q r sat bulk

d Cu

= -

- -

1 0063 0 3998

0 0123 50 0 0029

. .

. log( ) . log( )

log( ) . .

. log( ) . log( )

y r o bulk

d Cu

= - +

- -

1 4080 1 5344

0 8394 50 0 1510 (12)

log( ) . .

. log( ) . log( )

l = - + r

+ -

0 0411 0 0974

0 1925 50 0 4910

bulk

d Cu

The Ksat model equation is:

log( ) . .

. log( ) . log( )

K

d C

sat bulk

u

= - +

+ -

1 7690 0 0794

1 7507 50 0 3274

r

(13)

For comparison, the coefficients for the explanatory variables 

of models A and B are listed in tables 12 through 15 for qsat, 

yo, l, and Ksat.

Property-Transfer Model Application at the 

INEEL

Although multiple-linear regression models are empirical, 

with results depending on data types and quality used in 

calibration rather than on physical relationships or processes, 

their nature does not preclude their usefulness, but may 

limit their universal applicability. A model that is capable of 

estimating measurements over great depths and large areas 

where none were previously available, even if the results are 

not optimal, is better than having no model at all. Statistical 

model application is constrained to the data types used in 

calibration. Because the core sample textures included in 

the regression models ranged from sands to clays, with most 

samples classified as silt loams, q(y) prediction for sediment 

textures not used in calibration, such as clays or samples with 

more gravel, may be poor. Measurements of hydraulic and 

42  Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
bulk-physical properties on core samples with more diverse 

textures are desirable. Development of a PTM that relies 

more on physical than statistical relationships will be useful 

because it is not possible to characterize every point in space. 

A theoretical model developed from the statistical models 

could be more general in its application to other areas of the 

INEEL or other geographic locations. Because the calibration 

data used in this study represent only a small fraction of the 

total population of sediment properties, the PTMs can only be 

generalized to the southwestern part of the INEEL until further 

testing has been completed using samples from other locations 

at the INEEL. Examining variation in sediment texture and 

thickness across the INEEL may provide another way to 

determine locations where the PTMs may be applied.

rbulk was the most important bulk-physical property for 

developing the qsat model. Inclusion of rbulk as an explanatory 

variable in the PTMs requires continued collection of intact 

core samples. However, it may be possible to use average rbulk

values for each textural class or remote sensing of rbulk (using 

neutron, gamma-gamma, or acoustic-velocity logs) in place 

of core sample collection. Care must be taken in applying 

remote-sensing techniques because the methods respond 

differently in the presence of certain minerals, bound water, 

and high salinity water (Keys, 1988). Additionally, methods 

that provide total porosity estimates rather than effective 

porosity would be desirable to use because the rbulk values 

were determined using the total volumes of the core samples. 

Total volume includes pores that may not be interconnected. 

Measurements of rpart could be obtained from borehole 

cuttings and used with estimates of total porosity to calculate 

rbulk values. Further testing is needed to improve applicability 

and ease of use of the PTMs presented in this report.

The moderate goodness-of-fit values observed for 

the qsat, yo, l, and Ksat models, as measured by the R2

adj

values, might indicate missing information in the regression 

variates. Including other bulk-physical-property data, such as 

mineralogy or specific surface area, in the regression variates 

could provide better predictions of qsat, yo, l, and Ksat than 

the use of rbulk and particle-size data alone. Future adjustments 

to the PTMs may need to incorporate more diverse types of 

bulk-physical-property data. As more information is added to 

the PTMs, the model fits will likely improve. The moderate 

R2

adj values also could be due to significant errors in the 

hydraulic and bulk-physical-property measurements. Some 

experimental methods may have more uncertainty associated 

with the measurements than other methods, or sparse data may 

increase the uncertainty of calculations made from the data. 

Data limitations that could potentially affect the model results 

include sparseness of particle-size distributions, erroneous 

measurements (such as qsat exceeding total porosity), 

sparseness of q(y) measurements, and repacking samples prior 

to hydraulic property measurements. For some core samples 

used in calibrating the regression models, errors in qsat and 

particle size were observed. The approach used in dealing with 

anomalous measurements was to remove the core samples 

from the regression analyses. However, weighted least-squares 

regression could be used as an alternative approach, where 

more weight could be given to samples judged higher quality 

than others.

Most widely-used unsaturated flow and transport 

models use the van Genuchten (1980) model rather than 

the Rossi-Nimmo (1994) junction model to represent q(y)

measurements for a particular layer or texture. A conversion 

between the Rossi-Nimmo (1994) and van Genucthen (1980) 

parameters is presented in figure 14. The van Genuchten 

equation is parameterized by qsat, qr, ho, and N, where the 

scaling parameter for y is ho (analogous to yo) and the 

curve-shape parameter is N (analogous to l). In this study qsat

represents a physical measure­ment rather than an optimized 

parameter, and because of this, its regression equation can be 

used directly in the van Genuchten model. For the 58 potential 

samples used to calibrate the q(y) regression models, the 

relation between log(ho) and log(yo) is shown in figure 14A

and that between N and l is shown in figure 14B. The R2 was 

0.911 and 0.837, respectively. Setting qr equal to zero in the 

van Genuchten equation and assuming M = 1 – 1/N allows 

conversion between the Rossi-Nimmo (1994) junction and 

van Genuchten (1980) model parameters. An alternative way 

of converting between functional forms of the q(y) curve 

involves estimating q at several y values using the RossiNimmo (1994) junction model and the regression equations 

for each q(y) parameter, and then fitting the van Genuchten 

(1980) model (or any other curve-fit model) to these estimated 

q(y) points. 

Development of a PTM for K(q) is needed because 

both q(y) and K(q) are required to describe unsaturated flow 

using Richards’ equation. Mualem’s (1976) model is most 

commonly used for estimating K(q), in combination with 

van Genuchten’s (1980) q(y) equation. A PTM for K(q) can 

be created by (1) developing regression equations from the 

existing K(q) measurements for 40 core samples (Perkins 

and Nimmo, 2000; Perkins, 2003; and Winfield, 2003), or (2) 

combining regression models developed in this study for Ksat

and q(y) with the K(q) equation of Mualem (1976), or other 

K models in the literature (Gardner, 1958; Campbell, 1974). 

Andraski and Jacobson (2000), Andraski (1996), Rossi and 

Nimmo (1994), and Fayer and others (1992) discuss how 

to combine the Rossi-Nimmo (1994) junction model with 

Mualem’s (1976) equation. One unsaturated K measurement 

is better to use as the scaling parameter for K than Ksat in 

Mualem’s (1976) equation. Ksat includes the influence of 

macropores, which drain first during desaturation. The shape 

of the K(q) curve does not depend on drainage from the largest 

pores, but from the pores of the sediment matrix. Using Ksat as 

a scaling parameter for K can give a wrong shape to the K(q) 

curve if macropores are present. Therefore, the first approach, 

developing regression equations from parameters fit to K(q), 

may be more desirable than an approach that uses Ksat as the 

scaling parameter. 

Property-Transfer Model Analyses  43
Figure 14. Relation between water-retention parameters of the Rossi-Nimmo (1994) junction model 

and the van Genuchten (1980) model. 

ID20055115_fig14

y = 1.2039x - 0.1220

R2 = 0.911

y = 0.2147x + 0.2599

R2 = 0.837

LOG(ROSSI-NIMMO JUNCTION MODEL SCALING

PARAMETER, IN CENTIMETERS OF WATER)

LOG(VAN GENUCHTEN MODEL SCALING 

PARAMETER, IN CENTIMETERS OF WATER)

A. Scaling parameter for matric pressure

LOG(ROSSI-NIMMO JUNCTION MODEL

CURVE-SHAPE PARAMETER)

B. Curve-shape parameter

LOG(VAN GENUCHTEN MODEL 

CURVE-SHAPE PARAMETER)

0

0.1

0.2

0.3

0.4

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2

0

1

2

3

4

0 1 2 3 4

44  Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments, INEEL, Idaho
Testing of the regression models in this report is needed 

to determine whether the results can be generalized to a wide 

population of sediments. Model testing can be accomplished 

by (1) comparing regression models in this study to existing 

PTMs, (2) completing a new set of core sample measurements, 

and (3) splitting the original set of measurements into two 

parts, one for development of the regression equations and 

one for testing. Other available data sets (Borghese, 1991; 

Barraclough and others, 1976) and future data sets can be used 

to test the PTMs developed in this report for q(y) and Ksat, 

or to adjust the calibration of the models. New measurements 

can be used to validate the PTMs in this report by comparing 

the predictions of qsat, yo, l, and Ksat using equations 12 

and 13 to the observed values. Additionally, the predictions 

can be compared to results from other PTMs not calibrated 

for the INEEL site to evaluate the need for continued PTM 

development and the accuracy of the site-specific PTMs. A 

second set of measurements also could be used to develop 

separate PTMs that could be compared to the original PTMs 

in terms of predictive accuracy, and the magnitude and sign of 

the coefficients. These steps will help ensure that the original 

PTMs are appropriate for a diverse range of sediment textures 

and for multiple locations throughout the INEEL. Another 

way of testing the PTMs involves subdividing the original set 

of measurements to create one set of observations for model 

development and one for testing. The limiting factor in this 

approach, at least for the q(y) models, is the small number of 

measurements available. Splitting 58 or fewer observations 

into two parts leaves only a small data set to work with, which 

may cause problems in meeting the assumptions of multiple 

linear regression. 

Summary and Conclusions

In this study, multiple-linear regression models were 

developed for sedimentary interbeds near two facilities 

at the Idaho National Engineering and Environmental 

Laboratory, the Radioactive Waste Management Complex 

and the Vadose Zone Research Park, using available core 

sample measurements. Separate regression equations were 

developed for parameters defining the water-retention 

curve (q(y)) and for saturated hydraulic conductivity (Ksat). 

Three representations of the particle-size distribution were 

considered as possible input to these models, in addition 

to bulk density (rbulk) and particle density (rpart). These 

representations involved the calculation of textural-class 

percentages (gravel, sand, silt, and clay), geometric statistics 

(mean and standard deviation), and graphical statistics 

(median and uniformity coefficient) from the particle-size 

data. The Rossi-Nimmo junction model was used to represent 

the q(y) measurements. Three parameters define this model: 

(1) saturated water content (qsat), (2) a scaling parameter for 

matric pressure (yo), and (3) a curve-shape parameter (l). 

The bulk-physical-property data and optimized hydraulic 

parameter values were used to develop separate regression 

models for qsat, yo, and l. The predicted parameters then 

were used to calculate the q(y) curve from saturation to oven 

dryness. 

The individual regression models for qsat, yo, l, and Ksat

were evaluated based on goodness-of-fit and model behavior, 

in terms of meeting the assumptions of multiple linear 

regression. Because rpart was poorly correlated with qsat, yo, 

l, and Ksat, and affected the R2

adj values only marginally, it 

was not included as an explanatory variable in the regression 

analyses. The model fits, assessed using the adjusted 

coefficient of determination (R2

adj), ranged from 0.491 to 0.568 

for qsat, from 0.667 to 0.727 for yo, and from 0.192 to 0.667 

for l. The R2

adj values for Ksat were moderate, ranging from 

0.589 to 0.632. For qsat and Ksat, the highest R2

adj values were 

achieved using textural-class percentages and rbulk as input. 

The highest R2

adj for yo

 and l were obtained using geometric 

particle-size statistics and rbulk as input. Models involving 

geometric particle-size parameters tended to have better model 

behavior, in terms of residual normality and homoscedasticity, 

than those involving textural-class percentages or graphical 

particle-size statistics. The geometric statistics are appealing 

to use because they describe the breadth of particle sizes and 

average particle size, which may be related directly to similar 

features of the pore-size distribution. The best models for qsat

and Ksat used textural-class percentages and rbulk as input, and 

the best models for yo and l used geometric statistics and 

rbulk as input. The sensitivity of the model results to the form 

of the particle-size distribution used as input was tested by 

examining the combined effect of the qsat, yo, and l regression 

models in predicting the entire q(y) curve.

The root-mean-square-error (RMSE) distributions for four 

different model combinations were compared, involving: (1) 

highest R2

adj for qsat, yo

, and l, (2) textural-class percentages, 

(3) geometric statistics, and (4) graphical statistics. Because 

the goodness-of-fit values for the Ksat models were similar 

for each of the three sets of particle-size parameters, the final 

Ksat model was selected based on the best model combination 

to use for estimating q(y). For each core sample, the q(y)

curve was estimated using the predicted values of qsat, yo, 

and l from the individual regression equations in the RossiNimmo junction model. This estimated curve was compared 

to the curve fit to the measured q(y) points using RMSE as 

a goodness-of-fit indicator. These steps were completed for 

Summary and Conclusions  45
each of the four model combinations. RMSE distributions 

then were compared for each competing model to select the 

best particle-size parameter set to implement. RMSE values, 

in units of volumetric q, ranged from 0.001 to 0.064 among 

the four distributions. The RMSE distribution for the model 

combination involving the highest R2

adj values for qsat, yo

, 

and l was similar to the distributions that used geometric 

or graphical statistics as input. Median values for the four 

distributions were quite similar, ranging from 0.015 to 0.021. 

The RMSE distribution with textural-class percentages used 

to estimate qsat, yo, and l had a slightly larger median than the 

other combinations, likely due to the low R2

adj of 0.192 for the 

l regression model. The RMSE distribution based on graphical 

statistics was further subdivided according to core sample 

texture and data source for model development. The range of 

RMSE values for sands compared to loams was quite similar, 

indicating lack of model bias toward a particular texture. 

Variations in RMSE distributions among the source data 

sets were found to be mainly due to the textures of the core 

samples analyzed and the over- or underestimation of q near 

saturation. From comparisons of RMSE distributions among 

the four model combinations, the predicted q(y) curves were 

insensitive to the particle-size distribution form used as input 

to the regression equations. 

Given the similarity of the results for each model 

combination, other criteria were used for final model selection, 

including: (1) ease of model use or ease of determining 

the particle-size parameters, (2) optimal regression model 

behavior and fit, and (3) ability to develop a theoretical model 

from the statistical results. Ease of model use was deemed 

the most important criterion for selecting the form of the 

particle-size distribution to use in the q(y) and Ksat models. 

Although textural-class percentages are the easiest particlesize parameters to determine, multicollinearity among the 

textural-class parameters limits the applicability of this model. 

Even though the models with geometric statistics as input 

generally performed better than the other two particle-size 

models, geometric statistics are more difficult to calculate 

and require higher resolution of particle-size distribution 

than textural-class percentages or graphical statistics. Median 

particle diameter and uniformity coefficient, which can be 

calculated directly from a cumulative particle-size distribution, 

ranked second for ease of determination. The final regression 

equations for qsat, yo, l, and Ksat involved rbulk, the median 

particle diameter, and the uniformity coefficient as explanatory 

variables.

Moderate R2

adj values for qsat, yo

, l, and Ksat may indicate 

that the bulk-physical-property data used in calibrating the 

regression models are not sufficient to predict completely the 

hydraulic parameters or may indicate significant measurement 

errors in the response or explanatory variables. Other bulkphysical-property data not available for calibrating the 

property-transfer functions, such as mineralogy or specific 

surface area, might correlate more strongly with the hydraulic 

parameters, and may be useful in future regression analyses. 

The regression models in this report represent the first 

attempt to develop a site-specific model for estimating the 

unsaturated hydraulic properties of sediments at the INEEL. 

Because the models are the first of their kind for the INEEL, 

adjustments or recalibration likely will be required in the 

future as new data become available. These models allow 

easy determination of hydraulic properties without need for 

direct measurement. Although collection of core samples 

still likely will be needed to estimate rbulk and particle size, 

these measurements are less time-intensive to complete than 

hydraulic-property measurements. The regression models 

will be useful in estimating hydraulic properties of sediments 

over large areas, especially as input to unsaturated flow 

and contaminant transport models. The property-transfer 

models from this study provide a basis for development of a 

theoretical model that relies on physical relationships between 

the pore-size distribution and the bulk-physical properties of 

the media. A theoretical model based on physical relationships 

should be more universal in its application throughout the 

INEEL and other geographic locations. 

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Manuscript approved for publication, June 2, 2005

Prepared by the U.S. Geological Survey Publishing staff, 

Washington Water Science Center, Tacoma, Washington

Bill Gibbs

Debra Grillo

Bobbie Jo Richey

Ginger Renslow

For more information concerning the research in this report, contact the 

Idaho Water Science Center Director, 

U.S. Geological Survey, 230 Collins Road 

Boise, Idaho 83702-4520

http://id.water.usgs.gov
Development of Property-Transfer Models for Estimating the Hydraulic Properties of Deep Sediments at the 

Idaho National Engineering and Environmental Laboratory, Idaho SIR 2005–5114 Winfield